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

Abstract

We develop an equilibrium model of a closed-loop supply chain (CLSC) network with multiproducts, uncertain demands, and returns. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework under a stochastic environment. To satisfy the demands, we use two different channels: manufacturing new products and remanufacturing returned products through recycling used components. Since both the demands and product returns are uncertain, we consider two types of risks: overstocking and understocking of multiproducts in the forward supply chain. Then we set up the Cournot-Nash equilibrium conditions of the CLSC network whereby we maximize every oligopolistic firm's expected profit by deciding the production quantities of each new product as well as the path flows of each product on the forward supply chain. Furthermore, we formulate the Cournot-Nash equilibrium conditions of the CLSC network as a variational inequality and prove the existence and the monotonicity of the variational inequality. Finally, numerical examples are presented to illustrate the efficiency of our model.

1. Introduction

The competitive supply chain network models have been widely studied in the past decade. For instance, Nagurney et al. [1] first considered an equilibrium model of a competitive forward supply chain network which consisted of a lot of decision makers. Dong et al. [2] constructed an equilibrium model of a forward supply chain network under the assumption that the demands were uncertain. Nagurney and Toyasaki [3] constructed an equilibrium model of a reverse supply chain network for the optimal management of the electronic wastes. Hammond and Beullens [4] expanded the work of [1, 3] by presenting some insights of how closed-loop supply chain (CLSC) network equilibrium was achieved under WEEE (the waste electrical and the waste electrical and electronic equipment) directive legislation. Yang et al. [5] constructed an equilibrium model of a CLSC network consisting of raw materials, suppliers, and recovery centers besides manufacturers and retailers. In all the above papers [1–5], the equilibrium conditions were obtained by the theory of variational inequality; the variational inequalities in these papers were solved by the modified projection method proposed by Korpelevich [6].

The CLSC integrating the forward and reverse supply chain is important in real world applications due to government legislations (such as the paper recycling directive and WEEE within the European Union [7]). Moreover, the process used in the CLSC to recycle used products (such as papers, glass, building wastes, electric, and electronic equipment) for minimizing resource wastage also leads to people’s understanding of the important concept of the green supply chain management (GSCM). A comprehensive review of the CLSC management and the GSCM can be found in Gupta and Palsule-Desai [8], Sheu and Talley [9], and Seuring [10].

Competition and monopoly are two opposing market forms in economics. In a competition, there are numerous firms supplying their goods to the markets and the market price is determined exogenously. In a monopoly, only one firm supplies its goods to the markets. Oligopoly is a market form between perfect competition and monopoly, in which the market is dominated by 𝑛 firms (oligopolists), where 𝑛 is a small number. The 𝑛 firms compete in the oligopolistic market by producing the same product or perfect substitutes for the maximization of their profits, considering the best reply of the other firms. (When 𝑛=2, the oligopoly is called a duopoly which was first proposed by Cournot [11].) Recently, Barbagallo and Cojocaru [12] considered the dynamic equilibrium of an oligopolistic market. Based on the time-dependent variational inequality, Barbagallo and Mauro [13] investigated an equilibrium model of an oligopolistic market dealing with production and demand excesses.

From the work of Brown et al. [14], it is clear that more firms nowadays are aware of the importance of integrating the supply chain as a whole, consisting of all the marketing activities of all the competitors. Thus, several forward supply chain networks involving oligopolistic competition among firms have been developed. For instance, Nagurney [15] considered a design problem in which oligopolistic firms competed in a Cournot-Nash [16, 17] framework. Masoumi et al. [18], Nagurney and Yu [19], and Yu and Nagurney [20] developed equilibrium models involving oligopolistic competition among pharmaceutical firms, fashion firms, and fresh food firms, respectively. The equilibrium conditions of [15, 18–20] were also obtained by the theory of variational inequality; the variational inequalities in these papers were solved by the Euler method [21]. However, there are very few papers in the literature dealing with CLSC network involving oligopolistic competition among firms and none of these papers consider oligopolistic competition in a stochastic environment.

The major weakness of the literature concerning the CLSC network model is that only deterministic demands are considered. Research work on stochastic CLSC network model began only in the last few years. For instance, Qiang et al. [22] investigated an equilibrium model of a CLSC network involving perfect competition and uncertainties in demands, but not involving uncertainties in returns. Shi et al. [23, 24] constructed mathematical models of closed-loop manufacturing systems, in which both the demands and the returns were uncertain and price sensitive. However, the closed-loop systems in [23, 24] were not CLSC networks and did not involve oligopolistic competition among firms.

In this paper, we provide an innovative framework to study the effects of oligopolistic competition on a CLSC network and construct an equilibrium model of a multiproduct CLSC network involving oligopolistic competition among firms, by allowing uncertain demands and returns for all products. To satisfy these demands, we use two different channels: manufacturing new products and remanufacturing returned products through recycling used components. Since the demands and returns of all products are uncertain, we consider two types of risks: overstocking and understocking of multiproducts in the forward supply chain. Then we can set up the Cournot-Nash equilibrium conditions of the CLSC network whereby we maximize every oligopolistic firm’s expected profit by deciding the production quantities of each new product as well as the path flows of each product on the forward supply chain. Since the CLSC network model constructed in the above manner takes into consideration all those important issues such as competition among oligopolistic firms with multiproduct as well as uncertainty in both demands and returns, the CLSC model developed in this paper has more practical use than most of them in the literature, such as those developed in [4, 5, 22].

The rest of this paper is organized as follows. Section 2 develops the oligopolistic CLSC network model with multiproducts and uncertain demands and returns and constructs the Cournot-Nash equilibrium conditions. Section 3 gives the variational inequality formulations of our model. We then prove the existence and the monotonicity results for our model in Section 4. Section 5 presents an algorithm for finding the Cournot-Nash equilibrium. In Section 6, numerical examples are presented to test the effectiveness of the proposed algorithm. Sensitivity analysis is also conducted to illustrate how the change in a parameter can affect the firms’ strategies and their respective expected profits. Section 7 presents our summary and conclusion.

2. The Multiproduct CLSC Network Model under Oligopolistic Competition among Firms

We consider a CLSC network consisting of 𝐼 oligopolistic firms who compete noncooperatively. Each firm 𝑖 (𝑖=1,…,𝐼) manufactures 𝐽 products and has two different channels to meet the demands: manufacturing new products and remanufacturing returned products through recycling used components. Both the demands and their returns for all products are uncertain. It is assumed that all the demands and all the returns for each product are random variables with known probability distributions. The problem is to maximize all the oligopolistic firms’ expected profits by deciding the production quantities of each new product as well as the path flows of each product on the forward supply chain.

The topology of this network is shown in Figure 1. From Figure 1, the CLSC network consists of the forward supply chain network and the reverse supply chain network. Each link in Figure 1 represents an activity of the CLSC network. Consider any firm 𝑖 (𝑖=1,…,𝐼) in the CLSC network. The economic activities in the forward supply chain network can be described as follows.(1)Firm 𝑖 distributes raw materials or recycled products to each of its manufacturers (denoted by 𝑀^𝑖₁,…,𝑀^𝑖_𝑛𝑖𝑀).(2)The manufacturers then produce new products or remanufactured products and distribute them to each of the distribution centers of firm 𝑖 (denoted by 𝐷^𝑖₁,…,𝐷^𝑖_𝑛𝑖𝐷).(3)The distribution centers then distribute these products to each of the demand markets (denoted by 𝑅₁,…,𝑅_𝑛𝑅).

Figure 1 

The CLSC network topology.

The economic activities of firm 𝑖 in the reverse supply chain network can be described as follows.(1)The demand markets send a fraction of the used products to each of the recovery centers of firm 𝑖 (denoted by 𝐶^𝑖₁,…,𝐶^𝑖_𝑛𝑖𝐶). (The remainder of the used products will be sent to the landfill for disposition.)(2)The recovery centers then recycle the used products and send the recycled products to firm 𝑖 for remanufacturing.

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 loop in Figure 1.

We first give a few basic assumptions, which are common assumptions in the CLSC literature (see [4, 23] for details).

