Using game theory for analysing pricing models in closed-loop supply chain from short- and long-term perspectives

Closed-loop supply chain (CLSC) management is an environmental approach to supply chain management that aims to prevent hazardous material from entering the nature by means of creating a reverse flow. This paper studies the short- and long-term behaviour of agents in implementing the appropriate collecting strategy in a two-echelon CLSC. In short-term, based on the Stackelberg game, several novel pricing models for different collecting strategies are proposed and compared. Then, the optimal policies of the pricing decisions are determined for each model. The long-term behaviour of companies in implementing collecting process is examined by evolutionary game theory and the most stable strategy is selected. Furthermore, a numerical example is presented to compare the different collecting structures. Finally, a managerial insight is provided to indicate the effect of key parameters such as remanufacturing rate, marketing elasticity and government subsidies on selecting the appropriate strategy.


Introduction
Nowadays, with the rapid industrial growth, environmental issues such as global warming and depletion of natural resources have become extremely highlighted. Governments attempt to take the various policies and legislations such as imposing green tax in order to make manufacturers produce eco-friendly products (Chen and Hao forthcoming). Green supply chain management (GSCM) has emerged as a managerial principle to consider both environmental and economic issues during a product's life cycle (Min and Kim 2012;Sarkis 2014;Fahimnia et al. forthcoming). Reducing carbon emission during the production (Nagurney and Yu 2012;Zhao et al. 2012) and reducing waste materials are two of the practical examples of GSCM in the form of forward and reverse logistics, respectively. Closed-loop supply chain (CLSC), as a concept in GSCM, is extensively used to maximise the usage of materials in hand and also to prevent the excessive remnants of hazardous material entering the environment by creating a reverse flow (Yuan and Gao 2010;Yoo, Kim, and Park forthcoming).
CLSC has evolved from an individual process to a global system design (Guide and Van Wassenhove 2009). Primary studies are on remanufacturing products and reverse logistics. Traditionally, in the early work, planning of CLSCs has been through network design (Krikke 2001) and inventory control (van der Laan 1997) using a central decision-maker to optimise the overall performance of the system. Over time, CLSC has shifted to management processes and coordination in which adopting game theory models helps for a more realistic decision-making by supply chain players. There are a growing number of works on CLSC utilising game theory to model remanufacturing decisions. Downstream channel design issues are highlighted by Majumder and Groenevelt (2001). Savaskan, Bhattacharya, and Van Wassenhove (2004) propose three scenarios in which the used products can be collected by the manufacturer, the retailer or the third party. Several similar works have been presented in the literature in which the retailer (Xiang-yun and Jian-jun 2008) or the third party (Chung, Wee, and Yang 2008) performs the collection process. The interactions between new and remanufactured products are discussed extensively by Ferrer and Swaminathan (2006). Huang et al. (2013) investigate the competition between the retailer and the third party in remanufacturing the used products in a dual channel. All the aforementioned works consider the annual demand as a linear function of the price. However, in the real world the demand is sensitive to the marketing effort as well (Esmaeili et al. 2009;Xie and Wei 2009;Karray 2011). The effort in marketing is an expenditure spent in activities such as advertising, sales promotions and marketing research aiming at increasing the consumers' demand (Kotler 1994). By increasing the consumers' ecological awareness, the marketing paradigm has shifted to satisfying the consumers need by green activities such as green promotions, green labelling and remanufacturing (Ferrer and Whybark 2000;Peattie and Charter 2003).

Assumptions and notations
This section presents the assumptions and notations to facilitate understanding the model formulations presented in the paper.

Input parameters
k Constant value in the demand function (k > 0) α Price elasticity of the demand (α > 1) β Marketing expenditure elasticity of the demand (0 < β < 1, β + 1 < α) i The percentage of unit cost of an item held in inventory per year. This is further multiplied by 0.5 to use an average of products in inventory. τ Remanufacturing rate (0 ≤ τ ≤ 1) O M Manufacturer's set-up cost ($/set-up) O R Retailer's ordering and logistics cost ($/order) C L The cost of remanufacturing logistics ($/shipment) C m Unit cost of manufacturing a new product ($/unit) C r Unit cost of remanufacturing a used product ($/unit) C s Unit sales cost for the retailer ($/unit) b The cost charged by the retailer from the manufacturer for collecting ($/unit) t Green tax rate per unit of products that are not remanufactured ($/unit) λ Green subsidies by the government per unit of products ($/unit) D The annual demand E Green marketing expenditure to attract consumers to the remanufacturing process.

