Supply Chain Contracts with Multiple Retailers in a Fuzzy Demand Environment

Abstract

This study investigates supply chain contracts with a supplier and multiple competing retailers in a fuzzy demand environment. The market demand is considered as a positive triangular fuzzy number. The models of centralized decision, return contract, and revenue-sharing contract are built by the method of fuzzy cut sets theory, and their optimal policies are also proposed. Finally, an example is given to illustrate and validate the models and conclusions. It is shown that the optimal total order quantity of the retailers fluctuates at the center of the fuzzy demand. With the rise of the number of retailers, the optimal order quantity and the fuzzy expected profit for each retailer will decrease, and the fuzzy expected profit for supplier will increase.

1. Introduction

Over the last decade or so, supply chain management has emerged as a key area of research among the practitioners of operations research. In recent years, coordination mechanism of supply chain contracts has become one of the most challenging issues facing both practitioners and scholars. Supply chain contracts such as return contract and revenue-sharing contract are instruments for supply chain coordination, which shift the uncertain demand from the retailer to the supplier, thus encouraging the retailer to increase order quantities.

A large body of literature has explored to coordinate the supply chain with return contract and revenue-sharing contract during the last two decades. Pasternack [1] first claimed that an appropriate return policy can fully coordinate a single-supplier single-retailer supply chain, which was then extended by Mantrala and Raman [2] to the situation where the retailer had several stores. Taylor [3] and Lee et al. [4] studied the return contract with effort-dependant demand. They showed that in this problem, it attained supply chain coordination combined with feedback policy. Yao et al. [5] analyzed the profits of both actors when the manufacturer and retailer shared or did not share the forecast information in returns policy. Yue and Raghunathan [6] discussed the impact of a full return policy as well as information sharing on the manufacturer and the retailer under information asymmetry. Bose and Anand [7] considered the wholesale price as an exogenous price to study returns policies for coordinating the supply chain. They showed that, in general, an equilibrium returns policy was not Pareto efficient with respect to a price-only contract, but when the wholesale price was sufficiently high, the equilibrium returns policy was Pareto efficient. These conclusions were consistent with those of Yao et al. [8]. Ding and Chen [9] studied the return contract issues of a three-level supply in a single-period model. Yao et al. [10] analyzed the impact of price-sensitivity factors on characteristics of return contract in a single-period product supply chain. Mollenkopf et al. [11] used an empirical study to explore how internet product returns management systems affect loyalty intentions. Chen and Bell [12, 13] showed that the customer returns affect the firm’s pricing and inventory decision and proposed an agreement between the manufacturer and the retailer that includes two buyback prices. Chen [14] proposed a returns policy with a wholesale-price-discount scheme that can achieve supply chain coordination. Ai et al. [15] analyzed the implementation of full returns policies in the chain-to-chain competition.

Revenue-sharing contract has been applied in the video cassette rental and movie industry with much success. Giannoccaro and Pontrandolfo [16] showed that revenue-sharing could coordinate members in the newsboy channel with three stages: supplier, manufacturer, and retailer. Cachon and Lariviere [17] intensively discussed a revenue-sharing contract between a single supplier and a single retailer in a single-period newsboy problem. Gupta and Weerawat [18] designed a revenue-sharing contract to maximize the centralized revenue by choosing an appropriate inventory level. Yao et al. [19] investigated a revenue-sharing contract for coordinating a supply chain comprising one manufacturer and two competing retailers. Linh and Hong [20] studied a revenue-sharing contract in a two-period newsboy problem. Van Der Rhee et al. [21] proposed a revenue-sharing mechanism in multi-echelon supply chains. Ouardighi and Kim [22] considered a single supplier collaborating with two manufacturers on designing quality improvements for their respective products under a revenue-sharing contract. Krishnan and Winter [23] studied the role of revenue-sharing contracts in supply chains and established a foundation in aligning incentives. Sheu [24] explored revenue-sharing contracts under price promotion to end customers with three types of promotional demand patterns. Zhang et al. [25] investigated a revenue-sharing contract with demand disruptions in a supply chain comprising one manufacturer and two competing retailers. Palsule-Desai [26] studied revenue-dependent contracts and revenue-independent contracts in a two-period model, and they showed that both types of revenue-sharing contracts could coordinate the supply chain; however, there existed situations in which revenue-dependent contracts outperformed revenue-independent contracts. The conventional studies have focused on the cases in which the demands are probabilistic. In other words, the demands follow certain distribution function. However, in practice, especially for new products, the probabilities are not known due to lack of history data. In this case the demands are suitably described subjectively by linguistic terms, such as “high,” “low,” or “approximately equal 𝑑, but definitely not less than 𝑎 and not greater than 𝑏.” Thus, the uncertain theory, rather than the traditional probability theory, is well suited to the supply chain models problem. Therefore, we assume that the external demand can be approximately forecasted and expressed as a triangular membership function.

In this paper, the demands are approximately estimated by experts and regarded as fuzzy numbers. Return contract and revenue-sharing contract with multiple competing retailers in a fuzzy demand environment will be discussed, and the impact of the supplier’s production cost and the number of retailers on the models will be analyzed.

The rest of the paper is organized as follows. Section 2 introduces some definitions and propositions about fuzzy set theory and notations related to this paper. Section 3 develops three fuzzy supply models with multiple competing retailers. Section 4 provides a numerical example to illustrate the result of the proposed contracts. Section 5 summarizes the work done in this paper.

2. Preliminaries

2.1. Fuzzy Set Theory

Definition 1. The fuzzy set 𝐴=(𝑎₁,𝑎₂,𝑎₃), where 𝑎₁<𝑎₂<𝑎₃ and defined on 𝑅, is called the triangular fuzzy number, if the membership function of 𝐴 is given by 𝜇𝐴(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁,if𝑎₁≤𝑥≤𝑎₂,𝑎₃−𝑥𝑎₃−𝑎₂,if𝑎₂<𝑥≤𝑎₃,0,𝑥∉𝑎₁,𝑎₃, (1) where 𝑎₁ and 𝑎₃ are the lower limit and upper limit, respectively, of the triangular fuzzy number 𝐴. For 𝑥∈[𝑎₁,𝑎₂], the left membership function 𝐴_𝐿(𝑥)=(𝑥−𝑎₁)/(𝑎₂−𝑎₁) is an increase function of 𝑥. For 𝑥∈(𝑎₂,𝑎₃], the right membership function 𝐴_𝑅(𝑥)=(𝑎₃−𝑥)/(𝑎₃−𝑎₂) is a decrease function of 𝑥.

