Intelligent Optimization Algorithms: A Stochastic Closed-Loop Supply Chain Network Problem Involving Oligopolistic Competition for Multiproducts and Their Product Flow Routings

Abstract

Recently, the first oligopolistic competition model of the closed-loop supply chain network involving uncertain demand and return has been established. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework. In this paper, we modify the above model in two different directions. (i) For each returned product from demand market to firm in the reverse logistics, we calculate the percentage of its optimal product flows in each individual path connecting the demand market to the firm. This modification provides the optimal product flow routings for each product in the supply chain and increases the optimal profit of each firm at the Cournot-Nash equilibrium. (ii) Our model extends the method of finding the Cournot-Nash equilibrium involving smooth objective functions to problems involving nondifferentiable objective functions. This modification caters for more real-life applications as a lot of supply chain problems involve nonsmooth functions. Existence of the Cournot-Nash equilibrium is established without the assumption of differentiability of the given functions. Intelligent algorithms, such as the particle swarm optimization algorithm and the genetic algorithm, are applied to find the Cournot-Nash equilibrium for such nonsmooth problems. Numerical examples are solved to illustrate the efficiency of these algorithms.

Yan Zhou 1 and Chi Kin Chan 2 and Kar Hung Wong 3 and Y. C. E. Lee 2

Academic Editor:Giovanni Falsone

1, Department of Management Science and Engineering, Qingdao University, Qingdao 266071, 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 2050, South Africa

Received 29 March 2015; Revised 20 July 2015; Accepted 26 July 2015; 26 October 2015

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 work is properly cited.

1. Introduction

In the past decade, the perfect competition equilibrium models of the supply chain network (SCN) have been widely studied. For instance, a perfect competition deterministic equilibrium model and a stochastic equilibrium model involving a lot of decision-makers were first established by Nagurney et al. [1] and Dong et al. [2], respectively. On the other hand, a perfect competition equilibrium model of a reverse SCN was constructed by Nagurney and Toyasaki [3] for the optimal management of the electronic wastes. Moreover, perfect competition equilibrium models of a closed-loop supply chain (CLSC) network were established by Hammond and Beullens [4] and Yang et al. [5] for the optimal management of waste electrical/electronic equipment and raw material, respectively. Qiang et al. [6] were the first to investigate a stochastic equilibrium model of a CLSC network, which involved uncertainties in demands, but not uncertainties in returns. The equilibrium conditions of all the above papers were obtained by the theory of variational inequality and were solved by the modified projection method (Korpelevich [7]).

Nowadays, more firms are aware of the importance of integrating the supply chain as a whole, consisting of all the marketing activities of all the competitors. The integration of the entire supply chain generates oligopolistic competition among firms. As a consequence, research works were recently extended from the perfect competition markets model to the oligopolistic firms model in the forward SCN. The oligopolistic competition among firms in the forward SCN was considered in a lot of real-life situations. For instance, oligopolistic competition equilibrium models of a forward SCN were established by Masoumi et al. [8] and Yu and Nagurney [9] for the optimal management of perishable products such as pharmacies and fresh foods and by Nagurney and Yu [10] for the minimization of emission-generations. The equilibrium conditions of these models were also obtained by the theory of variational inequality and were solved by the Euler method (Dupuis and Nagurney [11]).

However, for both perfect competition and oligopolistic competition, the CLSC integrating the forward and reverse supply chain is more important than the forward supply chain alone due to the government legislation (such as the paper recycling directive and WEEE within the European Union [12]). Moreover, the process used in the CLSC to recycle used products (such as papers, glass, building wastes, and electronic and electrical equipment) for minimizing resource wastage also leads to people's understanding of the green supply chain management (GSCM) (Sheu and Talley [13] and Seuring [14]). Research work on oligopolistic competition model of the CLSC network only began very recently. The first oligopolistic competition equilibrium model of the CLSC network was established recently by Zhou et al. [15].

The model developed by Zhou et al. [15] belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework in a stochastic environment. Since the demands and returns are uncertain, two types of risk, namely, the overstocking and understocking of multiproducts in the forward supply chain, are considered. By using the quantities of each new product and the path flows of each product on the forward supply chain as the decision variables, every oligopolistic firm's expected profit can be maximized at the Cournot-Nash equilibrium. The equilibrium condition of this model was also obtained by the theory of variational inequality; the variational inequalities in this paper were solved by the logarithmic quadratic proximal prediction-correction method (He et al. [16]).

As mentioned in the abstract, we modify the above model in two different directions as follows:

(i) The model of Zhou et al. [15] can calculate the optimal product flow in each path from firms to demand markets in the forward logistics, but not in paths from demand markets to firms in the reverse logistics because the quantity of the returned products from each demand market is a random variable. In this paper, for each returned product [figure omitted; refer to PDF] from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] in the reverse logistics, we can calculate the percentage of its optimal product flows in each individual path connecting the above demand market to the above firm. This modification provides the optimal product flow routings for each product in the entire supply chain and hence can increase the optimal profit of each of the firms at the Cournot-Nash equilibrium.

(ii) Our proposed model extends the method of finding the Cournot-Nash equilibrium for CLSC problems involving smooth objective functions to problems involving nondifferentiable objective functions. This modification caters for more real-life applications as a lot of supply chain problems involve nonsmooth objective functions.

In this paper, we establish the Cournot-Nash equilibrium without the assumption of differentiability on the given functions and use intelligent algorithms, such as the genetic algorithm and the particle swarm optimization (PSO) algorithm, for finding the Cournot-Nash equilibrium for nonsmooth CLSC problems.

Genetic algorithm (Holland [17]) is a common intelligent optimization algorithm, which can find the Nash (Nash [18, 19]) equilibrium effectively. For instance, genetic algorithm has been used for finding the Stackelberg- [20] Nash equilibrium of a nonlinear, nonconvex, nondifferentiable multilevel programming model (Liu [21]). Complicated SNC problems involving the design of the hierarchical spanning tree network (Kim et al. [22]) and that of a vendor managed inventory SNC network (Yu and Huang [23]) were also successfully solved by the genetic algorithm. In these papers, the efficiency of the genetic algorithm for solving complicated combinational problems, in terms of both speed and accuracy, has been demonstrated.

The PSO algorithm, originally proposed by Eberhart and Kennedy [24], is a member of the swarm intelligence methods (Kennedy and Eberhart [25]) for solving global optimization problems. Similar to a lot of intelligent optimization algorithms, the PSO algorithm does not require the gradient information of both the objective functions and the constraint functions, but only their values. Thus, it is easily implemented and computationally inexpensive and has been successfully applied to solve continuous optimization problems as well as discrete optimization problems (Goksal et al. [26]). Numerical results have shown that the PSO algorithm is more efficient than the genetic algorithm, especially for solving problems involving continuous solution space. For instance, Kadadevaramath et al. [27] and Govindan et al. [28] solved, respectively, a three-echelon SCN problem and an optimization problem involving both the economic and environmental benefits of a perishable food SCN by both the genetic algorithm and the PSO algorithm. Numerical results indicated that the PSO algorithm is more efficient than the genetic algorithm, in terms of the accuracy of the optimal solutions.

In Section 5 of this paper, two numerical examples are solved to compare the efficiencies of the PSO algorithm, the genetic algorithm, and an algorithm based on variational inequalities for finding the Cournot-Nash equilibrium. In one numerical example (Example 1), the results show that when all the given functions are differentiable, the efficiencies of the PSO algorithm, the genetic algorithm, and the algorithm based on variational inequality are almost the same, in terms of the accuracy of the computed equilibrium. However, the PSO algorithm is just as efficient as the algorithm based on variational inequality but more efficient than the genetic algorithm, in terms of the total computational time required to obtain the equilibrium. In another numerical example (Example 2), the results show that, for problems involving nonsmooth objective functions, the PSO algorithm and the genetic algorithm can still find the Cournot-Nash equilibrium efficiently, but the algorithm based on variational inequality is not efficient.

The rest of this paper is as follows. Section 2 develops the oligopolistic CLSC network model with multiproducts and uncertain demands and returns and constructs the Cournot-Nash equilibrium conditions for our model. Section 3 proves the existence of the Cournot-Nash equilibrium for our model. Section 4 presents a PSO algorithm and a genetic algorithm to find the Cournot-Nash equilibrium. In Section 5, numerical examples are solved to compare the effectiveness of the PSO algorithm, the genetic algorithm, and an algorithm based on variational inequality for finding the Cournot-Nash equilibrium. Section 6 presents our summary and conclusion.

