Pricing decisions of a twoechelon supply chain with one manufacturer and duopolistic retailers in fuzzy environment are considered in this paper. The manufacturer produces a product and sells it to the two retailers, who in turn retail it to end customers. The fuzziness is associated with the customers’ demand and the manufacturing cost. The purpose of this paper is to analyze the effect of two retailers’ different pricing strategies on the optimal pricing decisions of the manufacturer and the two retailers themselves in MS Game scenario. As a reference model, the centralized decision scenario is also considered. The closedform optimal pricing decisions of the manufacturer and the two retailers are derived in the above decision scenarios. Some insights into how pricing decisions vary with decision scenarios and the two retailers’ pricing strategies in fuzzy environment are also investigated, which can serve as the basis for empirical study in the future.
There is abundant literature on the pricing/ordering policies for twoechelon supply chain management. Most of them focused on the twoechelon supply chain with one manufacturer/supplier and one retailer/buyer and adopted the following assumption on the channel structure: the manufacturer/supplier wholesales a product to the retailer/buyer who in turn retails it to end consumers. The retail market demand varies with the retail price according to a deterministic/stochastic demand function that is assumed to be known to both the manufacturer and the retailer, and the costs incurred in the manufacturing and inventory process are positive constant numbers. Moreover, a most common gaming assumption on the pricing/ordering decision process is that the manufacturer is a Stackelberg leader and the retailer is a Stackelberg follower (hereafter “MS Game”) in the existing twoechelon supply chain literature. For example, A. H. L. Lau and H. S. Lau [
In fact, in order to make effective supply chain management, uncertainties that happen in the real world cannot be ignored. Those uncertainties are usually associated with product supply, manufacturing cost, customer demand, and so on. The quantitative demand forecasts based on manager’s judgements, intuitions, and experience seem to be more appropriate, and the fuzzy theory rather than probability theory should be applied to model this kind of uncertainties [
Many researchers have already adopted fuzzy theory to depict uncertainties in the supply chain model. Zhao et al. [
This paper extends the current model related to twoechelon supply chain pricing issue from two aspects: one is considering fuzziness associated with customer’s demands and the manufacturing cost; the other is analyzing the effect of the two retailers’ different pricing strategies (e.g., Bertrand, Cooperation and Stackelberg) on the optimal pricing decisions of the manufacturer and the duopolistic retailers in MS Game scenario. First, as a benchmark model, one centralized pricing model (namely, assume that the manufacturer and the duopolistic retailers behave as part of a unified system) is established. Second, based on the two retailers’ different pricing strategies, three decentralized pricing models are constructed in fuzzy environment (e.g., the MSB model where the two retailers implement the Bertrand competition, the MSC model where the two retailers implement the cooperation strategy, and the MSS model where the two retailers implement the Stackelberg competition) and the effect of the two retailers’ different pricing strategies on the pricing decisions of the manufacturer and the two retailers is considered. Third, the closedform solutions for these models are provided. Finally, we provide numerical examples to show the difference among each firm’s optimal pricing decisions, the difference among each firm’s maximum expected profits, and the variation of each firm’s optimal pricing strategy and maximum expected profit with the two retailers’ pricing strategies and these decision scenarios in fuzzy environment.
