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Submitted tomanuscript (Please provide the mansucript number)
Inventory Routing Problem under Uncertainty
Zheng CUI Daniel Zhuoyu LONGDepartment of Systems Engineering and Engineering Management Chinese University of Hong Kong
New Territories Hong Kong zcuisecuhkeduhk zylongsecuhkeduhk
Jin QIDepartment of Industrial Engineering and Decision Analytics Hong Kong University of Science and Technology
Kowloon Hong Kong jinqiusthk
Lianmin ZHANGDepartment of Management Science and Engineering Nanjing University Nanjing zhanglmnjueducn
We study a stochastic inventory routing problem with a finite horizon The supplier acts as a central planner
who determines replenishment quantities and also the times and routes for delivery to all retailers We
allow ambiguity in the probability distribution of each retailerrsquos uncertain demand Adopting a service-level
viewpoint we minimize the risk of uncertain inventory levels violating a pre-specified acceptable range We
quantify that risk using a novel decision criterion the Service Violation Index that accounts for how often
and how severely the inventory requirement is violated The solutions proposed here are adaptive in that
they vary with the realization of uncertain demand We provide algorithms to solve the problem exactly and
then demonstrate the superiority of our solutions by comparing them with several benchmarks
Key words inventory routing distributionally robust optimization risk management adjustable robust
optimization decision rule
1 Introduction
A key concept in supply chain management is the coordination of different stakeholders to minimize
the overall cost In that regard the vendor-managed inventory (VMI) system as a relatively new
business model has attracted extensive attention in both academia and industry In the VMI
system the supplier (vendor) behaves like a central decision maker determining not only the timing
and quantities of the replenishment for all retailers but also the route on which they are visited
Under this arrangement retailers benefit from saving efforts to control inventory Meanwhile the
supplier can improve the service level and reduce its costs by using transportation capacity more
1
Z Cui et al Inventory Routing Problem under Uncertainty2 Article submitted to manuscript no (Please provide the mansucript number)
efficiently For a more detailed discussion of the VMI conceptrsquos advantages see Waller et al (1999)
Cetinkaya and Lee (2000) Cheung and Lee (2002) and Dong and Xu (2002)
This joint management of inventory and vehicle routing gives rise to the inventory routing
problem (IRP) The term IRP was first used by Golden et al (1984) when defining a routing
problem with an explicit inventory feature Federgruen and Zipkin (1984) study a single-period
IRP and describe the economic benefits of coordinating the decisions related to distribution and
inventory Since then many variants of the IRP have been proposed over the past three decades
(Coelho et al 2014a) planning horizons can be finite or infinite excess demand can be either lost
or back-ordered and there can be either just one or more than one vehicle Scholars have used
these formulations on many real-world applications of the IRP which include maritime logistics
(Christiansen et al 2011 Papageorgiou et al 2014) the transportation of gas and oil (Bell et al
1983 Campbell and Savelsbergh 2004 Groslashnhaug et al 2010) groceries (Gaur and Fisher 2004
Custodio and Oliveira 2006) perishable products (Federgruen et al 1986) and bicycle sharing
(Brinkmann et al 2016) We refer interested readers to Andersson et al (2010) for more on IRP
applications
Since the basic IRP incorporates a classical vehicle routing problem that is already NP-hard
it follows that the stochastic IRP must be even more complex We next focus our review on
the stochastic version of the IRP especially in the case of uncertain demand A classical way
to formulate the stochastic IRP is using Markov decision process which assumes that the joint
probability distribution of retailersrsquo demands is known Campbell et al (1998) introduce a dynamic
programming model in which the state is the current inventory level at each retailer and the Markov
transition matrix is obtained from the known probability distribution of demand in their model
the objective is to minimize the expected total discounted cost over an infinite horizon This work is
extended by Kleywegt et al (2002 2004) who solve the problem by constructing an approximation
to the optimal value function Adelman (2004) decomposes the optimal value function as the sum
of single-customer inventory value functions which are approximated by the optimal dual prices
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 3
derived under a linear program Hvattum and Loslashkketangen (2009) and Hvattum et al (2009) solve
the same problem using heuristics based on finite scenario trees
Another way to solve the stochastic IRP is via stochastic programming Federgruen and Zipkin
(1984) accommodate inventory and shortage cost in the vehicle routing problem and offer a heuristic
for that problem in a scenario-based random demand environment Coelho et al (2014b) address
the dynamic and stochastic inventory routing problem in the case where the distribution of demand
changes over time They give the algorithms for four solution policies whose use depends on whether
demand forecasts are used and whether emergency trans-shipments are allowed Adulyasak et al
(2015) consider the stochastic production routing problem with demand uncertainty under both
two-stage and multi-stage decision processes They minimize average total cost by enumerating all
possible scenarios and solve the problem by deriving some valid inequalities and taking a Benders
decomposition approach
The stochastic IRP literature is notable however for not adequately addressing two impor-
tant issues The first of these involves demand uncertainty Both the Markov decision process
and stochastic programming methods assume a known probability distribution for the uncertain
demand but full distributional information is seldom available in practice Owing to various rea-
sons such as estimation error and the lack of historical data we may have only partial informa-
tion on the probability distribution To cope with this shortcoming classical robust optimization
methodology has been applied to the IRP (see eg Aghezzaf 2008 Solyalı et al 2012 Bertsimas et
al 2016) These authors use a polyhedron to characterize the realization of uncertain demand and
then minimize the cost for the worst-case scenario In particular Aghezzaf (2008) assumes nor-
mally distributed demands and minimizes the worst-case cost for possible realizations of demands
within certain confidence level Solyalı et al (2012) consider a robust IRP by using the ldquobudget
of uncertaintyrdquo approach proposed by Bertsimas and Sim (2003 2004) and solve the problem via
a branch-and-cut algorithm Bertsimas et al (2016) consider the robust IRP from the perspective
of scalability they propose a robust and adaptive formulation that can be solved for instances of
Z Cui et al Inventory Routing Problem under Uncertainty4 Article submitted to manuscript no (Please provide the mansucript number)
around 6000 customers However traditional robust optimization does not capture any frequency
information and focuses only on the worst-case realization mdash an approach that some consider to be
unnecessarily conservative We therefore propose a distributionally robust optimization framework
that does exploit the frequency information The resulting model allows us to make robust inventory
replenishment and routing decisions that can immune against the effect of data uncertainty
Furthermore since this is a multi-period problem with uncertain demand each periodrsquos opera-
tional decisions should be adjusted in response to previous demand realization in order to achieve
appealing performance For such multi-stage problems Ben-Tal et al (2004) propose an adjustable
decision rule approach that models decisions as functions of the uncertain parametersrsquo realizations
Yet because finding the optimal decision rule is computationally intractable various decision rules
have been proposed to solve the problem approximately by restricting the set of feasible adaptive
decisions to some particular and simple functional forms (see eg Ben-Tal et al 2004 Chen et al
2008 Goh and Sim 2010 Bertsimas et al 2011 Kuhn et al 2011 Georghiou et al 2015) readers
are referred to the tutorial by Delage and Iancu (2015) To reduce computational complexity we
will use a linear decision rule that restricts the decisions in each period to be affine functions of
previous periodsrsquo uncertain demands The effectiveness of this decision rule has been established
both computationally (eg Ben-Tal et al 2005 Bertsimas et al 2018) and analytically (eg Kuhn
et al 2011 Bertsimas and Goyal 2012) To the best of our knowledge Bertsimas et al (2016) is
the only paper that incorporates the linear decision rule in IRP and hence is able to solve large-
scale instances However our paper differs from theirs in three ways we account for the frequency
information of demands we incorporate an enhanced decision rule and we propose a new decision
criterion that is consistent with risk aversion
The second issue that is largely ignored in the literature arises from the stochastic IRPrsquos objective
function Most research on this topic minimizes the expectation of operational cost which is the sum
of holding cost shortage cost and transportation cost However the calibration of cost parameters
especially the per-unit shortage cost is challenging in many practical cases (Brandimarte and
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 5
Zotteri 2007) And even if cost parameters could be precisely estimated the inventory cost is a
piecewise linear function and so calculating its expectation involves the sort of multi-dimensional
integration that increases the computational complexity (Ardestani-Jaafari and Delage 2016) Last
but not least expected cost is a risk-neutral criterion and thus fails to account for decision makersrsquo
attitudes toward risk Yet risk aversion is well documented empirically (as in the St Petersburg
paradox) and applied in inventory literatures (eg Chen et al 2007) So instead of optimizing the
expected cost we present an alternative approach that resolves the issues previously described
Our objective function is related to the concept of service level which is a prevalent requirement
in supply chain management (Miranda and Garrido 2004 Bernstein and Federgruen 2007 Bertsi-
mas et al 2016) From a practical standpoint it is extremely important for retailers to maintain
inventory levels within a certain range On the one hand the upper limit corresponds to a retailerrsquos
capacity eg storage capacity On the other hand running out of inventory will damage brand
image and also impose substantial cost due to emergent replenishments It follows that establishing
a lower limit will also help control inventory levels The rationale of such a service-level require-
ment also relates to the target-driven decision making It can be traced back to Simon (1955)
who suggests that the main goal for most decision making is not to maximize returns or minimize
costs but rather to achieve certain targets Since then a rich body of descriptive literatures has
confirmed that the decision making is often driven by targets (Mao 1970) Indeed with the absence
of inventory control Jaillet et al (2016) and Zhang et al (2018) have investigated the target-driven
decision making in the vehicle routing problem It is therefore natural to assume that in stochastic
IRP the objective is to maintain the inventory within certain target levels defined by an upper
and a lower bound
Given such a range imposed on inventory there are two classical ways to model the service-level
requirement The first approach is to ensure that the inventory level remains within the desired
range for all possible demand realizations however this approach is often viewed as being too
conservative The second way of modeling the service-level requirement is to minimize the violation
Z Cui et al Inventory Routing Problem under Uncertainty6 Article submitted to manuscript no (Please provide the mansucript number)
probability via chance constrained optimization Yet chance constraints are well known to be
nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach
has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van
de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion
that we use to evaluate the risk of violating the service-level requirement and to address the issues
discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet
et al (2016) when studying the stochastic vehicle routing problem It can be represented from
classical expected utility theory and can therefore be constructed using any utility function In
the case of single period the SVI is associated with the satisficing measure axiomatized by Brown
and Sim (2009)
We summarize our main contributions as follows
bull Instead of minimizing the expectation of inventory and transportation cost we study robust
IRP from the service-level perspective and aim to keep the inventory level within a certain range
The existence of uncertainty in demand motivates us to propose a multi-period decision criterion
that allows decision makers to account for the risk that the uncertain inventory level of each retailer
in each period will violate the stipulated service requirement window
bull We use a distributionally robust optimization framework to calibrate uncertain demand
by way of descriptive statistics to derive solutions that protect against demand variation This
approach incorporates more distributional information and will yield less conservative results than
does the classical robust optimization approach
bull We derive adaptive inventory replenishment solutions that can be updated with observed
demand by implementing a decision rule approach Then the robust IRP can be formulated as a
large-scale mixed-integer linear optimization problem
bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-
tion approach
This rest of our paper proceeds as follows Section 2 defines the problem and discusses how
we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
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Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
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Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
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Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
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Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
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optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty2 Article submitted to manuscript no (Please provide the mansucript number)
efficiently For a more detailed discussion of the VMI conceptrsquos advantages see Waller et al (1999)
Cetinkaya and Lee (2000) Cheung and Lee (2002) and Dong and Xu (2002)
This joint management of inventory and vehicle routing gives rise to the inventory routing
problem (IRP) The term IRP was first used by Golden et al (1984) when defining a routing
problem with an explicit inventory feature Federgruen and Zipkin (1984) study a single-period
IRP and describe the economic benefits of coordinating the decisions related to distribution and
inventory Since then many variants of the IRP have been proposed over the past three decades
(Coelho et al 2014a) planning horizons can be finite or infinite excess demand can be either lost
or back-ordered and there can be either just one or more than one vehicle Scholars have used
these formulations on many real-world applications of the IRP which include maritime logistics
(Christiansen et al 2011 Papageorgiou et al 2014) the transportation of gas and oil (Bell et al
1983 Campbell and Savelsbergh 2004 Groslashnhaug et al 2010) groceries (Gaur and Fisher 2004
Custodio and Oliveira 2006) perishable products (Federgruen et al 1986) and bicycle sharing
(Brinkmann et al 2016) We refer interested readers to Andersson et al (2010) for more on IRP
applications
Since the basic IRP incorporates a classical vehicle routing problem that is already NP-hard
it follows that the stochastic IRP must be even more complex We next focus our review on
the stochastic version of the IRP especially in the case of uncertain demand A classical way
to formulate the stochastic IRP is using Markov decision process which assumes that the joint
probability distribution of retailersrsquo demands is known Campbell et al (1998) introduce a dynamic
programming model in which the state is the current inventory level at each retailer and the Markov
transition matrix is obtained from the known probability distribution of demand in their model
the objective is to minimize the expected total discounted cost over an infinite horizon This work is
extended by Kleywegt et al (2002 2004) who solve the problem by constructing an approximation
to the optimal value function Adelman (2004) decomposes the optimal value function as the sum
of single-customer inventory value functions which are approximated by the optimal dual prices
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 3
derived under a linear program Hvattum and Loslashkketangen (2009) and Hvattum et al (2009) solve
the same problem using heuristics based on finite scenario trees
Another way to solve the stochastic IRP is via stochastic programming Federgruen and Zipkin
(1984) accommodate inventory and shortage cost in the vehicle routing problem and offer a heuristic
for that problem in a scenario-based random demand environment Coelho et al (2014b) address
the dynamic and stochastic inventory routing problem in the case where the distribution of demand
changes over time They give the algorithms for four solution policies whose use depends on whether
demand forecasts are used and whether emergency trans-shipments are allowed Adulyasak et al
(2015) consider the stochastic production routing problem with demand uncertainty under both
two-stage and multi-stage decision processes They minimize average total cost by enumerating all
possible scenarios and solve the problem by deriving some valid inequalities and taking a Benders
decomposition approach
The stochastic IRP literature is notable however for not adequately addressing two impor-
tant issues The first of these involves demand uncertainty Both the Markov decision process
and stochastic programming methods assume a known probability distribution for the uncertain
demand but full distributional information is seldom available in practice Owing to various rea-
sons such as estimation error and the lack of historical data we may have only partial informa-
tion on the probability distribution To cope with this shortcoming classical robust optimization
methodology has been applied to the IRP (see eg Aghezzaf 2008 Solyalı et al 2012 Bertsimas et
al 2016) These authors use a polyhedron to characterize the realization of uncertain demand and
then minimize the cost for the worst-case scenario In particular Aghezzaf (2008) assumes nor-
mally distributed demands and minimizes the worst-case cost for possible realizations of demands
within certain confidence level Solyalı et al (2012) consider a robust IRP by using the ldquobudget
of uncertaintyrdquo approach proposed by Bertsimas and Sim (2003 2004) and solve the problem via
a branch-and-cut algorithm Bertsimas et al (2016) consider the robust IRP from the perspective
of scalability they propose a robust and adaptive formulation that can be solved for instances of
Z Cui et al Inventory Routing Problem under Uncertainty4 Article submitted to manuscript no (Please provide the mansucript number)
around 6000 customers However traditional robust optimization does not capture any frequency
information and focuses only on the worst-case realization mdash an approach that some consider to be