2. An Equilibrium Model of a CLSC Network under Oligopolistic Competition among Firms Involving Product Flow Routings in Both Forward and Reverse Logistics

Zhou et al. [15] have established the first oligopolistic competition model of the closed-loop supply chain network (CLSC) involving uncertain demand and return. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework. In this model, they maximize every oligopolistic firm's expected profit by deciding the optimal production qualities of each new product as well as the amount of product flows in each individual path containing firms to demand markets in the forward logistics. Due to the fact that the quantities of returned products are random variables, they are unable to calculate the optimal amount of product flows in the reverse logistics. In this paper, we modify the above model by including the percentage of product flows in each path in the reverse logistics as a decision variable.

The topology of the network of our model is shown in Figure 1. Each firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) produces [figure omitted; refer to PDF] products. In order to satisfy the demand, the firms either manufacture new products or remanufacture used products through recycling used components obtained from the previous production period. Both the demands and returns are random variables having uniform distributions. As mentioned in the previous paragraph, each firm determines the optimal production quantities of the new products, the amount of product flows in each path in the forward logistics, and the percentage of product flows in each path in the reverse logistics to maximize its profit.

Figure 1: The CLSC network topology.

[figure omitted; refer to PDF]

In the closed-loop supply chain, each firm is represented as a network of its economic activities. In the forward logistics, firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) has [figure omitted; refer to PDF] manufacturing faculties/plants and [figure omitted; refer to PDF] distribution centers and serves the same [figure omitted; refer to PDF] demand markets. In the reverse logistics, firm [figure omitted; refer to PDF] has [figure omitted; refer to PDF] recovery centers. The activities in the forward supply chain involve the production, remanufacturing, delivery, and distribution of products. The activities in the reverse supply chain involve the disposal, recycling, and returning of products.

In Figure 1, the links from the top-tiered [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of the forward logistics representing the individual firm are connected to the manufacturing nodes of the respective firm [figure omitted; refer to PDF] , which are denoted by [figure omitted; refer to PDF] , respectively. The links from the manufacturing nodes are, in turn, connected to the distribution center nodes of firm [figure omitted; refer to PDF] , which are denoted by [figure omitted; refer to PDF] , respectively. The links from the distribution centers nodes are, in turn, connected to the demand market nodes, denoted by [figure omitted; refer to PDF] , respectively; each of these demand markets is served by all the firms.

Also, in Figure 1, the links from every demand market node [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of the reverse logistics are connected to all the recovery center nodes of firm [figure omitted; refer to PDF] , which are denoted by [figure omitted; refer to PDF] , respectively. Finally, the links from the recovery centers nodes are connected to firm [figure omitted; refer to PDF] to complete a single loop.

Thus, our CLSC network 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 closed-loop network.

We wish to emphasize that the network topology in Figure 1 is for demonstration purpose only. In fact, the model can handle any prospective closed-loop chain network topology, provided that there are top-tiered nodes to represent each firm and bottom-tiered nodes to represent the demand markets with two sequences of directed links containing each top-tiered node to each bottom-tiered node in the forward and reverse directions, respectively.

In the forward supply chain, any set of correlated links that can connect a firm to a demand market via a manufacturer and a distribution center form a forward path [figure omitted; refer to PDF] . Similarly, in the reverse supply chain, any set of correlated links that can connect a demand market to a firm via the recovery center form a reverse path [figure omitted; refer to PDF] . From the discussion in the previous paragraph, the structure of our model can be completed specified by its forward and reverse paths.

We now give a few assumptions together with their justifications. These are common assumptions in the CLSC literature, except possibly for (A8). (See Hammond and Beullens [4] and Zhou et al. [15] for details.)

Assumption A.

(A1) The demand for product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) at demand market [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) in one production period, denoted by [figure omitted; refer to PDF] , is a random variable. Any two demands [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] are independent of each other. This assumption is due to the fact that all the buyers are independent.

(A2) A fraction of the used products which is in good condition will be returned to the firms for remanufacturing; the remainder of used products which is not in good condition will be sent to the landfill for disposal. Thus, we assume that, for any used product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) which is in good condition, a cost of [figure omitted; refer to PDF] per item will be charged to the firms for the purchase of these used products. For any used product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) which is not in good condition, a cost of [figure omitted; refer to PDF] (independent of [figure omitted; refer to PDF] ) per item will be charged to the firms for the disposal of these used products at the landfall.

(A3) All the returned products are remanufacturable. This assumption is a direct consequence of (A2).

(A4) The return of used product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) from demand market [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) to firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) in one production period, denoted by [figure omitted; refer to PDF] , is a random variable. Any two returns [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] are independent of each other. This assumption is due to the fact that all the customers are independent.

(A5) Due to the existence of modern manufacturing technologies, manufacturers can easily transform the remanufactured products into as-new products. Thus, we assume that there is no distinction between the qualities of the brand new products and the remanufactured products and they are sold to the markets at the same price.

(A6) The operational cost of product [figure omitted; refer to PDF] on any link [figure omitted; refer to PDF] in the forward logistics, denoted by [figure omitted; refer to PDF] , is a continuous function of all the link flows of this product in the forward supply chain. In other words, [figure omitted; refer to PDF] depends continuously not only on the product flow in link [figure omitted; refer to PDF] , but also on the product flows in all the manufacturer links and all the shipment links and all the distribution links. Consequently, the operation cost of product [figure omitted; refer to PDF] on any path [figure omitted; refer to PDF] in the forward supply chain, denoted by [figure omitted; refer to PDF] , is also a continuous function of all the path flows of its product in the forward supply chain. This assumption is due to the fact that competition for business exists among all the companies involving economic activities in the forward logistics.

(A7) The operational cost of product [figure omitted; refer to PDF] on any path [figure omitted; refer to PDF] in the reverse logistics, denoted by [figure omitted; refer to PDF] , is a continuous function of the path flow of this product on [figure omitted; refer to PDF] only.

(A8) For product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) of firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), the cost of manufacturing [figure omitted; refer to PDF] items of new product [figure omitted; refer to PDF] by firm [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , and the cost of remanufacturing [figure omitted; refer to PDF] items of returned product [figure omitted; refer to PDF] by firm [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are given continuous functions. Due to the operational advantages, the manufacturing costs are higher than the remanufacturing costs; that is, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

(Note that, due to the logistics advantages, the optimal strategies of any firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) at the Cournot-Nash equilibrium must automatically satisfy the following relationship: total reverse logistics costs of firm [figure omitted; refer to PDF] (i.e., total purchase costs of returned products + total transportation costs of returned products) + total remanufacturing costs of firm [figure omitted; refer to PDF] [figure omitted; refer to PDF] total manufacturing costs of firm [figure omitted; refer to PDF] .)

(A9) Due to the limitation of the facilities at the manufacturers' factories, the quantity of new product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) manufactured by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) in one production period cannot exceed [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a positive integer.

In order to formulate the problem of finding the Cournot-Nash equilibrium for our CLSC network, we first formulate three types of decision variables, namely, the quantity of new manufactured product, the amount of product flows in each path in the forward logistics, and the percentage of product flows in each path in the reverse logistics.

For any firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), there are [figure omitted; refer to PDF] manufacturers and [figure omitted; refer to PDF] distribution centers served for firm [figure omitted; refer to PDF] in the forward logistics. Thus, there are altogether [figure omitted; refer to PDF] different paths which can connect firm [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) via the manufacturers and distribution centers. Let [figure omitted; refer to PDF] denote the set of all the forward paths associated with firm [figure omitted; refer to PDF] and demand market [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] denote the set of all the forward paths associated with all the firms and all the demand markets. For each [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be the amount of product flows of product [figure omitted; refer to PDF] in the forward path [figure omitted; refer to PDF] .

For any firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), there are [figure omitted; refer to PDF] recovery centers served for firm [figure omitted; refer to PDF] in the reverse logistics. Thus, there are altogether [figure omitted; refer to PDF] different paths which can connect demand market [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) to firm [figure omitted; refer to PDF] via the recovery centers. Let [figure omitted; refer to PDF] denote the set of all the reverse paths associated with firm [figure omitted; refer to PDF] and demand market [figure omitted; refer to PDF] . For each returned product [figure omitted; refer to PDF] from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] in the reverse logistics, let [figure omitted; refer to PDF] be the percentage of its optimal product flows in the path [figure omitted; refer to PDF] connecting the above demand market to the above firm.

