# Supply Chains Competition under Uncertainty Concerning Player’s Strategies and Customer Choice Behavior: A Generalized Nash Game Approach

Authors: A. Hafezalkotob; A. Makui
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

Decision makers in a supply chain confront two main sources of uncertainty in market environment including uncertainty about customers purchasing behaviors and rival chains strategies. Focusing on competition between two supply chains, it is considered that each customer as an independent player selects products of these chains based on random utility model. Similar to quantal response equilibrium approach, we take account of customer rationality as an exogenous parameter. Moreover, it is assumed that decision makers in a supply chain can perceive an estimation of rival strategies about price and service level formulated in the model by fuzzy strategies. In the competition model, chain’s decision makers consider a subjective probability for wining each customer which is formulated by coupled constraints. These constraints connect chains strategies regarding to each customer and yield a generalized Nash equilibrium problem. Since price cutting and increasing service level are main responses to rival supply chain, after calculating optimal strategies, we show that more efficient responses depend on customer preferences.


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## Body

## 1. Introduction

The supply chain (SC) is typically decentralized which implies that participants are independent firms with their own conflicting goals. Collaboration and coordination of these firms will hopefully result in adding benefits [1]. Decision makers in each SC encounter two sources of uncertainty in a market environment. In the real world competition, gaining a clear perception of rival strategies is not often possible for decision makers. That is, the competitor’s strategies are sensed with a level of indistinctness and vagueness. On the other hand, precise determination of customer’s responses to products is often not possible for decision makers in SCs as well. Thus, the second source of uncertainty is related to customers’ behaviors and their product selections procedure.In many practical situations, players share some resources available in limited amount. Moreover, in many real game applications in the economic systems, players expect a minimum level of payoff from actual transactions. Both these situations lead to a feasible set for players’ strategies. In the noncooperative game, if each player’s feasible strategies depend on the rival players’ strategies, we speak of generalized Nash equilibrium problems (GNEPs) [2]. GNEPs have received an increasing amount of attention in recent times because GNEPs naturally arise in modeling of complex and important economic systems [3]. In practical instances of GNEPs, the feasible sets of the players are usually defined by inequality constraints (or coupled constraints). Customer segments can be considered as shared and limited resources among competitive firms in a market. We concentrate on the competition between two independent SCs for winning customer segments of a specific product type. It is assumed that decision makers in each SC consider a minimum probability (risk) of losing each customer segment as a rule. This assumption connects SCs strategies together and results in inequality constraints in the competition model.In the marketing literature, the concept of competitive responsiveness argues that the changing strategy by one party in a market will be responded by the other competitors, and the elasticity measures capture the effect of competitive reactions [4]. Two distinguishing types exist regarding to competitive reactions: simple or multiple competitive relations. In the simple case, changing in one type of strategy will be responded by a same strategy. For instance, competitors respond to a change in price for a party by changing their prices and to a change in service level by a service level response, and so forth. However, our competition model is consistent with the concept of the multiple competitive reactions (the concept of marketing mix) such that an SC may react to the price change not only by changing its price but also by changing its service level. In the model, the combination of the profit maximization objective and the minimum risk of losing customer constraints specify the efficient reactions against rival chain changing strategies.Markets often are not homogenous. In the marketing segmentation models, the customers’ populations are divided into groups which have similar behaviors and utilities [5]. It is assumed that a market consists of finite and independent customer segments (or customers) that each of them has distinguishing utility and demand function of the products. Therefore, customer and customer segment are used interchangeably throughout the paper. When SCs perceive requirements of various segments, they will focus on satisfying them. Loss of a customer segment causes erosion of market share, which is often dramatic for SCs [6]. On the other hand, by taking the advantage of this concept, managers concentrate on each customer segment attractions to increase their market shares. The risk of losing a customer segment or customer harms the trade of all partners in an SC.We use fuzzy set theory to deal with vagueness of other competitor strategies. It is assumed that decision makers in an SC can perceive an estimation of rival’s strategies about price and service level formulated in the model by fuzzy strategies. Moreover, it is considered that each customer as an independent player selects products of these chains based on random utility model (RUM) [7]. Therefore, we use stochastic programming based on RUM to involve customer selection behavior in the model. Similar to quantal response equilibrium (QRE) approach [8, 9], the customer rationality is taken into account. We investigate how customer rationality affects competition between two chains and how this factor influences optimal utilities of chains’ products from customer point of view. The integration of fuzzy logic and stochastic programming give the research an original contribution to define SCs competition model based on mathematical element of game theory which can manage vagueness and uncertainty of decision makers.The paper is organized as follows. The literature related to SC competitions, noncooperative fuzzy game, and customer purchasing behaviors are separately reviewed in Section2. Section 3 includes a discussion of the problem and related notations. Section 4 concerns basic models of competition between two SCs for a single customer segment, then the model is developed for all customer segments in a market. Finally, the paper concludes in Section 5 with the results of the research.


## 2. Literature Review

Several researches reviewed and studied the applications of game theory in order to model conflicting goals in SCs management [1, 10, 11]. Beside competition and collaboration within an SC, in a specific product market, there are some kinds of competition among SCs to attract final customers. Some papers explicitly modeled SCs interactions in a market [12–25].Bernstein and Federgruen [14] developed a stochastic inventory model for an oligopoly where demand was a function of all retailers’ prices and service levels. Three scenarios were studied in the paper: only price competition, simultaneous price and service level competition, and two stage competition. Wu et al. [21] considered two manufactures; each produced a substitutable product and sold it by either a decentralized retail store or an integrated one. The problem was modeled as a price-setting newsvendor. The effects of demand uncertainty as well as product substitutability on SC configuration were also investigated.Bernstein and Federgruen [15] assumed a general model of two-echelon SCs with several competing retailers served by a common supplier. In the study, the demand of retailers is a stochastic function and depends on all of the firm’s strategies. Xiao and Yang [22] developed an information revelation mechanism model of two-echelon SC facing an outside competitor to investigate the effect of the risk-sharing rule on revelation mechanism under demand uncertainty, where the risk sensitivity of the retailer is private information. Furthermore, Xiao and Yang [23] developed a competition model of two SCs to study the optimal decisions of the players under demand uncertainty. They fully analyzed the effect of the retailer risk sensitivity and service investment efficiency on the optimal decisions.Product price and service level offered by SCs are two significant factors affecting the purchasing decisions of customers [23]. Several papers considered price and service competition [12, 14, 15, 22–26]. Among these researches, Bernstein and Federgruen [14, 15] developed a price and service competition model based on market share computed by attractions and linear demand function, as well. On the other hand, Allon and Federgruen [12] as well as Xiao and Yang [22, 23] proposed models where retailer’s demand is a linear combination of product price and service level offered by a retailer and his/her rival retailer. Hafezalkotob et al. [24] investigated a network design problem in a competition of two SCs where uncertain markets’ demands depend on price, service level, and marketing expenditure of chains. They assumed the risk of participants is derived from market’s demands uncertainty. However, our decision structure in the present paper is different because we assume that decision makers in an SC can perceive an estimation of rival strategies. Hafezalkotob and Makui [25] considered that the risk of losing customers can be defined based on products utility from customers’ viewpoint. They proposed a noncooperative competition model between an SC and rival manufacturer to investigate trade-offs of responsiveness and efficiency in an SC. They found that the higher the responsiveness to customers’ needs, the lower the risk of losing the customers will be. Tsay and Agrawal [26] investigated a one-manufacture and two-retailer SC which offers a common product to customers. The competition between the retailers was modeled based on linear customer demand function in service level and price which each retailer offered to the market. To the best of the authors’ knowledge, no research was found in the context of competition among SCs to investigate the risk of SCs with respect to the customers purchasing behavior.According to this gap in the literature, there are three main contributions in the research. First, we study customers’ purchasing behaviors based on random utility of SC’s products. The higher the utility of products for the customer, the higher the probability of purchasing will be. Second, by applying RUM, we investigate the effect of customer rationality on SC’s competition. Finally, contrary to the literature, we assume the strategies of rival SC are perceived by chain’s participants with some degree of uncertainty which are formulated by fuzzy strategies. The related literature to RUM and fuzzy strategies are briefly reviewed through the following subsections.


### 2.1. Noncooperative Fuzzy Games

One of the major deficiencies of the traditional noncooperative game theory is the assumption of all players to be completely aware of all data concerning the game structures [27]. In the real-world circumstances, it often happens that the players are not able to estimate exactly the outcomes of different strategies and/or their preferences of the other players. In other words, each player regularly has a vague and heuristic knowledge of the other players [28]; thus, a level of uncertainty exists about his/her choices. Fuzzy set theory [29, 30] is an excellent tool which can be used for softening the crisp noncooperative game theory. The degree of uncertainty in fuzzy set theory represents the value of vagueness of players in the game environment.Cooperative fuzzy games have been initially introduced by Aubin [31]. Afterward, Butnariu [32–35] for the first time incorporated fuzzy approach into noncooperative games. The work of Butnariu was complemented by other researchers [28, 36–39]. Recently, fuzzy sets approach has been intensively utilized to model noncooperative games in the uncertain environment [27].In general, there exist two approaches treating to fuzziness in games: the approach based on fuzzy preferences, strategies, and perception or belief of players and the approach based on fuzzy evaluation of the payoffs [27]. In the first approach, the players use partly crisp and partly fuzzy strategies as well as partly crisp and partly fuzzy preferences to play the game. However, the second approach is an extension of the traditional theory of games against nature, and players choose strategies with regard to both the behavior of other players and possible realization of fuzzy parameters of the nature or game situation [40].Garagic and Cruz [28] developed Nash equilibrium concept for the first approach of games. For a bimatrix game, they used fuzzy control to model situations where each strategy available to the player is defined by a membership function. They illustrated their model for planning and conducting a military operation. Similarly, fuzzy strategies in a noncooperative game are also considered by Song and Kandel [41]. Francesco and Gravio [42] proposed fuzzy logic to represent the characteristic aspect of a bilateral bargaining with incomplete information. Under imperfect information condition, the seller and buyer have only imprecise information, preferring to keep private the real value of opportunity cost and acceptable price, respectively. An intermediary repeatedly negotiates with supplier and customer to reach an agreement.In the real competition among SCs, an agent often is not able to recognize or estimate the precise strategies of other competitors. Generally, noise or vagueness exists about other SCs actions when it comes to decision making. We use fuzzy set theory to deal with the uncertainty in regard to rival SC’s strategies.


