Research Article Closed-Loop Supply Chain Network under Oligopolistic Competition with Multiproducts, Uncertain Demands, and Returns

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1 Mathematical Problems in Engineering, Article D , 15 pages Research Article Closed-Loop Supply Chain Networ under Oligopolistic Competition with Multiproducts, Uncertain Demands, and Returns Yan Zhou, 1 Chi Kin Chan, 2 Kar Hung Wong, 3 andy.c.e.lee 2 1 Department of Management Science and Engineering, Qingdao University, Qingdao, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong 3 School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa Correspondence should be addressed to Yan Zhou; yanyanz22@hotmail.com Received 29 January 214; Revised 2 April 214; Accepted 27 April 214; Published 27 May 214 Academic Editor: Changzhi Wu Copyright 214 Yan Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. We develop an equilibrium model of a closed-loop supply chain (CLSC networ with multiproducts, uncertain demands, and returns. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framewor under a stochastic environment. To satisfy the demands, we use two different channels: manufacturing new products and remanufacturing returned products through recycling used components. Since both the demands and product returns are uncertain, we consider two types of riss: overstocing and understocing of multiproducts in the forward supply chain. Then we set up the Cournot-Nash equilibrium conditions of the CLSC networ whereby we maximize every oligopolistic firm s expected profit by deciding the production quantities of each new product as well as the path flows of each product on the forward supply chain. Furthermore, we formulate the Cournot-Nash equilibrium conditions of the CLSC networ as a variational inequality and prove the existence and the monotonicity of the variational inequality. Finally, numerical examples are presented to illustrate the efficiency of our model. 1. ntroduction The competitive supply chain networ models have been widely studied in the past decade. For instance, Nagurney et al. [1 first considered an equilibrium model of a competitive forward supply chain networ which consisted of a lot of decision maers. Dong et al. [2 constructed an equilibrium model of a forward supply chain networ under the assumption that the demands were uncertain. Nagurney and Toyasai [3 constructed an equilibrium model of a reverse supply chain networ for the optimal management of the electronic wastes. Hammond and Beullens [4 expanded the wor of [1, 3 by presenting some insights of how closedloop supply chain (CLSC networ equilibrium was achieved under WEEE (the waste electrical and the waste electrical and electronic equipment directive legislation. Yang et al. [5 constructed an equilibrium model of a CLSC networ consisting of raw materials, suppliers, and recovery centers besides manufacturers and retailers. n all the above papers [1 5, the equilibrium conditions were obtained by the theory of variational inequality; the variational inequalities in these papers were solved by the modified projection method proposed by Korpelevich [6. The CLSC integrating the forward and reverse supply chain is important in real world applications due to government legislations (such as the paper recycling directive and WEEE within the European Union [7. Moreover, the process used in the CLSC to recycle used products (such as papers, glass, building wastes, electric, and electronic equipment for minimizing resource wastage also leads to people s understanding of the important concept of the green supply chain management (GSCM. A comprehensive review of the CLSC management and the GSCM can be found in Gupta and Palsule-Desai [8, Sheu and Talley [9, and Seuring [1. Competition and monopoly are two opposing maret forms in economics. n a competition, there are numerous firms supplying their goods to the marets and the maret price is determined exogenously. n a monopoly, only one

2 2 Mathematical Problems in Engineering firm supplies its goods to the marets. Oligopoly is a maret form between perfect competition and monopoly, in which the maret is dominated by n firms (oligopolists, where n is a small number. The n firms compete in the oligopolistic maret by producing the same product or perfect substitutes for the maximization of their profits, considering the best reply of the other firms. (When n=2, the oligopoly is called a duopoly which was first proposed by Cournot [11. Recently, Barbagallo and Cojocaru [12 considered the dynamic equilibrium of an oligopolistic maret. Based on the timedependent variational inequality, Barbagallo and Mauro [13 investigated an equilibrium model of an oligopolistic maret dealing with production and demand excesses. From the wor of Brown et al. [14, it is clear that more firms nowadays are aware of the importance of integrating the supply chain as a whole, consisting of all the mareting activities of all the competitors. Thus, several forward supply chain networs involving oligopolistic competition among firms have been developed. For instance, Nagurney [15 considered a design problem in which oligopolistic firms competed in a Cournot-Nash [16, 17framewor.Masoumietal. [18, Nagurney and Yu [19, and Yu and Nagurney [2developed equilibrium models involving oligopolistic competition among pharmaceutical firms, fashion firms, and fresh food firms, respectively. The equilibrium conditions of [15, 18 2 were also obtained by the theory of variational inequality; the variational inequalities in these papers were solved by the Euler method [21. However, there are very few papers in the literature dealing with CLSC networ involving oligopolistic competition among firms and none of these papers consider oligopolistic competition in a stochastic environment. The major weaness of the literature concerning the CLSC networ model is that only deterministic demands are considered. Research wor on stochastic CLSC networ model began only in the last few years. For instance, Qiang et al. [22 investigated an equilibrium model of a CLSC networ involving perfect competition and uncertainties in demands, but not involving uncertainties in returns. Shi et al. [23, 24 constructed mathematical models of closedloop manufacturing systems, in which both the demands and thereturnswereuncertainandpricesensitive.however,the closed-loop systems in [23, 24were notclscnetworsand did not involve oligopolistic competition among firms. nthispaper,weprovideaninnovativeframeworto study the effects of oligopolistic competition on a CLSC networ and construct an equilibrium model of a multiproduct CLSC networ involving oligopolistic competition among firms, by allowing uncertain demands and returns for all products. To satisfy these demands, we use two different channels: manufacturing new products and remanufacturing returned products through