Sustainable Supply Chain and Transportation Networks

International Journal o Sustainable Transportation ISSN: 1556-8318 (Print) 1556-8334 (Online) Journal homepage: http://www.tandonline.com/loi/ujst20 Sustainable Supply Chain and Transportation Networks Anna Nagurney, Zugang Liu & Trisha Woolley To cite this article: Anna Nagurney, Zugang Liu & Trisha Woolley (2007) Sustainable Supply Chain and Transportation Networks, International Journal o Sustainable Transportation, 1:1, 29-51, DOI: 10.1080/15568310601060077 To link to this article: https://doi.org/10.1080/15568310601060077 Published online: 24 Feb 2007. Submit your article to this journal Article views: 1734 View related articles Citing articles: 27 View citing articles Full Terms & Conditions o access and use can be ound at http://www.tandonline.com/action/journalinormation?journalcode=ujst20 Download by: [37.44.207.223] Date: 22 November 2017, At: 08:06

International Journal o Sustainable Transportation, 1:29 51, 2007 Copyright # Taylor & Francis Group, LLC ISSN: 1556-8318 print=1556-8334 online DOI: 10.1080/15568310601060077 Sustainable Supply Chain and Transportation Networks Anna Nagurney, Zugang Liu, and Trisha Woolley Department o Finance and Operations Management, Isenberg School o Management, University o Massachusetts, Amherst, Massachusetts, USA ABSTRACT In this paper, we show how sustainable supply chains can be transormed into and studied as transportation networks. Speciically, we develop a new supply chain model in which the manuacturers can produce the homogeneous product in dierent manuacturing plants with associated distinct environmental emissions. We assume that the manuacturers, the retailers with which they transact, as well as the consumers at the demand markets or the product are multicriteria decision-makers with the environmental criteria weighted distinctly by the dierent decision-makers. We derive the optimality conditions and the equilibrium conditions, which are then shown to satisy a variational inequality problem. We prove that the supply chain model with environmental concerns can be reormulated and solved as an elastic demand transportation network equilibrium problem. Numerical supply chain examples are presented or illustration purposes. This paper, hence, begins the construction o 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 oundation or the linking o economic activities. Without transportation, inputs to production processes do not arrive, nor can inished goods reach their destinations. In today s globalized economy, inputs to production processes may lie continents away rom assembly points and consumption locations, urther emphasizing the critical inrastructure o transportation in product supply chains. Received 11 May 2006; revised 28 September 2006; accepted 29 September 2006. Address correspondence to Anna Nagurney, Department o Finance and Operations Management, Isenberg School o Management, University o Massachusetts, Amherst, MA 01003, USA. E-mail: nagurney@gbin.umass.edu 29

A. Nagurney et al. At the same time that supply chains have become increasingly globalized, environmental concerns due to global warming and associated security risks regarding energy supplies have drawn the attention o numerous constituencies (c. Cline, 1992; Poterba, 1993; Painuly, 2001). Indeed, companies are increasingly being held accountable not only or their own perormance in terms o environmental accountability but also or that o their suppliers, subcontractors, joint venture partners, distribution outlets, and, ultimately, even or the disposal o their products. Consequently, poor environmental perormance at any stage o the supply chain may damage the most important asset that a company has, which is its reputation. In this paper, a signiicant extension o the supply chain network model o Nagurney and Toyasaki (2003), which introduced environmental concerns into a supply chain network equilibrium ramework [see also Nagurney, Dong, and Zhang (2002)], is made through the introduction o alternative manuacturing plants or each manuacturer with distinct associated environmental emissions. In addition, we demonstrate that the new supply chain network equilibrium model can be transormed into a transportation network equilibrium model with elastic demands over an appropriately constructed abstract network or supernetwork. We also illustrate how this theoretical result can be exploited in practice through the computation o numerical examples. This paper is organized as ollows. Section 2 develops the multitiered, multicriteria supply chain network model with distinct manuacturing plants and associated emissions and presents the variational inequality ormulation o the governing equilibrium conditions. We also establish that the weights associated with the environmental criteria o the various decision-makers can be interpreted as taxes. Section 3 then recalls the well-known transportation network equilibrium model o Daermos (1982). Section 4 demonstrates how the new supply chain network model with environmental concerns can be transormed into a transportation network equilibrium model over an appropriately constructed abstract network or supernetwork. This equivalence provides us with a new interpretation o the equilibrium conditions governing sustainable supply chains in terms o path lows. In Section 5, we apply an algorithm developed or the computation o solutions to elastic demand transportation network equilibrium problems to solve numerical supply chain network problems in which there are distinct manuacturing plants available or each manuacturer, and emissions associated with production as well as with transportation=transaction and the operation o the retailers are included. The numerical examples illustrate the potential power o this approach or policy analyses. The contributions in this paper urther demonstrate the generality o the concepts o transportation network equilibrium, originally proposed in the seminal book o Beckmann, McGuire, and Winsten (1956) [see also Boyce, Mahmassani, and Nagurney (2005)]. Indeed, recently, it has been shown by Nagurney (2006a) that supply chains can be reormulated and solved as transportation network problems. Moreover, the papers by Nagurney and Liu (2005) and Wu et al. (2006) demonstrate, as hypothesized by Beckmann, McGuire, and Winsten (1956), that electric power generation and distribution networks can be reormulated and solved as transportation network equilibrium problems. See also the book by Nagurney (2006b) or a variety o transportationbased supply chain network models and applications and the book by Nagurney (2000) on sustainable transportation networks. 