### 2.2. Customer Behavior and Brand Choice Models

Customer behavior theory investigates the customers purchasing behavior and their decision to buy or ignore a product. Similarly, the main aim of brand choice models is to answer the question that why customers shift from one brand to another and in which fashion the changes in brands’ market shares are occurring [43]. Therefore, these models analyze the previous data to estimate the customer’s choice process. A well-known brand choice model is RUM or in its most used form, the multinomial logit model (MNL) which was developed by McFadden [7]. The MNL market share represents an aggregate concept that the consumer choice-based utility theory may serve as a foundation for individual rational decision making [44]. Cooper [45] investigated the relationship between different market share models and individual choice probabilities. Choice models founded on RUM make a connection between the attributes that characterize different competing products to the probability that they will be selected by customers [46].According to the structure defined by the theory of RUM [47], the utility of customer i obtain from product j (Vij) is made of two parts, one deterministic Uij and the other random εij. The deterministic part can be considered as a function of a {Xkj}k∈K of attributes that characterize the choice procedure for customer i. The random component is stochastic and reflects the idiosyncrasies of customer i in taste for product j. Taking into account additive structure for the deterministic part, the customer utility function can be written as follows:(2.1)Vij=Uij+εij=θ0j+∑k∈KθkijXkj+εij,
where {θkij}k∈K represents the response parameters for attributes of set K, and θ0j denotes an additional parameter related to the intrinsic utility of product of brand j. The customer chooses the alternative which maximizes the utility. Therefore, the probability Pi(j∣θ0,θkij) of customer i selecting product of brand j from set M of possible products (where n stands for the number of elements in set M) is as follows:(2.2)Pi(j∣θ0,θ)=P(Vij≥Vim∀m∈M)=P(εij≤Uij-Uim+εim∀m∈M).Similar to the approach of logistic quintal response equilibria (Logistic-QRE) introduced by McKelvey and Palfrey [8, 9], we concentrate on a particular version of the RUM, where each εij is dependently and identically distributed according to the type I extreme value (or Weibull) distribution with cumulative density F(εij)=exp(-exp(-τεij)). This distribution has a variance of π2/6τ2; therefore, the parameter τ determines the precision of F. McFadden [7] showed that this distribution function yields the following choice probability(2.3)Pi(j∣θ0,θ)=exp⁡(τUij)∑m∈Mexp⁡(τUim)=exp⁡(τ(θ0j+∑k∈KθkijXkj))∑m∈Mexp⁡(τ(θ0m+∑k∈KθkimXkm)).With having samples data of each customer segment, the response parameters can be computed by maximum likelihood estimation [48].QRE model has recently attracted a great deal of attention. The main motivation of this model is assuming that players are not “perfectly rational.” Instead, their choices are noisy, and strategies with higher utility are chosen with higher probability [49]. Many experimental results showed that partial rationality of players modeled by QRE can account for players’ behavior in the real situations [8]. One of the particular aspects of the logit QRE is nonnegative parameter τ which can be considered as the rationality parameter. When τ→∞, player becomes “perfectly rational”; therefore, he/she certainly chooses the most favorable alternative. As this parameter declines, the rationality of the player reduces and his/her choice becomes increasingly based on randomness. τ→0 represents that the player is “completely irrational,” and he/she does not distinguish favorable products. We apply this approach for customer (or customer segment) i; thus, it can be concluded that with growing parameter τ, the customer is more responsive to deterministic part of the utility. Therefore, he/she selects more favorable product with higher probability. Obviously, the value 1 for the parameter provides the RUM introduced by McFadden [7], and τ=0 yields Pi(j∣θ0,θkij)=1/n for all j, as well.


## 2.1. Noncooperative Fuzzy Games

One of the major deficiencies of the traditional noncooperative game theory is the assumption of all players to be completely aware of all data concerning the game structures [27]. In the real-world circumstances, it often happens that the players are not able to estimate exactly the outcomes of different strategies and/or their preferences of the other players. In other words, each player regularly has a vague and heuristic knowledge of the other players [28]; thus, a level of uncertainty exists about his/her choices. Fuzzy set theory [29, 30] is an excellent tool which can be used for softening the crisp noncooperative game theory. The degree of uncertainty in fuzzy set theory represents the value of vagueness of players in the game environment.Cooperative fuzzy games have been initially introduced by Aubin [31]. Afterward, Butnariu [32–35] for the first time incorporated fuzzy approach into noncooperative games. The work of Butnariu was complemented by other researchers [28, 36–39]. Recently, fuzzy sets approach has been intensively utilized to model noncooperative games in the uncertain environment [27].In general, there exist two approaches treating to fuzziness in games: the approach based on fuzzy preferences, strategies, and perception or belief of players and the approach based on fuzzy evaluation of the payoffs [27]. In the first approach, the players use partly crisp and partly fuzzy strategies as well as partly crisp and partly fuzzy preferences to play the game. However, the second approach is an extension of the traditional theory of games against nature, and players choose strategies with regard to both the behavior of other players and possible realization of fuzzy parameters of the nature or game situation [40].Garagic and Cruz [28] developed Nash equilibrium concept for the first approach of games. For a bimatrix game, they used fuzzy control to model situations where each strategy available to the player is defined by a membership function. They illustrated their model for planning and conducting a military operation. Similarly, fuzzy strategies in a noncooperative game are also considered by Song and Kandel [41]. Francesco and Gravio [42] proposed fuzzy logic to represent the characteristic aspect of a bilateral bargaining with incomplete information. Under imperfect information condition, the seller and buyer have only imprecise information, preferring to keep private the real value of opportunity cost and acceptable price, respectively. An intermediary repeatedly negotiates with supplier and customer to reach an agreement.In the real competition among SCs, an agent often is not able to recognize or estimate the precise strategies of other competitors. Generally, noise or vagueness exists about other SCs actions when it comes to decision making. We use fuzzy set theory to deal with the uncertainty in regard to rival SC’s strategies.


## 2.2. Customer Behavior and Brand Choice Models

Customer behavior theory investigates the customers purchasing behavior and their decision to buy or ignore a product. Similarly, the main aim of brand choice models is to answer the question that why customers shift from one brand to another and in which fashion the changes in brands’ market shares are occurring [43]. Therefore, these models analyze the previous data to estimate the customer’s choice process. A well-known brand choice model is RUM or in its most used form, the multinomial logit model (MNL) which was developed by McFadden [7]. The MNL market share represents an aggregate concept that the consumer choice-based utility theory may serve as a foundation for individual rational decision making [44]. Cooper [45] investigated the relationship between different market share models and individual choice probabilities. Choice models founded on RUM make a connection between the attributes that characterize different competing products to the probability that they will be selected by customers [46].According to the structure defined by the theory of RUM [47], the utility of customer i obtain from product j (Vij) is made of two parts, one deterministic Uij and the other random εij. The deterministic part can be considered as a function of a {Xkj}k∈K of attributes that characterize the choice procedure for customer i. The random component is stochastic and reflects the idiosyncrasies of customer i in taste for product j. Taking into account additive structure for the deterministic part, the customer utility function can be written as follows:(2.1)Vij=Uij+εij=θ0j+∑k∈KθkijXkj+εij,
where {θkij}k∈K represents the response parameters for attributes of set K, and θ0j denotes an additional parameter related to the intrinsic utility of product of brand j. The customer chooses the alternative which maximizes the utility. Therefore, the probability Pi(j∣θ0,θkij) of customer i selecting product of brand j from set M of possible products (where n stands for the number of elements in set M) is as follows:(2.2)Pi(j∣θ0,θ)=P(Vij≥Vim∀m∈M)=P(εij≤Uij-Uim+εim∀m∈M).Similar to the approach of logistic quintal response equilibria (Logistic-QRE) introduced by McKelvey and Palfrey [8, 9], we concentrate on a particular version of the RUM, where each εij is dependently and identically distributed according to the type I extreme value (or Weibull) distribution with cumulative density F(εij)=exp(-exp(-τεij)). This distribution has a variance of π2/6τ2; therefore, the parameter τ determines the precision of F. McFadden [7] showed that this distribution function yields the following choice probability(2.3)Pi(j∣θ0,θ)=exp⁡(τUij)∑m∈Mexp⁡(τUim)=exp⁡(τ(θ0j+∑k∈KθkijXkj))∑m∈Mexp⁡(τ(θ0m+∑k∈KθkimXkm)).With having samples data of each customer segment, the response parameters can be computed by maximum likelihood estimation [48].QRE model has recently attracted a great deal of attention. The main motivation of this model is assuming that players are not “perfectly rational.” Instead, their choices are noisy, and strategies with higher utility are chosen with higher probability [49]. Many experimental results showed that partial rationality of players modeled by QRE can account for players’ behavior in the real situations [8]. One of the particular aspects of the logit QRE is nonnegative parameter τ which can be considered as the rationality parameter. When τ→∞, player becomes “perfectly rational”; therefore, he/she certainly chooses the most favorable alternative. As this parameter declines, the rationality of the player reduces and his/her choice becomes increasingly based on randomness. τ→0 represents that the player is “completely irrational,” and he/she does not distinguish favorable products. We apply this approach for customer (or customer segment) i; thus, it can be concluded that with growing parameter τ, the customer is more responsive to deterministic part of the utility. Therefore, he/she selects more favorable product with higher probability. Obviously, the value 1 for the parameter provides the RUM introduced by McFadden [7], and τ=0 yields Pi(j∣θ0,θkij)=1/n for all j, as well.


## 3. Problem Description and Notation

Two SCs are considered that each is comprised of one manufacturer and one retailer. SCs compete for customer segments in the market. Each SC only provides a single type of product for customers. These products are completely substitutive and differ only in price and offered service levels by retailers.The competition between two SCs constitutes a noncooperative game to attract customer segments. Due to assumption of independency of customer segments, the competition between retailers can be considered for each segment separately. Considering a minimum acceptable risk of losing each customer segment, decision makers of both SCs attempt to raise possible profit from selling products to the market. The noncooperative game among SCs involves that they simultaneously declare price and service level to the customers. On the other hand, the customer selects favorable products with higher probability.


### 3.1. Assumptions

#### 3.1.1. Uncertainty concerning Customers’ Behaviors

Customers independently choose products of SCs based on utility maximization. Their behaviors conform to the RUM assumptions introduced by McFadden [7]. The random component of utility of each customer segment is not revealed to decision makers of SCs. Nevertheless, this component is identically distributed according to the type I extreme value (or Weibull). The deterministic parts of the customers’ utilities are common knowledge for all decision makers. The RUM model has been widely employed to investigate customer preferences in many industries such as food [50], automotive [51], and electronics industries [52].


#### 3.1.2. Customer Rationality

Several economics models assume that humans are on average rational due to finite resources available for making decisions [53]. Similar to Logit-QRE approach [8, 9], we assume that customers as players are partially rational. The rationality is indicated by parameter τ in the RUM of customer behavior. Our assumption about partial (bounded) rationality of customer is more general and realistic than prefect rationality assumption.


#### 3.1.3. Customer Demand

When a customer segment selects one of the SCs products, his/her demand is proportional to the offered price and service level. Several researchers investigated price and service level impacts on customers’ demand and purchasing behavior in oligopolistic markets [15, 23, 24].


#### 3.1.4. Uncertainty concerning Competitor Strategies

Each agent in an SC perceives the price and service level of the other competitor chain with a noise and vagueness. Similar to fuzzy strategies introduced by Garagic and Cruz, [28], we assume that fuzzy price and service level strategies of rival chain are L-R type fuzzy numbers. Fuzzy-based game theory was also investigated by some researchers in oligopolistic markets when vagueness exists about competitors [54, 55].


