The optimal pricing and remanufacturing decisions problem of a fuzzy closedloop supply chain is considered in this paper. Particularly, there is one manufacturer who has incorporated a remanufacturing process for used products into her original production system, so that she can manufacture a new product directly from raw materials or from collected used products. The manufacturer then sells the new product to two different competitive retailers, respectively, and the two competitive retailers are in charge of deciding the rates of the remanufactured products in their consumers’ demand quantity. The fuzziness is associated with the customer’s demands, the remanufacturing and manufacturing costs, and the collecting scaling parameters of the two retailers. The purpose of this paper is to explore how the manufacturer and the two retailers make their own decisions about wholesale price, retail prices, and the remanufacturing rates in the expected value model. Using game theory and fuzzy theory, we examine each firm’s strategy and explore the role of the manufacturer and the two retailers over three different game scenarios. We get some insights into the economic behavior of firms, which can serve as the basis for empirical study in the future.
In recent years, the management of closedloop supply chains has gained growing attention from both business and academic research because of environmental consciousness, environmental concerns, and stringent environmental laws, for example, the legislation on producer responsibility, requiring companies to take back products from customers and to organize for proper recovery and disposal. This legislation is partially due to increased awareness of environmental issues. However, smart companies have also understood that used products often contain lots of value to be recovered. They manage closedloop supply chains simply because it is a profitable business proposition. It is said that the costs derived from reverselogistics activities in the USA exceed
Without a doubt, closedloop supply chains has become a matter of strategic importance: an element that companies must consider in decisionmaking processes concerning the design and development of their supply chains [
In fact, in order to make effective closedloop supply chain management, the uncertainties that happen in the real world cannot be ignored. Those uncertainties are usually associated with the product supply, used product collecting, the customer demand, and so on. Traditional probabilistic concepts have been used to model the various parameters among today’s many studies published on the reverse logistics [
In recent supply chain studies, some researchers have already adopted fuzzy theory to depict uncertainties in supply chain models [
Although some researches on the forward supply chain have been given through considering the supply chain’s fuzzy uncertainties, little researches on the reverse supply chain considering the fuzzy uncertainties has been established to our knowledge. So, in this paper, we consider a fuzzy manufacturing and remanufacturing closedloop supply chain with one manufacturer and two competitive retailers; the fuzziness is associated with the consumer demand, the manufacturing and remanufacturing costs of new product, and the collecting cost of the used product. In the forward supply chain, the manufacturer has incorporated a remanufacturing process for used products into her original production system, so that she can manufacture a new product directly from raw materials, or remanufacture part or whole of a collected unit, and wholesales the new products to the two competitive retailers who then sell them to the end consumers. For the the reverse supply chain, the two competitive retailers are in charge of collecting the used products from the consumers, respectively. Using game theory and fuzzy theory, the optimal decisions for each supply chain participant are explored in the expected value model. Some management insights are given in this paper.
The rest of the paper is organized as follows. Section
Consider a closedloop supply chain in a fuzzy environment with one manufacturer and two competitive retailers, labeled retailer 1 and retailer 2. In the following discussion, “he” represents one of the two manufacturers, and “she” represents the retailer. In the forward supply chain, similar to Savaskan et al. [
We define the retailer
In our models, the manufacturer can influence the demand by setting the new product’s wholesale price; the two competitive retailers can independently decide the retail price of the new product and the collecting rate of the used product. We do not assume any collusion or cooperation among firms; this assumption is typical in analytical model, although it overstates the information climate of the real world. The logistic cost components of the manufacturer and two retailers (e.g., carrying cost inventory cost, etc.) are without consideration for analytical convenience.
