Pricing decisions of two complementary products in a fuzzy environment are considered in this paper. The purpose of this paper is to analyze the changes of the optimal retail pricing of two complementary products under two different decentralized decision scenarios (e.g., Nash game case and Stackelberg game case). As a reference model, the centralized pricing model is also established. The closedform optimal pricing decisions of the two complementary products are obtained in the above three decision scenarios. Some interesting management insights into how pricing decisions vary with decision scenarios are given.
Pricing plays an important role for a product because the long run survival of this product depends upon the firm's ability to increase its sales and obtain the maximum profit from the existing and new capital investment on this product. Furthermore, customers will inevitably apply price as a clue to form their attitude about a product if they lack complete information [
In the last several years, a considerable amount of research has studied the optimal pricing problem. Among these studies, some results are established by considering a single product or substitutable products. For example, Cai et al. [
On the other hand, some results are established by considering complementary products (products are complementary if a reduction in the price of one leads to an increase in demand for the other, which are the opposite of substitutable products). For example, Yue et al. [
In fact, in order to make pricing schemes more effective, the uncertainties that happen in the real world cannot be neglected. Those uncertainties are usually associated with the product supply, the customer demand, and so on. There is an abundant literature that describes uncertainty in demand and/or lead time using probability distributions with known parameters, for example, Khan et al. [
In these situations, the uncertainty parameters can be approximately estimated by manager's judgements, intuitions, and experience and can be characterized as fuzzy variables [
However, as far as we know, no research has considered the pricing problem of complementary products in a fuzzy supply chain environment. In this paper, we consider the pricing decisions of two complementary products in a fuzzy environment. The fuzziness is associated with the manufacturing costs and customer demands that are sensitive to the retail prices of two complementary products. The closedform optimal pricing decisions of the two complementary products under two different decentralized decision scenarios (e.g., Nash game case and Stackelberg game case) are established. As a reference model, the centralized pricing model is also considered and the corresponding closedform optimal pricing decisions of the two complementary products are also given. Some interesting management insights into how pricing decisions vary with decision scenarios are given.
The rest of the paper is organized as follows. Section
Consider a supply chain of two complementary products with two manufacturers (labeled manufacturer 1 and manufacturer 2). The manufacturer
Similar to Mukhopadhyay et al. [
The total demand for product 1 (denoted as
Similarly, the total demand for product 2 (denoted as
The profits of manufacturer 1 (denoted as
The total profit (denoted as
As a benchmark to evaluate pricing decisions under different decision cases, we first examine the centralized pricing model (namely, assume that there is one entity who aims to optimize the total profit of manufacturer 1 and manufacturer 2). The two manufacturers' pricing decisions are fully coordinated in this case.
According to the above description, we know that the objective of the centralized pricing case is to maximize the expected total profit of manufacturer 1 and manufacturer 2, which can be denoted as
In the centralized pricing case, the optimal retail prices (denoted as
Using (
The first and second order partial derivatives of
With (
In this section, we assume that the two manufacturers are independent with each other in the supply chain system. They make their own decision with the objective to maximize their own expected profit. The gametheoretical approach is used to analyze the pricing decision models established in the following. The assumption regarding which firm possesses the bargaining power in the supply chain system can influence how the pricing models are solved.
The NG model represents a market in which there are relatively small to mediumsized manufacturers. The two manufacturers have the identical market power; they determine their strategies independently and simultaneously. Because manufacturer 1 cannot dominate the market over manufacturer 2, his price decision is conditioned on the retail price of product 2. On the contrary, manufacturer 2 must make his own retail price, simultaneously, conditional on the product 1’s retail price. This situation is called a Nash game and the solution to this structure is the Nash equilibrium. Therefore, we have the following results expressed in Proposition
In Nash game case, the two manufacturers' optimal retail prices (denoted as
Using (
The first and second order derivatives of
With (
By setting (
The SG model arises in a market where one manufacturer's size is smaller compared to the other manufacturers. So, the two manufacturers have different market powers, they determine their strategies independently and sequentially. Variation in the bargaining power can create one of the following two scenarios besides the above Nash game: (1) manufacturer 1 lead Stackelberg game, manufacturer 1 holds more bargaining power than manufacturer 2 and thus is the Stackelberg leader and (2) Manufacturer 2 lead Stackelberg game, manufacturer 2 holds more bargaining power than manufacturer 1 and thus is the Stackelberg leader. Without loss of generality, in this section, we assume that manufacturer 1 acts as the Stackelberg leader and manufacturer 2 acts as the Stackelberg follower.
When manufacturer 1 acts as the Stackelberg leader and manufacturer 2 acts as the Stackelberg follower. Manufacturer 1 first announces the retail price of the product 1. Having the information about the decision of manufacturer 1, manufacturer 2 would then use it to maximize his expected profit. So, we have the following result given as in Proposition
When the two manufacturers play Stackelberg game and manufacturer 1 is the game leader, manufacturer 2’s optimal retail price (denoted as
Using (
When the two manufacturers play Stackelberg game and manufacturer 1 is the game leader, manufacturer 1’s optimal retail price (denoted as
It follows from (
Using (
When the two manufacturers play Stackelberg game and manufacturer 1 is the game leader, manufacturer 2’s optimal retail price (denoted as
By Propositions
Because the optimal pricing strategy obtained in this paper is in a very complicated form, especially the
In this section, we compare the analytical results obtained from the above three different decision scenarios using numerical approach and study the behavior of the two firms facing changing decision environment. Using the analytical results obtained from the above three different decision scenarios, we can easily see that the expressions of the optimal retail prices and maximum expected profits under different decision scenarios.
Here, we consider two manufacturers of a consumer electronics manufacturing industry in China, the data used in the following numerical example are from them, which satisfies or closely satisfies the assumptions of this paper. These data have been appropriately manipulated, for example, standardization and nondimensionalization, before being employed, which satisfies or closely complies with certain assumptions of this research. We think these data can represent the realworld condition as closely as possible due to the difficulty of accessing the actual industry data.
The relationship between linguistic expressions and triangular fuzzy variables for manufacturing cost, primary demand, and price elasticity are often determined by experts' experiences, as shown in Table
Relation between linguistic expression and triangular fuzzy variable.
Linguistic expression  Triangular fuzzy variable  

