View
220
Download
0
Category
Preview:
Citation preview
IntroductionModels
Numerical Results
An Available-to-Promise Production-InventorySystem with Pseudo Orders
Long Gao
joint work with Susan Xu
University of DaytonPenn State University
MSOM Conference, June 6, 2008
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Outline
1 IntroductionMotivationResearch Questions
2 ModelsPseudo Order ModelOrder Promising Model
3 Numerical ResultsPolicy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
Outline
1 IntroductionMotivationResearch Questions
2 ModelsPseudo Order ModelOrder Promising Model
3 Numerical ResultsPolicy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
What is a pseudo order?
Pseudo OrdersIntended purchase orders, short-term forecasts, (imperfect)advance demand information in a B2B environmentSubject to change and cannot be enforcedContain information of the likelihood of becoming actualorders, due date, requested quantity, etc.Maintained and revised by sales personnelLumpy, nonstationary, volatile, highly uncertain
Trade-offThe presence of pseudo orders makes confirmed orders lesslikely to be accepted because it is more desirable to reservelimited resources for future higher value pseudo orders.
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
A Pseudo Order Example
Siebel.com, (Forbes, January 21, 2002)Acting upon the information of the sudden cancellation ofhundreds of potential deals in February 2000, Thomas Siebel(CEO) anticipated the recession months ahead of rivals andeconomists. He realigned his sales force, readjusted resourceallocation decisions, and avoided the worst in 2001.
However,Pseudo order information is not well-integrated intobusiness planning and control systems.The cost of ignoring such information can be very high.
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
Available-To-Promise System
What is an ATP system?A business function that matches incoming customerorders to planned resourcesDifferent from traditional planning, scheduling andinventory management processesOperate within a short-term operational environmentMost resources are considered fixed because ofprocurement leadtime limitationsDeal with multiple customer classes
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
An ATP System with Pseudo Orders
ATP
System
Inventory
Mgmt
System
Production
Capacity
Component
Availability
Comitted
Orders
Pseudo
Orders
modify supplier orders reserve
comp avail
& prod cap
accept/process
orders
supplier order
lead time
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
ATP Examples: Dell and Toshiba
Dell two-stage order promising practice
Customer Differentiation: Home, Small Business, Medium& Large Business, Government, etc.Provide initial soft confirmation via emailGenerate hard confirmation after checking resourceavailability, based on batch ATP
Toshiba electronic product ATP systemOrders for several thousand models are collected andprocessed by a single central order processing systemATP execution every 1/4 ∼ 1/2 hourBook pseudo orders up to 10 weeks in advance of delivery
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
MotivationResearch Questions
Research Questions
Research QuestionsHow to model the lumpy, non-stationary and volatilenatures of pseudo order information?What is the optimal order promising policy in an ATPsystem with pseudo order information?How robust is the optimal policy?What are the costs of using suboptimal policies?What is the value of the pseudo order information?
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Outline
1 IntroductionMotivationResearch Questions
2 ModelsPseudo Order ModelOrder Promising Model
3 Numerical ResultsPolicy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Pseudo Order Model
Three major characteristics of pseudo ordersLumpiness: non-negligible probability of cancellationNon-stationarity: demands are not identically distributedVolatility: attributes change before either confirmed orcancelled.
For each future pseudo order,
Random demand distribution: Ykt (ek) ∼ Fk
ek, where ek ∈ Ek
is a distribution state, evolving according to a Markovchain, qk
t (e′k|ek).
Random confirmation date: sk evolves according tohk
t (s′k|sk).
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
An Example: Zero-Inflated Poisson Distribution
Two dist. states: E = { 0, 1 }, cancellation or PP(λ)
Time-homogeneous transition probabilities of distributionstates
[qk(·|·)] =
[1 0πk (1 − πk)
]However, if cancellation information is unknown, thedemand distribution is the mix of mass 0 and PP(λ),resulting in Zero-Inflated Poisson (ZIP) distribution
P(Yk = j) =
{πk + (1 − πk)e−λ, if j = 0,
(1 − πk)e−λλj/j!, if j > 0.
ConclusionInformation updates can remove one source of uncertainties.