Assume each channel member has the same goal: to maximize his/her own expected profit. From the above descriptions, the two competitive retailers’ objectives are to maximize their own expected profits (denoted as
The manufacturer’s objective is to maximize his own expected profit (denoted as
Note that so far we have not made any assumptions regarding the bargaining power possessed by each channel member. The assumption regarding bargaining power possessed by each firm can influence how the pricing game is solved in our model. Variation in bargaining power in a particular supply chain can create one of the following three scenarios:
To analyze our model, we follow a game theory approach. The leader in each scenario makes his decision to maximize his/her own expected profit, conditioned on the follower’s response. The problem can be solved backwards. We begin by first solving for the decision of the follower of the game, given that he/she has observed the leader’s decision. For example, in Manufacturer Stackelberg, the two competitive retailers’ decisions are derived first, given that the two competitive retailers have observed the decision made by the manufacturer (on wholesale price). Then, the manufacturer solves his problem given that he knows how the two competitive retailers would react to his decision.
In the Manufacturer Stackelberg game case, the manufacturer first announces his wholesale prices of the new product. The two competitive retailers observe the wholesale price and then simultaneously decide the retail prices they are going to charge for their own product and the collecting rates of the used products. Note that the two competitive retailers move simultaneously. Therefore, we need to calculate the Nash decisions between them first.
The two competitive retailers’ optimal retail prices and optimal collecting rates of used products, given earlier decision
Using (
Solving (
The manufacturer in this game is the Stackelberg leader. He announces his new product’s wholesale price
In the Manufacturer Stackelberg game case, the manufacturer’s optimal decision (denoted as
With some manipulations, the expected value
With (
Therefore, by setting (
In the Manufacturer Stackelberg game case, the two competitive retailers’ optimal retail prices (denoted as
By Propositions
The Retailer Stackelberg scenario arises in markets where the two competitive retailers’ sizes are larger compared to their manufacturer. Because of their sizes, the two competitive retailers can maintain their margin on sales while squeezing profit from their suppliers. Similar gametheoretic framework as applied in the Manufacturer Stackelberg case is implemented to solve this problem. First, the manufacturer’s problem is solved to derive the decision conditional on the retail prices and collecting rates chosen by the two competitive retailers. The two competitive retailers’ problems are then solved given that the two competitive retailers know how the manufacturer would react to their retail prices and collecting rates.
Without loss of generality, let
Since the two competitive retailers move first in this game, we need to calculate the manufacturer’s decision. The manufacturer is trying to maximize his own expected profit
In the Retailer Stackelberg game case, the manufacturer’s optimal decision, given retail prices
Using (
We can easily see that Proposition
Having the information about the decision of the manufacturer, each retailer would then use it to maximize her own expected profit
Note that the two competitive retailers move simultaneously. Therefore, we need to calculate the Nash decisions between them first.
In the Retailer Stackelberg game case, the optimal retail price and collecting rate (denoted as
By (
We can get the first order conditions as follows:
In the Retailer Stackelberg game case, the manufacturer’s optimal decision (denoted as
By Propositions
In the Vertical Nash model, every firm has equal bargaining power and thus they make their decisions simultaneously. This scenario arises in a market in which there are relatively small to mediumsized manufacturers and retailers. Since a manufacturer cannot dominate the market over the two competitive retailers, his price decision is conditioned on how the two competitive retailers price the new product. On the other hand, the two competitive retailers must also condition their own retail price and own collecting rate decisions on the wholesale price.
Consider that the decisions of the two competitive retailers and the manufacturer are already derived in the Manufacturer Stackelberg and Retailer Stackelberg game cases, respectively. From the Manufacturer Stackelberg game, the two competitive retailers’ decisions for given wholesale price
Solving (
In the Vertical Nash case, the optimal retail prices (denoted as
Solving (
In this section, we compare the results obtained from the above three different decision scenarios using numerical approach and study the behavior of firms facing changing environment. By the results obtained from the above three different decision scenarios, we can easily see the expressions of the optimal wholesale price, retail prices, collected rates, and optimal expected profits under different decision scenarios.
Here, assume that the relationship between linguistic expressions and triangular fuzzy variables for manufacturing cost, remanufacturing cost, market base, scaling parameter, collecting transfer cost, and price elasticity is determined by experts’ experiences as shown in Table
Relation between linguistic expression and triangular fuzzy variable.
