Sustainable Supply Chain and Transportation Networks

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Sustainable Supply Chain andTransportation NetworksAnna Nagurney a , Zugang Liu a & Trisha Woolley aa Department of Finance and Operations Management , IsenbergSchool of Management, University of Massachusetts , Amherst,Massachusetts, USAPublished online: 24 Feb 2007.

To cite this article: Anna Nagurney , Zugang Liu & Trisha Woolley (2007) Sustainable Supply Chainand Transportation Networks, International Journal of Sustainable Transportation, 1:1, 29-51, DOI:10.1080/15568310601060077

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Sustainable Supply Chain and Transportation

Networks

Anna Nagurney, Zugang Liu, and Trisha WoolleyDepartment of Finance and Operations Management, Isenberg School ofManagement, University of Massachusetts, Amherst, Massachusetts, USA

ABSTRACT

In this paper, we show how sustainable supply chains can be transformed into andstudied as transportation networks. Specifically, we develop a new supply chainmodel in which the manufacturers can produce the homogeneous product in differ-ent manufacturing plants with associated distinct environmental emissions. We as-sume that the manufacturers, the retailers with which they transact, as well as theconsumers at the demand markets for the product are multicriteria decision-makerswith the environmental criteria weighted distinctly by the different decision-makers.We derive the optimality conditions and the equilibrium conditions, which are thenshown to satisfy a variational inequality problem. We prove that the supply chainmodel with environmental concerns can be reformulated and solved as an elasticdemand transportation network equilibrium problem. Numerical supply chain exam-ples are presented for illustration purposes. This paper, hence, begins the construc-tion of a bridge between sustainable supply chains and transportation networks.Key Words: environmental concerns, multicriteria decision-making, supply

chains, transportation network equilibrium, variational inequalities.

1. INTRODUCTION

Transportation provides the foundation for the linking of economic activities. With-out transportation, inputs to production processes do not arrive, nor can finishedgoods reach their destinations. In today’s globalized economy, inputs to productionprocesses may lie continents away from assembly points and consumption locations,further emphasizing the critical infrastructure of transportation in product supplychains.

Received 11 May 2006; revised 28 September 2006; accepted 29 September 2006.Address correspondence to Anna Nagurney, Department of Finance and OperationsManagement, Isenberg School of Management, University of Massachusetts, Amherst, MA01003, USA. E-mail: [email protected]

29

International Journal of Sustainable Transportation, 1:29–51, 2007Copyright # Taylor & Francis Group, LLCISSN: 1556-8318 print=1556-8334 onlineDOI: 10.1080/15568310601060077

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At the same time that supply chains have become increasingly globalized, en-vironmental concerns due to global warming and associated security risks regardingenergy supplies have drawn the attention of numerous constituencies (cf. Cline,1992; Poterba, 1993; Painuly, 2001). Indeed, companies are increasingly beingheld accountable not only for their own performance in terms of environmentalaccountability but also for that of their suppliers, subcontractors, joint venture part-ners, distribution outlets, and, ultimately, even for the disposal of their products.Consequently, poor environmental performance at any stage of the supply chainmay damage the most important asset that a company has, which is its reputation.

In this paper, a significant extension of the supply chain network model ofNagurney and Toyasaki (2003), which introduced environmental concerns into asupply chain network equilibrium framework [see also Nagurney, Dong, andZhang (2002)], is made through the introduction of alternative manufacturingplants for each manufacturer with distinct associated environmental emissions.In addition, we demonstrate that the new supply chain network equilibriummodel can be transformed into a transportation network equilibrium modelwith elastic demands over an appropriately constructed abstract network or super-network. We also illustrate how this theoretical result can be exploited in practicethrough the computation of numerical examples.

This paper is organized as follows. Section 2 develops the multitiered, multicri-teria supply chain network model with distinct manufacturing plants and associatedemissions and presents the variational inequality formulation of the governingequilibrium conditions. We also establish that the weights associated with the en-vironmental criteria of the various decision-makers can be interpreted as taxes. Sec-tion 3 then recalls the well-known transportation network equilibrium model ofDafermos (1982). Section 4 demonstrates how the new supply chain networkmodel with environmental concerns can be transformed into a transportation net-work equilibrium model over an appropriately constructed abstract network orsupernetwork. This equivalence provides us with a new interpretation of the equi-librium conditions governing sustainable supply chains in terms of path flows. InSection 5, we apply an algorithm developed for the computation of solutions to elas-tic demand transportation network equilibrium problems to solve numerical supplychain network problems in which there are distinct manufacturing plants availablefor each manufacturer, and emissions associated with production as well as withtransportation=transaction and the operation of the retailers are included. The nu-merical examples illustrate the potential power of this approach for policy analyses.

The contributions in this paper further demonstrate the generality of the conceptsof transportation network equilibrium, originally proposed in the seminal book ofBeckmann, McGuire, and Winsten (1956) [see also Boyce, Mahmassani, and Nagurney(2005)]. Indeed, recently, it has been shown by Nagurney (2006a) that supply chainscan be reformulated and solved as transportation network problems. Moreover, thepapers by Nagurney and Liu (2005) and Wu et al. (2006) demonstrate, as hypothesizedby Beckmann, McGuire, and Winsten (1956), that electric power generation and dis-tribution networks can be reformulated and solved as transportation network equilib-rium problems. See also the book by Nagurney (2006b) for a variety of transportation-based supply chain network models and applications and the book by Nagurney(2000) on sustainable transportation networks.

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2. THE SUPPLY CHAIN MODEL WITH ALTERNATIVE MANUFACTURINGPLANTS AND ENVIRONMENTAL CONCERNS

In this section, we develop the supply chain network model that includes manu-facturing plants as well as multicriteria decision-making associated with environ-mental concerns. We consider I manufacturers, each of which generally ownsand operates M manufacturing plants. Each manufacturing plant is associatedwith a different primary production process and energy consumption combinationwith associated environmental emissions. There are also J retailers, T transporta-tion=transaction modes between each retailer and demand market, with a totalof K demand markets, as depicted in Figure 1. The majority of the needednotation is given in Table 1. An equilibrium solution is denoted by ‘‘�’’. All vectorsare assumed to be column vectors, except where noted otherwise.

The top-tiered nodes in the supply chain network in Figure 1, enumerated by1, . . . , i . . . , I, represent the I manufacturers, who are the decision makers whoown and operate the manufacturing plants denoted by the second tier of nodesin the network. The manufacturers produce a homogeneous product using the dif-ferent plants and sell the product to the retailers in the third tier.

Node im in the second tier corresponds with manufacturer i’s plant m, with thesecond tier of nodes enumerated as: 11, . . . ,IM. We assume that each manufacturerseeks to determine his optimal production portfolio across his manufacturingplants and his sales allocations of the product to the retailers in order to maximizehis own profit. We also assume that each manufacturer seeks to minimize the totalemissions associated with production and transportation to the retailers.

Retailers, which are represented by the third-tiered nodes in Figure 1, functionas intermediaries. The nodes corresponding with the retailers are enumerated as

Figure 1. The supply chain network with manufacturing plants.