recycling used components. Since the demands and returns of all products are uncertain, we consider two types of riss: overstocing and understocing of multiproducts in the forward supply chain. Then we can set up the Cournot-Nash equilibrium conditions of the CLSC networ whereby we maximize every oligopolistic firm s expected profit by deciding the production quantities of each new product as well as the path flows of each product on M 1 1 D 1 1 Firm 1 1 M 1 n 1 M D 1 n 1 D C 1 1 M 1 C 1 n C D1 R 1 R nr R 1 Demand maret Forward supply chain networ Reverse supply chain networ Firm M n M D n D C 1 Demand maret Figure 1: The CLSC networ topology. theforwardsupplychain.sincetheclscnetwormodel constructed in the above manner taes into consideration all those important issues such as competition among oligopolisticfirmswithmultiproductaswellasuncertainty in both demands and returns, the CLSC model developed in this paper has more practical use than most of them in the literature, such as those developed in [4, 5, 22. The rest of this paper is organized as follows. Section 2 develops the oligopolistic CLSC networ model with multiproducts and uncertain demands and returns and constructs the Cournot-Nash equilibrium conditions. Section 3 gives the variational inequality formulations of our model. We then prove the existence and the monotonicity results for our model in Section 4. Section 5 presents an algorithm for finding the Cournot-Nash equilibrium. n Section 6, numerical examples are presented to test the effectiveness of the proposed algorithm. Sensitivity analysis is also conducted to illustrate how the change in a parameter can affect the firms strategies and their respective expected profits. Section7 presents our summary and conclusion. 2. The Multiproduct CLSC Networ Model under Oligopolistic Competition among Firms We consider a CLSC networ consisting of oligopolistic firms who compete noncooperatively. Each firm i (i = 1,..., manufactures J products and has two different channels to meet the demands: manufacturing new products and remanufacturing returned products through recycling used components. Both the demands and their returns for all products are uncertain. t is assumed that all the demands and all the returns for each product are random variables with nown probability distributions. The problem is to maximize all the oligopolistic firms expected profits by deciding the production quantities of each new product as well as the path flows of each product on the forward supply chain. The topology of this networ is shown in Figure 1. From Figure 1, the CLSC networ consists of the forward supply R nr C n C

3 Mathematical Problems in Engineering 3 chain networ and the reverse supply chain networ. Each lin in Figure 1 represents an activity of the CLSC networ. Consider any firm i (i = 1,..., in the CLSC networ. The economic activities in the forward supply chain networ can be described as follows. (1 Firm i distributes raw materials or recycled products to each of its manufacturers (denoted by M i 1,...,M i nm. i (2 The manufacturers then produce new products or remanufactured products and distribute them to each of the distribution centers of firm i (denoted by D i 1,...,D i nd. i (3 The distribution centers then distribute these products to each of the demand marets (denoted by R 1,...,R nr. The economic activities of firm i in the reverse supply chainnetworcanbedescribedasfollows. (1 The demand marets send a fraction of the used products to each of the recovery centers of firm i (denoted by C i 1,...,Ci n (The remainder of the used C. i products will be sent to the landfill for disposition. (2 The recovery centers then recycle the used products andsendtherecycledproductstofirmi for remanufacturing. Thus, our CLSC networ is one which incorporates the forward supply chain processes (such as production, remanufacturing, delivery, and distribution with the reverse supply chain processes (such as disposal, recycling, and returning to form a loop in Figure 1. We first give a few basic assumptions, which are common assumptions in the CLSC literature (see [4, 23 for details. Assumption A. (A1 The demand for product j (j = 1,...,J of firm i (i = 1,..., at demand maret R ( = 1,...,n R, denoted by d, is a random variable. Any two demands are independent of each other. (A2 The return of product j (j = 1,...,J from demand maret R ( = 1,...,n R tofirmi (i = 1,...,, denoted by r ji, is a random variable. Any two returns are independent of each other. (A3 The qualities of new products and remanufactured products are exactly the same. (A4 A fraction of the used products will be collected by the firms for remanufacturing; the uncollected used products will be sent from the maret centers to the landfill for disposition. The firm will incur a fee of ρ per unit for the disposal of these used products at the landfill site. (A5 For the collected products mentioned in (A4, the peritemforthepurchaseofthe returned product j (j = 1,...,J. t is also assumed that all returns are remanufacturable. (A6 The quantity of new product j (j = 1,...,J manufactured by firm i (i=1,..., cannot exceed x,where x is a positive integer. firm will incur a cost of ρ Re j n order to formulate the problem of finding the Cournot- Nash equilibrium for our CLSC networ, we need to first describe the decision variables. For each i (i = 1,...,andj (j = 1,...,J, let x New denote the quantity of new product j manufactured by firm i. n the forward supply chain, any set of correlated lins connectingthefirmstothedemandmaretsviathemanufacturers and the distribution centers form a path p f.foreachi (i=1,...,and (=1,...,n R, let P i denote the set of all the forward paths from firm i to demand maret R.Thenfor each p f P i,letx jpf denote the nonnegative product flow of product j (j=1,...,jontheforwardpathp f. Hencetherearetwotypesofdecisionvariablesforour problem, namely, the quantity of new product x New and the product flow on the forward path x jp f. Let X (x jp f,x New p f P i } R n Pi +1 + (1 denote all the decision variables associated with firm i and product j,where n R P i P i, (2 =1 and n P i denotes the cardinality of P i. Let X i (X i1,...,x ij T K i (3 be the strategy vector representing the overall decision variables associated with firm i,wherek i R (n Pi +1J +.Then X (X 11,...,X 1J,...,X i1,...,x ij,...,x 1,...,X J T K (4 is the overall decision variables for the entire CLSC networ, which is the strategies vector of all firms, where K R i=1 (n Pi +1J +. Now, we consider all the different types of constraints for our CLSC networ. Byvirtueof(A6,wehaveforeachi (i = 1,...,andj (j=1,...,j x New x. (5 Moreover, in the forward supply chain, the following inequality involving product j (j = 1,...,J manufactured by firm i (i=1,...,musthold: n R x jp f =1 p f P i x New +E(x Re, (6 where n R x Re = r ji (7 =1 and r ji is as defined in (A2.