For any firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), let [figure omitted; refer to PDF] be the quantity of new product [figure omitted; refer to PDF] manufactured by firm [figure omitted; refer to PDF] in one production period.

Thus, there are altogether [figure omitted; refer to PDF] decision variables associated with firm [figure omitted; refer to PDF] and product [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where the [figure omitted; refer to PDF] decision variables are of the form [figure omitted; refer to PDF] , which represent the amount of the flows of product [figure omitted; refer to PDF] on all the forward paths [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), the [figure omitted; refer to PDF] decision variables are of the form [figure omitted; refer to PDF] , which represent the percentage of the flows of product [figure omitted; refer to PDF] in all the reverse paths [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), and the other decision variable is of the form [figure omitted; refer to PDF] , which represents the quantity of new product [figure omitted; refer to PDF] manufactured by firm [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the vector representing all the decision variables associated with firm [figure omitted; refer to PDF] and product [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the strategy vector representing the overall decision variables associated with firm [figure omitted; refer to PDF] .

Then [figure omitted; refer to PDF] is the overall decision variables for the entire CLSC network, where [figure omitted; refer to PDF] .

We now formulate all the constraints in the reverse logistics. From the definition of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Thus (5) provides upper and lower bounds for the product flow [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ) in the reverse logistics.

Since the total amount of returned product [figure omitted; refer to PDF] from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , it is clear that the amount of product flow in the reverse path [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is equal to [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

We now formulate all the constraints in the forward logistics.

In view of (A9), we have [figure omitted; refer to PDF] Constraint (8) provides an upper bound for the quantity of new product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) manufactured by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Let [figure omitted; refer to PDF] be the random variable representing the quantity of product [figure omitted; refer to PDF] remanufactured by firm [figure omitted; refer to PDF] in one production period. Then in view of (A3) and (A4), we have [figure omitted; refer to PDF]

In view of (9), the following flow inequality, which provides an upper bound for the total amount of flows of product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) from firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) to all the demand markets in the forward supply chain, must hold: [figure omitted; refer to PDF]

Furthermore, the following flow inequality, which provides a lower bound for the amount of flows of product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) from firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) to the demand market [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) in the forward supply chain, must also hold: [figure omitted; refer to PDF]

We now formulate different cost functions for our CLSC network. We first formulate the expected penalty cost (due to excessive or insufficient supply).

Consider the forward supply chain involving firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), and demand market [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). Let [figure omitted; refer to PDF] denote the quantity of product [figure omitted; refer to PDF] supplied by firm [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] in one production period. Then the above quantity is equal to the total amount of flows of product [figure omitted; refer to PDF] on paths connecting firm [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be the total demand associated with firm [figure omitted; refer to PDF] , product [figure omitted; refer to PDF] , and demand market [figure omitted; refer to PDF] in one production period. Let [figure omitted; refer to PDF] be the probability density function of [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denote the probability. For each product [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] denote the vector consisting of all the demands of product [figure omitted; refer to PDF] in the entire CLSC network.

Now, the quantity of product [figure omitted; refer to PDF] supplied by firm [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] cannot exceed the minimum of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In other words, the actual sale of these products is equal to [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] denote, respectively, the quantity of the overstocking and the understocking of product [figure omitted; refer to PDF] associated with firm [figure omitted; refer to PDF] and demand market [figure omitted; refer to PDF] . The expected values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF]

Assume that the unit penalty incurred on firm [figure omitted; refer to PDF] due to excessive supply of product [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] is [figure omitted; refer to PDF] and the unit penalty incurred on firm [figure omitted; refer to PDF] due to insufficient supply of product [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then, the total expected penalty incurred on firm [figure omitted; refer to PDF] associated with product [figure omitted; refer to PDF] and demand market [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

Apart from the penalty costs given by (18) (due to excessive or insufficient supply), the following additional costs are also charged for firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ):

(i) From (A6), the total operational cost of product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) on the forward path is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the vector representing the amount of product flows of product [figure omitted; refer to PDF] on all the forward paths.

(ii) From (A7) and (7), the total operational cost of product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) on all the reverse path is [figure omitted; refer to PDF]

(iii) From (A2), the total cost related to the purchase of returned product [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

(iv) Also from (A2), the total cost related to the disposal of uncollected product [figure omitted; refer to PDF] at the landfill site is [figure omitted; refer to PDF]

(v) From (A8) and (9), the cost of manufacturing new product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] and the cost of remanufacturing the returned product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF]

By virtue of (18) and (19)-(24), the total expected cost incurred on firm [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF]

Now, we formulate the total revenue received by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). In order to capture competition for demand in the entire CLSC network, we assumed that, for each product [figure omitted; refer to PDF] , the demand price function [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ) is a continuous function of all the demands associated with product [figure omitted; refer to PDF] in the entire CLSC network; that is, [figure omitted; refer to PDF] In other words, the price function [figure omitted; refer to PDF] depends continuously not only on [figure omitted; refer to PDF] , but also on all the demand associated with product [figure omitted; refer to PDF] . Masoumi et al. [8] and Nagurney and Yu [10] used such demand price function in the study of forward supply chain network involving oligopolistic competition among firms. Then the expected revenue received by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is given by [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), denoted by [figure omitted; refer to PDF] , can be expressed as follows: [figure omitted; refer to PDF] where (12) holds for all decision variables defined by (4). By virtue of (16), (17), (19)-(24), and (26), we know that [figure omitted; refer to PDF] is a function of the strategy vector of all firms in the entire CLSC network. That is, [figure omitted; refer to PDF]

Now, in order to define the Cournot-Nash equilibrium of the CLSC network, we need to consider the following constraints involving firm [figure omitted; refer to PDF] and product [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ): [figure omitted; refer to PDF] where constraints (30), (31), (32), (33), and (34) are directly obtained from (8), (1), (11), (6), and (5), respectively; constraints (34) and (35) ensure that all the decision variables are nonnegative.

In this CLSC network, there are [figure omitted; refer to PDF] oligopolistic firms competing noncooperatively to maximize their own expected profits and selecting their strategy vectors till an equilibrium is established. So the game is based on oligopolistic Cournot pricing in a Cournot-Nash framework.

Now, we can define the Cournot-Nash equilibrium of the CLSC network according to Definitions 1, 2, and 3 given below.

Definition 1 (a feasible strategy vector).

For each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), a strategy vector [figure omitted; refer to PDF] is said to be a feasible strategy vector, if [figure omitted; refer to PDF] satisfies constraints (30)-(35).

Definition 2 (the set of all feasible strategy pattern).

Let [figure omitted; refer to PDF] be the set of all feasible strategy pattern defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is a feasible strategy vector defined by Definition 1.

Definition 3 (the Cournot-Nash equilibrium of the CLSC network).

A feasible strategy pattern [figure omitted; refer to PDF] constitutes a Cournot-Nash equilibrium of the CLSC network, if the following inequality holds for all [figure omitted; refer to PDF] and for all feasible strategy vectors [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Thus, an equilibrium is established if no firm in the CLSC network 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.

3. Existence Results

In this section, we establish the existence of the Cournot-Nash equilibrium of the CLSC network defined by Definition 3.

To guarantee the existence of the Cournot-Nash equilibrium of the CLSC network, the following additional assumption is held for [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Assumption B.

(B1) The operational cost of product [figure omitted; refer to PDF] in any path [figure omitted; refer to PDF] in the forward logistics, denoted by [figure omitted; refer to PDF] , is both a continuous and a convex function of [figure omitted; refer to PDF] , the vector consisting of all forward path flows of product [figure omitted; refer to PDF] .

(B2) The operational cost of product [figure omitted; refer to PDF] in any path [figure omitted; refer to PDF] in the reverse logistics, denoted by [figure omitted; refer to PDF] , is both a continuous and a convex function of [figure omitted; refer to PDF] , the product flow of this product in the reverse path [figure omitted; refer to PDF] .

(B3) The production cost [figure omitted; refer to PDF] is a continuous and convex function of [figure omitted; refer to PDF] , the new product [figure omitted; refer to PDF] manufactured by firm [figure omitted; refer to PDF] .

