Author's Accepted Manuscript

Multi-product multi-period inventory routingproblem with a transshipment option: Agreen approach

S.M.J. Mirzapour Al-e-hashem, Yacine Rekik

PII: S0925-5273(13)00391-5DOI: http://dx.doi.org/10.1016/j.ijpe.2013.09.005Reference: PROECO5570

To appear in: Int. J. Production Economics

Received date: 30 September 2012Accepted date: 4 September 2013

Cite this article as: S.M.J. Mirzapour Al-e-hashem, Yacine Rekik, Multi-productmulti-period inventory routing problemwith a transshipment option: A greenapproach, Int. J. Production Economics, http://dx.doi.org/10.1016/j.ijpe.2013.09.005

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Multi-product multi-period inventory routing problem with a transshipment option: a green approach

S.M.J. Mirzapour Al-e-hashem* and Yacine Rekik**

mirzapour@em-lyon.com, rekik@em-lyon.com

EMLYON Business School, 23 ave. Guy de Collongue, 69134 Ecully, Lyon, France

                                                            * Corresponding author: S.M.J. Mirzapour Al‐e‐hashem, Email: mirzapour@em‐lyon.com, Tel.: +33 6 27 414 439, Address: No. 403, 54 Rue Georges Courteline, 69100 Villeurbanne, Lyon, France. ** Y. Rekik, Emil: rekik@em‐lyon.com  

Abstract

This paper addresses a multi-product multi-period Inventory Routing Problem (IRP) where

multiple capacitated vehicles distribute products from multiple suppliers to a single plant to

meet the given demand of each product over a finite planning horizon.

The demand associated with each product is assumed to be deterministic and time varying. In

this supply chain, the products are assumed to be ready for collection at the supplier site

when the vehicle arrives. A transshipment option is considered as a possible solution to

increase the performance of the supply chain and shows the impact of this solution on the

environment. A green logistic issue is also incorporated into the model by considering the

interrelationship between the transportation cost and the greenhouse gas emission level. The

proposed model is a mixed-integer linear program and solved by CPLEX. We provide a

numerical study showing the applicability of the model and underlining the impact of the

transshipment option on improved supply chain performance.

Keywords: Inventory routing problem; green supply chain; transshipment.

1. Introduction

Because global warming is recognized as one of the greatest challenges of this century,

Greenhouse gas (GHG) emissions are increasingly becoming a focus of attention. Global

warming results from increased GHG concentrations in the atmosphere. In response to this

challenge, a number of organizations are applying ‘green’ principles such as using

environmentally friendly raw materials and recycled paper for packaging and reducing their

use of fossil fuels. These green principles have been expanded to many areas, including

supply chains ([Chung and Wee, 2008], [Zhu et al., 2008], [Lin et al., 2011] and [Wang et al.,

2012]). Adding the ‘green’ concept to the ‘supply chain’ concept creates a new paradigm

where the supply chain has a direct relation to the environment (e.g., [Diabat and Govindan,

2011], [Wang et al. 2011], [Zhu and Sarkis, 2011] and [Eltayeb et al., 2011]).

Globalization, with its increasing industrial trend towards outsourcing, has caused

transportation to become the most visible sector that has increased GHG emissions over the

last two decades. Transportation activities are therefore one of the primary contributors to

global warming (Figure 1), leading to the recent expansion of green logistics investigating as

a subset of the green supply chain ([Srivastava, 2007], [Sheu, 2008], [Bai and Sarkis, 2010]

and [Yeh and Chuang, 2011]). A comprehensive review of the studies on green logistics can

be found in Dekker et al. (2012). Logistics are now widely recognized as value-adding

components in organizations. The primary objective of logistics is to coordinate activities

such as freight transport, storage, inventory management and materials handling. One of the

well-known topics typically addressed in this regard is the Inventory Routing Problem (IRP).

Figure 1. Total GHG emissions by sector in the EU-27, 2011. (European Environment Agency, 2013)

The IRP in a supply chain simultaneously determines the optimal inventory levels, delivery

routes, and vehicle scheduling based on the minimal cost criterion (Moin et al., 2011). In the

past, this cost has been assessed solely in economic terms. Due to the increasing of

environmental concerns, companies must better account for the external costs of logistics

associated with global warming such as air pollution, noise, vibrations and accidents

(Quariguasi, et al., 2009). This study attempts a novel approach of reducing GHG emissions

in IRPs to achieve a balance between economic and environmental objectives.