#### 3.1.5. Independent Strategies for Customers

In the real markets, customers are almost independent when there are weak social relationships among them. Bekal et al. [56] considered customers’ independency when they were sufficiently geographically dispersed. For independent customers, a retailer is able to pursue customer-specific strategies.


#### 3.1.6. Maximum Risk of Supply Chains

Decision makers in each SC consider a minimum probability for attracting each customer segment (or equivalently, they take account of a maximum risk of losing each segment). That is to say, the decision makers do not permit their rival chain to reduce the probability of attracting customer segmenti lower than this limit. In the case of two echelons SCs, manufacturer and his/her retailer cooperatively determine this risk based on the importance and value of each customer segment for SC. Maximum acceptable risk condition associates chain’s strategies with the rival ones. This condition involves considering inequality constraints (coupled constraints) in the problem which leads to a GNEP.


### 3.2. Input Parameters

Assume that SCj offers product type j to customer segment i. Furthermore, Mj and Rj denote manufacturer and retailer in jth SC, respectively.θ0j:
the normalized attraction denotes intrinsic utility of productj.θ1ij:
the response parameter of customer segmentith for price of product j, (θ1ij>0).θ2ij:
the response parameter of customer segmentith for service level offered by SC j, (θ2ij>0).τi:
the rationality parameter of customer segmenti, (τi>0).αi:
the base demand of customer segmenti, (αi>0).βi:
the demand sensitivity of customer segmenti to offered service level, (βi>0).γi:
the demand sensitivity of customer segmenti to product price, (γi>0).cMj:
the unit production cost of manufacturerj, j=1,2, (cMj≥0).ηRj:
the service investment efficiency coefficient of retailerj, j=1,2. The larger the coefficient, the lower the service investment efficiency (1/ηRj) of retailer i will be, (ηRj>0).ϕij:
the maximum acceptable risk of losing customer segmenti for decision makers in SC j, equivalently 1-ϕij represents the minimum acceptable probability of attracting customer segment i, (0≤ϕij≤1).FCz:
the fixed operation cost of playersz, z=R1,R2,M1,M2,(FCz≥0).μP̃ij-(Pij-):
the fuzzy membership function corresponding to the vagueness of SCj about the price of product j- offered to customer segment i which is an L-R type fuzzy number (Pij-,Pij--,Pij-+)LPij-jRPij-j.μS̃Rj-i(y):
the fuzzy membership function corresponding to the vagueness of SCj about the service level of retailer j- offered to customer segment i which is an L-R type fuzzy number (SRj-i,SRj-i-,SRj-i+)LSij-jRSij-j,(SRj-i-,SRj-i+≥0).Note thatj- represents the rival SC. For instance, μS̃R132(y)=(SR13,3,7)LSR132RSR132 states the vagueness of decision makers in chain 2 about the service level of retailer 1 offered to the 3rd customer segment.


### 3.3. Decision Variables

wMj:
the wholesale price of manufacturerj offered to his/her retailer, j=1,2, (wMj≥cMj).mRji:
the profit margin of retailerj from selling product to customer segment i, j=1,2, (mRji≥0).Pij:
the retail price of retailerj offered to customer segment i, j=1,2; thus, we have Pij=wMj+mRji.SRji:
the service level of retailerj offered to customer segment i, j=1,2(SRji≥0).In SCj, the manufacturer decides on wholesale price wMj, and the retailer determines retail price Pij as well as service level investment SRji for each customer segment. Figure 1 illustrates the competition schema between two SCs.Figure 1
SCs competition schema for the customer segments.


## 3.1. Assumptions

### 3.1.1. Uncertainty concerning Customers’ Behaviors

Customers independently choose products of SCs based on utility maximization. Their behaviors conform to the RUM assumptions introduced by McFadden [7]. The random component of utility of each customer segment is not revealed to decision makers of SCs. Nevertheless, this component is identically distributed according to the type I extreme value (or Weibull). The deterministic parts of the customers’ utilities are common knowledge for all decision makers. The RUM model has been widely employed to investigate customer preferences in many industries such as food [50], automotive [51], and electronics industries [52].


### 3.1.2. Customer Rationality

Several economics models assume that humans are on average rational due to finite resources available for making decisions [53]. Similar to Logit-QRE approach [8, 9], we assume that customers as players are partially rational. The rationality is indicated by parameter τ in the RUM of customer behavior. Our assumption about partial (bounded) rationality of customer is more general and realistic than prefect rationality assumption.


### 3.1.3. Customer Demand

When a customer segment selects one of the SCs products, his/her demand is proportional to the offered price and service level. Several researchers investigated price and service level impacts on customers’ demand and purchasing behavior in oligopolistic markets [15, 23, 24].


### 3.1.4. Uncertainty concerning Competitor Strategies

Each agent in an SC perceives the price and service level of the other competitor chain with a noise and vagueness. Similar to fuzzy strategies introduced by Garagic and Cruz, [28], we assume that fuzzy price and service level strategies of rival chain are L-R type fuzzy numbers. Fuzzy-based game theory was also investigated by some researchers in oligopolistic markets when vagueness exists about competitors [54, 55].


### 3.1.5. Independent Strategies for Customers

In the real markets, customers are almost independent when there are weak social relationships among them. Bekal et al. [56] considered customers’ independency when they were sufficiently geographically dispersed. For independent customers, a retailer is able to pursue customer-specific strategies.


### 3.1.6. Maximum Risk of Supply Chains

Decision makers in each SC consider a minimum probability for attracting each customer segment (or equivalently, they take account of a maximum risk of losing each segment). That is to say, the decision makers do not permit their rival chain to reduce the probability of attracting customer segmenti lower than this limit. In the case of two echelons SCs, manufacturer and his/her retailer cooperatively determine this risk based on the importance and value of each customer segment for SC. Maximum acceptable risk condition associates chain’s strategies with the rival ones. This condition involves considering inequality constraints (coupled constraints) in the problem which leads to a GNEP.


## 3.1.1. Uncertainty concerning Customers’ Behaviors

Customers independently choose products of SCs based on utility maximization. Their behaviors conform to the RUM assumptions introduced by McFadden [7]. The random component of utility of each customer segment is not revealed to decision makers of SCs. Nevertheless, this component is identically distributed according to the type I extreme value (or Weibull). The deterministic parts of the customers’ utilities are common knowledge for all decision makers. The RUM model has been widely employed to investigate customer preferences in many industries such as food [50], automotive [51], and electronics industries [52].


## 3.1.2. Customer Rationality

Several economics models assume that humans are on average rational due to finite resources available for making decisions [53]. Similar to Logit-QRE approach [8, 9], we assume that customers as players are partially rational. The rationality is indicated by parameter τ in the RUM of customer behavior. Our assumption about partial (bounded) rationality of customer is more general and realistic than prefect rationality assumption.


## 3.1.3. Customer Demand

When a customer segment selects one of the SCs products, his/her demand is proportional to the offered price and service level. Several researchers investigated price and service level impacts on customers’ demand and purchasing behavior in oligopolistic markets [15, 23, 24].


## 3.1.4. Uncertainty concerning Competitor Strategies

Each agent in an SC perceives the price and service level of the other competitor chain with a noise and vagueness. Similar to fuzzy strategies introduced by Garagic and Cruz, [28], we assume that fuzzy price and service level strategies of rival chain are L-R type fuzzy numbers. Fuzzy-based game theory was also investigated by some researchers in oligopolistic markets when vagueness exists about competitors [54, 55].


## 3.1.5. Independent Strategies for Customers

In the real markets, customers are almost independent when there are weak social relationships among them. Bekal et al. [56] considered customers’ independency when they were sufficiently geographically dispersed. For independent customers, a retailer is able to pursue customer-specific strategies.


## 3.1.6. Maximum Risk of Supply Chains

Decision makers in each SC consider a minimum probability for attracting each customer segment (or equivalently, they take account of a maximum risk of losing each segment). That is to say, the decision makers do not permit their rival chain to reduce the probability of attracting customer segmenti lower than this limit. In the case of two echelons SCs, manufacturer and his/her retailer cooperatively determine this risk based on the importance and value of each customer segment for SC. Maximum acceptable risk condition associates chain’s strategies with the rival ones. This condition involves considering inequality constraints (coupled constraints) in the problem which leads to a GNEP.


## 3.2. Input Parameters

Assume that SCj offers product type j to customer segment i. Furthermore, Mj and Rj denote manufacturer and retailer in jth SC, respectively.θ0j:
the normalized attraction denotes intrinsic utility of productj.θ1ij:
the response parameter of customer segmentith for price of product j, (θ1ij>0).θ2ij:
the response parameter of customer segmentith for service level offered by SC j, (θ2ij>0).τi:
the rationality parameter of customer segmenti, (τi>0).αi:
the base demand of customer segmenti, (αi>0).βi:
the demand sensitivity of customer segmenti to offered service level, (βi>0).γi:
the demand sensitivity of customer segmenti to product price, (γi>0).cMj:
the unit production cost of manufacturerj, j=1,2, (cMj≥0).ηRj:
the service investment efficiency coefficient of retailerj, j=1,2. The larger the coefficient, the lower the service investment efficiency (1/ηRj) of retailer i will be, (ηRj>0).ϕij:
the maximum acceptable risk of losing customer segmenti for decision makers in SC j, equivalently 1-ϕij represents the minimum acceptable probability of attracting customer segment i, (0≤ϕij≤1).FCz:
the fixed operation cost of playersz, z=R1,R2,M1,M2,(FCz≥0).μP̃ij-(Pij-):
the fuzzy membership function corresponding to the vagueness of SCj about the price of product j- offered to customer segment i which is an L-R type fuzzy number (Pij-,Pij--,Pij-+)LPij-jRPij-j.μS̃Rj-i(y):
the fuzzy membership function corresponding to the vagueness of SCj about the service level of retailer j- offered to customer segment i which is an L-R type fuzzy number (SRj-i,SRj-i-,SRj-i+)LSij-jRSij-j,(SRj-i-,SRj-i+≥0).Note thatj- represents the rival SC. For instance, μS̃R132(y)=(SR13,3,7)LSR132RSR132 states the vagueness of decision makers in chain 2 about the service level of retailer 1 offered to the 3rd customer segment.


## 3.3. Decision Variables

wMj:
the wholesale price of manufacturerj offered to his/her retailer, j=1,2, (wMj≥cMj).mRji:
the profit margin of retailerj from selling product to customer segment i, j=1,2, (mRji≥0).Pij:
the retail price of retailerj offered to customer segment i, j=1,2; thus, we have Pij=wMj+mRji.SRji:
the service level of retailerj offered to customer segment i, j=1,2(SRji≥0).In SCj, the manufacturer decides on wholesale price wMj, and the retailer determines retail price Pij as well as service level investment SRji for each customer segment. Figure 1 illustrates the competition schema between two SCs.Figure 1
SCs competition schema for the customer segments.