Linguistic expression  Triangular fuzzy variable  

Low (about 7)  (6, 7, 9)  
Remanufacturing cost 
Medium (about 11)  (9, 11, 14) 
High (about 16)  (14, 16, 19)  
Low (about 17)  (15, 17, 20)  
Manufacturing cost 
Medium (about 23)  (20, 23, 25) 
High (about 29)  (25, 29, 35)  
Market base 
Large (about 400)  (300, 400, 450) 
Small (about 200)  (150, 200, 280)  
Price elasticity 
Very sensitive (about 0.8)  (0.6, 0.8, 0.9) 
Sensitive (about 0.5)  (0.3, 0.5, 0.6)  
Low (about 2)  (1, 2, 3)  
Taking back transfer cost 
Medium (about 4)  (3, 4, 5) 
High (about 6)  (5, 6, 8)  
Low (about 500)  (450, 500, 650)  
Scaling parameter 
Medium (about 800)  (700, 800, 1000) 
High (about 1100)  (1000, 1100, 1300)  
Low (about 550)  (400, 550, 650)  
Scaling parameter 
Medium (about 850)  (650, 850, 1000) 
High (about 1200)  (1000, 1200, 1300) 
Consider the case that the remanufacturing and manufacturing costs
Optimal expected profits of the manufacturer and the two retailers.
Game scenario 




Manufacturer Stackelberg  95194  15161  15159 
Retailer Stackelberg  76463  26996  27519 
Vertical Nash  91005  21868  21865 
Optimal decisions of retail prices, wholesale price, and collecting rates.
Game scenario 






Manufacturer Stackelberg  503.0028  503.0108  380.8939  0.3296  0.3247 
Retailer Stackelberg  484.3593  481.2045  279.4503  0.6861  0.6988 
Vertical Nash  455.7510  455.7604  309.0691  0.3883  0.3825 
From Tables
For the three decentralized decision cases, the firm who is the leader in the supply chain has the advantage to get the higher profit; for example, the manufacturer’s profit under Manufacturer Stackelberg game scenario is higher than that under Retailer Stackelberg game scenario and the Vertical Nash game case. For the two competitive retailers, they have their own minimal expected profits under Manufacturer Stackelberg game scenario.
The new product’s optimal retail prices charged by the two competitive retailers, respectively, under Vertical Nash decision case are lower than those under the Manufacturer Stackelberg and Retailer Stackelberg decision cases, and the optimal retail prices achieve the biggest value under Manufacturer Stackelberg game scenario.
The new product achieves the highest wholesale price in the Manufacturer Stackelberg game, followed by the Vertical Nash game and then the Retailer Stackelberg game case.
The optimal collecting rates of the used products charged by the two competitive retailers, respectively, achieve the highest wholesale price in the Retailer Stackelberg game, followed by the Vertical Nash game and then the Manufacturer Stackelberg game case.
Different from the conventional studies, this paper explores the roles of the two competitive retailers and the manufacturer and their bargaining powers by examining the supply chain in a fuzzy environment over three different game scenarios. We derive the expressions for optimal retail prices, wholesale price, and collecting rates with expected value model. By analyzing a numerical example, we further analyze the analytical solutions and give some managerial analysis.
Compared to the traditional approach used in the study of closedloop supply chain, the proposed approach in this paper requires less data to model the fuzziness which is associated with the consumer demand, the manufacturing and remanufacturing costs of new product, and the collecting cost of the used product and can make use of the subjective estimation based on decision maker’s judgment, experience, and intuitions. It is appropriate when the situation is ambiguous and lacks historical data.
However, we have made some assumptions that may be relaxed to improve the model in the future research. One assumption is that the demand function is linear; further work is desirable to test whether our conclusions extend to other forms of demand function. The other assumptions are that the closedloop supply chains only with one period and competition only existing in retail process. Thus, the supply chain with competitive manufacturers and/or competitive retailers, and the model over multiple periods can be considered in the future.
The authors wish to express their sincerest thanks to the editors and anonymous referees for their constructive comments and suggestions on this paper. This research was supported in part by National Natural Science Foundation of China, nos. 71001106, 70971069, and 71002106.