RETAIL PRICE COMPETITION OF DOMESTIC AND INTERNATIONAL COMPANIES: A BI-LEVEL GAME THEORETICAL OPTIMIZATION APPROACH

. Drawing on the Stackelberg game approach to solving the pricing problem in a supply chain, this paper develops a bi-level model whereby a domestic company and a foreign manufacturer compete to gain more profit from the market of a retailer. The domestic company acts as the leader and the retailer as the follower. The domestic company has two manufacturers each of whom produces and sells a different quality of the product. The retailer decides to purchase products based on the prices offered by the low-quality manufacturer, the high-quality manufacturer, and the foreign manufacturer, known as an exogenous factor. In fact, the first level seeks to maximize its profits and the second level seeks to reduce the cost of purchasing. In this paper, the price of the products of each manufacturer is considered a contributing factor to the retailer’s tendency to buy from each manufacturer. This assumption is designed by the multinomial logit model. As the proposed model has binary variables in its follower segment, a novel hybrid exact method based on explicit enumeration method and Lambert-W function is applied to solve it. In other words, to calculate the optimal selling price of domestic products and their profit first by using the explicit enumeration method, the bi-level model is transformed into a single-level problem. The problem is, then, solved precisely by applying the Lambert-W function. The efficiency of the proposed model is proven by the results obtained from solving the model and the sensitivity analysis of the main parameters of the model. Moreover, to have a detailed managerial analysis of each manufacturer’s profit on the competitive market environment, the market is studied in view of three different scenarios: (1) when there is a sense of patriotism regarding domestic manufacturers; (2) when customers have low incomes; and (3) when customers have high incomes. Finally, the study results conclude that if the domestic company has two manufacturers that produce a different type of quality can lead to an increase in the profit of the domestic company. Indeed, the proposed model can increase the competitive power of the domestic company against imported products by providing appropriate pricing on its products.


Introduction
Today's fast-evolving marketplace often means that quality-differentiated brands of a certain product are being introduced onto the market by leaps and bounds, and retailers can, consequently, sell multiple brands of the same product with differing quality as supplied by different manufacturers.On the other hand, the emergence of global supply chains (SC), mostly as a natural corollary of ever-increasing connectivity in today's globalized world, also means that all manner of foreign manufacturers have carved a niche in the domestic market to compete with domestic manufacturers.As a result, considering the increasing number of manufacturers producing a specific product with differing quality and brands, retailers are torn between foreign and domestic manufacturers.Market globalization has inexorably given rise to increasingly fierce competition between domestic and foreign manufacturers.In such a competitive market, wherein the market share and profit of a given product are highly influenced by the price offered by other competing manufacturers, the importance of pricing decisions can hardly be overemphasized.Not for nothing have competitive pricing models received significant attention in both academic and practical contexts recently.One case in point can be the competition between domestic and foreign manufacturers in the cosmetics industry.Yoon et al. [1] stated that to gain more market share and profit in local and international markets vis-a-vis their foreign competitors, South Korean cosmetics companies improve their competitive edge by manufacturing products with different qualities and prices to better cater to their prospective customers' needs and tastes.Pursuing this policy has helped these Korean companies increase their share of the global and the local market and profitability.
In this paper, we consider a domestic company with two domestic products competing over price with a foreign manufacturer, with each trying to maximize their market share.In real-world situations, retailers consider various factors before making their purchase decision.Among others, the price of the products, the product quality level, and lead time contributing to such decisions.Some social factors such as the propensity for buying domestic goods or the average income level of the community are also among the contributing factors.For instance, when customers set great store by buying domestic products for patriotic reasons, domestic manufacturers are likely to take over from foreign manufacturers in terms of market share.As a further illustration, taking the average income of a community into consideration, if the average income level is high, the manufacturer producing products of higher quality stands a better chance of gaining the market share.
To show the effect of different factors contributing to a given manufacturer's market share, researchers widely use the concept of the utility function.In the literature on the topic, different types of utility functions such as multinomial logit (MNL) function, Huff function, linear utility function, etc. are used.In this paper, we consider two local manufacturers competing with a foreign manufacturer.Of the two local manufacturers, one produces items of high quality, and the other produces the same product but with low quality.The competition between these two manufacturers and a foreign manufacturer is quite understandable.In this paper, a multinomial logit (MNL) function determines the market share of each manufacturer.In a multi-echelon SC, the members of that SC may adopt different approaches to decision-making.These approaches are mainly classified as centralized and decentralized decision-making.In centralized decision-making, a unique decision-maker decides on the optimal solutions, while in decentralized decision-making, individual members of the SC separately make their own optimal decisions.This paper assumes that the retailer and domestic manufacturers operate in a decentralized system where the domestic manufacturers act as the leaders and the retailer the follower.
In a decentralized system, the SC's members have different ways to address their problem.One common approach to addressing constrained decentralized problems, where two SC participants are non-cooperative, is bi-level optimization where decisions are made hierarchically.In fact, this type of mathematical model consists of two levels name upper (leader) and lower (follower) level [2,3].Each level of this problem is able to make the decision on some part of decision variables based on its objective function.Also, there may be some common constraints for both levels [4].Furthermore, the lower level makes the optimal decision based on the decision of the upper level.This means that these two levels based on their objective functions and interaction can affect the decision of each other [5].A bi-level model could be generally expressed as follows: ( This research defines an SC problem as one involving a retailer, one foreign manufacturer offering imported products, and two domestic manufacturers offering a single product under different brands and varying product quality (low and high quality).The MNL model is utilized to express the market share of each manufacturer, which is assumed to depend on their proposed prices.The retailer supplies the market demand based on the manufacturers' offered prices.Each manufacturer's market share is a function of the price they quote for their commodities.Drawing on a bi-level programming approach, a manufacturer-Stackelberg game model is established and solved.Regarding the leader-follower competitive pricing model in an SC operating on a global scale, the following questions are addressed: -What is the optimal pricing strategy of the domestic manufacturers to maximize their profit, against the pricing of an imported product offered by the foreign manufacturer?-How do the foreign manufacturer's pricing decisions affect the pricing strategy of the domestic manufacturers and the retailer's decisions?-How does the retailer's sensitivity to manufacturer's price affect the optimal decisions of each manufacturer, their market share, and profit?-How does customers' sensitivity to the price of the product influence the optimal pricing decision of the domestic manufacturers and the retailer's decision?-What is the optimal pricing decision for each manufacturer when faced with strongly patriotic customers?-How does the manufacturing technology affect the market share of each manufacturer and their optimal pricing?-How does the customers' average income level affect the optimal pricing decisions of the domestic manufacturers and, subsequently, their profits and market shares?-What is the optimal strategy of the retailer to minimize costs, against the pricing decisions of the domestic and foreign manufacturers?
The most important contribution of the present research is designing a competitive pricing model for a global market scenario, wherein a domestic company, with two manufacturers, each with different brands, is competing with a foreign manufacturer to gain the maximum market share and profit.This contribution takes on added significance against this backdrop that despite the advancement of globalization by leaps and bounds in recent decades and foreign manufacturers featuring more prominently than ever in local markets, little scholarship has been done on pricing problems either on a global or local scale.To fill this gap, this study focuses on the issue of competitive pricing between the domestic and foreign manufacturers for selling their products to the retailer.Here, we assume the price offered by each brand, besides the retailer's inclination to buy from either the foreign or domestic manufacturers, to be the most important factors, influencing each manufacturer's market share.Previous studies have mainly used linear utility functions to capture the effect of these factors on manufacturers' market shares; this study differs in that it has adopted a logit model to express the utility of each manufacturer from the retailer's point of view.Furthermore, to the best of our knowledge, few studies have been done on the bi-level programming approach to addressing the pricing model in a constrained Stackelberg game.In this research, we have applied a bi-level optimization model to address this leader-follower pricing model.Also, to solve this bi-level problem, we have used a novel hybrid exact method based on the explicit enumeration method and Lambert-W function.More specifically, we have used the explicit enumeration method to transform the proposed model into a single-level problem.Then, we have applied the Lambert-W function to solve our problem exactly.
The rest of the present study is structured as follows.Section 2 reviews the relevant literature.Section 3 offers a definition and formulation of the problem.It is followed by a presentation of a solution procedure to the developed problem in Section 4. Section 5, then, illustrates a numerical example and computational results.Section 6 explains managerial insights, with Section 7 drawing conclusions.Finally, all proofs of the propositions are shown in the Appendix A.