## 4. The Basics Models

For the purpose of model explanation, we first consider the competition between SCs for a single customer segment; afterwards, the model will be extended forn customer segments in a market.


### 4.1. Manufacturers Problem for a Single Customer Segment

Losing each customer segment diminishes market share which is often horrible for all partners in an SC. Although partners of an SC compete to gain more profit, they need to maintain a level of cooperation to sell product to the customer segments in a competitive market. The manufacturer injth SC attempts to increase possible profit with considering a minimum probability of selling products of SC to customer i. Thus, manufacturer j considers the following problem:(4.1)max⁡wMj⁡E(πMj)=(1-ϕij)(wMj-cMj)(αi-γPij+βSRji)-FCMj,
subject to(4.2)Pi(j∣θ0j,θkij)=exp⁡(τi(θ0j-θ1ijPij+θ2ijSRji))exp⁡(τi(θ0j-θ1ijPij+θ2ijSRji))+exp⁡(τi(θ0j--θ1ij-P̃ij-+θ2ij-S̃Rj-i))≥1-ϕij,(4.3)Pij=wMj+mRji,
where 1-ϕij is a minimum probability which is acceptable for managers of SC j to sell its products to customer segment i. By applying maximin concept [57], manufacturer j in objective function (4.1) decides on wholesale price to maximize the minimum possible profit from selling products to customer segment i. Since the exact strategies of the rival SC are not revealed for manufacturer j, he makes his/her decision based on an estimation of rival price (P̃ij-) and service level (S̃Rj-i). Taking the RUM into account for the behavior of customer segment i, constraint (4.2) ensures that the probability of selling products to customer segment i is higher than the minimum acceptable probability 1-ϕij. Combination of objective function (4.1) and constraint (4.2) expresses that the manufacturer relates his possible profit from selling products to the customer purchasing decisions. On the other hand, these decisions depend upon the rival strategies which only estimations of them are available for the manufacturer. The manufacturer problem is a nonlinear GNEP which is not appropriate for the optimization. Lemmas 4.1 and 4.2 can be applied for linearizing the problem.Lemma 4.1.
By consideringδ level cut concept for fuzzy constraint [58, 59], constraint (4.2) will be equivalent to
(4.4)θ0j-θ0j--θ1ijPij+θ1ij-Pij-+θ2ijSij-θ2ij-S2ij-≥ln⁡⁡(1-ϕijϕij)1/τi-(1-δij-j)ρij-j,
where ρij-j=θij1Pij-++θij2SRj-i- and 0≤δij-j≤1.Proofs of all propositions and lemmas are given in the Appendix. In Lemma4.1, δij-j level cut can be interpreted as a confidence level of decision makers of SC j in the strategies of SC j- corresponding to customer segment i. Therefore, in constraint (4.4), δij-j equals to 1 means that DMs of SC j are sure that the probability of attracting customer i is higher than 1-ϕij. This confidence level declines as δij-j approaches zero. It is straightforward from the right hand side of constraint (4.4) that the uncertainty about rival strategies measured by the confidence level reduces the minimum probability of attracting the customer segment. Thus, in objective function (4.1), 1-ϕij has to be changed with respect to the confidence level. In Lemma 4.2, the minimum probability of attracting customer segment i is updated.Lemma 4.2.
When vagueness about competitor’s strategies exists, the minimum probability of attracting customer segmentj, that is, 1-ϕij, is changed as follows:
(4.5)1-ϕij′=kij(1-ϕij),
where kij=[1-ϕij+ϕijexp(ρij-jτi(1-δij-j))]-1.Note that in the case of absolute confidence, or no vagueness about competitor strategies, or quite irrational customer (i.e.,δij-j=1 or ρij-j=0 or τi=0), we have 1-ϕij′=1-ϕij. By applying the results of Lemmas 4.1 and 4.2 to manufacturer problem (4.1)–(4.3), the linear GNEP is as follows:(4.6)max⁡wMj⁡E(πMj)=kij(1-ϕij)(wMj-cMj)[αi-γi(wMj+mRji)+βiSRji]-FCMj,
subject to(4.7)θ0j-θ0j--θ1ij(wMj+mRji)+θ1ij-Pij-+θ2ijSRji-θ2ij-SRj-i≥1τiln⁡(1-ϕijϕij)-(1-δij-j)ρij-j.