In Section 2, we review previous studies on the IRP in existing literature. We describe the

Inventory Routing Problem under study in Section 3; its mathematical formulation is then

provided in Section 4. A numerical study, as well as the managerial insights, is provided in

Section 5, and Section 6 concludes the paper and proposes further research in this field.

2. Literature review

Transportation and inventory management are two key logistic drivers of Supply Chain

Management (SCM). The coordination of these two drivers, often known as the IRP, is

typically the issue at hand in vendor-managed inventory systems (VMI) (Zachariadis, et al.,

2009).

VMI is the state of the art in value-added logistics. This practice constitutes a win/win

approach to inventory management; the suppliers make replenishment decisions based on

specific inventory and supply chain policies (saving on distribution and production costs by

combining and coordinating demands and shipments for different customers), while the

buyers gain by not allocating resources to controlling and managing inventories (Coelho et

al., 2012).

Andersson, et al. (2010) presented a classification and comprehensive literature review of

inventory routing problems. Another review of studies on IRPs can be found in Moin and

Salih (2007). IRPs can be broadly categorized according to the following criteria: finite or

infinite planning horizons ([Anily and Federgruen, 1990] and [Archetti et al., 2007]), single

or multiple periods (Moin, et al., 2011), single or multiple customers ([Bertazzi and Speranza,

2002], [Sindhuchao, et al., 2005] and [Archetti, et al., 2007]), single or multiple items

([Sindhuchao et al. 2005] and [Huang and Lin, 2010]), identical (homogeneous) or non-

identical vehicles (Persson and Gothe-Lundgren, 2005), and deterministic or stochastic

demand ([Kleywegt et al., 2002], [Kleywegt et al., 2004], [Bertazzi et al., 2011] and [Chen

and Lin, 2009]). Several other variants of IRPs can also be found, depending on the

underlying assumptions in the models such as IRPs with direct deliveries (Mishra and

Raghunathan, 2004) or with transshipment options ([Nonas and Jornsten, 2005], [Nonas and

Jornsten, 2007] and [Coelho et al., 2012]).

To the best of our knowledge, this study is among the first to consider the concept of “green

logistics” in IRP models. We consider green logistics through incorporating a decision

variable that would enable the proposed model to select the appropriate vehicle by

considering the greenhouse gas emission levels, vehicle capacity and transportation cost.

This paper also considers a transshipment option within the proposed inventory routing

problem. Under this policy, a vehicle may either provide a specific product for an assembly

plant directly from the supplier which produces the product or from other suppliers which

temporarily stored this product from previous trips (Nonas and Jornsten, 2007). More

discussion about transshipment option can be found in the studies of Herer, Tzur & Yücesan

(2002), Burton & Banerjee (2005), Lee, Jung & Jeon (2007), Tiacci & Saetta ( 2011), Chen,

et al., (2012) and Hochmuth & Köchel (2012). From a practical point of view, the use of a

transshipment option improves the performance of a supply chain through lead time

reduction; in this study, the impact of transshipment on GHG emissions is also discussed.

3. Problem Description

Assume that a company consists of one assembly plant (Node F) and a set of suppliers {1, 2,

…, N}; each supplier provides one product type for the assembly plant. The company has an

internal contract with a rental truck company (Depot) that ships the products from the

suppliers to the assembly plant in each period. This rental truck company has several types of

trucks, each one is characterized by its own capacity, fixed and variable transportation cost

rate and its GHG emission index.

The optimization problem must find the best configuration of the vehicle types, routes,

pickups, deliveries and transshipments in each period in a manner that minimizes the total

cost of the supply chain, including the inventory holding cost and transportation cost, while

satisfying all constraints.

Allowing the vehicles to temporarily store pickups during their trips at a supplier storage area

located along their itinerary is known as transshipment-enabled IRP. As previously

mentioned, our research is developed on the premise that the use of a transshipment option

improves supply chain performance through lead time reduction. To explain this premise we

consider a simple illustrative example.

Figure 2 illustrates the case of 3 suppliers and 2 periods to discuss the possible reduction in

travel distance by transshipment-enabled IRP.