Assumption A. (A1) The demand for product 𝑗 (𝑗=1,…,𝐽) of firm 𝑖 (𝑖=1,…,𝐼) at demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅), denoted by 𝑑_𝑖𝑗𝑘, is a random variable. Any two demands are independent of each other.
(A2) The return of product 𝑗 (𝑗=1,…,𝐽) from demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅) to firm 𝑖 (𝑖=1,…,𝐼), denoted by 𝑟̂_𝑘𝑗𝑖, is a random variable. Any two returns are independent of each other.
(A3) The qualities of new products and remanufactured products are exactly the same.
(A4) A fraction of the used products will be collected by the firms for remanufacturing; the uncollected used products will be sent from the market centers to the landfill for disposition. The firm will incur a fee of 𝜌 per unit for the disposal of these used products at the landfill site.
(A5) For the collected products mentioned in (A4), the firm will incur a cost of 𝜌_𝑗^Re per item for the purchase of the returned product 𝑗 (𝑗=1,…,𝐽). It is also assumed that all returns are remanufacturable.
(A6) The quantity of new product 𝑗 (𝑗=1,…,𝐽) manufactured by firm 𝑖 (𝑖=1,…,𝐼) cannot exceed 𝑥, where 𝑥 is a positive integer.
In order to formulate the problem of finding the Cournot-Nash equilibrium for our CLSC network, we need to first describe the decision variables.
For each 𝑖 (𝑖=1,…,𝐼) and 𝑗 (𝑗=1,…,𝐽), let 𝑥^New_𝑖𝑗 denote the quantity of new product 𝑗 manufactured by firm 𝑖.
In the forward supply chain, any set of correlated links connecting the firms to the demand markets via the manufacturers and the distribution centers form a path 𝑝^𝑓. For each 𝑖 (𝑖=1,…,𝐼) and 𝑘 (𝑘=1,…,𝑛_𝑅), let 𝑃^𝑖_𝑘 denote the set of all the forward paths from firm 𝑖 to demand market 𝑅_𝑘. Then for each 𝑝^𝑓∈𝑃^𝑖_𝑘, let 𝑥_𝑗𝑝𝑓 denote the nonnegative product flow of product 𝑗 (𝑗=1,…,𝐽) on the forward path 𝑝^𝑓.

Hence there are two types of decision variables for our problem, namely, the quantity of new product 𝑥^New_𝑖𝑗 and the product flow on the forward path 𝑥_𝑗𝑝𝑓.

Let 𝑋_𝑖𝑗≡𝑥_𝑗𝑝𝑓,𝑥^New_𝑖𝑗∣𝑝^𝑓∈𝑃^𝑖∈ℝ^𝑛𝑃𝑖₊⁺¹ (1) denote all the decision variables associated with firm 𝑖 and product 𝑗, where 𝑃^𝑖≡^𝑛𝑅_𝑘=1𝑃^𝑖_𝑘, (2) and 𝑛_𝑃^𝑖 denotes the cardinality of 𝑃^𝑖.

Let 𝑋_𝑖≡𝑋_𝑖1,…,𝑋_𝑖𝐽^𝑇∈𝐾_𝑖 (3) be the strategy vector representing the overall decision variables associated with firm 𝑖, where 𝐾_𝑖≡ℝ^{(𝑛𝑃𝑖}₊^+1)𝐽. Then 𝑋≡𝑋₁₁,…,𝑋_1𝐽,…,𝑋_𝑖1,…,𝑋_𝑖𝐽,…,𝑋_𝐼1,…,𝑋_𝐼𝐽^𝑇∈𝐾 (4) is the overall decision variables for the entire CLSC network, which is the strategies vector of all firms, where 𝐾≡ℝ^{∑𝐼𝑖=1(𝑛𝑃𝑖}₊^+1)𝐽.

Now, we consider all the different types of constraints for our CLSC network.

By virtue of (A6), we have for each 𝑖 (𝑖=1,…,𝐼) and 𝑗 (𝑗=1,…,𝐽) 𝑥^New_𝑖𝑗≤𝑥. (5)

Moreover, in the forward supply chain, the following inequality involving product 𝑗 (𝑗=1,…,𝐽) manufactured by firm 𝑖 (𝑖=1,…,𝐼) must hold: ^𝑛𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓≤𝑥^New_𝑖𝑗+𝐸𝑥^Re_𝑖𝑗, (6) where 𝑥^Re_𝑖𝑗=^𝑛𝑅_𝑘=1𝑟̂_𝑘𝑗𝑖 (7) and 𝑟̂_𝑘𝑗𝑖 is as defined in (A2).

Furthermore, in the reverse supply chain, the following flow inequality must hold: 𝐸𝑟̂_𝑘𝑗𝑖≤_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓. (8) The above inequality ensures that the expected value of the returned product 𝑗 (𝑗=1,…,𝐽) from demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅) to firm 𝑖 (𝑖=1,…,𝐼) cannot exceed the total amount of forward flows of the above product on any path connecting firm 𝑖 to demand market 𝑅_𝑘.

Now, we formulate the different cost functions for our CLSC network. We first formulate the expected penalty cost (due to excessive or insufficient supply) and the operational cost.

Consider the forward supply chain involving firm 𝑖 (𝑖=1,…,𝐼), product 𝑗 (𝑗=1,…,𝐽), and demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅). Let 𝑣_𝑖𝑗𝑘 denote the quantity of product 𝑗 supplied by firm 𝑖 to demand market 𝑅_𝑘. Then the following flow equation must hold: _{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓=𝑣_𝑖𝑗𝑘. (9)

Let 𝑑_𝑖𝑗𝑘 denote the demand associated with firm 𝑖, product 𝑗, and demand market 𝑅_𝑘. Let ℱ_𝑖𝑗𝑘 be the probability density function of 𝑑_𝑖𝑗𝑘. Then Pr𝑑_𝑖𝑗𝑘≤𝑥=^𝑥₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘, (10) where Prdenotes the probability.

Now, the quantity of product 𝑗 supplied by firm 𝑖 to demand market 𝑅_𝑘 cannot exceed the minimum of 𝑣_𝑖𝑗𝑘 and 𝑑_𝑖𝑗𝑘. In other words, the actual sale of these products is equal to min{𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘}.

Let Δ⁺_𝑖𝑗𝑘=max0,𝑣_𝑖𝑗𝑘−𝑑_𝑖𝑗𝑘,Δ⁻_𝑖𝑗𝑘=max0,𝑑_𝑖𝑗𝑘−𝑣_𝑖𝑗𝑘, (11) denote, respectively, the quantity of the overstocking and the understocking of goods associated with firm 𝑖, product 𝑗 and demand market 𝑅_𝑘.

The expected values of Δ⁺_𝑖𝑗𝑘, and Δ⁻_𝑖𝑗𝑘 are given by 𝐸Δ⁺_𝑖𝑗𝑘=^{𝑣𝑖𝑗𝑘}₀𝑣_𝑖𝑗𝑘−𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘,𝐸Δ⁻_𝑖𝑗𝑘=_𝑣^+∞_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘−𝑣_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘. (12)

Assume that the unit penalty incurred on firm 𝑖 due to excessive supply of product 𝑗 to demand market 𝑅_𝑘 is 𝜃⁺_𝑖𝑗𝑘 and the unit penalty incurred on firm 𝑖 due to insufficient supply of product 𝑗 to demand market 𝑅_𝑘 is 𝜃⁻_𝑖𝑗𝑘, where 𝜃⁺_𝑖𝑗𝑘≥0 and 𝜃⁻_𝑖𝑗𝑘≥0. Then, the total expected penalty cost incurred on firm 𝑖 associated with product 𝑗 and demand market 𝑅_𝑘 is given by 𝐸𝜃⁺_𝑖𝑗𝑘Δ⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘Δ⁻_𝑖𝑗𝑘=𝜃⁺_𝑖𝑗𝑘𝐸Δ⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘𝐸Δ⁻_𝑖𝑗𝑘. (13)

Let 𝐿^𝑓 be the set of all correlated links in the forward supply chain. Let 𝑢^𝑓_𝑗𝑎 denote the product flow of product 𝑗 on any link 𝑎 in the forward supply chain. Then, in the forward supply chain involving product 𝑗, the relationship between the product flow in a link and the product flow in a path is as follows: 𝑢^𝑓_𝑗𝑎=^𝐼𝑛_𝑖=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓𝛿_𝑎𝑝𝑓,∀𝑎∈𝐿^𝑓, (14) where the binary parameter 𝛿_𝑎𝑝𝑓 is used to indicate whether link 𝑎 is included in forward path 𝑝^𝑓 (𝛿_𝑎𝑝𝑓=1) or not included in forward path 𝑝^𝑓 (𝛿_𝑎𝑝𝑓=0). Let 𝑢^𝑓_𝑗 be the vector consisting of all the forward link flows for each product 𝑗.