4 4 Mathematical Problems in Engineering Furthermore, in the reverse supply chain, the following flow inequality must hold: E( r ji x jp f. p f P i The above inequality ensures that the expected value of the returned product j (j = 1,...,J from demand maret R ( = 1,...,n R tofirmi (i = 1,..., cannot exceed the total amount of forward flows of the above product on any path connecting firm i to demand maret R. Now, we formulate the different cost functions for our CLSC networ. We first formulate the expected penalty cost (due to excessive or insufficient supply and the operational cost. Consider the forward supply chain involving firm i (i = 1,...,, product j (j = 1,...,J, and demand maret R ( = 1,...,n R. Let V denote the quantity of product j supplied by firm i to demand maret R.Thenthefollowing flow equation must hold: x jp f = V. p f P i Let d denote the demand associated with firm i,product j, and demand maret R.LetF be the probability density function of d.then Pr ( d x x= F ( d d d, (1 where Pr denotes the probability. Now, the quantity of product j supplied by firm i to demand maret R cannot exceed the minimum of V and d. n other words, the actual sale of these products is equal to minv, d }. Let Δ + = max, V d }, Δ = max, d V }, (8 (9 (11 denote, respectively, the quantity of the overstocing and the understocing of goods associated with firm i,productj and demand maret R. The expected values of Δ +,andδ are given by E(Δ + V = (V d F ( d d d, E(Δ = V ( d V F ( d d d. (12 Assume that the unit penalty incurred on firm i due to excessive supply of product j to demand maret R is θ + and the unit penalty incurred on firm i due to insufficient supply of product j to demand maret R is θ,whereθ+ and. Then, the total expected penalty cost incurred on θ firm i associated with product j and demand maret R is given by E (θ + Δ+ +θ Δ =θ+ E (Δ+ +θ E (Δ. (13 Let L f be the set of all correlated lins in the forward supply chain. Let u f ja denote the product flow of product j on any lin a in the forward supply chain. Then, in the forward supply chain involving product j, therelationshipbetween theproductflowinalinandtheproductflowinapathis as follows: u f n R ja = x jp fδ ap f, a L f, (14 i=1 =1p f P i where the binary parameter δ ap f is used to indicate whether lin a is included in forward path p f (δ ap f = 1ornot included in forward path p f (δ ap f =. Let u f j be the vector consisting of all the forward lin flows for each product j. Then, in the forward supply chain, the total operational cost of product j on a lin a is a continuous function of all the forward lin flows of product j given by c ja = c ja (u f j, a Lf. (15 Apart from the penalty costs given by (13(duetoexcessive or insufficient supply and the operational costs given by (15, firm i (i = 1,..., also incurs the following additional costs: (i the cost related to the production of new product j (j=1,...,j: f (x New, (16 where f is a continuous function, (ii the cost related to the production of the returned product j (j=1,...,j: where f Re f Re (xre is a continuous function,, (17 (iii the cost related to the purchase of returned product j (j=1,...,j from demand maret R (=1,...,n R to firm i: ρ Re j r ji, (18 where ρ Re j is the purchase cost per item of these returned products, (iv the shipping cost related to the transportation of returned product j (j = 1,...,J from demand maret R (=1,...,n R tofirmi: c Re ji r ji, (19 where c Re ji is the transportation cost per item of these returned products,

5 Mathematical Problems in Engineering 5 (vthecostrelatedtothedisposalofuncollectedproduct associated with firm i, productj (j = 1,...,J, and demand maret R (=1,...,n R atthelandfillsite: ρ( x jp f r ji. (2 p f P i By virtue of (13, (15, and (16 (2, the total cost incurred on firm i is given by j=1 =1 + E(θ + Δ+ +θ Δ J j=1 + f (x New J + j=1 a L f i ρ Re J j E( r ji+ j=1 =1 j=1 c Re j=1 =1 + ji E( r ji+ρ c ja (u f j E(f Re j=1 =1 (xre E( x jp f r ji, p f P i (21 where L f i isthesetofallcorrelatedlinsintheforwardsupply chain of firm i (i=1,...,. Now, we formulate the total revenue received by firm i (i = 1,...,. n order to capture competition in the demands in theentireclscnetwor,weassumedthat,foreachproduct j, the demand price function ρ (j=1,...,j, =1,...,n R is a continuous function of all the demands of product j in the entire CLSC networ; that is, where ρ =ρ ( d j, (22 d j =( d 1j1,..., d 1jnR,..., d j1,..., d jnr T R n R + (23 denotes the vector consisting of all the demands of product j in the entire CLSC networ. Masoumi et al. [18 andnagurney and Yu [19 used such demand price function in the study of forward supply chain networ involving oligopolistic competition among firms. Then the expected revenue received by firm i (i=1,...,isgivenby j=1 =1 E(ρ ( d j min V, d }. (24 By virtue of the revenue, the cost of the forward supply chain, and the cost of the reverse supply chain, the expected profit function of firm i (i = 1,...,, denoted by U i,canbe expressed as follows: j=1 =1 U i = j=1 =1 J j=1 E(ρ ( d j min V, d } E(θ + Δ+ +θ Δ f (x New J j=1 a L f i ρ Re J j E( r ji j=1 =1 j=1 c Re j=1 =1 ji E( r ji ρ c ja (u f j E(f Re j=1 =1 (xre E( x jp f r ji, p f P i (25 where (7, (9, and (14 hold for all decision variables defined by (4. By virtue of (12, (15 (2, and (22, it is clear that U i is a function of the strategies of all firms in the entire CLSC networ. That is, U i =U i (X. (26 Now, in order to define the Cournot-Nash equilibrium of the CLSC networ, we need to consider the following constraints involving firm i (i = 1,...,: n R x jp f =1 p f P i x jp f ( p f x New x, (27 x New E( r ji x jp f, i=1 p f P i P i, +E(x Re, (28 (29 x New. (3 Constraints (27, (28, and (29 are directly obtained from (5, (6, and (8, respectively; constraint (3 is due to the nonnegativity requirement imposed on all the decision variables. There are oligopolistic firms in the maret to form the CLSC networ with perfect information shared by all the firms throughout the networ. So the game is based on the oligopolistic Cournot pricing in a Cournot-Nash framewor. Each firm in the CLSC networ competes noncooperatively to maximize its own expected profit and selects its strategy vector till an equilibrium is established. Now, we can define the Cournot-Nash equilibrium of theclscnetworaccordingtodefinitions1, 2, and3 given below.