Solution (a) Solution (b) Demand

Node period1 period2 Trips in period 1 Load i - di Trips in period 2 Unload j dj - Supplier k dk1 dk2

Figure 2. How transshipment can reduce travel distances

In solution (a), nodes j and k are visited by the vehicle in period 1 (solid arrows). The vehicle

picks up dj and dk1 (cf. the table) units of product type j and k, respectively. In the next period

(dashed arrows), node j has no demand, so the vehicle only visits nodes i and k and picks up

di and dk2 units of product type i and k, respectively.

In solution (b), the vehicle is allowed to arbitrarily store pickups at every node on its trip

(transshipment). In this manner, the vehicle can pick up products from one node and store

them temporarily at another node to reduce the total travel distance while meeting the current

demand of the assembly plant. In Figure 2 (b), despite the fact that there is no current demand

for product type i in period 1; the vehicle visits node i and picks up di units (thus meeting the

demand for product i in the next period), it then visits node j and picks up the required

number of product type j (dj), then goes to node k and stores di units at this node (meaning

that the di units are transshipped to node k) while picking up dk1 units of product type k. In the

next period (dashed arrows), the vehicle directly goes to node k to pick up the previously

stored products (di) and the new demand for product type k (dk2). By comparing these two

solutions, it is apparent that transshipment can reduce the total travel distance if the distance

(i, k) is greater than the distance (i, j).

The proposed transshipment-enabled IRP framework uses the following notation:

Sets 1}N,… {0,1, +=Ω set of all nodes

N} ,… 2, {1,=ω set of suppliers {0}=O Depot (rental vehicle company)

1}{N+=F assembly plant

Parameters

ptD demand for product type p (1, 2, …,P) in period t (1, 2, …, T)

kv variable transportation cost per unit distance for vehicle type k (1, 2, …, K)

ku fixed transportation cost for vehicle type k per trip

ktNT the number of vehicle type k available in period t

kcap capacity of vehicle type k

iph inventory holding cost in node i for product type p per unit product per period

ijc length of arc (i, j)

ip0I initial inventory level of product type p in node i

tGHL allowed level of GHG emission in each period

kGHG GHGs produced by vehicle type k per unit distance Decision variables

iptI the inventory level of product type p at supplier i ( ω∈i ) or at assembly plant ( Fi∈ )

in period t ijktx a binary variable that determines if arc (i, j) is visited by vehicle type k in period t

ikty a binary variable that determines if supplier i is visited by vehicle type k in period t

ijpktQ the quantity of product type p transported by vehicle type k through arc (i, j) in period

t ipta the quantity of product type p picked up from supplier i in period t

iptb the quantity of product type p transshipped to supplier i in period t

4. Mathematical formulation

The mixed integer programming for the transshipment-enabled IRP is modeled as follows:

∑∑∑ ∑∈∪∈Ω∈

++=tki

0iktktpFi

iptipji tk

ijktijk xuIhxcvZMin,,,,),( , ωω

(1)

subject to:

iptipttipipt abII −+= − )1( tipi ,, ≠∈∀ ω (2)

ptki

1)pkti(N1)p(tNptN DQII −+= ∑∈

+−++,

)1()1(ω

tp,∀ (3)

∑ ∑Ω∈ Ω∈

==j

iktj

jiktijkt yxx tki ,,ω∈∀ (4)

1≤∑k

ikty ti ,ω∈∀ (5)

∑ ∑∪∈ ∪∈

=−+kOj kFj

ijpktiptiptjipkt QbaQ, ,ω ω

tpi ,,ω∈∀ (6)

ijktkp

ijpkt xcapQ ≤∑ tkji ,,),( Ω∈∀ (7)

)1( −≤ tipipt Ia tipi ,, ≠∈∀ ω (8)

kti

0ikt NTx ≤∑∈ω

tk,∀ (9)

1,

≥∑∈ ki

0iktxω

t∀ (10)

1)1( ≥∑∈

+ωi

ktNix tk,∀ (11)

tji k

ijktijk GHLxcGHG ≤∑ ∑∈ω),(

t∀ (12)

0=i0ktx tki ,,ω∈∀ (13)

0)1( =+ iktNx tki ,,ω∈∀ (14)

0=iiktx tki ,,Ω∈∀ (15)

0=+1)kt0(Nx tk,∀ (16)

0=0ipktQ tkpi ,,,ω∈∀ (17)

{0,1}xy ijktikt ∈, tkji ,,),( Ω∈∀ (18)

integerbaQ iptiptijpkt ,0,, ≥

Equation (1) is the objective function of the proposed model that aims to minimize the total

supply chain cost, including the inventory holding costs as well as the transportation costs.