Sustainable Supply Chain and Transportation Networks

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1, . . . , j, . . . , J with node j corresponding with retailer j. They purchase the productfrom the manufacturers and sell the product to the consumers at the different de-mand markets. We assume that the retailers compete with one another in a non-cooperative manner. Also, we assume that the retailers are assumed to bemulticriteria decision-makers with environmental concerns and they also seek tominimize the emissions associated with transacting (which can include transpor-tation) with the consumers as well as in operating their retail outlets.

The bottom-tiered nodes in Figure 1 represent the demand markets, which canbe distinguished from one another by their geographic locations or the type ofassociated consumers such as whether they correspond, for example, withbusinesses or with households. There are K bottom-tiered nodes with node k cor-responding with demand market k.

The retailers need to cover the direct costs and to decide which transportation=transaction modes should be used and how much product should be delivered.The structure of the network in Figure 1 guarantees that the conservation of

Table 1. Notation for the supply chain model

Notation Definition

qim Quantity of product produced by manufacturer i using plant m, wherei ¼ 1; . . . ; I ; m ¼ 1; . . . ;M .

qm I-dimensional vector of the product generated by manufacturers usingplant m with components q1m ; . . . ; qIm .

q IM-dimensional vector of all the production outputs generated by themanufacturers at the plants.

Q 1 IM J-dimensional vector of flows between the plants of the manufacturersand the retailers with component imj denoted by qimj.

Q 2 JTK-dimensional vector of product flows between retailers and demandmarkets with component jtk denoted by qt

jk and denoting the flowbetween retailer j and demand market k via transportation=transactionmode t.

d K-dimensional vector of market demands with component k denoted by dk.fimðqmÞ Production cost function of manufacturer i using plant m with marginal

production cost with respect to qim denoted by @[email protected] ðqimjÞ Transportation=transaction cost incurred by manufacturer i using plant m

in transacting with retailer j with marginal transaction cost denoted by@cimjðqimj Þ=@qimj .

h J-dimensional vector of the retailers’ supplies of the product with

component j denoted by hj, with hj �PI

i¼1

PMm¼1 qimj .

cjðhÞ � cjðQ 1Þ Operating cost of retailer j with marginal operating cost with respect to hj

denoted by @cj=@hj and the marginal operating cost with respect to qimj

denoted by @cjðQ 1Þ[email protected]

jkðqtjkÞ The transportation=transaction cost associated with the transaction

between retailer j and demand market k via transportation=transaction t.cct

jkðQ 2Þ Unit transportation=transaction cost incurred by consumers at demandmarket k in transacting with retailer j via mode t.

q3kðdÞ Demand market price function at demand market k.

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flow equations associated with the production and distribution are satisfied. Theflows on the links joining the manufacturers in Figure 1 to the plant nodes are re-spectively q11, . . . ,qim, . . . ,qIM; the flows on the links from the plant nodes to the re-tailer nodes are given, respectively, by the components of the vector Q1, whereasthe flows on the links joining the retailer nodes with the demand markets aregiven by the respective components of the vector Q2.

Of course, if a particular manufacturer does not own M manufacturingplants, then the corresponding links (and nodes) can just be removed fromthe supply chain network in Figure 1 and the notation reduced accordingly.Similarly, if a mode of transportation=transaction is not available for a retailer=demand market pair, then the corresponding link may be removed from thesupply chain network in Figure 1 and the notation changed accordingly. Onthe other hand, multiple modes of transportation=transaction from the plantsto the retailers can easily be added as links to the supply chain network in Fig-ure 1 joining the plant nodes with the retailer nodes (with an associated in-crease in notation).

We now describe the behavior of the manufacturers, the retailers, and the con-sumers at the demand markets. We then state the equilibrium conditions of thesupply chain network and provide the variational inequality formulation.

2.1. Multicriteria Decision-Making Behavior of the Manufacturers and TheirOptimality Conditions

Let q�1imj denote the unit price charged by manufacturer i for the transactionwith retailer j for the product produced at plant m. q�1imj is an endogenous variableand can be determined once the complete supply chain network equilibriummodel is solved. Because we have assumed that each individual manufacturer i,i ¼ 1, . . . ,I, is a profit maximizer, the profit-maximization objective function ofmanufacturer i can be expressed as follows:

MaximizeXMm¼1

XJ

j¼1

q�1imj qimj �XMm¼1

fimðqmÞ �XMm¼1

XJ

j¼1

cimjðqimj Þ: ð1aÞ

The first term in the objective function (1a) represents the revenue and the nexttwo terms represent the production cost and transportation=transaction costs,respectively.

In addition, we assume that manufacturer i is concerned with the total amountof emissions generated both in production of the product at the various manufac-turing plants as well as in transportation of the product to the various retailers. Let-ting eim denote the amount of emissions generated per unit of product producedat plant m of manufacturer i, and eimj the amount of emissions generated in trans-porting the product from plant m of manufacturer i to retailer j, we have that thesecond objective function of manufacturer i is given by:

MinimizeXMm¼1

eimqim þXMm¼1

XJ

j¼1

eimj qimj : ð1bÞ

We assign now a non-negative weight of ai to the emissions-generation criterion(1b) with the weight associated with profit maximization [cf. (1a)] being set equal



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to 1. Thus, we can construct a value function for each manufacturer using aconstant additive weight value function [see, e.g., Nagurney and Dong (2002),Nagurney and Toyasaki (2003), and the references therein]. Consequently, themulticriteria decision-making problem for manufacturer i is transformed into:

MaximizeXMm¼1

XJ

j¼1

q�1imj qimj �XMm¼1

fimðqmÞ �XMm¼1

XJ

j¼1

cimj ðqimjÞ

� ai

XMm¼1

eimqim þXMm¼1

XJ

j¼1

eimj qimj

!ð1cÞ

subject to: XJ

j¼1

qimj ¼ qim ; m ¼ 1; . . . ;M ; ð2Þ

qimj � 0; m ¼ 1; . . . ;M ; j ¼ 1; . . . ; J : ð3ÞConservation of flow equation (2) states that the amount of product produced at aparticular plant of a manufacturer is equal to the amount of product transacted bythe manufacturer from that plant with all the retailers (and this holds for each ofthe manufacturing plants). Expression (3) guarantees that the quantities of theproduct produced at the various manufacturing plants are nonnegative.

We assume that the production cost and the transportation cost functions foreach manufacturer are continuously differentiable and convex [cf. (1c), subjectto (2) and (3)], and that the manufacturers compete in a noncooperative mannerin the sense of Nash (1950, 1951). The optimality conditions for all manufacturerssimultaneously, under the above assumptions [see also Gabay and Moulin (1980),Bazaraa, Sherali, and Shetty (1993), and Nagurney (1999)] coincide with the sol-ution of the following variational inequality: determine ðq�;Q 1�Þ 2 K1 satisfying

XI

i¼1

XMm¼1

@fimðq�mÞ@qim

þ ai eim

� �� ½qim � q�im � þ

XI

i¼1

XMm¼1

XJ

j¼1

@cimjðq�imjÞ@qimj

þ ai eimj � q�1imj

� �

� ½qimj � q�imj � � 0; 8ðq;Q 1Þ 2 K1; ð4Þ

where K1 �nðq;Q 1Þjðq;Q 1Þ 2 R

IMþIMJþ and ð2Þ holds

o.