Decision variables
W Wholesale price charged by the manufacturer to the retailer G j Green costs related to the manufacturer and the retailer ($/unit), j = M (manufacturer), R (retailer) x Participation rate of the manufacturer in the collecting process (0 ≤ x ≤ 1) P Selling price charged by the retailer Q Lot size determined by the retailer (unit)

Assumptions
The following assumptions are considered in the proposed models: (1) Planning horizon is infinite.
(2) Supply and demand shortages are not allowed.
(3) There is no difference between the quality of the new and the remanufactured products (Savaskan, Bhattacharya, and Van Wassenhove 2004;He 2015). (4) Remanufacturing process is less costly than the manufacturing (Sheu 2011). (5) Government applies subsidies and penalties to the manufacturer as a motivation for being green (Mitra and Webster 2008). (6) The manufacturer has power over the retailer to act as a Stackelberg leader. (7) The annual demand is a function of the selling price (P) and the green marketing expenditure (E) (Peattie and Charter 2003;Xie and Wei 2009;Karray 2011). Moreover, green marketing expenditure depends on green costs spent by the manufacturer or the retailer.

Model formulation
In this section, three models (A, B and C) for implementing CLSC and one model (D) without the remanufacturing process are proposed ( Figure 1). Moreover, the optimal policies for each model are investigated based on the Stackelberg game.

Model A
In this model, only the manufacturer is responsible for collecting the used products in remanufacturing. The real examples of the model are cartridges of Canon (Canon, Inc., Accessed August 5, 2014, http://www.usa.canon.com/cusa/consumer/ standard_display/ink_recycle) and televisions and digital cameras of Samsung (Samsung, Inc., Accessed August 5, 2014, http://www.samsung.com/us/aboutsamsung/citizenship/usactivities/environmentalinitiatives/). The manufacturer's objective is to determine the optimal wholesale price and the collecting cost by maximising his annual profit. The manufacturer's annual profit is: The manufacturer's Profit = Revenue − Production cost of new products − Production cost of used products − Set-up cost − Holding cost − Green cost − Remanufacturing logistics cost − Green tax + Green subsidy The retailer's objective is to determine the optimal selling price and the lot size by maximising his annual profit. The retailer's annual profit is: The retailer's Profit = Revenue − Purchase cost − Ordering cost − Sales cost − Holding cost The demand function in Equations (3) and (4) is considered as follows: According to assumption 6, first the retailer's optimal policies are obtained. It is shown that R (P, Q) is concave with respect to P and Q in Appendix 1. Therefore, the first-order condition with respect to P determines the optimal policy that maximises R (P, Q) for a fixed Q.
By substituting Equation (6) into Equation (4) and using the first-order condition with respect to Q, Equation (7) is deduced which has two roots.
According to Appendix 1, R (P, Q) is a concave function with respect to Q when the roots of Equation (7) is satisfied the condition function in (8).
The manufacturer-Stackelberg game is modelled as follows: S.T: By substituting Equation (10) and the roots of Equation (11) into Equation (9), the problem transforms to a model with nonlinear function and one constraint and the solution is obtained using of Lingo software. After obtaining the optimal policies of the manufacturer, the optimal policies of the retailer are simply calculated by Equations (10) and (11).

Model B
In some companies, due to the direct interaction of the retailer and the consumers, the green cost of the retailer is marginal so that the manufacturer can avoid excessive expenses by paying little cost to the retailer. For instance, Eastman Kodak Company collects single-use cameras from consumers by large retailers (Savaskan, Bhattacharya, and Van Wassenhove 2004). In this model, the collecting is performed only by the retailer who receives b from the manufacturer for each unit collected. The manufacturer's and retailer's profits are: The manufacturer's Profit = Revenue − Production cost of new products − Production cost of used products − Set-up cost − Holding cost − Green tax + Green subsidy The retailer's Profit = Revenue − Purchase cost − Ordering cost − Sales cost − Holding cost − Green cost − Remanufacturing logistics cost In this model, the demand function is transformed as follows: It is also shown that R (P, G R , Q) is concave with respect to P and G R for a fixed Q (Appendix 2). By applying the first-order condition with respect to P, the optimal P * for a fixed G R and Q is obtained.
Substituting Equation (16) in Equation (14) and applying the first-order condition with respect to G R results in: By substituting Equation (17) and Equation (18) into Equation (14) and using the first-order condition with respect to Q, The Stackelberg problem formulation is as follows: S.T: Similar to the previous model, by substituting Equation (22), Equation (23) and the roots of Equation (24) into Equation (21), a nonlinear model is deduced. The solution is obtained using of Lingo software. After obtaining the optimal policies of the manufacturer, the optimal policies of the retailer are simply calculated by Equations (22)-(24).