Constraint (2) is an inventory balance equation at the suppliers and determines that the

inventory level for product type p at the supplier i in period t is equal to its previous inventory

level (period t-1) plus the quantity made available in period t (transshipped by the vehicles)

minus the quantity picked up by the vehicle in period t.

Constraint (3) is an inventory balance equation at the assembly plant that implies the

inventory level for product p in current period is equal to its previous inventory level in

addition to the total quantity delivered by the vehicles minus its demand in the current period.

Constraints (4 and 5) guarantee that each supplier should not be visited by the vehicles more

than once in each period. Constraint (6) is an inventory balance equation for the arc (i, j)

visited during period t and ensures that the quantity of product type p shipped from supplier i

in period t is equal to the quantity of that product shipped to this supplier plus the quantity of

that product picked up by the vehicle minus the quantity transshipped to this supplier in the

current period. Constraint (7) guarantees that the vehicle's capacity should not be exceeded. It

also implies that the quantity of product type p transported by vehicle type k through arc (i, j)

in period t ( ijpktQ ) can be positive if only the arc (i, j) is visited by this vehicle at this period (

ijktx =1). Constraint (8) ensures that the vehicles cannot pick up from the suppliers that do not

produce that product, a quantity of products more than was transshipped to them in previous

periods. Constraint (9) limits the number of type k vehicles available in period t to a given

quantity. Constraints (10 and 11) are sub-tour elimination constraints that ensure a trip begins

at depot (node O) and finishes at assembly plant (node N+1). Constraint (12) limits the

greenhouse gas emissions of a logistic issue to a given level (GHG constraint). Constraints

(13-16) determine the impossible arcs. Constraint (17) specifies that the vehicles should not

return any quantity to the depot (Node O). Finally, Constraint (18) defines the variable types.

The introduced GHG limit can be interpreted either as an ethical boundary (fixed by

corporate strategy) or as a threshold over which the firm might pay extra taxes or fees

because of its emissions ratio.

5. Experimental result

The aim of this section is twofold:

• To show that the theoretical framework presented in the previous section can be

applied in a straightforward manner.

• To provide a sensitivity analysis and derive managerial insights around the proposed

framework.

5.1. Small-sized test problem

A typical company is willing to plan its IRP. The planning time horizon is assumed to be 2

periods. This company assembles a product consisting of 5 parts in its assembly plant. This

company also owns 5 suppliers S1, …, S5. Each supplier produces only one product type. The

company has a subcontract with a rental truck company (Depot) that ships the products from

the suppliers to the assembly plant in each period. The rental truck company has two different

truck types. The information on the capacity and cost rate for these truck types, as well as

other data, are summarized in Table 1.

Table 1. vehicle characteristics

vehicle type k kv ku ktNT

kcap GHGk t=1 t=2

1 13 1000 3 3 500 1.3

2 11 3000 3 3 1000 5.1

The travel distances are provided in Table 2. We also assume that the unit inventory holding

cost per period is the same for all suppliers (5), and we also assume a higher unit holding cost

(20) for the assembly plant. As previously mentioned, the products cannot be stored at the

depot. The initial inventories at all nodes are assumed to be zero. Table 3 shows the demand

for each product in each period. Finally, the average permitted GHG emission levels among

all periods is assumed to be 950.

Table 2. Travel distances between nodes ( ijc )

Depot S1 S2 S3 S4 S5 Assembly plant

Depot 0 30 25 50 60 90 90

S1 30 0 35 50 45 70 65

S2 25 35 0 30 60 70 95

S3 50 50 30 0 50 45 120

S4 60 45 60 50 0 40 45

S5 90 70 70 45 40 0 60

Assembly plant 90 65 95 120 45 60 0

Table 3. Demand for each product in each period

Period t

Product type p 1 2

1 0 500

2 500 0

3 0 100

4 200 200

5 300 100

All computations were performed using the Branch and Bound algorithm accessed through an

IBM ILOG CPLEX 12.2 on a PC Pentium IV-3.2 GHz i3 with 2GB RAM operating under

Windows XP SP3. The subsequent solutions rely on the above-mentioned data.

