Simultaneous Decentralized Competitive Supply Chain Network Design under Oligopoly Competition

Introduction

Today’s international business and open markets promote developing countries to omit monopoly and enroll in the World Trade Organization (WTO) to achieve benefits from open world trading, so they ratify different foreign investment strategies and policies to observe international investors. In these situations, the investors come across good opportunities to design their networks domestically and obtain intact markets encountered by simultaneous competitions. On the other hand, competition is promoted by firms against firms and supply chains versus supply chains. So, investors have questions such as the following: What is the best network design in this competitive mode? How many market shares can be obtained? What is their equilibrium condition? This paper aims to provide solutions to these questions.

According to the Supply Chain Network Design (SCND) literature many studies on monopoly assumptions (Altiparmak et al., 2006; Badri et al., 2013; Özceylan et al., 2014; Vahdani & Mohamadi, 2015; Yang et al., 2015 b; Ardalan et al., 2016; Keyvanshokooh et al., 2016; Özceylan et al., 2016; Jeihoonian et al., 2017; Varsei & Polyakovskiy, 2017) have been conducted. Several examples of supply chain (SC) competition can be found in maritime shipping, the automotive and retail industries, and online bookstores (see Farahani et al., 2014 for a review on competitive supply chain network design CSCND).

Players and customers are two integral elements in CSCND. Based on players’ reactions, a newcomer encounters monopoly competition (i.e., no rival exists, or existing rivals show no reactions to the newcomer), duopoly competition (i.e., just one rival shows reactions to the newcomer), and oligopoly competition (i.e., more than one rival show reactions to the newcomer). Based on players’ reactions, three types of competition have been discussed in the literature: Static competition (Berman & Krass, 1998; Aboolian et al., 2007a, 2007b; Revelle et al., 2007), dynamic competition (Sinha & Sarmah, 2010; Friesz et al., 2011; Jain et al., 2014; Chen et al., 2015; Nagurney et al., 2015; Santibanez-Gonzalez & Diabat, 2016; Hjaila et al., 2016 b; Lipan et al., 2017), and competition with foresight (Zhang & Liu, 2013; Yue & You, 2014; Zhu, 2015; Drezner et al., 2015; Yang et al., 2015 a; Hjaila et al., 2016 a; Aydin et al., 2016; Genc & Giovanni, 2017).

Customer behavior and customer demand functions are important factors in CSCND. Customer demand can be elastic or inelastic, and in the case of elastic demand, it can depend on price, service, price and service, or price and distance (Farahani et al., 2014). Customer utility functions are mostly categorized into deterministic, introduced by Hotelling (1929), and random utility, introduced by Huff (1964, 1966) models. Moreover, three types of competition exist in the SC competition literature: Horizontal competition (Nagurney et al., 2002; Cruz, 2008; Zhang & Zhou, 2012; Qiang et al., 2013; Huseh, 2015; Qiang, 2015; Li & Nagurney, 2016; Nagurney et al., 2016), vertical competition (Chen et al., 2013; Wu, 2013; Zhao & Wang, 2015; Zhang et al., 2015; Bai et al., 2016; Bo & Li, 2016; Li & Nagurney, 2016; Huang et al., 2016; Wang et al., 2017; Genc & Giovanni, 2017; Chaeb & Rasti-Barzoki, 2017), and SC versus SC (Li et al., 2013; Chung & Kwon, 2016).

Contribution of This Paper

This paper addresses a simultaneous decentralized supply chain network design problem (SD-SCND) in which  decentralized SCs simultaneously enter a virgin market, shape their networks, set the wholesale and retail prices, and specify flows of products among the tiers in dynamic competition without any cooperation. This problem and this paper’s proposed approach to solve it, is most like that described in Rezapour and Farahani (2010), Nagurney (1999), and a subsequent paper of Nagurney. However, this paper’s main contribution is the addition of the location decision as 0-1 variables, which puts the problem in the class of mixed-integer, nonlinear programming models. Therefore, the proposed problem cannot be solved with the explicit solution algorithm. Thus, we propose a three-step algorithm to solve the problem and find an equilibrium solution. As the chains enter the dynamic competition, we use variational inequality formulation to find equilibrium results. However, VI is only applicable to models with continuous variables and convex functions; but, because our problem has 0-1 variables, we use the Wilson algorithm and specify some strategies related to the 0-1 variables to handle this matter. We define all the possible strategies based on location variables of the chains in the first stage of our algorithm and then use the VI formulation and modified projection method to obtain equilibrium results of continuous variables in the second stage. The third stage is constructed with the help of the Wilson algorithm, and we select the equilibrium locations in this step. With this three-step algorithm we are able to solve the problem of SD-SCND in dynamic competition.