unnecessarily conservative We therefore propose a distributionally robust optimization framework
that does exploit the frequency information The resulting model allows us to make robust inventory
replenishment and routing decisions that can immune against the effect of data uncertainty
Furthermore since this is a multi-period problem with uncertain demand each periodrsquos opera-
tional decisions should be adjusted in response to previous demand realization in order to achieve
appealing performance For such multi-stage problems Ben-Tal et al (2004) propose an adjustable
decision rule approach that models decisions as functions of the uncertain parametersrsquo realizations
Yet because finding the optimal decision rule is computationally intractable various decision rules
have been proposed to solve the problem approximately by restricting the set of feasible adaptive
decisions to some particular and simple functional forms (see eg Ben-Tal et al 2004 Chen et al
2008 Goh and Sim 2010 Bertsimas et al 2011 Kuhn et al 2011 Georghiou et al 2015) readers
are referred to the tutorial by Delage and Iancu (2015) To reduce computational complexity we
will use a linear decision rule that restricts the decisions in each period to be affine functions of
previous periodsrsquo uncertain demands The effectiveness of this decision rule has been established
both computationally (eg Ben-Tal et al 2005 Bertsimas et al 2018) and analytically (eg Kuhn
et al 2011 Bertsimas and Goyal 2012) To the best of our knowledge Bertsimas et al (2016) is
the only paper that incorporates the linear decision rule in IRP and hence is able to solve large-
scale instances However our paper differs from theirs in three ways we account for the frequency
information of demands we incorporate an enhanced decision rule and we propose a new decision
criterion that is consistent with risk aversion
The second issue that is largely ignored in the literature arises from the stochastic IRPrsquos objective
function Most research on this topic minimizes the expectation of operational cost which is the sum
of holding cost shortage cost and transportation cost However the calibration of cost parameters
especially the per-unit shortage cost is challenging in many practical cases (Brandimarte and
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 5
Zotteri 2007) And even if cost parameters could be precisely estimated the inventory cost is a
piecewise linear function and so calculating its expectation involves the sort of multi-dimensional
integration that increases the computational complexity (Ardestani-Jaafari and Delage 2016) Last
but not least expected cost is a risk-neutral criterion and thus fails to account for decision makersrsquo
attitudes toward risk Yet risk aversion is well documented empirically (as in the St Petersburg
paradox) and applied in inventory literatures (eg Chen et al 2007) So instead of optimizing the
expected cost we present an alternative approach that resolves the issues previously described
Our objective function is related to the concept of service level which is a prevalent requirement
in supply chain management (Miranda and Garrido 2004 Bernstein and Federgruen 2007 Bertsi-
mas et al 2016) From a practical standpoint it is extremely important for retailers to maintain
inventory levels within a certain range On the one hand the upper limit corresponds to a retailerrsquos
capacity eg storage capacity On the other hand running out of inventory will damage brand
image and also impose substantial cost due to emergent replenishments It follows that establishing
a lower limit will also help control inventory levels The rationale of such a service-level require-
ment also relates to the target-driven decision making It can be traced back to Simon (1955)
who suggests that the main goal for most decision making is not to maximize returns or minimize
costs but rather to achieve certain targets Since then a rich body of descriptive literatures has
confirmed that the decision making is often driven by targets (Mao 1970) Indeed with the absence
of inventory control Jaillet et al (2016) and Zhang et al (2018) have investigated the target-driven
decision making in the vehicle routing problem It is therefore natural to assume that in stochastic
IRP the objective is to maintain the inventory within certain target levels defined by an upper
and a lower bound
Given such a range imposed on inventory there are two classical ways to model the service-level
requirement The first approach is to ensure that the inventory level remains within the desired
range for all possible demand realizations however this approach is often viewed as being too
conservative The second way of modeling the service-level requirement is to minimize the violation
Z Cui et al Inventory Routing Problem under Uncertainty6 Article submitted to manuscript no (Please provide the mansucript number)
probability via chance constrained optimization Yet chance constraints are well known to be
nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach
has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van
de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion
that we use to evaluate the risk of violating the service-level requirement and to address the issues
discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet
et al (2016) when studying the stochastic vehicle routing problem It can be represented from
classical expected utility theory and can therefore be constructed using any utility function In
the case of single period the SVI is associated with the satisficing measure axiomatized by Brown
and Sim (2009)
We summarize our main contributions as follows
bull Instead of minimizing the expectation of inventory and transportation cost we study robust
IRP from the service-level perspective and aim to keep the inventory level within a certain range
The existence of uncertainty in demand motivates us to propose a multi-period decision criterion
that allows decision makers to account for the risk that the uncertain inventory level of each retailer
in each period will violate the stipulated service requirement window
bull We use a distributionally robust optimization framework to calibrate uncertain demand
by way of descriptive statistics to derive solutions that protect against demand variation This
approach incorporates more distributional information and will yield less conservative results than
does the classical robust optimization approach
bull We derive adaptive inventory replenishment solutions that can be updated with observed
demand by implementing a decision rule approach Then the robust IRP can be formulated as a
large-scale mixed-integer linear optimization problem
bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-
tion approach
This rest of our paper proceeds as follows Section 2 defines the problem and discusses how
we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 3
derived under a linear program Hvattum and Loslashkketangen (2009) and Hvattum et al (2009) solve
the same problem using heuristics based on finite scenario trees
Another way to solve the stochastic IRP is via stochastic programming Federgruen and Zipkin
(1984) accommodate inventory and shortage cost in the vehicle routing problem and offer a heuristic
for that problem in a scenario-based random demand environment Coelho et al (2014b) address
the dynamic and stochastic inventory routing problem in the case where the distribution of demand
changes over time They give the algorithms for four solution policies whose use depends on whether
demand forecasts are used and whether emergency trans-shipments are allowed Adulyasak et al
(2015) consider the stochastic production routing problem with demand uncertainty under both
two-stage and multi-stage decision processes They minimize average total cost by enumerating all
possible scenarios and solve the problem by deriving some valid inequalities and taking a Benders
decomposition approach
The stochastic IRP literature is notable however for not adequately addressing two impor-
tant issues The first of these involves demand uncertainty Both the Markov decision process
and stochastic programming methods assume a known probability distribution for the uncertain
demand but full distributional information is seldom available in practice Owing to various rea-
sons such as estimation error and the lack of historical data we may have only partial informa-
tion on the probability distribution To cope with this shortcoming classical robust optimization
methodology has been applied to the IRP (see eg Aghezzaf 2008 Solyalı et al 2012 Bertsimas et
al 2016) These authors use a polyhedron to characterize the realization of uncertain demand and
then minimize the cost for the worst-case scenario In particular Aghezzaf (2008) assumes nor-
mally distributed demands and minimizes the worst-case cost for possible realizations of demands
within certain confidence level Solyalı et al (2012) consider a robust IRP by using the ldquobudget
of uncertaintyrdquo approach proposed by Bertsimas and Sim (2003 2004) and solve the problem via
a branch-and-cut algorithm Bertsimas et al (2016) consider the robust IRP from the perspective
of scalability they propose a robust and adaptive formulation that can be solved for instances of
Z Cui et al Inventory Routing Problem under Uncertainty4 Article submitted to manuscript no (Please provide the mansucript number)
around 6000 customers However traditional robust optimization does not capture any frequency
information and focuses only on the worst-case realization mdash an approach that some consider to be
unnecessarily conservative We therefore propose a distributionally robust optimization framework
that does exploit the frequency information The resulting model allows us to make robust inventory
replenishment and routing decisions that can immune against the effect of data uncertainty
Furthermore since this is a multi-period problem with uncertain demand each periodrsquos opera-
tional decisions should be adjusted in response to previous demand realization in order to achieve
appealing performance For such multi-stage problems Ben-Tal et al (2004) propose an adjustable
decision rule approach that models decisions as functions of the uncertain parametersrsquo realizations
Yet because finding the optimal decision rule is computationally intractable various decision rules
have been proposed to solve the problem approximately by restricting the set of feasible adaptive
decisions to some particular and simple functional forms (see eg Ben-Tal et al 2004 Chen et al
2008 Goh and Sim 2010 Bertsimas et al 2011 Kuhn et al 2011 Georghiou et al 2015) readers
are referred to the tutorial by Delage and Iancu (2015) To reduce computational complexity we
will use a linear decision rule that restricts the decisions in each period to be affine functions of
previous periodsrsquo uncertain demands The effectiveness of this decision rule has been established
both computationally (eg Ben-Tal et al 2005 Bertsimas et al 2018) and analytically (eg Kuhn
et al 2011 Bertsimas and Goyal 2012) To the best of our knowledge Bertsimas et al (2016) is
the only paper that incorporates the linear decision rule in IRP and hence is able to solve large-
scale instances However our paper differs from theirs in three ways we account for the frequency
information of demands we incorporate an enhanced decision rule and we propose a new decision
criterion that is consistent with risk aversion
The second issue that is largely ignored in the literature arises from the stochastic IRPrsquos objective
function Most research on this topic minimizes the expectation of operational cost which is the sum
of holding cost shortage cost and transportation cost However the calibration of cost parameters
especially the per-unit shortage cost is challenging in many practical cases (Brandimarte and
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 5
Zotteri 2007) And even if cost parameters could be precisely estimated the inventory cost is a
piecewise linear function and so calculating its expectation involves the sort of multi-dimensional
integration that increases the computational complexity (Ardestani-Jaafari and Delage 2016) Last
but not least expected cost is a risk-neutral criterion and thus fails to account for decision makersrsquo
attitudes toward risk Yet risk aversion is well documented empirically (as in the St Petersburg
paradox) and applied in inventory literatures (eg Chen et al 2007) So instead of optimizing the
expected cost we present an alternative approach that resolves the issues previously described
Our objective function is related to the concept of service level which is a prevalent requirement
in supply chain management (Miranda and Garrido 2004 Bernstein and Federgruen 2007 Bertsi-
mas et al 2016) From a practical standpoint it is extremely important for retailers to maintain
inventory levels within a certain range On the one hand the upper limit corresponds to a retailerrsquos
capacity eg storage capacity On the other hand running out of inventory will damage brand
image and also impose substantial cost due to emergent replenishments It follows that establishing
a lower limit will also help control inventory levels The rationale of such a service-level require-
ment also relates to the target-driven decision making It can be traced back to Simon (1955)
who suggests that the main goal for most decision making is not to maximize returns or minimize
costs but rather to achieve certain targets Since then a rich body of descriptive literatures has
confirmed that the decision making is often driven by targets (Mao 1970) Indeed with the absence
of inventory control Jaillet et al (2016) and Zhang et al (2018) have investigated the target-driven
decision making in the vehicle routing problem It is therefore natural to assume that in stochastic
IRP the objective is to maintain the inventory within certain target levels defined by an upper
and a lower bound
Given such a range imposed on inventory there are two classical ways to model the service-level
requirement The first approach is to ensure that the inventory level remains within the desired
range for all possible demand realizations however this approach is often viewed as being too
conservative The second way of modeling the service-level requirement is to minimize the violation
Z Cui et al Inventory Routing Problem under Uncertainty6 Article submitted to manuscript no (Please provide the mansucript number)
probability via chance constrained optimization Yet chance constraints are well known to be
nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach
has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van
de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion
that we use to evaluate the risk of violating the service-level requirement and to address the issues
discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet
et al (2016) when studying the stochastic vehicle routing problem It can be represented from
classical expected utility theory and can therefore be constructed using any utility function In
the case of single period the SVI is associated with the satisficing measure axiomatized by Brown
and Sim (2009)
We summarize our main contributions as follows
bull Instead of minimizing the expectation of inventory and transportation cost we study robust
IRP from the service-level perspective and aim to keep the inventory level within a certain range
The existence of uncertainty in demand motivates us to propose a multi-period decision criterion
that allows decision makers to account for the risk that the uncertain inventory level of each retailer
in each period will violate the stipulated service requirement window
bull We use a distributionally robust optimization framework to calibrate uncertain demand
by way of descriptive statistics to derive solutions that protect against demand variation This
approach incorporates more distributional information and will yield less conservative results than
does the classical robust optimization approach
bull We derive adaptive inventory replenishment solutions that can be updated with observed
demand by implementing a decision rule approach Then the robust IRP can be formulated as a
large-scale mixed-integer linear optimization problem
bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-
tion approach
This rest of our paper proceeds as follows Section 2 defines the problem and discusses how
we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
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Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
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Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
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Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
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optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
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Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty4 Article submitted to manuscript no (Please provide the mansucript number)
around 6000 customers However traditional robust optimization does not capture any frequency
information and focuses only on the worst-case realization mdash an approach that some consider to be
unnecessarily conservative We therefore propose a distributionally robust optimization framework
that does exploit the frequency information The resulting model allows us to make robust inventory
replenishment and routing decisions that can immune against the effect of data uncertainty
Furthermore since this is a multi-period problem with uncertain demand each periodrsquos opera-
tional decisions should be adjusted in response to previous demand realization in order to achieve
appealing performance For such multi-stage problems Ben-Tal et al (2004) propose an adjustable
decision rule approach that models decisions as functions of the uncertain parametersrsquo realizations
Yet because finding the optimal decision rule is computationally intractable various decision rules
have been proposed to solve the problem approximately by restricting the set of feasible adaptive
decisions to some particular and simple functional forms (see eg Ben-Tal et al 2004 Chen et al
2008 Goh and Sim 2010 Bertsimas et al 2011 Kuhn et al 2011 Georghiou et al 2015) readers
are referred to the tutorial by Delage and Iancu (2015) To reduce computational complexity we
will use a linear decision rule that restricts the decisions in each period to be affine functions of
previous periodsrsquo uncertain demands The effectiveness of this decision rule has been established
both computationally (eg Ben-Tal et al 2005 Bertsimas et al 2018) and analytically (eg Kuhn
et al 2011 Bertsimas and Goyal 2012) To the best of our knowledge Bertsimas et al (2016) is
the only paper that incorporates the linear decision rule in IRP and hence is able to solve large-
scale instances However our paper differs from theirs in three ways we account for the frequency
information of demands we incorporate an enhanced decision rule and we propose a new decision
criterion that is consistent with risk aversion
The second issue that is largely ignored in the literature arises from the stochastic IRPrsquos objective
function Most research on this topic minimizes the expectation of operational cost which is the sum
of holding cost shortage cost and transportation cost However the calibration of cost parameters
especially the per-unit shortage cost is challenging in many practical cases (Brandimarte and
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 5
Zotteri 2007) And even if cost parameters could be precisely estimated the inventory cost is a
piecewise linear function and so calculating its expectation involves the sort of multi-dimensional
integration that increases the computational complexity (Ardestani-Jaafari and Delage 2016) Last
but not least expected cost is a risk-neutral criterion and thus fails to account for decision makersrsquo
attitudes toward risk Yet risk aversion is well documented empirically (as in the St Petersburg