2.2. Multicriteria Decision-Making Behavior of the Retailers and TheirOptimality Conditions

The retailers, in turn, are involved in transactions both with the manufacturersand with the consumers at demand markets.

It is reasonable to assume that the total amount of product sold by a retailer j,j ¼ 1, . . . , J, is equal to the total amount of the product that he purchased from themanufacturers and that was produced via the different manufacturing plants avail-able to the manufacturers. This assumption can be expressed as the following con-servation of flow equations:XK

k¼1

XT

t¼1

qtjk ¼

XI

i¼1

XMm¼1

qimj ; j ¼ 1; . . . ; J : ð5Þ

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Let qt�2jk denote the price charged by retailer j to demand market k via transporta-

tion=transaction mode t. This price is determined endogenously in the modelonce the entire network equilibrium problem is solved. As noted above, it isassumed that each retailer seeks to maximize his own profit. Hence, the profit-maximization objective function faced by retailer j may be expressed as follows:

MaximizeXK

k¼1

XT

t¼1

qt�2jkqt

jk � cjðQ 1Þ �XI

i¼1

XMm¼1

q�1imj qimj �XK

k¼1

XT

t¼1

ctjkðqt

jkÞ: ð6aÞ

The first term in (6a) denotes the revenue of retailer j, the second term denotesthe operating cost of the retailer, and the third term denotes the paymentsfor the product to the various manufacturers. The last term in (6a) denotes thetransportation=transaction costs. Note that here we have assumed imperfect com-petition in terms of the operating cost but, of course, if the operating cost func-tions cj, j ¼ 1, . . . , J, depend only on the product handled by j (and not also onthe product handled by the other retailers), then the dependence of these func-tions on Q1 can be simplified accordingly (and this is a special case of themodel). The latter would reflect perfect competition.

In addition, for notational convenience, we let

hj �XI

i¼1

XMm¼1

qimj ; j ¼ 1; . . . ; J : ð7Þ

As defined in Table 1, the operating cost of retailer j, cj, is a function of the totalproduct inflows to the retailer, that is,

cjðhÞ � cjðQ 1Þ; j ¼ 1; . . . ; J : ð8Þ

Hence, his marginal cost with respect to hj is equal to the marginal cost withrespect to qimj :

@cjðhÞ@hj

�@cjðQ 1Þ@qimj

; j ¼ 1; . . . ; J ; m ¼ 1; . . . ;M : ð9Þ

In addition, we assume that each retailer seeks to minimize the emissions asso-ciated with managing his retail outlet and with transacting with consumers at thedemand markets. Let ej denote the amount of emissions generated by the retailer j,j ¼ 1, . . . ,J, and let et

jk denote the amount of emissions per unit of producttransacted between k and j via t, for j ¼ 1, . . . , J; k ¼ 1, . . . , K; and t ¼ 1, . . . ,T.Then we have that the second objective function of retailer j is given by

Minimize ej hj þXK

k¼1

XT

t¼1

etjkqt

jk : ð6bÞ

We associate the nonnegative weight bj with the environmental objective (cri-terion) function (6b) and we construct retailer j’s multicriteria decision-makingproblem, given by

MaximizeXK

k¼1

XT

t¼1

qt�2jkqt

jk � cjðQ 1ÞXI

t¼1

XMm¼1

q�1imj qimj

�XK

k¼1

XT

t¼1

ctjkðqt

jkÞ � bj ej hj þXK

k¼1

XT

t¼1

etjkqt

jk

!ð6cÞ



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subject to (7) and XK

k¼1

XT

t¼1

qtjk ¼

XI

i¼1

XMm¼1

qimj ð10Þ

qimj � 0; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; ð11Þ

qtjk � 0; k ¼ 1; . . . ;K ; t ¼ 1; . . . ;T : ð12Þ

We assume that the transaction costs and the operating costs [cf. (6a)] are allcontinuously differentiable and convex and that the retailers compete in a non-cooperative manner.

Hence, the optimality conditions for all retailers, simultaneously, under theabove assumptions [see also Dafermos and Nagurney (1987) and Nagurney,Dong, and Zhang (2002)] can be expressed as the following variational inequality:determine ðh�;Q 2�;Q 1�Þ 2 K3 such that

XJ

j¼1

@cjðh�Þ@hj

þ bj ej

� �� ½hj � h�j � þ

XJ

j¼1

XK

k¼1

XT

t¼1

@ctjkðqt�

jk Þ@qt

jk

þ bj etjk � qt�

2jk

" #

� ½qtjk � qt�

jk � þXI

i¼1

XMm¼1

XJ

j¼1

½q�1imj � � ½qimj � q�imj � � 0; 8ðh;Q 1;Q 2; Þ 2 K3; ð13Þ

where K3 �nðh;Q 2;Q 1Þjðh;Q 2;Q 1Þ 2 R

J ð1þTKþIM Þþ and ð7Þ and ð10Þ hold

o.

2.3. Equilibrium Conditions for the Demand Markets

At each demand market k, k ¼ 1, . . . ,K, the following conservation of flowequation must be satisfied:

dk ¼XJ

j¼1

XT

t¼1

qtjk : ð14Þ

We also assume that the consumers at the demand markets may be environmen-tally conscious in choosing their modes of transaction with the retailer with anassociated non-negative weight of gk for demand market k. Because the demandmarket price functions are given, the market equilibrium conditions at demandmarket k then take the form: for each retailer j, j ¼ 1, . . . ,J, and transportation=transaction mode t, t ¼ 1, . . . ,T,

qt�2jk þ cct

jkðQ 2�Þ þ gketjk

¼ q3kðd�Þ; if qt�jk > 0;

� q3kðd�Þ; if qt�jk ¼ 0.

�ð15Þ

Nagurney and Toyasaki (2003) [see also Nagurney and Toyasaki (2005)] consideredsimilar demand market equilibrium conditions but in the case in which the demandfunctions, rather than the demand price functions as above, were given.

The interpretation of conditions (15) is as follows: Consumers at a demand mar-ket will purchase the product from a retailer via a transportation=transaction mode,provided that the purchase price plus the unit transportation=transaction cost plus

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the marginal cost of emissions associated with that transaction is equal to the pricethat the consumers are willing to pay at that demand market. If the purchase priceplus the unit transportation=transaction cost plus the marginal cost of emissionsassociated with that transaction exceeds the price the consumers are willing topay, then there will be no transaction between that retailer and demand marketvia that transportation=transaction mode. The equivalent variational inequality gov-erning all the demand markets takes the form: determine ðQ 2�; d�Þ 2 K4, such that

XJ

j¼1

XK

k¼1

XT

t¼1

½qt�2jk þ cct

jkðQ 2�Þ þ gketjk � � ½qt

jk � qt�jk � �

XK

k¼1

q3kðd�Þ

� ½dk � d�k � � 0; 8ðQ 2; dÞ 2 K4; ð16Þ

where K4 �nðQ 2; dÞjðQ 2; dÞ 2 R

K ðJTþ1Þþ and ð14Þ holds

o.