Remark 4.

Note that Assumption (B1) and Assumption (B3) in this paper are similar to Assumption (B1) and Assumption (B2) by Zhou et al. [15], except that there is no assumption of differentiability of the given functions in this paper.

In the following discussion, Lemma 5 provides us with sufficient conditions under which the set of all feasible strategy pattern [figure omitted; refer to PDF] has a Cournot-Nash equilibrium, whereas in Lemmas 6-8 we prove that, under Assumptions A and B, the set of all feasible strategy pattern [figure omitted; refer to PDF] indeed satisfies the sufficient condition for the existence of the Cournot-Nash equilibrium.

Lemma 5.

Suppose the set of all feasible strategy pattern [figure omitted; refer to PDF] defined by Definition 2 is both a compact and convex set of [figure omitted; refer to PDF] . Suppose the expected profit function [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) defined by (28) is continuous and concave with respect to the decision variable [figure omitted; refer to PDF] defined by (4). Then there exists a Cournot-Nash equilibrium of the CLSC network defined by Definition 3.

Proof.

By virtue of the fact that all the conditions imposed in this lemma are almost the same as those imposed in Theorem 1, page 639 of Nishimura and Friedman's paper [29], except that the concavity condition in this lemma is more restrictive than that imposed in Theorem [figure omitted; refer to PDF] of Nishimura and Friedman's paper [29] (see condition (A5) on page 639 of Nishimura and Friedman's paper [29]), the proof of this lemma follows easily from that of Theorem [figure omitted; refer to PDF] of Nishimura and Friedman's paper [29].

Lemma 6.

Suppose that Assumption A is satisfied; then the set of all feasible strategy pattern [figure omitted; refer to PDF] defined by Definition 2 is both a compact and a convex set of [figure omitted; refer to PDF] .

Proof.

The fact that [figure omitted; refer to PDF] is bounded follows easily from (35), (31), (30), and (34). The fact that [figure omitted; refer to PDF] is closed and convex follows easily from (30), (31), (32), (33), (34), and (35). Hence, [figure omitted; refer to PDF] is both a compact and a convex set of [figure omitted; refer to PDF] .

Lemma 7.

Suppose that Assumptions A and B are satisfied; then the expected profit function [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) defined by (28) is continuous with respect to the decision variable [figure omitted; refer to PDF] being defined by (4).

Proof.

Firstly, from (28), (12), (16)-(19), and (B1), we know that the first, second, and fourth terms of [figure omitted; refer to PDF] are continuous with respect to [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). From (B3), we know that the third term of [figure omitted; refer to PDF] is a continuous function of [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). The sixth, seventh, and eighth terms of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are constants, because [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are random variables, for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). By virtue of the fact that [figure omitted; refer to PDF] is a random variable, we know that the last term of [figure omitted; refer to PDF] is also a continuous function of [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From (B2), we know that the fifth term of [figure omitted; refer to PDF] is a continuous function of [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). Thus, every term of [figure omitted; refer to PDF] is continuous with respect to the decision variable [figure omitted; refer to PDF] defined by (4). The proof is complete.

Lemma 8.

Suppose that Assumptions A and B are satisfied; then the expected profit function [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) defined by (28) is concave with respect to the decision variable [figure omitted; refer to PDF] defined by (4).

Proof.

In order to prove that the first and second terms of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are concave with respect to the decision variable [figure omitted; refer to PDF] defined by (4), we calculate the second partial derivative of these terms with respect to the decision variable [figure omitted; refer to PDF] as follows.

By virtue of the fact that any two demands are independent of each other (from (A1)), the first term of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] From (38), we have [figure omitted; refer to PDF] But, from (12), we know that [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Thus, from (41) and (42), the second partial derivative of the first term of [figure omitted; refer to PDF] becomes [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

From (16), we have [figure omitted; refer to PDF]

From (17), we have [figure omitted; refer to PDF]

Thus, from (18), (42), (45), and (47), the second partial derivative of the second term of [figure omitted; refer to PDF] becomes [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

By virtue of (B1), we know that [figure omitted; refer to PDF] is concave with respect to [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Now, by virtue of (43), (48), the fact that the third, fifth, sixth, and seventh terms of (28) are independent of [figure omitted; refer to PDF] , the fact that the fourth term of (28) is concave with respect to [figure omitted; refer to PDF] , and the fact that the last term of (28) is a linear function of [figure omitted; refer to PDF] , we conclude that [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is concave with respect to [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

From (B3), we know that [figure omitted; refer to PDF] is concave with respect to [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is the only term in (28) involving [figure omitted; refer to PDF] , we conclude that [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is concave with respect to [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

By virtue of (B2), we know that [figure omitted; refer to PDF] is concave with respect to [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Hence, we conclude that the expected profit function [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) defined by (28) is concave with respect to the decision variable [figure omitted; refer to PDF] defined by (4).

Theorem 9.

Suppose that Assumptions A and B are satisfied; then there exists a Cournot-Nash equilibrium of the CLSC network defined by Definition 3.

Proof.

The proof follows easily from Lemmas 5, 6, 7, and 8.

4. Computing the Cournot-Nash Equilibrium

In this section, we develop two intelligent optimization algorithms, namely, the particle swarm algorithm (proposed by Eberhart and Kennedy [24]) and the Nash genetic algorithm (proposed by Sefrioui and Periaux [30]), denoted by PSO algorithm and GA, respectively, for finding the Cournot-Nash equilibrium of the CLSC network defined by Definition 3. For this purpose, we first need to transform the problem of finding the Cournot-Nash equilibrium into solving sequences of optimization problems with a continuous solution space as follows.

Let [figure omitted; refer to PDF] denote the potential strategy vector of firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) obtained at iteration [figure omitted; refer to PDF] of the algorithm (to be presented in the next paragraph). Let [figure omitted; refer to PDF] be the strategy vector of all firms obtained at iteration [figure omitted; refer to PDF] . In order to obtain the Cournot-Nash equilibrium as defined in Definition 3, at iteration [figure omitted; refer to PDF] of the algorithm, we need to find a feasible strategy vector [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) which satisfies [figure omitted; refer to PDF] for all feasible strategy vector [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is chosen arbitrarily and [figure omitted; refer to PDF] denotes the best strategy of firm [figure omitted; refer to PDF] obtained at iteration [figure omitted; refer to PDF] of the algorithm.

Thus, we need to find the feasible strategy vector [figure omitted; refer to PDF] which maximizes the expected profit function [figure omitted; refer to PDF] over the set of all the feasible vectors [figure omitted; refer to PDF] . For this purpose, we need to define a fitness function for any strategy vector [figure omitted; refer to PDF] , denoted by [figure omitted; refer to PDF] , which is obtained by appending a penalty function [figure omitted; refer to PDF] to the expected profit function [figure omitted; refer to PDF] . The penalty function [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a large number and [figure omitted; refer to PDF] denotes the total amount of constraints violations for all the constraints (30)-(34) of the strategy vector [figure omitted; refer to PDF] . (Note that when [figure omitted; refer to PDF] is a feasible strategy vector, then [figure omitted; refer to PDF] .) Thus, the fitness function for the strategy vector [figure omitted; refer to PDF] at iteration [figure omitted; refer to PDF] of the algorithm can be defined as follows: [figure omitted; refer to PDF] Thus, (53) states that, at iteration [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), the fitness of the strategy vector [figure omitted; refer to PDF] for firm [figure omitted; refer to PDF] is defined as "the expected profit obtained by firm [figure omitted; refer to PDF] by using the strategy [figure omitted; refer to PDF] , under the condition that all the other firms are using their best strategies obtained at iteration [figure omitted; refer to PDF] minus the amount of penalty due to the constraints violations of the strategy vector [figure omitted; refer to PDF] ."

Thus, the problem of finding the Cournot-Nash equilibrium of the CLSC network is equivalent to solving the following sequences of optimization problems with a continuous solution space, denoted by [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

: [figure omitted; refer to PDF] is as follows: [figure omitted; refer to PDF]

where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are as defined in (51). For the sake of simplicity, we replace [figure omitted; refer to PDF] by [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the error function, where [figure omitted; refer to PDF] . Suppose that [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] is the Cournot-Nash equilibrium of the CLSC network. In other words, for all [figure omitted; refer to PDF] , when the optimal solutions of two consecutive optimization problems [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are sufficiently close to each other, then we arrive at the Cournot-Nash equilibrium of the CLSC network.