Manufacturing cost 
High (about 8)  (6, 8, 9) 
Medium (about 5)  (4, 5, 6)  
Low (about 2)  (1, 2, 4)  


Manufacturing cost 
High (about 16)  (14, 16, 19) 
Medium (about 9)  (7, 9, 13)  
Low (about 4)  (2, 4, 6)  


Primary demand 
Large (about 850)  (650, 850, 950) 
Small (about 300)  (200, 300, 350)  


Price elasticity 
Very sensitive (about 11)  (9, 11, 13) 
Sensitive (about 6)  (3, 6, 8)  


Price elasticity 
Very sensitive (about 10)  (8, 10, 11) 
Sensitive (about 7)  (5, 7, 9)  


Price elasticity 
Very sensitive (about 14)  (12, 14, 16) 
Sensitive (about 9)  (6, 9, 11)  


Price elasticity 
Very sensitive (about 10)  (8, 10, 11) 
Sensitive (about 7)  (5, 7, 8)  


Price elasticity 
Very sensitive (about 9)  (7, 9, 12) 
Sensitive (about 6)  (4, 6, 8)  


Price elasticity 
Very sensitive (about 11)  (8, 11, 14) 
Sensitive (about 8)  (7, 8, 9) 
Consider the case where the manufacturing costs
The optimal expected profits under different decision cases.
Scenario 




CD model  370580  
NG model  26494  13915  12580 
SG model  25292  13297  11995 
Optimal decisions of retail prices under different decision cases.
Scenario 



CD model  116.2764  41.8494 
NG model  38.4315  47.2720 
SG model  33.0081  48.3567 
The expected profit of the whole supply chain in CD model is the biggest one among the three pricing models, and the expected profit of the whole supply chain in NG model is the bigger one between the two decentralized pricing models. Moreover, both manufacturer 1 and manufacturer 2 achieve their own higher expected profits in NG model followed by SG model. This indicates that the NG game involving two manufacturers is more beneficial to the whole supply chain than their Stackelberg game; similarly, the NG game involving two manufacturers is more beneficial to each manufacturer than their Stackelberg game. From which one can see that the leader does not have the advantage to achieve higher expected profit in SG model, the whole supply chain and each manufacturer are better off when no channel member in a dominant position. This means that the leadership between the two manufacturers reduces profits of the whole supply chain and each manufacturer.
Table
We analyze the pricing decisions for two complementary products under three different decision cases. As a benchmark to evaluate channel decision in different decision cases, we first develop the pricing model in centralized decision case and derive the optimal retail prices. We then establish the pricing models in two decentralized decision cases (e.g., NG model and SG model) and give the optimal retail prices. Finally, we provide comparison of the expected profits and optimal pricing decisions of the whole supply chain and each supply chain members. Some interesting insights into the economic behavior of firms are established in this paper.
However, our results are based upon some assumptions about the pricing models of complementary products. Thus, several extensions to the analysis in this paper are possible. First, this paper considers a case with lineardemand function in a singleperiod, one can study the case with other forms of demand function in multiple periods. Second, we assumed that both the two manufacturers have symmetric information about costs and demands. So, an extension would be to consider the supply chain with information asymmetry, such as asymmetry in cost information and demand information. Third, we can also consider the coordination of the supply chain under linear or isoelastic demand with symmetric and asymmetric information.
For the preliminary on fuzzy set used in this paper, see Appendix 1 in [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the support of the (i) National Natural Science Foundation of China (NSFC), Research Fund nos. 71001106, 71371186, and 91024002 for J. Wei; (ii) jointly funded projects of National Natural Science Foundation Committee and Civil Aviation Administration of China no. U1333111 and Fundamental Research Project of Central Universities no. ZXB 2011A003, for L. Wang; and (iii) NSFC Research Fund no. 71301116, for J. Zhao.