2.4. The Equilibrium Conditions for the Supply Chain Network withManufacturing Plants and Environmental Concerns

In equilibrium, the optimality conditions for all the manufacturers, the optim-ality conditions for all the retailers, and the equilibrium conditions for all thedemand markets must be simultaneously satisfied so that no decision maker hasany incentive to alter his transactions.

Definition 1: Supply Chain Network Equilibrium with Manufacturing Plants and

Environmental ConcernsThe equilibrium state of the supply chain network with manufacturing plants and en-

vironmental concerns is one where the product flows between the tiers of the network coincideand the product flows and prices satisfy the sum of conditions (4), (13), and (16).

We now state and prove:

Theorem 1: Variational Inequality Formulation of the Supply Chain Network Equi-

librium with Manufacturing Plants and Environmental ConcernsThe equilibrium conditions governing the supply chain network according to Definition 1

coincide with the solution of the variational inequality given by: determineðq�; h�;Q 1�;Q 2�; d�Þ 2 j5 satisfying

XI

i¼1

XMm¼1

@fimðq�mÞ@qim

þ ai eim

� �� ½qim � q�im � þ

XJ

j¼1

@cjðh�Þ@hj

þ bj ej

� �� ½hj � h�j �

þXI

i¼1

XMm¼1

XJ

j¼1

@cimj ðq�imjÞ@qimj

þ ai eimj

� �� ½qimj � q�imj �

þXJ

j¼1

XK

k¼1

XT

t¼1

@ctjkðqt�

jk Þ@qt

jk

þ cctjkðQ 2�Þ þ ðbj þ gkÞet

jk

" #� ½qt

jk � qt�jk �

�XK

k¼1

q3kðd�Þ � ½dk � d�k � � 0; 8ðq; h; Q 1; Q 2; dÞ 2 K5; ð17Þ

where K5 � fðq; h; Q1; Q2 ; dÞjðq; h; Q1; Q2 ; dÞ 2 RIMþJþIMJþTJKþKþ and (2),

(5), and (7) hold}.



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Proof: We first prove that an equilibrium according to Definition 1 coincides withthe solution of variational inequality (17). Indeed, summation of (4), (13), and(16), after algebraic simplifications, yields (17).

We now prove the converse, that is, a solution to variational inequality (17) satis-fies the sum of conditions (4), (13), and (16), and is, therefore, a supply chain net-work equilibrium pattern according to Definition 1.

First, we add the term q�1imj � q�1imj to the first term in the third summand ex-pression in (17). Then, we add the term qt�

2jk � qt�2jk the first term in the fourth sum-

mand expression in (17). Because these terms are all equal to zero, they do notchange (17). Hence, we obtain the following inequality:

XI

i¼1

XMm¼1

@fimðq�mÞ@qim

þ ai eim

� �� ½qim � q�im � þ

XJ

j¼1

@cjðh�Þ@hj

þ bj ej

� �� ½hj � h�j �

þXI

i¼1

XMm¼1

XJ

j¼1

@cimjðq�imj Þ@qimj

þ ai eimj þ q�1imj � q�1imj

� �� ½qimj � q�imj �

þXJ

j¼1

XK

k¼1

XT

t¼1

@ctjkðqt�

jk Þ@qt

jk

þ cctjkðqt�

jk Þ þ ðbj þ gkÞetjk þ qt�

2jk � qt�2jk

" #� ½qt

jk � qt�jk �

�XK

k¼1

q3kðd�Þ � ½dk � d�k � � 0; 8ðq; h; Q 1; Q 2; dÞ 2 K5; ð18Þ

which can be rewritten as

XI

i¼1

XMm¼1

@fimðq�mÞ@qim

þ ai eim

� �� ½qim � q�im � þ

XI

i¼1

XMm¼1

XG

g¼1

@cimjðq�imjÞ@qimj

� q�1imj þ ai eimj

��

� ½qimj � q�imj � þXJ

j¼1

@cjðh�Þ@hj

þ bj ej

� �� ½hj � h�j �

þXJ

j¼1

XK

k¼1

XT

t¼1

@ctjkðqt�

jk Þ@qt

jk

� qt�2jk þ bj e

tjk

" #� ½qt

jk � qt�jk �

þXJ

j¼1

XMm¼1

XI

i¼1

½q�1imj � � ½qimj � q�imj � þXJ

j¼1

XK

k¼1

XT

t¼1

½qt�2jk þ cct

jkðqt�jk Þ þ gket

jk �

� ½qtjk � qt�

jk � �XK

k¼1

q3kðd�Þ � ½dk � d�k � � 0; 8ðq; h; Q 1; Q 2; dÞ 2 K5: ð19Þ

Clearly, (19) is the sum of the optimality conditions (4) and (13), and the equi-librium conditions (16), and is, hence, according to Definition 1 a supply chainnetwork equilibrium. &

Remark

Note that, in the above model, we have assumed that the various decision-makers are environmentally conscious (to a certain degree) depending uponthe weights that they assign to the respective environmental criteria denoted byai ; i ¼ 1; . . . ; I ; for the manfacturers; by bj ; j ¼ 1; . . . ; J ; for the retailers, and by

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gk ; k ¼ 1; . . . ;K ; for the consumers at the respective demand markets. Theseweights are associated with the environmental emissions generated in production,transportation=transaction, and the operation of the retail outlets as the product‘‘moves’’ through the supply chain, driven by the demand for the product at thedemand markets. This implies (assuming all weights are not identically equal tozero) environmentally conscious decision-makers. It is worth emphasizing thatthe weights can also be interpreted as taxes, for example, carbon taxes [cf. Wuet al. (2006) and Nagurney, Liu, and Woolley (2006)], which would be assignedby a governmental authority. Such a framework was devised by Wu et al. (2006)in the case of electric power supply chains. However, in that model, the carbonemissions only occurred in the production of electric power using alternativepower-generation plants, which could utilize different forms of energy (renewableor not, for example). Hence, the carbon taxes were only associated with the man-ufacturers and the power-generating plants. In the case of the supply chain net-work model in this paper, in contrast, pollution can be emitted not only at theproduction stage but also in the transportation of the product, as well as duringthe operation of the retail outlets. In order to construct sustainable supply chains,it is essential to have a system-wide view of pollution generation.

We now describe how to recover the prices associated with the first and third tiersof nodes in the supply chain network. Clearly, the components of the vector q�3 canbe directly obtained from the solution to variational inequality (17). We now de-scribe how to recover the prices q�1imj , for all i, m, j, and qt�

2jk for all j, k, t, from thesolution of variational inequality (17). The prices associated with the retailers canbe obtained by setting [cf. (15)] qt�

2jk ¼ q�3k � gketjk � cct

jkðQ 2�Þ for any j, t, k suchthat qt�

sk > 0. The top-tiered prices, in turn, can be recovered by setting [cf. (4)]q�1imj ¼ @fimðq�mÞ=@qimj þ @cimjðq�imjÞ=@qimj þ ai eimj for any i, m, j such that q�imj > 0.