Model C
In this model, both the manufacturer and the retailer are responsible for collecting the used products. One example of this model is Apple (Apple, Inc., Accessed August 5, 2014, http://www.apple.com/recycling/ipod-cell-phone/) who provides a system for receiving the old iPods by both the Apple retail store and the manufacturer. The participation rate of the manufacturer in the collecting process is considered as a decision variable which is denoted by x. The annual profits of both agents and the demand function used in the model are as follows: The manufacturer's Profit = Revenue − Production cost of new products − Production cost of used products − Set-up cost − Holding cost − Green cost − Remanufacturing logistics cost − Green tax + Green subsidy The retailer's Profit = Revenue − Purchase cost − Ordering cost − Sales cost − Holding cost − Green cost − Remanufacturing logistics cost The concavity of R (P, Q, G R ) with respect to P and G R for a fixed Q is presented in Appendix 2. The optimal P and G R for a fixed Q and the optimal Q are obtained by the first-order condition. Finally, the stackelberg game is as follows: S.T: The obtained model is nonlinear when Equations (30) and (31) and the roots of Equation (32) are substituted into Equation (29). Lingo software is used to find the solution. Similar to previous models, by obtaining the optimal policies of the manufacturer, the optimal policies of the retailer are calculated by Equations (30)-(32).

Model D
There is no reverse channel in this model. Therefore, the manufacturer has to pay the complete green tax to the government because of neglecting the green supply chain. The annual profits are: The manufacturer's Profit = Revenue − Production cost − Set-up cost − Holding cost − Green tax The retailer's Profit = Revenue − Purchase cost − Ordering cost − Sales cost − Holding cost In this model, the demand function in Equations (34) and (35) is: The Stackelberg game is modelled by imitating the procedures for the three previous models. Note that the concavity of S.T: Similarly, in this model, the solution of the nonlinear model is obtained using of Lingo software. The optimal policies of the retailer are calculated by substituting the optimal policies of the manufacturer in to Equations (38) and (39).

Evolutionary game analysis
This section seeks to consider the proposed models from a holistic perspective by EGT. Hence, the CLSC models are assumed in a population. This population consists of several manufacturers and retailers who play as a two-player game. First, the behaviour of the agents is modelled by replicator dynamics. Finally, the optimal strategy set (ESS) by which the players attain maximum benefits in long-term is obtained by solving the replicator dynamic equations. The evolutionary game of the models mentioned in the previous section is presented in Table 1. Note that the notations G and N G denote the strategy of performing the collecting process and not performing the collecting process, respectively. Suppose that S M and S R denotes the proportions of the manufacturers and the retailers who perform strategy G in longterm, respectively. Moreover, E M n (t) is the expected pay-off to the manufacturer using strategy n(G, N G) at time t and E M(t) is the average pay-off of the manufacturer in the population at time t. Similarly, E R n (t) and E R(t) are the notations used for the retailer. The calculations are as follows: The replicator dynamic equations of the manufacturers and the retailers for strategy G are (F(S M , S R )) and (H (S M , S R )) as follows. For notational simplicity, M z is the manufacturer's profit of model z and R z is the notation for the retailer.
The ESS (S M , S R ) are obtained by equating the replicator dynamic equations to zero. The fixed points derived from Equations (47) and (48) are as follow: To analyse all the possible situations in the long-term, the fixed points in Equations (49) and (50) form a set of pairs. The Jacobian matrix in Equation (51) is applied to check the stability of the pairs. The conditions for stability of fixed points are: (i) determinant of the Jacobian should be positive, and (ii) trace of the Jacobian should be negative. If a fixed point satisfies these conditions, it is said to be asymptotically stable and, hence, the solution is evolutionary stable (Bukowski and Miekisz 2004;Barari et al. 2012). It is obvious that the proportions of the manufacturers and retailers who perform strategy N G in long-term are simply obtained by 1 − S M and 1 − S R , respectively.
Table 2 presents the profits' intervals in which the fixed points (S M , S R ) are stable derived using Jacobian matrix (Appendix 3). Based upon the game theoretical analysis, Table 2 demonstrates the behaviour of manufacturers and retailers in longterm. For instance, strategy (0,0) which means none of the manufacturers and retailers perform green activities, will be stable when manufacturers' and retailers' profits in not green situation (model D) is greater than the green situation respectively in models A and B. The mentioned rule can be generalised to the other fixed points in Table 2. As it is seen, the behaviour of the population in adopting the remanufacturing strategy is determined by the profits of the both manufacturers and retailers. Although remanufacturing process increases consumers' demand for the products, it leads agents to spend more on green activities. Therefore, a strategy will remain in long-term that offers the best economic gains to the agents in interacting with each other.