The rest of the paper is organized as follows. Section
A possibility space is defined as a triplet
Pos
Pos
Pos
Let
Suppose that
A fuzzy variable is defined as a function from the possibility space
A fuzzy variable
Let
The independence of fuzzy variables was discussed by several researchers, such as Zadeh [
The fuzzy variables
Let
Let
The triangular fuzzy variable
Let
Let
Let
The triangular fuzzy variable
Let
Let
Let
Consider a twoechelon supply chain with one monopolistic manufacturer and two duopolistic retailers (retailer 1 and retailer 2) in fuzzy environment. The monopolistic manufacturer manufactures products and sells them to the duopolistic retailers, who in turn retail them to end customers. The manufacturer produces products with unit manufacturing cost
Similar to McGuire and Staelin [
In our model, the manufacturer can influence the market demand by setting his wholesale price, and the retailers can also influence the market demands by making their retail prices, respectively. We assume that the chain members are independent, risk neutral, and profit maximizing. The chain members choose their decisions sequentially in a manufacturerStackelberg game (namely, the manufacturer acts as the Stackelberg leader and the duopolistic retailers act as the followers), and they have complete information about the other members. Moreover, the logistic cost components of the chain members (i.e
From the above problem description, the manufacturer’s objective is to maximize his expected profit
The objectives of the retailers are to maximize their respective expected profits
As a benchmark to evaluate channel decisions under different decision cases, we first give the centralized pricing model; namely, there is one entity that aims to optimize the whole supply chain system performance, so both the duopolistic retailers’ and the manufacturer’s decisions are fully coordinated in the centralized decision case. The wholesale price charged by the manufacturer is seen as inner transfer price and thus will be neglected. The total profit is determined by the production cost and retail prices.
Let
To maximize the system expected profit
In the CD model, the optimal retail prices
From (
Then, the firstorder derivatives of
By (
So, solving (
In this section, we assume that the manufacturer acts as the Stackelberg leader and the duopolistic retailers act as the followers. The gametheoretical approach is used to analyze the models established in the following. For this case, the manufacturer chooses the wholesale prices of the product using the response functions of both the retailers. Then, given the wholesale prices made by the manufacturer, the duopolistic retailers determine their retail prices.
When the two retailers pursue the Bertrand solution, the manufacturer first announces the wholesale price and the two retailers observe the wholesale price and then decide the retail prices simultaneously. Then the MSB model is formulated as
We first derive the retailers’ Bertrand decisions as follows.
When the duopolistic retailers pursue the Bertrand solution, the optimal retail prices (denoted by
Using (
From (
Similarly, from (
Therefore, solving (
Having the information about the decisions of the two retailers, the manufacturer would then use them to maximize his expected profit
When the duopolistic retailers pursue the Bertrand solution, the manufacturer’s optimal wholesale price (denoted by
By (
When the duopolistic retailers pursue the Bertrand solution, their optimal retail prices (denoted by
By Propositions
In this decision case where the two retailers adopt the cooperation strategy, we assume that the retailers recognize their interdependence and agree to act in union in order to maximize the total expected profit of the downstream retail market. So, the manufacturer first announces the wholesale price and the retailers observe the wholesale price and then decide their retail prices with the objective to maximize the total expected profit of the downstream retail market. Thus, the MSC model is formulated as
When the two retailers adopt the cooperation strategy, their optimal retail prices
By (
From (
Having the information about the decisions of the retailers, the manufacturer would then use them to maximize his expected profit
When the two retailers adopt the cooperation strategy, the manufacturer’s optimal wholesale price (denoted as
By (
By (
When the two retailers adopt the cooperation strategy, their optimal retail prices
By Propositions
In this decision case when the duopolistic retailers play Stackelberg Game, we assume that one of the duopolistic retailers (e.g., retailer 1) acts as a Stackelberg leader and the other (i.e., retailer 2) acts as a Stackelberg follower. The manufacturer first announces the wholesale price of the product, and retailer 1 then decides the retail price to maximize her expected profit and retailer 2 finally decides the retail price when knowing both the manufacturer and retailer 1 decisions. So, we first need to derive retailer 2’s decision (as the Stackelberg game’s follower). The MSS model is formulated as follows:
We first derive retailer 2’s decision as follows.
When the duopolistic retailers play Stackelberg Game, retailer 2’s optimal decision (denoted as
Using (
By (
When the duopolistic retailers play Stackelberg Game, retailer 1’s optimal decision (denoted as
Using (
By (
When the duopolistic retailers play Stackelberg Game, retailer 2’s optimal decision (denoted as
Using Propositions
When the duopolistic retailers play Stackelberg Game, the manufacturer’s optimal decision is (denoted by
Using (
By (
When the duopolistic retailers play Stackelberg Game, their optimal retail prices (denoted by
By Propositions
In this section, we compare analytical results obtained from the above different decision scenarios using numerical approach and study the behavior of firms facing changing fuzzy environment. Here, we assume that the fuzzy variables used in this paper are all triangular fuzzy variables which take values as follows: the manufacturing cost
Maximum expect profit of total system and every firm under different pricing models.