30 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks 2. THE SUPPLY CHAIN MODEL WITH ALTERNATIVE MANUFACTURING PLANTS AND ENVIRONMENTAL CONCERNS In this section, we develop the supply chain network model that includes manuacturing plants as well as multicriteria decision-making associated with environmental concerns. We consider I manuacturers, each o which generally owns and operates M manuacturing plants. Each manuacturing plant is associated with a dierent primary production process and energy consumption combination with associated environmental emissions. There are also J retailers, T transportation=transaction modes between each retailer and demand market, with a total o K demand markets, as depicted in Figure 1. The majority o the needed notation is given in Table 1. An equilibrium solution is denoted by. All vectors are assumed to be column vectors, except where noted otherwise. The top-tiered nodes in the supply chain network in Figure 1, enumerated by 1,..., i..., I, represent the I manuacturers, who are the decision makers who own and operate the manuacturing plants denoted by the second tier o nodes in the network. The manuacturers produce a homogeneous product using the dierent plants and sell the product to the retailers in the third tier. Node im in the second tier corresponds with manuacturer i s plant m, with the second tier o nodes enumerated as: 11,...,IM. We assume that each manuacturer seeks to determine his optimal production portolio across his manuacturing plants and his sales allocations o the product to the retailers in order to maximize his own proit. We also assume that each manuacturer seeks to minimize the total emissions associated with production and transportation to the retailers. Retailers, which are represented by the third-tiered nodes in Figure 1, unction as intermediaries. The nodes corresponding with the retailers are enumerated as Figure 1. The supply chain network with manuacturing plants. International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 31

A. Nagurney et al. Table 1. Notation Notation or the supply chain model Deinition q im Quantity o product produced by manuacturer i using plant m, where i ¼ 1;...; I ; m ¼ 1;...; M. q m I-dimensional vector o the product generated by manuacturers using plant m with components q 1m ;...; q Im. q IM-dimensional vector o all the production outputs generated by the manuacturers at the plants. Q 1 IM J-dimensional vector o lows between the plants o the manuacturers and the retailers with component imj denoted by q imj. Q 2 JTK-dimensional vector o product lows between retailers and demand markets with component jtk denoted by qjk t and denoting the low between retailer j and demand market k via transportation=transaction mode t. d K-dimensional vector o market demands with component k denoted by d k. im ðq m Þ Production cost unction o manuacturer i using plant m with marginal production cost with respect to q im denoted by @ im =@q im. c imj ðq imj Þ Transportation=transaction cost incurred by manuacturer i using plant m in transacting with retailer j with marginal transaction cost denoted by @c imj ðq imj Þ=@q imj. h J-dimensional vector o the retailers supplies o the product with component j denoted by hj, with h j P I P M i¼1 m¼1 q imj. c j ðhþ c j ðq 1 Þ Operating cost o retailer j with marginal operating cost with respect to h j denoted by @c j =@h j and the marginal operating cost with respect to q imj denoted by @c j ðq 1 Þ=@q imj. cjk t ðqt jkþ The transportation=transaction cost associated with the transaction between retailer j and demand market k via transportation=transaction t. ^c jk t ðq 2 Þ Unit transportation=transaction cost incurred by consumers at demand market k in transacting with retailer j via mode t. q 3k ðdþ Demand market price unction at demand market k. 1,..., j,..., J with node j corresponding with retailer j. They purchase the product rom the manuacturers and sell the product to the consumers at the dierent demand markets. We assume that the retailers compete with one another in a noncooperative manner. Also, we assume that the retailers are assumed to be multicriteria decision-makers with environmental concerns and they also seek to minimize the emissions associated with transacting (which can include transportation) with the consumers as well as in operating their retail outlets. The bottom-tiered nodes in Figure 1 represent the demand markets, which can be distinguished rom one another by their geographic locations or the type o associated consumers such as whether they correspond, or example, with businesses or with households. There are K bottom-tiered nodes with node k corresponding 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 o the network in Figure 1 guarantees that the conservation o 32 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks low equations associated with the production and distribution are satisied. The lows on the links joining the manuacturers in Figure 1 to the plant nodes are respectively q 11,...,q im,...,q IM ; the lows on the links rom the plant nodes to the retailer nodes are given, respectively, by the components o the vector Q 1, whereas the lows on the links joining the retailer nodes with the demand markets are given by the respective components o the vector Q 2. O course, i a particular manuacturer does not own M manuacturing plants, then the corresponding links (and nodes) can just be removed rom the supply chain network in Figure 1 and the notation reduced accordingly. Similarly, i a mode o transportation=transaction is not available or a retailer= demand market pair, then the corresponding link may be removed rom the supply chain network in Figure 1 and the notation changed accordingly. On the other hand, multiple modes o transportation=transaction rom the plants to the retailers can easily be added as links to the supply chain network in Figure 1 joining the plant nodes with the retailer nodes (with an associated increase in notation). We now describe the behavior o the manuacturers, the retailers, and the consumers at the demand markets. We then state the equilibrium conditions o the supply chain network and provide the variational inequality ormulation. 