In this article, we have focused on the development of a supply chain networkmodel with a view toward sustainability in which the weights (equivalently, taxes)are known=assigned a priori. In order to achieve a particular environmental goal[see also Nagurney (2000)], for example, in the case of a bound on the totalemissions in the entire supply chain, one could conduct simulations associatedwith the different weights in order to achieve the desired policy result. An interest-ing extension would be to construct a model in which the weights=taxes are en-dogenous, as was done in the case of electric power supply chains and carbontaxes by Nagurney, Liu, and Woolley (2006). However, as also discussed therein,the transportation network equilibrium reformulation may be lost for the full sup-ply chain (although still exploited computationally during the iterative algorithmicprocess).

3. THE TRANSPORTATION NETWORK EQUILIBRIUM MODEL WITHELASTIC DEMANDS

In this section, we recall the transportation network equilibrium model withelastic demands, due to Dafermos (1982), in which the travel disutility functionsare assumed known and given. In Section 4, we establish that the supply chain net-work model in Section 2 can be reformulated as such a transportation networkequilibrium problem but over a specially constructed network topology.



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We consider a network G with the set of links L with nL elements, the set of pathsP with nP elements, and the set of origin=destination (O=D) pairs W with nW ele-ments. We denote the set of paths joining O=D pair w by Pw. Links are denoted bya, b, and so forth; paths by p, q, and so forth; and O=D pairs by w1, w2, and so forth.

We denote the flow on path p by xp and the flow on link a by fa. The user travelcost on a link a is denoted by ca and the user travel cost on a path p by Cp. We de-note the travel demand associated with traveling between O=D pair w by dw and thetravel disutility by kw.

The link flows are related to the path flows through the following conservationof flow equations:

fa ¼Xp2P

xpdap ; 8a 2 L; ð20Þ

where dap ¼ 1 if link a is contained in path p, and dap ¼ 0 otherwise. Hence, theflow on a link is equal to the sum of the flows on paths that contain that link.

The user costs on paths are related to user costs on links through the followingequations:

Cp ¼Xa2L

cadap ; 8p 2 P ; ð21Þ

that is, the user cost on a path is equal to the sum of user costs on links that makeup the path.

For the sake of generality, we allow the user cost on a link to depend upon theentire vector of link flows, denoted by f, so that

ca ¼ caðf Þ; 8a 2 L: ð22Þ

We have the following conservation of flow equations:Xp2Pw

xp ¼ dw ; 8w: ð23Þ

Also, we assume, as given, travel disutility functions, such that

kw ¼ kwðdÞ; 8w; ð24Þ

where d is the vector of travel demands with travel demand associated with O=Dpair w being denoted by dw .

Definition 2: Transportation Network EquilibriumIn equilibrium, the following conditions must hold for each O=D pair w2W and each

path p2 Pw:

Cpðx�Þ � kwðd�Þ¼ 0; if x�p > 0;� 0; if x�p ¼ 0:

�ð25Þ

The interpretation of conditions (25) is as follows: only those paths connecting anO=D pair are used that have minimal travel costs, and those costs are equal to the tra-vel disutility associated with traveling between that O=D pair. As proved in Dafermos(1982), the transportation network equilibrium conditions (25) are equivalent to the

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following variational inequality in path flows: determine ðx�; d�Þ 2 K6 such that

Xw2W

Xp2Pw

Cpðx�Þ � ½xp � x�p � �Xw2W

kwðd�Þ � ½dw � d�w � � 0; 8ðx; dÞ 2 K6; ð26Þ

where K6 �nðx; dÞjðx; dÞ 2 R

npþnWþ and dw ¼

Pp2Pw

xp ; 8wo:

We now recall the equivalent variational inequality in link form due to Dafermos(1982).

Theorem 2: Variational Inequality Formulation of Transportation NetworkEquilibrium

A link flow pattern and associated travel demand pattern is a transportation network equilibriumif and only if it satisfies the variational inequality problem: determine ðf �; d�Þ 2 K7 satisfyingX

a2L

caðf �Þ � ðfa � f �a Þ �Xw2W

kwðd�Þ � ðdw � d�wÞ � 0; 8ðf ; dÞ 2 K7; ð27Þ

where K7 � fðf ;dÞ 2 RnLþnWþ jthere exists an x satisfying ð20Þ and dw ¼

Pp2Pw

xp ;8wg:

Beckmann, McGuire, and Winsten (1956) were the first to formulate rigorouslythe transportation network equilibrium conditions (25) in the context of user linkcost functions and travel disutility functions that admitted symmetric Jacobianmatrixes so that the equilibrium conditions (25) coincided with the Kuhn–Tuckeroptimality conditions of an appropriately constructed optimization problem. Thevariational inequality formulation, in turn, allows for asymmetric functions [seealso, e.g., Nagurney (1999) and the references therein].

4. TRANSPORTATION NETWORK EQUILIBRIUM REFORMULATIONOF THE SUPPLY CHAIN NETWORK EQUILIBRIUM MODEL WITHMANUFACTURING PLANTS AND ENVIRONMENTAL CONCERNS

In this section, we show that the supply chain network equilibrium model presentedin Section 2 is isomorphic to a properly configured transportation network equilib-rium model through the establishment of a supernetwork equivalence of the former.

We now establish the supernetwork equivalence of the supply chain networkequilibrium model to the transportation network equilibrium model with knowntravel disutility functions described in Section 3. This transformation allows us,as we will demonstrate in Section 5, to apply algorithms developed for the latterclass of problems to solve the former.

Consider a supply chain network with manufacturing plants as discussed in Sec-tion 2 with given manufacturers, i ¼ 1, . . . , I; given manufacturing plants for eachmanufacturer, m ¼ 1, . . . , M; retailers, j ¼ 1, . . . J; transportation=transactionmodes, t ¼ 1, . . . ,T; and demand markets, k ¼ 1, . . . ,K. The supernetwork, GS,of the isomorphic transportation network equilibrium model is depicted inFigure 2 and is constructed as follows.

It consists of six tiers of nodes with the origin node 0 at the top or first tier andthe destination nodes at the sixth or bottom tier. Specifically, GS consists of a singleorigin node 0 at the first tier and K destination nodes at the bottom tier, denoted,



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respectively, by z1, . . . , zK. There are K O=D pairs in GS denoted by w1 ¼ (0, z1), . . . ,wk ¼ (0, zk), . . . , wK ¼ (0, zK). Node 0 is connected to each second-tiered node xi,i ¼ 1, . . . , I, by a single link. Each second-tiered node xi, in turn, is connected toeach third-tiered node xim, i ¼ 1, . . . , I, m ¼ 1, . . . ,M, by a single link, and eachthird-tiered node is then connected to each fourth-tiered node yj, j ¼ 1, . . . , J, bya single link. Each fourth-tiered node yj is connected to the corresponding fifth-tiered node yj 0 by a single link. Finally, each fifth-tiered node yj 0 is connected toeach destination node zk, k ¼ 1, . . . , K, at the sixth tier by T parallel links.