6 6 Mathematical Problems in Engineering Definition 1 (a feasible strategy vector. For each i (i = 1,...,, a strategy vector X i K i is said to be a feasible strategy vector, if X i satisfies constraints (27 (3. Definition 2 (the set of all feasible strategy patterns. Let S be the set of all feasible strategy patterns defined by S (X 1,X 2,...,X X i K i and constraints (27 (3 holds}. (31 Definition 3 (the Cournot-Nash equilibrium of the CLSC networ. A feasible strategy pattern X S constitutes a Cournot-Nash equilibrium of the CLSC networ, if the following inequality holds for all i andforallfeasiblestrategy vectors X i : U i (X i, X i U i (X i, X i, (32 where X i (X 1,...,X i 1,X i+1,...,x. Thus, an equilibrium is established if no firm in the CLSC networ can unilaterally increase its expected profit (without violating feasibility by changing any of its strategy, given that the strategies of the other firms do not change. By virtue of the fact that any two demands are independent of each other (from (A1, the first term of U i ( can be expressed as where G( d = E(ρ ( d j min V, d } V = d G( d F ( d d d + V G( d F ( d d d, V (33 ρ ( d 111,..., d ( 1, d, d (+1,..., d jnr F 111 ( d 111 F ( 1 ( d ( 1 F (+1 ( d (+1 F jnr ( d jnr d d 111 d d ( 1 d d (+1 3. The Variational nequality Formulation To guarantee the existence of the Cournot-Nash equilibrium of the CLSC networ, the following additional assumption is held. d d jnr. Thus, we obtain from (33that (34 Assumption B. (B1 The total operational cost c ja (a L f, j = 1,...,J defined by (15 isaconvexandcontinuously differentiable function of u f j, the vector consisting of all the forward lin flows of product j. (B2 The production cost f (i = 1,...,, j = 1,...,J is a convex and continuously differentiable function of x New, the new product j manufactured by firm i. Lemma 4. Suppose that Assumption A is satisfied, then the set of all feasible strategy patterns defined by Definition 2 is both a compact and convex subset of K. Proof. The fact that S is bounded follows easily from (3, (27, and (28. The fact that S isclosedandconvexfollows easily from (27, (28, (29, and (3. Hence, S is both a compact and convex subset of K. E(ρ ( d j min V, d } V But, from (9, it is clear that = G( d F ( d d d. V V x jp f (35 =1, (36 for each j (j = 1,...,Jandp f P i ( = 1,...,n R. Thus, from (35and(36, the first partial derivative of the first term of U i ( becomes Lemma 5. Suppose that Assumption A and Assumption B are satisfied, then the expected profit function U i ( (i = 1,..., defined by (25 is continuously differentiable with respect to the decision variable X defined by (4. Proof. Firstly, from (9, (12, (13, (B1, and (14, it is clear that thefirst,second,andfourthtermsofu i ( defined by (25are continuous functions of x jp f,foreachj (j = 1,...,Jand p f P i (=1,...,n R. E(ρ ( d j min V, d } x jp f = G( d F ( d d d, p f P i x jp f for each j (j=1,...,jandp f P i (=1,...,n R. (37

7 Mathematical Problems in Engineering 7 From (12, we have E(Δ + = V (V V V d F ( d d d V = F ( d d d, E(Δ = ( V V d V F ( d d d V = F ( d d d. V (38 Thus, from (13, (36, and (38, the first partial derivative of the second term of U i ( becomes E(θ + Δ+ +θ Δ x jp f =θ + p f P i x jp f F ( d d d θ p f P i x jp f F ( d d d, (39 for each j (j=1,...,jandp f P i (=1,...,n R. From (37 and(39, it is clear that the first and second terms of U i ( are continuously differentiable with respect to x jp f,foreachj (j=1,...,jandp f P i (=1,...,n R. From (B1 and (14, the fourth term of U i ( is continuously differentiable with respect to x jp f,foreachj (j = 1,...,Jandp f P i (=1,...,n R. The third, fifth, sixth, and seventh terms of U i ( are independent of x jp f and the last term of U i ( is a linear function of x jp f.thus,allthetermsin(25 are continuously differentiable with respect to x jp f,foreachj (j = 1,...,Jand p f P i (=1,...,n R. From (B2, it is clear that the third term of U i ( is continuously differentiable with respect to x New andalltheother terms are independent of x New.Thus,allthetermsin(25are also continuously differentiable with respect to x New,foreach j (j=1,...,j. The proof is complete. Lemma 6. The expected profit function U i ( (i = 1,..., defined by (25 is concave with respect to the decision variable X defined by (4. Proof. n order to prove that the first and second terms of U i ( are concave with respect to the decision variable X defined by (4, we calculate the second partial derivative of these terms with respect to the decision variable x jp f (j = 1,...,J, p f P i, =1,...,n R as follows. We obtain from (35that 2 E(ρ ( d j min V, d } = G(V F (V. V 2 (4 Thus, from (36and(4, for each j (j=1,...,jandp f P i ( = 1,...,n R, the second partial derivative of the first term of U i ( becomes 2 E(ρ ( d j min V, d } = G(V F (V. x 2 jp f From (38, we have 2 E(Δ + = F (V, V 2 2 E(Δ = F (V. V 2 (41 (42 Thus, from (13, (36, and (42, for each j (j = 1,...,Jand p f P i ( = 1,...,n R,thesecondpartialderivativeofthe second term of U i ( becomes 2 E(θ + Δ+ +θ Δ =(θ + +θ F (V. x 2 jp f (43 By virtue of (14, (15, and (B1, it is clear that ( c ja (u f j is concave with respect to x jp f,foreachj (j = 1,...,Jand p f P i (=1,...,n R. Now, by virtue of (41, (43, the fact that the third, fifth, sixth, and seventh terms of U i ( are independent of x jp f,the fact that the fourth term of U i ( is concave with respect to x jp f,andthefactthatthelasttermofu i ( is a linear function of x jp f,weconcludethatu i ( is concave with respect to x jp f, for each j (j=1,...,jandp f P i (=1,...,n R. From (B2, it is clear that ( f (x New is concave with respect to x New.Since( f (x New istheonlytermin(25 involving x New,weconcludethatU i ( is concave with respect to x New,foreachj (j=1,...,j. Hence, we conclude that the expected profit function U i ( (i = 1,..., defined by (25 is concave with respect to the decision variable X defined by (4. The proof is complete. We now give the variational inequality formulations of the CLSC networ Cournot-Nash equilibrium in the following theorem. Theorem 7. X Sis a CLSC networ Cournot-Nash equilibrium according to Definition 3 if and only if it satisfies the variational inequality: i=1 Xi U i (X,X i X i, X S, (44