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Order Aggregation Scheme
Less volatile, aid ATP decision making, increaseoperations and computation efficiencyAggregate demands with order confirmation dates s:
Xt,s =∑
{ k:sk=s }
Ykt (ek) ∼ ⊗k Fk
ek
Aggregated demands are temporally dependent, governedby
P { Et−1 | Et } =∏k∈Kt
hkt (s
′k|sk) · qk
t (e′k|ek)
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Order Aggregation Scheme: Poisson Example
Two dist. states E = {0, 1}, cancellation or PP(λ)
System state can be simplified to the total number ofuncancelled orders Et = (nt,1, . . . , nt,t−1).Aggregated demand Xt,s follows PP(nt,sλ).Transition probability of Pt {Et−1 | Et } is given by
Pt ((nt−1,1, nt−1,2, . . . , nt−1,t−1)|(nt,1, nt,2, . . . , nt,t−1))
=∑
n
∏t−1s=1
nt,s!n(0|s)!(nt,s−n(0|s))!π
n(0|s)(1 − π)nt,s−n(0|s)
×∏t−1
s=1
((nt,s−n(0|s))!
n(1|s)!···n(t−1|s)!∏t−1
s′=1(ht(s′|s))n(s′|s)) .
ConclusionOur Markov chain model completely describes the evolution ofpseudo orders at both the individual and the aggregate level.
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Assumptions for the Order Promising Model
T-period ATP system with MTO manufacturing strategyMultiple classes of orders bring in revenue r1 > · · · > rI
Each order consumes one unit of production capacity,takes a single production periodPseudo order forecast Et is updated by P(Et−1|Et)
Newly confirmed orders Nt
Accepted orders xt = {xit} must be fulfilled within L periods
Production: first-accepted, first-served
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Sequence of Events in Each Period t
Nt orders confirmedFuture pseudo orders updated to Et
Observe net capacity Qt
Planned capacity Kt becomes availableDecision: accept orders xt = {xi
t : i ∈ I}
ObjectiveMaximize the expected total profit over the ATP executionhorizon T
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
A Markov Decision Process Formulation
Vt(Qt, Nt, Et) = maxxt∈At
r · xt − p(Qt + Kt − |xt|)++
∑Et−1
∑Nt−1
p(Nt−1|Et)P(Et−1|Et)
×Vt−1 (Qt−1, Nt−1, Et−1)
,(1)
The action space At is defined by
0 ≤ xt ≤ Nt, (2)|xt| ≤ Qt + [K]tt−L. (3)
Nonlinear system dynamics
Qt−1 = [Qt + Kt − |xt|] ∧ 0. (4)
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Characterization of the Optimal Policy
Optimal order acceptance policyAccept in an increasing order of the indexReject class i if class i − 1 are not fully acceptedAccept class i until
1 all N it are accepted (demand)
2 cumulative leadtime capacity for i is exhausted (supply)3 the net capacity rationing level is reached (rationing)
Formally, for class i ∈ I, the optimal acceptance is
x̂it = min
Ni
t ,[Qt + [K]tt−L − [N]i−1
1
]+,[
Qt + Kt − [N]i−11 − ηi
t−1(Et)]+
. (5)
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Pseudo Order ModelOrder Promising Model
Contributions of the Characterization
Explicitly reveal the dependence on demand quantity, leadtime capacity, and capacity rationing level in a simple form
Result in (Qt, Nt)-state independent threshold ηit−1(Et),
depending on forecast only
Ease the “Curse of Dimensionality” for such multi-dim MDP
O(Q × E × I) v.s. O(Q × E × NI)
For example, if N = 100, I = 3, save 0.3 × 106 times incomputation efforts!
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Outline
1 IntroductionMotivationResearch Questions
2 ModelsPseudo Order ModelOrder Promising Model
3 Numerical ResultsPolicy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Numerical Results
When are rationing and pseudo order informationnecessary?What are the costs of using suboptimal policies?Is it beneficial to use short term volatile forecast, or justuse long term forecast?How robust is the optimal policy?
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Experiment Design
Two-class inventory ATP system, horizon T = 10
4 demand settings: { SL, SH, NL, NH }3 resource availability ρ levels: scarce, intermediate, ampleρ = S/[EX1
t + EX2t ]
3 profit ratio γ = r1/r2 levels, r1 + r2 = 10
Holding cost: h = 0.5
2 lead time levels: L = { 0, 2 }72 scenarios, each generates 100 instances, total 7200instances
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Policy Comparison
OPT: rationing with complete pseudo order informationMVE: rationing with mean demand, ignore stochasticityPRO: priority rule only, ignore pseudo order informationFS: fair share or first-come first-served, ignore bothprioritization and pseudo order information
Performance Gap: percentage difference of total profits,e.g., ∆VM = [V∗ − VM]/V∗ × 100%
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
What are the costs of using suboptimal policies?