Then, in the forward supply chain, the total operational cost of product 𝑗 on a link 𝑎 is a continuous function of all the forward link flows of product 𝑗 given by 𝑐̂_𝑗𝑎=𝑐̂_𝑗𝑎𝑢^𝑓_𝑗,∀𝑎∈𝐿^𝑓. (15)

Apart from the penalty costs given by (13) (due to excessive or insufficient supply) and the operational costs given by (15), firm 𝑖 (𝑖=1,…,𝐼) also incurs the following additional costs:(i)the cost related to the production of new product 𝑗 (𝑗=1,…,𝐽): 𝑓_𝑖𝑗𝑥^New_𝑖𝑗, (16) where 𝑓_𝑖𝑗 is a continuous function,(ii)the cost related to the production of the returned product 𝑗 (𝑗=1,…,𝐽): 𝑓^Re_𝑖𝑗𝑥^Re_𝑖𝑗, (17) where 𝑓^Re_𝑖𝑗 is a continuous function,(iii)the cost related to the purchase of returned product 𝑗 (𝑗=1,…,𝐽) from demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅) to firm 𝑖: 𝜌_𝑗^Re𝑟̂_𝑘𝑗𝑖, (18) where 𝜌_𝑗^Re is the purchase cost per item of these returned products,(iv)the shipping cost related to the transportation of returned product 𝑗 (𝑗=1,…,𝐽) from demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅) to firm 𝑖: 𝑐̂^Re_𝑘𝑗𝑖𝑟̂_𝑘𝑗𝑖, (19) where 𝑐̂^Re_𝑘𝑗𝑖 is the transportation cost per item of these returned products,(v)the cost related to the disposal of uncollected product associated with firm 𝑖, product 𝑗 (𝑗=1,…,𝐽), and demand market 𝑅_𝑘 (𝑘=1,…,𝑛_𝑅) at the landfill site: 𝜌_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓−𝑟̂_𝑘𝑗𝑖. (20)

By virtue of (13), (15), and (16)–(20), the total cost incurred on firm 𝑖 is given by ^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝐸𝜃⁺_𝑖𝑗𝑘Δ⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘Δ⁻_𝑖𝑗𝑘+^𝐽_𝑗=1𝑓_𝑖𝑗𝑥^New_𝑖𝑗+^𝐽_{𝑗=1𝑎∈𝐿𝑓𝑖}𝑐̂_𝑗𝑎𝑢^𝑓_𝑗+^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝜌_𝑗^Re𝐸𝑟̂_𝑘𝑗𝑖+^𝐽_𝑗=1𝐸𝑓^Re_𝑖𝑗𝑥^Re_𝑖𝑗+^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝑐̂^Re_𝑘𝑗𝑖𝐸𝑟̂_𝑘𝑗𝑖+𝜌^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝐸_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓−𝑟̂_𝑘𝑗𝑖, (21) where 𝐿^𝑓_𝑖 is the set of all correlated links in the forward supply chain of firm 𝑖 (𝑖=1,…,𝐼).

Now, we formulate the total revenue received by firm 𝑖 (𝑖=1,…,𝐼). In order to capture competition in the demands in the entire CLSC network, we assumed that, for each product 𝑗, the demand price function 𝜌_𝑖𝑗𝑘 (𝑗=1,…,𝐽, 𝑘=1,…,𝑛_𝑅) is a continuous function of all the demands of product 𝑗 in the entire CLSC network; that is, 𝜌_𝑖𝑗𝑘=𝜌_𝑖𝑗𝑘𝑑_𝑗, (22) where 𝑑_𝑗=𝑑_1𝑗1,…,𝑑_{1𝑗𝑛𝑅},…,𝑑_𝐼𝑗1,…,𝑑_{𝐼𝑗𝑛𝑅}^𝑇∈ℝ^{𝐼×𝑛𝑅}₊ (23) denotes the vector consisting of all the demands of product 𝑗 in the entire CLSC network. Masoumi et al. [18] and Nagurney and Yu [19] 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 𝑖 (𝑖=1,…,𝐼) is given by ^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘. (24)

By virtue of the revenue, the cost of the forward supply chain, and the cost of the reverse supply chain, the expected profit function of firm 𝑖 (𝑖=1,…,𝐼), denoted by 𝑈_𝑖, can be expressed as follows: 𝑈_𝑖=^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘−^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝐸𝜃⁺_𝑖𝑗𝑘Δ⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘Δ⁻_𝑖𝑗𝑘−^𝐽_𝑗=1𝑓_𝑖𝑗𝑥^New_𝑖𝑗−^𝐽_{𝑗=1𝑎∈𝐿𝑓𝑖}𝑐̂_𝑗𝑎𝑢^𝑓_𝑗−^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝜌_𝑗^Re𝐸𝑟̂_𝑘𝑗𝑖−^𝐽_𝑗=1𝐸𝑓^Re_𝑖𝑗𝑥^Re_𝑖𝑗−^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝑐̂^Re_𝑘𝑗𝑖𝐸𝑟̂_𝑘𝑗𝑖−𝜌^𝐽𝑛_𝑗=1^𝑅_𝑘=1𝐸_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓−𝑟̂_𝑘𝑗𝑖, (25) where (7), (9), and (14) hold for all decision variables defined by (4). By virtue of (12), (15)–(20), and (22), it is clear that 𝑈_𝑖 is a function of the strategies of all firms in the entire CLSC network. That is, 𝑈_𝑖=𝑈_𝑖(𝑋). (26)

Now, in order to define the Cournot-Nash equilibrium of the CLSC network, we need to consider the following constraints involving firm 𝑖 (𝑖=1,…,𝐼): 𝑥^New_𝑖𝑗≤𝑥,_{𝑛𝑅𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓≤𝑥^New_𝑖𝑗+𝐸𝑥^Re_𝑖𝑗,𝐸𝑟̂_𝑘𝑗𝑖≤_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓,𝑥_𝑗𝑝𝑓≥0∀𝑝^𝑓∈^𝐼_𝑖=1𝑃^𝑖,𝑥^New_𝑖𝑗≥0. (27)(28)(29)(30) Constraints (27), (28), and (29) are directly obtained from (5), (6), and (8), respectively; constraint (30) is due to the nonnegativity requirement imposed on all the decision variables.

There are 𝐼 oligopolistic firms in the market to form the CLSC network with perfect information shared by all the firms throughout the network. So the game is based on the oligopolistic Cournot pricing in a Cournot-Nash framework. Each firm in the CLSC network competes noncooperatively to maximize its own expected profit and selects its strategy vector till an equilibrium is established.

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

Definition 1 (a feasible strategy vector). For each 𝑖 (𝑖=1,…,𝐼), a strategy vector 𝑋_𝑖∈𝐾_𝑖 is said to be a feasible strategy vector, if 𝑋_𝑖 satisfies constraints (27)–(30).

Definition 2 (the set of all feasible strategy patterns). Let 𝑆 be the set of all feasible strategy patterns defined by 𝑆≡𝑋₁,𝑋₂,…,𝑋_𝐼∣𝑋_𝑖∈𝐾_𝑖andconstraints(27)--(30)holds}. (31)

Definition 3 (the Cournot-Nash equilibrium of the CLSC network). A feasible strategy pattern 𝑋^∗∈𝑆 constitutes a Cournot-Nash equilibrium of the CLSC network, if the following inequality holds for all 𝑖 and for all feasible strategy vectors 𝑋_𝑖: 𝑈_𝑖𝑋^∗_𝑖,𝑋^∗_𝑖≥𝑈_𝑖𝑋_𝑖,𝑋^∗_𝑖, (32) where 𝑋^∗_𝑖≡(𝑋^∗₁,…,𝑋^∗_𝑖−1,𝑋^∗_𝑖+1,…,𝑋^∗_𝐼).

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. The Variational Inequality Formulation

To guarantee the existence of the Cournot-Nash equilibrium of the CLSC network, the following additional assumption is held.

Assumption B. (B1) The total operational cost 𝑐̂_𝑗𝑎 (𝑎∈𝐿_𝑓, 𝑗=1,…,𝐽) defined by (15) is a convex and continuously differentiable function of 𝑢^𝑓_𝑗, the vector consisting of all the forward link flows of product 𝑗.
(B2) The production cost 𝑓_𝑖𝑗 (𝑖=1,…,𝐼, 𝑗=1,…,𝐽) is a convex and continuously differentiable function of 𝑥^New_𝑖𝑗, the new product 𝑗 manufactured by firm 𝑖.

Lemma 4. Suppose that Assumption A is satisfied, then the set of all feasible strategy patterns defined by Definition 2 is both a compact and convex subset of 𝐾.

Proof. The fact that 𝑆 is bounded follows easily from (30), (27), and (28). The fact that 𝑆 is closed and convex follows easily from (27), (28), (29), and (30). Hence, 𝑆 is both a compact and convex subset of 𝐾.

Lemma 5. Suppose that Assumption A and Assumption B are satisfied, then the expected profit function 𝑈_𝑖(⋅) (𝑖=1,…,𝐼) defined by (25) is continuously differentiable with respect to the decision variable 𝑋 defined by (4).