Literature review
Recent decades have seen many analytical and research-based studies on different aspects of SC such as inventory management, network design, and pricing.Among these, pricing and revenue management have received more attention.This paper mainly focuses on the pricing decision in a global SC framework.Our research pertains to four streams of literature including pricing competition in an SC, global SC, bi-level programming, and considering multi-brands of a single product.
One critical aspect of marketing strategies, which considerably affects customers' preferences, is pricing decisions.Nowadays, the emergence of various manufacturers, both domestic and foreign, has fueled competition in terms of pricing decisions.Recently, many studies have investigated the competitive pricing issue in SC, among which the following works are more related to our study.By applying one Nash game theory and two Stackelberg models, Choi [6] analyzed a competitive SC where one retailer was selling two products of two competing manufacturers.Their results indicate that the less differentiated the products, the greater the prices and the profits of the SC members.Porteus et al. [7] studied a competitive pricing model characterized by quality differentiation and limited capacity.Their results indicate that the leaders use the low-pricing strategy to lower competition among followers and increase their own profits.Anderson and Bao [8] examined the effect of the level of price competition on the profits of the SC members and demonstrated that this factor can determine whether a decentralized system can outdo a centralized one.Li et al. [9] also developed a competitive pricing model composed of two suppliers and one retailer to discuss the optimal sourcing and pricing decisions made by suppliers and the retailer, respectively.Similarly, Lin et al. [10] introduced a competition SC problem with three levels where the manufacturer acts as the main decision-maker and has three integration strategies: forward, backward, and no integration.Ba et al. [11] presented an oligopoly-pricing model in online retailing competitive markets to explore the effect of adverse prices.Xiao et al. [12] also explored the outsourcing decision-making model for two manufacturers competing on retail prices and product quality.They examined the impact of various factors such as production cost on the optimal outsourcing decisions.On the other hand, by considering order size constraints, Ekici et al. [13] established a two-stage SC in a pricing model, composed of a retailer and two suppliers, to optimize pricing decisions of the two suppliers who compete for the market share.Xiao et al. [14] presented a model based on game theory to examine pricing and lead-time where an SC and an outside manufacturer compete with each other.They considered the presence of brand differentiation between two chains, leading to fierce competition for gaining better market shares.More recently, by considering two quality-differentiated brands, Li and Chen [15] have discussed a backward integration model in a retailer SC where the retailer has three options for integration with low-quality and high-quality manufacturers.
In view of globalization and the introduction of foreign products into the domestic market in today's market, an increasing number of foreign manufacturers have emerged in national markets.Meixell and Gargeya [16] proposed an exhaustive review of global SC.Research has shown that patriotic sentiments can influence the perception of customers towards imported products.Therefore, consumer sentiments of nationalism or patriotism can considerably affect the manufacturers' market share [17][18][19].In addition, individuals who are intensively consumer-ethnocentric perceive the purchase of foreign products as a wrong decision, believing it to be detrimental to the domestic economy and stoking unemployment.Elliott and Cameron [20] examined the effect of the quality of products and the country of origin on consumers' decision-making.Although the emergence of foreign manufacturers can significantly affect pricing strategies and, consequently, the profits of domestic manufacturers, little scholarship has been done on the pricing issue in the global SC.The following papers are noteworthy.Seppälä et al. [21] studied the transfer-pricing issue in an SC operating on a global scale.By considering a monopoly market, He and Xiao [22] developed a pricing model in a non-cooperative global SC to optimize the pricing decisions of two foreign manufacturers who compete for the market share.Matta and Miller [23] also established a multinational SC to optimize transfer prices and the structure of the SC.Nagurney and Li [24] extended a quality and pricing competition game model characterized by globalization.In their model, they assumed products to be differentiated by different brands with a possibility for outsourcing.Recently, using a game-theoretic approach in a three-level SC, Noori-daryan et al. [25] have presented a joint optimization model of pricing, lead-time, ordering, and supplier selection.Song et al. [26] discussed optimal pricing decisions in the supply chain by developing a distribution channel in a Stackelberg game structure with seller and independent buyer frameworks.Mukherjee et al. [27] examined pricing problems by considering some important parameters like crisis and recovery likelihood that comes from product recall of manufacturers.
The third stream of the related literature focuses on the bi-level programming approach discussed by many researchers, a limited number of whom have considered this approach to pricing models.Gao et al. [28] examined two nonlinear bi-level programming approaches to investigate the optimal pricing decisions in a vendor-buyer setting.By considering the price and advertising-dependent demand expansion, Sadigh et al. [29] optimized inventory, pricing, and advertising strategies in an SC, manufacturing different products.To address their problem, they proposed two Stackelberg game models solved through a bi-level programming approach.Mokhlesian and Zegordi [30] extended a bi-level inventory-pricing optimization model in a multi-product SC with multiple retailers and a single manufacturer.Zhang et al. [31] also established a bi-level optimization approach to investigate the best price and replenishment cycle in a leader-follower SC for high-tech products.Similarly, using two bi-level models, Ma et al. [32] developed a joint lot sizing and pricing model in two leader-follower SCs.Based on a bi-level programming approach, Parvasi et al. [33] addressed a school bus routing problem with a possibility for outsourcing.Wang et al. [34] established two Stackelberg models consisting of a vender-led SC and buyer-led SC in a multi-product, eco-friendly SC.They applied a bi-level programming model to obtain the optimal decisions on ordering, pricing, advertising, and environmental efforts.Recently, Amirtaheri et al. [35] have extended a manufacturer-led SC to investigate optimal pricing and advertising solutions by applying a bi-level optimization approach.Recently, Zhang et al. [36] implemented a bi-level pricing structure in energy generation systems.They considered in the supply side the generation is stochastic.Also, they used a multi-agent learning algorithm to get the optimal price.
From the consumers' perspective, products of different brands and, consequently, different product quality are distinguishable [37,38].Therefore, the differences among various brands highly affect manufacturers' market share.For example, selling products with higher quality or at lower prices can help the manufacturer secure more market share.As a result, different features of the various brands of a given product contribute to each brand's market share.By considering the quality and product features, Choi and Coughlan [39] discussed the retailer's decision-making problem regarding brand differentiation.By considering one manufacturer and two retailers, Rajagopalan and Xia [40] developed a pricing model in an SC to probe the impact of product variety and differentiation on the optimal decisions of the SC's participants.Pang and Tan [41] proposed four different game-theoretic models of pricing and quality of a single product with different brands produced by two competing manufacturers.Xiao et al. [42] also discussed the optimal strategy of the channel format and product variety in a retailer-led SC.Xiao et al. [14] explored the impact of brand differentiation on the lead-time and optimal price strategies of a competitive environment wherein an SC competes with an outside manufacturer.Zhou and Lin [43] extended an advertising model in a single-retailer multiple-manufacturer SC to investigate the issue of brand competition.Similarly, Giri et al. [44] developed a centralized and manufacturer-Stackelberg model to examine the optimal pricing and quality decisions of an SC containing multiple manufacturers and one retailer who make a single product under multiple brands with varying quality.More recently, by considering brand differentiation and profits, Giri et al. [45] have optimized the quality and pricing decisions of multiple manufacturers and a single retailer competing to differentiate their brand and gain more market share.Li and Chen [15] discussed the optimal price and quality strategies in a competitive retailer-led SC with two manufacturers producing two different high-or low-quality brands.Recently, Taleizadeh et al. [46] extended a model by studying optimal decisions and operational strategies in an SC network.Their model considered two manufacturers that produce goods of different quality and compete with each other to gain more market share and profit from a retailer that has fixed demand.
To capture the effect of different factors such as pricing on the market share of each member of the SC, researchers have drawn on various types of utility functions including linear function, logit function, and Huff function.This paper has adopted a logit model to introduce the price effect into each manufacture's market share.Considering the historical perspective of utility functions, we found the following works more relevant to our study.Huang [47] developed three transport pricing models where the mode choice behavior was formulated using the logit model.Lüer-Villagra and Marianov [48] studied the optimal pricing and hub location decisions in a competitive atmosphere between two companies.In their research, they used a logit discrete choice model to formulate the customers' behavior.Čvokić et al. [49] extended a joint pricing-hub location model with competition raging between the leader and follower with market shares represented by logit functions.More recently, Zhang [50] has developed an optimization model of pricing and location in a competitive atmosphere between retailers.To express the impact of travel cost and mill price on each retailer's market share, he has used the multinomial logit (MNL) function.Mahmoodjanloo et al. [51] by using the MNL model developed a hub location pricing model by considering customer loyalty and assuming the demands are elastic.Parvasi and Taleizadeh [52] recently developed a bi-level programming model where the domestic manufacturer is the leader, and the foreign manufacturer is the follower.In their model, manufacturers intend to achieve the highest profit and market share in three different local markets by setting appropriate pricing strategies, where the customers of each market have different sales levels than the customers of the other markets.Accordingly, the authors determined the profit of each manufacturer for all markets by developing a desirability function depending on the price and quality of the goods.
Table 1 presents a concise review of the related literature.A close review of the literature reveals that no study has drawn on the bi-level programming approach to address pricing competition between two domestic manufacturers with different brands and a foreign manufacturer selling products to a common retailer.Furthermore, very few studies have looked into the effect of patriotic sentiments on the purchase decision.We have considered this issue by incorporating the effect of each manufacturer's price and the fact that the MNL model can determine the retailer's inclination to buy products from each of these manufacturers.In other words, the MNL model is used to find out how much the retailer buys from each manufacturer.Also, it bears noting that the retailer's tendency to buy from each manufacturer is determined by the price offered by the domestic manufacturer and foreign manufacturer as well as the customers' preference.