Definition 2. The triangular fuzzy number 𝐴 is called the positive triangular fuzzy number if 𝑎₁>0.

Definition 3. For any 𝛼∈[0,1], the set 𝐴𝐴(𝛼)={𝑥∣𝜇(𝑥)≥𝛼} is called the 𝛼 cut set of 𝐴. 𝐴(𝛼) is a nonempty bounded closed interval contained in the set of real numbers, and it can be denoted by 𝐴(𝛼)=𝐴_𝐿⁻¹(𝛼),𝐴_𝑅⁻¹(𝛼), (2) where 𝐴_𝐿⁻¹(𝛼) and 𝐴_𝑅⁻¹(𝛼) are, respectively, the left and right boundaries of 𝐴(𝛼), with 𝐴_𝐿⁻¹(𝛼)𝐴=inf𝑥∈𝑅∶𝜇_(𝑥),𝐴≥𝛼_𝑅−1(𝛼)𝐴=sup𝑥∈𝑅∶𝜇_(𝑥)≥𝛼. (3)

Example 4. For any 𝛼∈[0,1], the 𝛼 cut set of a triangular fuzzy number 𝐴=(𝑎₁,𝑎₂,𝑎₃) is 𝐴_𝐿⁻¹(𝛼)=𝑎₁+𝑎₂−𝑎₁𝐴𝛼,_𝑅−1(𝛼)=𝑎₃−𝑎₃−𝑎₂𝛼. (4)

Based on the extension principle in fuzzy sets, we have the following Propositions 5 and 6.

Proposition 5. For any 𝛼∈[0,1], let 𝐴 be a positive triangular fuzzy number and let 𝑘 be a nonzero real number; then 𝑘𝐴(𝛼)=𝑘𝐴_𝐿⁻¹(𝛼),𝑘𝐴_𝑅⁻¹(𝛼),𝑘∈𝑅⁺,𝑘𝐴_𝑅⁻¹(𝛼),𝑘𝐴_𝐿⁻¹(𝛼),𝑘∈𝑅⁻. (5)

Proposition 6. For any 𝛼∈[0,1], let 𝐵(𝛼)=[𝐵_𝐿⁻¹(𝛼),𝐵_𝑅⁻¹(𝛼)] and 𝐶(𝛼)=[𝐶_𝐿⁻¹(𝛼),𝐶_𝑅⁻¹(𝛼)], respectively, be the 𝛼 cut set of the positive triangular fuzzy numbers 𝐵 and 𝐶; then 𝐵(𝛼)+𝐶(𝛼)=𝐵_𝐿⁻¹(𝛼)+𝐶_𝐿⁻¹(𝛼),𝐵_𝑅⁻¹(𝛼)+𝐶_𝑅⁻¹(𝛼,𝐵(𝛼)−𝐶(𝛼)=𝐵_𝐿−1)(𝛼)−𝐶_𝑅⁻¹(𝛼),𝐵_𝑅⁻¹(𝛼)−𝐶_𝐿⁻¹(𝛼). (6)

Proposition 7 (see B. Liu and Y.-K. Liu [27]). Let 𝐴 be a positive triangular number; the expected value of 𝐴 is 𝐸𝐴=12¹₀𝐴_𝐿⁻¹(𝛼)+𝐴_𝑅⁻¹(𝛼)d𝛼. (7)

Proposition 8 (see Y.-K. Liu and B. Liu [28]). Let 𝐴 and 𝐵 be two independent positive triangular fuzzy numbers with finite expected values. Then for any real numbers 𝑎 and 𝑏, one has 𝐸𝑎𝐴+𝑏𝐵=𝑎𝐸𝐴+𝑏𝐸𝐵. (8)

2.2. Problem Descriptions

Consider a single-period setting for a two-echelon supply, consisting of a supplier and multiple competing retailers with fuzzy demand. We assume that at the beginning of the selling season, the retailer 𝑖  (𝑖=1,2,…,𝑛) has no inventory on hand and must decide the order quantity 𝑞_𝑖 from the supplier. Then, the retailer 𝑖 sells his order of short-life products, such as personal computers, consumer electronics, or fashion items, with high uncertain demand. The products are sold only in one period. As the lead times of such goods are much longer than their selling season, the actors have no chance to place a second order.

We consider the total uncertain demand faced by the retailers as a positive triangular fuzzy variable 𝐷=(𝑑₁,𝑑₂,𝑑₃) with the most possible value 𝑑₂, where 0<𝑑₁<𝑑₂<𝑑₃. The fuzzy demand 𝐷 means that the total demand is about 𝑑₂. 𝑑₁ and 𝑑₃ are the lower limit and upper limit, respectively, of the fuzzy demand 𝐷 and described by a general membership function 𝜇𝐷(𝑥): 𝜇𝐷(𝑥)=𝐿(𝑥),𝑥∈𝑑₁,𝑑₂,𝑅(𝑥),𝑥∈𝑑₂,𝑑₃,0,𝑥∉𝑑₁,𝑑₃. (9)

Let the total retail demand be divided between the 𝑛 retailers proportional to their stocking quantity; that is, retailer 𝑖’s demand, 𝐷_𝑖, is 𝐷_𝑖=𝑞_𝑖𝑞𝐷, (10) where 𝑞=∑^𝑛_𝑖=1𝑞_𝑖.

The following notations are used for a product in the models: 𝑝: the retail price; 𝑤: the wholesale price;  𝑐: the per unit product cost incurred to the supplier;  𝑏: the return price offered by supplier in return contract; Φ: the fraction revenue of the retailer in revenue-sharing contract and 0<Φ<1; Π_𝑆: the fuzzy profit of the supplier;  Π_𝑅𝑖: the fuzzy profit of the retailer 𝑖; Π_SC: the fuzzy profit of the supply chain.

The supplier and the retailer 𝑖 are assumed to be risk neutral and pursued maximization of their fuzzy expected profits.