Hence, in GS, there are Iþ IMþ 2 JþKþ 1 nodes; Iþ IMþ IMJþ Jþ JTK links,K O=D pairs, and IMJTK paths. We now define the link and link flow notation. Let ai

denote the link from node 0 to node xi with associated link flow fai, for i ¼ 1, . . . , I. Let

am, denote the link from node 0 to node xi to node xim with link flow faimfor i ¼ 1, . . . , I,

m ¼ 1, . . . , M. Also, let aimj denote the link from node xim to node yj with associated linkflow faimj

for i ¼ 1, . . . , I, m ¼ 1, . . . , M, and j ¼ 1, . . . , J. Let ajj 0 denote the link connect-ing node yj with node yj 0 with associated link flow fajj 0 for jj 0 ¼ 110, . . . , J J 0. Finally, let at

j 0kdenote the tth link joining node yj 0 with node zk for j 0 ¼ 10, . . . ,J 0, t ¼ 1, . . . , T, andk ¼ 1, . . . , K and with associated link flow f t

at0 k. We group the link flows into the vectors

as follows: we group the ffaig into the vector f1, the ffaim

g into the vector f 2, the ffaimjg

into the vector f3, the ffa0jjg into the vector f4, and the ff taj 0kg into the vector f5.

Figure 2. The GS supernetwork representation of supply chain network equilib-

rium with manufacturing plants.

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Thus, a typical path connecting O=D pair wk ¼ (0, zk) is denoted by ptimjj 0k and

consists of five links: ai, aim, aimj, ajj 0, and atj 0k . The associated flow on the path is

denoted by xtpimjj 0k

. Finally, we let dwkbe the demand associated with O=D pair wk

where kwkdenotes the travel disutility for wk.

Note that the following conservation of flow equations must hold on the networkGS:

fai¼XMm¼1

XJ

j¼1

XJ 0

j 0¼1

XK

k¼1

XT

t¼1

xptimjj0k

; i ¼ 1; . . . ; I ; ð28Þ

faim ¼XJ

j¼1

XJ 0

j 0¼10

XK

k¼1

XT

t¼1

xptimjj 0k

; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; ð29Þ

faimj ¼XJ 0

j 0¼10

XK

k¼1

XT

t¼1

xptimjj 0k

; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; j ¼ 1; . . . ; J ; ð30Þ

fajj 0 ¼XI

i¼1

XMm¼1

XK

k¼1

XT

t¼1

xptimjj0k

; jj 0 ¼ 110; . . . ; JJ 0; ð31Þ

fatj 0k¼XI

i¼1

XMm¼1

XJ

j¼1

xptimjj0k

; j 0 ¼ 10; . . . ; J 0; t ¼ 1; . . . ;T ; k ¼ 1; . . . ;K : ð32Þ

Also, we have that

dwk¼XI

i¼1

XMm¼1

XJJ 0jj 0¼110

XT

t¼1

xptimjj 0k

; k ¼ 1; . . . ;K : ð33Þ

If all path flows are non-negative and (28)–(33) are satisfied, the feasible pathflow pattern induces a feasible link flow pattern.

We can construct a feasible link flow pattern for GS based on the correspondingfeasible supply chain flow pattern in the supply chain network model, (q, h, Q1, Q2,d) 2 K5, in the following way:

qi � fai ; i ¼ 1; . . . ; I ; ð34Þ

qim � fa im ; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; ð35Þ

qimj � fa imj ; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; j ¼ 1; . . . ; J ; ð36Þ

hj � fajj 0 ; jj 0 ¼ 110; . . . ; JJ 0; ð37Þ

qtjk ¼ fat j 0k ; j ¼ 1; . . . ; J ; j 0 ¼ 10; . . . ; J 0; t ¼ 1; . . . ;T ; k ¼ 1; . . . ;K ; ð38Þ

dk ¼XJ

j¼1

XT

t¼1

qtjk ; k ¼ 1; . . . ;K : ð39Þ

Observe that although qi is not explicitly stated in the model in Section 2, it isinferred in that

qi ¼XMm¼1

qim ; i ¼ 1; . . . ; I ; ð40Þ

and simply represents the total amount of product produced by manufacturer i.



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Note that if (q, Q1, h, Q2, d) is feasible, then the link flow and demand patternconstructed according to (34)–(39) is also feasible, and the corresponding pathflow pattern that induces this link flow (and demand) pattern is also feasible.

We now assign user (travel) costs on the links of the network GS as follows: witheach link ai we assign a user cost cai defined by

cai � 0; i ¼ 1; . . . ; I ; ð41Þ

caim �@fim@qim

þ ai eim ; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; ð42Þ

with each link aimj we assign a user cost caimjdefined by

caimj�@fimj

@qimjþ ai eimj ; i ¼ 1; . . . ; I ; m ¼ 1; . . . ;M ; j ¼ 1; . . . ; J ; ð43Þ

with each link jj 0 we assign a user cost defined by

cajj 0 �@cj

@hjþ bj ej ; jj 0 ¼ 110; . . . ; JJ 0: ð44Þ

Finally, for each link atj 0k we assign a user cost defined by

catj 0k �

@ctjk

@qtjk

þ cctjk þ ðbj þ gkÞet

jk ; j 0 ¼ j ¼ 1; . . . ; J ; t ¼ 1; . . . ;T ; k ¼ 1; . . . ;K : ð45Þ

Then a user of path ptimjj 0k ; for i ¼ 1; . . . ; I ; m ¼ 1; . . . ; M ; jj 0 ¼ 110; . . . ; JJ 0;

t ¼ 1; . . . ;T ; k ¼ 1; . . . ;K ; on network GS in Figure 2 experiences a path travelcost Cpt

imjj 0kgiven by

Cptimjj 0k¼ cai þ caim þ caimj þ cajj0 þ cat

j 0k¼ @fim@qim

þ ai eim þ@cimj

@qimj

þ ai eimj þ@cj

@hjþ bj ej þ

@ctjk

@qtjk

þ cctjk þ ðbj þ gkÞet

jk : ð46Þ

Also, we assign the (travel) demands associated with the O=D pairs as follows:

dwk � dk ; k ¼ 1; . . . ;K ; ð47Þ

and the (travel) disutilities:

kwk � q3k ; k ¼ 1; . . . ;K : ð48Þ

Consequently, the equilibrium conditions (25) for the transportation networkequilibrium model on the network GS state that for every O=D pair wk and everypath connecting the O=D pair wk:

Cptimjj 0k� kwk ¼

@fim@qim

þ ai eim þ@cimj

@qimjþ aieimj þ

@cj

@hjþ bj ej þ

@ctjk

@qtjk

þ cctjk

þ ðbj þ gkÞetjk � kwk

¼ 0; if x�ptimjj 0k

> 0,

� 0; if x�ptimjj 0k¼ 0

8<: ð49Þ

We now show that the variational inequality formulation of the equilibrium con-ditions (49) in link form as in (27) is equivalent to the variational inequality (17)governing the supply chain network equilibrium with manufacturing plants andenvironmental concerns.