We will use both PSO algorithm and GA to solve the above sequences of optimization problems. We now define the following terms for PSO algorithm.

Let [figure omitted; refer to PDF] be the individual best strategy vector found by a particle, where [figure omitted; refer to PDF] is the best strategy vector for firm [figure omitted; refer to PDF] found by this particle. Let [figure omitted; refer to PDF] be the global best strategy vector among all the particles in the swarm, where [figure omitted; refer to PDF] is the best strategy vector for firm [figure omitted; refer to PDF] among all the particles in the swarm. Let [figure omitted; refer to PDF] be the velocity vector which represents both the distance and the direction that should be traveled by the particle from its current position. Let [figure omitted; refer to PDF] denote the swarm size of the particle swarm. Let [figure omitted; refer to PDF] denote the weight representing the trade-off between global exploration and local exploitation abilities of the swarm. Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the weight representing the stochastic acceleration terms that pull each particle toward [figure omitted; refer to PDF] and [figure omitted; refer to PDF] positions of PSO algorithm, respectively. Let [figure omitted; refer to PDF] be a small number. Let [figure omitted; refer to PDF] be the number of successive iterations that criteria [figure omitted; refer to PDF] needed to be satisfied before we can ensure the convergence of PSO algorithm. PSO algorithm can now be formally stated as follows.

PSO Algorithm. Input parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) with [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a large number.

Iteration 0. Choose initial velocity vectors [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). For each [figure omitted; refer to PDF] , choose initial [figure omitted; refer to PDF] strategy vectors to form the initial swarm [figure omitted; refer to PDF] . For each particle [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be the best strategy vector found by the above particle. Then [figure omitted; refer to PDF] is the individual best strategy vector found by the above particle.

Compute [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] to obtain the best strategy vector [figure omitted; refer to PDF] for firm [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is the global best strategy vector among all particles in the swarm.

Let [figure omitted; refer to PDF]

Iteration [figure omitted; refer to PDF]

Step 1.

For each [figure omitted; refer to PDF] , update each particle in the swarm [figure omitted; refer to PDF] by using the following: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are pseudorandom number. The new swarm obtained is denoted by [figure omitted; refer to PDF] .

Step 2.

For each [figure omitted; refer to PDF] , compute [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] . Update the individual best vector found by each particle in the swarm as follows: [figure omitted; refer to PDF]

Step 3.

For each [figure omitted; refer to PDF] , compare [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] to obtain the best strategy vector [figure omitted; refer to PDF] . Hence obtain the best strategy vector [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] . Update the global best strategy vector found by all particles in the swarm as follows: [figure omitted; refer to PDF]

Step 4.

If [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] and repeat iteration [figure omitted; refer to PDF] ; otherwise go to Step 5.

Step 5.

Compute [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , the Cournot-Nash equilibrium of the CLSC network is reached, which is given by [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] , compute [figure omitted; refer to PDF] to obtain the optimal expected profit of firm [figure omitted; refer to PDF] , stop; otherwise, let [figure omitted; refer to PDF] and repeat iteration [figure omitted; refer to PDF] .

We now define the following parameters for GA.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote the population size, the crossover probability, and the mutation probability of the genetic algorithm, respectively. Let [figure omitted; refer to PDF] be a small number. Let [figure omitted; refer to PDF] be the number of successive iterations that criteria [figure omitted; refer to PDF] needed to be satisfied before we can ensure the convergence of GA. GA can now be formally stated as follows.

GA. Input parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Iteration 0. For each [figure omitted; refer to PDF] , choose initial [figure omitted; refer to PDF] strategy vectors to form the initial population [figure omitted; refer to PDF] based on real-coded genetic algorithm. Compute [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] to obtain the best strategy vector [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] .

Iteration [figure omitted; refer to PDF]

Step 1.

For each [figure omitted; refer to PDF] , update the population [figure omitted; refer to PDF] by spinning the roulette wheel with a bias towards selecting fitter strategy vectors to form the new population.

Step 2.

For each [figure omitted; refer to PDF] , update the population [figure omitted; refer to PDF] by using the crossover operation and the mutation operation given in Appendix A. The new population obtained is denoted by [figure omitted; refer to PDF] .

Step 3.

For each [figure omitted; refer to PDF] , compute [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] to obtain the best strategy vector [figure omitted; refer to PDF] . Hence obtain the best strategy vector for all the firms [figure omitted; refer to PDF] .

Step 4.

If [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] and repeat iteration [figure omitted; refer to PDF] ; otherwise go to Step 5.

Step 5.

Compute [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , the Cournot-Nash equilibrium of the CLSC network is reached, which is given by [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] , compute [figure omitted; refer to PDF] , which is the optimal expected profit of firm [figure omitted; refer to PDF] , stop; otherwise, let [figure omitted; refer to PDF] and repeat iteration [figure omitted; refer to PDF] .

Note that PSO algorithm and GA developed in this paper can always find the Cournot-Nash equilibrium in a certain number of iterations, even when the expected profit function [figure omitted; refer to PDF] defined by (28) is nondifferentiable.

5. Numerical Examples

In this section, two numerical examples are used to compare the efficiencies of the PSO algorithm, the genetic algorithm, and an algorithm based on variational inequalities for finding the Cournot-Nash equilibrium of the CLSC network. (The theory of variational inequality method is given in Appendix B.) In the first example (Example 1), all the given cost functions are differentiable. Thus, we can find the Cournot-Nash equilibrium by PSO algorithm, GA, and Euler algorithm (Dupuis and Nagurney [11]) based on variational inequality (Euler algorithm is given in Appendix C). In the second example (Example 2), the given cost functions corresponding to the manufacturing of the new products are not everywhere differentiable. We use this example to illustrate that both PSO algorithm and GA can still solve nonsmooth optimization problem efficiently, but the algorithm based on variational inequality is not efficient.

We consider a CLSC network involving oligopolistic competition among four firms. Each firm manufactures two products and has two manufacturers and two distribution centers to supply goods to three demand markets. 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 as those given by Zhou et al. [15].

Example 1 (comparison of the efficiencies of PSO algorithm, GA, and the Euler algorithm based on variational inequality for solving problems with smooth cost functions).

In this example, the demand price functions are as given by Zhou et al. [15].

The operation cost of product [figure omitted; refer to PDF] in the forward logistics (which is a function of the product flow of product [figure omitted; refer to PDF] on all the paths in the forward logistics) is as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The above operation cost is modified from that of Zhou et al.'s paper [15] by adding the last two terms of (66) to capture the competition among firms for the optimal product flows in the forward logistics.

The operation cost of product [figure omitted; refer to PDF] in the reverse logistics is as follows: [figure omitted; refer to PDF] for any path [figure omitted; refer to PDF] connecting demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] via the recovery center [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for any path [figure omitted; refer to PDF] connecting demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] via the recovery center [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

The manufacturing cost of new products and the remanufacturing cost of returned products, which are exactly the same as those in Zhou et al.'s paper [15], are as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Assume that [figure omitted; refer to PDF] is uniformly distributed in [figure omitted; refer to PDF] with probability density function given by [figure omitted; refer to PDF] where, for product [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and, for product [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

Assume that [figure omitted; refer to PDF] is uniformly distributed in [figure omitted; refer to PDF] with probability density function given by [figure omitted; refer to PDF] where, for product [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and, for product [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ).

The values of the parameters are as follows:

: [figure omitted; refer to PDF] (unit penalty incurred on firm [figure omitted; refer to PDF] due to excessive supply of product [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] ) [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] (unit penalty incurred on firm [figure omitted; refer to PDF] due to insufficient supply of product [figure omitted; refer to PDF] to demand market [figure omitted; refer to PDF] ) [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] (purchase cost per item of returned product [figure omitted; refer to PDF] from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] ) [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] (disposal fee per item of the used products at the landfill site) [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] (the maximum quantity of new product [figure omitted; refer to PDF] manufactured by firm [figure omitted; refer to PDF] ) [figure omitted; refer to PDF] ,

where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

We have solved Example 1 by PSO algorithm, GA, and the Euler algorithm based on the variational inequality. All the optimal strategies in the forward logistics and the reverse logistics at the Cournot-Nash equilibrium are depicted in Tables 1 and 2, respectively.