3. Model Analysis

3.1. Centralized Decision Making with Fuzzy Demand

Consider a supply chain occupied by an integrated actor, which can also be regarded as the retailers and the supplier-making cooperation. The fuzzy profit of two-stage supply chain can be expressed as Π_SC𝐷=𝑝min𝑞,−𝑐𝑞s.t.𝑑₁≤𝑞≤𝑑₃. (11)

Since the fuzzy demand 𝐷=(𝑑₁,𝑑₂,𝑑₃) in (11) is a positive triangular fuzzy number, we know that the order quantity 𝑞 has two cases; that is, 𝑞∈[𝑑₁,𝑑₂] or 𝑞∈(𝑑₂,𝑑₃].

Theorem 9. When 𝑐<𝑝≤2𝑐, the optimal total order quantity 𝑞^∗ of the retailers is 𝑞^∗=𝐿⁻¹2𝑝−𝑐𝑝. (12)

Proof. If 𝑞∈[𝑑₁,𝑑₂], then the 𝛼 cut set of min{𝑞,𝐷} is 𝐷𝛼)=𝐿min𝑞,(⁻¹(𝛼)𝑞,𝐿𝑞.,𝑞,𝛼∈0,𝐿𝑞,𝑞,𝛼∈,1 (13)(a)For𝛼∈[0,𝐿(𝑞)], the 𝛼 cut set of the supply chain’s fuzzy profit is Π_SC(𝛼)=𝑝𝐿⁻¹(𝛼)−𝑐𝑞,𝑝𝑞−𝑐𝑞. (14)(b)For𝛼∈(𝐿(𝑞),1], the result turns to Π_SC(𝛼)=𝑝𝑞−𝑐𝑞,𝑝𝑞−𝑐𝑞. (15)By (7), we can get the fuzzy expected profit 𝐸[Π_SC] as 𝐸Π_SC=12₀^𝐿(𝑞)𝑝𝐿⁻¹(𝛼)+1−𝑐𝑞+𝑝𝑞−𝑐𝑞d𝛼2¹_𝐿𝑞=1𝑝𝑞−𝑐𝑞+𝑝𝑞−𝑐𝑞d𝛼2𝑝₀^𝐿(𝑞)𝐿⁻¹(𝛼)d𝛼−𝑞𝐿𝑞+𝑞−𝑐𝑞. (16) The first and second derivatives of 𝐸[Π_SC] in (16) can be obtained as follows: d𝐸Π_SC=1d𝑞2𝑝𝑞d1−𝐿−𝑐,₂𝐸Π_SCd𝑞²1=−2𝑝𝐿^′𝑞. (17) Since 𝐿(𝑞) is an increasing function with 𝐿^′(𝑞)>0, therefore d²𝐸[Π_SC]/d𝑞² is negative and 𝐸[Π_SC] is concave in 𝑞.
Hence, the optimal total order quantity of the retailers can be obtained by solving d𝐸[Π_SC]/d𝑞=0, which gives 𝑞^∗=𝐿⁻¹2𝑝−𝑐𝑝. (18) Since 0<𝐿(𝑞^∗)≤1, thus we can get 𝑐<𝑝≤2𝑐.
The proof of Theorem 9 is completed.

Theorem 10. When 𝑝>2𝑐, the optimal total order quantity 𝑞^∗ of the retailers is 𝑞^∗=𝑅⁻¹2𝑐𝑝. (19)

Proof. If 𝑞∈(𝑑₂,𝑑₃], then the 𝛼 cut set of min{𝑞,𝐷} is 𝐷𝛼)=𝐿min𝑞,(⁻¹(𝛼)𝑞,𝐿,𝑞,𝛼∈0,𝑅₋₁(𝛼),𝑅⁻¹(𝛼),𝛼∈𝑅𝑞,1. (20)(a)For𝛼∈[0,𝑅(𝑞)], the 𝛼 cut set of the supply chain’s fuzzy profit is Π_SC(𝛼)=𝑝𝐿⁻¹(𝛼)−𝑐𝑞,𝑝𝑞−𝑐𝑞. (21)(b)For𝛼∈(𝑅(𝑞),1], the result turns to Π_SC(𝛼)=𝑝𝐿⁻¹(𝛼)−𝑐𝑞,𝑝𝑅⁻¹(𝛼)−𝑐𝑞. (22)By (7), we can get the fuzzy expected profit 𝐸[Π_SC] as 𝐸Π_SC=12₀^𝑅(𝑞)𝑝𝐿⁻¹(𝛼)+1−𝑐𝑞+𝑝𝑞−𝑐𝑞d𝛼2¹_𝑅𝑞𝑝𝐿⁻¹(𝛼)−𝑐𝑞+𝑝𝑅⁻¹(𝛼)=1−𝑐𝑞d𝛼2𝑝¹_𝑅𝑞𝑅⁻¹(𝛼)𝑞+d𝛼+𝑞𝑅¹₀𝐿⁻¹(𝛼)d𝛼−𝑐𝑞. (23) The first and second derivatives of 𝐸[Π_SC] in (23) can be obtained as follows: d𝐸Π_SC=1𝑑𝑞2𝑞d𝑝𝑅−𝑐,₂𝐸Π_SCd𝑞²=12𝑝𝑅^′𝑞. (24) Since 𝑅(𝑞) is a decreasing function with 𝑅^′(𝑞)<0, therefore d²𝐸[Π_SC]/d𝑞² is negative and 𝐸[Π_SC] is concave in 𝑞.
Hence, the optimal total order quantity of the retailers can be obtained by solving d𝐸[Π_SC]/d𝑞=0, which gives 𝑞^∗=𝑅⁻¹2𝑐𝑝. (25) Since 0<𝑅(𝑞^∗)≤1, thus we can get 𝑝>2𝑐.
The poof of Theorem 10 is completed.