Computational result
In this section, a numerical study is presented to the models discussed in the previous section in order to provide managerial insights on the models and the behaviour of the agents.

Fixed points (S M , S R )
Stability conditions

Numerical example
Consider firms with the remanufacturing process for the used products. The set-up, ordering and logistics costs for the manufacturer (O M ) and the retailer (O R ) are $25 and $27. The percentage of the inventory holding cost (i) is 0.4% per unit per year. The unit cost of manufacturing (C m ) and remanufacturing (C r ) are $30 and $2, respectively. The cost of sales for the retailer is $3. The remanufacturing rate in the firm (τ ) is 0.8. Moreover, government imposes $5 as the green tax and $ as the subsidy for the manufacturers. The retailer receives $1 from the manufacturer for collecting one unit of used products (b). The logistics cost for each unit of used products transported to the firm (T ) is $8. The consumer demand for the firm's products is sensitive to the price and the marketing expenditure. The price elasticity (α) and the marketing elasticity (β) are 2.3 and 1, respectively. Furthermore, the constant value in the demand function is assumed to be 10. The corresponding results for the optimal policies of each model and the ESS are listed in Table 3. As it is seen in Table 3, all companies tend to perform remanufacturing processes and model D is not acceptable or profitable for both manufacturers and retailers in the long-term. Moreover, strategy (1,1) is the stable strategy which means both manufacturers and retailers tend to perform green activities in the long-term. The reason is that the profit of the manufacturer in model C is greater than model B and the profit of the retailer in model C is greater than model A. Converging to the ESS can be clearly seen by plotting the trajectories of the replicator dynamics (Figure 2).

Managerial insight
In this section, the behaviour of the agents in implementing the remanufacturing process with respect to green marketing elasticity, the remanufacturing rate and the green subsidy is shown. In fact, to illustrate some significant features of the models, the effect of some green parameters such as (β), (τ ) and (λ) is studied.
• The effect of green marketing elasticity on the long-term behaviour The sensitivity analysis with respect to green marketing elasticity (β) is presented in Table 4. By increasing β in the models, the consumers' demand is increased exponentially which leads to a rise in the manufacturer's and the retailer's profits. As it is seen in Table 4, in lower amount of (β), model A is not profitable for both the manufacturer and the retailer and as β increases, it is profitable for the manufacturers to singly perform collecting and green activities. The long-term behaviour of the agents is shown in Figure 3. As it is seen, by increasing green marketing elasticity, the trajectories converge to the stable strategy (1,0) which means M D < M A , R C < R A . Note that although there are multiple ESS found in each β, only one of them is the dominant strategy in the long-term due to Figure 3 which is underlined in Table 4. High marketing elasticity (β) means the consumers are very sensitive to the marketing paradigm or green activities that usually happens in the development countries. In such society, the manufacturers do not need to spent more money to convince the consumers. Therefore, they obtain maximum profit using the advantage of government motivation plans with less money. Generally, the collecting programs are performed by the manufacturers only by incurring lower cost which is logical. However, when (β) is low, the manufacturers prefer to assign the collecting responsibility on the retailer for the higher green activities cost. • The effect of remanufacturing rate on the long-term behaviour As it is seen in Table 5, models A, B and C are sensitive to the remanufacturing rate (τ ). By increasing τ , due to low remanufacturing process cost (assumption (4)), the production cost and consequently the wholesale price is decreased. Therefore, the profits of the manufacturer in models A, B and C increase. Moreover, by decreasing the wholesale price, the selling price decreases due to Equation (10). This leads to the higher consumer demand and higher profit for the retailer in Model A. In models B and C, the retailer's profit decreases due to the increment of price in Equations (22) and (30). Therefore, by increasing the number of products collected, the behaviour will shift to strategy (1,0). In fact, it is logical that in long-term, by increasing the remanufacturing rate, the manufacturers tend to perform collecting and green activities. The effect of remanufacturing rate on the trajectories converging to the stable strategy is also shown in Figure 4. • The effect of government subsidy on the long-term behaviour The results of the sensitivity analysis with respect to green subsidy are presented in Table 6. As it is seen, the profits are improved which means the government subsidies have a positive effect on promoting green activities. By increasing green subsidy, the manufacturers tend to invest more on green activities and consequently, in long-term the stable strategy turned to (1,0). However, in low amount of government subsidy, it is profitable for the manufacturer to assign the responsibility to the retailer. The process of converging the trajectories is shown in Figure 5.