paradox) and applied in inventory literatures (eg Chen et al 2007) So instead of optimizing the
expected cost we present an alternative approach that resolves the issues previously described
Our objective function is related to the concept of service level which is a prevalent requirement
in supply chain management (Miranda and Garrido 2004 Bernstein and Federgruen 2007 Bertsi-
mas et al 2016) From a practical standpoint it is extremely important for retailers to maintain
inventory levels within a certain range On the one hand the upper limit corresponds to a retailerrsquos
capacity eg storage capacity On the other hand running out of inventory will damage brand
image and also impose substantial cost due to emergent replenishments It follows that establishing
a lower limit will also help control inventory levels The rationale of such a service-level require-
ment also relates to the target-driven decision making It can be traced back to Simon (1955)
who suggests that the main goal for most decision making is not to maximize returns or minimize
costs but rather to achieve certain targets Since then a rich body of descriptive literatures has
confirmed that the decision making is often driven by targets (Mao 1970) Indeed with the absence
of inventory control Jaillet et al (2016) and Zhang et al (2018) have investigated the target-driven
decision making in the vehicle routing problem It is therefore natural to assume that in stochastic
IRP the objective is to maintain the inventory within certain target levels defined by an upper
and a lower bound
Given such a range imposed on inventory there are two classical ways to model the service-level
requirement The first approach is to ensure that the inventory level remains within the desired
range for all possible demand realizations however this approach is often viewed as being too
conservative The second way of modeling the service-level requirement is to minimize the violation
Z Cui et al Inventory Routing Problem under Uncertainty6 Article submitted to manuscript no (Please provide the mansucript number)
probability via chance constrained optimization Yet chance constraints are well known to be
nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach
has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van
de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion
that we use to evaluate the risk of violating the service-level requirement and to address the issues
discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet
et al (2016) when studying the stochastic vehicle routing problem It can be represented from
classical expected utility theory and can therefore be constructed using any utility function In
the case of single period the SVI is associated with the satisficing measure axiomatized by Brown
and Sim (2009)
We summarize our main contributions as follows
bull Instead of minimizing the expectation of inventory and transportation cost we study robust
IRP from the service-level perspective and aim to keep the inventory level within a certain range
The existence of uncertainty in demand motivates us to propose a multi-period decision criterion
that allows decision makers to account for the risk that the uncertain inventory level of each retailer
in each period will violate the stipulated service requirement window
bull We use a distributionally robust optimization framework to calibrate uncertain demand
by way of descriptive statistics to derive solutions that protect against demand variation This
approach incorporates more distributional information and will yield less conservative results than
does the classical robust optimization approach
bull We derive adaptive inventory replenishment solutions that can be updated with observed
demand by implementing a decision rule approach Then the robust IRP can be formulated as a
large-scale mixed-integer linear optimization problem
bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-
tion approach
This rest of our paper proceeds as follows Section 2 defines the problem and discusses how
we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Zotteri 2007) And even if cost parameters could be precisely estimated the inventory cost is a
piecewise linear function and so calculating its expectation involves the sort of multi-dimensional
integration that increases the computational complexity (Ardestani-Jaafari and Delage 2016) Last
but not least expected cost is a risk-neutral criterion and thus fails to account for decision makersrsquo
attitudes toward risk Yet risk aversion is well documented empirically (as in the St Petersburg
paradox) and applied in inventory literatures (eg Chen et al 2007) So instead of optimizing the
expected cost we present an alternative approach that resolves the issues previously described
Our objective function is related to the concept of service level which is a prevalent requirement
in supply chain management (Miranda and Garrido 2004 Bernstein and Federgruen 2007 Bertsi-
mas et al 2016) From a practical standpoint it is extremely important for retailers to maintain
inventory levels within a certain range On the one hand the upper limit corresponds to a retailerrsquos
capacity eg storage capacity On the other hand running out of inventory will damage brand
image and also impose substantial cost due to emergent replenishments It follows that establishing
a lower limit will also help control inventory levels The rationale of such a service-level require-
ment also relates to the target-driven decision making It can be traced back to Simon (1955)
who suggests that the main goal for most decision making is not to maximize returns or minimize
costs but rather to achieve certain targets Since then a rich body of descriptive literatures has
confirmed that the decision making is often driven by targets (Mao 1970) Indeed with the absence
of inventory control Jaillet et al (2016) and Zhang et al (2018) have investigated the target-driven
decision making in the vehicle routing problem It is therefore natural to assume that in stochastic
IRP the objective is to maintain the inventory within certain target levels defined by an upper
and a lower bound
Given such a range imposed on inventory there are two classical ways to model the service-level
requirement The first approach is to ensure that the inventory level remains within the desired
range for all possible demand realizations however this approach is often viewed as being too
conservative The second way of modeling the service-level requirement is to minimize the violation
Z Cui et al Inventory Routing Problem under Uncertainty6 Article submitted to manuscript no (Please provide the mansucript number)
probability via chance constrained optimization Yet chance constraints are well known to be
nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach
has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van
de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion
that we use to evaluate the risk of violating the service-level requirement and to address the issues
discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet
et al (2016) when studying the stochastic vehicle routing problem It can be represented from
classical expected utility theory and can therefore be constructed using any utility function In
the case of single period the SVI is associated with the satisficing measure axiomatized by Brown
and Sim (2009)
We summarize our main contributions as follows
bull Instead of minimizing the expectation of inventory and transportation cost we study robust
IRP from the service-level perspective and aim to keep the inventory level within a certain range
The existence of uncertainty in demand motivates us to propose a multi-period decision criterion
that allows decision makers to account for the risk that the uncertain inventory level of each retailer
in each period will violate the stipulated service requirement window
bull We use a distributionally robust optimization framework to calibrate uncertain demand
by way of descriptive statistics to derive solutions that protect against demand variation This
approach incorporates more distributional information and will yield less conservative results than
does the classical robust optimization approach
bull We derive adaptive inventory replenishment solutions that can be updated with observed
demand by implementing a decision rule approach Then the robust IRP can be formulated as a
large-scale mixed-integer linear optimization problem
bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-
tion approach
This rest of our paper proceeds as follows Section 2 defines the problem and discusses how
we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty6 Article submitted to manuscript no (Please provide the mansucript number)
probability via chance constrained optimization Yet chance constraints are well known to be
nonconvex and computationally intractable in most cases (eg Ben-Tal et al 2009) that approach
has also been criticized for being unable to capture a violationrsquos magnitude (Diecidue and van
de Ven 2008) Hence we introduce the Service Violation Index (SVI) a general decision criterion
that we use to evaluate the risk of violating the service-level requirement and to address the issues
discussed so far Our SVI decision criterion generalizes the performance measure proposed by Jaillet
et al (2016) when studying the stochastic vehicle routing problem It can be represented from
classical expected utility theory and can therefore be constructed using any utility function In
the case of single period the SVI is associated with the satisficing measure axiomatized by Brown
and Sim (2009)
We summarize our main contributions as follows
bull Instead of minimizing the expectation of inventory and transportation cost we study robust
IRP from the service-level perspective and aim to keep the inventory level within a certain range
The existence of uncertainty in demand motivates us to propose a multi-period decision criterion
that allows decision makers to account for the risk that the uncertain inventory level of each retailer
in each period will violate the stipulated service requirement window
bull We use a distributionally robust optimization framework to calibrate uncertain demand
by way of descriptive statistics to derive solutions that protect against demand variation This
approach incorporates more distributional information and will yield less conservative results than
does the classical robust optimization approach
bull We derive adaptive inventory replenishment solutions that can be updated with observed
demand by implementing a decision rule approach Then the robust IRP can be formulated as a
large-scale mixed-integer linear optimization problem
bull We provide algorithms that can solve our model efficiently by way of the Benders decomposi-
tion approach
This rest of our paper proceeds as follows Section 2 defines the problem and discusses how
we model and evaluate uncertainty in the multi-period case In Section 3 we formulate adaptive
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
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248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
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Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
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optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
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Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 7
decisions using our decision rule approach and Section 4 proposes a solution procedure under
various information sets Section 5 reports results from several computational studies We conclude
in Section 6 with a brief summary
Notations A boldface lowercase letter such as x represents a column vector row vectors are
given within parentheses and with a comma separating each element xprime = (x1 xn) We use
(yxminusi) to denote the vector with all elements equal to those in x except the ith element which
is equal to y that is (yxminusi) = (x1 ximinus1 y xi+1 xT ) Boldface uppercase letters such as
Y represent matrices thus Y i is the ith column vector of Y and yprimei is its ith row vector For
two matrices XY isin Rmtimesn the standard inner product is given by 〈XY 〉 =m983123i=1
n983123j=1
XijYij We
use ldquolerdquo to represent an element-wise comparison for example X le Y means Xij le Yij for all
i= 1 m j = 1 n We denote uncertain quantities by characters with ldquo˜rdquo sign such as d and
the corresponding characters themselves such as d as the realization We denote by V the space
of random variables and let the probability space be (ΩF P) We assume that full information
about the probability distribution P is not known Instead we know only that P belongs to an
uncertainty set P which is a set of probability distributions characterized by particular descriptive
statistics An inequality between two uncertain parameters t ge v implies statewise dominance
that is it implies t(ω) ge v(ω) for all ω isin Ω In addition the strict inequality t gt v implies that
there exists an 983171gt 0 for which tge v+ 983171 We let em isinRT be the mth standard basis vector We use
[middot] to represent a set of running indices eg [T ] = 1 T and denote the cardinality of the set
S as |S|
2 Model
Section 21 defines the inventory routing problem In Section 22 we introduce a new service
measure to evaluate the uncertain outcome from any inventory routing decision
21 Problem statement
We consider a basic stochastic inventory routing problem in which there is one supplier andN retail-
ers over T periods The problem is defined formally on a directed network G = ([N ]cup0A) where
Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
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Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
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inventory systems Management Science 46(2) 217-232
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Operations Research 63(5) 1117-1130
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Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
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Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
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Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
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and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
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Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
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Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
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ations Research 58(4-part-1) 902-917
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Z Cui et al Inventory Routing Problem under Uncertainty8 Article submitted to manuscript no (Please provide the mansucript number)
node 0 is the supplier node and A is the set of arcs Given any subset of nodes S sube ([N ]cup 0) we
define A(S) as the set of all arcs that connect two nodes in S That is A(S) = (i j)isinA i j isin S
We let D isin VNtimesT denote the matrix of uncertain demand where Dnt is retailer nrsquos uncertain
demand in period t Also Dt=
983059D1 Dt
983060isin VNtimest is the matrix of uncertain demand from
period 1 to period t hence we have DT= D We assume that D has an upper and a lower bound
which are denoted (respectively) D and D The support of D we denote by W = D|DleDleD
Any excess demand is assumed to be backlogged For the sake of simplicity we assume that the
transportation cost from node i to node j is deterministic and denoted by cij for (i j)isinA
We assume that the supplier uses only a single vehicle but one of unconstrained capacity to
deliver products In each period the supplier observes the inventory levels of all retailers and
then makes three decisions jointly which subset of retailers to visit which route to use and what
quantities of product to replenish
The decisions made in period tisin [T ] are affected by the demand realization in previous periods
we therefore represent those decisions as functions of previous uncertain demands (ie Dtminus1
) The
routing decisions are denoted by ytn z
tij isinBt where Bt is the space of all measurable functions that
map from RNtimes(tminus1) to 01 More specifically in period t we have ytn
983059D
tminus1983060= 1 if retailer n is
visited and yt0
983059D
tminus1983060= 1 if the supplierrsquos vehicle is used to replenish a subset of retailers
Moreover for (i j) isinA we have ztij
983059D
tminus1983060= 1 if that vehicle travels directly from node i to
node j Retailer nrsquos replenishment decision is denoted by qtn
983059D
tminus1983060isinRt where Rt is the space of
measurable functions that map from RNtimes(tminus1) to R For retailer n in period t we use xtn
983059D
t983060isinRt+1
to represent the ending inventory Without loss of generality (WLOG) we assume that the system
has no inventory at the start of the planning horizon x0n = 0 Furthermore we put D
0= empty so that
any function on D0is essentially a constant The inventory dynamics can therefore be calculated
as
xtn
983059D
t983060= xtminus1
n
983059D
tminus1983060+ qtn
983059D
tminus1983060minus Dnt =
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060 foralltisin [T ]
We summarize the decision variables and parameters in Table 1
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 9
Parameters Description
cij (i j)isinA Transportation cost from node i to node j
Dnt nisin [N ] tisin [T ] Uncertain demand of retailer n in period t
Decision variables Description
yt0 isinBt tisin [T ] yt
0
983059D
tminus1983060= 1 if the vehicle is used in period t (= 0 otherwise)
ytn isinBt nisin [N ] tisin [T ] yt
n
983059D
tminus1983060= 1 if retailer n is replenished in period t (= 0 otherwise)
ztij isinBt (i j)isinA ztij
983059D
tminus1983060= 1 if the arc (i j) is traversed in period t (= 0 otherwise)
qtn isinRt nisin [N ] tisin [T ] qtn
983059D
tminus1983060is the replenishment quantity of retailer n in period t
xtn isinRt+1 nisin [N ] tisin [T ] xt
n
983059D
t983060is the inventory level of retailer n at the end of period t
Table 1 Parameters and decisions
We now present the constraints on the routing and replenishment decisions with a feasible set
Z as
Z =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn isinBt
ztij isinBt
qtn isinRt
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
0le qtn
983059D
tminus1983060leMyt
n
983059D
tminus1983060 forallnisin [N ] tisin [T ] (a)
ytn
983059D
tminus1983060le yt
0
983059D
tminus1983060 forallnisin [N ] tisin [T ] (b)
983131
j(nj)isinA
ztnj
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (c)
983131
j(jn)isinA
ztjn
983059D
tminus1983060= yt
n
983059D
tminus1983060 forallnisin [N ]cup 0 tisin [T ] (d)
983131
(ij)isinA(S)
ztij
983059D
tminus1983060le983131
nisinS
ytn
983059D
tminus1983060minus yt
k
983059D
tminus1983060
forallS sube [N ] |S|ge 2 k isin S tisin [T ] (e)
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(1)
With M being a large number constraint (1a) uses the Big-M method to infer that the ordering
decision can be made only when retailer n is served Constraint (1b) ensures that if any retailer
n is served in period t then the route in that period must ldquovisitrdquo the supplier at node 0 The
constraints (1c) and (1d) guaranteen that each visited node n has only one arc entering n and one
arc exiting n Constraint (1e) uses a two-index vehicle flow formulation which has been shown to
be computationally attractive to eliminate subtours (see eg Solyalı et al 2012)
Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty10 Article submitted to manuscript no (Please provide the mansucript number)
We require the period-t inventory level xtn
983059D
t983060
to be within a pre-specified interval known
as the requirement window [τ tn τ
tn] The requirement window can be determined by practical
considerations For example τ tn = 0 indicates a strong preference for avoiding stockouts and an
upper bound τ tn can be specified in accordance with the storage capacity which if exceeded would