From (16) and (23), Theorems 9 and 10, we can easily obtain the optimal fuzzy expected value of the profit for the integrated supply chain, which is given by 𝐸Π_SC^∗=12𝑝₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝜆)1d𝜆,𝑐<𝑝≤2𝑐,2𝑝¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝜆+¹₀𝐿⁻¹(𝛼)d𝛼,𝑝>2𝑐. (26)

3.2. Return Contract with Fuzzy Demand

In a return contract, the supplier sets a wholesale price 𝑤 and gives the retailer 𝑖  (𝑖=1,2,…,𝑛) a return price 𝑏 for unsold products at the end of the season. The fuzzy profit of the retailer 𝑖 can be expressed as follows: Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=𝑝min𝑞_𝑖,𝐷_𝑖+𝑏max𝑞_𝑖−𝐷_𝑖,0−𝑤𝑞_𝑖=𝑝min𝑞_𝑖,𝑞_𝑖𝑞𝐷+𝑏max𝑞_𝑖−𝑞_𝑖𝑞𝐷,0−𝑤𝑞_𝑖, (27) where 𝑞=∑^𝑛_𝑖=1𝑞_𝑖 and 𝑞_−𝑖=𝑞−𝑞_𝑖.

The retailer 𝑖  (𝑖=1,2,…,𝑛) tries to maximize its fuzzy expected profit 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] in return contract by choosing the optimal order quantity 𝑞_𝑖, which solves the following model: 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=𝐸𝑝min𝑞_𝑖,𝑞_𝑖𝑞𝐷+𝑏max𝑞_𝑖−𝑞_𝑖𝑞𝐷,0−𝑤𝑞_𝑖𝑞s.t.^𝑖𝑞𝑑₁≤𝑞_𝑖≤𝑞_𝑖𝑞𝑑₃. (28)

Theorem 11. When 𝑐<𝑝≤2𝑐, the optimal wholesale price 𝑤^∗ in return contract is 𝑤^∗1=𝑝−22𝑝−𝑏𝑝−𝑐𝑝−𝑛−1𝑛×∫₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝛼)d𝛼𝐿⁻¹2.𝑝−𝑐/𝑝 (29)

Proof. If 𝑞_𝑖∈[(𝑞_𝑖/𝑞)𝑑₁,(𝑞_𝑖/𝑞)𝑑₂], that is, 𝑞∈[𝑑₁,𝑑₂], then the 𝛼 cut sets of min{𝑞_𝑖,(𝑞_𝑖/𝑞)𝐷} and max{𝑞_𝑖−(𝑞_𝑖/𝑞)𝐷,0} are min𝑞_𝑖,𝑞_𝑖𝑞𝐷𝛼)=𝑞_𝑖(𝑞𝐿⁻¹(𝛼),𝑞_𝑖𝑞,𝑞,𝛼∈0,𝐿_𝑖,𝑞_𝑖𝐿𝑞,𝑞,𝛼∈,1max_𝑖−𝑞_𝑖𝑞𝛼)=𝐷,0(0,𝑞_𝑖−𝑞_𝑖𝑞𝐿⁻¹(𝛼𝑞,[]𝐿𝑞.),𝛼∈0,𝐿0,0,𝛼∈,1 (30)(a)For𝛼∈[0,𝐿(𝑞)], the 𝛼 cut set of the retailer 𝑖’s fuzzy profit is Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝛼)=𝑝𝑞_𝑖(𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖,𝑝𝑞_𝑖+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖. (31)(b)For𝛼∈(𝐿(𝑞),1], the result turns to Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖(𝛼)=𝑝𝑞_𝑖−𝑤𝑞_𝑖,𝑝𝑞_𝑖−𝑤𝑞_𝑖. (32)By (7), we can get the fuzzy expected profit 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=12₀^𝐿(𝑞)𝑝𝑞_𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖+𝑝𝑞_𝑖+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖+1d𝛼2¹_𝐿𝑞𝑝𝑞_𝑖−𝑤𝑞_𝑖+𝑝𝑞_𝑖−𝑤𝑞_𝑖=1d𝛼2𝑞𝑝−𝑏^𝑖𝑞₀^𝐿(𝑞)𝐿⁻¹(𝛼)d𝛼−𝑞_𝑖𝐿𝑞+𝑝−𝑤𝑞_𝑖. (33) The first and second derivatives of 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] in (33) can be obtained as follows: d𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞_𝑖=12𝑝−𝑏𝑞−𝑞_𝑖𝑞²₀^𝐿(𝑞)𝐿⁻¹(𝛼)𝑞dd𝛼−𝐿+𝑝−𝑤,₂𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝑑²_𝑞𝑖1=−2𝑝−𝑏𝑞−𝑞_𝑖𝑞³₀^𝐿(𝑞)𝐿⁻¹(𝛼)𝑞d𝛼+^𝑖𝑞𝐿^′𝑞. (34) Since 𝐿(𝑞) is an increasing function with 𝐿^′(𝑞)>0 and 𝑝>𝑏, therefore d²𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞²_𝑖 is negative and 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] is concave in 𝑞_𝑖. Hence, there exists an optimal order quantity for retailer 𝑖 for each 𝑞_−𝑖, where 𝑞_−𝑖=∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑞_𝑗. A set of order quantity 𝑞^∗=(𝑞^∗₁,…,𝑞^∗_𝑛) is a Nash equilibrium of the decentralized system if each retailer’s order quantity is a best response. Thus, any Nash equilibrium must satisfy each retailer’s first-order condition. Let d𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞_𝑖=0; we can get 𝑞^∗_𝑖=𝑞^∗2𝑝−𝑤+1𝑝−𝑏𝑞^{∗𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)d𝛼−𝐿𝑞^∗×1𝑞^{∗𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)d𝛼⁻¹. (35)
Equation (35) gives each retailer’s equilibrium order conditional on 𝑞^∗ being the equilibrium total order quantity. Hence, (35) describes an equilibrium only if 𝑞^∗=𝑛𝑞^∗_𝑖.
Substitute (35) into 𝑞^∗=𝑛𝑞^∗_𝑖 and simplify 𝐿𝑞^∗−𝑛−1𝑛1𝑞^{∗𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)=2d𝛼𝑝−𝑤.𝑝−𝑏 (36) In order to fully coordinate the supply chain, let 𝐿(𝑞^∗)=2(𝑝−𝑐)/𝑝; we can obtain 𝑤^∗1=𝑝−2×𝐿𝑞𝑝−𝑏_∗−𝑛−1𝑛1𝑞^{∗𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)1d𝛼=𝑝−22𝑝−𝑏𝑝−𝑐𝑝−𝑛−1𝑛×∫₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝛼)d𝛼𝐿⁻¹2.𝑝−𝑐/𝑝 (37) The poof of Theorem 11 is completed.