2.1. Multicriteria Decision-Making Behavior o the Manuacturers and Their Optimality Conditions Let q 1imj denote the unit price charged by manuacturer i or the transaction with retailer j or the product produced at plant m. q 1imj is an endogenous variable and can be determined once the complete supply chain network equilibrium model is solved. Because we have assumed that each individual manuacturer i, i ¼ 1,...,I, is a proit maximizer, the proit-maximization objective unction o manuacturer i can be expressed as ollows: Maximize X J m¼1 j¼1 q 1imj q imj XM m¼1 im ðq m Þ XM X J m¼1 j¼1 c imj ðq imj Þ: The irst term in the objective unction (1a) represents the revenue and the next two terms represent the production cost and transportation=transaction costs, respectively. In addition, we assume that manuacturer i is concerned with the total amount o emissions generated both in production o the product at the various manuacturing plants as well as in transportation o the product to the various retailers. Letting e im denote the amount o emissions generated per unit o product produced at plant m o manuacturer i, and e imj the amount o emissions generated in transporting the product rom plant m o manuacturer i to retailer j, we have that the second objective unction o manuacturer i is given by: Minimize m¼1 e im q im þ XM X J m¼1 j¼1 e imj q imj : We assign now a non-negative weight o a i to the emissions-generation criterion (1b) with the weight associated with proit maximization [c. (1a)] being set equal ð1aþ ð1bþ International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 33

A. Nagurney et al. to 1. Thus, we can construct a value unction or each manuacturer using a constant additive weight value unction [see, e.g., Nagurney and Dong (2002), Nagurney and Toyasaki (2003), and the reerences therein]. Consequently, the multicriteria decision-making problem or manuacturer i is transormed into: subject to: Maximize a i m¼1 X J m¼1 j¼1 e im q im þ XM X J j¼1 q 1imj q imj XM X J m¼1 j¼1 m¼1 e imj q imj! im ðq m Þ XM X J m¼1 j¼1 c imj ðq imj Þ ð1cþ q imj ¼ q im ; m ¼ 1;...; M; ð2þ q imj 0; m ¼ 1;...; M; j ¼ 1;...; J : ð3þ Conservation o low equation (2) states that the amount o product produced at a particular plant o a manuacturer is equal to the amount o product transacted by the manuacturer rom that plant with all the retailers (and this holds or each o the manuacturing plants). Expression (3) guarantees that the quantities o the product produced at the various manuacturing plants are nonnegative. We assume that the production cost and the transportation cost unctions or each manuacturer are continuously dierentiable and convex [c. (1c), subject to (2) and (3)], and that the manuacturers compete in a noncooperative manner in the sense o Nash (1950, 1951). The optimality conditions or all manuacturers simultaneously, under the above assumptions [see also Gabay and Moulin (1980), Bazaraa, Sherali, and Shetty (1993), and Nagurney (1999)] coincide with the solution o the ollowing variational inequality: determine ðq ; Q 1 Þ2K 1 satisying X I @ im ðqm Þ X þ a i e im ½q M X J @c im imj ðqimj im q Þ ŠþXI þ a i e imj q 1imj @q im @q imj i¼1 m¼1 i¼1 m¼1 j¼1 ½q imj qimj Š0; 8ðq; Q 1 Þ2K 1 ; ð4þ n o where K 1 ðq; Q 1 Þjðq; Q 1 IM þimj Þ2Rþ and ð2þ holds. 2.2. Multicriteria Decision-Making Behavior o the Retailers and Their Optimality Conditions The retailers, in turn, are involved in transactions both with the manuacturers and with the consumers at demand markets. It is reasonable to assume that the total amount o product sold by a retailer j, j ¼ 1,..., J, is equal to the total amount o the product that he purchased rom the manuacturers and that was produced via the dierent manuacturing plants available to the manuacturers. This assumption can be expressed as the ollowing conservation o low equations: X K k¼1 t¼1 q t jk ¼ XI i¼1 m¼1 q imj ; j ¼ 1;...; J : ð5þ 34 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks Let q t 2jk denote the price charged by retailer j to demand market k via transportation=transaction mode t. This price is determined endogenously in the model once the entire network equilibrium problem is solved. As noted above, it is assumed that each retailer seeks to maximize his own proit. Hence, the proitmaximization objective unction aced by retailer j may be expressed as ollows: Maximize X K k¼1 t¼1 q t 2jk qt jk c jðq 1 Þ XI i¼1 m¼1 q 1imj q imj XK k¼1 t¼1 c t jk ðqt jk Þ: The irst term in (6a) denotes the revenue o retailer j, the second term denotes the operating cost o the retailer, and the third term denotes the payments or the product to the various manuacturers. The last term in (6a) denotes the transportation=transaction costs. Note that here we have assumed imperect competition in terms o the operating cost but, o course, i the operating cost unctions c j, j ¼ 1,..., J, depend only on the product handled by j (and not also on the product handled by the other retailers), then the dependence o these unctions on Q 1 can be simpliied accordingly (and this is a special case o the model). The latter would relect perect competition. In addition, or notational convenience, we let h j XI i¼1 m¼1 ð6aþ q imj ; j ¼ 1;...; J : ð7þ As deined in Table 1, the operating cost o retailer j, c j, is a unction o the total product inlows to the retailer, that is, c j ðhþ c j ðq 1 Þ; j ¼ 1;...; J : ð8þ Hence, his marginal cost with respect to h j is equal to the marginal cost with respect to q imj : @c j ðhþ @h j @c jðq 1 Þ @q imj ; j ¼ 1;...; J ; m ¼ 1;...; M: ð9þ In addition, we assume that each retailer seeks to minimize the emissions associated with managing his retail outlet and with transacting with consumers at the demand markets. Let e j denote the amount o emissions generated by the retailer j, j ¼ 1,...,J, and let ejk t denote the amount o emissions per unit o product transacted between k and j via t, or j ¼ 1,..., J; k ¼ 1,..., K; and t ¼ 1,...,T. Then we have that the second objective unction o retailer j is given by Minimize e j h j þ XK k¼1 t¼1 e t jk qt jk : We associate the nonnegative weight b j with the environmental objective (criterion) unction (6b) and we construct retailer j s multicriteria decision-making problem, given by ð6bþ Maximize XK k¼1 t¼1 X K k¼1 t¼1 c t jk ðqt jk Þ b j q t 2jk qt jk c jðq 1 Þ XI e j h j þ XK t¼1 m¼1 ejk t qt jk k¼1 t¼1 q 1imj q imj! ð6cþ International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 35