Conclusion
With the rapid industrial growth and emergence of ecological problems, green supply chain has arisen as an important subject. CLSC, as a concept in green supply chain, is extensively used since it has the potential to prevent the detrimental materials from getting in to the environment. In this paper, novel pricing models in a two-echelon CLSC are presented regarding various strategies for collecting the used products. The annual demand was shown to be sensitive to the marketing effort as well as the selling price which is a function of green activities. The different government policies are also considered to motivate the manufacturer for being green. In each model, the interaction between a manufacturer and a retailer is considered based on the Stackelberg game. Moreover, with the increasing attention of companies to include CLSC in their planning horizon, conducting a holistic analysis of the models seemed logical. Therefore, using the evolutionary game approach, the models were considered as a population to examine the behaviour of companies in the remanufacturing process. In other words, both individual and population-based aspects of the problem were investigated in this paper. Numerical examples were presented to illustrate and compare the models. Finally, sensitivity analyses were used to show the behaviour of the agents in implementing the remanufacturing process with respect to some key parameters. Results showed that in the long-term the remanufacturing process is more profitable for companies compared to a process without remanufacturing. Furthermore, as the consumers need for green products increases, the manufacturers tend to perform green activities in long-term. Otherwise, the green responsibility will shift to the retailer. The remanufacturing rate and government subsidies are also key factors to motivate manufacturers for green activities.
There is a broad scope to extend the present research. For instance, it is assumed that the remanufacturing rate is known for both the manufacturer and the retailer. However, the appropriate value for this rate can be variable and obtained by a model. The proposed models can become more realistic by extending them to a case where some information of the agents is hidden from their opponent. This condition leads to games with incomplete information.

Disclosure statement
No potential conflict of interest was reported by the authors.

Appendix 1.
In this section, a complete proofs for the concavity of the retailer's profit function with respect to P and Q is presented. For notational simplicity we let R (P, Q) ≡ R . For R to be concave with respect to P and Q, the Hessian matrix must be negative definite. Hence, the following conditions should be satisfied: Condition (i) is satisfied as follows: To prove condition (ii), we have: As it is seen, Equation (A2) is positive when: By substituting (6) in (A3), the following condition is deduced. Therefore, R is concave with respect to P and Q when condition (A3) is satisfied.

Appendix 2.
This section presents the proofs for concavity of retailer's profit function with respect to the decision variables. First, the concavity of retailer's profit in model C is presented and the procedure can be used for other models in the paper. For notational simplicity we let R (P, G R , Q) ≡ R . For R to be concave with respect to P, G R and Q, the Hessian matrix must be negative definite. Hence, the following conditions should be satisfied: Since [k + xG M + (1 − x)G R ] = β P * α is satisfied, by substituting Equation (30) in the second-order condition for R with respect to P, condition(i) is satisfied as follows: Similarly, conditions (ii) is calculated using condition [k + xG M + (1 − x)G R ] = β P * α and Equation (30) as follows: To prove condition (iii), we have: Therefore, substituting Equations (B3)-(B7) in condition (iii): By substituting Equation (30) in equation(B8), the following equation is negative when condition in (33) is satisfied.

Appendix 3.
To find the ESS of the evolutionary game, the fixed points derived from dynamic replicator equations have to satisfy two conditions: (i) determinant of the Jacobian in Equation (51) Partial derivatives of the above equations with respect to S M and S R are: The fixed points are tested by the two conditions as follows: For (0,0) to be stable: For (0,1) to be stable: For (1,0) to be stable: For (1,1) to be stable: , 0 to be stable: , 1 to be stable: For 1, is unstable because the determinant and the trace of the jacobian matrix is zero.