### 4.2. Retailers Problem for a Single Customer Segmenti

Now we study the retailer problem in the case of a single customer segmenti. Retailer j determines the margin profit as well as service level. Similar to the manufacturer, the retailer assumes a lower bound for the probability of selling products to the customer segment. The retailer maximizes the minimum possible profit from trade with the customer segment. Therefore, the following model can be considered for the retailer(4.8)max⁡mRj,SRjiE(πRj)=(1-ϕij)(mRj)(αi-γiPij+βiSRji)-12ηRjSRji2-FCRj,
subject to constraints (4.2) and (4.3).As suggested in the extensive literature of SC game models [22, 23, 26, 60], we assume that when a player provides service level SRji, the service cost of the player is 0.5ηRjSRji2, that is, improving service level has a diminishing return on service expenditure. Both of the chain’s partners by choosing proper value for their controllable variables endeavor to maintain this probability within an acceptable limit. The retailer problem is also a nonlinear GNEP which is not appropriate for computing Nash equilibrium. By applying Lemmas 4.1 and 4.2 for the retailer problem, we have(4.9)max⁡mRji,SRjE(πRj)=kij(1-ϕij)(mRji)[αi-γi(wMj+mRji)+βiSRj]-12ηRjSRj2-FCRj,
subject to constraint (4.7).Constraint (4.7) is a “coupled constraint” which relates SCs’ strategies to rival ones to guarantee the minimum probability of attracting customer segment i. Therefore, manufacturer profit (4.6) and retailer profit (4.9) along with coupled constraint (4.7) constitute GNEPs. To solve these problems, we associate a multiplier λij with coupled constraint (4.7). Hessian matrix of E(πRj)is(4.10)HRj=[-2kij(1-ϕij)γikij(1-ϕij)βikij(1-ϕij)βi-ηRj].The expected profitE(πRj) is a concave function on (mRji,SRji) if and only if the Hessian matrix is negatively definite [61]. Let us define Bij′=2γiηRj-βi2 and Bij=3γiηRj-kij(1-ϕij)βi2. From the manufacturer and retailer problems, we drive the following.Proposition 4.3.
IfBij′>0, then the optimum values for wholesale price of the manufacturer, as well as the marginal profit and service level of the retailer, are as follows:(i)
ifλij=0, then one has(4.11)mRji*=(αi-γicMj)ηRjBij,wMj*=cMj+(αi-γicMj)ηRjBij,SRji*=βikij(1-ϕij)(αi-γicMj)Bij,(ii)
ifλij=Λij≠0, then one has
(4.12)mRji*=kij(1-ϕij)θ2ij2(αi-γicMj)2θ1ijEij-θ2ijkij(1-ϕij)Fij+Eij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-]2θ1ijEij-θ2ijkij(1-ϕij)Fij,wMj*=cMj+kij(1-ϕij)θ2ij2(αi-γicMj)2θ1ijEij-θ2ijkij(1-ϕij)Fij+Eij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-]2θ1ijEij-θ2ijkij(1-ϕij)Fij,SRji*=kij(1-ϕij)2θ1ijEij-θ2ijkij(1-ϕij)Fij×[2θ1ijθ2ij(αi-γicMj)+Fij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-i]],Λij=kij(1-ϕij)2θ1ijEij-θ2ijkij(1-ϕij)Fij×[(αi-γicMj)[2ηRjθ1ij-kij(1-ϕij)βiθ2ij]-Bij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-i]],
where Eij=θ1ijηRj-θ2ijkij(1-ϕij)βi, Fij=θ1ijβi-3θ2ijγi, and Qij = (1/τi)ln((1-ϕij)/ϕij)-(1-δij-j)ρij-j.Retailerj excessively invests in service if Bij′<0, which incurs a negative profit to him. The condition Bij′>0 means that the service investment should not be too inexpensive, which is consistent with those considered in [23, 26, 53]. Therefore, we assume that Bij′>0 throughout this paper. Moreover, the negative values for optimal profit margin and service level are not feasible for the retailer, that is, mRji*,SRji*>0. Since from Bij′>0 and 0≤kij≤1, it follows that Bij>0, and we assume that αi>γicMj to ensure mRji*,SRji*>0. The condition αi>γicMj indicates that unit production cost should not be too high. The multiplier λij associated with KKT condition may be thought as a shadow price for coupled constraint (4.7); furthermore, complementary slackness condition holds for the multiplier and the coupled constraint. That is to say, coupled constraint (4.7) is binding if λij=Λij≠0.Proposition4.3 presents the optimal strategies of SC j regarding customer segment i which depends on the strategies of rival chain (j-). In other words, if Xij and Xij- denote vectors of the decisions or strategy variables of SCs j and its rival j-, respectively, then the optimal values in Proposition 4.3 are Xij*(Xij-). This proposition gives the following insights.(i)
For a single customer segmenti, the optimal marginal profits of manufacturer and retailer are equal. Therefore, if Pij*-cMj is the optimal profit margin of a product unit for SC j, it is optimal that the manufacturer and the retailer consent to the equal profit margin (Pij*-cMj)/2. This situation holds for both the conditions λij=0 and λij≠0.(ii)
Theλij=0 states that the SC jth will sell product to customer segment i with probability higher than the least acceptable probability, that is, Pi(j∣θ0j,θkij)≥1-ϕij. In this situation, the rival chain strategies (i.e., Xij-) have no effects on the optimal price and service level of SCj.(iii)
Adversely,λij=Λij≠0 expresses that the probability of attracting customer segment i by SC jth decreases to the threshold of the minimum acceptable probability, that is, Pi(j∣θ0j,θkij)=1-ϕij. In this situation, SC’s strategies depend on the rival strategies. In other words, any variation in price and service level of the rival needs to be responded by the SC to ensure the least probability of selling products to customer segment i. In Proposition 4.4, the SC responses to variations in strategies of the rival are investigated.Proposition 4.4.
In the case ofλij=Λij≠0, for the optimal strategies values in Proposition 4.3, one has(1)
∂mRji*/∂wMj-=∂mRji*/∂mRj-i=∂wMj*/∂wMj-=∂wMj*/∂mRj-i=Eijθ2ij-/[2θ1ijEij-θ2ijkij(1-ϕij)Fij], ∂mRji*/∂SRj-i = ∂wMj*/∂SRj-i = -Eijθ2ij-/[2θ1ijEij-θ2ijkij(1-ϕij)Fij],(2)
∂SRji*/∂wMj- = ∂SRji*/∂mRj-i = [kij(1-ϕij)Fijθ1ij-]/[2θ1ijEij-θ2ijkij(1-ϕij)Fij], ∂SRji*/∂SRj-i = [-kij(1-ϕij)Fijθ2ij-]/[2θ1ijEij-θ2ijkij(1-ϕij)Fij].Proposition4.4 analyzes how the optimal wholesale and retail price as well as service level of one SC are affected by the retail price and service level of the rival chain. For instance, assume that the rival retailer decides to reduce ΔmRj-i from his/her retail price to attract customer segment i. From constraint (2.3), it is obvious that the product utility of the rival chain increases ΔmRj-iθ1ij- from the customer point of view. The SC j considers the following changes in its optimal strategies to react to his/her rival:(4.13)ΔmRji=Eijθ1ij-2θ1ijEij-θ2ijkij(1-ϕij)Fij(-ΔmRj-i),ΔwMj=Eijθ1ij-2θ1ijEij-θ2ijkij(1-ϕij)Fij(-ΔmRj-i),ΔSRj=-kij(1-ϕij)Fijθ2ij-2θ1ijEij-θ2ijkij(1-ϕij)Fij(-ΔmRj-i).Therefore, the product utility of SC increases from the customer’s viewpoint as follows:(4.14)ΔUij=-θ1ij(ΔmRji+ΔwMj)+θ2ij(ΔSRj)=ΔmRj-iθ1ij-=ΔUij-.
In other words, SC jth increases its product utility just equal to the rival’s one to hold coupled constraint (4.2), that is, if λij=Λij≠0, then ΔUij=ΔUij-. Now in Proposition 4.5, we investigate how an SC specifies its new strategies.Proposition 4.5.
FromBij>0, it follows that kij(1-ϕij)βi/ηRj<3γi/βi, and for the price changes of the rival chain, one has(i)
∂mRji*/∂wMj- = ∂mRji*/∂mRj-i = ∂wMj*/∂wMj- = ∂wMj*/∂mRj-i > 0 if θ1ij/θ2ij > kij(1-ϕij)βi/ηRj, ∂SRj*/∂wMj- = ∂SRj*/∂mRj-i < 0 if θ1ij/θ2ij < 3γi/βi.Similarly, for service level changes of the rival chain, one has(ii)
∂mRji*/∂SRj- = ∂wMj*/∂SRj- < 0 if θ1ij/θ2ij > kij(1-ϕij)βi/ηRj, and ∂SRj*/∂SRj- = ∂SRj*/∂SRj- > 0 if θ1ij/θ2ij < 3γi/βi.Two valueskij(1-ϕij)βi/ηRj and 3γi/βi are key thresholds which determine SC’s efficient strategies against changes in the price and service level of the rival. Part (i) of Proposition 4.5 gives us the following insights.(i)
Ifθ1ij/θ2ij>3γi/βi, then declining in the product price of the rival chain j- will be responded by decreasing in price and increasing in service level of SC jth. However, since the customer is more responsive to price, price cutting is a more efficient strategy for rising the product’s utility from the customer segment viewpoint.(ii)
Ifθ1ij/θ2ij<kij(1-ϕij)βi/ηRj, then the customer segment is more responsive to the service level. Therefore, if the decision makers in chain j want to maintain the segment with a specific minimum probability, it is needed to compensate for a price increase with service level which is more effective.(iii)
Ifkij(1-ϕij)βi/ηRj<θ1ij/θ2ij<3γi/βi, then both strategies of cutting price and increasing service level are efficient.(iv)
Similarly, part (ii) states the SC’s reactions to service level changes of rival chain. Cutting price and increasing service level by rival chain lead to rise in product utility. Figure2 illustrates the efficient strategies of an SC against rising product utility by the rival.Figure 2
SC’s efficient strategies for customer segmenti against variation in rival product utility.When manufacturers and retailers in both chains consider coupled constraint (4.7), and if ∑j=12(1-ϕij)≤1, then multipliers λ corresponding to the coupled constraints must satisfy λ=0 or λ>0 (the KKT condition). Therefore, in attempting to find Nash equilibrium solution, it is sufficient to take the cases of Table 1 into account.Table 1
Four possible cases for competing chains associated with customer segmenti.
λij=0λij>0λij-=0Case1Case2λij->0Case3Case4Therefore, Nash equilibrium solution of the competition is obtained from one of the following cases.Case 1 (λij=0 and λij-=0).
Strategies of SC and rival chain are obtained by part (i) of Proposition4.3. Here, both coupled constraints are satisfied at (wMj*,mRji*,SRj*) and (wMj-*,mRj-i*,SRj-*).Case 2 (λij=0 and λij->0).
The coupled constraint (4.7) is binding only for rival SC. Therefore, the optimal strategies of SC j are achieved by part (i) of Proposition 4.3. However, rival chain takes the optimal strategies of the first SC into account and computes optimal strategies by part (ii) of Proposition 4.3.Case 3 (λij>0 and λij-=0).
In this case, the situation is contrary to Case2 and the coupled constraint is binding only for SC j.Case 4 (λij>0 and λij->0).
Here, both coupled constraints for competing SCs are binding. This rare condition takes place only if∑j=12(1-ϕij)=1. The optimal strategies of SC and rival are obtained by part (ii) of Proposition 4.3.Note that the values of1-ϕij are subjective probabilities which decision makers in one SC specify it independently of rival chain. This probability is not revealed for rival. The value of 1-ϕij is determined based on importance and worth of customer segment i and expresses that SC partners are content to sacrifice profit to win customer segment i with the least probability of 1-ϕij. The assumption ∑j=12(1-ϕij)≤1 insures that a permanent equilibrium point exists in the competition, and resulting strategies Xij(Xij-) and Xij-(Xij) satisfy the least probability conditions for both SCs. Otherwise, SCs involve eternal competition to present the customer more favorable products, and no equilibrium exists in a one-shot competition. However, in a multiperiod competition, presenting more favorable products to the customers to increase business is the very essence of competition which brings about innovation and improvement in technology and knowledge. For instance, consider the dynamics of the microprocessor market which drives severe competition between two main suppliers, that is, AMD and Intel. The main customers of microprocessors are major original equipment manufacturers (OEMs) such as Dell, IBM, and HP which enjoy this rivalry. Throughout years of competition, Intel and AMD contest to win customers’ business not just on price and service level, but on all aspects of their products including quality, performance, and reliability.Consider an example with the following default values for parameters corresponding to customer segmenti:(4.15)θ01=150,θ1i1=0.3,θ2i1=0.7,θ02=200,θ1i2=0.25,θ2i2=0.75,αi=5000,γi=10,βi=0.5.Additionally, assume the following values for parameters of competing SCs:(4.16)ηR1=ηR2=0.1,cM1=cM2=100,FCM1=FCM2=20000,FCR1=FCR2=10000,Pi1-,Pi1+=5,SR1i-,SR1i+=5,Pi2-,Pi2+=5,SR2i-,SR2i+=5,1-ϕi1=0.4,1-ϕi2=0.3.Now for the RUM for customer segmenti behavior (i.e., τi=1) and with 100% confidence level about rival strategies (i.e., δij=δij-=1), we can compute four cases in Table 1 as follows in Table 2. In Case 1, coupled constraint for SC 1 is not satisfied, thus the solution is not acceptable. Since negative value for λi2 is not permitted, Case 3 is not feasible, as well. Case 4 is impossible because two coupled constraints cannot be binding simultaneously. Therefore, the single equilibrium solution for GNEP is obtained in Case 2 where λi1>0 and λi2=0. Case 2 expresses that SC 1 will win customer segment i with the least acceptable probability (i.e., 0.4), whereas rival chain will win the customer with higher probability than 0.3.Table 2
An example for competing chains associated with customer segmenti.
λi1=0λi1>0wM1*mR1i*SR1i*wM2*mR2i*SR2i*wM1*mR1i*SR1i*wM2*mR2i*SR2i*237.9137.9275.9236.8136.8205.1238.5138.5318.8236.8136.8205.1λi2=0Ui1=230.34,Ui1=260.06,Ui2=260.47Ui2=260.47 which is not feasibleΛi1=5.9727  which is feasible237.9137.9275.9236.2136.2163.5Not feasibleλi2>0Ui1=230.34,Ui2=231.19Λi2=-5.44  which is not feasibleTable2 represents a Nash solution for SCs competition. However, in the real-world competition, one SC may decide to cut prices or increase service level to affect the other party (or parties). For instance, assume that in the solution of Table 2, retailer in SC 2 increases his products utility by declining profit margin. Since θ1i1/θ2i1<ki1(1-ϕi1)βi/ηR1, it is obvious from Proposition 4.5 that rising service level is a more efficient strategy for SC i against the rival retailer. Figure 3 shows that SC 1 increases its products utility along with rival’s ones to hold customer segment i with the least probability of 0.4. Another important point in Figure 3 is illustrating the impact of customer rationality on SCs competition. Like QRE concept, τ can be interpreted as a rationality parameter which specifies customer responsiveness to utility of products in which τi=1 leads to RUM of McFadden [7]. When τi=0, the customer segment is totally indifferent about product utility and chooses the brand of products completely by chance. Herein, coupled constraints (4.7) are nonbinding for both rivals and optimal SCs strategies certainly are obtained from Case 1 of Table 1. Adversely, τi→∞ yields deterministic utility model. Here, in the case of certainty about rival strategies (i.e., δij-j=1 or ρij-j=0), both coupled constraints are binding, and rivals require to offer product with the same utility to the customer; therefore, the optimal strategies of SCs are achieved from Case 4 of Table 1.Figure 3
The products utility of SCs versus the profit margin of retailer 2 with different customer rationality parameters (withδi1=1).We can conclude that, when a customer segment is highly responsive to utility, each SC needs to present products with utility close to rival’s ones which brings about more severe competition. Therefore, customer purchasing behavior dictates severity of competition in the market which is consistent with the real-world competition.Figures4 and 5 illustrate profit variation of the retailer and manufacturer in SC 1 against price cutting strategy of retailer 2. Since rising service level investment and product price are reactive strategies of SC 1, profit of retailer 1 declines due to higher service level. However, increasing product price leads to rise in profit of manufacturer 1.Figure 4
The minimum possible profit of retailer 1 versus the profit margin of retailer 2 with different customer rationality parameters (withδi1=1).Figure 5
The minimum possible profit of manufacturer 1 versus the profit margin of retailer 2 with different customer rationality parameters (withδi1=1).Figure6 shows the wholesale and retail-sale prices, and service level investment in the case of existence of vagueness about rival chain strategies. In Figure 5, lower confidence level δi1 brings about lesser price and service level investment. From Figure 7, it can be concluded that lower δi1 makes offering products with lesser utility for the customer. Therefore, by decreasing δi1, there is more possibility to lose the customer. Figure 8 demonstrates variation in the possible profit of the retailer and manufacturer in SC 1 against different confidence level. When δi1=0, the confidence level to win the customer segment approaches zero; thus, fixed cost and service level cost cause a negative profit for the retailer and manufacturer.Figure 6
Strategies of SC 1 versus the confidence level of rival fuzzy strategy(withτi=1).Figure 7
The products utility of SCs versus the confidence level of rival fuzzy strategy(withτi=1).Figure 8
The minimum possible profit of partners in SC 1 versus the confidence level of rival fuzzy strategy(withτi=1).