incur a high holding cost Similarly we set a budget Bt for the transportation cost in each period
tisin [T ]
Therefore our major concern now is to evaluate the risk that the uncertain inventory level fails to
satisfy the requirement or that the transportation cost violates the budget constraint We propose
a new performance measure to evaluate this risk and then present a comprehensive model
22 Service Violation Index
We now introduce a service quality measure to evaluate the risk of service violation in terms of
inventory level and transportation cost Inspired by Chen et al (2015) and Jaillet et al (2016)
we propose a new and more general index to evaluate this risk Our construction of this index is
based on classical expected utility theory and risk measures
In particular given an I-dimensional random vector x isin VI we must evaluate the risk that x
realizes to be outside of the required window [τ τ ]subeRI Toward that end we propose the following
notion of Service Violation Index We first define a function vτ τ (middot) as vτ τ (x) = maxxminus τ τ minus
x= (maxxi minus τ i τ i minusxi)iisin[I] which is the violation of x with regard to the requirement window
[τ τ ] Specifically a negative value of violation eg v03(2) =minus1 implies that a change can still
be made in any arbitrary direction without violating the requirement window
Definition 1 A class of functions ρτ τ (middot) VI rarr [0infin] is a Service Violation Index (SVI) if for
all x y isin VI it satisfies the following properties
1 Monotonicity ρτ τ (x)ge ρτ τ (y) if vτ τ (x)ge vτ τ (y)
2 Satisficing
(a) Attainment content ρτ τ (x) = 0 if vτ τ (x)le 0
(b) Starvation aversion ρτ τ (x) =infin if there exists an iisin [I] such that vτ iτ i(xi)gt 0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
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Research 55(5) 828-842
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Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
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Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
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Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 11
3 Convexity ρτ τ (λx+(1minusλ)y)le λρτ τ (x)+ (1minusλ)ρτ τ (y) for any λisin [01]
4 Positive homogeneity ρλτ λτ (λx) = λρτ τ (x) for any λge 0
5 Dimension-wise additivity ρτ τ (x) =983123I
i=1 ρτ τ ((xiwminusi)) for any w isin [τ τ ]
6 Order invariance ρτ τ (x) = ρPτ Pτ (P x) for any permutation matrix P
7 Left continuity limadarr0
ρminusinfin0(xminus a1) = ρminusinfin0(x) where (minusinfin) is the vector all of whose com-
ponents are minusinfin
We define the SVI such that a low value indicates a low risk of violating the requirement window
Monotonicity states that if a given violation is always greater than another then the former will
be associated with higher risk and hence will be less preferred Satisficing specifies the riskiness
in two extreme cases Under attainment content if any realization of the uncertain attributes lies
within the requirement window then there is no risk of violation In contrast starvation aversion
implies that risk is highest when there is at least one attribute that always violates the requirement
window Our convexity property mimics risk managementrsquos preference for diversification Positive
homogeneity enables the cardinal nature the motivation of which can be found in Artzner et al
(1999) Dimension-wise additivity and order invariance imply that overall risk is the aggregation
of individual risk and the insensitivity to the sequence of dimensions respectively Both properties
are justified in a multi-stage setting by Chen et al (2015) Finally for solutions to be tractable
we need left continuity together with other properties it also allows us to represent the SVI in
terms of a convex risk measure as follows
Theorem 1 A function ρτ τ (middot) VI rarr [0infin] is an SVI if and only if it has the representation as
ρτ τ (x) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτ iτ i(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070 (2)
where we define inf empty=infin by convention and micro(middot) V rarrR is a normalized convex risk measure ie
micro(middot) satisfies the following properties for all x y isin V
1 Monotonicity micro(x)ge micro(y) if xge y
2 Cash invariance micro(x+w) = micro(x)+w for any w isinR
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
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on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
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Transportation Research Procedia 19 316-327
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55(1) 71-84
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Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
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Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
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Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
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Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
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Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty12 Article submitted to manuscript no (Please provide the mansucript number)
3 Convexity micro(λx+(1minusλ)y)le λmicro(x)+ (1minusλ)micro(y) for any λisin [01]
4 Normalization micro(0) = 0
Conversely given an SVI ρτ τ (middot) the underlying convex risk measure is given by
micro(x) =mina |ρminusinfin0((xminus a)e1)le 1 (3)
For the sake of brevity we present all proofs in the appendix
To calculate the SVI value efficiently and to illustrate some insights into its connection with the
classical expected utility criterion we focus on the SVI whose underlying convex risk measure is the
shortfall risk measure defined by Follmer and Schied (2002) In order to incorporate distributional
ambiguity we redefine it as follows
Definition 2 A shortfall risk measure is any function microu(middot) V rarrR defined by
microu(x) = infη
983069η
983055983055983055983055supPisinP
EP [u(xminus η)]le 0
983070 (4)
where u is an increasing convex normalized utility function such that u(0) = 0
The operation supPisinP allows us to take the robustness into account Hence the shortfall risk can
be interpreted as the minimum amount that needs to be subtracted from the uncertain attribute
so that this attributersquos expected utility is less than 0 ie the risk becomes acceptable We now
use the dual representation described in Theorem 1 to show that when shortfall risk measure is
the underlying risk measure the corresponding SVI has the following representation
Proposition 1 The shortfall risk measure is a normalized convex risk measure Its corresponding
SVI which we refer to as the utility-based SVI can be represented as follows
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055supPisinP
EP
983061u
983061vτ iτ i(xi)
αi
983062983062le 0αi gt 0foralliisin [I]
983100983104
983102 (5)
According to the equation (5) the utility-based SVI is the aggregation of the smallest positive
scalar factors αi over all indexes i isin [I] such that vτ iτ i(xi)αi is within the acceptable set or
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
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and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
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Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 13
equivalently vτ iτ i(xi)αi is with nonpositive expected utility This particular index allows the
utility function to take different forms For instance if the utility function is exponential as u(x) =
exp(x)minus 1 then the index can be reformulated as
ρτ τ (x) = inf
983099983103
983101983131
iisin[I]
αi
983055983055983055983055983055983055Cαi
(xi)le 0αi ge 0foralliisin [I]
983100983104
983102
where
Cαi(xi) = αi ln
983061supPisinP
EP
983061exp
983061vτ iτ i(xi)
αi
983062983062983062
Function Cαi(xi) is the worst-case certainty equivalent of the uncertain violation of xi with respect
to the requirement window [τ i τ i] when the parameter of absolute risk aversion is 1αi (Gilboa
and Schmeidler 1989) More details can be found in Hall et al (2015) and Jaillet et al (2016)
For the sake of computational tractability we focus on the SVI whose construction is based on
the piecewise linear utility function u In particular we let u(x) =maxkisin[K]akx+ bk with ak ge 0
for all k isin [K] Adopting this piecewise linear utility u preserves the optimization problemrsquos linear
structure which eases the computational burden even while accounting for risk We remark that
practically speaking a piecewise linear utility can be used to approximate any type of increasing
convex utility Because αi gt 0 for the piecewise linear function considered here we multiply both
the left-hand side (LHS) and the right-hand side (RHS) of the constraint by αi Then the SVI can
be simplified as
ρτ τ (x) = inf
983099983105983105983103
983105983105983101
983131
iisin[I]
αi
983055983055983055983055983055983055983055983055
supPisinP
EP
983061maxkisin[K]
983051akvτ iτ i(xi)+ bkαi
983052983062le 0
αi gt 0foralliisin [I]
983100983105983105983104
983105983105983102
Having specified the objective function we can now present our stochastic IRP model as follows
Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty14 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn +
983131
tisin[T ]
βt
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minus xt
n
983059D
t983060 xt
n
983059D
t983060minus τ t
n
983154+ bkα
tn
983154983062le 0 forallnisin [N ] tisin [T ]
supPisinP
EP
983091
983107maxkisin[K]
983099983103
983101ak
983091
983107983131
(ij)isinA
cijztij
983059D
tminus1983060minusBt
983092
983108+ bkβt
983100983104
983102
983092
983108le 0 foralltisin [T ]
xtn
983059D
t983060=
t983131
m=1
(qmn
983059D
mminus1983060minus Dnm) forallnisin [N ] tisin [T ]
αtn gt 0 forallnisin [N ] tisin [T ]
(qtn yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZ
(6)
3 Adaptive Decisions
We facilitate the optimization by adopting a decision rule approach to derive adaptive decisions
In particular each periodrsquos decision is assumed to be a specific mapping from previous demand
realizations At each stage the decisions involve both binary ones (ie routing) and continuous
ones (ie replenishment) Although decision rules with respect to continuous decisions have been
investigated by a rich body of literatures the rules related to binary decisions have attracted
much less attention and is still computationally challenging (eg Bertsimas and Caramanis 2007
Bertsimas and Georghiou 2014 2015) Since this study is the first to consider such a complex
stochastic IRP we simplify the model by assuming that the binary decisions are nonadaptable
That is we consider the problem in two steps In the first step the visiting decisions and routing
decisions for all periods are determined at the beginning of planning horizon Thus all routing
decisions are made a priori and irrespective of demand realizations This approach approximates the
optimal solution and indeed is a reasonable one in practice Visiting plans must be communicated
to retailers in advance so that they can adequately prepare besides scheduling the vehicle and
informing the staff on short notice is not always practically feasible (Bertsimas et al 2016) In the
second step the supplier makes replenishment decisions at the start of each period after demand
realizations in previous periods In this sense then inventory decisions are made adaptively Figure
1 summarizes the timeline applicable to these decisions
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
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Transportation Research Procedia 19 316-327
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55(1) 71-84
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Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
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inventory systems Management Science 46(2) 217-232
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Operations Research 63(5) 1117-1130
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Research 55(5) 828-842
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tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
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Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 15
Period 1 Period 2 Period Thellip
Routing decisions
Order quantities Order quantities
Actual demand Actual demand
Decisionmaking
Uncertaintyrealization
Order quantities
Actual demand
Figure 1 Sequence of routing decisions and inventory replenishment decisions
For the sake of tractability we now introduce a special case of decision rules mdash namely a
linear decision rule mdash whereby each decision is assumed to be an affine function of uncertainty
realizations Following this linear decision rule we can write the order quantity as
qtn
983059D
tminus1983060= qtn0 + 〈Qt
nD〉 nisin [N ] tisin [T ]
where qtn0 isin R and Qtn isin RNtimesT are decision variables In particular we let the lth column of Qt
n
denoted by (Qtn)l be the zero vector 0 when lge t If t= 1 then Q1
n = 0 That captures the ldquonon-
anticipativerdquo property since there is no way for the order quantity to depend on future demand
realization
So at the end of each period the inventory level is also an affine function of demand realization
This follows because for all tisin [T ]
xtn
983059D
t983060=
t983131
m=1
983059qmn
983059D
mminus1983060minus Dnm
983060=
t983131
m=1
983059qmn0 + 〈Qm
n D〉minus Dnm
983060= xt
n0 + 〈XtnD〉 (7)
here xtn0 =
t983123m=1
qmn0 and Xtn =
t983123m=1
(Qmn minus ene
primem)
Since now the routing decisions ztij for (i j) isinA and t isin [T ] are nonadaptable the transporta-
tion cost983123
(ij)isinA cijztij is independent of the demand realizations Hence the constraint in model
(6) supPisinP EP
983059u983059983059983123
(ij)isinA cijztij minusBt
983060βt
983060983060le 0 reduces to the deterministic budget constraint
983123(ij)isinA cijz
tij leBt because the utility function u is non-decreasing and u(0) = 0 Under this deci-
sion rule approach our model can be formulated as follows
Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty16 Article submitted to manuscript no (Please provide the mansucript number)
inf983131
tisin[T ]
983131
nisin[N ]
αtn
st supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062le 0forallnisin [N ] tisin [T ] (a)
qtn0 + 〈QtnD〉 ge 0 forallD isinW nisin [N ] tisin [T ] (b)
qtn0 + 〈QtnD〉 leMyt
n forallD isinW nisin [N ] tisin [T ] (c)
xtn0 =
t983131
m=1
qmn0 forallnisin [N ] tisin [T ]
Xtn =
t983131
m=1
(Qmn minus ene
primem) forallnisin [N ] tisin [T ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 forallnisin [N ] tisin [T ]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
(8)
Here ZR is the feasible set for static routing decisions
ZR =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
yt0 y
tn z
tij isin 01
forallnisin [N ] tisin [T ]
(i j)isinA
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
983131
(ij)isinA
cijztij leBt foralltisin [T ]
ytn le yt
0 forallnisin [N ] tisin [T ]
983131
j(nj)isinA
ztnj = ytn forallnisin [N ]cup 0 tisin [T ]
983131
j(jn)isinA
ztjn = ytn forallnisin [N ]cup 0 tisin [T ]
983131
(ij)isinA(S)
ztij le983131
nisinS
ytn minus yt
k forallS sube [N ] |S|ge 2 k isin S tisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
(9)
Instead of directly using the constraint αtn gt 0 as in the original problem (6) here we use αt
n ge 983171
so that the feasible set will be closed Indeed we can always choose a positive 983171 small enough that
optimality is not compromised (Chen et al 2015)
4 Solution Procedure
The main problem (8) involves an inner optimization over the set of possible distributions ie
the operation of supPisinP in the first constraint Therefore the optimization procedure inevitably
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
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Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
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Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
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inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
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selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
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Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
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and robust optimization Mathematical Programming 130(1) 177-209
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349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
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and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
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Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
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69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
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Journal of Business Logistics 20(1) 183-203
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Research 62(6) 1358-1376
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uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 17
depends on the information available about uncertain demand In this section we describe different
types of information sets P and then give the corresponding reformulations and solution techniques
41 Stochastic approach
We first consider the case when uncertain demand follows a known distribution ie P is a singleton
Thus we assume that D takes the value ofD(m) with probability pmmisin [Ms] where983123
misin[Ms]pm =
1 In this case
P =
983099983103
983101P
983055983055983055983055983055983055P983059D=D(m)
983060= pmmisin [Ms]
983131
misin[Ms]
pm = 1
983100983104
983102 (10)
and the LHS of (8a) becomes
supPisinP
EP
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
=983131
misin[Ms]
pm
983061maxkisin[K]
983153akmax
983153τ tn minusxt
n0 minus〈XtnD
(m)〉 xtn0 + 〈Xt
nD(m)〉minus τ t
n
983154+ bkα
tn
983154983062
Hence we can now reformulate the overall problem as a mixed-integer linear programming (MILP)
Proposition 2 Given P as defined by the equation (10) the constraints (8a)ndash(8c) are equivalent
to these constraints
983131
misin[Ms]
pmνmnt le 0 forallnisin [N ] tisin [T ]
νmnt ge ak
983059τ tn minusxt
n0 minus〈XtnD
(m)〉983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
νmnt ge ak
983059xtn0 + 〈Xt
nD(m)〉minus τ t
n
983060+ bkα
tn forallmisin [Ms] k isin [K] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 ge 0 forallmisin [Ms] nisin [N ] tisin [T ]
qtn0 + 〈QtnD
(m)〉 leMytn forallmisin [Ms] nisin [N ] tisin [T ]
It follows that problem (8) can be reformulated as an MILP
42 Robust approach with mean absolute deviation
When knowledge about distribution is incomplete we can also characterize uncertain demand using
a distributionally robust optimization framework (see eg Delage and Ye 2010 Goh and Sim 2010
Wiesemann et al 2014) Because the deterministic version of IRP is already difficult to solve we
need to choose the ambiguity set P so as not to significantly increase computational complexity
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
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Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
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Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
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Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
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Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
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Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
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Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
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Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
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Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