A. Nagurney et al. subject to (7) and X K k¼1 t¼1 q t jk ¼ XI q imj i¼1 m¼1 ð10þ q imj 0; i ¼ 1;...; I ; m ¼ 1;...; M; ð11þ q t jk 0; k ¼ 1;...; K ; t ¼ 1;...; T : ð12þ We assume that the transaction costs and the operating costs [c. (6a)] are all continuously dierentiable and convex and that the retailers compete in a noncooperative manner. Hence, the optimality conditions or all retailers, simultaneously, under the above assumptions [see also Daermos and Nagurney (1987) and Nagurney, Dong, and Zhang (2002)] can be expressed as the ollowing variational inequality: determine ðh ; Q 2 ; Q 1 Þ2K 3 such that X J j¼1 @c j ðh Þ j þ b @h j e j ½h j h ŠþXJ j X K j¼1 k¼1 t¼1 " # @cjk t ðqt jk Þ @qjk t þ b j e t jk qt 2jk X J ½qjk t qt jk ŠþXI ½q 1imj Š½q imj qimj Š0; 8ðh; Q 1 ; Q 2 ; Þ2K 3 ; ð13þ i¼1 m¼1 j¼1 n o where K 3 ðh; Q 2 ; Q 1 Þjðh; Q 2 ; Q 1 J ð1þtk þim Þ Þ2Rþ and ð7þ and ð10þ hold. 2.3. Equilibrium Conditions or the Demand Markets At each demand market k, k ¼ 1,...,K, the ollowing conservation o low equation must be satisied: d k ¼ XJ j¼1 t¼1 We also assume that the consumers at the demand markets may be environmentally conscious in choosing their modes o transaction with the retailer with an associated non-negative weight o g k or demand market k. Because the demand market price unctions are given, the market equilibrium conditions at demand market k then take the orm: or each retailer j, j ¼ 1,...,J, and transportation= transaction mode t, t ¼ 1,...,T, q t 2jk þ ^c t jk ðq 2 Þþg k ejk t ¼ q 3k ðd Þ; i qjk t > 0; q 3k ðd Þ; i qjk t ¼ 0. ð15þ q t jk : ð14þ Nagurney and Toyasaki (2003) [see also Nagurney and Toyasaki (2005)] considered similar demand market equilibrium conditions but in the case in which the demand unctions, rather than the demand price unctions as above, were given. The interpretation o conditions (15) is as ollows: Consumers at a demand market will purchase the product rom a retailer via a transportation=transaction mode, provided that the purchase price plus the unit transportation=transaction cost plus 36 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks the marginal cost o emissions associated with that transaction is equal to the price that the consumers are willing to pay at that demand market. I the purchase price plus the unit transportation=transaction cost plus the marginal cost o emissions associated with that transaction exceeds the price the consumers are willing to pay, then there will be no transaction between that retailer and demand market via that transportation=transaction mode. The equivalent variational inequality governing all the demand markets takes the orm: determine ðq 2 ; d Þ2K 4, such that X J X K j¼1 k¼1 t¼1 ½q t 2jk þ ^c t jk ðq 2 Þþg k ejk t Š½qt jk qt jk Š XK q 3k ðd Þ k¼1 ½d k dk Š0; 8ðQ 2 ; dþ 2K 4 ; ð16þ n o where K 4 ðq 2 ; dþjðq 2 K ðjt þ1þ ; dþ 2Rþ and ð14þ holds. 2.4. The Equilibrium Conditions or the Supply Chain Network with Manuacturing Plants and Environmental Concerns In equilibrium, the optimality conditions or all the manuacturers, the optimality conditions or all the retailers, and the equilibrium conditions or all the demand markets must be simultaneously satisied so that no decision maker has any incentive to alter his transactions. Deinition 1: Supply Chain Network Equilibrium with Manuacturing Plants and Environmental Concerns The equilibrium state o the supply chain network with manuacturing plants and environmental concerns is one where the product lows between the tiers o the network coincide and the product lows and prices satisy the sum o conditions (4), (13), and (16). We now state and prove: Theorem 1: Variational Inequality Formulation o the Supply Chain Network Equilibrium with Manuacturing Plants and Environmental Concerns The equilibrium conditions governing the supply chain network according to Deinition 1 coincide with the solution o the variational inequality given by: determine ðq ; h ; Q 1 ; Q 2 ; d Þ2j 5 satisying X I i¼1 m¼1 þ XI þ XJ @ im ðqm Þ im þ a i e im ½q im q ŠþXJ @q im X J i¼1 m¼1 j¼1 X K j¼1 k¼1 t¼1 @c imj ðqimj Þ þ a i e imj @q imj @cjk t ðqt jk Þ @qjk t j¼1 ½q imj q imj Š " # þ ^c t jk ðq 2 Þþðb j þ g k Þe t jk @c j ðh Þ þ b @h j e j j ½q t jk ½h j h j Š qt jk Š XK k¼1 q 3k ðd Þ½d k d k Š0; 8ðq; h; Q 1 ; Q 2 ; dþ 2K 5 ; ð17þ where K 5 ðq; h; Q 1 ; Q 2 ; dþjðq; h; Q 1 ; Q 2 ; dþ 2R IMþJþIMJþTJKþK þ and (2), (5), and (7) hold}. International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 37