Table 1: Comparison of the optimal strategies obtained by the Euler algorithm with the optimal strategies obtained by PSO algorithm and GA in the forward logistics of Example 1.

			PSO	GA	Euler
			Mean	Stdev.	Mean	Stdev.
Firm 1	Optimal quantity of product 1 supplied by firm 1 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	15.4812	0.1763	15.2751	0.3265	15.2807
[figure omitted; refer to PDF]	15.3944	0.2543	15.8214	0.3163	15.4540
[figure omitted; refer to PDF]	15.1247	0.3744	14.9799	0.5525	15.2798
Optimal quantity of new manufactured product 1		34.0004	0.2496	34.0874	0.0301	34.0124
Optimal quantity of product 2 supplied by firm 1 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	11.8935	0.2569	11.2828	0.7107	12.0464
[figure omitted; refer to PDF]	11.6125	0.3073	12.3592	0.5675	12.1290
[figure omitted; refer to PDF]	12.5527	0.3890	12.5719	0.7039	12.0301
Optimal quantity of new manufactured product 2		24.0588	0.4450	24.2513	0.0708	24.2025
Firm 2	Optimal quantity of product 1 supplied by firm 2 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	14.8362	0.4743	14.4733	0.3827	15.0069
[figure omitted; refer to PDF]	14.4819	0.5303	14.7227	0.2718	15.0784
[figure omitted; refer to PDF]	14.8862	0.3080	14.9882	0.2961	14.8107
Optimal quantity of new manufactured product 1		32.2042	0.2463	32.1870	0.0825	32.8937
Optimal quantity of product 2 supplied by firm 2 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	11.8696	0.2370	11.4820	0.7778	11.8671
[figure omitted; refer to PDF]	11.6569	0.2564	11.9422	0.4798	11.8944
[figure omitted; refer to PDF]	11.4962	0.3879	11.7069	0.8136	11.7576
Optimal quantity of new manufactured product 2		23.0228	0.3726	23.3518	0.2146	23.5153
Firm 3	Optimal quantity of product 1 supplied by firm 3 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	15.2002	0.3042	14.7245	0.6152	14.5303
[figure omitted; refer to PDF]	13.6609	0.5233	14.3239	0.7908	14.5972
[figure omitted; refer to PDF]	14.1390	0.3938	14.5945	0.8268	14.3372
Optimal quantity of new manufactured product 1		31.0001	0.1009	31.6438	0.4290	31.4625
Optimal quantity of product 2 supplied by firm 3 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	11.7620	0.3609	10.7958	0.4134	11.5539
[figure omitted; refer to PDF]	11.9596	0.3486	11.8798	0.4157	11.5755
[figure omitted; refer to PDF]	11.5913	0.3544	11.4596	0.7349	11.4546
Optimal quantity of new manufactured product 2		23.3130	0.4206	22.1509	0.0989	22.5796
Firm 4	Optimal quantity of product 1 supplied by firm 4 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	13.7444	0.3978	13.5301	0.4678	14.1712
[figure omitted; refer to PDF]	14.1173	0.2761	14.1522	0.8036	14.1888
[figure omitted; refer to PDF]	14.2150	0.4706	13.7095	0.7046	14.0771
Optimal quantity of new manufactured product 1		30.0767	0.5645	30.4098	0.1273	30.4346
Optimal quantity of product 2 supplied by firm 4 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	11.1741	0.2702	10.2537	0.8623	11.2665
[figure omitted; refer to PDF]	11.2078	0.2615	11.8534	0.7726	11.2646
[figure omitted; refer to PDF]	12.0043	0.2985	11.4950	0.7313	11.2136
Optimal quantity of new manufactured product 2		22.3862	0.5094	21.6579	0.2226	21.7392

Table 2: Comparison of the optimal strategies obtained by the Euler algorithm with the optimal strategies obtained by PSO algorithm and GA in the reverse logistics of Example 1.

			PSO	GA	Euler
			Mean	Stdev.	Mean	Stdev.
Firm 1	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 1 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.5014	0.0289	0.5036	0.0187	0.5000
[figure omitted; refer to PDF]	0.5240	0.0243	0.5545	0.0121	0.5937
[figure omitted; refer to PDF]	0.6922	0.0339	0.6201	0.0172	0.6875
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 1 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.4938	0.0313	0.5258	0.0183	0.5000
[figure omitted; refer to PDF]	0.6306	0.0212	0.6133	0.0253	0.6250
[figure omitted; refer to PDF]	0.7990	0.0300	0.7686	0.0242	0.7500
Firm 2	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 2 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.4124	0.0281	0.4216	0.0161	0.4063
[figure omitted; refer to PDF]	0.5515	0.0162	0.5235	0.0103	0.5000
[figure omitted; refer to PDF]	0.5958	0.0331	0.5309	0.0090	0.5937
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 2 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.3626	0.0235	0.3118	0.0213	0.3750
[figure omitted; refer to PDF]	0.5039	0.0339	0.5100	0.0159	0.5000
[figure omitted; refer to PDF]	0.6709	0.0338	0.6086	0.0219	0.6250
Firm 3	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 3 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.3098	0.0296	0.3770	0.0175	0.3125
[figure omitted; refer to PDF]	0.3992	0.0285	0.4551	0.0108	0.4063
[figure omitted; refer to PDF]	0.5000	0.0237	0.5092	0.0122	0.5000
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 3 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.2012	0.0236	0.2647	0.0182	0.2500
[figure omitted; refer to PDF]	0.3534	0.0321	0.3434	0.0156	0.3750
[figure omitted; refer to PDF]	0.5028	0.0258	0.5078	0.0154	0.5000
Firm 4	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 4 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.2128	0.0170	0.2099	0.0186	0.2188
[figure omitted; refer to PDF]	0.3749	0.0231	0.3623	0.0199	0.3125
[figure omitted; refer to PDF]	0.4598	0.0315	0.4481	0.0100	0.4063
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 4 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.1238	0.0317	0.1118	0.0317	0.1250
[figure omitted; refer to PDF]	0.2219	0.0224	0.2399	0.0133	0.2500
[figure omitted; refer to PDF]	0.3221	0.0224	0.3182	0.0182	0.3750

Note: optimal percentage of product [figure omitted; refer to PDF] returning from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] via recovery center [figure omitted; refer to PDF] - optimal percentage of product [figure omitted; refer to PDF] returning from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] via recovery center [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ).

The fourth column of Tables 1 and 2 represents the mean values of the optimal strategies of all the firms obtained by PSO algorithm, using the following input parameters: [figure omitted; refer to PDF]

The mean values of the expected profits obtained by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) at the Cournot-Nash equilibrium, denoted by [figure omitted; refer to PDF] , are [figure omitted; refer to PDF] Hence, the mean value of the total expected profit of all firms is [figure omitted; refer to PDF] .

Figure 2 shows the values of the error function [figure omitted; refer to PDF] and Figure 3 shows all firms' fitness (i.e., [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ) from iteration [figure omitted; refer to PDF] to [figure omitted; refer to PDF] for a particular result generated by PSO algorithm.

Figure 2: Values of the error function for Example 1 by PSO algorithm.

[figure omitted; refer to PDF]

Figure 3: All firms' fitness for Example 1 by PSO algorithm.

[figure omitted; refer to PDF]

The sixth column of Tables 1 and 2 represents the mean values of the optimal strategies of all the firms obtained by GA, using the following input parameters: [figure omitted; refer to PDF]

The mean values of the expected profits obtained by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) at the Cournot-Nash equilibrium, denoted by [figure omitted; refer to PDF] , are [figure omitted; refer to PDF] Hence, the mean value of the total expected profit of all firms is [figure omitted; refer to PDF] .

Figure 4 shows the values of the error function [figure omitted; refer to PDF] and Figure 5 shows all firms' fitness (i.e., [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ) from iteration [figure omitted; refer to PDF] to [figure omitted; refer to PDF] for a particular result generated by GA.

Figure 4: Values of the error function for Example 1 by GA.

[figure omitted; refer to PDF]

Figure 5: All firms' fitness for Example 1 by GA.