VFS ≤ VPRO ≤ VMVE ≤ V∗
Prioritization is effectiveregardless of the capacitylevelRationing with mean valueis necessary when thecapacity level is low tointermediateStochasticity of pseudoorders cannot be ignoredwhen capacity is atintermediate level
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Policy selection: partitioning of the parameter space
0.2 0.4 0.6 0.8 1.0 1.21.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
22
2
2
22
2
22
Resource Availability: ρ
Pro
fit R
atio
: γ
MVE
OPT
PRO / MVE
FS / PRO / MVE
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Are later due dates always beneficial?
Figure: Impact of Lead Time L OPT, MVE and FSbenefit fromincreased lead timeresource availabilityPRO may sufferfrom later due date!Customercannibalization:larger percentage ofclass-2 acceptancedue to increasedresource availability
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Value of Pseudo Order Information Updates
Volatility and dynamic information availability
Question: Given the volatile nature of pseudo orders, is itbeneficial to use the short term forecast, or just use longterm forecast?
The percentage difference of the systems with and withoutupdating quantifies the value of pseudo order updating
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Value of Pseudo Order Information Updates2
2
22
2
2
44
4
4
4
6
6
6
Resource Availability ρ
Pro
fit R
atio
γ
0 0.2 0.4 0.6 0.8 11.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
significant region: ∆ V*S > 2%
Updating is always beneficialSignificant region: ∆V ≥ 2%,scarce capacity,heterogenous customersIn this region, updating canfurther strengthen theeffectiveness of rationing by2% ∼ 7%
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
How robust is the optimal policy?
RobustnessWhat if the the forecast is inaccurate?What if the underlying distributions changed?Is OPT still better than others, especially forecastindependent policies?
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Robustness Comparison
Table: Robustness for Forecast Errors over 4200 Instances
Forecast Errors Dominance over suboptimalsType (ε2, ε3) ∆µ1% ∆cv1% PMVE% PPRO % PFS%
I (+3, +3) 25.42 2.32 64.74 71.85 85.45II (−3, +3) −5.08 26.67 76.73 94.42 100.00III (−3,−3) −25.42 0.97 87.35 100.00 100.00IV (+3,−3) 5.08 −3.61 83.08 93.76 98.85
Overall (±3,±3) ±15.25 ±8.58 77.98 90.01 96.08
OPT is robust for small to moderate forecast errors.OPT should be implemented with forecast updatingmechanisms.
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Conclusions
We quantify lumpy volatile pseudo order information, andcharacterize the optimal order acceptance policy.Commonly used policies may suffer severe losses due toignoring pseudo order information and rationing.Prioritization without rationing may reduce the profitabilitywith extended due dates!OPT is fairly robust and should be implemented withpseudo order updating mechanisms.
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Future Research
Multiple components, class-specific lead timeImpact of pseudo order information on strategic or tacticalresources planningRandom supply and production processesOther Applications: Hub Group, Inc. intermodal shippingload acceptance
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Characterization of the Value Function: Proof
Sketch of the ProofDifficulty: nonlinear dynamics of capacityInduction for three cases: both positive, both negative, andone eachObserve that: r1 ≥ ∆Vt−1(Qt−1|Et) ≥ −p
Use complementary property of max{x, 0} and min{x, 0}:at least one of them is 0Regarding Nt, there is no lost sale penalty and the actionspace is convex
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Literature Review: Rationing Models
Continuous time rationing modelsHa. (1997)Benjaafar & ElHafsi (2006)
Discrete time rationing modelsTopkis. (1968)Only one nonperishable resource, available at thebeginning, no pseudo ordersWang and Gupta. (2007)Two classes, single resource, no pseudo ordersOur model deals with multi-period, both perishable andnonperishable resources, incorporating pseudo orders
Long Gao An ATP System with Pseudo Orders
IntroductionModels
Numerical Results
Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy
Characterization of the Optimal Value Function
Lemma(i) Vt(Qt, Nt, Et) is concave in net capacity Qt.(ii) Vt(Qt, Nt, Et) is increasing concave in realized demand Nt.
Managerial InsightsMarginal value of unit capacity diminishes when capacityincreases.Carefully plan and allocate capacity over time [T, 1], usingpseudo order information.Marketing activities on demand management, such asorder expedition and postpone, need to be coordinatedwith the planned resources.
Long Gao An ATP System with Pseudo Orders
Recommended