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a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
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Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
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uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty18 Article submitted to manuscript no (Please provide the mansucript number)
We accommodate that consideration by introducing a P that allows us to capture the correlation
effect among uncertain demands Formally
P =
983099983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059Dle DleD
983060= 1 (a)
EP
983059D983060=Ξ (b)
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ (c)
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lhforalll isin [L] hisin [H] (d)
983100983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983102
(11)
In this set P the equation (11a) specifies the uncertain demandrsquos bound support while the
equation (11b) specifies its mean The inequality (11c) specifies the parameter Σnt as the bound of
uncertain demandrsquos mean absolute deviation Similar to the standard deviation this mean absolute
deviation provides a direct measure of uncertain demandrsquos dispersion about its mean in addition
it can be obtained in closed form for most common distributions (see eg Pham-Gia and Hung
2001) For example if Dnt is distributed uniformly on [01] then the mean absolute deviation
takes the value 14 The mean absolute deviation of a normal distribution amounts to the standard
deviation multiplied by9831552π
In constraint (11d) fl(middot) is a piecewise linear convex function that can be represented byKl pieces
as fl(x) = maxkisin[Kl] olkx+ plk This information set is intended to capture the correlation of
uncertain demand among different retailers over several periods In practice the respective demand
of these retailers may be correlated since retailers in general are geographically dispersed within a
nearby region and serve common customers As an illustrative example we consider a special case of
(11d) in which f1(x) =maxxminusx= |x| If S1 = (i τ)iisin[N ] then we have EP
983075983055983055983055983055983055983123
iisin[N ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le
98317111 in essence this is a bound on the variation of total normalized demand over all retailers in
period τ Similarly for S2 = (i τ)τisin[T ] we have EP
983075983055983055983055983055983055983123
τisin[T ]
DiτminusΞiτΣiτ
983055983055983055983055983055
983076le 98317112 which bounds the
variation of the total normalized demand over all periods at node i We remark that with typical
characterization of correlation effect such as the covariance matrix the inventory management
problem is computationally challenging even when there is just one retailer Nonetheless we shall
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 19
later show that our problem remains tractable with the joint dispersion information described by
constraint (11d)
We now demonstrate that for the distributional uncertainty set P defined by the equation (11)
the constraints in problem (8) can be transformed into linear ones
Proposition 3 Let P be as defined in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8a) can be reformulated as the following set of linear constraints
stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh le 0
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0 forallk isin [K]
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0 forallk isin [K]
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ] k isin [K]
Ft
nk +Gt
nk = T tn forallk isin [K]
F tnk +Gt
nk = T tn forallk isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H] k isin [K]
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 forallk isin [K]
wtnklhkprime w
tnklhkprime ge 0 foralll isin [L] hisin [H] k isin [K] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty20 Article submitted to manuscript no (Please provide the mansucript number)
A similar technique can be used to transform constraint (8b) as follows
Proposition 4 Let P be defined as in the equation (11) Then for any n isin [N ] and t isin [T ] the
constraint (8b) can be written as
minusqtn0 + 〈DLtn〉minus 〈DH t
n〉 le 0
Qtn +Lt
n minusH tn = 0
LtnH
tn ge 0
We remark that much as in Proposition 4 the constraint (8c) can also be represented equivalently
by a set of linear constraints Now we can achieve the following reformulation of the overall problem
Corollary 1 Given P as defined in the equation (11) the problem (8) can be formulated as an
MILP
A full description of the reformulation is given in the Appendix F
The solution quality can be further improved by incorporating a more sophisticated decision
rule For instance Bertsimas et al (2018) define the lifted ambiguity set by introducing certain
auxiliary random variables and then incorporating them into an approximation of a linear decision
rule To differentiate that approach from the traditional one we call it the enhanced linear decision
rule (ELDR) We can use this ELDR in the case of the ambiguity set P considered previously For
that purpose we introduce the auxiliary random variables U and V and then define the following
lifted ambiguity set Q
Q=
983099983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983101
Q
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
Q983059983059
D U V983060isin 983142W
983060= 1
EQ
983059D983060=Ξ
EQ
983059U983060leΣ
EQ [vlh]le 983171lhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983102
(12)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
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Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
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Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
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Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
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of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
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Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
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Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
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Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
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Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
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Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
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Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
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Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
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Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
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Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
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349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
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Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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69(1) 99-118
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uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 21
here 983142W is the lifted support defined formally as
983142W =
983099983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983101
983059D U V
983060isinRNtimesT timesRNtimesT timesRLtimesH
983055983055983055983055983055983055983055983055983055983055983055983055983055
Dle DleD983055983055983055DminusΞ
983055983055983055le U
fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108le vlhforalll isin [L] hisin [H]
983100983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983102
where vlh denotes the element located at the lth row and hth column of matrix V Wiesemann
et al (2014) show that the ambiguity set P is equivalent to the set of marginal distributions of
D under Q for all Q isin Q Under ELDR the ordering quantity qtn n isin [N ] t isin [T ] is restricted
to depend linearly not only on the primary random variable D but also on the auxiliary random
variables U and V Then the order quantity can be written as
qtn
983059D U V
983060= qtn0 + 〈Qt
0nD〉+ 〈Qt1n U〉+ 〈Qt
2n V 〉 nisin [N ] tisin [T ]
here qtn0 isinR Qt0n isinRNtimesT Qt
1n isinRNtimesT and Qt2n isinRLtimesH are decision variables Since the adaptive
decisions cannot be anticipated both983043Qt
0n
983044land
983043Qt
1n
983044lare set to 0 for all l ge t we also put
983043Qt
2n
983044h= 0 when there exists some τ ge t such that (i τ)isin Sh Now constraint (8a) can be written
as
supQisinQ
EQ
983061maxkisin[K]
983153akmax
983153τ tn minus qtn0 minus〈Qt
0nD〉minus 〈Qt1n U〉minus 〈Qt
2n V 〉
qtn0 + 〈Qt0nD〉+ 〈Qt
1n U〉+ 〈Qt2n V 〉minus τ t
n
983154+ bkα
tn
983154983060le 0
(13)
Here problem (8) can be solved by the corresponding reformulation method This claim is formal-
ized in our next proposition
Proposition 5 Let the distribution ambiguity set Q be as defined in the equation (12) and let
constraint (8a) be replaced by (13) Then problem (8) can be equivalently formulated as an MILP
43 Robust approach with Wasserstein distance
Considerable interest has recently been shown in ambiguity sets whose construction is based on the
notion of statistical distance In particular by measuring the proximity of probability distributions
with statistical distance measures the ambiguity sets are defined as the set of probability distri-
butions which are ldquocloserdquo to the reference distribution The examples of such statistical distance
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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ations Research 58(4-part-1) 902-917
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Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
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stochastic inventory routing problem Transportation Science 38(1) 42-70
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and robust optimization Mathematical Programming 130(1) 177-209
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Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
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69(1) 99-118
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Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
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Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty22 Article submitted to manuscript no (Please provide the mansucript number)
are φ-divergence (Ben-Tal et al 2013 Wang et al 2016 Jiang and Guan 2016) and Wasserstein
distance (Esfahani and Kuhn 2017 Gao and Kleywegt 2016) Among others Wasserstein distancersquos
increasing popularity is due to its powerful out-of-sample performance Here we show that our
model also enables use of an ambiguity set based on Wasserstein distance when the matrix norm
is chosen as entrywise 1-norm
We denote by P0(W) the space of all probability distributions supported on W The reference
probability distribution Pdagger is given for the random variable Ddaggerwhich takes values D(1) D(Mr)
with equal likelihood that is
Pdagger983059D
dagger= D
(m)983060=
1
Mr
forallmisin [Mr]
Given any distribution PisinP0(W) on D the Wasserstein distance between P and Pdagger is defined as
dW (PPdagger) = inf EP
983059983056983056983056Dminus Ddagger983056983056983056983060
st (DDdagger)sim P
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
Here P is the joint distribution of D and Ddagger the terms ΠDP and Π
DdaggerP stand for the marginal
distribution of P on D and Ddagger respectively and 983042 middot 983042 represents an arbitrary norm Now the
ambiguity set can be constructed as the Wasserstein ball of radius θgt 0 centered at the reference
probability distribution Pdagger
P =983051PisinP0(W)|dW (PPdagger)le θ
983052 (14)
Esfahani and Kuhn (2017) and Gao and Kleywegt (2016) show that strong duality holds while
studying the distributionally robust optimization problem with Wasserstein distance We now
incorporate that result into our problem In this case reformulating constraints (8b)ndash(8c) to a finite
set of linear constraints can proceed as described in Proposition 4 hence we focus on constraints
(8a)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 23
Proposition 6 Let P be as defined in the equation (14) and let the norm 983042 middot 983042 be chosen as the
entrywise 1-norm Then for any given nisin [N ] and tisin [T ] the constraints (8a) are equivalent to
rtnθ+983131
misin[Mr ]
stnm le 0
〈DV tnmk〉minus 〈DU t
nmk〉+ 〈D(m)
Rtnmk〉 leMrs
tnm minus ak (τ
tn minusxt
n0)minus bkαtnforallmisin [Mr] k isin [K]
〈DVt
nmk〉minus 〈DUt
nmk〉+ 〈D(m)
Rt
nmk〉 leMrstnm minus ak (x
tn0 minus τ t
n)minus bkαtnforallmisin [Mr] k isin [K]
minusU tnmk +V t
nmk =minusakXtn minusRt
nmk forallmisin [Mr] k isin [K]
minusUt
nmk +Vt
nmk = akXtn minusR
t
nmk forallmisin [Mr] k isin [K]
983043Rt
nmk
983044iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983043Rt
nmk
983044iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτle rtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
983059R
t
nmk
983060
iτgeminusrtn foralliisin [N ] τ isin [T ]misin [Mr] k isin [K]
U tnmkV
tnmkU
t
nmkVt
nmk ge 0 forallmisin [Mr] k isin [K]
rtn ge 0stn isinRMs
Therefore problem (8) can be formulated as an MILP
Remark 1 After accounting for the risk described in Section 22 and for the uncertainty sets
introduced in Sections 41-43 we find that our problemrsquos final formulation is still an MILP Com-
pared with the deterministic inventory routing formulation only a set of linear constraints and
continuous random variables are added
Note that the computation can be expedited by applying in our stochastic framework meth-
ods designed for the deterministic IRP Most valid cuts developed for deterministic problem to
strengthen the routing decisions remain applicable in this context example include ztij le yti z
tij le yt
j
and983123
nisin[N ] qtn
983059D
tminus1983060le yt
0 (Archetti et al 2007) We can also facilitate computation by limiting
the feasible routes to a pre-determined subset such that instances of much larger scale can be solved
(Bertsimas et al 2016)
Finally there are many possible directions in which this framework can be extended From the
operational perspective one could forgo specifying the budget for each period and instead stipulate
Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty24 Article submitted to manuscript no (Please provide the mansucript number)
a total budget that controls the transportation cost within a given planning horizon Furthermore
it would be straightforward to consider the case of multiple vehicles with capacity constraints The
model we propose also allows for incorporating the production decisions (Adulyasak et al 2013
2015)
44 Benders decomposition
In Sections 41ndash43 we showed that our IRP under uncertainty can be formulated as an MILP
Although the size of the resulting MILP is large we can solve it using the Benders (1962) decom-
position approach (for additional details see Rahmaniani et al 2016) For completeness we also
describe how our own model can incorporate the Benders decomposition and hence be solved more
efficiently
Our preceding reformulations of the inventory routing problem as an MILP differ since they use
different information set to characterize uncertainty Here we express all those formulations in the
following general form
min gprimeφ
st aprimeiφ+ bprimeiy0 + 〈Cy
i Y 〉+ 〈Czi Z〉 le si foralliisin [I]
(y0Y Z)isinZR
(15)
where φ is used to represent all continuous decision variables the terms y0 isin 01T Y isin 01NtimesT
and Z isin 01|A|timesT are the routing decisions all of which are binary So if we now define a function
f ZR rarrRcup infin as
f(y0Y Z) =min gprimeφ
st aprimeiφle si minus bprimeiy0 minus〈Cy
i Y 〉minus 〈Czi Z〉 foralliisin [I]
(16)
then the main problem (15) is equivalent to min(y0Y Z)isinZRf(y0Y Z) In defining f(y0Y Z) we
consider the minimization problem to be a primal problem its dual problem can be easily obtained
as
fd(y0Y Z) =max983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi
st983131
iisin[I]
aiψi = g
(17)
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 25
We note that problem (15) is bounded from below by NT 983171 Hence the primal problem (16)
must be either finite or infeasible from which it follows that the dual problem (17) must be either
finite or unbounded In the former case by strong duality we have f(y0Y Z) = fd(y0Y Z)
The latter case occurs if and only if983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 for some ψ with
983123iisin[I]aiψi = 0 Define an optimization problem as follows
min η
st983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le η forallψ isin G1
983131
iisin[I]
(si minus bprimeiy0 minus〈Cyi Y 〉minus 〈Cz
i Z〉)ψi le 0 forallψ isin G2
(y0Y Z)isinZR
(18)
If G1 = ψ|983123
iisin[I]aiψi = g and G2 = ψ|983123
iisin[I]aiψi = 0 then the main problem (15) is equivalent
to (18)
We incorporate the Benders decomposition as described in the following algorithm
Algorithm Benders Decomposition
1 Initialize G1 as a singleton containing an arbitrary extreme point of the polyhedron
ψ|983123
iisin[I]aiψi = g and initialize G2 = empty
2 Solve the problem (18) and denote the optimal solution by (ηlowastylowast0Y
lowastZlowast)
3 Solve problem (17) with (y0Y Z) = (ylowast0Y
lowastZlowast) If fd(ylowast0Y
lowastZlowast) is finite then let ψlowast
denote an optimal extreme-point solution and update G1 = G1 cup ψlowast Otherwise let ψlowast signify
the direction with983123
iisin[I](si minus bprimeiy0 minus 〈Cyi Y 〉 minus 〈Cz
i Z〉)ψi gt 0 and983123
iisin[I]aiψi = 0 and update
G2 = G2 cup ψlowast
4 If ηlowast = fd(ylowast0Y
lowastZlowast) then terminate the algorithm and return (xlowastylowast0Y
lowastZlowast) as the optimal
solution otherwise return to Step 2
It is easy to show that in theory the algorithm can be terminated within a finite number of
iterations since in Step 3 there is only a finite number of extreme pointsrays On a practical
level it has been well recognized that this approach can help solve MILP with greater efficiency
(Rahmaniani et al 2016)
Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty26 Article submitted to manuscript no (Please provide the mansucript number)
5 Computational Study
In this section we carry out computational studies to show that our proposed model is practically
solvable and can yield an appealing solution under uncertainty The program is coded in C++ and
run on an Intel Core i5 PC with a 270-GHz CPU and 8 GB of RAM using CPLEX 125 Unless
otherwise stated the maximum CPU time is set to two hours
To assess the solvability of our model and the quality of its solutions we test instances with
N = 8 retailers on a randomly generated graph with T = 4 planning periods The transportation
cost between any two nodes is proportional to the distance between them
Taking the network as given we calculate the minimum cost to visit all nodes based on the
traditional traveling salesman problem We denote this minimum cost as TSP Since the initial
inventory level of each retailer is set as zero we let the budget for the first period be the TSP
such that all retailers can be served For subsequent periods we let the budget be κtimes TSP in
this setup κlt 1 indicates that the supplierrsquos vehicle cannot visit all retailers in a single period
Our experiment considers two cases κ = 07 and κ = 08 We put Dnt = 10 + 30σnt where σnt
follows the distribution of Beta(24) for all n isin [N ] and t isin [T ] In all periods all retailers share
a common inventory level requirement window [0 τ ] we vary τ to capture the effects of different
requirement windows For simplicity we choose u(x) =maxminus1 x as the utility function We also
conduct these computational studies for other distributions The results are quite consistent across
distributions so we only report the case with a Beta distribution
51 Comparison between our approaches and benchmark approach