Proof. Firstly, from (9), (12), (13), (B1), and (14), it is clear that the first, second, and fourth terms of 𝑈_𝑖(⋅) defined by (25) are continuous functions of 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
By virtue of the fact that any two demands are independent of each other (from (A1)), the first term of 𝑈_𝑖(⋅) can be expressed as 𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘=^{𝑣𝑖𝑗𝑘}₀𝑑_𝑖𝑗𝑘𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘+_𝑣^+∞_𝑖𝑗𝑘𝑣_𝑖𝑗𝑘𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘, (33) where 𝐺𝑑_𝑖𝑗𝑘=₀^+∞⋯₀^+∞𝜌_𝑖𝑗𝑘𝑑₁₁₁,…,𝑑_{𝑖𝑗(𝑘−1)},𝑑_𝑖𝑗𝑘,𝑑_{𝑖𝑗(𝑘+1)},…,𝑑_{𝐼𝑗𝑛𝑅}×ℱ₁₁₁𝑑₁₁₁⋯ℱ_{𝑖𝑗(𝑘−1)}×𝑑_{𝑖𝑗(𝑘−1)}ℱ_{𝑖𝑗(𝑘+1)}𝑑_{𝑖𝑗(𝑘+1)}⋯ℱ_{𝐼𝑗𝑛𝑅}𝑑_{𝐼𝑗𝑛𝑅}d𝑑₁₁₁⋯d𝑑_{𝑖𝑗(𝑘−1)}d𝑑_{𝑖𝑗(𝑘+1)}⋯d𝑑_{𝐼𝑗𝑛𝑅}. (34) Thus, we obtain from (33) that 𝜕𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘𝜕𝑣_𝑖𝑗𝑘=_𝑣^+∞_𝑖𝑗𝑘𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘. (35) But, from (9), it is clear that 𝜕𝑣_𝑖𝑗𝑘𝜕𝑥_𝑗𝑝𝑓=1, (36) for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅). Thus, from (35) and (36), the first partial derivative of the first term of 𝑈_𝑖(⋅) becomes 𝜕𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘𝜕𝑥_𝑗𝑝𝑓=_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘, (37) for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
From (12), we have 𝜕𝐸Δ⁺_𝑖𝑗𝑘𝜕𝑣_𝑖𝑗𝑘=𝜕𝜕𝑣_𝑖𝑗𝑘^{𝑣𝑖𝑗𝑘}₀𝑣_𝑖𝑗𝑘−𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘=^{𝑣𝑖𝑗𝑘}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘,𝜕𝐸Δ⁻_𝑖𝑗𝑘𝜕𝑣_𝑖𝑗𝑘=𝜕𝜕𝑣_{𝑖𝑗𝑘𝑣}^+∞_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘−𝑣_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘=−_𝑣^+∞_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘. (38) Thus, from (13), (36), and (38), the first partial derivative of the second term of 𝑈_𝑖(⋅) becomes 𝜕𝐸𝜃⁺_𝑖𝑗𝑘Δ⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘Δ⁻_𝑖𝑗𝑘𝜕𝑥_𝑗𝑝𝑓=𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁻_{𝑖𝑗𝑘∑}^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘, (39) for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
From (37) and (39), it is clear that the first and second terms of 𝑈_𝑖(⋅) are continuously differentiable with respect to 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
From (B1) and (14), the fourth term of 𝑈_𝑖(⋅) is continuously differentiable with respect to 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
The third, fifth, sixth, and seventh terms of 𝑈_𝑖(⋅) are independent of 𝑥_𝑗𝑝𝑓 and the last term of 𝑈_𝑖(⋅) is a linear function of 𝑥_𝑗𝑝𝑓. Thus, all the terms in (25) are continuously differentiable with respect to 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
From (B2), it is clear that the third term of 𝑈_𝑖(⋅) is continuously differentiable with respect to 𝑥^New_𝑖𝑗 and all the other terms are independent of 𝑥^New_𝑖𝑗. Thus, all the terms in (25) are also continuously differentiable with respect to 𝑥^New_𝑖𝑗, for each 𝑗 (𝑗=1,…,𝐽). The proof is complete.

Lemma 6. The expected profit function 𝑈_𝑖(⋅) (𝑖=1,…,𝐼) defined by (25) is concave with respect to the decision variable 𝑋 defined by (4).

Proof. In order to prove that the first and second terms of 𝑈_𝑖(⋅) are concave with respect to the decision variable 𝑋 defined by (4), we calculate the second partial derivative of these terms with respect to the decision variable 𝑥_𝑗𝑝𝑓 (𝑗=1,…,𝐽, 𝑝^𝑓∈𝑃^𝑖_𝑘, 𝑘=1,…,𝑛_𝑅) as follows.
We obtain from (35) that 𝜕²𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘𝜕𝑣²_𝑖𝑗𝑘=−𝐺𝑣_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑣_𝑖𝑗𝑘. (40) Thus, from (36) and (40), for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅), the second partial derivative of the first term of 𝑈_𝑖(⋅) becomes 𝜕²𝐸𝜌_𝑖𝑗𝑘𝑑_𝑗min𝑣_𝑖𝑗𝑘,𝑑_𝑖𝑗𝑘𝜕𝑥²_𝑗𝑝𝑓=−𝐺𝑣_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑣_𝑖𝑗𝑘≤0. (41)
From (38), we have 𝜕²𝐸Δ⁺_𝑖𝑗𝑘𝜕𝑣²_𝑖𝑗𝑘=ℱ_𝑖𝑗𝑘𝑣_𝑖𝑗𝑘,𝜕²𝐸Δ⁻_𝑖𝑗𝑘𝜕𝑣²_𝑖𝑗𝑘=ℱ_𝑖𝑗𝑘𝑣_𝑖𝑗𝑘. (42) Thus, from (13), (36), and (42), for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅), the second partial derivative of the second term of 𝑈_𝑖(⋅) becomes 𝜕²𝐸𝜃⁺_𝑖𝑗𝑘Δ⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘Δ⁻_𝑖𝑗𝑘𝜕𝑥²_𝑗𝑝𝑓=𝜃⁺_𝑖𝑗𝑘+𝜃⁻_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑣_𝑖𝑗𝑘≥0. (43)
By virtue of (14), (15), and (B1), it is clear that (−𝑐̂_𝑗𝑎(𝑢^𝑓_𝑗)) is concave with respect to 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
Now, by virtue of (41), (43), the fact that the third, fifth, sixth, and seventh terms of 𝑈_𝑖(⋅) are independent of 𝑥_𝑗𝑝𝑓, the fact that the fourth term of 𝑈_𝑖(⋅) is concave with respect to 𝑥_𝑗𝑝𝑓, and the fact that the last term of 𝑈_𝑖(⋅) is a linear function of 𝑥_𝑗𝑝𝑓, we conclude that 𝑈_𝑖(⋅) is concave with respect to 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑘=1,…,𝑛_𝑅).
From (B2), it is clear that (−𝑓_𝑖𝑗(𝑥^New_𝑖𝑗)) is concave with respect to 𝑥^New_𝑖𝑗. Since (−𝑓_𝑖𝑗(𝑥^New_𝑖𝑗)) is the only term in (25) involving 𝑥^New_𝑖𝑗, we conclude that 𝑈_𝑖(⋅) is concave with respect to 𝑥^New_𝑖𝑗, for each 𝑗 (𝑗=1,…,𝐽).
Hence, we conclude that the expected profit function 𝑈_𝑖(⋅) (𝑖=1,…,𝐼) defined by (25) is concave with respect to the decision variable 𝑋 defined by (4). The proof is complete.

We now give the variational inequality formulations of the CLSC network Cournot-Nash equilibrium in the following theorem.

Theorem 7. 𝑋^∗∈𝑆 is a CLSC network Cournot-Nash equilibrium according to Definition 3 if and only if it satisfies the variational inequality: −^𝐼_𝑖=1∇_𝑋𝑖𝑈_𝑖𝑋^∗,𝑋_𝑖−𝑋^∗_𝑖≥0,∀𝑋∈𝑆, (44) where ⟨⋅,⋅⟩ denotes the inner product in the corresponding Euclidean space and ∇_𝑋𝑖𝑈_𝑖(𝑋) denotes the gradient of 𝑈_𝑖(𝑋) with respect to 𝑋_𝑖.

Proof. In view of the fact that the set of all feasible patterns 𝑆 is both a compact and convex subset of 𝐾 and 𝑈_𝑖(⋅) is concave and continuously differentiable with respect to the decision variable 𝑋 defined by (4) (from Lemmas 4–6), the proof of this theorem follows easily from [25].