### 4.3. Extension of Retailers and Manufacturers Problems in a Market

Consider a product marketM which consists of n customer segments with different demands and utility functions. In the real-world competitions, SCs often present different prices and service levels to the customer segments. Each segment selects one product type based on RUM. In this case, the SC competition model can be extended for this market. The possible profit of the manufacturer in a market is obtained from the aggregate possible profit from customer segments. Therefore, the problem of the manufacturer in SC j is defined as follows:(4.17)max⁡wMjE(πMj)=∑i∈Mkij(1-ϕij)(wMj-cMj)[αi-γi(wMj+mRji)+βiSRji]-FCMj,
subject to(4.18)θ0j-θ0j--θ1ij(wMj+mRji)+θ1ij-Pij-+θ2ijSij-θ2ij-S2ij-≥Qij,∀i∈M.
Additionally, the mathematical formulation of the retailer in SC j is as follows:(4.19)max⁡mRji,SRjiE(πRj)=∑i∈M[kij(1-ϕij)mRji[αi-γi(wMj+mRji)+βiSRji]-12ηRjSRji2]-FCMj,
subject to constraint (4.18), where kij=[1-ϕij+ϕijexp(ρij-jτi(1-δij-j))]-1 and Qij=(1/τi)ln[(1-ϕij)/ϕij]-(1-δij-j)ρij-j.Objective functions (4.17) and (4.19) indicate possible manufacturer and retailer from purchasing products to customer segments set n, respectively. Coupled constraint (4.18) assures that the least probability of winning each customer segment is not less than the minimum acceptable probability. Since the coupled constraints for both chains can be binding or nonbinding with regard to each customer segment, 4n cases can occurr. Proposition 4.5 states that a single Nash equilibrium solution of the model will exist.Proposition 4.6.
When∑j=12(1-ϕij)≤1 and Bij′>0 hold for each customer segment i and SC j, a single Nash equilibrium for SCs strategies is found from 4n cases of KKT condition.Parametric solutions of KKT condition for even two customer segments are too large to summarize in the paper; however, the numerical examples can easily be solved by numerical solvers. For instance, assume that two customer segments exist in the previous numerical example with the following parameters:(4.20)θ01=150,θ02=200,θ111=0.3,θ211=0.7,θ112=0.25,θ212=0.75,θ121=0.2,θ221=0.8,θ122=0.3,θ222=0.7,α1=5000,α2=6000,γ1,γ2=10,β1=0.5,β2=0.8.Additionally, assume the following parameters for competing chains:(4.21)Pij-,Pij+,SRji-,SRji+=5,∀i,j∈{1,2},1-ϕ11=0.4,1-ϕ12=0.3,1-ϕ21=0.2,1-ϕ22=0.25.Therefore, in the case ofτi=1and δij-j=1 for all i,j∈{1,2}, we have Q11=-4.055, Q12=-8.473, Q21=-1.386, and Q22=-1.099. Since SCs compete for two customer segments, 42 cases in Table 2 should be investigated. Based on KKT conditions, the feasible and optimal case is needed to hold λij≥0 for all i,j∈{1,2}, 4.055≤U11-U12≤8.473, and 1.386≤U21-U22≤1.099.In the solutions of Table3, the single feasible and optimal solution is related to λ11=λ21=0, λ12=11.22, and λ22=14.00. Therefore, in the Nash equilibrium, coupled constraint for customer segment 1 and SC 2 and coupled constraint for customer 2 and SC 1 are binding as well. Consequently, SC 2 will win customer segment 1 with the least probability of 0.3; however, SC 1 will attract the customer segment 1 with probability more than 0.4. Similarly, SC 1 (and 2) will win customer segment 2 with probability of 0.2 (more than 0.25). Note that confidence levels about rival strategies can be varied or equal for different customer segments. Moreover, the degree of customer responsiveness to utility (rationality) will affect the degree of severity of the competition between chains for that customer.Table 3
An example for competing chains for two customer segments.
λ11=0λ11>0λ21=0λ21>0λ21=0λ21>0λij=0λ21=-11.10λ11=-15.19λ11=-15.31, λ21=-11.12λ22=0U11=269.71U12=197.78NFSU11=270.32U12=197.78NFSU11=193.72U12=197.78NFSU11=193.72U12=197.78NFSλ12=0U21=351.90U22=279.96U21=278.57U22=279.97U21=352.65U22=279.97U21=278.57U22=279.97λ22=13.89λ11=-15.33,λ22=14.04λ22>0U11=269.71U12=197.08NFSNFU11=193.01U12=197.07NFSNFU21=351.90U22=350.80U21=352.65U22=351.65λ12=11.10λ21=-11.09,λ12=-11.20λ22=0U11=269.71U12=261.71NFSU11=270.33U12=261.85NFSNFNFλ12>0U21=351.90U22=279.41U21=278.03U22=279.41λ22>0λ12=11.22,λ22=14.00U11=269.71U12=261.24FSNFNFNFU21=351.90U22=350.80FS: feasible solution, NF: nonfeasible, and NFS: nonfeasible solution.


## 4.1. Manufacturers Problem for a Single Customer Segment

Losing each customer segment diminishes market share which is often horrible for all partners in an SC. Although partners of an SC compete to gain more profit, they need to maintain a level of cooperation to sell product to the customer segments in a competitive market. The manufacturer injth SC attempts to increase possible profit with considering a minimum probability of selling products of SC to customer i. Thus, manufacturer j considers the following problem:(4.1)max⁡wMj⁡E(πMj)=(1-ϕij)(wMj-cMj)(αi-γPij+βSRji)-FCMj,
subject to(4.2)Pi(j∣θ0j,θkij)=exp⁡(τi(θ0j-θ1ijPij+θ2ijSRji))exp⁡(τi(θ0j-θ1ijPij+θ2ijSRji))+exp⁡(τi(θ0j--θ1ij-P̃ij-+θ2ij-S̃Rj-i))≥1-ϕij,(4.3)Pij=wMj+mRji,
where 1-ϕij is a minimum probability which is acceptable for managers of SC j to sell its products to customer segment i. By applying maximin concept [57], manufacturer j in objective function (4.1) decides on wholesale price to maximize the minimum possible profit from selling products to customer segment i. Since the exact strategies of the rival SC are not revealed for manufacturer j, he makes his/her decision based on an estimation of rival price (P̃ij-) and service level (S̃Rj-i). Taking the RUM into account for the behavior of customer segment i, constraint (4.2) ensures that the probability of selling products to customer segment i is higher than the minimum acceptable probability 1-ϕij. Combination of objective function (4.1) and constraint (4.2) expresses that the manufacturer relates his possible profit from selling products to the customer purchasing decisions. On the other hand, these decisions depend upon the rival strategies which only estimations of them are available for the manufacturer. The manufacturer problem is a nonlinear GNEP which is not appropriate for the optimization. Lemmas 4.1 and 4.2 can be applied for linearizing the problem.Lemma 4.1.
By consideringδ level cut concept for fuzzy constraint [58, 59], constraint (4.2) will be equivalent to
(4.4)θ0j-θ0j--θ1ijPij+θ1ij-Pij-+θ2ijSij-θ2ij-S2ij-≥ln⁡⁡(1-ϕijϕij)1/τi-(1-δij-j)ρij-j,
where ρij-j=θij1Pij-++θij2SRj-i- and 0≤δij-j≤1.Proofs of all propositions and lemmas are given in the Appendix. In Lemma4.1, δij-j level cut can be interpreted as a confidence level of decision makers of SC j in the strategies of SC j- corresponding to customer segment i. Therefore, in constraint (4.4), δij-j equals to 1 means that DMs of SC j are sure that the probability of attracting customer i is higher than 1-ϕij. This confidence level declines as δij-j approaches zero. It is straightforward from the right hand side of constraint (4.4) that the uncertainty about rival strategies measured by the confidence level reduces the minimum probability of attracting the customer segment. Thus, in objective function (4.1), 1-ϕij has to be changed with respect to the confidence level. In Lemma 4.2, the minimum probability of attracting customer segment i is updated.Lemma 4.2.
When vagueness about competitor’s strategies exists, the minimum probability of attracting customer segmentj, that is, 1-ϕij, is changed as follows:
(4.5)1-ϕij′=kij(1-ϕij),
where kij=[1-ϕij+ϕijexp(ρij-jτi(1-δij-j))]-1.Note that in the case of absolute confidence, or no vagueness about competitor strategies, or quite irrational customer (i.e.,δij-j=1 or ρij-j=0 or τi=0), we have 1-ϕij′=1-ϕij. By applying the results of Lemmas 4.1 and 4.2 to manufacturer problem (4.1)–(4.3), the linear GNEP is as follows:(4.6)max⁡wMj⁡E(πMj)=kij(1-ϕij)(wMj-cMj)[αi-γi(wMj+mRji)+βiSRji]-FCMj,
subject to(4.7)θ0j-θ0j--θ1ij(wMj+mRji)+θ1ij-Pij-+θ2ijSRji-θ2ij-SRj-i≥1τiln⁡(1-ϕijϕij)-(1-δij-j)ρij-j.