As a means of further evaluating and understanding the benefit of our approach we compare the
quality of solutions from three approaches Two of them are variants of our SVI method and one
is a benchmark approach based on minimizing the probability of violation
511 Robust approach with mean absolute deviation (SVI-R) The first approach
which we dub the SVI-R model is from our robust optimization using information of mean absolute
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 27
deviation (Section 42) When solving for the optimal solution we use the information characterized
as follows
P =
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983101
P
983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055983055
P983059dle Dnt le d
983060= 1
EP
983059Dnt
983060= micro
EP
983059983055983055983055Dnt minusmicro983055983055983055983060le σ
EP
983091
983107
983055983055983055983055983055983055
983131
tisin[T ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171oforallnisin [N ]
EP
983091
983107
983055983055983055983055983055983055
983131
nisin[N ]
Dnt minusmicro
σ
983055983055983055983055983055983055
983092
983108le 983171lowastforalltisin [T ]
983100983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983104
983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983105983102
Here d= 10+30times 0 = 10 d= 10+30times 1 = 40 and the mean value is micro= 10+30times 22+4
= 20 The
values of σ 983171o and 983171lowast can similarly be obtained from the underlying distribution of Dnt Observe
that the penultimate constraint is to incorporate the correlation effect among uncertain demand
of all periods for each retailer the last constraint is for the correlation among uncertain demand
of all retailers for each period We lift the ambiguity set P to Q as described in the equation (12)
after which we solve the problem using ELDR as described in Proposition 5
512 Stochastic approach (SVI-S) In the second approach called the SVI-S model we
have exact information on the distribution of D (Section 41) Following the underlying distribu-
tion on D we generate Ms samples of D and use the sampling distribution as P in Section 41 As
expected performance and computation time depend strongly on the number Ms of training sam-
ples A large number of samples will describe the distribution better and lead to a good solution
but computation time will increase significantly because there are more constraints involved We
test the performance of our SVI-S model in two cases Ms = 100 and Ms = 500
Note that as the feature of distributional robustness has been reflected by the SVI-R model
here we do not incorporate the model based on Wasserstein distance (Section 43) to avoid the
specification of θ which might be abstract in the computational study Indeed the SVI-S is its
special case with θ= 0
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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52(4) 499-514
Adulyasak Y Cordeau J F Jans R (2013) Formulations and branch-and-cut algorithms for mul-
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Adulyasak Y Cordeau J F Jans R (2015) Benders decomposition for production routing under
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Aghezzaf E H (2008) Robust distribution planning for supplier-managed inventory agreements
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Andersson H Hoff A Christiansen M Loslashkketangen A (2010) Industrial aspects and literature
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1515-1536
Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
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Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
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Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
9(3) 203-228
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Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
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Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
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Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
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248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty28 Article submitted to manuscript no (Please provide the mansucript number)
513 Minimizing violation probability (MVP) Since we focus on the service level a
natural benchmark is to minimize the probability of an inventory requirement violation that
is to minimize 1NT
983123nisin[N ]
983123tisin[T ] P(v0τ (xnt) gt 0) Because this so-called MVP model is highly
intractable we can use the sampling average approximation to derive only static decisions not
adjustable ones More specifically we generate Mc sample paths for uncertain demands as D(m)
misin [Mc] Then the problem of MVP can be approximated as
min1
McNT
983131
misin[Mc]
983131
tisin[T ]
983131
nisin[N ]
I(m)nt
st I(m)nt ge x
(m)nt minus τ
Mforallmisin [Mc] nisin [N ] tisin [T ]
I(m)nt ge minusx
(m)nt
M forallmisin [Mc] nisin [N ] tisin [T ]
qtn leMytn forallnisin [N ] tisin [T ]
qtn ge 0 forallnisin [N ] tisin [T ]
I(m)nt isin 01 forallmisin [Mc] nisin [N ] tisin [T ]
983043yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA
983044isinZR
Here ZR is as defined in the equation (9) M is a sufficiently large scalar for the use of big-M
method If the sample size Mc is large enough then this formulation closely approximates the MVP
model however a large sample size will also result in long computation times In this experiment
we set Mc = 100
514 Comparison For each approach we first derive the optimal solution and then generate
Mo = 10000 sample paths of demand D to test the quality of solutions We simplify the perfor-
mance analysis by aggregating the uncertain violation at all nodes and all periods into a single
uncertain violation v In particular v is the violation at node n in period t with equal likelihood
for all n isin [N ] and t isin [T ] formally P(v = maxxnt minus τ minusxnt0) = 1NT So for each solution
we have Mo timesN timesT instances of v Using this sample set we report each modelrsquos performance as
measured by five criteria a probability of there being a violation P(v gt 0) the violationrsquos expected
value E(v) and standard deviation STD(v) and the Value at Risk (VaR) at the 95th and 99th
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
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Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
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Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
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Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
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Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
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Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
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Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
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Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
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Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
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Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
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Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
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Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
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Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
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Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 29
percentiles denoted VaR095(v) and VaR099(v) For each criterion a smaller value corresponds to
better performance We also report how long in seconds it takes to obtain the solution as ldquoCPU
timerdquo
The performance of our SVI-R model is presented in Table 2 As the transportation budget
increases (higher κ) the supplier has more flexibility in choosing routes which reduces the risk of
violation If the transportation budget is held constant then when τ increases the ending inventory
requirement is less restrictive once again violations are thus less likely The SVI-R model entails
reasonable computation time fewer than 100 seconds are needed to solve for an eight-node network
over four periods
κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU time07 27 1313 053 192 389 1044 56
30 911 042 185 265 1044 9235 553 027 148 064 849 86
08 27 1123 046 180 338 990 6530 778 036 171 190 975 6235 472 023 136 000 779 62
Table 2 Performance of the SVI-R model
We summarize the performance of other models in Table 3 For purposes of comparison we
normalize the performance of these models by that of our SVI-R model Hence a value greater
than 1 in Table 3 indicates that the risk is higher than under the SVI-R model We observe that
for Ms = 100 the computation times for solving the SVI-S model and SVI-R models are similar
Yet performance is poor as reflected by most values exceeding unity When the sample size Ms
increases to 500 computation time increases significantly but performance is only slightly better
than under SVI-S with Ms = 100 and is still worse than the SVI-R modelrsquos performance Recall
that performance of the MVP model is also affected by the training sample size But even for 100
samples this approach cannot deliver an optimal solution within the two-hour time limit The MVP
performance is the worst among all these models in terms not only of the violation probability but
also of the violation magnitude We therefore conclude that our proposed SVI is a good measure
for capturing the risk of a requirement violation and we know that it can be computed efficiently
Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
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butionally robust linear optimization problems Management Science (forthcoming)
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Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
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Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
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Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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utility International Economic Review 49 683-700
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Part E Logistics and Transportation Review 38(2) 75-95
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Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
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Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
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Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
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Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
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Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
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Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
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Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
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Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
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Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
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Z Cui et al Inventory Routing Problem under Uncertainty30 Article submitted to manuscript no (Please provide the mansucript number)
For the case κ= 08 and τ = 35 since VaR095(v) = 0 we can only report ldquonardquo (ie not applicable)
in the table Indeed all of the VaR095(v) values for models SVI-S and MVP are positive
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v) CPU timeSVI-S 07 27 132 130 110 129 102 159
(Ms = 100) 30 172 145 108 167 099 09135 172 122 100 356 093 105
08 27 139 135 112 136 104 16530 185 158 115 223 106 14535 201 143 109 na 102 155
SVI-S 07 27 113 106 098 107 093 2230(Ms = 500) 30 127 090 081 109 076 1243
35 124 070 068 145 062 112808 27 114 107 099 110 096 752
30 127 092 084 129 078 53735 125 074 071 na 064 395
MVP 07 27 158 223 170 214 155 1285730 181 214 157 253 143 782635 195 200 151 611 141 8372
08 27 170 228 171 227 155 1107730 196 225 160 324 145 1161335 215 217 158 na 148 11613
Table 3 Comparison of performance for different benchmarks
52 Robustness
We shall now explain the need to incorporate distributional ambiguity From a practical standpoint
given a set of historical data we can extract certain information and then derive the corresponding
optimal solution For the SVI-R model we use information of mean absolute deviation for both the
SVI-S and MVP models we use the sampling distribution Nevertheless the samples may deviate
from the true distribution We test the performance of different models under this scenario
In particular we assume that the true distribution of demand is as given in Section 51 Dnt =
10 + 30σnt where σnt follows the Beta(24) distribution For this test however the samples are
drawn from the ldquowrongrdquo distribution Dwnt = 10+30σw
nt where σwnt follows the Beta(48) distribution
Note that the true and wrong distributions do not deviate too much in the sense that Dnt and
Dwnt have the same mean and bound We derive the optimal solution for all models based on the
samples drawn from the wrong distribution Dwnt and then test the solution using samples drawn
from the true distribution Dnt Because the mean absolute deviation information we extract from
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 31
both Dwnt and Dnt is the same the SVI-R modelrsquos solution here is as the same as in Section 51
And since the testing samples are both from Dnt it follows that the SVI-R modelrsquos performance
is identical to that reported in Table 2 In Table 4 we summarize the other modelsrsquo performances
which are again normalized by that of the SVI-R model
Model κ τ P(v gt 0) E(v) STD(v) VaR095(v) VaR099(v)SVI-S 07 27 175 213 158 186 143
(Ms = 100) 30 199 186 130 204 11735 226 174 122 534 110
08 27 194 220 153 201 13630 227 214 142 283 12635 268 235 155 NA 141
SVI-S 07 27 129 121 104 119 097(Ms = 500) 30 150 112 090 132 083
35 162 100 086 273 08008 27 137 124 104 125 098
30 161 117 093 168 08535 177 113 090 NA 085
MVP 07 27 166 223 168 209 15130 192 207 148 238 13235 210 200 153 558 140
08 27 184 228 164 220 14630 207 217 149 306 13535 234 213 149 NA 139
Table 4 Performance of different models when given the wrong distribution information
Since the performance of models SVI-S and MVP are highly dependent on the training samples
the solutions these models derive from the wrong distribution can easily lead to performance far
inferior to that of SVI-R This claim is confirmed by the Table 4 values exceeding those reported
in Table 3 So in comparison with the other models evaluated here adopting our SVI-R approach
will better immunize the decision maker against the effects of data ambiguity
6 Conclusions
We take a distributionally robust optimization approach to exploring the stochastic inventory
routing problem After proposing a new decision criterion for evaluating the associated risk we
develop methods geared to solve for the optimal affine policies Our model can easily be extended to
incorporate various operational features (eg production) and its computational efficiency can be
Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty32 Article submitted to manuscript no (Please provide the mansucript number)
improved by employing some well-developed heuristics (eg restricting to only a subset of routes)
We undertake computational studies to illustrate the strength of our method
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 33
Appendix A Proof of Theorem 1
We first prove the ldquoifrdquo direction If micro is a convex risk measure we prove that the ρτ τ (x) defined in the
equation (2) is an SVI by showing its satisfaction of the properties in Definition 1
1 Monotonicity If vτiτi(xi)ge vτiτi
(yi)foralliisin [I] then by the monotonicity of micro we have
micro
983061vτiτi
(xi)
αi
983062ge micro
983061vτiτi
(yi)
αi
983062 forallαi gt 0
Hence ρτ τ (x)ge ρτ τ (y)
2(a) Attainment content If vτiτi(xi)le 0foralliisin [I] we have for any αi gt 0
micro
983061vτiτi
(xi)
αi
983062le micro(0) = 0
Hence ρτ τ (x) = 0
2(b) Starvation aversion If existiisin [I] such that vτiτi(xi)gt 0 then for any αi gt 0 we have that vτiτi
(xi)αi gt
0 Hence exist983171gt 0 such that vτiτi(xi)αi ge 983171 and
micro
983061vτiτi
(xi)
αi
983062ge micro(983171) = micro(0)+ 983171= 983171gt 0
Therefore ρτ τ (x) =infin
3 Convexity Given any xisin VI we define a set of α as
S(x) =
983069α
983055983055983055983055micro983061vτiτi
(xi)
αi
983062le 0αi gt 0 foralliisin [I]
983070
We consider any αx isin S(x) αy isin S(y) and define αλ = λαx + (1minus λ)αy for any λ isin [01] Based on the
definition of vτ τ (x) =maxxminus τ τ minus x the function vτiτi(middot) is convex Hence
micro
983061vτiτi
(λxi +(1minusλ)yi)
αλi
983062le micro
983061λvτiτi
(xi)+ (1minusλ)vτiτi(yi)
αλi
983062
= micro
983061λαx
i
αλi
vτiτi(xi)
αxi
+(1minusλ)αy
i
αλi
vτiτi(yi)
αyi
983062
le omicro
983061vτiτi
(xi)
αxi
983062+(1minus o)micro
983061vτiτi
(yi)
αyi
983062
le 0
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
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Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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cation to data-driven problems Operations Research 58(3) 595-612
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utility International Economic Review 49 683-700
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Part E Logistics and Transportation Review 38(2) 75-95
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Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
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Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
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Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
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Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
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a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
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inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
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uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty34 Article submitted to manuscript no (Please provide the mansucript number)
where o= λαxi α
λi isin [01] The first inequality holds because of the convexity of vτiτi
(middot) and the monotonicity
of micro(middot) the second inequality is due to the convexity of micro and the last inequality holds based on the definition
of S(x) and S(y) Hence for any λisin [01]αx isin S(x)αy isin S(y) αλ isin S(λx+(1minusλ)y) and we get
λρτ τ (x)+ (1minusλ)ρτ τ (y) = λ inf
983069I983123
i=1
αxi α
x isin S(x)
983070+(1minusλ) inf
983069I983123
i=1
αyi α
y isin S(y)
983070
= inf
983069I983123
i=1
(λαxi +(1minusλ)αy
i ) αx isin S(x)αy isin S(y)
983070
ge inf
983069I983123
i=1
αiαisin S(λx+(1minusλ)y)
983070
= ρτ τ (λx+(1minusλ)y)
4 Positive homogeneity For all λgt 0
ρλτ λτ (λx) = inf
983083I983131
i=1
αi
983055983055983055983055micro983061vλτiλτi
(λxi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061λvτiτi
(xi)
αi
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
αi
983055983055983055983055micro983061vτiτi
(xi)
αiλ
983062le 0αi gt 0foralliisin [I]
983084
= inf
983083I983131
i=1
λβi
983055983055983055983055micro983061vτiτi
(xi)
βi
983062le 0βi gt 0foralliisin [I]
983084
= λρτ τ (x)
5 Dimension-wise additivity It is obvious due to representation (2)
6 Order invariance It is obvious due to representation (2)
7 Left continuity Denote αlowast = ρminusinfin0(x) αlowasti = ρminusinfin0(xiei) i isin [I] By dimension-wise additivity we
have αlowast =983123
iisin[I]αlowasti We next prove the left continuity in two cases