Theorem 12. When 𝑝>2𝑐, the optimal wholesale price 𝑤^∗ in return contract is 𝑤^∗1=𝑏+2×𝑝−𝑏2𝑐𝑝+𝑛−1𝑛×∫¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝛼+∫¹₀𝐿⁻¹(𝛼)d𝛼𝑅⁻¹.2𝑐/𝑝 (38)

Proof. If 𝑞_𝑖∈((𝑞_𝑖/𝑞)𝑑₂,(𝑞_𝑖/𝑞)𝑑₃], that is, 𝑞∈(𝑑₂,𝑑₃], then the 𝛼 cut sets of min{𝑞_𝑖,(𝑞_𝑖/𝑞)𝐷} and max{𝑞_𝑖−(𝑞_𝑖/𝑞)𝐷,0} are min𝑞_𝑖,𝑞_𝑖𝑞𝐷𝛼)=𝑞_𝑖(𝑞𝐿⁻¹(𝛼),𝑞_𝑖𝑞,𝑞,𝛼∈0,𝑅_𝑖𝑞𝐿⁻¹(𝛼),𝑞_𝑖𝑞𝑅⁻¹(𝛼𝑅𝑞,𝑞),𝛼∈,1max_𝑖−𝑞_𝑖𝑞𝛼)=𝐷,0(0,𝑞_𝑖−𝑞_𝑖𝑞𝐿⁻¹(𝛼𝑞,𝑞),𝛼∈0,𝑅_𝑖−𝑞_𝑖𝑞𝑅⁻¹(𝛼),𝑞_𝑖−𝑞_𝑖𝑞𝐿⁻¹(𝛼),𝛼∈𝑅𝑞,1. (39)(a)For𝛼∈[0,𝑅(𝑞)], the 𝛼 cut set of the retailer 𝑖’s fuzzy profit is Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝛼)=𝑝𝑞_𝑖(𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖,𝑝𝑞_𝑖+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖. (40)(b)For𝛼∈(𝑅(𝑞),1], the result turns to Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝛼)=𝑝𝑞(^𝑖𝑞𝐿⁻¹(𝛼)+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝑅⁻¹(𝛼)−𝑤𝑞_𝑖𝑞,𝑝^𝑖𝑞𝑅⁻¹(𝛼)+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖. (41)By (7), we can get the fuzzy expected profit 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=12₀^𝑅(𝑞)𝑝𝑞_𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖+𝑝𝑞_𝑖+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖+1d𝛼2¹_𝑅(𝑞)𝑝𝑞_𝑖𝑞𝐿⁻¹(𝛼)+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝑅⁻¹(𝛼)−𝑤𝑞_𝑖𝑞+𝑝^𝑖𝑞𝑅⁻¹(𝛼)+𝑏𝑞_𝑖𝑞−𝑏^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖=1d𝛼2𝑞𝑝−𝑏^𝑖𝑞¹_𝑅(𝑞)𝑅⁻¹(𝛼)+𝑞d𝛼_𝑖𝑞¹₀𝐿⁻¹(𝛼)d𝛼+𝑞_𝑖𝑅𝑞−(𝑤−𝑏)𝑞_𝑖. (42) The first and second derivatives of 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] in (42) can be obtained as follows: d𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞_𝑖=12𝑝−𝑏𝑞−𝑞_𝑖𝑞²¹_𝑅𝑞𝑅⁻¹(𝛼)−d𝛼₁₀𝐿⁻¹(𝛼)𝑞−(),dd𝛼+𝑅𝑤−𝑏₂𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞_𝑖²1=−2𝑞𝑝−𝑏^𝑖𝑞³¹_𝑅𝑞𝑅⁻¹(𝛼)−d𝛼₁₀𝐿⁻¹(𝛼)−𝑞d𝛼^𝑖𝑞𝑅^′𝑞. (43) Since 𝑅(𝑞) is a decreasing function with 𝑅^′(𝑞)<0 and 𝑝>𝑏, therefore d²𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞²_𝑖 is negative and 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] is concave in 𝑞_𝑖.
Any Nash equilibrium must satisfy each retailer’s first-order condition. Let d𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞_𝑖=0; we can get 𝑞^∗_𝑖=𝑞^∗1𝑞^∗1_𝑅𝑞^∗𝑅⁻¹(𝛼)1d𝛼+𝑞^∗1₀𝐿⁻¹(𝛼)𝑞d𝛼+𝑅^∗−2(𝑤−𝑏)×1𝑝−𝑏𝑞^∗1_𝑅𝑞^∗𝑅⁻¹(𝛼)1d𝛼+𝑞^∗1₀𝐿⁻¹(𝛼)d𝛼⁻¹. (44) Substitute (44) into 𝑞^∗=𝑛𝑞^∗_𝑖 and simplify 𝑅𝑞^∗+𝑛−1𝑛×1𝑞^∗1_𝑅𝑞^∗𝑅⁻¹(𝛼)1d𝛼+𝑞^∗1₀𝐿⁻¹(𝛼)=2()d𝛼𝑤−𝑏.𝑝−𝑏 (45) In order to fully coordinate the supply chain, let 𝑅(𝑞^∗)=2𝑐/𝑝; we can obtain𝑤^∗1=𝑏+2×𝑅𝑞𝑝−𝑏_∗+𝑛−1𝑛×1𝑞^∗1_𝑅𝑞^∗𝑅⁻¹(𝛼)1d𝛼+𝑞^∗1₀𝐿⁻¹(𝛼)1d𝛼=𝑏+2×𝑝−𝑏2𝑐𝑝+𝑛−1𝑛×∫¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝛼+∫¹₀𝐿⁻¹(𝛼)d𝛼𝑅⁻¹.2𝑐/𝑝 (46) The poof of Theorem 12 is completed.