Lemma 8. Variational inequality (44) for our model is equivalent to 𝐹𝑖𝑛𝑑𝑋^∗∈𝑆,𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐹𝑋^∗,𝑋−𝑋^∗≥0,∀𝑋∈𝑆, (45) where 𝑋∈ℝ^{(∑𝐼𝑖=1𝑛𝑝𝑖𝐽+𝐼𝐽)}, 𝐹(𝑋)=(𝐹₁(𝑋),𝐹₂(𝑋)), 𝐹₁(𝑋)∈ℝ^{(∑𝐼𝑖=1𝑛𝑝𝑖𝐽)}, 𝐹₂(𝑋)∈ℝ^𝐼𝐽 such that the component of 𝐹₁(𝑋) corresponding to the variable 𝑥_𝑗𝑝𝑓 (∀𝑝^𝑓∈𝑃^𝑖_𝑘,∀𝑖,∀𝑗,∀𝑘) is _{𝑎∈𝐿𝑓𝑖}𝜕𝑐̂_𝑗𝑎𝑢^𝑓_𝑗𝜕𝑥_𝑗𝑝𝑓+𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁻_{𝑖𝑗𝑘∑}^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘+𝜌−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘, (46) where 𝐺(𝑑_𝑖𝑗𝑘) is as defined in (34), and the component of 𝐹₂(𝑋) corresponding to the variable 𝑥^𝑁𝑒𝑤_𝑖𝑗 (∀𝑖,∀𝑗) is 𝜕𝑓_𝑖𝑗𝑥^𝑁𝑒𝑤_𝑖𝑗𝜕𝑥^𝑁𝑒𝑤_𝑖𝑗. (47)

Proof. Since the first, second, fourth, and last terms of (25) are the only terms involving 𝑥_𝑗𝑝𝑓, we obtain from (25), (37), and (39) that 𝜕𝑈_𝑖𝜕𝑥_𝑗𝑝𝑓=_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘+𝜃⁻_{𝑖𝑗𝑘∑}^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−_{𝑎∈𝐿𝑓𝑖}𝜕𝑐̂_𝑗𝑎𝑢^𝑓_𝑗𝜕𝑥_𝑗𝑝𝑓−𝜌 (48) for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑖=1,…,𝐼, 𝑘=1,…,𝑛_𝑅).
Since the third term of (25) is the only term involving 𝑥^New_𝑖𝑗, we have 𝜕𝑈_𝑖𝜕𝑥^New_𝑖𝑗𝜕=−𝑓_𝑖𝑗𝑥^New_𝑖𝑗𝜕𝑥^New_𝑖𝑗 (49) for each 𝑥^New_𝑖𝑗 (𝑖=1,…,𝐼, 𝑗=1,…,𝐽).
By virtue of (44), (48), and (49), we obtain variational inequality (45). Thus, the proof is complete.

4. Existence and Monotonicity Results

In this section, we establish the existence and monotonicity results of variational inequality (45), the solution of which is a CLSC network Cournot-Nash equilibrium of our model.

From Lemma 4, the feasible set of variational inequality (45) is both a compact and convex subset of 𝐾. From Lemma 5, the expected profit function 𝑈_𝑖(⋅) (𝑖=1,…,𝐼) defined by (25) is continuously differentiable with respect to the decision variable 𝑋 defined by (4). Thus, we have the following theorem.

Theorem 9 (existence). Variational inequality (45) admits at least one solution.

Proof. Since 𝑆 is compact and convex and 𝐹(𝑋) is continuous, Theorem 9 follows from [26, Theorem 3.1, page 12].

Theorem 10 (monotonicity). Suppose that Assumptions A and B are satisfied, then the vector function 𝐹(𝑋) of the variational inequality (45) is monotone; that is, 𝐹𝑋^′−𝐹𝑋^′′,𝑋^′−𝑋^′′≥0,∀𝑋^′,𝑋^′′∈𝑆. (50)

Proof. Let 𝑋^′∈𝑆 and 𝑋^′′∈𝑆. Variational inequality (50) has the following deduction: ⟨𝐹𝑋^′−𝐹𝑋^′′,𝑋^′−𝑋^′′⟩=^𝐼𝐽_𝑖=1^𝑛_𝑗=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘𝑎∈𝐿𝑓𝑖}𝜕𝑐̂_𝑗𝑎𝑢^𝑓′_𝑗𝜕𝑥^′_𝑗𝑝𝑓−_{𝑎∈𝐿𝑓𝑖}𝜕𝑐̂_𝑗𝑎𝑢^𝑓′′_𝑗𝜕𝑥^′′_𝑗𝑝𝑓×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓+^𝐼𝐽_𝑖=1^𝑛_𝑗=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥′𝑓𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓′′𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓+^𝐼𝐽_𝑖=1^𝑛_𝑗=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝜃⁻_𝑖𝑗𝑘−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥′𝑓𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁻_𝑖𝑗𝑘−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓′′𝑗𝑝}ℱ_𝑖𝑗𝑘×𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓+^𝐼𝐽_𝑖=1^𝑛_𝑗=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥′𝑓𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘+_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓′′𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘×ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓+^𝐼𝐽_𝑖=1𝑗=1𝜕𝑓_𝑖𝑗𝑥^New′_𝑖𝑗𝜕𝑥^New′_𝑖𝑗−𝜕𝑓_𝑖𝑗𝑥^New′′_𝑖𝑗𝜕𝑥^New′′_𝑖𝑗×𝑥^New′_𝑖𝑗−𝑥^New′′_𝑖𝑗, (51) where 𝐺(𝑑_𝑖𝑗𝑘) is as defined in (34).
By virtue of the fact that 𝑐̂_𝑗𝑎(𝑢^𝑓_𝑗) is a convex function (from (B1)), we obtain from (14) that _{𝐼𝐽𝑖=1}^𝑛_{𝑗=1𝑅𝑘=1𝑝𝑓∈𝑃𝑖𝑘𝑎∈𝐿𝑓𝑖}𝜕𝑐̂_𝑗𝑎𝑢^𝑓′_𝑗𝜕𝑥^′_𝑗𝑝𝑓−_{𝑎∈𝐿𝑓𝑖}𝜕𝑐̂_𝑗𝑎𝑢^𝑓′′_𝑗𝜕𝑥^′′_𝑗𝑝𝑓×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓≥0. (52) Since ^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘=𝑃𝑟𝑑_𝑖𝑗𝑘≤_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓,−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘=Pr𝑑_𝑖𝑗𝑘≤_{𝑝𝑓∈𝑃𝑖𝑘}𝑥_𝑗𝑝𝑓−1, (53) and the probability function Pr(𝑑_𝑖𝑗𝑘≤∑_{𝑝𝑓∈𝑃}^𝑖_𝑘𝑥_𝑗𝑝𝑓) is an increasing function of 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽), 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑖=1,…,𝐼 and 𝑘=1,…,𝑛_𝑅), we have ^𝐼𝐽_𝑖=1^𝑛_𝑗=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥′𝑓𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁺_𝑖𝑗𝑘^{∑𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓′′𝑗𝑝}₀ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓≥0,_{𝐼𝐽𝑖=1𝑛𝑗=1𝑅𝑘=1𝑝𝑓∈𝑃𝑖𝑘}𝜃⁻_𝑖𝑗𝑘−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥′𝑓𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘−𝜃⁻_𝑖𝑗𝑘−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓′′𝑗𝑝}ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓≥0. (54) From (37) and (41), we know that ∫_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓𝑗𝑝}𝐺(𝑑_𝑖𝑗𝑘)ℱ_𝑖𝑗𝑘(𝑑_𝑖𝑗𝑘)d𝑑_𝑖𝑗𝑘 is a decreasing function of 𝑥_𝑗𝑝𝑓, for each 𝑗 (𝑗=1,…,𝐽) and 𝑝^𝑓∈𝑃^𝑖_𝑘 (𝑖=1,…,𝐼, 𝑘=1,…,𝑛_𝑅). Hence, we have ^𝐼𝐽_𝑖=1^𝑛_𝑗=1^𝑅_{𝑘=1𝑝𝑓∈𝑃𝑖𝑘}−_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥′𝑓𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘+_∑^+∞_{𝑝𝑓𝑖𝑘∈𝑃𝑥𝑓′′𝑗𝑝}𝐺𝑑_𝑖𝑗𝑘ℱ_𝑖𝑗𝑘𝑑_𝑖𝑗𝑘d𝑑_𝑖𝑗𝑘×𝑥^′_𝑗𝑝𝑓−𝑥^′′_𝑗𝑝𝑓≥0. (55) By virtue of the fact that 𝑓_𝑖𝑗(𝑥^New_𝑖𝑗) is a convex function (from (B2)), we have _{𝐼𝐽𝑖=1𝑗=1}𝜕𝑓_𝑖𝑗𝑥^New′_𝑖𝑗𝜕𝑥^New′_𝑖𝑗−𝜕𝑓_𝑖𝑗𝑥^New′′_𝑖𝑗𝜕𝑥^New′′_𝑖𝑗×𝑥^New′_𝑖𝑗−𝑥^New′′_𝑖𝑗≥0. (56) Thus, we conclude that (51) is nonnegative. The proof is complete.