Problem definition and mathematical model
Nowadays, the rise in the number of manufacturers offering similar products with differing quality in a specific area has given rise to a competitive market.Foreign manufacturers release their products to local markets, with an inevitable decrease in profitability for domestic manufacturers operating on those local markets.Hence, to maintain their profit, domestic manufacturers and companies should offer an appropriate selling price to retailers.On the other hand, to preserve their market share in this fiercely competitive environment, retailers should cater to customers' needs.Therefore, in a given area, based on customers' preferences for each manufacturer's product, retailers determine the quantity of purchase from these manufacturers.
Against this backdrop, this study examines the competition between manufacturers and a retailer in an SC network (Fig. 1).As Figure 1 shows, there are three manufacturers named H, L, and F such that the domestic company has two manufacturers (Manufacturers H and L).The products of Manufacturers H and L are of high quality and low quality, respectively.Manufacturer F also sells imported products to the retailer.
By considering the bi-level programming application, the decision-making environment is designed based on a Stackelberg game (Fig. 2).In this regard, the domestic company and the retailer are introduced as the leader and follower, respectively.In this structure, the domestic company owns the two manufacturers so that the company makes a central decision for both domestic manufacturers.Regarding the product price of each domestic manufacturer (L and H) and the product price of the foreign manufacturer (F), the follower seeks to minimize its cost of purchasing products.Indeed, by estimating the customer's demand for each manufacturer's product (L, H, and F) along with each manufacturer's product price, the retailer decides on how much to buy It should be noted that the leader, as the level problem (ULP) decision-maker, has complete knowledge of the foreign manufacturer's product price and the retailer's demand for each manufacturer's product.Also, in this paper, the product price of the foreign manufacturer is considered as an exogenous factor.This is because we have assumed that the domestic company knows the price of foreign goods offered in the market to determine an appropriate price for their products.
More specifically, this paper primarily aims to determine a proper pricing strategy for domestic company manufacturers, so that they can compete with the foreign manufacturer and achieve the maximum share and profit of the retailer market.To this end, a mathematical model has been designed, using the concept of the Stackelberg game model based on bi-level programming.Also, to determine the share and the profit of each manufacturer, the MNL model has been used.
To maximize the profit of the domestic company, in view of the stated assumptions, a non-linear mixed integer-mathematical model is proposed as follows:

𝑁
Set of domestic manufacturers (,  ∈ {1, 2}) In this paper, the retailer can decide to buy their products from whichever manufacturer that it desires.The retailer's sensitivity to the price of each manufacturer (  and   ) is considered as the factor affecting the retailer's decision.Therefore, the desirability of the domestic and foreign manufacturer for the retailer in the market is defined as follows: where   and   are positive parameters.
Considering the factors contributing to the retailer's attraction to a manufacturer, the MNL function, which is applied to calculate the probability of the retailer's attraction to a manufacturer, is stated as: Expressions ( 4) and ( 5) are utility functions.Indeed, these expressions calculate the retailer's desire toward the products of domestic and foreign manufacturers, respectively.In other words,   and   show the manufacturers' market share.It is important to mention that,   and   are measured by the retailer's sensitivity to the price of each manufacturer (  and   ).
It is also important to note that the costs of manufacturing each domestic product depend on some important expenses such as manufacturing technology, labor costs, setup costs, and R&D unit-related costs (  ).Apart from these expenses, to export its product to the domestic manufacturers' market, the foreign manufacturer must consider the costs of globalization fees such as international shipping costs, tariffs, and customs fees (  ).Hence, the product price of the foreign manufacturer is calculated as follows: where   is greater than 1.
In fact,   is a coefficient that shows the marginal profit rate of the foreign manufacturer according to the costs of production and globalization fees of that product.For example, if  is equal to 1.1, it means the selling price of the product is 10% higher than the cost of making the product.Therefore, the marginal profit rate of the foreign manufacturer is 1.1.
Lastly, to find the optimal price for each domestic manufacturer and the retailer's optimal decision, the problem is formulated as: s.t. s.t.
Objective (7) and constraints ( 8) and ( 9) indicate the first level (ULP); objective (10) and constraints (11) depict the second level (LLP) of the model.The objective of the ULP () maximizes the gain of the domestic company.The objective function of the ULP is shaped by two sections.The first section involves the profits of Manufacturer L ( = 1).Similarly, the second section includes the profits of Manufacturer H ( = 2).It is important to note that, since the products' quality of Manufacturer H is better than that of Manufacturer L, the production cost of Manufacturer H would be greater than that of Manufacturer L ( 2 ≥  1 ).Constraint (8) guarantees that the price of the manufacturers' products cannot exceed a specific price.Indeed, this price ( max  ) is determined according to the governmental market control policy.Constraint (9) indicates that product price is a positive and continuous variable.The second level objective function ( ) aims to minimize the retailer's charges.In fact, being a fraction of the lower-level problem (LLP) regarding each manufacturer's (H, L, and F) price and total customer demand (), the retailer decides how much of the demand to source from each manufacturer.Finally, constraint (11) shows that   is considered to be the binary variable of the second level.