We first relax the greenhouse gas emission level limitation (the “Relaxed model”) and solve

the test problem. We then apply the GHG constraint (Eq. (12)) and report a feasible solution

(the “Green model”). The comparison results are reported in Tables 4-6.

Table 4. Greenhouse gas emission level (comparison)

Relaxed model Green model ∆%

Period 1 918.0 952.5 3.75

Period 2 1071.0 943.5 -11.9

Average 994.5 948 -4.67

In Table 4, the GHG emission level produced by the vehicles during the first and second

periods is compared for both the Relaxed and Green models. As seen in this Table, a 4.67%

average savings is achieved by applying the GHG limit in the Green model. Note that during

the first period, the GHG emissions for the Green model solution are actually greater than the

Relaxed model solution; in the succeeding period, however, the GHG emissions in the Green

model are 12% lower, more than covering the previous increase.

Table 5. Objective function components (comparison)

Relaxed model Green model ∆%

Inventory holding cost 00.0 500.0 -

Transportation cost 10290.0 10795.0 4.90

Total cost 10290.0 11295.0 9.77

These results show that when the average GHG emission level is decreased by 4.67% (see

Table 4), the total supply chain cost increases by approximately 9.77% (Table 5). This

increase can be interpreted as an extra charge incurred by making use of more fuel-efficient

(but more expensive) vehicles to meet the GHG limitations. In addition, this increase is a

rational result of the inventory holding costs of the products that must be temporarily held at

the suppliers (transshipment) to reduce the number of trips.

In Table 6, the decision variables x, a and b are reported for the two periods to compare the

Relaxed and Green model solutions.

Table 6. the visited arcs ( ijktx ), pickups ( ipta ) and transshipped quantities ( iptb )

Relaxed model Green model

Period 1 Period 2 Period 1 Period 2

Variable ( ijktx ) Value Variable ( ijktx ) Value Variable ( ijktx ) Value Variable ( ijktx ) Value

X( 0, 2, 2, 1) 1.00 X( 0, 1, 2, 2) 1.00 X( 0, 2, 2, 1) 1.00 X( 0, 1, 2, 2) 1.00

X( 2, 5, 2, 1) 1.00 X( 1, 3, 2, 2) 1.00 X( 0, 4, 1, 1) 1.00 X( 1, 5, 2, 2) 1.00

X( 5, 4, 2, 1) 1.00* X( 3, 5, 2, 2) 1.00 X( 2, 3, 2, 1) 1.00 X( 5, 4, 2, 2) 1.00

X( 4, 6, 2, 1) 1.00 X( 5, 4, 2, 2) 1.00 X( 3, 5, 2, 1) 1.00 X( 4, 6, 2, 2) 1.00

X( 4, 6, 2, 2) 1.00 X( 4, 6, 1, 1) 1.00

X( 5, 6, 2, 1) 1.00 * It means that a vehicle type II visits arc (5, 4) at period 1 in the Relaxed solution.

Relaxed model Green model

Variable ( ipta ) Value Variable ( ipta ) Value Variable (ipta ) Value Variable (

ipta ) Value

a( 2, 2, 1) 500.00 a( 1, 1, 2) 500.00 a( 2, 2, 1) 500.00 a( 1, 1, 2) 500.00

a( 4, 4, 1) 200.00 a( 3, 3, 2) 100.00 a( 3, 3, 1) 100.00 a( 4, 4, 2) 200.00

a( 5, 5, 1) 300.00 a( 4, 4, 2) 200.00 a( 4, 4, 1) 200.00 a( 5, 3, 2) 100.00*

a( 5, 5, 2) 100.00 a( 5, 5, 1) 300.00 a( 5, 5, 2) 100.00

* It means that 100 units of product type 3 are picked up from supplier 5 at period 2 in the Green model.

Relaxed model Green model

Variable ( iptb ) Value Variable ( iptb ) Value Variable ( iptb ) Value Variable ( iptb ) Value

- - - - b( 5, 3, 1) 100.00* - -

* It means that 100 units of product type 3 are transshipped to supplier 5 at period 1 in the Green model.