Case 1 If αlowasti isinR+foralli isin [I] then αlowast isinR+ We need to show that forall983171gt 0 there exists δ gt 0 such that forallδ isin
9830430 δ
983044 |ρminusinfin0(xminusδ1)minusρminusinfin0(x)|le 983171 Since vminusinfin0(xminusδ1) =xminusδ1lex= vminusinfin0(x) by Monotonicity we
have ρminusinfin0(xminusδ1)le ρminusinfin0(x) Hence we only need to prove ρminusinfin0(xminusδ1)ge ρminusinfin0(x)minus983171= αlowastminus983171 The
case of 983171ge αlowast is trivial For 983171isin (0αlowast) we define I = iisin [I]|αlowasti gt 0 and I = [I]I = iisin [I]|αlowast
i = 0 Then
we have 983171lt αlowast =983123
iisinI αlowasti Hence we can find a vector 983171iiisinI such that
983123iisinI 983171i = 983171 and 983171i isin (0αlowast
i ) foralliisin I
Choose δ =miniisinI(αlowasti minus 983171i)micro(
vminusinfin0(xi)
αlowastiminus983171i
) Since αlowasti = ρminusinfin0(xiei) = infαgt 0 |micro(vminusinfin0(xi)α)le 0 foralliisin I δ
is strictly positive Considering forallδ isin (0 δ) and iisin I we have
micro
983061vminusinfin0(xi minus δ)
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)minus δ
αlowasti minus 983171i
983062= micro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
gtmicro
983061vminusinfin0(xi)
αlowasti minus 983171i
983062minus δ
αlowasti minus 983171i
ge 0
where the first equality holds due to the fact that forallx isinR vminusinfin0(x) =maxminusinfinminus xxminus 0= x the second
equality holds because of cash invariance and the last inequality holds due to the definition of δ By Lemma
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
(1962) Numerische Mathematik 4(1) 238-252
Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
affected by uncertain probabilities Management Science 59(2) 341-357
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
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Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
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inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
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selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
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Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
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Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
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Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
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and robust optimization Mathematical Programming 130(1) 177-209
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Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
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Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 35
1 in Chen et al (2015) and the representation (2) we know ρminusinfin0((ximinus δ)ei)ge αlowasti minus 983171i For iisin I obviously
we have ρminusinfin0((xi minus δ)ei) = 0 Therefore
ρminusinfin0(xminus δ1) =983131
iisinI
ρminusinfin0((xi minus δ)ei)ge983131
iisinI
(αlowasti minus 983171i) = αlowast minus 983171
Case 2 If existiisin [I] such that αlowasti =infin then αlowast =infin Suppose lim
adarr0ρminusinfin0(xminusa1) is not infin ie lim
adarr0ρminusinfin0(xminus
a1) = αo isinR+ It implies that forall983171gt 0 existδgt 0 such that forallδ isin9830430 δ
983044 we have ρminusinfin0(xminus δ1)isin [αo minus 983171αo + 983171]
Since αlowasti = ρminusinfin0(xiei) =infin we have micro(
vminusinfin0(xi)
α)gt 0 forallαgt 0 Hence we are able to choose
δ=min
983069δ
2 α
983061micro
983061vminusinfin0(xi)
α
983062minus η
983062983070isin (0 δ)
where α= αo +2983171 and η is any real number in (0 micro(vminusinfin0(xi)α)) As δ isin9830430 δ
983044 we still have ρminusinfin0((xi minus
δ)ei)le ρminusinfin0(xminusδ1)le αo+983171 So by Lemma 1 in Chen et al (2015) micro (vminusinfin0(xi minus δ)α)le 0 since αgt αo+983171
We can also get that
micro
983061vminusinfin0(xi minus δ)
α
983062= micro
983061vminusinfin0(xi)minus δ
α
983062
= micro
983061vminusinfin0(xi)
α
983062minus δ
αge micro
983061vminusinfin0(xi)
α
983062minus983061micro
983061vminusinfin0(xi)
α
983062minus η
983062= ηgt 0
which contradicts with micro (vminusinfin0(xi minus δ)α)le 0 So the assumption is false limadarr0
ρminusinfin0(xminus a1) =infin
We next prove the ldquoonly ifrdquo direction First we need to show function micro defined by the equation (3) is a
convex risk measure Notice that forallxisinRI vminusinfin0(x) =maxminusinfinminusxxminus0=x
1 Monotonicity If xge y then vminusinfin0((xminus a)e1)ge vminusinfin0((yminus a)e1) for any a isinR Hence based on the
monotonicity property we have ρminusinfin0((xminus a)e1)ge ρminusinfin0((yminus a)e1) According to the definition of micro(middot)
we get micro(x)ge micro(y)
2 Cash invariance Notice that forallw isinR
micro(x+w) = mina983055983055ρminusinfin0((x+wminus a)e1)le 1
= mina+w983055983055ρminusinfin0((xminus a)e1)le 1
= micro(x)+w
3 Convexity
ρminusinfin0 ((λx+(1minusλ)yminusλmicro(x)minus (1minusλ)micro(y))e1)
le λρminusinfin0 ((xminusmicro(x))e1)+ (1minusλ)ρminusinfin0 ((yminusmicro(y))e1)
le λ+(1minusλ)
= 1
Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty36 Article submitted to manuscript no (Please provide the mansucript number)
where the first inequality follows from the convexity of ρτ τ (x) and the second inequality holds based on
the definition of micro Hence we have
micro (λx+(1minusλ)y) =min983051a983055983055ρminusinfin0 ((λx+(1minusλ)yminus a)e1)le 1
983052le λmicro(x)+ (1minusλ)micro(y)
4 Normalization ρminusinfin0 (minusae1) is 0 if age 0 and it is infin if alt 0 Therefore
micro(0) =min983051a983055983055ρminusinfin0 (minusae1)le 1
983052= 0
Finally we need to show for any SVI defined as (2) the equation (3) holds With Left continuity of SVI
the minimum in (3) is achievable Therefore
min983051a983055983055ρminusinfin0 ((xminus a)e1)le 1
983052
= min
983099983103
983101a
983055983055983055983055983055983055
983131
iisin[I]
αi le 1 micro
983061vminusinfin0(xminus a)
α1
983062le 0 micro
983061vminusinfin0(0)
αj
983062le 0 j = 2 I αi gt 0 iisin [I]
983100983104
983102
= min
983069a
983055983055983055983055αle 1 micro
983061vminusinfin0(xminus a)
α
983062le 0 αgt 0
983070
= mina |micro (vminusinfin0(xminus a))le 0
= mina |micro (xminus a)le 0
= mina |micro (x)le a
= micro(x)
where the second equality is due to micro(vminusinfin0(0)αj)le 0 forallαj gt 0 the fourth equality holds for the definition
of function v and the fifth equality holds for the cash invariance of micro In particular the third equality is due
to the equivalence between condition A) micro(w)le 0 and condition B) existαisin (01] such that micro(wα)le 0 which
we prove as follows The direction of ldquoArArrBrdquo is trivial To see the direction of ldquoBrArrArdquo we observe that for
any such αisin (01]
micro(w) = micro
983061αw
α+(1minusα)0
983062le αmicro
983061w
α
983062+(1minusα)micro(0) = αmicro
983061w
α
983062le 0
where inequalities are due to the convexity of micro and the condition B) respectively QED
Appendix B Proof of Proposition 1
The shortfall risk measure without supPisinP
has been shown to be a convex risk measure (Follmer and Schield
2002) Incorporating distributional ambiguity it is straightforward to show our shortfall risk measure to be
convex risk measure as well
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
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Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
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Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
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Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
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Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
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Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
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Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
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Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
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Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
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Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
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uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 37
By the definition of shortfall risk measure (4) the SVI defined by (2) can be formulated as
ρτ τ (x) = inf
983069I983123
i=1
αi
983055983055983055microu
983059vτiτi
(xi)
αi
983060le 0αi gt 0 foralliisin [I]
983154
= inf
983069I983123
i=1
αi
983055983055983055983055infηisinR
983069η
983055983055983055983055supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0
983070le 0αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055existηi le 0 supPisinP
EP
983147u983059
vτiτi(xi)
αiminus ηi
983060983148le 0 αi gt 0 foralliisin [I]
983070
= inf
983069I983123
i=1
αi
983055983055983055983055 supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le 0αi gt 0foralliisin [I]
983070
The last equation holds due to the equivalence between condition A) existη le 0 such that
supPisinP
EP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 and condition B) sup
PisinPEP
983147u983059
vτiτi(xi)
αi
983060983148le 0 ldquoBrArrArdquo is trivial by setting η= 0
For the direction ldquoArArrBrdquo we see that u983059
vτiτi(xi)
αi
983060le u
983059vτiτi
(xi)
αiminus η
983060for η le 0 because u is an increasing
function So supPisinP
EP
983147u983059
vτiτi(xi)
αi
983060983148le sup
PisinPEP
983147u983059
vτiτi(xi)
αiminus η
983060983148le 0 QED
Appendix C Proof of Proposition 2
It is straightforward QED
Appendix D Proof of Proposition 3
For any node nisin [N ] and period tisin [T ] consider
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
as a primal optimization problem where the decision variable is P This is a semi-infinite linear programming
problem with infinite number of decision variables and finite constraints The strong duality holds (Isii 1962
Z Cui et al Inventory Routing Problem under Uncertainty38 Article submitted to manuscript no (Please provide the mansucript number)
Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Wiesemann et al 2014 Bertsimas et al 2018) and we can write its dual form Then
supPisinP
EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
= sup EP
983061maxkisin[K]
983153ak max
983153τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983154+ bkα
tn
983154983062
st P983059Dle DleD
983060= 1
EP
983059D983060=Ξ
EP
983059983055983055983055DminusΞ983055983055983055983060leΣ
EP
983093
983095fl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983094
983096le 983171lh foralll isin [L] hisin [H]
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge maxkisin[K]
983051ak max
983051τ tn minusxt
n0 minus〈XtnD〉 xt
n0 + 〈XtnD〉minus τ t
n
983052+ bkα
tn
983052forallDleDleD
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
= min stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983131
lisin[L]
983131
hisin[H]
983171lhrtnlh
st stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043xtn0 + 〈Xt
nD〉minus τ tn
983044+ bkα
tn forallDleDleD k isin [K] (a)
stn0 + 〈StnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
ge ak
983043τ tn minusxt
n0 minus〈XtnD〉
983044+ bkα
tn forallDleDleD k isin [K] (b)
rtnlh ge 0 foralll isin [L] hisin [H]
T tn ge 0
The last optimization problem is a semi-infinite linear programming problem with infinite number of con-
straints For forallk isinK the constraint (a) is equivalent to
minDleDleD
983099983103
983101〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlhfl
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108
983100983104
983102ge ak (xtn0 minus τ t
n)
+bkαtn minus stn0
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 39
Since function fl(middot) is a piecewise linear function represented by fl(x) = maxkprimeisin[Kl] olkprimex+ plkprime the left
hand side of the above inequality can be written as a linear optimization problem
min 〈Stn minus akX
tnD〉+ 〈T t
n |DminusΞ|〉+983131
lisin[L]
983131
hisin[H]
rtnlh maxkprimeisin[Kl]
983091
983107olkprime
983091
983107983131
(iτ)isinSh
Diτ minusΞiτ
Σiτ
983092
983108+ plkprime
983092
983108
st DgeD
DleD
Since strong duality holds it is equivalent to
max 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
st983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Consequently constraint (a) is equivalent to the following constraints
〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (xtn0 minus τ t
n)+ bkαtn minus stn0
983059U
t
nk minusVt
nk minusFt
nk +Gt
nk
983060
iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n minus akXtn
983044iτ
foralliisin [N ] τ isin [T ]
Ft
nk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
Ut
nkVt
nkFt
nkGt
nk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Z Cui et al Inventory Routing Problem under Uncertainty40 Article submitted to manuscript no (Please provide the mansucript number)
Following the same analysis the constraint (b) is equivalent to the following constraints
〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
ge ak (τtn minusxt
n0)+ bkαtn minus stn0
983043U t
nk minusV tnk minusF t
nk +Gtnk
983044iτminus
983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime =
983043St
n + akXtn
983044iτ
foralliisin [N ] τ isin [T ]
F tnk +Gt
nk = T tn
983131
kprimeisin[Kl]
wtnklhkprime = rtnlh foralll isin [L] hisin [H]
U tnkV
tnkF
tnkG
tnk ge 0
wtnklhkprime ge 0 foralll isin [L] hisin [H] kprime isin [Kl]
Combining all of these analysis parts together we can derive the constraints stated in the proposition 3
QED
Appendix E Proof of Proposition 4
When the distributional uncertain set P is specified as the set (11) and W =983051D|DleDleD
983052 for any
nisin [N ] tisin [T ] we can transform the constraint (8b) as
qn0 + 〈DQtn〉 ge 0 forallD isinW
hArr qn0 + minDisinW
〈DQtn〉 ge 0
hArr qn0 + minDleDleD
〈DQtn〉 ge 0
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0
Qtn +Lt
n minusHtn = 0
LtnH
tn ge 0
The last equivalence holds due to strong duality for linear optimization problems The constraint (8c) can
be formulated as
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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tic programming Operations Research 56(2) 344-357
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in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
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Science 48(1) 1-19
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routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
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products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
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tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
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Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 41
qtn0 + 〈DQtn〉 leMyt
n forallD isinW
hArr qtn0 + maxDisinW
〈DQtn〉 leMyt
n
hArr qtn0 + maxDleDleD
〈DQtn〉 leMyt
n
hArr
983099983105983105983105983105983105983103
983105983105983105983105983105983101
qtn0 + 〈DLt
n〉minus 〈DHt
n〉 leMytn
Qtn minusL
t
n +Ht
n = 0
Lt
nHt
n ge 0
QED
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty42 Article submitted to manuscript no (Please provide the mansucript number)
Appendix F Proof of Corollary 1
inf983131
tisin[T ]
983131
nisin[N]
αtn
st stn0 + 〈StnΞ〉+ 〈T t
nΣ〉+983123
lisin[L]
983123hisin[H]
983171lhrtnlh le 0 foralltisin [T ] nisin [N ]
stn0 + 〈DUt
nk〉minus 〈DVt
nk〉+ 〈ΞGt
nk minusFt
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
minusak
983075t983131
m=1
qmn0
983076minus bkα
tn geminusakτ
tn foralltisin [T ] nisin [N ] k isin [K]
stn0 + 〈DU tnk〉minus 〈DV t
nk〉+ 〈ΞGtnk minusF t
nk〉+983131
lisin[L]
983131
hisin[H]
983131
kprimeisin[Kl]
983091
983107plkprime minus olkprime
983131
(iτ)isinSh
Ξiτ
Σiτ
983092
983108wtnklhkprime
+ak
983075t983131
m=1
qmn0
983076minus bkα
tn ge akτ
tn foralltisin [T ] nisin [N ] k isin [K]
983075U
t
nk minusVt
nk minusFt
nk +Gt
nk minusStn + ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=
983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
983075U t
nk minusV tnk minusF t
nk +Gtnk minusSt
n minus ak
983075t983131
m=1
Qmn
983076983076
iτ
minus983131
hisinhisin[H](iτ)isinSh
983131
lisin[L]
983131
kprimeisin[Kl]
olkprime
Σiτ
wtnklhkprime
=minus983075ak
t983131
m=1
(eneprimem)
983076
iτ
foralliisin [N ] τ isin [T ] tisin [T ] nisin [N ] k isin [K]
Ft
nk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
F tnk +Gt
nk minusT tn = 0 foralltisin [T ] nisin [N ] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
983131
kprimeisin[Kl]
wtnklhkprime minus rtnlh = 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K]
minusqtn0 + 〈DLtn〉minus 〈DHt
n〉 le 0 foralltisin [T ] nisin [N ]
qtn0 + 〈DLt
n〉minus 〈DHt
n〉minusMytn le 0 foralltisin [T ] nisin [N ]
Qtn +Lt
n minusHtn = 0 foralltisin [T ] nisin [N ]
Qtn minusL
t
n +Ht
n = 0 foralltisin [T ] nisin [N ]
983043Qt
n
983044l= 0 forallnisin [N ] lge t l tisin [T ]
αtn ge 983171 foralltisin [T ] nisin [N ]
rtnlh ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H]
T tnL
tnH
tnL
t
nHt
n ge 0 foralltisin [T ] nisin [N ]
Ut
nkVt
nkFt
nkGt
nkUtnkV
tnkF
tnkG
tnk ge 0 foralltisin [T ] nisin [N ] k isin [K]
wtnklhkprime wt
nklhkprime ge 0 foralltisin [T ] nisin [N ] l isin [L] hisin [H] k isin [K] kprime isin [Kl]
(yt0 y
tn z
tij nisin [N ] tisin [T ] (i j)isinA)isinZR
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
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inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
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Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 43
QED
Appendix G Proof of Proposition 5
It is similar to that for Proposition 4 QED
Appendix H Proof of Proposition 6
For completeness here we also present a full proof customized to our problem For any node n and period t
we reformulate the constraints (8a)ndash(8c) as follows With a slight abuse of notation we drop the subscript n
and superscript t in the rest of the proof We start from the constraint (8a) and consider the left-hand-side
(LHS) as a primal optimization problem where the decision variable is P and lies in the ambiguity set (14)
Then it can be formulated as follow
sup EP
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st EP
983059983056983056983056Dminus Ddagger9830569830569830561
983060le θ
ΠDP= P
ΠD
daggerP= Pdagger
P983059(DD
dagger)isinW timesW
983060= 1
(19)
Let Pm be the conditional probability distribution of P on D conditioning on that Ddagger= D
(m) m isin
[Mr] In another word Pm
983059D=D
983060= P
983059D=D|Ddagger
= D(m)
983060 With the law of total probability we have
P983059D=D
983060=983123
misin[Mr]Pm
983059D=D
983060Pdagger
983059D
dagger= D
(m)983060= 1
Mr
983123misin[Mr]
Pm
983059D=D
983060 Hence the problem (19)
can be reformulated as follows
sup1
Mr
983131
misin[Mr]
EPm
983061maxkisin[K]
983153ak max
983153τ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ
983154+ bkα
983154983062
st1
Mr
983131
misin[Mr]
EPm
983059983056983056983056Dminus D(m)
9830569830569830561
983060le θ
Pm
983059D isinW
983060= 1 misin [Mr]
Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
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Aghezzaf E H (2008) Robust distribution planning for supplier-managed inventory agreements
when demand rates and travel times are stationary Journal of the Operational Research Society
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1515-1536
Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
managed inventory-routing problem Transportation Science 41(3) 382-391
Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
with application to inventory problems Operations Research 64(2) 474-494
Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
9(3) 203-228
Bell W J Dalberto L M Fisher M L Greenfield A J Jaikumar R Kedia P Mack R G Prutzman
P J (1983) Improving the distribution of industrial gases with an on-line computerized routing
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Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