## 4.2. Retailers Problem for a Single Customer Segmenti

Now we study the retailer problem in the case of a single customer segmenti. Retailer j determines the margin profit as well as service level. Similar to the manufacturer, the retailer assumes a lower bound for the probability of selling products to the customer segment. The retailer maximizes the minimum possible profit from trade with the customer segment. Therefore, the following model can be considered for the retailer(4.8)max⁡mRj,SRjiE(πRj)=(1-ϕij)(mRj)(αi-γiPij+βiSRji)-12ηRjSRji2-FCRj,
subject to constraints (4.2) and (4.3).As suggested in the extensive literature of SC game models [22, 23, 26, 60], we assume that when a player provides service level SRji, the service cost of the player is 0.5ηRjSRji2, that is, improving service level has a diminishing return on service expenditure. Both of the chain’s partners by choosing proper value for their controllable variables endeavor to maintain this probability within an acceptable limit. The retailer problem is also a nonlinear GNEP which is not appropriate for computing Nash equilibrium. By applying Lemmas 4.1 and 4.2 for the retailer problem, we have(4.9)max⁡mRji,SRjE(πRj)=kij(1-ϕij)(mRji)[αi-γi(wMj+mRji)+βiSRj]-12ηRjSRj2-FCRj,
subject to constraint (4.7).Constraint (4.7) is a “coupled constraint” which relates SCs’ strategies to rival ones to guarantee the minimum probability of attracting customer segment i. Therefore, manufacturer profit (4.6) and retailer profit (4.9) along with coupled constraint (4.7) constitute GNEPs. To solve these problems, we associate a multiplier λij with coupled constraint (4.7). Hessian matrix of E(πRj)is(4.10)HRj=[-2kij(1-ϕij)γikij(1-ϕij)βikij(1-ϕij)βi-ηRj].The expected profitE(πRj) is a concave function on (mRji,SRji) if and only if the Hessian matrix is negatively definite [61]. Let us define Bij′=2γiηRj-βi2 and Bij=3γiηRj-kij(1-ϕij)βi2. From the manufacturer and retailer problems, we drive the following.Proposition 4.3.
IfBij′>0, then the optimum values for wholesale price of the manufacturer, as well as the marginal profit and service level of the retailer, are as follows:(i)
ifλij=0, then one has(4.11)mRji*=(αi-γicMj)ηRjBij,wMj*=cMj+(αi-γicMj)ηRjBij,SRji*=βikij(1-ϕij)(αi-γicMj)Bij,(ii)
ifλij=Λij≠0, then one has
(4.12)mRji*=kij(1-ϕij)θ2ij2(αi-γicMj)2θ1ijEij-θ2ijkij(1-ϕij)Fij+Eij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-]2θ1ijEij-θ2ijkij(1-ϕij)Fij,wMj*=cMj+kij(1-ϕij)θ2ij2(αi-γicMj)2θ1ijEij-θ2ijkij(1-ϕij)Fij+Eij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-]2θ1ijEij-θ2ijkij(1-ϕij)Fij,SRji*=kij(1-ϕij)2θ1ijEij-θ2ijkij(1-ϕij)Fij×[2θ1ijθ2ij(αi-γicMj)+Fij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-i]],Λij=kij(1-ϕij)2θ1ijEij-θ2ijkij(1-ϕij)Fij×[(αi-γicMj)[2ηRjθ1ij-kij(1-ϕij)βiθ2ij]-Bij[-θ1ijcMj+θ0j-θ0j--Qij+θ1ij-(wMj-+mRj-i)-θ2ij-SRj-i]],
where Eij=θ1ijηRj-θ2ijkij(1-ϕij)βi, Fij=θ1ijβi-3θ2ijγi, and Qij = (1/τi)ln((1-ϕij)/ϕij)-(1-δij-j)ρij-j.Retailerj excessively invests in service if Bij′<0, which incurs a negative profit to him. The condition Bij′>0 means that the service investment should not be too inexpensive, which is consistent with those considered in [23, 26, 53]. Therefore, we assume that Bij′>0 throughout this paper. Moreover, the negative values for optimal profit margin and service level are not feasible for the retailer, that is, mRji*,SRji*>0. Since from Bij′>0 and 0≤kij≤1, it follows that Bij>0, and we assume that αi>γicMj to ensure mRji*,SRji*>0. The condition αi>γicMj indicates that unit production cost should not be too high. The multiplier λij associated with KKT condition may be thought as a shadow price for coupled constraint (4.7); furthermore, complementary slackness condition holds for the multiplier and the coupled constraint. That is to say, coupled constraint (4.7) is binding if λij=Λij≠0.Proposition4.3 presents the optimal strategies of SC j regarding customer segment i which depends on the strategies of rival chain (j-). In other words, if Xij and Xij- denote vectors of the decisions or strategy variables of SCs j and its rival j-, respectively, then the optimal values in Proposition 4.3 are Xij*(Xij-). This proposition gives the following insights.(i)
For a single customer segmenti, the optimal marginal profits of manufacturer and retailer are equal. Therefore, if Pij*-cMj is the optimal profit margin of a product unit for SC j, it is optimal that the manufacturer and the retailer consent to the equal profit margin (Pij*-cMj)/2. This situation holds for both the conditions λij=0 and λij≠0.(ii)
Theλij=0 states that the SC jth will sell product to customer segment i with probability higher than the least acceptable probability, that is, Pi(j∣θ0j,θkij)≥1-ϕij. In this situation, the rival chain strategies (i.e., Xij-) have no effects on the optimal price and service level of SCj.(iii)
Adversely,λij=Λij≠0 expresses that the probability of attracting customer segment i by SC jth decreases to the threshold of the minimum acceptable probability, that is, Pi(j∣θ0j,θkij)=1-ϕij. In this situation, SC’s strategies depend on the rival strategies. In other words, any variation in price and service level of the rival needs to be responded by the SC to ensure the least probability of selling products to customer segment i. In Proposition 4.4, the SC responses to variations in strategies of the rival are investigated.Proposition 4.4.
In the case ofλij=Λij≠0, for the optimal strategies values in Proposition 4.3, one has(1)
∂mRji*/∂wMj-=∂mRji*/∂mRj-i=∂wMj*/∂wMj-=∂wMj*/∂mRj-i=Eijθ2ij-/[2θ1ijEij-θ2ijkij(1-ϕij)Fij], ∂mRji*/∂SRj-i = ∂wMj*/∂SRj-i = -Eijθ2ij-/[2θ1ijEij-θ2ijkij(1-ϕij)Fij],(2)
∂SRji*/∂wMj- = ∂SRji*/∂mRj-i = [kij(1-ϕij)Fijθ1ij-]/[2θ1ijEij-θ2ijkij(1-ϕij)Fij], ∂SRji*/∂SRj-i = [-kij(1-ϕij)Fijθ2ij-]/[2θ1ijEij-θ2ijkij(1-ϕij)Fij].Proposition4.4 analyzes how the optimal wholesale and retail price as well as service level of one SC are affected by the retail price and service level of the rival chain. For instance, assume that the rival retailer decides to reduce ΔmRj-i from his/her retail price to attract customer segment i. From constraint (2.3), it is obvious that the product utility of the rival chain increases ΔmRj-iθ1ij- from the customer point of view. The SC j considers the following changes in its optimal strategies to react to his/her rival:(4.13)ΔmRji=Eijθ1ij-2θ1ijEij-θ2ijkij(1-ϕij)Fij(-ΔmRj-i),ΔwMj=Eijθ1ij-2θ1ijEij-θ2ijkij(1-ϕij)Fij(-ΔmRj-i),ΔSRj=-kij(1-ϕij)Fijθ2ij-2θ1ijEij-θ2ijkij(1-ϕij)Fij(-ΔmRj-i).Therefore, the product utility of SC increases from the customer’s viewpoint as follows:(4.14)ΔUij=-θ1ij(ΔmRji+ΔwMj)+θ2ij(ΔSRj)=ΔmRj-iθ1ij-=ΔUij-.
In other words, SC jth increases its product utility just equal to the rival’s one to hold coupled constraint (4.2), that is, if λij=Λij≠0, then ΔUij=ΔUij-. Now in Proposition 4.5, we investigate how an SC specifies its new strategies.Proposition 4.5.
FromBij>0, it follows that kij(1-ϕij)βi/ηRj<3γi/βi, and for the price changes of the rival chain, one has(i)
∂mRji*/∂wMj- = ∂mRji*/∂mRj-i = ∂wMj*/∂wMj- = ∂wMj*/∂mRj-i > 0 if θ1ij/θ2ij > kij(1-ϕij)βi/ηRj, ∂SRj*/∂wMj- = ∂SRj*/∂mRj-i < 0 if θ1ij/θ2ij < 3γi/βi.Similarly, for service level changes of the rival chain, one has(ii)
∂mRji*/∂SRj- = ∂wMj*/∂SRj- < 0 if θ1ij/θ2ij > kij(1-ϕij)βi/ηRj, and ∂SRj*/∂SRj- = ∂SRj*/∂SRj- > 0 if θ1ij/θ2ij < 3γi/βi.Two valueskij(1-ϕij)βi/ηRj and 3γi/βi are key thresholds which determine SC’s efficient strategies against changes in the price and service level of the rival. Part (i) of Proposition 4.5 gives us the following insights.(i)
Ifθ1ij/θ2ij>3γi/βi, then declining in the product price of the rival chain j- will be responded by decreasing in price and increasing in service level of SC jth. However, since the customer is more responsive to price, price cutting is a more efficient strategy for rising the product’s utility from the customer segment viewpoint.(ii)
Ifθ1ij/θ2ij<kij(1-ϕij)βi/ηRj, then the customer segment is more responsive to the service level. Therefore, if the decision makers in chain j want to maintain the segment with a specific minimum probability, it is needed to compensate for a price increase with service level which is more effective.(iii)
Ifkij(1-ϕij)βi/ηRj<θ1ij/θ2ij<3γi/βi, then both strategies of cutting price and increasing service level are efficient.(iv)
Similarly, part (ii) states the SC’s reactions to service level changes of rival chain. Cutting price and increasing service level by rival chain lead to rise in product utility. Figure2 illustrates the efficient strategies of an SC against rising product utility by the rival.Figure 2
SC’s efficient strategies for customer segmenti against variation in rival product utility.When manufacturers and retailers in both chains consider coupled constraint (4.7), and if ∑j=12(1-ϕij)≤1, then multipliers λ corresponding to the coupled constraints must satisfy λ=0 or λ>0 (the KKT condition). Therefore, in attempting to find Nash equilibrium solution, it is sufficient to take the cases of Table 1 into account.Table 1
Four possible cases for competing chains associated with customer segmenti.
λij=0λij>0λij-=0Case1Case2λij->0Case3Case4Therefore, Nash equilibrium solution of the competition is obtained from one of the following cases.Case 1 (λij=0 and λij-=0).
Strategies of SC and rival chain are obtained by part (i) of Proposition4.3. Here, both coupled constraints are satisfied at (wMj*,mRji*,SRj*) and (wMj-*,mRj-i*,SRj-*).Case 2 (λij=0 and λij->0).
The coupled constraint (4.7) is binding only for rival SC. Therefore, the optimal strategies of SC j are achieved by part (i) of Proposition 4.3. However, rival chain takes the optimal strategies of the first SC into account and computes optimal strategies by part (ii) of Proposition 4.3.Case 3 (λij>0 and λij-=0).
In this case, the situation is contrary to Case2 and the coupled constraint is binding only for SC j.Case 4 (λij>0 and λij->0).
Here, both coupled constraints for competing SCs are binding. This rare condition takes place only if∑j=12(1-ϕij)=1. The optimal strategies of SC and rival are obtained by part (ii) of Proposition 4.3.Note that the values of1-ϕij are subjective probabilities which decision makers in one SC specify it independently of rival chain. This probability is not revealed for rival. The value of 1-ϕij is determined based on importance and worth of customer segment i and expresses that SC partners are content to sacrifice profit to win customer segment i with the least probability of 1-ϕij. The assumption ∑j=12(1-ϕij)≤1 insures that a permanent equilibrium point exists in the competition, and resulting strategies Xij(Xij-) and Xij-(Xij) satisfy the least probability conditions for both SCs. Otherwise, SCs involve eternal competition to present the customer more favorable products, and no equilibrium exists in a one-shot competition. However, in a multiperiod competition, presenting more favorable products to the customers to increase business is the very essence of competition which brings about innovation and improvement in technology and knowledge. For instance, consider the dynamics of the microprocessor market which drives severe competition between two main suppliers, that is, AMD and Intel. The main customers of microprocessors are major original equipment manufacturers (OEMs) such as Dell, IBM, and HP which enjoy this rivalry. Throughout years of competition, Intel and AMD contest to win customers’ business not just on price and service level, but on all aspects of their products including quality, performance, and reliability.Consider an example with the following default values for parameters corresponding to customer segmenti:(4.15)θ01=150,θ1i1=0.3,θ2i1=0.7,θ02=200,θ1i2=0.25,θ2i2=0.75,αi=5000,γi=10,βi=0.5.Additionally, assume the following values for parameters of competing SCs:(4.16)ηR1=ηR2=0.1,cM1=cM2=100,FCM1=FCM2=20000,FCR1=FCR2=10000,Pi1-,Pi1+=5,SR1i-,SR1i+=5,Pi2-,Pi2+=5,SR2i-,SR2i+=5,1-ϕi1=0.4,1-ϕi2=0.3.Now for the RUM for customer segmenti behavior (i.e., τi=1) and with 100% confidence level about rival strategies (i.e., δij=δij-=1), we can compute four cases in Table 1 as follows in Table 2. In Case 1, coupled constraint for SC 1 is not satisfied, thus the solution is not acceptable. Since negative value for λi2 is not permitted, Case 3 is not feasible, as well. Case 4 is impossible because two coupled constraints cannot be binding simultaneously. Therefore, the single equilibrium solution for GNEP is obtained in Case 2 where λi1>0 and λi2=0. Case 2 expresses that SC 1 will win customer segment i with the least acceptable probability (i.e., 0.4), whereas rival chain will win the customer with higher probability than 0.3.Table 2
An example for competing chains associated with customer segmenti.
λi1=0λi1>0wM1*mR1i*SR1i*wM2*mR2i*SR2i*wM1*mR1i*SR1i*wM2*mR2i*SR2i*237.9137.9275.9236.8136.8205.1238.5138.5318.8236.8136.8205.1λi2=0Ui1=230.34,Ui1=260.06,Ui2=260.47Ui2=260.47 which is not feasibleΛi1=5.9727  which is feasible237.9137.9275.9236.2136.2163.5Not feasibleλi2>0Ui1=230.34,Ui2=231.19Λi2=-5.44  which is not feasibleTable2 represents a Nash solution for SCs competition. However, in the real-world competition, one SC may decide to cut prices or increase service level to affect the other party (or parties). For instance, assume that in the solution of Table 2, retailer in SC 2 increases his products utility by declining profit margin. Since θ1i1/θ2i1<ki1(1-ϕi1)βi/ηR1, it is obvious from Proposition 4.5 that rising service level is a more efficient strategy for SC i against the rival retailer. Figure 3 shows that SC 1 increases its products utility along with rival’s ones to hold customer segment i with the least probability of 0.4. Another important point in Figure 3 is illustrating the impact of customer rationality on SCs competition. Like QRE concept, τ can be interpreted as a rationality parameter which specifies customer responsiveness to utility of products in which τi=1 leads to RUM of McFadden [7]. When τi=0, the customer segment is totally indifferent about product utility and chooses the brand of products completely by chance. Herein, coupled constraints (4.7) are nonbinding for both rivals and optimal SCs strategies certainly are obtained from Case 1 of Table 1. Adversely, τi→∞ yields deterministic utility model. Here, in the case of certainty about rival strategies (i.e., δij-j=1 or ρij-j=0), both coupled constraints are binding, and rivals require to offer product with the same utility to the customer; therefore, the optimal strategies of SCs are achieved from Case 4 of Table 1.Figure 3
The products utility of SCs versus the profit margin of retailer 2 with different customer rationality parameters (withδi1=1).We can conclude that, when a customer segment is highly responsive to utility, each SC needs to present products with utility close to rival’s ones which brings about more severe competition. Therefore, customer purchasing behavior dictates severity of competition in the market which is consistent with the real-world competition.Figures4 and 5 illustrate profit variation of the retailer and manufacturer in SC 1 against price cutting strategy of retailer 2. Since rising service level investment and product price are reactive strategies of SC 1, profit of retailer 1 declines due to higher service level. However, increasing product price leads to rise in profit of manufacturer 1.Figure 4
The minimum possible profit of retailer 1 versus the profit margin of retailer 2 with different customer rationality parameters (withδi1=1).Figure 5
The minimum possible profit of manufacturer 1 versus the profit margin of retailer 2 with different customer rationality parameters (withδi1=1).Figure6 shows the wholesale and retail-sale prices, and service level investment in the case of existence of vagueness about rival chain strategies. In Figure 5, lower confidence level δi1 brings about lesser price and service level investment. From Figure 7, it can be concluded that lower δi1 makes offering products with lesser utility for the customer. Therefore, by decreasing δi1, there is more possibility to lose the customer. Figure 8 demonstrates variation in the possible profit of the retailer and manufacturer in SC 1 against different confidence level. When δi1=0, the confidence level to win the customer segment approaches zero; thus, fixed cost and service level cost cause a negative profit for the retailer and manufacturer.Figure 6
Strategies of SC 1 versus the confidence level of rival fuzzy strategy(withτi=1).Figure 7
The products utility of SCs versus the confidence level of rival fuzzy strategy(withτi=1).Figure 8
The minimum possible profit of partners in SC 1 versus the confidence level of rival fuzzy strategy(withτi=1).