[figure omitted; refer to PDF]

The eighth column of Tables 1 and 2 represents the optimal strategies of all the firms obtained by the Euler algorithm based on the variational inequality. The expected profits obtained by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) at the Cournot-Nash equilibrium, denoted by [figure omitted; refer to PDF] , are [figure omitted; refer to PDF] Hence, the total expected profit of all firms is [figure omitted; refer to PDF] .

Figure 6 shows all firms' fitness (i.e., [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ) obtained by the Euler algorithm from iteration [figure omitted; refer to PDF] to [figure omitted; refer to PDF] .

Figure 6: All firms' fitness for Example 1 by Euler algorithm.

[figure omitted; refer to PDF]

From the values of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] given in the previous paragraph, we observe that, on the average, the optimal expected profits of each of the firms obtained by each of the three algorithms are almost the same, with the total optimal expected profit (of all firms) obtained by the Euler algorithm being marginally higher than that obtained by PSO algorithm (by [figure omitted; refer to PDF] ) and by GA (by [figure omitted; refer to PDF] ). Thus, on the average, the efficiency of the PSO algorithm, the genetic algorithm, and the Euler algorithm (based on the variational inequality) is almost the same, in terms of the accuracy of the computed equilibrium.

From Figure 6, we observe that the Euler algorithm (based on the variational inequality) converges to the Cournot-Nash equilibrium of the CLSC network in [figure omitted; refer to PDF] iterations. Such convergence of the sequence of solutions obtained by the Euler algorithm to the optimal solution of variational inequality has been proved by Dupuis and Nagurney [11] and Nagurney and Zhang [31]. On the other hand, the convergence of the sequence of solutions obtained by PSO algorithm (which converges in [figure omitted; refer to PDF] iterations) and by GA (which converges in [figure omitted; refer to PDF] iterations) to the Cournot-Nash equilibrium is illustrated in Figure 2 to Figure 5. In terms of computational effort, the computational time required by the Euler algorithm to obtain the equilibrium of the CLSC network is [figure omitted; refer to PDF] seconds, whereas the average computational time required by PSO algorithm and GA is [figure omitted; refer to PDF] seconds and [figure omitted; refer to PDF] seconds, respectively. Thus, on the average, the computational time required by PSO algorithm is slightly longer than that required by the Euler algorithm but is much shorter than that required by GA to obtain the equilibrium. Thus, the PSO algorithm is just as efficient as the Euler algorithm but is more efficient than the genetic algorithm, in terms of the computational time required to obtain the equilibrium.

From this example, the two heuristics (PSO and GA) are effective in searching for a near-optimal solution despite the additional computational time as compared to the Euler algorithm. However, in reality, the production cost functions are not everywhere differentiable and, hence, the Euler algorithm is not applicable. In the next example, Example 2, some given functions are not everywhere differentiable and therefore we cannot find the Cournot-Nash equilibrium of the CLSC network by the Euler algorithm. PSO algorithm and GA, which do not require the gradient information of both the objective functions and the constraint functions, can still obtain the Cournot-Nash equilibrium efficiently.

Example 2 (comparison of the efficiencies of PSO algorithm and GA for solving problems with nonsmooth cost functions).

It is the same as Example 1, except that the production cost functions [figure omitted; refer to PDF] in Example 1 are now being replaced by [figure omitted; refer to PDF] (In other words, the differentiable production cost functions in Example 1 are now being replaced by production cost functions which are not everywhere differentiable.) Thus, the main purpose of this example is to test the efficiency of PSO algorithm and GA for finding the Cournot-Nash equilibrium for problems involving nondifferentiable cost functions.

With the same input parameters as those used in Example 1, we run PSO algorithm and GA to obtain the near-optimal strategies. All the optimal strategies in the forward logistics and the reverse logistics at the Cournot-Nash equilibrium are depicted in Tables 3 and 4, respectively.

The mean values of the expected profits obtained by firm [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) at the Cournot-Nash equilibrium, denoted by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, are [figure omitted; refer to PDF] Hence, the mean value of the total expected profit of all firms obtained by PSO algorithm is [figure omitted; refer to PDF] and that obtained by GA is [figure omitted; refer to PDF] .

From the values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] given in the previous paragraph, we observe that, on the average, the optimal expected profits of each of the firms obtained by each of the two algorithms are almost the same, with the total optimal expected profit (of all the firms) obtained by PSO algorithm being marginally higher (by [figure omitted; refer to PDF] ) than that obtained by GA.

Table 3: Comparison of the optimal strategies obtained by PSO algorithm and GA in the forward logistics of Example 2.

			PSO	GA
			Mean	Stdev.	Mean	Stdev.
Firm 1	Optimal quantity of product 1 supplied by firm 1 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	13.0209	0.3023	14.4351	0.7513
[figure omitted; refer to PDF]	14.0660	0.3063	13.3885	0.4320
[figure omitted; refer to PDF]	13.2913	0.2571	13.5359	0.6898
Optimal quantity of new manufactured product 1		28.3783	0.3691	29.3675	0.2781
Optimal quantity of product 2 supplied by firm 1 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	10.6559	0.2370	10.6977	0.5947
[figure omitted; refer to PDF]	10.1936	0.3070	11.1715	0.7511
[figure omitted; refer to PDF]	11.7843	0.3214	10.9480	0.3460
Optimal quantity of new manufactured product 2		20.6338	0.3454	20.8259	0.2688
Firm 2	Optimal quantity of product 1 supplied by firm 2 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	13.0987	0.1915	12.7234	0.6973
[figure omitted; refer to PDF]	12.9814	0.2434	13.7705	0.0993
[figure omitted; refer to PDF]	12.9200	0.2846	13.5463	0.2621
Optimal quantity of new manufactured product 1		27.0000	0.4155	28.0405	0.6897
Optimal quantity of product 2 supplied by firm 2 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	11.3123	0.3327	9.8867	0.7544
[figure omitted; refer to PDF]	10.6753	0.3787	11.4666	0.9035
[figure omitted; refer to PDF]	10.3289	0.3804	11.2320	0.8300
Optimal quantity of new manufactured product 2		20.3165	0.3663	20.5914	0.5499
Firm 3	Optimal quantity of product 1 supplied by firm 3 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	12.5400	0.4467	12.9380	0.6420
[figure omitted; refer to PDF]	13.0791	0.4328	13.0966	0.8916
[figure omitted; refer to PDF]	12.7047	0.5142	12.0838	0.3959
Optimal quantity of new manufactured product 1		26.3238	0.3406	26.1193	0.7190
Optimal quantity of product 2 supplied by firm 3 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	10.4219	0.4377	10.7817	0.4922
[figure omitted; refer to PDF]	10.4076	0.3780	9.8720	0.3852
[figure omitted; refer to PDF]	10.6253	0.2417	10.7148	0.3356
Optimal quantity of new manufactured product 2		19.4547	0.3097	19.3687	0.1064
Firm 4	Optimal quantity of product 1 supplied by firm 4 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	12.5708	0.3300	11.2457	0.8963
[figure omitted; refer to PDF]	13.3946	0.5604	13.1968	0.4776
[figure omitted; refer to PDF]	12.1992	0.2689	13.2656	0.0695
Optimal quantity of new manufactured product 1		26.1646	0.5159	25.7109	0.7179
Optimal quantity of product 2 supplied by firm 4 to demand market [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	10.5663	0.3677	11.0721	0.6788
[figure omitted; refer to PDF]	10.0228	0.3526	10.3900	0.6832
[figure omitted; refer to PDF]	10.7935	0.3971	11.1403	0.8083
Optimal quantity of new manufactured product 2		19.3827	0.2282	19.6067	0.7179

Table 4: Comparison of the optimal strategies obtained by PSO algorithm and GA in the reverse logistics of Example 2.