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For the transportation network equilibrium problem on GS, according to The-orem 2, we have that a link flow and travel disutility pattern ðf �; d�Þ 2 K7 is an equi-librium [according to (49)], if and only if it satisfies the variational inequality:

XI

i¼1

cai ðf 1�Þ � ðfai � f �aiÞ þ

XI

i¼1

XMm¼1

caim ðf 2�Þ � ðfaim � f �aimÞ

þXI

i¼1

XMm¼1

XJ

j¼1

caimj ðf 3�Þ � ðfaimj � f �aimjÞ þ

XJJ 0jj 0¼110

cajj 0 ðf4�Þ � ðfajj0 � f �ajj0

Þ

þXJ 0

j 0¼10

XK

k¼1

XT

t¼1

catj 0kðf 5�Þ � ðfat

j 0k� f �at

j 0kÞ �

XK

k¼1

kwk ðd�Þ � ðdwk � d�wkÞ � 0;

8ðf ; dÞ2K7: ð50Þ

After the substitution of (34)–(45) and (47)–(48) into (50), we have the follow-ing variational inequality: determine ðq�; h�;Q 1�;Q 2�; d�Þ 2 K5 satisfying:

XI

i¼1

XMm¼1

@fimðq�i Þ@qim

þ ai eim

� �� ½qim � q�im � þ

XJ

j¼1

@cjðh�Þ@hj

þ bj ej

� �� ½hj � h�j �

þXI

i¼1

XMm¼1

XJ

j¼1

@cimjðq�imj Þ@qimj

þ ai eimj

� �� ½qimj � q�imj � þ

XJ

j¼1

XK

k¼1

XT

t¼1

"@ct

jkðqt�jk Þ

@qtjk

þ cctjk þ ðQ 2�Þ þ ðbj þ gkÞet

jk

#� ½qt

jk � qt�jk � �

XK

k¼1

q3kðd�Þ � ½dk � d�k � � 0;

8ðq; h;Q 1;Q 2; dÞ 2 K5: ð51Þ

Variational inequality (51) is precisely variational inequality (17) governing thesupply chain network equilibrium. Hence, we have the following result:

Theorem 3: Equivalence of Sustainable Supply Chain and TransportationNetwork Equilibrium

A solution ðq�; h�;Q 1�;Q 2�; d�Þ 2 K5 of the variational inequality (17) governing thesupply chain network equilibrium coincides with the [via (34)–(45) and (47)–(48)] feasiblelink flow and travel demand pattern for the supernetwork GS constructed above and satisfiesvariational inequality (50). Hence, it is a transportation network equilibrium according toTheorem 2.

We now further discuss the interpretation of the supply chain network equilib-rium conditions. These conditions define the supply chain network equilibrium interms of paths and path flows, which, as shown above, coincide with Wardrop’s(1952) first principle of user-optimization in the context of transportation net-works over the network given in Figure 2. Hence, we now have an entirely new in-terpretation of supply network equilibrium with environmental concerns, whichstates that only minimal cost paths will be used from the super source node 0 toany destination node. Moreover, the cost on the utilized paths for a particularO=D pair is equal to the disutility (or the demand market price) that the usersare willing to pay.



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In Section 5, we will show how Theorem 3 can be utilized to exploit algorithmi-cally the theoretical results obtained above when we compute the equilibrium pat-terns of numerical supply chain network examples using an algorithm previouslyused for the computation of elastic demand transportation network equilibria.Of course, existence and uniqueness results obtained for elastic demand transpor-tation network equilibrium models as in Dafermos (1982) as well as stability andsensitivity analysis results [see also Nagurney and Zhang (1996)] can now be trans-ferred to sustainable supply chain networks using the formalism=equivalenceestablished above.

5. COMPUTATIONS

In this section, we provide numerical examples to demonstrate how the theor-etical results in this paper can be applied in practice. We utilize the Euler methodfor our numerical computations. The Euler method is induced by the general iter-ative scheme of Dupuis and Nagurney (1993) and has been applied by Nagurneyand Zhang (1996) to solve variational inequality (26) in path flows [equivalently,variational inequality (27) in link flows]. Convergence results can be found inthe above references.

5.1. The Euler Method

For the solution of (26), the Euler method takes the form: at iteration s, com-pute the path flows for paths p 2 P (and the travel demands) according to:

xsþ1p ¼ maxf0; xs

p þ asðkwðdT Þ � CpðxsÞÞg: ð52Þ

The simplicity of (52) lies in the explicit formula that allows for the compu-tation of the path flows in closed form at each iteration. The demands at each iter-ation simply satisfy (23), and this expression can be substituted into the kwð�Þfunctions.

The Euler method was implemented in FORTRAN, and the computer system usedwas a Sun system at the University of Massachusetts at Amherst. The convergencecriterion utilized was that the absolute value of the path flows between two success-ive iterations differed by no more than 10� 4. The sequence fasg in the Eulermethod [cf. (52)] was set to f1; 1=2; 1=2; 1=3; 1=3; 1=3; . . .g. The Euler methodwas initialized by setting the demands equal to 100 for each O=D pair with the pathflows equally distributed. The Euler method was also used to compute solutions toelectric power supply chain network examples, reformulated as transportationnetwork equilibrium problems in Wu et al. (2006).

In all the numerical examples, the supply chain network consisted of two man-ufacturers, with two manufacturing plants each, two retailers, one transportation=transaction mode, and two demand markets as depicted in Figure 3. The supernet-work representation that allows for the transformation (as proved in Section 4) toa transportation network equilibrium problem is given also in Figure 3. Hence, inthe numerical examples (see also Fig. 2), we had that I ¼ 2, M ¼ 2, J ¼ 2, J 0 ¼ 20,K ¼ 2, and T ¼ 1.

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The notation is presented for the examples in the form of the supply chain networkequilibrium model of Section 2. The equilibrium solutions for the examples, along withthe translations of the computed equilibrium link flows and the travel demands (anddisutilities) into the equilibrium supply chain flows and prices are given in Table 2.

Example 1

The data for the first numerical example is given below. In order to construct abenchmark, we assumed that all the weights associated with the environmental cri-teria were equal to zero, that is, we set a1 ¼ a2 ¼ 0, b1 ¼ b2 ¼ 0, and g1 ¼ g2 ¼ 0.