Theorem 13. In fuzzy return contract, the retailer 𝑖  (𝑖=1,2,…,𝑛) and the supplier attain their optimal fuzzy expected value of the profits at 𝑤^∗ in fuzzy return contract, where 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗=𝑝−𝑏𝑝𝑛²𝐸Π_𝑆𝐶^∗𝐸Π,𝑖=1,2,…,𝑛,_𝑆∗=𝑝(𝑛−1)+𝑏𝐸Π𝑝𝑛^𝑆𝐶∗. (47)

Proof. Consider the following.
Case  1 ( 𝑑₁≤𝑞≤𝑑₂). When 𝑑₁≤𝑞≤𝑑₂, substituting 𝑤^∗ and 𝐿(𝑞^∗)=2(𝑝−𝑐)/𝑝 into (33), the fuzzy expected profit of the retailer 𝑖  (𝑖=1,2,…,𝑛) is given as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗=𝑝−𝑏2𝑛²₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝛼)=d𝛼𝑝−𝑏𝑝𝑛²𝐸Π_SC^∗. (48) The fuzzy expected profit of the supplier is 𝐸Π_𝑆^∗=𝐸Π_SC^∗−𝑛𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗=𝑝(𝑛−1)+𝑏𝐸Π𝑝𝑛^SC∗. (49)
Case  2  ( 𝑑₂≤𝑞≤𝑑₃). When 𝑑₂<𝑞≤𝑑₃, substituting 𝑤^∗ and 𝑅(𝑞^∗)=2𝑐/𝑝 into (42), the fuzzy expected profit of the retailer 𝑖  (𝑖=1,2,…,𝑛) is given as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗=𝑝−𝑏2𝑛²¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝛼+¹₀𝐿⁻¹(𝛼)=d𝛼𝑝−𝑏𝑝𝑛²𝐸Π_SC^∗. (50) The fuzzy expected profit of the supplier is 𝐸Π_𝑆^∗=𝐸Π_SC^∗−𝑛𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗=𝑝(𝑛−1)+𝑏𝐸Π𝑝𝑛^SC∗. (51) The poof of Theorem 13 is completed.

3.3. Revenue-Sharing Contract with Fuzzy Demand

In a revenue-sharing contract, the retailer 𝑖  (𝑖=1,2,…,𝑛) shares with the supplier a percentage of his revenue. Let (1−Φ) be the fraction the supplier earns, and then Φ is the fraction the retailer 𝑖 keeps.

Thus, we can express the fuzzy profit of the retailer 𝑖 as follows: Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=𝑝Φmin𝑞_𝑖,𝐷_𝑖−𝑤𝑞_𝑖=𝑝Φ_𝑖min𝑞_𝑖,𝑞_𝑖𝑞𝐷−𝑤𝑞_𝑖, (52) where 𝑞=∑^𝑛_𝑖=1𝑞_𝑖 and 𝑞_−𝑖=𝑞−𝑞_𝑖.

The retailer 𝑖  (𝑖=1,2,…,𝑛) tries to maximize its fuzzy expected profit 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] in revenue-sharing contract by choosing the optimal order quantity 𝑞_𝑖, which solves the following model: 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=𝐸𝑝Φmin𝑞_𝑖,𝑞_𝑖𝑞𝐷−𝑤𝑞_𝑖𝑞s.t.^𝑖𝑞𝑑₁≤𝑞_𝑖≤𝑞_𝑖𝑞𝑑₃. (53)

Theorem 14. When 𝑐<𝑝≤2𝑐, the optimal wholesale price 𝑤^∗∗ in revenue-sharing contract is 𝑤^∗∗1=𝑝Φ−2×2𝑝Φ𝑝−𝑐𝑝−𝑛−1𝑛×∫₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝛼)d𝛼𝐿⁻¹2.𝑝−𝑐/𝑝 (54)

Proof. If 𝑞_𝑖∈[(𝑞_𝑖/𝑞)𝑑₁,(𝑞_𝑖/𝑞)𝑑₂], that is, 𝑞∈[𝑑₁,𝑑₂], then the 𝛼 cut set of min{𝑞_𝑖,(𝑞_𝑖/𝑞)𝐷} is min𝑞_𝑖,𝑞_𝑖𝑞𝐷𝛼)=𝑞_𝑖(𝑞𝐿⁻¹(𝛼),𝑞_𝑖𝑞,𝑞,𝛼∈0,𝐿_𝑖,𝑞_𝑖,𝛼∈𝐿𝑞,1. (55)(a)For𝛼∈[0,𝐿(𝑞)], the 𝛼 cut set of the retailer 𝑖’s fuzzy profit is Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝛼)=𝑞(𝑝Φ^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖,𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖. (56)(b)For𝛼∈(𝐿(𝑞),1], the result turns to Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖(𝛼)=𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖,𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖. (57)By (7), we can get the fuzzy expected profit 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=12₀^𝐿(𝑞)𝑞𝑝Φ^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖+𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖+1d𝛼2¹_𝐿𝑞𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖+𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖=1d𝛼2𝑞𝑝Φ^𝑖𝑞₀^𝐿(𝑞)𝐿⁻¹(𝛼)d𝛼−𝑞_𝑖𝐿𝑞+𝑝Φ−𝑤𝑞_𝑖. (58) The first and second derivatives of 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] in (58) can be obtained as follows: d𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞_𝑖=12𝑝Φ𝑞−𝑞_𝑖𝑞²₀^𝐿(𝑞)𝐿⁻¹(𝛼)𝑞dd𝛼−𝐿+𝑝Φ−𝑤,₂𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞²_𝑖1=−2𝑝Φ𝑞−𝑞_𝑖𝑞³₀^𝐿(𝑞)𝐿⁻¹(𝛼)𝑞d𝛼+^𝑖𝑞𝐿^′𝑞. (59)
Since 𝐿(𝑞) is an increasing function with 𝐿^′(𝑞)>0 and Φ>0, therefore d²𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞²_𝑖 is negative and 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] is concave in 𝑞_𝑖. Hence, there exists an optimal order quantity for retailer 𝑖 for each 𝑞_−𝑖. A set of order quantity, 𝑞^∗=(𝑞^∗₁,…,𝑞^∗_𝑛) is a Nash equilibrium of the decentralized system if each retailer’s order quantity is a best response.
Any Nash equilibrium must satisfy each retailer’s first-order condition. Let d𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞_𝑖=0; we can get 𝑞₁^∗∗=𝑞^∗2𝑝Φ−𝑤/𝑝Φ+1/𝑞^∗∫^{𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)d𝛼−𝐿𝑞^∗1/𝑞^∗∫^{𝐿(𝑞∗)}₀𝐿⁻¹(𝛼).d𝛼 (60)
Equation (60) gives each retailer’s equilibrium order conditional on 𝑞^∗ being the equilibrium total order quantity. Hence, (60) describes an equilibrium only if 𝑞^∗=𝑛𝑞_𝑖^∗∗.
Substitute (60) into 𝑞^∗=𝑛𝑞_𝑖^∗∗ and simplify 𝐿𝑞^∗−𝑛−1𝑛1𝑞^{∗𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)=2d𝛼𝑝Φ−𝑤.𝑝Φ (61) In order to fully coordinate the supply chain, let 𝐿(𝑞^∗)=2(𝑝−𝑐)/𝑝; we can obtain 𝑤^∗∗1=𝑝Φ−2×𝐿𝑞𝑝Φ_∗−𝑛−1𝑛1𝑞^{∗𝐿(𝑞∗)}₀𝐿⁻¹(𝛼)1d𝛼=𝑝Φ−2×2𝑝Φ𝑝−𝑐𝑝−𝑛−1𝑛×∫₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝛼)d𝛼𝐿⁻¹2.𝑝−𝑐/𝑝 (62) The poof of Theorem 14 is completed.