According to Table 6, the GHG limitation considerably influences the trip configurations,

vehicle types, pickups and transshipments. These changes are depicted in Figure 3 for

clarification.

Figure 3 shows that the first period of the Relaxed solution only requires a single type II

truck, while the Green solution uses one truck from each type. In the Green solution, a

transshipment occurs at supplier 5, where 100 units of product type 3 are temporarily stored

and picked up by the vehicle in period 2. This transshipment creates considerable trip savings

during the second period, where the truck goes directly from node 1 to node 5 (in the Relaxed

solution, the truck must visit node 3).

Period 1 Period 2

Rel

axed

Mod

el

(Ave

. GH

G le

vel:

994.

5)

Gre

en m

odel

(Ave

. GH

G le

vel:

948)

*i.e., 500(1) means that 500 units of product type 1

Figure 3. Comparison between the solutions obtained from Relaxed and Green model

GHG emissions are reduced by omitting this trip. Although the trip distances in the first

period of the Green solution are actually greater than the Relaxed solution, the shorter trips

used in the second period compensate for this. The travelling time (lead time) in the Green

solution is also decreased because the number of vehicles used in the first period has doubled

and the travelled distances in the second period are meaningfully shortened. The extra type I

truck used in the first period of the Green solution, in addition to the inventory holding cost

for the quantity transshipped to supplier 5 (and stored for one period), led to a 10% increase

in the total cost of the supply chain.

The proposed model takes advantage of the transshipment option to save on excessive trips

and this consequently reduces the GHG emission level. Although the transshipment reduces

the travelling time, it increases the inventory holding cost at the suppliers, consequently

increasing the total cost of the supply chain.

This example shows that the transshipment option is not an expensive strategy for moderating

GHG emission levels. To further study this example, let us tighten the average right hand side

of the GHG constraint to 770. The resulting optimum solution is depicted in Figure 4.

Period 1 Period 2

Gre

en M

odel

(Ave

. GH

G le

vel:7

65)

Figure 4. The average right hand side of the GHG constraint is limited to 770.

Figure 4 shows a type II truck beginning its trip from the depot and reaching node 2, where it

picks up 500 units (1st period requirement) of product type 2; it then visits node 3 and picks

up 100 units (2nd period requirement) of product type 3. The truck then reaches supplier 5,

where it picks up 400 units (sum of the 1st and 2nd periods requirements) of product type 5. At

this point, the truck reaches its capacity (1000 units). The truck then visits node 4, where it

frees up space by offloading 100 units of product type 3 and 100 units of product type 5

(previously picked up at nodes 3 and 5, respectively). The truck then picks up 200 units (1st

period requirement) of product type 4 and finishes its trip to the assembly plant.

This optimal solution takes advantage of transshipment to reduce both the number of trucks

and trips. The average GHG level decreases to 765 (a 23% reduction in GHG levels in

comparison with the Relaxed model) and the total supply chain cost minimally increases to

10300 (~0.1% increase).

5.2. Medium- and large-sized test problems

To highlight the impact of the transshipment option on supply chain performance as well as

GHG levels, we generate a set of medium- and large-scale multi-period examples and analyze

the results. The demands and distances are presented in Tables 7 and 8; we assume that in

each problem, the first node is the depot and the last one is the assembly plant. Table 9 shows

the dimension of each problem and the associated results.

As shown in Table 9, considering the GHG constraint in the proposed model can reduce a

considerable amount of GHG emissions (13% on average), while the total supply chain cost

increases by 8.1% on average for the 5 test problems. Table 7. Distances between nodes in problems 1 to 5 (one block for each problem)

Node i

Node j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0 0 50 48 70 90 132 136 189 155 105 105 137 33 108 122 175 93