Lemma 11. Suppose that Assumptions A and B are satisfied. Furthermore, suppose that all the total operational costs 𝑐̂_𝑗𝑎 (𝑎∈𝐿^𝑓, 𝑗=1,…,𝐽) and all the production costs 𝑓_𝑖𝑗 (𝑖=1,…,𝐼, 𝑗=1,…,𝐽) are strictly convex functions, and then the vector function 𝐹(𝑋) of the variational inequality (45) is strictly monotone; that is, ∀𝑋^′, 𝑋^′′∈𝑆 with 𝑋^′≠𝑋^′′𝐹𝑋^′−𝐹𝑋^′′,𝑋^′−𝑋^′′>0. (57)

Proof. Let the strategy vector of all firms 𝑋 be written in the form: 𝑋=𝑋₁,𝑋₂, (58) where 𝑋₁=𝑥_𝑗𝑝𝑓∣𝑗=1,…,𝐽,𝑝^𝑓∈^𝐼_𝑖=1𝑃^𝑖∈ℝ^{∑𝐼𝑖=1(𝑛𝑃𝑖}₊^)𝐽,𝑋₂=𝑥^New_𝑖𝑗∣𝑖=1,…,𝐼,𝑗=1,…,𝐽∈ℝ₊^𝐼𝐽. (59)
Let 𝑋^′, 𝑋^′′ be the given strategy vectors such that 𝑋^′≠𝑋^′′. Then 𝑋^′₁≠𝑋₁^′′ or 𝑋^′₂≠𝑋₂^′′.
Case  1  (𝑋^′₁≠𝑋₁^′′). Since all the total operational costs 𝑐̂_𝑗𝑎 (𝑎∈𝐿^𝑓, 𝑗=1,…,𝐽) are strictly convex functions, it follows from (14) that the first term of (51) is positive.
Case  2 (𝑋^′₂≠𝑋₂^′′). Since all the production costs 𝑓_𝑖𝑗 (𝑖=1,…,𝐼, 𝑗=1,…,𝐽) are strictly convex functions, the last term of (51) is positive.
Thus, by using exactly the same argument as that given for the proof of Theorem 10, we obtain ⟨𝐹𝑋^′−𝐹𝑋^′′,𝑋^′−𝑋^′′⟩>0. (60) The proof is complete.

Theorem 12 (uniqueness). Under the conditions of Lemma 11, there is a unique solution of variational inequality (45).

Proof. Theorem 12 follows from [26, Theorem 1.4, page 84].

5. Computing the Cournot-Nash Equilibrium

In this section, we use the logarithmic quadratic proximal (LQP) prediction-correction method [27] for solving the variational inequality (45) (the LQP prediction-correction method is an iterative method). For the sake of convenience of the readers, we first give a brief description of the LQP prediction-correction method.

Consider the following variational inequality problem: Find𝑥^∗∈𝑆suchthat𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆, (61) with linear constraints: 𝑆=𝑥∈𝑅^𝑛∣𝐴𝑥≤𝑏,𝑥≥0, (62) where 𝑓 is a given continuously differentiable function in ℝ^𝑛 and 𝐴 is an 𝑚×𝑛 matrix and 𝑏∈ℝ^𝑚.

Then the above variational inequality problem can be transformed into the following variational inequality problem: Find𝑢^∗∈Ωsuchthat𝐹𝑢^∗,𝑢−𝑢^∗≥0,∀𝑢∈Ω, (63) where 𝑥𝜆(𝑢)=𝑓(𝑥)𝑢=,𝐹+𝐴^𝑇𝜆−𝐴𝑥+𝑏,Ω=ℝ^𝑛₊×ℝ^𝑚₊, (64) and 𝜆 is the Lagrange multiplier vector of the constraints 𝐴𝑥≤𝑏.

Let ℝ^𝑛₊₊ and ℝ^𝑚₊₊ denote the interior of ℝ^𝑛₊ and ℝ^𝑚₊, respectively. For any given 𝑥^𝑘∈ℝ^𝑛₊₊, 𝜆^𝑘∈ℝ^𝑚₊₊, 𝛽^𝑘>0, the new iterate (𝑥^𝑘+1,𝜆^𝑘+1) can be obtained by solving the following system of equations: 𝛽^𝑘𝑓(𝑥)+𝐴^𝑇𝜆+𝑥−𝑥^𝑘+𝜇𝑥^𝑘−𝑋²_𝑘𝑥⁻¹=0,𝜆=𝑃_ℝ^𝑚₊𝜆^𝑘+𝛽^𝑘𝜈(𝐴𝑥−𝑏), (65)(66) where 0<𝜇<1 is a given fixed parameter, 𝜈>0 is a positive number, 𝑋_𝑘=diag(𝑥^𝑘₁,𝑥^𝑘₂,…,𝑥^𝑘_𝑛), 𝑥⁻¹=(1/𝑥₁,1/𝑥₂,…,1/𝑥_𝑛), and 𝑃_ℝ⁺_𝑚[𝜆] denote the projection of 𝜆 on ℝ⁺_𝑚; that is, 𝑃_ℝ⁺_𝑚[𝜆]=argmin‖𝑌−𝜆‖∣𝑌∈ℝ⁺_𝑚. (67) Here the term (𝑥−𝑥^𝑘)+𝜇(𝑥^𝑘−𝑋²_𝑘𝑥⁻¹) is used to ensure that the new iterate 𝑥^𝑘+1 is not too far from 𝑥^𝑘. Thus the purpose of (65) is to keep 𝑥^𝑘+1 not too far from 𝑥^𝑘 and at the same time keep 𝑓(𝑥^𝑘+1)+𝐴^𝑇𝜆^𝑘+1 close to zero, whereas the purpose of (66) is to keep both 𝜆^𝑘+1≥0 and 𝐴𝑥^𝑘+1−𝑏≥0. However, system (65) and (66) cannot be easily solved simultaneously. To overcome this difficulty, in the prediction step of the LQP prediction-correction method, we first replace 𝑥 by the current iterate 𝑥^𝑘 in (66) so that an approximation of 𝜆, denoted by 𝜆^𝑘, can be easily obtained by solving 𝜆^𝑘=𝑃_ℝ⁺_𝑚𝜆^𝑘+𝛽^𝑘𝜈𝐴𝑥^𝑘−𝑏. (68) Then we substitute 𝜆=𝜆^𝑘 into (65) to obtain an approximation of 𝑥, denoted by 𝑥^𝑘. In other words, the prediction (𝑥^𝑘,𝜆^𝑘) can be obtained by solving the following system: 𝛽Prediction^𝑘𝑓(𝑥)+𝐴^𝑇𝜆+𝑥−𝑥^𝑘−𝜇𝑥^𝑘−𝑋²_𝑘𝑥⁻¹≈0,𝜆=𝑃_ℝ^𝑚₊𝜆^𝑘+𝛽^𝑘𝜈𝐴𝑥^𝑘−𝑏. (69) After we have obtained the prediction (𝑥^𝑘,𝜆^𝑘), the new iterate (𝑥^𝑘+1,𝜆^𝑘+1), called the corrector, can be obtained by solving the following system: Correction𝛼𝛽^𝑘𝑓𝑥^𝑘+𝐴^𝑇𝜆^𝑘+𝑥−𝑥^𝑘−𝜇𝑥^𝑘−𝑋²_𝑘𝑥⁻¹=0,𝜆=𝑃_ℝ^𝑚₊𝜆^𝑘+𝛼𝜈𝛽^𝑘𝐴𝑥^𝑘−𝑏, (70) where 𝛼 is the step size.

From the above discussion, it is obvious that there are two main advantages for choosing the LQP prediction-correction method for solving the variational inequality (45). From the computational point of view, each iteration of the LQP prediction-correction method consists of a prediction and a correction, both of which require very tiny computational load; thus the method is effectively applicable in practice. From the mathematical point of view, there are only two conditions to guarantee the convergence of the LQP prediction-correction method; namely, the function 𝐹(𝑋) in the variational inequality (45) is continuous and monotone and the solution set of the variational inequality (45) is nonempty. Since the existence and the monotonicity of the variational inequality (45) for our CLSC model have been proved in Theorems 9 and 10, respectively, we can easily use the LQP prediction-correction method for finding the Cournot-Nash equilibrium defined by Definition 3, which is equivalent to the solution of the variational inequality (45).

For the sake of using the above LQP prediction-correction method for solving the variational inequality (45), we need to reformulate the set of all feasible strategy patterns defined by Definition 2 as follows: 𝑆=𝑋∈ℝ^{(∑𝐼𝑖=1𝑛𝑝𝑖𝐽+𝐼𝐽)}∣𝐴𝑋≤𝑏,𝑋≥0, (71) where 𝐴∈ℝ^{(2+𝑛𝑅)𝐼𝐽×(∑𝐼𝑖=1𝑛𝑝𝑖𝐽+𝐼𝐽)}, 𝑏∈ℝ^{(2+𝑛𝑅)𝐼𝐽} are obtained from constraints (27)–(29).