Solution method
When the decision of actors interacts and can affect each other, usually game theory approach is often implemented to design this competition.Indeed, game theory structure establishes a theoretical basis to track how actors affect each other and what through interaction, they can and are trying to acquire something [33].Since, this research wants to study the interaction between components of SC and reach the optimal decision, use the game theory framework.In addition, based on the problem structure of this study (hierarchy decision between the domestic manufacturer and the retailer), this paper applied the Stackelberg game.Since the bi-level programming approach can design the Stackelberg game structure mathematically, this research implemented this programming method to develop the problem [52].Therefore, this paper addresses a pricing model in a game theory framework based on a bi-level optimization problem (BOP) for a competitive SC network.BOP problems have been used in various applications and theories such as Stackelberg game, Hierarchical planning, SC, Pricing, Electricity, Transportation, Environmental Economics, and Machine Learning [55].Many researchers have attempted to solve this problem.In general, approaches developed to solve this problem can be classified into two major groups [55].
A. Classical approaches: they consist of four types of approaches, including single-level reduction, descent methods, penalty function methods, and trust-region methods.
-The single-level reduction is used when the LLP is convex.By utilizing the Karush-Kuhn-Tucker (KKT) conditions in the LLP, the goal is to transform the BOP to a single-level problem and, afterward, handle the single-level formulation.-A descent method in a bi-level optimization problem causes a decrease in the upper-level objective function for as long as it takes to maintain the feasibility of the new point.Since the feasibility of a point is assured only when its lower level is optimal, finding a proper the descent method can be extremely difficult.To solve the BOP, scientists have proposed several methods to approximately calculate the gradient of the objective of the ULP.To decide the orientation of descent, they have also considered formulating auxiliary programs [56].-The penalty function methods generally involve the KKT conditions used for the LLP and then, a penalizing approach is applied to resolve the single-level formulation.In fact, in this method, a penalizing value is added to the objective function of the ULP to fulfill the LLP's optimization [57].-In the trust-region method, the algorithm based on an iterative technique estimates a certain area of the objective function using a model function.In fact, the algorithm optimizes the current iteration and the radius of the trust region simultaneously, with the process continuing until convergence occurs [58].
B. Evolutionary approaches: apart from the above methods, BOPs have been addressed using some evolutionary algorithms.One of the more successful and popular methods is the nested evolutionary algorithm.The nested method uses two common approaches to solve the BOPs.One of them uses an evolutionary algorithm at the ULP and a classical algorithm at the LLP.The other one utilizes evolutionary algorithms at the ULP and LLP, simultaneously.Some examples of this method's successful application for solving the BOPs are found in the works of [33,[59][60][61][62][63].Besides, the evolutionary approaches addressing BOPs include other ways such as single-level reduction, meta-modeling-based methods, and the auxiliary bi-level meta-model [55].In this paper, applying an exact solution to address the proposed model involves the determination of the best strategy for the ULP, carried out by considering all LLP responses against the upper-level decisions.In other  words, we are faced with two decisions here.The first one is the pricing decision for two domestic manufacturers in the ULP.Then, in the LLP, we should determine which manufacturer the retailer buys their product from and how much of it.To solve this problem, the first single-level reduction has been applied.More specifically, since the LLP has a binary variable, it can be regarded as an integer programming challenge at this level.Consequently, the exact solution of the model can be denoted by straightforward enumeration in the LLP.
In other words, to solve the problem, we examine all response scenarios for the LLP vis-a-vis the upper-level decisions.A mathematical proof is used to confirm that one of these scenarios will always provide the best answer (profit) to the upper level of the model.Finally, to address the single-level problem, the Lambert-W function has been used.It should be noted that the Lambert-W function is an exact method that is practical and commonly used in solving problems containing the MNL function.

Reformulation of the bi-level competitive pricing problem
As the lower level has a binary variable, there are four response scenarios in LLP.Table 2 presents these response scenarios.
Proposition 1 shows that for each decision in the ULP, there is a specified response scenario at the LLP.
Proposition 1.The retailer's optimal response strategies are as follows: (A) When  1 ≥   and  2 ≥   .The retailer's optimal response strategy is Strategy 1.
The retailer's optimal response strategy is Strategy 2.
The retailer's optimal response strategy is Strategy 4.