Pricing model 





CD model  7696.5  
MSB model  7388.7  6301.3  701.1  386.3 
MSC model  6163.1  4604.8  1058.1  500.2 
MSS model  6778.0  5492.1  546.5  739.4 
Optimal retail prices and wholesale price under different pricing models.
Pricing model 






CD model  17.3190  16.9190  
MSB model  19.0790  1.9600  18.5740  1.4549  17.1190 
MSC model  21.1405  4.0214  20.7405  3.6214  17.1190 
MSS model  20.5076  3.4869  19.0337  2.0129  17.0208 
From Tables
The expected profit of the total supply chain in the centralized decision case is higher than that in all decentralized decision cases.
One can observe directly from Table
From Table
From Table
From Table
From Table
We observe from Table
In order to see how the two retailers’ different competitive behaviors affect the optimal pricing policy and the total expected profits of the manufacturer and the two retailers, we further assume that the retailers have the same market bases (here we set
Maximum profit of total system and every firm under different pricing models.
Pricing model 





CD model  5790.5  
MSB model  5572.3  4813.9  379.2  379.2 
MSC model  4697.6  3604.8  546.4  546.4 
MSS model  5212.3  4311.1  289.1  612.1 
Optimal retail prices and wholesale price under different pricing models.
Pricing model 






CD model  15.9286  15.9286  
MSB model  17.3701  1.4415  17.3701  1.4415  15.9286 
MSC model  19.1548  3.2262  19.1548  3.2262  15.9286 
MSS model  18.4646  2.5361  17.7599  1.8313  15.9286 
From Tables
The expected profit of the whole supply chain system in the centralized decision is higher than that in all decentralized decisions. This is consistent with the general case when the two retailers have different market bases.
One can observe directly from Table
From Table
From Table
One can observe directly from Table
From Table
We have analyzed the duopolistic retailers’ and the manufacturer’s pricing decisions by considering the duopolistic retailers’ three kinds of pricing strategies: Bertrand, Cooperation, and Stackelberg in fuzzy environment. As a benchmark to evaluate channel decision in different decision case, we first developed the pricing model in centralized decision case and derived the optimal retail prices. We then established the pricing models in decentralized decision cases by considering the duopolistic retailers’ three kinds of pricing strategies and obtained the analytic equilibrium decisions. Finally, we provided comparison of the expected profits and optimal pricing decisions of the whole supply chain and every supply chain members in both the general case (namely, the two retailers have different market bases) and the special case (viz., the two retailers have the same market bases). The analytical and numerical results revealed some insights into the economic behavior of firms.
Our results, however, are based upon some assumptions about the twoechelon supply chain models. Thus, several extensions to the analysis in this paper are possible by considering the duopolistic retailers’ three kinds of pricing strategies. First, as opposed to the risk neutral twoechelon supply chain members considered in this paper, one could study the case where the supply chain members with different attitudes toward risk and could also examine the influence of their attitudes toward risk on individual expected profits and the expected profit of the whole supply chain. This would add complications to the analysis of the twoechelon supply chain members’ decisions. Second, we assumed that both the duopolistic retailers and the manufacturer have symmetric information about costs and demands. So, an extension would be to consider the twoechelon supply chain models in information asymmetry, such as asymmetry in cost information and demand information. Finally, we can also consider the coordination of the twoechelon supply chain under linear or isoelastic demand with symmetric and asymmetric information.
The authors wish to express their sincerest thanks to the editors and anonymous referees for their constructive comments and suggestions on the paper. This research was supported in part by the National Natural Science Foundation of China, nos.: 71001106, 70971069.