Let 𝑒(𝑋,𝜆)≡𝑋−𝑃_ℝ^𝐼_{𝑛𝑝𝑖+(∑𝑖=1𝐽+𝐼𝐽)}𝑋−𝐹(𝑋)+𝐴^𝑇𝜆𝜆−𝑃_ℝ𝑅+^{(2+𝑛)𝐼𝐽}{[,𝜆+𝐴𝑋−𝑏]} (72) where the vector function 𝐹(𝑋) is as defined in (45), 𝜆∈ℝ^(2+𝑛𝑅₊^)𝐼𝐽 is the Lagrange multiplier vector of the linear inequality constraints 𝐴𝑋−𝑏≤0, and 𝑃_𝐾 denotes the projection on 𝐾. Choose ‖𝑒(𝑋,𝜆)‖<𝜀 as the stopping criterion.

Now we are in a position to describe Algorithm 13, which is a modified version of the LQP prediction-correction method described above.

Algorithm 13.  
Step  0. Let 𝛽⁰=1, 𝜈=1(>0), 𝜂=0.9(<1), 𝜇=0.1, 𝜎=2.0(>1) and 𝜀=10⁻⁸. Initialize 𝑋⁰∈ℝ^{(∑𝐼𝑖=1𝑛𝑝𝑖𝐽+𝐼𝐽)}, 𝜆⁰∈ℝ^(2+𝑛𝑅₊^)𝐼𝐽. Let 𝑘=0.
Step  1. If ‖𝑒(𝑋^𝑘,𝜆^𝑘)‖_∞≤𝜀, then stop; else, go to Step 2.
Step  2 (prediction step). Produce the predictor 𝑋^𝑘,𝜆^𝑘.
Step  2.1. Calculate 𝜆^𝑘 and 𝑋^𝑘. 𝜆^𝑘=𝑃_ℝ𝑅^{(2+𝑛)𝐼𝐽}𝜆^𝑘+𝛽^𝑘𝜈𝐴𝑋^𝑘,̂𝑋−𝑏𝑠≡1−𝜇^𝑘−𝛽^𝑘𝐹𝑋^𝑘+𝐴^𝑇𝜆^𝑘,𝑋_𝑙^𝑘=𝑠_𝑙̂+𝑠̂_𝑙²+4𝜇𝑋^𝑘_𝑙²2,𝑙=1,…,^𝐼_𝑖=1𝑛_𝑝𝑖𝐽+𝐼𝐽. (73)
Step  2.2. Update 𝛽^𝑘.
Let 𝑟^𝑘=𝜈𝜉^𝑘₁²+1+𝜇𝜉^𝑘₂²𝜈1−𝜇²𝜋^𝑘₁²+𝜈²1−𝜇𝜋^𝑘₂², (74) where 𝜉^𝑘₁=𝛽^𝑘(𝐹(𝑋^𝑘)−𝐹(𝑋^𝑘))∈ℝ^{∑𝐼𝑖=1𝑛𝑝𝑖𝐽+𝐼𝐽}, 𝜉^𝑘₂=𝛽^𝑘𝐴(𝑋^𝑘−𝑋^𝑘)∈ℝ^{(2+𝑛𝑅)𝐼𝐽}, 𝜋^𝑘₁=𝑋^𝑘−𝑋^𝑘∈ℝ^{∑𝐼𝑖=1𝑛𝑝𝑖𝐽+𝐼𝐽},𝜋^𝑘₂=𝜆^𝑘−𝜆^𝑘∈ℝ^{(2+𝑛𝑅)𝐼𝐽}. (75)  If 𝑟^𝑘>𝜂, then 𝛽^𝑘=𝛽^𝑘×0.8/𝑟^𝑘; go back to Step 2.1; else, go to Step 3.
Step  3. Adjust 𝛽^𝑘 and 𝜈 as follows: 𝛽^𝑘+1=𝛽^𝑘×0.7𝑟^𝑘,if𝑟^𝑘𝛽≤0.5,_𝑘𝜉,otherwise,𝜈=0.5𝜈,if^𝑘1>4𝜉1+𝜇√𝜈^𝑘2,4𝜉2𝜈,if^𝑘1<𝜉1+𝜇√^𝑘2𝜈,𝜈,otherwise. (76)
Step  4. Calculate the step-size 𝛼^𝑘 in the correction step. 𝛼^𝑘=𝛾𝛼^𝑘∗𝛽^𝑘1−𝜇,1+𝜇 (77)  where 𝛼^𝑘∗=𝜋^𝑘₁+𝜉^𝑘₁^𝑇𝜋^𝑘₁+𝜈𝜋^𝑘₂+𝜉^𝑘₂^𝑇𝜋^𝑘₂×1+𝜇𝜋^𝑘₁+𝜉^𝑘₁^𝑇𝜋^𝑘₁+1+𝜇⁻¹𝜉^𝑘₁+𝜈𝜋^𝑘₂+𝜉^𝑘₂^𝑇𝜋^𝑘₂+𝜈⁻¹𝜉^𝑘₂⁻¹. (78)
Step  5 (correction step). Calculate the new iterate 𝑋^𝑘+1, 𝜆^𝑘+1. 𝑠̂≡1−𝜇𝑋^𝑘−𝛼^𝑘𝐹𝑋^𝑘+𝐴^𝑇𝜆^𝑘,𝑋_𝑙^𝑘+1=𝑠_𝑙̂+𝑠̂_𝑙²+4𝜇𝑋^𝑘_𝑙²2,𝑙=1,…,^𝐼_𝑖=1𝑛_𝑝𝑖𝜆𝐽+𝐼𝐽,_𝑘+1=𝑃_ℝ𝑅+^{(2+𝑛)𝐽𝐼}𝜆^𝑘+𝛼^𝑘𝜈𝐴𝑋^𝑘−𝑏. (79) Let 𝑘=𝑘+1; go to Step  1.
The convergence result for Algorithm 13 is stated in the following theorem.

Theorem 14 (convergence). Suppose that Assumptions A and B are satisfied, and then the sequence {𝑋^𝑘} generated by Algorithm 13 converges to some 𝑋^∗, where 𝑋^∗ is the solution of the variational inequality (45).

Proof. The proof follows easily from Theorems 9 and 10 and Theorem 4.2 of [27].

6. Numerical Examples

In this section, numerical examples are solved to illustrate the efficiency of Algorithm 13 for finding the Cournot-Nash equilibrium of the CLSC network.

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, which are modified from [18].

For product 1, the corresponding demand price functions associated with firm 𝑖 (𝑖=1,2,3,4) and demand market 𝑅_𝑘 (𝑘=1,2,3) are as follows: Firm1:𝜌_1𝑗1=−1.1𝑑_1𝑗1−0.9𝑑_2𝑗1−0.9𝑑_3𝑗1−0.9𝑑_4𝑗1𝜌+500,_1𝑗2=−0.9𝑑_1𝑗2−0.8𝑑_2𝑗2−0.8𝑑_3𝑗2−0.8𝑑_4𝑗2𝜌+500,_1𝑗3=−1.0𝑑_1𝑗3−0.9𝑑_2𝑗3−0.9𝑑_3𝑗3−0.9𝑑_4𝑗3+500;Firm2:𝜌_2𝑗1=−0.7𝑑_1𝑗1−1.1𝑑_2𝑗1−0.7𝑑_3𝑗1−0.7𝑑_4𝑗1𝜌+490,_2𝑗2=−0.7𝑑_1𝑗2−0.9𝑑_2𝑗2−0.7𝑑_3𝑗2−0.7𝑑_4𝑗2𝜌+490,_2𝑗3=−0.8𝑑_1𝑗3−1.2𝑑_2𝑗3−0.8𝑑_3𝑗3−0.8𝑑_4𝑗3+490;Firm3:𝜌_3𝑗1=−0.8𝑑_1𝑗1−0.8𝑑_2𝑗1−1.1𝑑_3𝑗1−0.8𝑑_4𝑗1𝜌+480,_3𝑗2=−0.8𝑑_1𝑗2−0.8𝑑_2𝑗2−0.9𝑑_3𝑗2−0.8𝑑_4𝑗2𝜌+480,_3𝑗3=−0.9𝑑_1𝑗3−0.9𝑑_2𝑗3−1.2𝑑_3𝑗3−0.9𝑑_4𝑗3+480;Firm4:𝜌_4𝑗1=−0.7𝑑_1𝑗1−0.7𝑑_2𝑗1−0.7𝑑_3𝑗1−1.0𝑑_4𝑗1𝜌+470,_4𝑗2=−0.7𝑑_1𝑗2−0.7𝑑_2𝑗2−0.7𝑑_3𝑗2−0.9𝑑_4𝑗2𝜌+470,_4𝑗3=−0.7𝑑_1𝑗3−0.7𝑑_2𝑗3−0.7𝑑_3𝑗3−1.1𝑑_4𝑗3+470. (80)