Problem Definition

This paper considers the problem of simultaneous decentralized competitive supply chain network design (SD-CSCND) in which n SCs plan to enter into virgin market. Each chain has two independent tiers called plant and distribution center (DC) levels, which try to maximize the chain’s profits by selecting the best locations for their facilities and setting wholesale and retail prices. All the entities make decisions simultaneously in a noncooperative manner. The chains produce either identical or highly substitutable products, and customer demand is elastic and price-dependent.

Fig. 1. Simultanous decentralized competitive supply chain network design problem.

Before modelling the formulation, imagine there are incoming SCs indexed by ; then, th SC has  potential locations for opening plants, indexed by  and , and potential locations for opening DCs, indexed by . There exist  demand points indexed by . Similar to the description provided by Tsay and Agrawal (2000), demand functions for th SC in market k can be defined as follows):

The term  is the potential market size and  refers to self-price sensitivity. The term  represents the cross-price sensitivity. Since demand cannot be negative, we assume the following:

Now we can introduce parameters and variables as follows:

Parameters and variables

 Modelling Framework

Plants model

Term (3) represents the objective function of the plants for each chain that includes total revenue from selling the product to the DCs of the chain minus total location and transaction costs. Constraint (4) specifies the number of opened plants in the chain, and constraint (5) is related to the binary and non-negativity restriction on the corresponding decision variables.

DC’s model

Term (6) represents the objective function of DCs of the chain which includes profits captured by selling the product to the customers minus total location and transaction costs. Constraint (7) is related to flow balance; constraint (8) specifies the number of opened DCs in each chain and constraint (9) is related to the binary and non-negativity restriction on the corresponding decision variables.

Solution Approach

This section presents our proposed algorithm for solving the SD-CSCND problem. The algorithm is essentially based on the chains’ decision variables. Each SC has two different types of decision variables, including continuous and discrete (0-1) variables. Continuous variables  are related to wholesale and retail prices and the amount of shipments among the tiers while discrete variables  are related to the locations of the facilities. Also, they are intrinsically different decisions, as location is strategic while the other variables are related to operational decisions. On the other hand, they are related to each other, as each opened location has its own costs that affect the chains’ prices, market shares, demands, and profits.

With the help of the Wilson algorithm, Wilson (1971), the variational inequality formulation, the modified projection method, and Nagurney et al. (2002), and references therein, we propose a three-step algorithm in which the first step defines basic strategies based on location variables. Each strategy contains a potential network design structure for the chains, as their location is fixed. In the second step, we use variational inequalities and the modified projection method to calculate the payoff for each potential network structure. After steps one and two, the payoff for all the possible structures of the chains is calculated. In step three and with the help of the Wilson algorithm, the Nash equilibrium locations can be found, and the chains’ equilibrium network design is obtained. The algorithm procedure is as follows:

1. Initialize the whole strategies for the players:

1.1. Construct an empty poly-matrix by considering all pure strategies of the players (any combination of the facilities to be opened from all the potential facilities of each player).

1.  Calculate Nash equilibrium prices and flows for all players in the chains in the defied strategies:

2.1. Develop VI formulation of the players’ problems in each strategy and solve it with the help of the modified projection method.

1.  Find the best response of all the players:

3.1. Fill the empty poly-matrix with the obtained payoffs from the previous stage and find the best network structure using the Wilson algorithm.

To clarify the proposed algorithm, consider an example in which one SC, composed of plant and DC levels, is planning to enter one virgin market in a decentralized manner, shape its network, and set wholesale and retail prices and flows. Imagine both plant and DC have two potential locations, and they intend to open one facility to capture the market demand. Table 1 shows the cost functions of the entities related to the places.

Table 1. Cost functions

According to step one, each player has  pure strategies; consequently,  potential strategies are available. The opened plant and DC can be as follows:  . Now, the algorithm can be applied to step two in which it can calculate the equilibrium prices and flows with the help of the VI formulation and the modified projection method. The results of this step are presented in Table 2.

Table 2. Nash equilibrium solution of each potential strategy

Step one trough step two result in an equilibrium solution for the continuous variables in each strategy. The algorithm can then be applied to step three in which it can select the equilibrium locations of the facilities in order to finalize the network design with the help of the Wilson algorithm. It is worth noting that in the case of two existing players, this step can also be conducted using the Lemke and Howson’s (1964) algorithm. Table 3 presents this step. According to the constructed matrix, strategy  is selected as the Nash equilibrium solution of the game.

Table 3. Final Nash equilibrium solution

It is worth noting that since steps one and three are based on the players’ potential strategies, and step two is based on variational inequality and the modified projection method. With respect to the fact that pure strategies are finite and the modified projection method has a convergence criterion, the proposed procedure converges to an equilibrium solution (see the convergence proof of the Wilson algorithm; Wilson (1971); the variational inequality formulation and the modified projection method; and Nagurney et al. (2002) and references therein).