Proof. See Appendix A.
The optimal strategy is demonstrated in Figure 3. Proposition 1 and Figure 3 show that when the product prices of the domestic manufacturers are higher than those of the foreign manufacturer (A), the retailer does not purchase from the domestic manufacturer.Noticeably, if the product prices of the domestic manufacturers are lower than those of the imported products (D), owing to the utility function of each manufacturer, the retailer decides on the quantity of purchase from each manufacturer.Finally, modes B and C show that if the prices of Manufacturers H or L are higher than those of the foreign manufacturer, the domestic manufacturer has no chance of selling its products to the retailer.
Regarding Proposition 1, to solve the original problem, we can decompose the original model into four singlelevel models as follows: (1) If the LLP selects the  1 strategy, the original model objective function would be as follows: Figure 3.The retailer's optimal response strategy vis-a-vis the pricing decision of each domestic manufacturer.
(2) If the LLP selects the  2 strategy, the original model would be as follows: Equations ( 8) and (9).
1 = exp( 1 ) exp( 1 ) + exp(  ) ( 14) Note that constraint  2 ≥   is not added to the above model since the retailer does not purchase from Manufacture H. Thus, practically, Manufacture H does not produce any product for the retailer.(3) If the LLP selects the  3 strategy, the original model would be as follows: Equations ( 8) and (9).
Also, because the retailer does not purchase from Manufacture L, constraint  1 ≥   must not be added to the above model.Practically, Manufacture L does not produce any product for the retailer.(4) If the LLP selects the  4 strategy, the original model would be as follows: Equations ( 4), ( 8), ( 9), ( 15) and (18).
Since the ULP has complete information about the LLP response strategies, we can decompose the space problem into four smaller spaces based on the LLP response strategies, such that each response strategy for the LLP is added as a constraint to the upper-level objective function.As a result, to determine the best decision strategy for the BOP, we are faced with four single-level math problems ( 1 −  4 ).
Then, among these single-level problems, the model that would provide the most profit for the ULP should be selected.Indeed, that single-level model should identify the best leader decision and the best follower response strategy to the ULP decision.Proposition 2 shows that the most profit for the domestic company is always found in Space III (Fig. 3).
Proposition 2. If the domestic company can search the entire decision space according to the model parameters and constraints, then the best profit of the domestic company will always be found in Space III.In other words: Proposition 2 shows that diversification in product quality can attract a wider range of customers.Obviously, absorbing more customers will increase the system's profit, too.Indeed, according to Proposition 2, to attract more customers, the domestic company should always seek to sell both of its products to be able to compete with the foreign manufacturer.
Proof.See Appendix A.

Optimizing the single-level problem
Given that   is an exogenous and constant parameter, if  is defined as: By using equations ( 19) and ( 21), the objective function of the single-level problem is as follows: Therefore, by performing differentiation on expression (22) with respect to   and putting the expression equal to zero, the following expressions for the problem are obtained: According to the exponential behavior of the domestic manufacturers' market share, it is clear that expression (25) can only show a function of the price and cannot show the price quantitatively.However, it is possible to obtain an optimum solution by using the Lambert-W function [64].The domestic manufacturers' optimum price at each manufacturer is stated in Proposition 3.
Proposition 3. The optimal price of the domestic manufacturers is given by the following closed expression: where the Lambert-W function can be defined as the inverse function of  () =   [48].Therefore, based on Proposition 3, the optimal price and the objective function can be calculated if the proposed single level does not have any constraint.Given expressions 8, 15, and 18,  *  is not accurate if  *  exceeds the limitation of these expressions.Condition 1 can fix this problem.
Condition 1.The optimal price of the domestic manufacturers is calculated under this condition: Since in the MNL function, the objective function grows up to a certain level with an increase in the price only to decline later [64].Hence, based on Condition1, if Min(  ,  max  ) is smaller than  *  , the maximum value of the objective function is obtained where  *  is equal to Min(  ,  max  ).
Against this backdrop, in this section, first with the help of Propositions 1 and 2, the proposed bi-level problem is transformed into the single-level model.Then, by using the Lambert-W function, the optimal value of the price is obtained (Prop.3).Finally, the optimum profit of the domestic manufacturers is calculated in accordance with their respective prices.

Computational results
This section mainly aims to analyze the proposed model and the solution method.To offer a better understanding of the model behavior, a sensitivity analysis is conducted for different parameters of the proposed model so that we can offer some relevant managerial insights in this study at Section 6.Hence, this section analyzes the sensitivity of the cardinal parameters   ,   ,   , and   , to scrutinize the response of the proposed model and its far-reaching implications for the pricing policy, and total expected profit of the domestic company, and the foreign manufacturer.
Concerning the validation of the presented model, we run some tests.Given that in the literature there is no model sample, we have generated some random problems.For the experiments, we have used the following setting:  = 10 000,  1 = 3,  2 = 5,   = 6,   = 0.5,   = 20,  max 1 = 25, and  max 2 = 30.It should be mentioned that the unit of all parameters which are related to the expense and price is the same (i.e., Euro or Dollar, etc.).In addition, since the proposed model is a non-linear mixed integer, we have submitted the written model to MATLAB 2019a.The codes ran on a computer with the following configuration, a 2/90 GHz Intel Core i5 CPU and 8 GB RAM and Window10 OS.

Sensitivity analysis on the 𝑐 𝑖 parameter
In this part, the effects of different values of parameter   on the domestic company's profit, the manufacturers' profit (L and H), and the foreign manufacturer's profit are examined.Hence, to determine the foreign manufacturer's profit, we have stated equation (27) as follows: We generated five sample problems with different values of   to examine the effect of parameter   on the aforementioned variables accurately and from the different angles; moreover, we have studied these issues in 6 different sizes of   .It should be noted that   is calculated based on equation (28).
Table 3. Analysis of the effect of   on the domestic company and the foreign manufacturer.

Data set Θ Problem 1 Problem 2 Problem 3 Problem 4
Problem 5 1 = 0.5 2 = 0.5 Table 3 illustrates the value of Θ in all 6 samples.According to Table 3, as expected, with an increase in the manufacturing costs of each product unit, the profit of the domestic company decreases, resulting in an increased profit for the foreign manufacturer.To take a closer look at the results of Problems 1-5, you can refer to the appendix section (Tabs.A.1-A.5).
Figure 4 illustrates the trend of the domestic manufacturer's profit (L) in Problems 1 and 2. The results indicate that if the market is more interested in the product of a specific manufacturer (lower   ), an increase in the costs of manufacturing that product, and, consequently, an increase in the price of that commodity has a far more devastating impact on the loss of the domestic manufacturer's profit than when the market is less interested in that product.This is because the demand of the market is more sensitive to the price of that commodity.
Given that the behavior of profit reduction is linear, we used linear regression to show the accuracy of this claim (dotted line).As can be seen in Figure 4, the slope of the linear regression of Problem 2 is greater than that of Problem 1 (| − 7826.1|> | − 6229.7|).Let us note that in Problems 1 and 2, the only difference is in parameter  1 .In Figures 5 and 6, we have examined the slope of the profit reduction for manufacturer H and the domestic company in Problems 1, 2 and 1, 4, and 5, respectively.The results reveal that the asserted claim is correct.

Sensitivity analysis on the 𝜃 𝐹 parameter
This section deals with the profit of the domestic company, the foreign manufacturer's profit, and the foreign manufacturer's market share in view of the impact of patriotism on the purchase of domestic commodities.This study defines the patriotic behavior of the market as either the market's tendency to buy domestic goods (less   ) or imported goods (more   ).Therefore, we have defined two problems with different values of parameter   (Problems 6 and 7) for accurately understanding the market behavior with respect to the above-mentioned conditions.We have also created 5 data sets with different values of   .The computational results of this data set are available in Tables 4 and 5.As expected, the market share of the domestic company increases when the market avoids purchasing the foreign commodity.On the other hand, this helps the domestic manufacturers raise the sale price of their products to retailers and, as a result, turn more profit.The profitability trend for the domestic and foreign companies in Problems 6 and 7 are depicted in Figures 7 and 8, respectively.
Figure 9 suggests that in addition to purchase restrictions of foreign commodities, if the market tends to show greater willingness to buy domestic commodities, the domestic company certainly makes more profit.In fact, as shown in Figure 9, in a scenario where the market is inclined towards local commodities (Problem 7), the  domestic company makes greater profit than otherwise (Problem 6), and the profit of the foreign manufacturer is much lower.As noted above, when a market refrains from buying goods, the market share of the foreign manufacturer starts shrinking.However, this decline in the market share is even more severe when the market shows a greater tendency toward purchasing domestic products.Indeed, if the market shows a greater preference for domestic commodities, the foreign manufacturer's product becomes less competitive than the domestic one's.Figure 10 illustrates the accuracy of this claim (| − 0.0574| > | − 0.044|).

Sensitivity analysis of the 𝜃 𝑖 parameter
In this section, to investigate the impact of the   parameter on the proposed model, we examined the market in two different scenarios: when customers have a low average income, and when they enjoy a high average income.Considering the two different conditions, we have also analyzed and evaluated the market share     profitability of each manufacturer.To this end, we have investigated Problem 1 in two different modes.Results of Tables 6 and 7 refer to customers with a low average income and a high average income, respectively.As seen here, we have examined Problem 1 for five different data sets.In Tables 6 and 7, parameter  1 and  2 have also changed, respectively.As expected and shown in Table 6, with a decrease in the average income level of the community, the market gradually tends to buy lower-quality goods.In other words, the retailer's tendency to buy from Manufacturer L  increases according to the market demand.As a result, the market share of Manufacturer L increases, in turn this manufacturer's profitability.It is also clear that as the Manufacturer L's profitability increases, the profitability of Manufacturer H and the foreign manufacturer decrease (Fig. 11a) -an adverse correlation, in fact.The same thing applies to Manufacturer H when the average income level of the community increases.Figure 11b substantiate the validity of the results.
Furthermore, in a situation where the consumers' economic condition dictates market behavior, the analysis findings show that when the retailer sources the product from one particular domestic manufacturer, that manufacturer increases its product price in comparison to other competitors.For example, assume that the average income level of a community is improving (i.e.,  2 decreases gradually).In this case, the slope in the price of Manufacturer H's products is higher than that in the price of Manufacturer L's products.The accuracy of this statement is depicted for both market conditions under consideration in Figure 12.

Model analysis in the presence of one domestic manufacturer
This part aims to show how a domestic company with two manufacturers can make more profit compared to a domestic company with only one manufacturer that produces goods with a quality on a par with that of imported ones.Hence, in this part, the domestic company's profit is investigated in two different scenarios: (1) with two domestic manufacturers (two different types of product quality) and (2) one domestic manufacturer (one level of product quality).
To this end, we have examined Problems 1, 2, and 3 in the two above-mentioned modes.We selected these samples because we can properly investigate each aspect of this segment with them.As seen in Table 8, Problem 1 depicts a state where the manufacturers have the same conditions in terms of sensitivity to the product price.Problem 2 also shows a state where the domestic manufacturer with a low-quality product has more desirable conditions than the other manufacturers in the aforementioned parameters.In contrast, Problem 3 represents a state where the domestic manufacturer with a high-quality product has more favorable conditions than the other manufacturers in the above-mentioned parameters.
It should be noted that in this section  (the sensitivity parameter of the retailer to the product price of the domestic manufacturer) and  (the unit cost of producing one product by the domestic manufacturer) for the domestic manufacturer in a mode with one manufacturer is calculated through the averaging method.For example, the parameters mentioned in Problem 2 are calculated follows: As illustrated in Figure 13, the domestic company's profit in all instances improves when the company decides to produce products with different qualities.Also, the foreign manufacturer's profit significantly decreases when the domestic company has two manufacturers (Problem 2 and 3).

Managerial insights
This study can be used in various applications in the real world especially in the auto parts industry and electronic home appliances industry.The reason behind this is that in these industries the foreign and domestic manufacturers compete to acquire more profit from the retail market, as a result, the optimal pricing strategy   for the domestic manufacturer to keep their market share would be inevitable [65].Hence, this study can help domestic manufacturer by determining the best competitive pricing strategy by identifying customer preferences.
In addition, our result reveals the fact that one way by which a domestic company can enhance its competitive advantage is to increase the level of response to customer preferences by producing products of different qualities.
In other words, by producing products with different qualities, the domestic company can meet the needs of high-income customers as well as low-income customers (Fig. 13).Following are some important points in which managers and beneficiaries can benefit from the results of this study.