8 8 Mathematical Problems in Engineering where, denotes the inner product in the corresponding Euclidean space and Xi U i (X denotes the gradient of U i (X with respect to X i. Proof. nviewofthefactthatthesetofallfeasiblepatternss is both a compact and convex subset of K and U i ( is concave and continuously differentiable with respect to the decision variable X defined by (4 (fromlemmas4 6, the proof of this theorem follows easily from [25. Lemma 8. Variational inequality (44 for our model is equivalent to Find X S, such that F (X,X X, X S, (45 where X R ( i=1 n p i J+J, F(X = (F 1 (X, F 2 (X, F 1 (X R ( i=1 n p i J, F 2 (X R J such that the component of F 1 (X corresponding to the variable x jp f ( p f P i, i, j, is c ja (u f j +θ + p f P i x jp f F ( d d d a L f i x jp f θ p f P i x jp f F ( d d d + ρ G( d F ( d d d, p f P i x jp f (46 where G( d is as defined in (34, and the component of F 2 (X corresponding to the variable x New ( i, jis f (x New x New. (47 Proof. Since the first, second, fourth, and last terms of (25 are the only terms involving x jp f,weobtainfrom(25, (37, and (39that U i = G( d x F ( d d d jp f p f P i x jp f θ + p f P i x jp f F ( d d d +θ p f P i x jp f c ja (u f j ρ a L f i x jp f F ( d d d (48 for each j (j = 1,...,Jandp f P i (i = 1,...,, = 1,...,n R. Sincethethirdtermof(25 is the only term involving,wehave x New for each x New U i x New = f (xnew x New (i=1,...,, j=1,...,j. (49 By virtue of (44, (48, and (49, we obtain variational inequality (45. Thus, the proof is complete. 4. Existence and Monotonicity Results n this section, we establish the existence and monotonicity results of variational inequality (45, the solution of which is a CLSC networ Cournot-Nash equilibrium of our model. From Lemma 4, the feasible set of variational inequality (45 is both a compact and convex subset of K. From Lemma 5, the expected profit function U i ( (i = 1,..., defined by (25 is continuously differentiable with respect to thedecisionvariablex defined by (4. Thus, we have the following theorem. Theorem 9 (existence. Variational inequality (45 admits at least one solution. Proof. Since S is compact and convex and F(X is continuous, Theorem 9 follows from [26, Theorem 3.1, page 12. Theorem 1 (monotonicity. Suppose that Assumptions A and B are satisfied, then the vector function F(X of the variational inequality (45 is monotone; that is, F (X F(X,X X, X,X S. (5 Proof. Let X Sand X S. Variational inequality (5has the following deduction: F (X F(X,X X i=1 j=1 =1 p f P i = + [ i=1 j=1 =1 p f P i i=1 j=1 =1 p f P i + a L [ f i c ja (u f j x jp f [x jpf x jp f a L f i c ja (u f j x jp f [θ + p f P i x jp f F ( d d d θ + p f P i x jp f [x jpf x jp f F ( d d d [ θ ( F ( d d d [ p f P i x jp f θ ( [x jpf x jp f p f P i x jp f F ( d d d

9 Mathematical Problems in Engineering 9 i=1 j=1 =1 p f P i + J + [ i=1 j=1 [ [ G( d F ( d d d [ p f P i x jp f + G( d p f P i x jp f [x jpf x jp f f (x New [x New x New x New, f (x New F ( d d d x New (51 where G( d is as defined in (34. By virtue of the fact that c ja (u f j is a convex function (from (B1, we obtain from (14that i=1 j=1 =1 p f P i Since [ a L [ f i c ja (u f j x jp f [x jpf x jp f. a L f i c ja (u f j x jp f p f P i x jp f F ( d d d = Pr ( d x jp f, p f P i (52 F ( d d d = Pr ( d x jp f 1, p f P i x jp f p f P i (53 and the probability function Pr( d p f P x i jp f is an increasing function of x jp f,foreachj (j = 1,...,J, p f P i (i=1,...,and =1,...,n R, we have i=1 j=1 =1 p f P i [θ + p f P i x jp f F ( d d d θ + p f P i x jp f [x jpf x jp f, F ( d d d i=1j=1=1p f P i [ θ ( F ( d d d [ p f P i x jp f θ ( F ( d d d p f P i x jp f [x jpf x jp f. (54 From (37 and(41, we now that G( d p f P i x jp f F ( d d d is a decreasing function of x jp f,foreachj (j = 1,...,Jandp f P i (i = 1,...,, = 1,...,n R. Hence, we have i=1j=1=1p f P i [ G( d F ( d d d [ p f P i x jp f + G( d F ( d d d p f P i x jp f [x jpf x jp f. (55 By virtue of the fact that f (x New is a convex function (from (B2, we have J [ i=1 j=1 [ f (x New [x New x New f (x New x New. x New (56 Thus, we conclude that (51 is nonnegative. The proof is complete. Lemma 11. Suppose that Assumptions A and B are satisfied. Furthermore, suppose that all the total operational costs c ja (a L f, j = 1,...,J and all the production costs f (i = 1,...,, j = 1,...,J are strictly convex functions, and then the vector function F(X of the variational inequality (45 is strictly monotone; that is, X, X Swith X =X F (X F(X,X X >. (57 Proof. Let the strategy vector of all firms X be written in the form: X=(X 1,X 2, (58 where X X 1 = (x jp f j=1,...,j,p f i=1 P i } R i=1 (n P i J +, X 2 = (x New i=1,...,,j=1,...,j} R J +. (59 Let X, X be the given strategy vectors such that =X.ThenX 1 =X 1 or X 2 =X 2.