A. Nagurney et al. Proo: We irst prove that an equilibrium according to Deinition 1 coincides with the solution o variational inequality (17). Indeed, summation o (4), (13), and (16), ater algebraic simpliications, yields (17). We now prove the converse, that is, a solution to variational inequality (17) satisies the sum o conditions (4), (13), and (16), and is, thereore, a supply chain network equilibrium pattern according to Deinition 1. First, we add the term q 1imj q 1imj to the irst term in the third summand expression in (17). Then, we add the term q t 2jk qt 2jk the irst term in the ourth summand expression in (17). Because these terms are all equal to zero, they do not change (17). Hence, we obtain the ollowing inequality: X I i¼1 m¼1 þ XI þ XJ @ im ðqm Þ ½q þ a i e im @q im q im @c j ðh Þ ŠþXJ þ b im @h j e j j¼1 j X J @c imj ðqimj Þ þ a i e imj þ q 1imj q 1imj i¼1 m¼1 j¼1 X K j¼1 k¼1 t¼1 XK k¼1 which can be rewritten as X I i¼1 m¼1 @q imj @cjk t ðqt jk Þ @qjk t ½q imj q imj Š " # þ ^c jk t ðqt jk Þþðb j þ g k Þejk t þ qt 2jk qt 2jk ½h j h j Š ½q t jk qt jk Š q 3k ðd Þ½d k d k Š0; 8ðq; h; Q 1 ; Q 2 ; dþ 2K 5 ; ð18þ @ im ðqm Þ im þ a i e im ½q im q ŠþXI @q im ½q imj qimj @c j ðh Þ ŠþXJ þ b @h j e j j¼1 j " # þ XJ þ XJ X K j¼1 k¼1 t¼1 X I j¼1 m¼1 i¼1 @cjk t ðqt jk Þ @qjk t q t 2jk þ b je t jk X G i¼1 m¼1 g ¼1 ½h j h j Š ½q t jk qt jk Š @c imj ðqimj Þ q 1imj þ a ie imj @q imj ½q 1imj Š½q X K imj qimj ŠþXJ ½q t 2jk þ ^c jk t ðqt jk Þþg kejk t Š j¼1 k¼1 t¼1 ½q t jk qt jk Š XK q 3k ðd Þ½d k dk Š0; 8ðq; h; Q 1 ; Q 2 ; dþ 2K 5 : ð19þ k¼1 Clearly, (19) is the sum o the optimality conditions (4) and (13), and the equilibrium conditions (16), and is, hence, according to Deinition 1 a supply chain network equilibrium. & Remark Note that, in the above model, we have assumed that the various decisionmakers are environmentally conscious (to a certain degree) depending upon the weights that they assign to the respective environmental criteria denoted by a i ; i ¼ 1;...; I ; or the manacturers; by b j ; j ¼ 1;...; J ; or the retailers, and by 38 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks g k ; k ¼ 1;...; K ; or the consumers at the respective demand markets. These weights are associated with the environmental emissions generated in production, transportation=transaction, and the operation o the retail outlets as the product moves through the supply chain, driven by the demand or the product at the demand markets. This implies (assuming all weights are not identically equal to zero) environmentally conscious decision-makers. It is worth emphasizing that the weights can also be interpreted as taxes, or example, carbon taxes [c. Wu et al. (2006) and Nagurney, Liu, and Woolley (2006)], which would be assigned by a governmental authority. Such a ramework was devised by Wu et al. (2006) in the case o electric power supply chains. However, in that model, the carbon emissions only occurred in the production o electric power using alternative power-generation plants, which could utilize dierent orms o energy (renewable or not, or example). Hence, the carbon taxes were only associated with the manuacturers and the power-generating plants. In the case o the supply chain network model in this paper, in contrast, pollution can be emitted not only at the production stage but also in the transportation o the product, as well as during the operation o the retail outlets. In order to construct sustainable supply chains, it is essential to have a system-wide view o pollution generation. We now describe how to recover the prices associated with the irst and third tiers o nodes in the supply chain network. Clearly, the components o the vector q 3 can be directly obtained rom the solution to variational inequality (17). We now describe how to recover the prices q 1imj, or all i, m, j, and qt 2jk or all j, k, t, rom the solution o variational inequality (17). The prices associated with the retailers can be obtained by setting [c. (15)] q t 2jk ¼ q 3k g kejk t ^c jk t ðq 2 Þ or any j, t, k such that qsk t > 0. The top-tiered prices, in turn, can be recovered by setting [c. (4)] q 1imj ¼ @ im ðqm Þ=@q imj þ @c imj ðqimj Þ=@q imj þ a i e imj or any i, m, j such that qimj > 0. In this article, we have ocused on the development o a supply chain network model 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)], or example, in the case o a bound on the total emissions in the entire supply chain, one could conduct simulations associated with the dierent weights in order to achieve the desired policy result. An interesting extension would be to construct a model in which the weights=taxes are endogenous, as was done in the case o electric power supply chains and carbon taxes by Nagurney, Liu, and Woolley (2006). However, as also discussed therein, the transportation network equilibrium reormulation may be lost or the ull supply chain (although still exploited computationally during the iterative algorithmic process). 3. THE TRANSPORTATION NETWORK EQUILIBRIUM MODEL WITH ELASTIC DEMANDS In this section, we recall the transportation network equilibrium model with elastic demands, due to Daermos (1982), in which the travel disutility unctions are assumed known and given. In Section 4, we establish that the supply chain network model in Section 2 can be reormulated as such a transportation network equilibrium problem but over a specially constructed network topology. International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 39

A. Nagurney et al. We consider a network G with the set o links L with n L elements, the set o paths P with n P elements, and the set o origin=destination (O=D) pairs W with n W elements. We denote the set o paths joining O=D pairw by P w. Links are denoted by a, b, and so orth; paths by p, q, and so orth; and O=D pairsbyw 1, w 2,andsoorth. We denote the low on path p by x p and the low on link a by a. The user travel cost on a link a is denoted by c a and the user travel cost on a path p by C p. We denote the travel demand associated with traveling between O=D pair w by d w and the travel disutility by k w. The link lows are related to the path lows through the ollowing conservation o low equations: a ¼ X x p d ap ; 8a 2 L; ð20þ p2p where d ap ¼ 1 i link a is contained in path p, and d ap ¼ 0 otherwise. Hence, the low on a link is equal to the sum o the lows on paths that contain that link. The user costs on paths are related to user costs on links through the ollowing equations: C p ¼ X c a d ap ; 8p 2 P; ð21þ a2l that is, the user cost on a path is equal to the sum o user costs on links that make up the path. For the sake o generality, we allow the user cost on a link to depend upon the entire vector o link lows, denoted by, so that We have the ollowing conservation o low equations: X c a ¼ c a ð Þ; 8a 2 L: ð22þ p2p w x p ¼ d w ; 8w: ð23þ Also, we assume, as given, travel disutility unctions, such that k w ¼ k w ðdþ; 8w; ð24þ where d is the vector o travel demands with travel demand associated with O=D pair w being denoted by d w. Deinition 2: Transportation Network Equilibrium In equilibrium, the ollowing conditions must hold or each O=D pair w 2 W and each path p 2 P w : C p ðx Þ k w ðd ¼ 0; i xp Þ > 0; 0; i xp ¼ 0: ð25þ The interpretation o conditions (25) is as ollows: only those paths connecting an O=D pair are used that have minimal travel costs, and those costs are equal to the travel disutility associated with traveling between that O=D pair. As proved in Daermos (1982), the transportation network equilibrium conditions (25) are equivalent to the 40 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks ollowing variational inequality in path lows: determine ðx ; d Þ2K 6 such that X X C p ðx Þ½x p xp Š X k w ðd Þ½d w dw Š0; w2w p2p w w2w 8ðx; dþ 2K6 ; ð26þ n where K 6 ðx; dþjðx; dþ 2R npþnw þ and d w ¼ P o p2p w x p ; 8w : We now recall the equivalent variational inequality in link orm due to Daermos (1982). Theorem 2: Variational Inequality Formulation o Transportation Network Equilibrium A link low pattern and associated travel demand pattern is a transportation network equilibrium i and only i it satisies the variational inequality problem: determine ð ; d Þ2K 7 satisying X a2l c a ð Þð a a Þ X k w ðd Þðd w dw Þ0; 8ð ; dþ 2K7 ; ð27þ w2w where K 7 ð ; dþ2r n Lþn W þ jthere exists an x satisying ð20þ and d w ¼ P p2p w x p ; 8wg: Beckmann, McGuire, and Winsten (1956) were the irst to ormulate rigorously the transportation network equilibrium conditions (25) in the context o user link cost unctions and travel disutility unctions that admitted symmetric Jacobian matrixes so that the equilibrium conditions (25) coincided with the Kuhn Tucker optimality conditions o an appropriately constructed optimization problem. The variational inequality ormulation, in turn, allows or asymmetric unctions [see also, e.g., Nagurney (1999) and the reerences therein]. 4. TRANSPORTATION NETWORK EQUILIBRIUM REFORMULATION OF THE SUPPLY CHAIN NETWORK EQUILIBRIUM MODEL WITH MANUFACTURING PLANTS AND ENVIRONMENTAL CONCERNS In this section, we show that the supply chain network equilibrium model presented in Section 2 is isomorphic to a properly conigured transportation network equilibrium model through the establishment o a supernetwork equivalence o the ormer. We now establish the supernetwork equivalence o the supply chain network equilibrium model to the transportation network equilibrium model with known travel disutility unctions described in Section 3. This transormation allows us, as we will demonstrate in Section 5, to apply algorithms developed or the latter class o problems to solve the ormer. Consider a supply chain network with manuacturing plants as discussed in Section 2 with given manuacturers, i ¼ 1,..., I; given manuacturing plants or each manuacturer, m ¼ 1,..., M; retailers, j ¼ 1,...J; transportation=transaction modes, t ¼ 1,...,T; and demand markets, k ¼ 1,...,K. The supernetwork, G S, o the isomorphic transportation network equilibrium model is depicted in Figure 2 and is constructed as ollows. It consists o six tiers o nodes with the origin node 0 at the top or irst tier and the destination nodes at the sixth or bottom tier. Speciically, G S consists o a single origin node 0 at the irst tier and K destination nodes at the bottom tier, denoted, International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 41

A. Nagurney et al. Figure 2. The G S supernetwork representation o supply chain network equilibrium with manuacturing plants. respectively, by z 1,..., z K. There are K O=D pairs in G S denoted by w 1 ¼ (0, z 1 ),..., w k ¼ (0, z k ),..., w K ¼ (0, z K ). Node 0 is connected to each second-tiered node x i, i ¼ 1,..., I, by a single link. Each second-tiered node x i, in turn, is connected to each third-tiered node x im, i ¼ 1,..., I, m ¼ 1,...,M, by a single link, and each third-tiered node is then connected to each ourth-tiered node y j, j ¼ 1,..., J, by a single link. Each ourth-tiered node y j is connected to the corresponding ithtiered node y j 0 by a single link. Finally, each ith-tiered node y j 0 is connected to each destination node z k, k ¼ 1,..., K, at the sixth tier by T parallel links. Hence, in G S, there are I þ IM þ 2JþKþ1 nodes; I þ IM þ IMJ þ J þ JTK links, K O=D pairs,andimjtk paths. We now deine the link and link low notation. Let a i denote the link rom node 0 to node x i with associated link low ai,ori ¼ 1,..., I.Let a m, denote the link rom node 0 to node x i to node xim with link low aim or i ¼ 1,..., I, m ¼ 1,..., M.Also,leta imj denote the link rom node x im to node y j with associated link low aimj or i ¼ 1,..., I, m ¼ 1,..., M,andj ¼ 1,..., J.Leta jj 0 denote the link connecting node y j with node y j 0 with associated link low ajj 0 or jj 0 ¼ 11 0,..., JJ 0. Finally, let aj0k t denote the tth link joining node y j 0 with node z k or j 0 ¼ 1 0,...,J 0, t ¼ 1,..., T, and k ¼ 1,..., K and with associated link low t a t 0 k. We group the link lows into the vectors as ollows: we group the ai g into the vector 1,the aim g into the vector 2,the aimj g into the vector 3,the a 0 jj g into the vector 4,andthea t j 0 k g into the vector 5. 42 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks Thus, a typical path connecting O=D pair w k ¼ (0, z k ) is denoted by pimjj t 0 k and consists o ive links: a i, a im, a imj, a jj 0, and aj t 0 k. The associated low on the path is denoted by xp t imjj 0 k. Finally, we let d wk be the demand associated with O=D pair w k where k wk denotes the travel disutility or w k. Note that the ollowing conservation o low equations must hold on the network G S : ai ¼ XM X J X J 0 X K m¼1 j¼1 j 0 ¼1 k¼1 t¼1 aim ¼ XJ X J 0 X K j¼1 j 0 ¼1 0 k¼1 t¼1 x p t ; i ¼ 1;...; I ; ð28þ imjj 0 k x p t ; i ¼ 1;...; I ; m ¼ 1;...; M; ð29þ imjj 0 k aimj ¼ XJ 0 X K j 0 ¼1 0 k¼1 t¼1 x p t ; i ¼ 1;...; I ; m ¼ 1;...; M; j ¼ 1;...; J ; ð30þ imjj 0 k ajj 0 ¼ XI X K i¼1 m¼1 k¼1 t¼1 x p t ; jj 0 ¼ 11 0 ;...; JJ 0 ; ð31þ imjj 0 k a t ¼ XI X J x j 0 k p t ; j 0 ¼ 1 0 ;...; J 0 ; t ¼ 1;...; T ; k ¼ 1;...; K : ð32þ imjj 0 k i¼1 m¼1 j¼1 Also,wehavethat