Managerial decisions on the cost parameter
Due to analysis of Section 5.1, with an increase in the manufacturing costs of each domestic product, the domestic company becomes less competitive in pricing products against the foreign manufacturer.As a result, to maintain the profit, the domestic company increases the price of its products; this price rise decreases the desirability of the products, and as its utility function declines, so, too, does its profit.Therefore, the results of Section 5.1 indicate that domestic companies need to upgrade their manufacturing technology to maintain the profitability of their products and their market share and they can manufacture products at a lower cost.Thus, they will be able to increase the pricing ability of their products in today's competitive environment.Practically, the obtained results reveal that in case of purchasing manufacturing technology, companies should shift their perspective from a cost-oriented one to an investment-oriented one.Of course, it is crucial to bear in mind that the decision to increase the level of technology for the production depends on several factors such as the size of the company, market needs, and investment costs; therefore, given these factors, a company should choose the optimum level of technology.

Managerial decisions on price sensitivity parameter
It is important to note that in analyzing the results of Section 5.3, we hit upon an interesting managerial implication.In a market where the average income is low, customers prefer to buy products of Manufacturer L, thus the market share of Manufacturer L increases (Fig. 14).This phenomenon leads to a monopoly in the market and results in Manufacturer L increasing the price of its goods to increase profitability.At some point in time, the price of Manufacturer L's products is even likely to be close to that of Manufacturer H's products.Hence, one can conclude that an excessive tendency toward a certain manufacturer's product can be detrimental to the consumers of a society.Based on the results, to prevent market monopolization, retailers should control the level of utility (  and   ) and not allow it to exceed a certain level.As a result, a competitive environment between the manufacturers for the sale of the product prevails in the market.In other words, this practice prevents an aggressive selling price of a commodity relative to its value (the quality of that commodity) since it prevents market monopolization.

Conclusions
The competitive pricing problem in the SC plays an instrumental role in the pricing of manufacturers' commodities.The competitive environment governing the market forces manufacturers to price their products properly to maximize their market share and profit.Against this backdrop, we developed a bi-level model considering the competition between a domestic company and foreign manufacturer for selling their products to a retailer.
Taking the upper level of the model into consideration, with the participation of two different manufacturers (low-quality manufacturer and high-quality manufacturer), the domestic company seeks to maximize its total profit and compete with the foreign manufacturer through the pricing of its goods.Indeed, with suitable pricing strategies for low-quality and high-quality goods, it seeks to increase its market share and profit.Subsequently, considering the selling price of the domestic and foreign manufacturers' products as well as the needs of the market, the retailer at the lower level seeks to minimize the purchasing cost.In fact, the upper level and the lower level of the model are in conflict with each other.Hence, the proposed model acts as a Stackelberg game.The principal innovation of this study is employing a bi-level modeling proposal to capture the competition among the domestic company and the foreign manufacturer.To render the presented study more realistic, the concept of the utility function is also used to express the market share of each manufacturer.
To solve the proposed model, a new hybrid exact method based on the explicit enumeration method and Lambert-W function is used.Considering all response scenarios of the lower level, the solution space of the bi-level model is divided into four sub-single-level models.The optimal answer can always be obtained from one of the sub-problems.Then, to exactly solve this single-level model, the Lambert-W function is applied.To validate the presented model, sensitivity analysis is carried out on three main parameters of the model (the unit cost of producing one product, sensitivity parameter of the retailer to the price of the imported product, )) is equal or greater than the objective functions  1 ,  2, and  3 (Eqs.( 12), (13), and ( 16)).Indeed, we show that the difference between the objective function  4 and other objective functions is always positive.Consequently, we have: (1)  4 vs.

Figure 2 .
Figure 2. The perceptual structure of bi-level SC network.

Figure 4 .
Figure 4. Consequence analysis of   on Manufacturer L's profit.

Figure 5 .
Figure 5. Consequence analysis of   on Manufacturer H's profit.

Figure 6 .
Figure 6.Consequence analysis of   on the domestic company's profit.

Figure 7 .
Figure 7. Effect analysis of   on the profit of the domestic company and the foreign manufacturer (Problem 6).

Figure 8 .
Figure 8.Effect analysis of   on the profit of the domestic company and the foreign manufacturer (Problem 7).

Figure 9 .
Figure 9.Effect analysis of   on the profit of the domestic company and the foreign manufacturer (Problem 6 and 7).

Figure 10 .
Figure 10.Effect analysis of   on the foreign manufacturer's market share (Problems 6 and 7).

Figure 11 .
Figure 11.Analysis of the effect of   on the profit of Manufacturer L, Manufacturer H, and the foreign manufacturer.(a) Customers' low income.(b) Customers' high income.

Figure 12 .
Figure 12. Analysis of the effect of   on the price of Manufacturer L and Manufacturer H. (a) Customers' low income.(b) Customers' high income.

Figure 13 .
Figure 13.Comparison of the domestic company with two manufacturers and one manufacturer.(a) Domestic company profit.(b) Foreign manufacturer profit.

Figure 14 .
Figure 14.Consequence analysis of  1 on the market share of Manufacturer L, Manufacturer H, and the foreign manufacturer.

Table 1 .
A brief review of the literature.

Table 4 .
Analysis of the effect of   on the domestic company and the foreign manufacturer (Problem 6).

Table 5 .
Analysis of the effect of   on the domestic company and the foreign manufacturer (Problem 7).

Table 6 .
Analysis of the effect of  1 on the domestic company and the foreign manufacturer.

Table 7 .
Analysis of the effect of  2 on the domestic company and the foreign manufacturer.

Table 8 .
Comparison between the domestic company with two manufacturers and one manufacturer.

Table A .
Table A.2. Computational results for the problem 2. Table A.3.Computational results for the problem 3. Table A.4. Computational results for the problem 4. 5. Computational results for the problem 5.