10 1 Mathematical Problems in Engineering Case 1 (X 1 1.Sinceallthetotaloperationalcosts c ja (a L f, j=1,...,j are strictly convex functions, it follows from (14thatthefirsttermof(51ispositive. =X =X Case 2 (X 2 2. Since all the production costs f (i = 1,...,, j = 1,...,J are strictly convex functions, the last term of (51ispositive. Thus, by using exactly the same argument as that given for the proof of Theorem 1,weobtain The proof is complete. F (X F(X,X X >. (6 Theorem 12 (uniqueness. Under the conditions of Lemma 11, there is a unique solution of variational inequality (45. Proof. Theorem 12 follows from [26, Theorem 1.4,page Computing the Cournot-Nash Equilibrium n this section, we use the logarithmic quadratic proximal (LQP prediction-correction method [27 forsolvingthe variational inequality (45 (the LQP prediction-correction method is an iterative method. For the sae of convenience of the readers, we first give a brief description of the LQP prediction-correction method. Consider the following variational inequality problem: Find x S such that f (x,x x, x S, (61 with linear constraints: S=x R n Ax b,x }, (62 where f is a given continuously differentiable function in R n and A is an m nmatrix and b R m. Then the above variational inequality problem can be transformed into the following variational inequality problem: Find u Ω such that F (u,u u, u Ω, (63 where u=( x λ, F(u =(f (x +AT λ Ax + b, Ω=Rn + Rm +, (64 and λ isthelagrangemultipliervectoroftheconstraints Ax b. Let R n ++ and Rm ++ denote the interior of Rn + and Rm +, respectively. For any given x R n ++, λ R m ++, β >, the new iterate (x +1,λ +1 can be obtained by solving the following system of equations: β (f (x +A T λ + (x x +μ(x X 2 x 1 =, (65 λ=p R m + [λ + β (Ax b, (66 where < μ < 1 is a given fixed parameter, > is a positive number, X = diag(x 1,x 2,...,x n, x 1 = (1/x 1, 1/x 2,...,1/x n,andp R + m [λ denote the projection of λ on R + m ; that is, P R + m [λ = arg min Y λ Y R + m }. (67 Here the term (x x +μ(x X 2 x 1 is used to ensure that the new iterate x +1 is not too far from x.thusthepurpose of (65istoeepx +1 not too far from x and at the same time eep f(x +1 +A T λ +1 close to zero, whereas the purpose of (66 istoeepbothλ +1 and Ax +1 b.however, system (65and(66 cannot be easily solved simultaneously. To overcome this difficulty, in the prediction step of the LQP prediction-correction method, we first replace x by the current iterate x in (66 so that an approximation of λ, denoted by λ, can be easily obtained by solving λ =P R + m [λ + β (Ax b. (68 Then we substitute λ = λ into (65 toobtainanapproximation of x, denoted by x. n other words, the prediction ( x, λ can be obtained by solving the following system: β (f (x +A T λ + (x x μ (x X 2 Prediction x 1, λ=p R m + [λ + β (Ax b. (69 After we have obtained the prediction ( x, λ, the new iterate (x +1,λ +1,calledthecorrector,canbeobtainedbysolving the following system: αβ (f ( x +A T λ +(x x Correction μ (x X 2 x 1 =, λ=p R m + [λ + α β (A x b, (7 where α is the step size. From the above discussion, it is obvious that there are two main advantages for choosing the LQP predictioncorrection method for solving the variational inequality (45. From the computational point of view, each iteration of the LQP prediction-correction method consists of a prediction and a correction, both of which require very tiny computational load; thus the method is effectively applicable in practice. From the mathematical point of view, there are only two conditions to guarantee the convergence of the LQP prediction-correction method; namely, the function F(X in the variational inequality (45 is continuous and monotone and the solution set of the variational inequality (45 is

11 Mathematical Problems in Engineering 11 nonempty. Since the existence and the monotonicity of the variational inequality (45 forourclscmodelhavebeen proved in Theorems 9 and 1, respectively,wecaneasily use the LQP prediction-correction method for finding the Cournot-Nash equilibrium defined by Definition 3,whichis equivalent to the solution of the variational inequality (45. For the sae of using the above LQP prediction-correction method for solving the variational inequality (45, we need to reformulate the set of all feasible strategy patterns defined by Definition2 as follows: S=X R ( i=1 n p i J+J AX b,x }, (71 where A R (2+n RJ ( i=1 n p i J+J, b R (2+nRJ are obtained from constraints (27 (29. Let X P X [F(X +A T λ} ( i=1 n pi J+J R+ e (X, λ (, (72 λ P (2+n R R J λ+[ax b} + where the vector function F(X is as defined in (45, λ R (2+n RJ + is the Lagrange multiplier vector of the linear inequality constraints AX b,andp K denotes the projection on K.Choose e(x, λ < ε as the stopping criterion. Now we are in a position to describe Algorithm 13, which is a modified version of the LQP prediction-correction method described above. Algorithm 13. Step. Let β =1, =1(>, η =.9(< 1, μ =.1, σ = 2.