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Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
affected by uncertain probabilities Management Science 59(2) 341-357
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
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Z Cui et al Inventory Routing Problem under Uncertainty44 Article submitted to manuscript no (Please provide the mansucript number)
The strong duality of this problem holds (see Esfahani and Kuhn 2017 Gao and Kleywegt 2016) and its
dual problem is
inf rθ+983131
misin[Mr]
sm
str
Mr
983056983056983056Dminus D(m)
9830569830569830561+ sm ge 1
Mr
983061maxkisin[K]
ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα983062
forallD isinWmisin [Mr]
rge 0sisinRMr
= inf rθ+1primes
st r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak maxτ minusx0 minus〈XD〉 x0 + 〈XD〉minus τ+ bkα
forallD isinWmisin [Mr] k isin [K]
rge 0sisinRMr
For anymisin [Mr] k isin [K] the first constraint in the above optimization problem is equivalent to the following
two inequalities
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (τ minusx0 minus〈XD〉)+ bkα forallD isinW
r983056983056983056Dminus D
(m)9830569830569830561+Mrsm ge ak (x0 + 〈XD〉minus τ)+ bkα forallD isinW
Then can be further equivalently reformulated as
supDisinW
983153minusak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (τ minusx0)minus bkα (a)
supDisinW
983153ak〈XD〉minus r
983056983056983056Dminus D(m)
9830569830569830561
983154le Mrsm minus ak (x0 minus τ)minus bkα (b)
(20)
By the definition of the dual norm the 1-norm in (20) can be written as
r983056983056983056Dminus D
(m)9830569830569830561= max983042Rmk983042infin
ler
983111983059Dminus D
(m)983060Rmk
983112
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
References
Adelman D (2004) A price-directed approach to stochastic inventoryrouting Operations Research
52(4) 499-514
Adulyasak Y Cordeau J F Jans R (2013) Formulations and branch-and-cut algorithms for mul-
tivehicle production and inventory routing problems INFORMS Journal on Computing 26(1)
103-120
Adulyasak Y Cordeau J F Jans R (2015) Benders decomposition for production routing under
demand uncertainty Operations Research 63(4) 851-867
Aghezzaf E H (2008) Robust distribution planning for supplier-managed inventory agreements
when demand rates and travel times are stationary Journal of the Operational Research Society
59(8) 1055-1065
Andersson H Hoff A Christiansen M Loslashkketangen A (2010) Industrial aspects and literature
survey Combined inventory management and routing Computers amp Operations Research 37(9)
1515-1536
Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
managed inventory-routing problem Transportation Science 41(3) 382-391
Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
with application to inventory problems Operations Research 64(2) 474-494
Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
9(3) 203-228
Bell W J Dalberto L M Fisher M L Greenfield A J Jaikumar R Kedia P Mack R G Prutzman
P J (1983) Improving the distribution of industrial gases with an on-line computerized routing
and scheduling optimizer Interfaces 13(6) 4-23
Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
(1962) Numerische Mathematik 4(1) 238-252
Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
affected by uncertain probabilities Management Science 59(2) 341-357
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 45
with the auxiliary variable Rmk isin RNtimesT and || middot ||infin being the entrywise infinity-norm Therefore the con-
straint (20a) is equivalent to
supDisinW
983083minusak〈XD〉minus max
983042Rmk983042infinler
983111983059Dminus D
(m)983060Rmk
983112983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr supDisinW
983083min
983042Rmk983042infinler
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983084leMrsm minus ak (τ minusx0)minus bkα
lArrrArr min983042Rmk983042infin
ler
983069supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154983070leMrsm minus ak (τ minusx0)minus bkα
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
983153minusak〈XD〉minus
983111983059Dminus D
(m)983060Rmk
983112983154leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
lArrrArr
983099983105983105983103
983105983105983101
supDisinW
〈minusakX minusRmkD〉+983111D
(m)Rmk
983112leMrsm minus ak (τ minusx0)minus bkα
983042Rmk983042infin le r
Here the second equivalance follows from Esfahani and Kuhn (2017) which proves that the maximization
over D and minimization over Rmk are interchangable By strong duality we have
supDisinW
〈minusakX minusRmkD〉
= sup 〈minusakX minusRmkD〉
st DgeD
DleD
= inf 〈DV mk〉minus 〈DUmk〉
st minusUmk +V mk =minusakX minusRmk
UmkV mk ge 0
where Umk V mk are the dual variables Moreover 983042Rmk983042infin = maxiisin[N]τisin[T ]
|(Rmk)iτ | Therefore the constraint
983042Rmk983042infin le r is equivalent to minusrle (Rmk)iτ le r foralliisin [N ] τ isin [T ] So the constraint (20a) is equivalent to
983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (τ minusx0)minus bkα
minusUmk +V mk =minusakX minusRmk
(Rmk)iτ le r foralliisin [N ] τ isin [T ]
(Rmk)iτ geminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
References
Adelman D (2004) A price-directed approach to stochastic inventoryrouting Operations Research
52(4) 499-514
Adulyasak Y Cordeau J F Jans R (2013) Formulations and branch-and-cut algorithms for mul-
tivehicle production and inventory routing problems INFORMS Journal on Computing 26(1)
103-120
Adulyasak Y Cordeau J F Jans R (2015) Benders decomposition for production routing under
demand uncertainty Operations Research 63(4) 851-867
Aghezzaf E H (2008) Robust distribution planning for supplier-managed inventory agreements
when demand rates and travel times are stationary Journal of the Operational Research Society
59(8) 1055-1065
Andersson H Hoff A Christiansen M Loslashkketangen A (2010) Industrial aspects and literature
survey Combined inventory management and routing Computers amp Operations Research 37(9)
1515-1536
Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
managed inventory-routing problem Transportation Science 41(3) 382-391
Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
with application to inventory problems Operations Research 64(2) 474-494
Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
9(3) 203-228
Bell W J Dalberto L M Fisher M L Greenfield A J Jaikumar R Kedia P Mack R G Prutzman
P J (1983) Improving the distribution of industrial gases with an on-line computerized routing
and scheduling optimizer Interfaces 13(6) 4-23
Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
(1962) Numerische Mathematik 4(1) 238-252
Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
affected by uncertain probabilities Management Science 59(2) 341-357
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty46 Article submitted to manuscript no (Please provide the mansucript number)
By the same analysis the constraint (20b) can be formulated as983099983105983105983105983105983105983105983105983105983105983105983105983105983105983103
983105983105983105983105983105983105983105983105983105983105983105983105983105983101
〈DV mk〉minus 〈DUmk〉+ 〈D(m)
Rmk〉 leMrsm minus ak (x0 minus τ)minus bkα
minusUmk +V mk = akX minusRmk
983043Rmk
983044iτle r foralliisin [N ] τ isin [T ]
983043Eτ
i Rmk
983044iτgeminusr foralliisin [N ] τ isin [T ]
UmkV mk ge 0
By combining the analysis above we get the results of proposition 6 QED
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
References
Adelman D (2004) A price-directed approach to stochastic inventoryrouting Operations Research
52(4) 499-514
Adulyasak Y Cordeau J F Jans R (2013) Formulations and branch-and-cut algorithms for mul-
tivehicle production and inventory routing problems INFORMS Journal on Computing 26(1)
103-120
Adulyasak Y Cordeau J F Jans R (2015) Benders decomposition for production routing under
demand uncertainty Operations Research 63(4) 851-867
Aghezzaf E H (2008) Robust distribution planning for supplier-managed inventory agreements
when demand rates and travel times are stationary Journal of the Operational Research Society
59(8) 1055-1065
Andersson H Hoff A Christiansen M Loslashkketangen A (2010) Industrial aspects and literature
survey Combined inventory management and routing Computers amp Operations Research 37(9)
1515-1536
Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
managed inventory-routing problem Transportation Science 41(3) 382-391
Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
with application to inventory problems Operations Research 64(2) 474-494
Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
9(3) 203-228
Bell W J Dalberto L M Fisher M L Greenfield A J Jaikumar R Kedia P Mack R G Prutzman
P J (1983) Improving the distribution of industrial gases with an on-line computerized routing
and scheduling optimizer Interfaces 13(6) 4-23
Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
(1962) Numerische Mathematik 4(1) 238-252
Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
affected by uncertain probabilities Management Science 59(2) 341-357
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 47
References
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52(4) 499-514
Adulyasak Y Cordeau J F Jans R (2013) Formulations and branch-and-cut algorithms for mul-
tivehicle production and inventory routing problems INFORMS Journal on Computing 26(1)
103-120
Adulyasak Y Cordeau J F Jans R (2015) Benders decomposition for production routing under
demand uncertainty Operations Research 63(4) 851-867
Aghezzaf E H (2008) Robust distribution planning for supplier-managed inventory agreements
when demand rates and travel times are stationary Journal of the Operational Research Society
59(8) 1055-1065
Andersson H Hoff A Christiansen M Loslashkketangen A (2010) Industrial aspects and literature
survey Combined inventory management and routing Computers amp Operations Research 37(9)
1515-1536
Archetti C Bertazzi L Laporte G Speranza M G (2007) A branch-and-cut algorithm for a vendor-
managed inventory-routing problem Transportation Science 41(3) 382-391
Ardestani-Jaafari A Delage E (2016) Robust optimization of sums of piecewise linear functions
with application to inventory problems Operations Research 64(2) 474-494
Artzner P Delbaen F Eber J M Heath D (1999) Coherent measures of risk Mathematical Finance
9(3) 203-228
Bell W J Dalberto L M Fisher M L Greenfield A J Jaikumar R Kedia P Mack R G Prutzman
P J (1983) Improving the distribution of industrial gases with an on-line computerized routing
and scheduling optimizer Interfaces 13(6) 4-23
Benders J F (1962) Partitioning procedures for solving mixed-variables programming problems
(1962) Numerische Mathematik 4(1) 238-252
Ben-Tal A Den Hertog D DeWaegenaere A et al (2013) Robust solutions of optimization problems
affected by uncertain probabilities Management Science 59(2) 341-357
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty48 Article submitted to manuscript no (Please provide the mansucript number)
Ben-Tal A El Ghaoui L Nemirovski A (2009) Robust Optimization Princeton University Press
Ben-Tal A Golany B Nemirovski A Vial J P (2005) Retailer-supplier flexible commitments con-
tracts A robust optimization approach Manufacturing amp Service Operations Management 7(3)
248-271
Ben-Tal A Goryashko A Guslitzer E Nemirovski A (2004) Adjustable robust solutions of uncertain
linear programs Mathematical Programming 99(2) 351-376
Bernstein F Federgruen A (2007) Coordination mechanisms for supply chains under price and
service competition Manufacturing amp Service Operations Management 9(3) 242-262
Bertsimas D Caramanis C (2007) Adaptability via sampling Decision and Control 2007 46th
IEEE Conference on IEEE 2007 4717-4722
Bertsimas D Georghiou A (2015) Design of near optimal decision rules in multistage adaptive
mixed-integer optimization Operations Research 63(3) 610-627
Bertsimas D Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer
optimization Mathematical Programming 167(2) 395-433
Bertsimas D Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive
optimization Mathematical Programming 134(2) 491-531
Bertsimas D Gupta S Tay J (2016) Scalable Robust and Adaptive Inventory Routing Available
on Optimization Online
Bertsimas D Iancu D A Parrilo P A (2011) A hierarchy of near-optimal policies for multistage
adaptive optimization IEEE Transactions on Automatic Control 56(12) 2809-2824
Bertsimas D Sim M (2003) Robust discrete optimization and network flows Mathematical Pro-
gramming 98(1) 49-71
Bertsimas D Sim M (2004) The price of robustness Operations Research 52(1) 35-53
Bertsimas D Sim M Zhang M (2018) A practically efficient approach for solving adaptive distri-
butionally robust linear optimization problems Management Science (forthcoming)
Brandimarte P Zotteri G (2007) Introduction to Distribution Logistics John Wiley amp Sons
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 49
Brinkmann J Ulmer M W Mattfeld D C (2016) Inventory routing for bike sharing systems
Transportation Research Procedia 19 316-327
Brown D B Sim M (2009) Satisficing measures for analysis of risky positions Management Science
55(1) 71-84
Campbell A Clarke L Kleywegt A Savelsbergh M (1998) The inventory routing problem Fleet
Management and Logistics Springer Boston MA 95-113
Campbell A M Savelsbergh M W P (2004) A decomposition approach for the inventory-routing
problem Transportation Science 38(4) 488-502
Cetinkaya S Lee C Y (2000) Stock replenishment and shipment scheduling for vendor-managed
inventory systems Management Science 46(2) 217-232
Chen L G Long D Z Sim M (2015) On Dynamic Decision Making to Meet Consumption Targets
Operations Research 63(5) 1117-1130
Chen X Sim M Simchi-Levi D Sun P (2007) Risk aversion in inventory management Operations
Research 55(5) 828-842
Chen X Sim M Sun P Zhang J (2008) A linear decision-based approximation approach to stochas-
tic programming Operations Research 56(2) 344-357
Cheung K L Lee H L (2002) The inventory benefit of shipment coordination and stock rebalancing
in a supply chain Management Science 48(2) 300-306
Christiansen M Fagerholt K Flatberg T Haugen Oslash Kloster O Lund E H (2011) Maritime inven-
tory routing with multiple products A case study from the cement industry European Journal
of Operational Research 208(1) 86-94
Coelho L C Cordeau J F Laporte G (2014a) Thirty years of inventory routing Transportation
Science 48(1) 1-19
Coelho L C Cordeau J F Laporte G (2014b) Heuristics for dynamic and stochastic inventory-
routing Computers amp Operations Research 52 55-67
Custodio A L Oliveira R C (2006) Redesigning distribution operations a case study on integrating
inventory management and vehicle routes design International Journal of Logistics Research
and Applications 9(2) 169-187
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty50 Article submitted to manuscript no (Please provide the mansucript number)
Delage E Iancu D A (2015) Robust multistage decision making INFORMS Tutorials in Operations
Research 20-46
Delage E Ye Y (2010) Distributionally robust optimization under moment uncertainty with appli-
cation to data-driven problems Operations Research 58(3) 595-612
Diecidue E van de Ven J (2008) Aspiration level probability of success and failure and expected
utility International Economic Review 49 683-700
Dong Y Xu K (2002) A supply chain model of vendor managed inventory Transportation Research
Part E Logistics and Transportation Review 38(2) 75-95
Esfahani P M Kuhn D (2017) Data-driven distributionally robust optimization using the Wasser-
stein metric Performance guarantees and tractable reformulations Mathematical Programming
Federgruen A Prastacos G Zipkin P H (1986) An allocation and distribution model for perishable
products Operations Research 34(1) 75-82
Federgruen A Zipkin P (1984) A combined vehicle routing and inventory allocation problem
Operations Research 32(5) 1019-1037
Follmer H Schied A (2002) Convex measures of risk and trading constraints Finance and Stochas-
tics 6(4) 429-447
Gao R Kleywegt A J (2016) Distributionally robust stochastic optimization with Wasserstein
distance arXiv preprint arXiv160402199
Gaur V Fisher M L (2004) A periodic inventory routing problem at a supermarket chain Opera-
tions Research 52(6) 813-822
Georghiou A Wiesemann W Kuhn D (2015) Generalized decision rule approximations for stochas-
tic programming via liftings Mathematical Programming 152(1-2) 301-338
Gilboa I Schmeidler D (1989) Maxmin expected utility with non-unique prior Journal of Mathe-
matical Economics 18(2) 141-153
Goh J Sim M (2010) Distributionally robust optimization and its tractable approximations Oper-
ations Research 58(4-part-1) 902-917
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under UncertaintyArticle submitted to manuscript no (Please provide the mansucript number) 51
Golden B Assad A Dahl R (1984) Analysis of a large scale vehicle routing problem with an
inventory component Large Scale Systems 7(2-3) 181-190
Groslashnhaug R Christiansen M Desaulniers G Desrosiers J (2010) A branch-and-price method for
a liquefied natural gas inventory routing problem Transportation Science 44(3) 400-415
Hall N G Long D Z Qi J Sim M (2015) Managing underperformance risk in project portfolio
selectionOperations Research 63(3) 660-675
Hvattum L M Loslashkketangen A (2009) Using scenario trees and progressive hedging for stochastic
inventory routing problems Journal of Heuristics 15(6) 527-557
Hvattum L M Loslashkketangen A Laporte G (2009) Scenario tree-based heuristics for stochastic
inventory-routing problems INFORMS Journal on Computing 21(2) 268-285
Isii K (1962) On sharpness of chebycheff-type inequalities Annals of the Institute of Statistical
Mathematics 14(1) 185-197
Jaillet P Qi J Sim M (2016) Routing optimization under uncertainty Operations Research 64(1)
186-200
Jiang R Guan Y (2016) Data-driven chance constrained stochastic program Mathematical Pro-
gramming 158(1-2) 291-327
Kleywegt A J Nori V S Savelsbergh M W P (2002) The stochastic inventory routing problem
with direct deliveries Transportation Science 36(1) 94-118
Kleywegt A J Nori V S Savelsbergh M W P (2004) Dynamic programming approximations for a
stochastic inventory routing problem Transportation Science 38(1) 42-70
Kuhn D Wiesemann W Georghiou A (2011) Primal and dual linear decision rules in stochastic
and robust optimization Mathematical Programming 130(1) 177-209
Mao J C T (1970) Survey of capital budgeting Theory and practice The Journal of Finance 25(2)
349-360
Miranda P A Garrido R A (2004) Incorporating inventory control decisions into a strategic distri-
bution network design model with stochastic demand Transportation Research Part E Logistics
and Transportation Review 40(3) 183-207
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming
Z Cui et al Inventory Routing Problem under Uncertainty52 Article submitted to manuscript no (Please provide the mansucript number)
Papageorgiou D J Cheon M S Nemhauser G Sokol J (2014) Approximate dynamic programming
for a class of long-horizon maritime inventory routing problems Transportation Science 49(4)
870-885
Pham-Gia T Hung T L (2001) The mean and median absolute deviations Mathematical and
Computer Modelling 34(7-8) 921-936
Rahmaniani R Crainic T G Gendreau M et al (2017) The Benders decomposition algorithm A
literature review European Journal of Operational Research 259(3) 801-817
Simon H A (1955) A behavioral model of rational choice The Quarterly Journal of Economics
69(1) 99-118
Solyalı O Cordeau J F Laporte G (2012) Robust inventory routing under demand uncertainty
Transportation Science 46(3) 327-340
Waller M Johnson M E Davis T (1999) Vendor-managed inventory in the retail supply chain
Journal of Business Logistics 20(1) 183-203
Wang Z Glynn P W Ye Y (2013) Likelihood robust optimization for data-driven problems Com-
putational Management Science 13(2) 241-261
Wiesemann W Kuhn D Sim M (2014) Distributionally robust convex optimization Operations
Research 62(6) 1358-1376
Zhang Y Baldacci R Sim M Tang J (2018) Routing optimization with time windows under
uncertainty Mathematical Programming