			PSO	GA
			Mean	Stdev.	Mean	Stdev.
Firm 1	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 1 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.5709	0.0269	0.5151	0.0110
[figure omitted; refer to PDF]	0.5927	0.0250	0.5211	0.0226
[figure omitted; refer to PDF]	0.6977	0.0224	0.6428	0.0145
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 1 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.5790	0.0317	0.5413	0.0043
[figure omitted; refer to PDF]	0.6262	0.0218	0.6666	0.0135
[figure omitted; refer to PDF]	0.7497	0.0204	0.7562	0.0134
Firm 2	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 2 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.4084	0.0352	0.4199	0.0152
[figure omitted; refer to PDF]	0.5694	0.0244	0.5242	0.0190
[figure omitted; refer to PDF]	0.5859	0.0313	0.5452	0.0180
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 2 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.4000	0.0326	0.3517	0.0158
[figure omitted; refer to PDF]	0.5401	0.0231	0.5403	0.0182
[figure omitted; refer to PDF]	0.6085	0.0366	0.6510	0.0066
Firm 3	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 3 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.3107	0.0235	0.3166	0.0080
[figure omitted; refer to PDF]	0.4213	0.0175	0.4384	0.0138
[figure omitted; refer to PDF]	0.5590	0.0317	0.5069	0.0249
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 3 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.2819	0.0215	0.2287	0.0071
[figure omitted; refer to PDF]	0.3262	0.0318	0.3523	0.0152
[figure omitted; refer to PDF]	0.5463	0.0176	0.5463	0.0229
Firm 4	Optimal percentage of product 1 returning from demand market [figure omitted; refer to PDF] to firm 4 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.2110	0.0315	0.2177	0.0143
[figure omitted; refer to PDF]	0.3224	0.0293	0.3159	0.0152
[figure omitted; refer to PDF]	0.4405	0.0297	0.4358	0.0094
Optimal percentage of product 2 returning from demand market [figure omitted; refer to PDF] to firm 4 via recovery center [figure omitted; refer to PDF]	[figure omitted; refer to PDF]	0.1726	0.0214	0.1219	0.0080
[figure omitted; refer to PDF]	0.2561	0.0221	0.2699	0.0220
[figure omitted; refer to PDF]	0.3390	0.0334	0.3169	0.0162

Note: optimal percentage of product [figure omitted; refer to PDF] returning from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] via recovery center [figure omitted; refer to PDF] - optimal percentage of product [figure omitted; refer to PDF] returning from demand market [figure omitted; refer to PDF] to firm [figure omitted; refer to PDF] via recovery center [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ).

Figures 7 and 9 show the values of the error functions obtained by PSO algorithm and GA, respectively, whereas Figures 8 and 10 show all firms' fitness obtained by the above two algorithms, respectively. From Figures 7 and 9, we observe that PSO algorithm and GA converge to the Cournot-Nash equilibrium of the CLSC network in [figure omitted; refer to PDF] iterations and [figure omitted; refer to PDF] iterations, respectively. In terms of computational effort, the average computational time required by PSO algorithm and GA is [figure omitted; refer to PDF] seconds and [figure omitted; refer to PDF] seconds, respectively. Hence, the average computational time required by the PSO algorithm to obtain the Cournot-Nash equilibrium of the CLSC network is much shorter than that required by the genetic algorithm. Similar to the conclusion of Example 1, the PSO algorithm is more efficient than the genetic algorithm, in terms of computational effort.

Figure 7: Values of the error function for Example 2 by PSO algorithm.

[figure omitted; refer to PDF]

Figure 8: All firms' fitness for Example 2 by PSO algorithm.

[figure omitted; refer to PDF]

Figure 9: Values of the error function for Example 2 by GA.

[figure omitted; refer to PDF]

Figure 10: All firms' fitness for Example 2 by GA.

[figure omitted; refer to PDF]

Remark 3.

From Examples 1 and 2, we conclude that as long as the given functions are continuous (it does not matter whether they are differentiable everywhere or not), on the average, the efficiency of the PSO algorithm and the genetic algorithm is almost the same, in terms of the accuracy of the computed equilibrium; however, the PSO algorithm is more efficient than the genetic algorithm, in terms of the computational time required to obtain the equilibrium. This difference in efficiencies between the two algorithms in terms of speed is probably due to the fact that genetic algorithms are more suitable for solving combinatorial problems than problems with a continuous solution space. (Note that the problem of finding the Cournot-Nash equilibrium of the CLSC network in our paper is transformed into solving sequences of optimization problems with a continuous solution space in Section 4.)

Remark 4.

Since the production cost functions in this example are continuous but not everywhere differentiable, we cannot get the variational inequality formulation of this example and thus cannot obtain the Cournot-Nash equilibrium by the variational inequality method. Thus, PSO algorithm and GA of this paper are particularly useful for solving this nonsmooth problem, because they do not require the gradient information of both the objective functions and the constraint functions.

6. Conclusion and Suggestions for Further Studies

In this paper, we develop stochastic multiproducts closed-loop supply chain network equilibrium model involving oligopolistic competition for multiproducts and their product flow routings in the entire supply chain. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework under a stochastic environment. Since the problem of finding the Cournot-Nash for the oligopolistic competition CLSC network model in our paper can be transformed into solving sequences of optimization problems with a continuous solution space, both the PSO algorithm and the genetic algorithm can be used for finding the above equilibrium. The numerical results show that when all the given functions are differentiable, the efficiencies of the PSO algorithm, the genetic algorithm, and an algorithm based on variational inequality are almost the same, in terms of the accuracy of the computed equilibrium. However, the PSO algorithm is just as efficient as the algorithm based on variational inequality but more efficient than the genetic algorithm, in terms of the computational time required to obtain the equilibrium. For problems involving nondifferentiable cost functions, the numerical results show that the PSO algorithm and the genetic algorithm can still find the Cournot-Nash equilibrium efficiently, but the algorithm based on variational inequality is not efficient. Thus, we conclude that both the PSO algorithm and the genetic algorithm can solve optimization problems with a continuous solution space efficiently, but the PSO algorithm is faster than the genetic algorithm. In the future, we would like to apply the PSO algorithm and the genetic algorithm to solve problems involving a multiperiod CLSC network under oligopolistic competition among firms.

Acknowledgments

This research was supported by the General Research Fund (Project no. Poly U 5405/11 H) of the Research Grants Council, Hong Kong, and the Research committee of the Hong Kong Polytechnic University. It was also supported by the National Natural Science Foundation of China (no. 71371102) and the Natural Science Foundation of Shandong Province of China (ZR2013GQ007).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Appendix

Appendices

A. The Crossover Operation and the Mutation Operation

In the following crossover and mutation operation, the term "chromosome" is used to represent the strategy vector of firm [figure omitted; refer to PDF] (denoted by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] ).

Crossover Operation. Consider the following.

Step 1.

Among all the chromosomes in a given population of size [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] chromosome as parents (where [figure omitted; refer to PDF] ), with the probability of each chromosome being selected equal to [figure omitted; refer to PDF] .

Step 2.

We group all the selected parents in Step [figure omitted; refer to PDF] into pairs and denoted them by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then for each pair of parents, we generate a random number, denoted by [figure omitted; refer to PDF] , from the interval [figure omitted; refer to PDF] and create a pair of children according to these formulae: [figure omitted; refer to PDF] Replace these [figure omitted; refer to PDF] parents by their children to obtain the new population.

Mutation Operation. Consider the following.

Step 1.

Among all the chromosomes in a given population of size [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] chromosome as parents (where [figure omitted; refer to PDF] ), with the probability of each chromosome being selected equal to [figure omitted; refer to PDF] .

Step 2.

Denote each selected parent in Step [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, for each [figure omitted; refer to PDF] , we choose a feasible direction [figure omitted; refer to PDF] and create a child according to this formula: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a random number generated in the interval [figure omitted; refer to PDF] . Replace these [figure omitted; refer to PDF] parents by their children to obtain the new population.

B. The Theorem of Variational Inequality Method

Theorem C.

Suppose that Assumptions A and B are satisfied, and suppose that the expected profit function [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) defined by (28) is differentiable with respect to the decision variable [figure omitted; refer to PDF] defined by (4). Then the strategy vector of all firms [figure omitted; refer to PDF] is a Cournot-Nash equilibrium of the CLSC network defined by Definition 3 if there exist [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , such that the following variational inequality holds: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is as defined in (39) and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are Lagrangian multipliers of constraints (30), (31), (32), and (33), respectively.

Proof.

The proof of Theorem C follows from that of Theorem [figure omitted; refer to PDF] of [8].

Variational inequality has a standard form.

Find [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF] and the component of [figure omitted; refer to PDF] corresponding to the variable [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is [figure omitted; refer to PDF]

C. Euler Algorithm

Euler algorithm is induced by the general iterative scheme of [11]. At iteration [figure omitted; refer to PDF] of the Euler algorithm, one computes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the projection on the set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the function corresponding to the variational inequality in Appendix B.