For product 2, the corresponding demand price functions associated with firm 𝑖 (𝑖=1,2,3,4) and demand market 𝑅_𝑘 (𝑘=1,2,3) are as follows: Firm1:𝜌_1𝑗1=−1.1𝑑_1𝑗1−0.9𝑑_2𝑗1−0.9𝑑_3𝑗1−0.9𝑑_4𝑗1𝜌+420,_1𝑗2=−0.9𝑑_1𝑗2−0.8𝑑_2𝑗2−0.8𝑑_3𝑗2−0.8𝑑_4𝑗2𝜌+420,_1𝑗3=−1.0𝑑_1𝑗3−0.9𝑑_2𝑗3−0.9𝑑_3𝑗3−0.9𝑑_4𝑗3+420;Firm2:𝜌_2𝑗1=−0.7𝑑_1𝑗1−1.1𝑑_2𝑗1−0.7𝑑_3𝑗1−0.7𝑑_4𝑗1𝜌+430,_2𝑗2=−0.7𝑑_1𝑗2−0.9𝑑_2𝑗2−0.7𝑑_3𝑗2−0.7𝑑_4𝑗2𝜌+430,_2𝑗3=−0.8𝑑_1𝑗3−1.2𝑑_2𝑗3−0.8𝑑_3𝑗3−0.8𝑑_4𝑗3+430;Firm3:𝜌_3𝑗1=−0.8𝑑_1𝑗1−0.8𝑑_2𝑗1−1.1𝑑_3𝑗1−0.8𝑑_4𝑗1𝜌+440,_3𝑗2=−0.8𝑑_1𝑗2−0.8𝑑_2𝑗2−0.9𝑑_3𝑗2−0.8𝑑_4𝑗2𝜌+440,_3𝑗3=−0.9𝑑_1𝑗3−0.9𝑑_2𝑗3−1.2𝑑_3𝑗3−0.9𝑑_4𝑗3+440;Firm4:𝜌_4𝑗1=−0.7𝑑_1𝑗1−0.7𝑑_2𝑗1−0.7𝑑_3𝑗1−1.0𝑑_4𝑗1𝜌+450,_4𝑗2=−0.7𝑑_1𝑗2−0.7𝑑_2𝑗2−0.7𝑑_3𝑗2−0.9𝑑_4𝑗2𝜌+450,_4𝑗3=−0.7𝑑_1𝑗3−0.7𝑑_2𝑗3−0.7𝑑_3𝑗3−1.1𝑑_4𝑗3+450. (81)

Assume that 𝑑_𝑖𝑗𝑘 is uniformly distributed in [0,𝜏_𝑖𝑗𝑘] with probability density function given by ℱ_𝑖𝑗𝑘(𝑥)=1𝜏_𝑖𝑗𝑘,if𝑥∈0,𝜏_𝑖𝑗𝑘,0,if𝑥∈𝜏_𝑖𝑗𝑘,+∞, (82) where for product 𝑗=1, 𝜏_1𝑗𝑘=28, 𝜏_2𝑗𝑘=27, 𝜏_3𝑗𝑘=26, 𝜏_4𝑗𝑘=25 (𝑘=1,2,3) and for product 𝑗=2, 𝜏_1𝑗𝑘=20, 𝜏_2𝑗𝑘=19, 𝜏_3𝑗𝑘=18, 𝜏_4𝑗𝑘=17 (𝑘=1,2,3).

Assume that 𝑟̂_𝑘𝑗𝑖 (the returned product 𝑗 (𝑗=1,2) from demand market 𝑅_𝑘 (𝑘=1,2,3) to firm 𝑖 (𝑖=1,2,3,4)) is uniformly distributed in [0,8].

Example 15. In this example, the demand price functions are as given in (80)-(81). The cost functions, which are modified from those given in [4], are as follows: 𝑐̂_𝑗𝑎𝑢^𝑓_𝑗=2𝑢^𝑓_𝑗𝑎²+𝑢^𝑓_𝑗𝑎,∀𝑎∈𝐿^𝑖_𝑓;𝑓_𝑖𝑗𝑥^New_𝑖𝑗=2.5𝑥^New_𝑖𝑗²+2𝑥^New_𝑖𝑗,𝑓^Re_𝑖𝑗𝑥^Re_𝑖𝑗=𝑥^Re_𝑖𝑗²+0.5𝑥^Re_𝑖𝑗, (83) where 𝑖=1,2,3,4, 𝑗=1,2, and 𝑘=1,2,3.
The values of the parameters are as follows: 𝜃⁺_𝑖𝑗𝑘 (unit penalty incurred on firm 𝑖 due to excessive supply of product 𝑗 to demand market 𝑅_𝑘) =20, 𝜃⁻_𝑖𝑗𝑘 (unit penalty incurred on firm 𝑖 due to insufficient supply of product 𝑗 to demand market 𝑅_𝑘) =20, 𝑐̂^Re_𝑘𝑗𝑖 (transportation cost per item of returned product 𝑗 from demand market 𝑅_𝑘 to firm 𝑖) =0.5, 𝜌_𝑗^Re (purchase cost per item of returned product 𝑗 from demand market 𝑅_𝑘 to firm 𝑖) =10, and 𝜌 (disposal fee per item of the used products at the landfill site) =10, where 𝑖=1,2,3,4, 𝑗=1,2, and 𝑘=1,2,3.

In Example 15, all the conditions of Theorem 12 are satisfied. We solve this example by Algorithm 13 (using Matlab) to obtain the Cournot-Nash equilibrium of the CLSC network. The Cournot-Nash equilibrium representing the optimal strategies of all the firms is presented in the fourth column of Table 1. The expected profits of firm 𝑖 (𝑖=1,2,3,4), denoted by 𝑈^∗_𝑖, at the Cournot-Nash equilibrium are 𝑈^∗₁=13093,𝑈^∗₂𝑈=12988,_∗3=12536,𝑈^∗₄=12483. (84)

Table 1 

Optimal strategies for all firms for Examples 15 and 16.

Here is a summary of the numerical results for Example 15.

For both Product 1 and Product 2, the situation of the optimal strategies at Cournot-Nash equilibrium is as follows.

Firm 1 should manufacture the largest amount of new products as well as provide the largest amount of forward product flows to each of the demand markets 𝑅₁, 𝑅₂, and 𝑅₃, followed by Firm 2, Firm 3, and Firm 4. Furthermore, Firm 1 should receive the largest amount of profit, followed by Firm 2, Firm 3, and Firm 4.

In the next example, we use the parameter 𝜌 to perform sensitivity analysis. By fixing the other parameters, we study the effect of changing the parameter 𝜌 on the Cournot-Nash equilibrium together with the optimal expected profits of all firms.

Example 16. Same as Example 15, except that 𝜌=10 in Example 15 is first replaced by 𝜌=20 and then by 𝜌=30 in this example. (In other words, we now increase the disposal fee per item by 100% and 200%). Thus, the main purpose of this example is to test the effect of increasing the disposal fee per item on the firms’ optimal strategies and the firms’ expected profits.

The Cournot-Nash equilibrium representing the optimal strategies of all the firms when 𝜌=20 is presented in the fifth column of Table 1, and the expected profits of firm 𝑖 (𝑖=1,2,3,4), denoted by 𝑈^∗_𝑖, at the Cournot-Nash equilibrium are 𝑈^∗₁=12701,𝑈^∗₂𝑈=12606,_∗3=12170,𝑈^∗₄=12127. (85)

The Cournot-Nash equilibrium representing the optimal strategies of all the firms when 𝜌=30 is presented in the sixth column of Table 1, and the expected profits of firm 𝑖 (𝑖=1,2,3,4), denoted by 𝑈^∗_𝑖, at the Cournot-Nash equilibrium are 𝑈^∗₁=12323,𝑈^∗₂𝑈=12237,_∗3=11816,𝑈^∗₄=11783. (86)

From the fourth, fifth, and sixth columns of Table 1 and the value of the expected profits given above, we observe the following.(i)When 𝜌 is increased, both the optimal product flows and the new manufactured products of each firm decrease.(ii)When 𝜌 is increased, the expected profit of each firm also decreases.

Remark 17. From the above two examples, we conclude that the change in disposal fee per item of used products not only affects the firms’ optimal strategies, but also their expected profits.

7. Conclusion and Suggestions for Further Studies

In this paper, we develop a multiproduct CLSC network equilibrium model with uncertain demands and product returns. This model belongs to the context of oligopolistic firms that compete noncooperatively in a Cournot-Nash framework under a stochastic environment. The Cournot-Nash equilibrium conditions are solved by using the LQP prediction-correction method. Finally, numerical examples are presented to test the efficiency of the algorithm for finding the Cournot-Nash equilibrium of the CLSC network. In the future, we would like to extend our method to solve a multiperiod CLSC network equilibrium model involving oligopolistic competition among firms.

Conflict of Interests

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

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).