## 4.3. Extension of Retailers and Manufacturers Problems in a Market

Consider a product marketM which consists of n customer segments with different demands and utility functions. In the real-world competitions, SCs often present different prices and service levels to the customer segments. Each segment selects one product type based on RUM. In this case, the SC competition model can be extended for this market. The possible profit of the manufacturer in a market is obtained from the aggregate possible profit from customer segments. Therefore, the problem of the manufacturer in SC j is defined as follows:(4.17)max⁡wMjE(πMj)=∑i∈Mkij(1-ϕij)(wMj-cMj)[αi-γi(wMj+mRji)+βiSRji]-FCMj,
subject to(4.18)θ0j-θ0j--θ1ij(wMj+mRji)+θ1ij-Pij-+θ2ijSij-θ2ij-S2ij-≥Qij,∀i∈M.
Additionally, the mathematical formulation of the retailer in SC j is as follows:(4.19)max⁡mRji,SRjiE(πRj)=∑i∈M[kij(1-ϕij)mRji[αi-γi(wMj+mRji)+βiSRji]-12ηRjSRji2]-FCMj,
subject to constraint (4.18), where kij=[1-ϕij+ϕijexp(ρij-jτi(1-δij-j))]-1 and Qij=(1/τi)ln[(1-ϕij)/ϕij]-(1-δij-j)ρij-j.Objective functions (4.17) and (4.19) indicate possible manufacturer and retailer from purchasing products to customer segments set n, respectively. Coupled constraint (4.18) assures that the least probability of winning each customer segment is not less than the minimum acceptable probability. Since the coupled constraints for both chains can be binding or nonbinding with regard to each customer segment, 4n cases can occurr. Proposition 4.5 states that a single Nash equilibrium solution of the model will exist.Proposition 4.6.
When∑j=12(1-ϕij)≤1 and Bij′>0 hold for each customer segment i and SC j, a single Nash equilibrium for SCs strategies is found from 4n cases of KKT condition.Parametric solutions of KKT condition for even two customer segments are too large to summarize in the paper; however, the numerical examples can easily be solved by numerical solvers. For instance, assume that two customer segments exist in the previous numerical example with the following parameters:(4.20)θ01=150,θ02=200,θ111=0.3,θ211=0.7,θ112=0.25,θ212=0.75,θ121=0.2,θ221=0.8,θ122=0.3,θ222=0.7,α1=5000,α2=6000,γ1,γ2=10,β1=0.5,β2=0.8.Additionally, assume the following parameters for competing chains:(4.21)Pij-,Pij+,SRji-,SRji+=5,∀i,j∈{1,2},1-ϕ11=0.4,1-ϕ12=0.3,1-ϕ21=0.2,1-ϕ22=0.25.Therefore, in the case ofτi=1and δij-j=1 for all i,j∈{1,2}, we have Q11=-4.055, Q12=-8.473, Q21=-1.386, and Q22=-1.099. Since SCs compete for two customer segments, 42 cases in Table 2 should be investigated. Based on KKT conditions, the feasible and optimal case is needed to hold λij≥0 for all i,j∈{1,2}, 4.055≤U11-U12≤8.473, and 1.386≤U21-U22≤1.099.In the solutions of Table3, the single feasible and optimal solution is related to λ11=λ21=0, λ12=11.22, and λ22=14.00. Therefore, in the Nash equilibrium, coupled constraint for customer segment 1 and SC 2 and coupled constraint for customer 2 and SC 1 are binding as well. Consequently, SC 2 will win customer segment 1 with the least probability of 0.3; however, SC 1 will attract the customer segment 1 with probability more than 0.4. Similarly, SC 1 (and 2) will win customer segment 2 with probability of 0.2 (more than 0.25). Note that confidence levels about rival strategies can be varied or equal for different customer segments. Moreover, the degree of customer responsiveness to utility (rationality) will affect the degree of severity of the competition between chains for that customer.Table 3
An example for competing chains for two customer segments.
λ11=0λ11>0λ21=0λ21>0λ21=0λ21>0λij=0λ21=-11.10λ11=-15.19λ11=-15.31, λ21=-11.12λ22=0U11=269.71U12=197.78NFSU11=270.32U12=197.78NFSU11=193.72U12=197.78NFSU11=193.72U12=197.78NFSλ12=0U21=351.90U22=279.96U21=278.57U22=279.97U21=352.65U22=279.97U21=278.57U22=279.97λ22=13.89λ11=-15.33,λ22=14.04λ22>0U11=269.71U12=197.08NFSNFU11=193.01U12=197.07NFSNFU21=351.90U22=350.80U21=352.65U22=351.65λ12=11.10λ21=-11.09,λ12=-11.20λ22=0U11=269.71U12=261.71NFSU11=270.33U12=261.85NFSNFNFλ12>0U21=351.90U22=279.41U21=278.03U22=279.41λ22>0λ12=11.22,λ22=14.00U11=269.71U12=261.24FSNFNFNFU21=351.90U22=350.80FS: feasible solution, NF: nonfeasible, and NFS: nonfeasible solution.


## 5. Conclusion

In this paper, we develop a generalized Nash equilibrium model for the competition between two SCs where their customers are free to choose. We examine the situation where SC’s decision makers consider minimum probabilities for attracting the customers based on customers worth for the SC. Using coupled constraints concept in GNEPs, these probabilities are involved in competition model formulations. Finally, the equilibrium strategies for consequent GNEPs are obtained from exploring possible cases of KKT condition. We show that SCs competition is affected by customer purchasing behaviors due to two reasons. In the first place, higher rational customers dictate more severe competition between chains. Secondly, efficient response of an SC to the variation in rival strategies depends on the sensitivity of the customer segment to price and service level. Moreover, we investigate the impacts of vagueness regarding rival strategies on SC decisions. We find that decision makers with different confidence levels about rival strategies may achieve different retail price and service level. The less the confidence level about the rival strategies, the less the actual probability of wining customer will be.



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*Source: 421265-2012-06-10.xml*