d wk ¼ XI X JJ 0 i¼1 m¼1 jj 0 ¼11 0 t¼1 x p t ; k ¼ 1;...; K : ð33þ imjj 0 k I all path lows are non-negative and (28) (33) are satisied, the easible path low pattern induces a easible link low pattern. We can construct a easible link low pattern or G S based on the corresponding easible supply chain low pattern in the supply chain network model, (q, h, Q 1, Q 2, d) 2K 5, in the ollowing way: q i ai ; i ¼ 1;...; I ; ð34þ q im a im ; i ¼ 1;...; I ; m ¼ 1;...; M; ð35þ q imj a imj ; i ¼ 1;...; I ; m ¼ 1;...; M; j ¼ 1;...; J ; ð36þ h j ajj 0; jj 0 ¼ 11 0 ;...; JJ 0 ; ð37þ q t jk ¼ a t j 0 k ; j ¼ 1;...; J ; j0 ¼ 1 0 ;...; J 0 ; t ¼ 1;...; T ; k ¼ 1;...; K ; ð38þ d k ¼ XJ j¼1 t¼1 qjk t ; k ¼ 1;...; K : ð39þ Observe that although q i is not explicitly stated in the model in Section 2, it is inerred in that q i ¼ XM m¼1 q im ; i ¼ 1;...; I ; ð40þ and simply represents the total amount o product produced by manuacturer i. International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 43

A. Nagurney et al. Note that i (q, Q 1, h, Q 2, d) is easible, then the link low and demand pattern constructed according to (34) (39) is also easible, and the corresponding path low pattern that induces this link low (and demand) pattern is also easible. We now assign user (travel) costs on the links o the network G S as ollows: with each link a i we assign a user cost c ai deined by c ai 0; i ¼ 1;...; I ; ð41þ c aim with each link a imj we assign a user cost c aimj c aimj @ im @q im þ a i e im ; i ¼ 1;...; I ; m ¼ 1;...; M; ð42þ deined by @ imj @q imj þ a i e imj ; i ¼ 1;...; I ; m ¼ 1;...; M; j ¼ 1;...; J ; ð43þ with each link jj 0 we assign a user cost deined by c ajj 0 @c j @h j þ b j e j ; jj 0 ¼ 11 0 ;...; JJ 0 : ð44þ Finally, or each link aj t 0 k we assign a user cost deined by caj t 0 k @ct jk @qjk t þ ^c jk t þðb j þ g k Þejk t ; j0 ¼ j ¼ 1;...; J ; t ¼ 1;...; T ; k ¼ 1;...; K : ð45þ Then a user o path pimjj t 0 k ; or i ¼ 1;...; I ; m ¼ 1;...; M ; jj 0 ¼ 11 0 ;...; JJ 0 ; t ¼ 1;...; T ; k ¼ 1;...; K ; on network G S in Figure 2 experiences a path travel cost C p t given by imjj 0 k C p t ¼ c imjj 0 ai þ c aim þ c aimj þ c ajj 0 þ c k a t ¼ @ im þ a i e im þ @c imj j 0 k @q im @q imj þ a i e imj þ @c j þ b @h j e j þ @ct jk j @qjk t þ ^c jk t þðb j þ g k Þejk t : Also, we assign the (travel) demands associated with the O=D pairs as ollows: and the (travel) disutilities: ð46þ d wk d k ; k ¼ 1;...; K ; ð47þ k wk q 3k ; k ¼ 1;...; K : ð48þ Consequently, the equilibrium conditions (25) or the transportation network equilibrium model on the network G S state that or every O=D pair w k and every path connecting the O=D pair w k : C p t k wk ¼ @ im þ a i e im þ @c imj þ a imjj 0 ieimj k @q im @q imj 8 < þðb j þ g k Þejk t k wk : þ @c j þ b @h j e j þ @ct jk j @qjk t þ ^c jk t ¼ 0; i xp > 0, t imjj 0 k 0; i xp ¼ 0 t imjj 0 k ð49þ We now show that the variational inequality ormulation o the equilibrium conditions (49) in link orm as in (27) is equivalent to the variational inequality (17) governing the supply chain network equilibrium with manuacturing plants and environmental concerns. 44 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks For the transportation network equilibrium problem on G S, according to Theorem 2, we have that a link low and travel disutility pattern ð ; d Þ2K 7 is an equilibrium [according to (49)], i and only i it satisies the variational inequality: X I i¼1 c ai ð 1 Þð ai a i Þþ XI þ XI X J i¼1 m¼1 j¼1 þ XJ 0 X K j 0 ¼1 0 k¼1 t¼1 8ð ; dþ2k 7 : i¼1 m¼1 c aimj ð 3 Þð aimj c aim ð 2 Þð aim a im Þ a imj Þþ XJJ 0 jj 0 ¼11 0 c ajj 0 ð 4 Þð ajj 0 a jj 0 Þ c a t ð 5 Þð j 0 k a t j 0 k a Þ XK k t wk ðd Þðd wk d j 0 w k k Þ0; k¼1 ð50þ Ater the substitution o (34) (45) and (47) (48) into (50), we have the ollowing variational inequality: determine ðq ; h ; Q 1 ; Q 2 ; d Þ2K 5 satisying: X I i¼1 m¼1 þ XI @ im ðqi Þ im þ a i e im ½q im q ŠþXJ @q im j¼1 X J @c imj ðqimj Þ ½q þ a i e imj imj q i¼1 m¼1 j¼1 @q imj þ ^c t jk þðq 2 Þþðb j þ g k Þe t jk 8ðq; h; Q 1 ; Q 2 ; dþ 2K 5 : # ½q t jk @c j ðh Þ þ b @h j e j j X K imj ŠþXJ j¼1 k¼1 t¼1 ½h j h j Š " @c t jk ðq t jk Þ @q t jk qt jk Š XK q 3k ðd Þ½d k dk Š0; k¼1 Variational inequality (51) is precisely variational inequality (17) governing the supply chain network equilibrium. Hence, we have the ollowing result: Theorem 3: Equivalence o Sustainable Supply Chain and Transportation Network Equilibrium A solution ðq ; h ; Q 1 ; Q 2 ; d Þ2K 5 o the variational inequality (17) governing the supply chain network equilibrium coincides with the [via (34) (45) and (47) (48)] easible link low and travel demand pattern or the supernetwork G S constructed above and satisies variational inequality (50). Hence, it is a transportation network equilibrium according to Theorem 2. We now urther discuss the interpretation o the supply chain network equilibrium conditions. These conditions deine the supply chain network equilibrium in terms o paths and path lows, which, as shown above, coincide with Wardrop s (1952) irst principle o user-optimization in the context o transportation networks over the network given in Figure 2. Hence, we now have an entirely new interpretation o supply network equilibrium with environmental concerns, which states that only minimal cost paths will be used rom the super source node 0 to any destination node. Moreover, the cost on the utilized paths or a particular O=D pair is equal to the disutility (or the demand market price) that the users are willing to pay. ð51þ International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 45