Theorem 15. When 𝑝>2𝑐, the optimal wholesale price 𝑤^∗∗ in revenue-sharing contract is 𝑤^∗∗=12×𝑝Φ2𝑐𝑝+𝑛−1𝑛×∫¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝛼+∫¹₀𝐿⁻¹(𝛼)d𝛼𝑅⁻¹.2𝑐/𝑝 (63)

Proof. If 𝑞_𝑖∈((𝑞_𝑖/𝑞)𝑑₂,(𝑞_𝑖/𝑞)𝑑₃], that is, 𝑞∈(𝑑₂,𝑑₃], then the 𝛼 cut set of min{𝑞_𝑖,(𝑞_𝑖/𝑞)𝐷} is min𝑞_𝑖,𝑞_𝑖𝑞𝐷𝛼)=𝑞_𝑖(𝑞𝐿⁻¹(𝛼),𝑞_𝑖𝑞,𝑞,𝛼∈0,𝑅_𝑖𝑞𝐿⁻¹(𝛼),𝑞_𝑖𝑞𝑅⁻¹(𝛼),𝛼∈𝑅𝑞,1. (64)(a)For 𝛼∈[0,𝑅(𝑞)], the 𝛼 cut set of the retailer 𝑖’s fuzzy profit is Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝛼)=𝑞(𝑝Φ^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖,𝑝𝑞_𝑖−𝑤𝑞_𝑖. (65)(b)For𝛼∈(𝑅(𝑞),1], the result turns to Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖𝛼)=𝑞(𝑝Φ^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖𝑞,𝑝Φ^𝑖𝑞𝑅⁻¹(𝛼)−𝑤𝑞_𝑖. (66)By (7), we can get the fuzzy expected profit 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖=12₀^𝑅(𝑞)𝑞𝑝Φ^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖+𝑝Φ𝑞_𝑖−𝑤𝑞_𝑖+1d𝛼2¹_𝑅(𝑞)𝑞𝑝Φ^𝑖𝑞𝐿⁻¹(𝛼)−𝑤𝑞_𝑖𝑞+𝑝Φ^𝑖𝑞𝑅⁻¹(𝛼)−𝑤𝑞_𝑖=1d𝛼2𝑞𝑝Φ^𝑖𝑞¹_𝑅𝑞𝑅⁻¹(𝛼)+𝑞d𝛼_𝑖𝑞¹₀𝐿⁻¹(𝛼)d𝛼+𝑞_𝑖𝑅𝑞−𝑤𝑞_𝑖. (67) The first and second derivatives of 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] in (67) can be obtained as follows: d𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞_𝑖=12𝑝Φ𝑞−𝑞_𝑖𝑞²¹_𝑅𝑞𝑅⁻¹(𝛼)−d𝛼₁₀𝐿⁻¹(𝛼)𝑞dd𝛼+𝑅−𝑤,₂𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖d𝑞²_𝑖1=−2𝑞𝑝Φ^𝑖𝑞³¹_𝑅𝑞𝑅⁻¹(𝛼)−d𝛼₁₀𝐿⁻¹(𝛼)−𝑞d𝛼^𝑖𝑞𝑅^′𝑞. (68) Since 𝑅(𝑞) is a decreasing function with 𝑅^′(𝑞)<0 and Φ>0, therefore d²𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞²_𝑖 is negative and 𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)] is concave in 𝑞_𝑖.
Any Nash equilibrium must satisfy each retailer’s first-order condition. Let d𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]/d𝑞_𝑖=0; we can get 𝑞_𝑖^∗∗=𝑞^∗1𝑞^∗1_𝑅(𝑞^∗₎𝑅⁻¹(𝛼)+1d𝛼𝑞^∗1₀𝐿⁻¹(𝛼)d𝛼+𝑅𝑞^∗−2𝑤×1𝑝Φ𝑞^∗1_𝑅(𝑞^∗₎𝑅⁻¹(𝛼)1d𝛼+𝑞^∗1₀𝐿⁻¹(𝛼)d𝛼⁻¹. (69) Substitute (69) into 𝑞^∗=𝑛𝑞_𝑖^∗∗ and simplify 𝑅𝑞^∗+𝑛−1𝑛×1𝑞^∗1_𝑅(𝑞^∗₎𝑅⁻¹(𝛼)1d𝛼+𝑞^∗1₀𝐿⁻¹(𝛼)d𝛼=2𝑤.𝑝Φ (70) In order to fully coordinate the supply chain, let 𝑅(𝑞^∗)=2𝑐/𝑝; we can obtain 𝑤^∗∗=12𝑝Φ𝑅𝑞^∗+𝑛−1𝑛×1𝑞^∗1_𝑅𝑞^∗𝑅⁻¹(𝛼)+1d𝛼𝑞^∗1₀𝐿⁻¹(𝛼)=1d𝛼2𝑝Φ2𝑐𝑝+𝑛−1𝑛×∫¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝛼+∫¹₀𝐿⁻¹(𝛼)d𝛼𝑅⁻¹.2𝑐/𝑝 (71) The poof of Theorem 15 is completed.