The production cost functions for the manufacturers were given by

f11ðq1Þ ¼ 2:5q211 þ q11q21 þ 2q11; f12qð2Þ ¼ 2:5q2

12 þ q11q12 þ 2q22;

f21qð1Þ ¼ 0:5q221 þ 0:5q11q21 þ 2q21; f22qð2Þ ¼ 0:5q2

22 þ q12q22 þ 2q22:

The transportation=transaction cost functions faced by the manufacturers andassociated with transacting with the retailers were given by

cimjðqimj Þ ¼ 0:5q2imj þ 3:5qimj ; i ¼ 1; m ¼ 1; 2; j ¼ 1; 2;

cimjðqimj Þ ¼ 0:5q2imj þ 2qimj ; i ¼ 2; m ¼ 1; 2; j ¼ 1; 2:

The operating costs of the retailers, in turn, were given by

c1ðQ 1Þ ¼ 0:5�X2

i¼1

qi1

�2; c2ðQ 1Þ ¼ 0:5

�X2

i¼1

qi2

�2:

The demand market price functions at the demand markets were

q31ðdÞ ¼ �d1 þ 500; q32 ¼ �d2 þ 500;

Figure 3. Supply chain network and corresponding supernetwork GS for the

numerical examples.



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and the unit transportation=transaction costs between the retailers and the consu-mers at the demand markets were given by

cc1jkðq1

jkÞ ¼ q1jk þ 5; j ¼ 1; 2; k ¼ 1; 2:

All other transportation=transaction costs were assumed to be equal to zero. Weassumed that the manufacturing plants emitted pollutants where e11¼e12¼e21¼e22¼5.

We utilized the supernetwork representation of this example depicted in Figure 3with the links enumerated as in Figure 3 in order to solve the problem via the Eulermethod. Note that there are 13 nodes and 20 links in the supernetwork in Figure 3.Using the procedure outlined in Section 4, we defined O=D pair w1 ¼ ð0; z1Þ and O=Dpair w2 ¼ ð0; z2Þ, and we associated the O=D pair travel disutilities with the demandmarket price functions as in (48) and the user link travel cost functions as given in(41)–(45) (analogous constructions were done for the subsequent examples).

The Euler method converged in 56 iterations and yielded the equilibrium sol-ution given in Table 2 (cf. also the supernetwork in Fig. 3). In Table 2, we also pro-vide the translations of the computed equilibrium pattern(s) into the supply chainnetwork flow, demand, and price notation using (34)–(40) and (47)–(48).

Table 2. Equilibrium solutions of Examples 1, 2, 3, and 4

Equilibrium values Example 1 Example 2 Example 3 Example 4

f �a1 ¼ q�1 48.17 47.68 47.17 42.04

f �a2 ¼ q�2 169.62 167.89 166.20 109.34

f �a11 ¼ q�11 33.37 33.03 32.69 25.87

f �a12 ¼ q�12 14.80 14.65 14.48 16.37

f �a21 ¼ q�21 33.71 33.37 33.02 26.17

f �a22 ¼ q�22 135.91 134.53 133.17 83.17

f �a110 ¼ h�1 108.90 107.79 103.82 0.00

f �a220 ¼ h�2 108.90 107.79 109.54 151.58

f �a111 ¼ q�111 16.69 16.52 15.60 0.00

f �a112 ¼ q�112 16.69 16.52 17.09 25.87

f �a121 ¼ q�121 7.40 7.32 6.61 0.00

f �a122 ¼ q�122 7.40 7.32 7.87 16.37

f �a211 ¼ q�211 16.85 16.68 15.77 0.00

f �a212 ¼ q�212 16.85 16.68 17.25 26.17

f �a221 ¼ q�221 67.96 67.26 65.84 0.00

f �a222 ¼ q�222 67.96 67.26 67.33 83.17

f �a1

101¼ q1�

11 54.45 53.89 51.91 0.00

f �a1

102¼ q1�

12 54.45 53.89 51.91 0.00

f �a1

201¼ q1�

21 54.45 53.89 54.77 75.79

f �a1

202¼ q1�

22 54.45 53.89 54.77 75.79

d�w1 ¼ d�1 108.90 107.79 106.68 75.79

d�w2 ¼ d�2 108.90 107.79 106.68 75.79

kw1 ¼ q31 391.11 392.23 393.30 424.21

kw2 ¼ q32 391.11 392.23 393.30 424.21

A. Nagurney et al.


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We don’t report the path flows due to space limitations (there are eight pathsconnecting each O=D pair) but note that all paths connecting each O=D pairwere used, that is, had positive flow, and the travel costs for paths connectingeach O=D pair were equal to the travel disutility for that O=D pair. The optimali-ty=equilibrium conditions were satisfied with excellent accuracy. The total amountof emissions in this example was e11q�11 þ e12q�12 þ e21q�21 þ e22q�22 ¼ 1089:

Example 2

We then solved the following variant of Example 1. We kept the data identical tothat in Example 1 except that we assumed now that the weights associated with theenvironmental criteria of the manufacturers were a1 ¼ a2 ¼ 1, with all otherweights equal to zero. The complete computed solution is now given.

The Euler method converged in 56 iterations and yielded the equilibrium linkflows, travel demands and travel disutilities (cf. Fig. 3) given in Table 2. Althoughwe do not report the equilibrium path flows, due to space constraints, we note that,in this example, all paths were again used. The total emissions generated wereequal to 1077.85 and, hence, as expected, given that both manufacturers nowassociated positive weights with the environmental criteria, the total emissionswere reduced, relative to the amount emitted in Example 1.

Example 3

Example 3 was constructed as follows from Example 2. The data were identicalto the data in Example 2, except that we now assumed that the first retailer used apolluting mode of transportation to deliver the product to the consumers at thedemand markets so that e1

11 ¼ e112 ¼ 10. We also assumed that the consumers

were now environmentally conscious and that the weights associated with the en-vironmental criteria at the demand markets were g1 ¼ g2 ¼ 1.

The Euler method converged in 67 iterations and yielded the new equilibriumpattern given in Table 2. In this example (as in Examples 1 and 2), all pathsconnecting each O=D pair were used, that is, they had positive equilibriumflows. The total amount of pollution emitted was now e11q�11 þ e12q�12 þ e21q�21þe22q�22þ e1�

11q111 þ e1�

12q1�12 ¼ 1585:95:

Example 4

In Example 4, we set out to ask the question, how high would g1 and g2 have to be sothat the demand markets did not utilize retailer 1 at all and the associated link flowswould be zero on those transportation=transaction links? We conducted simulationsand found that with g1 ¼ g2 ¼ 32, the desired result was achieved (withg1 ¼ g2 ¼ 30, there were still positive flows on those polluting links).

The Euler method converged in 102 iterations with the computed equilibriumlink flows, travel demands, and travel disutilities given in Table 2, along with theequivalent equilibrium supply chain network flows=transactions, demands, andprices. There were four paths used (and four not used) in each O=D pair. Thetotal amount of emissions were now 756.70. Hence, environmentally consciousconsumers could significantly reduce the environmental emissions through theeconomics and the underlying decision-making behavior in the supply chainnetwork.



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ACKNOWLEDGMENTS

This research of the authors was supported, in part, by NSF grant no. IIS00026471. The research of the first author was also supported, in part, by theRadcliffe Institute for Advanced Study at Harvard University under its 2005–2006 Radcliffe Fellows Program. This support is gratefully acknowledged andappreciated. The authors also appreciate helpful comments and suggestions onan earlier version of this article.

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Sustainable Supply Chain and Transportation Networks

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