A. Nagurney et al. In Section 5, we will show how Theorem 3 can be utilized to exploit algorithmically the theoretical results obtained above when we compute the equilibrium patterns o numerical supply chain network examples using an algorithm previously used or the computation o elastic demand transportation network equilibria. O course, existence and uniqueness results obtained or elastic demand transportation network equilibrium models as in Daermos (1982) as well as stability and sensitivity analysis results [see also Nagurney and Zhang (1996)] can now be transerred to sustainable supply chain networks using the ormalism=equivalence established above. 5. COMPUTATIONS In this section, we provide numerical examples to demonstrate how the theoretical results in this paper can be applied in practice. We utilize the Euler method or our numerical computations. The Euler method is induced by the general iterative scheme o Dupuis and Nagurney (1993) and has been applied by Nagurney and Zhang (1996) to solve variational inequality (26) in path lows [equivalently, variational inequality (27) in link lows]. Convergence results can be ound in the above reerences. 5.1. The Euler Method For the solution o (26), the Euler method takes the orm: at iteration s, compute the path lows or paths p 2 P (and the travel demands) according to: x sþ1 p ¼ max0; x s p þ a sðk w ðd T Þ C p ðx s ÞÞg: ð52þ The simplicity o (52) lies in the explicit ormula that allows or the computation o the path lows in closed orm at each iteration. The demands at each iteration simply satisy (23), and this expression can be substituted into the k w ðþ unctions. The Euler method was implemented in FORTRAN, and the computer system used was a Sun system at the University o Massachusetts at Amherst. The convergence criterion utilized was that the absolute value o the path lows between two successive iterations diered by no more than 10 4. The sequence a s g in the Euler method [c. (52)] was set to 1; 1=2; 1=2; 1=3; 1=3; 1=3;...g. The Euler method was initialized by setting the demands equal to 100 or each O=D pair with the path lows equally distributed. The Euler method was also used to compute solutions to electric power supply chain network examples, reormulated as transportation network equilibrium problems in Wu et al. (2006). In all the numerical examples, the supply chain network consisted o two manuacturers, with two manuacturing plants each, two retailers, one transportation= transaction mode, and two demand markets as depicted in Figure 3. The supernetwork representation that allows or the transormation (as proved in Section 4) to a transportation network equilibrium problem is given also in Figure 3. Hence, in the numerical examples (see also Fig. 2), we had that I ¼ 2, M ¼ 2, J ¼ 2, J 0 ¼ 2 0, K ¼ 2, and T ¼ 1. 46 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007

Sustainable Supply Chain and Transportation Networks Figure 3. Supply chain network and corresponding supernetwork G S or the numerical examples. The notation is presented or the examples in the orm o the supply chain network equilibrium model o Section 2. The equilibrium solutions or the examples, along with the translations o the computed equilibrium link lows and the travel demands (and disutilities) into the equilibrium supply chain lows and prices are given in Table 2. Example 1 The data or the irst numerical example is given below. In order to construct a benchmark, we assumed that all the weights associated with the environmental criteria were equal to zero, that is, we set a 1 ¼ a 2 ¼ 0, b 1 ¼ b 2 ¼ 0, and g 1 ¼ g 2 ¼ 0. The production cost unctions or the manuacturers were given by 11 ðq 1 Þ¼2:5q11 2 þ q 11q 21 þ 2q 11 ; 12 q ð2þ ¼ 2:5q12 2 þ q 11q 12 þ 2q 22 ; 21 q ð1þ ¼ 0:5q21 2 þ 0:5q 11q 21 þ 2q 21 ; 22 q ð2þ ¼ 0:5q22 2 þ q 12q 22 þ 2q 22 : The transportation=transaction cost unctions aced by the manuacturers and associated with transacting with the retailers were given by c imj ðq imj Þ¼0:5qimj 2 þ 3:5q imj; i ¼ 1; m ¼ 1; 2; j ¼ 1; 2; c imj ðq imj Þ¼0:5qimj 2 þ 2q imj; i ¼ 2; m ¼ 1; 2; j ¼ 1; 2: The operating costs o the retailers, in turn, were given by X 2 2; c 1 ðq 1 Þ¼0:5 q i1 i¼1 X 2 2: c2 ðq 1 Þ¼0:5 q i2 i¼1 The demand market price unctions at the demand markets were q 31 ðdþ ¼ d 1 þ 500; q 32 ¼ d 2 þ 500; International Journal o Sustainable Transportation Vol. 1, No. 1, 2007 47

A. Nagurney et al. Table 2. Equilibrium solutions o Examples 1, 2, 3, and 4 Equilibrium values Example 1 Example 2 Example 3 Example 4 a 1 1 0 1 a1 ¼ q 1 48.17 47.68 47.17 42.04 a2 ¼ q 2 169.62 167.89 166.20 109.34 a11 ¼ q 11 33.37 33.03 32.69 25.87 a12 ¼ q 12 14.80 14.65 14.48 16.37 a21 ¼ q 21 33.71 33.37 33.02 26.17 a22 ¼ q 22 135.91 134.53 133.17 83.17 a11 ¼ 0 h 1 108.90 107.79 103.82 0.00 a22 ¼ 0 h 2 108.90 107.79 109.54 151.58 a111 ¼ q 111 16.69 16.52 15.60 0.00 a112 ¼ q 112 16.69 16.52 17.09 25.87 a121 ¼ q 121 7.40 7.32 6.61 0.00 a122 ¼ q 122 7.40 7.32 7.87 16.37 a211 ¼ q 211 16.85 16.68 15.77 0.00 a212 ¼ q 212 16.85 16.68 17.25 26.17 a221 ¼ q 221 67.96 67.26 65.84 0.00 a222 ¼ q 222 67.96 67.26 67.33 83.17 ¼ q11 1 54.45 53.89 51.91 0.00 ¼ q a 1 1 0 12 1 54.45 53.89 51.91 0.00 2 ¼ q a 1 2 0 21 1 54.45 53.89 54.77 75.79 1 ¼ q a 1 2 0 22 1 54.45 53.89 54.77 75.79 2 dw1 ¼ d 1 108.90 107.79 106.68 75.79 dw2 ¼ d 2 108.90 107.79 106.68 75.79 k w1 ¼ q 31 391.11 392.23 393.30 424.21 k w2 ¼ q 32 391.11 392.23 393.30 424.21 and the unit transportation=transaction costs between the retailers and the consumers at the demand markets were given by ^c jk 1 ðq1 jk Þ¼q1 jk þ 5; j ¼ 1; 2; k ¼ 1; 2: All other transportation=transaction costs were assumed to be equal to zero. We assumed that the manuacturing plants emitted pollutants where e 11 ¼e 12 ¼e 21 ¼e 22 ¼5. We utilized the supernetwork representation o this example depicted in Figure 3 with the links enumerated as in Figure 3 in order to solve the problem via the Euler method. Note that there are 13 nodes and 20 links in the supernetwork in Figure 3. Using the procedure outlined in Section 4, we deined O=Dpairw 1 ¼ð0; z 1 Þ and O=D pair w 2 ¼ð0; z 2 Þ, and we associated the O=D pair travel disutilities with the demand market price unctions as in (48) and the user link travel cost unctions as given in (41) (45) (analogous constructions were done or the subsequent examples). The Euler method converged in 56 iterations and yielded the equilibrium solution given in Table 2 (c. also the supernetwork in Fig. 3). In Table 2, we also provide the translations o the computed equilibrium pattern(s) into the supply chain network low, demand, and price notation using (34) (40) and (47) (48). 48 International Journal o Sustainable Transportation Vol. 1, No. 1, 2007