Stage one

This stage defines the number of strategies and shapes the matrix that should be filled in the next stage. By examining the 0-1 decision variables in each chain, it is understandable that each plant has  pure strategies, and each DC has  pure strategies. So the dimension of the matrix is .

Stage two

In this stage we should optimize the following models for the opened plants and DCs in each strategy to calculate payoff for the game.

Modified plants model

Term (10) represents the objective function of the opened plants for each chain that includes total revenue from selling the product to the DCs of the chain minus total location and transaction costs and; constraint (11) is related to the non-negativity restriction on the corresponding decision variables.

Modified DC’s model

Term (12) represents the objective function of opened DCs of the chain which includes profits captured by selling the product to the customers minus total transaction costs. Constraint 15 is related to flow balance between opened plants and DCs; constraint 16 is related to the non-negativity restrictions on the corresponding decision variables.

The modified plant and DC model should be formulated as VI model (according to Rezapour & Farahani (2010), Nagurney (1999), and a subsequent paper of Nagurney). The VI is solvable by several algorithms as modified projection method, like Nagurney and Toyasaki (2005), Rezapour and Farahani (2010), the Euler-type model (Nagurney et al., 2003; Santibanez-Gonzalez & Diabat, 2016), some evolutionary algorithm (Majig et al., 2007), and extended mathematical programming (EMP) of GAMS (Santibanez-Gonzalez & Diabat, 2016). In this article, we use modified projection method to solve the model.

Stage three

Now, we can find the Nash equilibrium locations and shape the network structure of the chains by the specified prices and amount of shipments. This step is conducted with the help of the Wilson algorithm (Wilson, 1971).

Computational Results

This section presents a real-world problem occurring in the Iranian capacitor industry and inspired us to propose the problem and solution described in this paper. The first subsection describes the problem environment, and the second subsection provides discussions of the results.

Case study

As a consulter group, we study a real-world problem in which two SCs are planning to enter the capacitor industry in the Iranian market. These two SCs want to produce a special type of capacitor that is used in refrigerators. This type of capacitor is solely imported, and there is no domestic producer for it, so they decided to enter this virgin market by shaping their SC domestically. They want to open one plant and two DCs from two and four potential locations and shape their networks in a decentralized manner in a dynamic competition and set the wholesale and retail prices. Table 4 represents their cost structures, and Table 5 represents the achieved results. Both chains open a plant at location one and DCs in location one and two to obtain Nash equilibrium locations. The proposed algorithm was implemented in Matlab 2014a. The convergence criterion used was that the absolute value of the flows and prices between two successive iterations differed by no more than , and the computational time is negligible.

Table 4. Cost structure of the Chains

Table 5. Computational results for SCs

Discussion

The case study presented in this paper reflects an SD-CSCND problem that has nonlinear, fixed production and transaction costs related to the producers and DCs. Moreover, the demand function at each market is related to the retail prices of the chains and the prices relate to the costs of the players; therefore, the chains can use different locations for their facilities or marketing activities to influence the costs of the chains and parameter values of the demand function. Here, we discuss the sensitivity analysis for SCs with respect to the cross-price and self-price parameters.

Tables 6 and 7 represent the sensitivity analysis for SCs with respect to cross price effect while the self-price parameter is set to 1.2. Tables 8 and 9 represent the sensitivity analysis for SCs with respect to self-price effect while the cross-price parameter is set to 1.5.

Table 6. Sensitivity analysis for SC 1 with respect to cross price effect

Table 7. Sensitivity analysis for SC 2 with respect to cross price effect

Table 8. Sensitivity analysis for SC 1 with respect to self-price effect

Table 9. Sensitivity analysis for SC 2 with respect to self-price effect

It is worth noting that in our case, the change of self-price and cross-price parameters have no effects on location decision variables, but changes in location decision variables by change in these parameters are possible, and in these circumstances the shape of the networks will change.

Conclusion

This paper presents an important real-world problem in which  decentralized SCs simultaneously enter the virgin market to shape their networks, set the wholesale and retail prices, and specify their market shares in dynamic competition. This problem is essential, as several developing countries are trying to omit monopoly and open their markets to international investors. These investors then encounter virgin markets and competition simultaneously.

We propose a three-step algorithm to reach a Nash equilibrium network design in which step one constructs all the potential network structures; step two computes the related decisions in dynamic competition for all the potential structures through VI formulation and the modified projection method, and step three determines the Nash structures for the SD-CSCND problem with the help of the Wilson algorithm.