1 50 0 81 49 103 113 94 151 106 73 56 89 70 104 93 130 48

2 48 81 0 67 48 111 139 183 184 149 136 151 76 76 112 210 107

3 70 49 67 0 65 66 72 121 127 119 85 86 101 56 53 160 44

4 90 103 48 65 0 76 124 156 191 176 149 147 121 36 89 225 108

5 132 113 111 66 76 0 67 82 157 176 130 102 165 40 30 197 84

6 136 94 139 72 124 67 0 57 93 134 81 37 163 96 39 135 47

7 189 151 183 121 156 82 57 0 133 189 135 83 218 121 70 175 105

8 155 106 184 127 191 157 93 133 0 82 50 56 167 174 127 42 83

9 105 73 149 119 176 176 134 189 82 0 54 107 101 175 150 80 93

10 105 56 136 85 149 130 81 135 50 54 0 95 118 138 102 76 47

11 137 89 151 86 147 102 37 83 56 107 95 0 158 124 72 98 45

12 33 70 76 101 121 165 163 218 167 101 118 158 0 141 154 178 117

13 108 104 76 56 36 40 96 121 174 175 138 124 141 0 58 212 93

14 122 93 112 53 89 30 39 70 127 150 102 72 154 58 0 167 57

15 175 130 210 160 225 197 135 175 42 80 76 98 178 212 167 0 119

16 93 48 107 44 108 84 47 105 83 93 47 45 117 93 57 119 0

The GHG level in test problem 2 decreased in comparison with test problem 1; as a result of

adding a very fuel-efficient truck type in test problem 2. For this truck, kv , ku and GHGk are

assumed to be 15, 700 and 1.1, respectively, with a capacity equal to 500. We consider this

truck type to show how improving environmental standards on trucks can directly reduce

GHG levels.

Table 8. Demand for problems 1 to 5 (one block for each problem)

period

product 1 2 3 4 5 6 7 8 9 10 11 12

1 100 0 200 0 100 100 0 100 100 0 100 100

2 200 200 0 100 0 200 200 0 200 200 100 0

3 0 100 100 0 200 200 0 100 200 100 0 200

4 100 0 200 200 0 200 100 0 0 200 200 100

5 200 200 0 200 200 100 0 100 200 200 100 100

6 100 300 0 300 0 0 200 0 100 0 100 0

7 300 0 100 100 100 100 100 200 0 300 200 100

8 200 100 300 0 0 200 100 0 100 100 0 200

9 100 0 100 200 200 0 100 200 200 200 300 0

10 0 100 300 100 0 0 200 300 100 0 0 200

11 100 200 100 0 0 100 200 200 0 100 100 100

12 100 0 100 0 100 0 100 0 100 200 0 100

13 200 200 0 100 200 200 0 100 200 200 0 100

14 100 0 100 100 0 100 200 0 200 0 200 0

15 200 300 0 100 100 100 0 0 0 300 300 100

Table 9. Comparison between Relaxed and Green model for medium- and large-sized problems

Number of Total cost Average GHG level

Problem

No.

periods suppliers vehicle

types

arcs Relaxed Green ∆% Relaxed Green -∆%

1 3 5 2 21 14880 15570 4.63 579.3 492.3 15.0

2 5 7 3 36 16155 18280 13.2 201.8 148.8 26.3

3* 7 10 4 66 63240 67514 6.8 3100.1 2974.2 4.06

4* 10 13 5 105 143228 157772 10.15 4522.3 4012.7 11.3

5* 12 15 5 135 175896 185940 5.71 6552.1 6004.4 8.36

* solving the large-scale problems was time consuming; we reported the best solution after one hour.

Figure 5 depicts the greenhouse gas emission levels for different vehicle types against their

capacities. These data are taken from the “Network for Transport and Environment” (NTM),

a nonprofit organization initiated in 1993 aimed at establishing a common base of values for

calculating the environmental performance of various modes of transport.

Figure 5. GHG emission levels for different vehicle types

A sensitivity analysis was performed to study the effect of varying GHG limitations in

objective function components; the results are depicted in Figure 6. We initially solved test

problem 2 by relaxing the greenhouse gas emission constraint (100%). This constraint, as

expected, incurred the minimum total cost. We then tightened the right-hand side of this

constraint (step by step) and analyzed the impact of the greenhouse gas emission level on the

inventory holding cost and the transportation cost.

* "100% Relaxation" means that the GHG constraint is fully relaxed. In other words, the right hand side of GHG

constraint is set to the upper bound of GHG emission. For lower percentage of relaxation; the right hand side of GHG constraint is tightened, accordingly.