(> 1 and ε=1 8.nitializeX R ( i=1 n p i J+J, λ R (2+n RJ +. Let =. Step 1. f e(x,λ ε,thenstop;else,gotostep2. Step 2 (prediction step. Produce the predictor X, λ. Step 2.1. Calculate λ and X. λ =P R (2+n R J [λ + β (AX b, s (1 μx β (F (X +A T λ, ( s l + ( s l 2 + 4μ(X X l = l 2, l=1,..., n 2 p ij+j. i=1 (73 Step 2.2. Update β. Let r = ξ 1 2 +(1+μ ξ 2 2 (1 μ 2 π (1 μ π 2 2, (74 where ξ 1 =β (F(X F( X R i=1 n p i J+J, ξ 2 =β A(X X R (2+nRJ, π 1 =X X R i=1 n p i J+J, π 2 =λ λ (75 R (2+nRJ. f r >η,thenβ =β.8/r ;gobactostep2.1;else,go to Step 3. Step 3. Adjust β and as follows: β β +1.7, = r if r.5, β, otherwise,.5, if ξ 1 1+μ > 4 ξ 2, = 2, if 4 ξ 1 1+μ < ξ 2,, otherwise. Step 4. Calculate the step-size α in the correction step. where (76 α = γα β (1 μ, (77 (1 + μ α = ((π 1 +ξ 1 T π 1 +(π 2 +ξ 2 T π 2 ([(1 + μ π 1 +ξ 1 T [π 1 +(1+μ 1 ξ 1 +(π 2 +ξ 2 T (π ξ 2 1. (78 Step 5 (correction step. Calculate the new iterate X +1, λ +1. X +1 l = s (1 μx α (F ( X +A T λ, ( s l + ( s l 2 + 4μ(X l 2 2, l=1,..., λ +1 =P R (2+n R J + i=1 [λ + α (A X b. n p ij+j, (79 Let =+1;gotoStep1. The convergence result for Algorithm 13 is stated in the following theorem. Theorem 14 (convergence. SupposethatAssumptionsAand B are satisfied, and then the sequence X } generated by Algorithm 13 converges to some X,whereX is the solution of the variational inequality (45. Proof. The proof follows easily from Theorems 9 and 1 and Theorem 4.2 of [ Numerical Examples n this section, numerical examples are solved to illustrate the efficiency of Algorithm 13 for finding the Cournot-Nash equilibrium of the CLSC networ. We consider a CLSC networ involving oligopolistic competition among four firms. Each firm manufactures two products and has two manufacturers and two distribution

12 12 Mathematical Problems in Engineering centers to supply goods to three demand marets. Each firm also has two recovery centers for recycling the used products. For each of the examples solved in this section, we use the same demand price functions, which are modified from [18. For product 1, the corresponding demand price functions associated with firm i (i = 1, 2, 3, 4 and demand maret R ( =1,2,3 are as follows: Firm 1: ρ 1j1 = 1.1 d 1j1.9 d 2j1.9 d 3j1.9 d 4j1 + 5, ρ 1j2 =.9 d 1j2.8 d 2j2.8 d 3j2.8 d 4j2 + 5, ρ 1j3 = 1. d 1j3.9 d 2j3.9 d 3j3.9 d 4j3 + 5; Firm 2: ρ 2j1 =.7 d 1j1 1.1 d 2j1.7 d 3j1.7 d 4j1 + 49, ρ 2j2 =.7 d 1j2.9 d 2j2.7 d 3j2.7 d 4j2 + 49, ρ 2j3 =.8 d 1j3 1.2 d 2j3.8 d 3j3.8 d 4j3 + 49; Firm 3: ρ 3j1 =.8 d 1j1.8 d 2j1 1.1 d 3j1.8 d 4j1 + 48, ρ 3j2 =.8 d 1j2.8 d 2j2.9 d 3j2.8 d 4j2 + 48, ρ 3j3 =.9 d 1j3.9 d 2j3 1.2 d 3j3.9 d 4j3 + 48; Firm 4: ρ 4j1 =.7 d 1j1.7 d 2j1.7 d 3j1 1. d 4j1 + 47, ρ 4j2 =.7 d 1j2.7 d 2j2.7 d 3j2.9 d 4j2 + 47, ρ 4j3 =.7 d 1j3.7 d 2j3.7 d 3j3 1.1 d 4j (8 For product 2, the corresponding demand price functions associated with firm i (i = 1, 2, 3, 4 and demand maret R ( =1,2,3 are as follows: Firm 1: ρ 1j1 = 1.1 d 1j1.9 d 2j1.9 d 3j1.9 d 4j1 + 42, ρ 1j2 =.9 d 1j2.8 d 2j2.8 d 3j2.8 d 4j2 + 42, ρ 1j3 = 1. d 1j3.9 d 2j3.9 d 3j3.9 d 4j3 + 42; Firm 2: ρ 2j1 =.7 d 1j1 1.1 d 2j1.7 d 3j1.7 d 4j1 + 43, ρ 2j2 =.7 d 1j2.9 d 2j2.7 d 3j2.7 d 4j2 + 43, ρ 2j3 =.8 d 1j3 1.2 d 2j3.8 d 3j3.8 d 4j3 + 43; Firm 3: ρ 3j1 =.8 d 1j1.8 d 2j1 1.1 d 3j1.8 d 4j1 + 44, ρ 3j2 =.8 d 1j2.8 d 2j2.9 d 3j2.8 d 4j2 + 44, ρ 3j3 =.9 d 1j3.9 d 2j3 1.2 d 3j3.9 d 4j3 + 44; Firm 4: ρ 4j1 =.7 d 1j1.7 d 2j1.7 d 3j1 1. d 4j1 + 45, ρ 4j2 =.7 d 1j2.7 d 2j2.7 d 3j2.9 d 4j2 + 45, ρ 4j3 =.7 d 1j3.7 d 2j3.7 d 3j3 1.1 d 4j (81 Assume that d is uniformly distributed in [, τ with probability density function given by 1, if x [,τ F (x = τ,, if x (τ,, (82 where for product j=1, τ 1j =28, τ 2j =27, τ 3j =26, τ 4j = 25 ( = 1, 2, 3 andforproductj = 2, τ 1j = 2, τ 2j =19, τ 3j =18, τ 4j =17( =1,2,3. Assume that r ji (the returned product j (j =1,2from demand maret R ( = 1, 2, 3 tofirmi (i = 1, 2, 3, 4 is uniformly distributed in [, 8. Example 15. n this example, the demand price functions are as given in (8-(81. The cost functions, which are modified fromthose given in [4, are as follows: c ja (u f j =2(uf ja 2 +(u f ja, a Li f ; f (x New = 2.5(x New 2 +2x New, f Re (xre =(xre 2 +.5x Re, (83 where i = 1, 2, 3, 4, j=1,2,and=1,2,3. The values of the parameters are as follows: θ + (unit penalty incurred on firm i due to excessive supply of product j to demand maret R =2,θ (unit penalty incurred on firm i duetoinsufficientsupplyofproductj to demand maret R =2, c Re ji (transportation cost per item of returned product j from demand maret R to firm