Theorem 16. In fuzzy revenue-sharing contract, the retailer 𝑖  (𝑖=1,2,…,𝑛) and the supplier attain their optimal fuzzy expected value of the profits at 𝑤^∗∗ in fuzzy return contract, where 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗=Φ𝑛²𝐸Π_𝑆𝐶^∗𝐸Π,𝑖=1,2,…,𝑛,_𝑆∗∗=𝑛−Φ𝑛𝐸Π_𝑆𝐶^∗. (72)

Proof. Consider the following.
Case  1  ( 𝑑₁≤𝑞≤𝑑₂). When 𝑑₁≤𝑞≤𝑑₂, substituting 𝑤^∗∗ and 𝐿(𝑞^∗)=2(𝑝−𝑐)/𝑝 into (35), the fuzzy expected profit of the retailer 𝑖  (𝑖=1,2,…,𝑛) is given as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗=𝑝Φ2𝑛²₀^{2(𝑝−𝑐)/𝑝}𝐿⁻¹(𝛼)=Φd𝛼𝑛²𝐸Π_SC^∗. (73) The fuzzy expected profit of the supplier is 𝐸Π_𝑆^∗∗=𝐸Π_SC^∗−𝑛𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗=𝑛−Φ𝑛𝐸Π_SC^∗. (74)
Case  2  ( 𝑑₂<𝑞≤𝑑₃). When 𝑑₂<𝑞≤𝑑₃, substituting 𝑤^∗∗ and 𝑅(𝑞^∗)=2𝑐/𝑝 into (44), the fuzzy expected profit of the retailer 𝑖  (𝑖=1,2,…,𝑛) is given as 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗=𝑝Φ2𝑛²¹_2𝑐/𝑝𝑅⁻¹(𝛼)d𝛼+¹₀𝐿⁻¹(𝛼)=Φd𝛼𝑛²𝐸Π_SC^∗. (75) The fuzzy expected profit of the supplier is 𝐸Π_𝑆^∗∗=𝐸Π_SC^∗−𝑛𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗=𝑛−Φ𝑛𝐸Π_SC^∗. (76) The poof of Theorem 16 is completed.

Theorem 17. If 𝑏=𝑝(1−Φ), then the fuzzy expected profits for retailer 𝑖  (𝑖=1,2,…,𝑛) and supplier in return contract are equal to those in revenue-sharing contract: 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗=𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗,𝐸Π_𝑆^∗∗=𝐸Π_𝑆^∗. (77)

Proof. Substituting  𝑏=𝑝(1−Φ)  into  𝐸[Π_𝑅𝑖(𝑞_𝑖,𝑞_−𝑖)]^∗=((𝑝−𝑏)/𝑝𝑛²)𝐸[Π_SC]^∗ and  𝐸[Π_𝑆]^∗=((𝑝(𝑛−1)+𝑏)/(𝑝𝑛))𝐸[Π_SC]^∗, we can obtain 𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗=Φ𝑛²𝐸Π_SC^∗=𝐸Π_𝑅𝑖𝑞_𝑖,𝑞_−𝑖^∗∗,𝐸Π_𝑆^∗=𝑛−Φ𝑛𝐸Π_SC^∗=𝐸Π_𝑆^∗∗. (78) The poof of Theorem 17 is completed.

4. Numerical Example

In this section, we tend to further elucidate the previously proposed two contracts with a numerical example. We will analyze the effect of the number of retailer 𝑛 on the other parameters.

Suppose the most possible value of the demand in market is 𝑑₂=2000 units and the maximum and minimum possible values of the demand are, respectively, 𝑑₁=1000 units and 𝑑₃=3000 units; that is to say, the fuzzy demand is 𝐷=(1000,2000,3000). Let 𝑝=100.00$ per unit.

The optimal order quantity, wholesale price, and fuzzy expected profit of the actors in return contract and revenue-sharing contract can be listed in Tables 1 and 2, respectively.

Table 1 

Equilibrium value of the parameters for different 𝑐 and 𝑛 in return contract (𝑏=30.00).

Table 2 

Equilibrium value of the parameters for different 𝑐 and 𝑛 in revenue sharing contract (Φ=0.60).

From Tables 1 and 2, we analyze the influence of parameters 𝑐 and 𝑛on the optimal equilibrium values as shown below.(a)It is obvious that the optimal order quantity for retailer 𝑖 will decrease along with the rise of the supplier’s production cost 𝑐 and the number of retailers 𝑛 when the other parameters are fixed. Particulary, in this numerical example, the optimal total order quantity 𝑞^∗ is equal to the most possible value of fuzzy demand when 𝑐=50. When 𝑐>50 and 𝑐<50, the optimal total order quantity of retailer is located at the left and right of the most possible value of fuzzy demand 𝐷, respectively. The optimal wholesale price will increase along with the supplier’s production cost 𝑐 and thenumber of retailers 𝑛 when the other parameters are fixed. (b)From Tables 1 and 2, it can be noted that when the other parameters are fixed in both contracts, the fuzzy expected profit of the retailer 𝑖 will decrease along with the rise of parameters 𝑐 and 𝑛. On the other hand, the fuzzy expected profit of the supplier will decrease along with the rise of 𝑐 and increase along with the rise of 𝑛.

5. Conclusions

This paper formulates supply chain contracts based on fuzzy set theory, where a supplier and multiple competing retailers adopt return contract and revenue-sharing contract. In order to examine models performance in fuzzy demand, we use fuzzy cut sets method to solve this problem. The advantage of the proposed method is that it removes the need for enumeration over alternative values and provides a better understanding of the relations among the decision parameters. The technique proposed in this paper is easier to implement and requires less data. It is appropriate when the environment is complex, ambiguous, or there is a lack of statistical data.

Acknowledgments

The author wishes to express his sincerest thanks to the editors and anonymous referees for their constructive comments and suggestions on this paper. This work was supported by the National Natural Science Foundation of China (70972005, 71071018) and the Doctoral Foundation of Heze University (XY12BS03).