Figure 6. GHG emission relaxation against transportation, inventory and total costs

0

1000

2000

3000

4000

5000

50 55 60 65 70 75 80 85 90 95 100

Inve

ntor

y c

ost

GHG relaxation*%

(a)

0

10000

20000

30000

40000

50000

60000

50 55 60 65 70 75 80 85 90 95 100

Tra

nspo

rtat

ion

cost

GHG relaxation%

(b)

0

10000

20000

30000

40000

50000

60000

50 55 60 65 70 75 80 85 90 95 100

Tot

al c

ost

GHG relaxation%

(c)

0

10000

20000

30000

40000

50000

60000

50 55 60 65 70 75 80 85 90 95 100

Tra

nspo

rtat

ion

/Tot

al c

ost

(d)

total inventory transportation

Figure 6 shows that relaxing the GHG constraint will minimize the incurred cost (GHG

relaxation=100%). However, when we impose an upper limit for greenhouse gas emissions,

the transportation costs slowly increase with a relatively fixed slope and the inventory

holding costs increase with a greater angle of slope. This indicates that the proposed model

attempts to lower GHG emissions by merging trips as much as possible through

transshipment. Because this strategy (transshipment) must make use of larger trucks, the total

number of trucks used to handle distribution decreases. In other words, the optimal solution is

a tradeoff between fixed transportation unit costs (a larger truck has greater fixed

transportation unit costs) and lower variable transportation costs (a larger truck has lower

variable transportation costs). It is also a tradeoff between the reduction of trips (as a result of

using larger trucks and the transshipment option) and increased GHG emissions (larger trucks

produce more GHG emissions per distance unit).

If we further tighten the GHG limitation (GHG relaxation=85%), the transportation costs still

increase with a gentle slope (Figure 6-b), but the inventory holding costs rapidly increase

(Figure 6-a). This implies that merging the trips (transshipment option) and using appropriate

vehicles and routes can still decrease GHG emissions. At 70% GHG relaxation, the

maximum values for inventory holding costs are obtained and the trend reverses; at this point,

the transshipment strategy cannot achieve further GHG reductions. Therefore, the model

attempts to make use of more fuel-efficient trucks (with lower capacities and more expensive

variable transportation costs). Consequently, this limits the number of merging trips and

transshipment options. At this point, the inventory holding costs collapse and the

transportation costs sharply increase.

Thereafter, further GHG limitation is not possible because the model produces infeasible

solution errors due to the lack of more fuel-efficient trucks. As previously mentioned, the

GHG limits can be interpreted either as an ethical boundary fixed by corporate strategy or as

a threshold over which the firm might pay extra taxes or fees because of its emission ratio. In

the former case, Figure 6 could act as a Pareto set for problem 2; the decision maker could

select the most preferred solution according to his/her preferences. For example, when the

GHG limit relaxation is assumed to be 70%, the transportation cost is equal to 17888 and

inventory holding cost is 4500.

6. Conclusion

In this study, a novel mathematical model was presented to address a multi-product multi-

period inventory routing problem in a many-to-one supply chain network. The proposed

model exhibited two distinct features. First, a transshipment option was considered as a

possible solution to reduce travel distances. Under this policy, a vehicle provided a specific

product for the assembly plant, either directly from the supplier which manufactured the

product or from the temporary storage of the other suppliers resulting from previous trips.

Second, various vehicle types with different capacities and GHG emission indices were

considered. These features enabled the model to select the appropriate transportation mode

(as well as the transportation route) to reduce the total supply chain costs and improve the

environmental health criteria (lowering GHG emissions). The results show that the model is

straightforward to use in practice; a sensitivity analysis was performed to prove that the

model could present more constructive solutions from a “green logistics” point of view.

Promising areas for further research include applying the proposed model to other supply

chain structures and other kinds of products (e.g. deteriorating items), developing multi-

objective models with respect to green logistics and developing models under uncertain

conditions.

Acknowledgements

We thank the Group TOUPARGEL for partially funding this research project and for the rich

and constructive discussion about the distribution framework developed in this paper. We

also thank the anonymous referees for their helpful comments on earlier versions of this

paper.

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Highlights

• A multi-period multi-product inventory routing problem is proposed. • Transport carrier selection is introduced as a guide to greener supply chain. • We use transshipment option to increase the performance of the supply chain. • Transshipment option enables the model to reduce GHG emissions. • The best configuration of the vehicles types, routes, pickups, deliveries and

transshipments in each period is determined.