i =.5,ρ Re j (purchase cost per item of returned product j from demand maret R to firm i=1,andρ (disposal fee per item of the used products at the landfill site =1, where i = 1, 2, 3, 4, j = 1,2,and =1,2,3. n Example 15, all the conditions of Theorem 12 are satisfied. We solve this example by Algorithm 13 (using Matlab to obtain the Cournot-Nash equilibrium of the CLSC networ. The Cournot-Nash equilibrium representing the optimal strategies of all the firms is presented in the fourth column of Table 1. The expected profits of firm i (i = 1, 2, 3, 4, denoted by U i, at the Cournot-Nash equilibrium are U 1 = 1393, U 2 = 12988, (84 U 3 = 12536, U 4 = Here is a summary of the numerical results for Example 15. For both Product 1 and Product 2, the situation of the optimal strategies at Cournot-Nash equilibrium is as follows. Firm 1 should manufacture the largest amount of new products as well as provide the largest amount of forward productflowstoeachofthedemandmaretsr 1, R 2,andR 3, followed by Firm 2, Firm 3, and Firm 4. Furthermore, Firm 1 should receive the largest amount of profit, followed by Firm 2, Firm 3, and Firm 4.

13 Mathematical Problems in Engineering 13 Table1: Optimal strategies for all firmsfor Examples15and 16. Optimal strategies involving product 1 Optimal strategies involving product 2 Example 15 Example 16 ρ=1 ρ=2 ρ=3 Optimal product supplied by firm 1 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 2 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 3 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 4 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 1 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 2 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 3 to demand maret R = = = New manufactured product Firm Optimal product supplied by firm 4 to demand maret R = = = New manufactured product Firm n the next example, we use the parameter ρ to perform sensitivity analysis. By fixing the other parameters, we study the effect of changing the parameter ρ on the Cournot-Nash equilibrium together with the optimal expected profits of all firms. Example 16. Same as Example 15, exceptthatρ = 1 in Example 15 is first replaced by ρ=2andthenbyρ=3in this example. (n other words, we now increase the disposal feeperitemby1%and2%.thus,themainpurposeof this example is to test the effect of increasing the disposal fee per item on the firms optimal strategies and the firms expected profits. The Cournot-Nash equilibrium representing the optimal strategies of all the firms when ρ = 2 is presented in the fifth column of Table 1, and the expected profits of firm i (i = 1, 2, 3, 4, denoted by U i, at the Cournot-Nash equilibrium are U 1 = 1271, U 2 = 1266, U 3 = 1217, U 4 = (85 The Cournot-Nash equilibrium representing the optimal strategies of all the firms when ρ = 3 is presented in the sixth column of Table 1, and the expected profits of

14 14 Mathematical Problems in Engineering firm i (i = 1, 2, 3, 4, denoted by U i,atthecournot-nash equilibrium are U 1 = 12323, U 2 = 12237, U 3 = 11816, U 4 = (86 From the fourth, fifth, and sixth columns of Table 1 and the value of the expected profits given above, we observe the following. (i When ρ is increased, both the optimal product flows and the new manufactured products of each firm decrease. (ii When ρ is increased, the expected profit of each firm also decreases. Remar 17. From the above two examples, we conclude that the change in disposal fee per item of used products not only affects the firms optimal strategies, but also their expected profits. 7. Conclusion and Suggestions for Further Studies n this paper, we develop a multiproduct CLSC networ equilibrium model with uncertain demands and product returns. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framewor under a stochastic environment. The Cournot- Nash equilibrium conditions are solved by using the LQP prediction-correction method. Finally, numerical examples are presented to test the efficiency of the algorithm for finding the Cournot-Nash equilibrium of the CLSC networ. n the future, we would lie to extend our method to solve a multiperiod CLSC networ equilibrium model involving oligopolistic competition among firms. Conflict of nterests The authors declare that there is no conflict of interests regarding the publication of this paper. Acnowledgments ThisresearchwassupportedbytheGeneralResearchFund (Projectno.PolyU545/11HoftheResearchGrantsCouncil, Hong Kong, and the Research Committee of the Hong Kong Polytechnic University. t was also supported by the National Natural Science Foundation of China (no and the Natural Science Foundation of Shandong Province of China (ZR213GQ7. 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