Cooperation mechanisms for a competitive, sustainable food supply chain to reduce greenhouse gas emissions

The food industry is a major source of greenhouse gases (GHG). Given that consumers in this industry are aware of the negative consequences of GHG emissions, such as global warming and climate change, members of the food supply chain should consider mechanisms to reduce GHG emissions. The purpose of this paper is to examine the effects of supply chain structure and cooperation methods on the objectives and decisions of a sustainable food supply chain composed of one manufacturer and two suppliers. In the first scenario, a single-level problem is solved for a centralized supply chain. Other scenarios consider the decentralized structure, in which supply chain members face constraints such as maximum additive use and GHG emission, budget, and capacity. The bi-level programming is used to model competition between members of the sustainable food supply chain. It is demonstrated first that the lower-level models in decentralized scenarios can be converted to a single-level model, and then the proposed bi-level model is converted to a single-level one using the KKT method. Computational results show that the alliance of the manufacturer and the first supplier yields the highest total profit for all decentralized scenarios. Because the lowest GHG emission rate among decentralized scenarios is achieved through alliance and cost-sharing mechanisms, the use of these mechanisms concurrently is appropriate for environmental purposes. In scenarios where supply chain members compete with each other, it is found to be cost-effective to increase the budget. Additional considerations can be made regarding the effects of other variables such as distance and advertising on demand as well as alternative forms of the demand function.


Introduction
Demand for food is rising due to the growing global population. Due to scarce resources, there is no longer a balance between food production and consumption, and thus sustainability should be considered a vital part of the food supply chain (Martínez-Guido et al., 2018). As one of the most important food products in any country's economy, dairy production contributes to air pollution and resource waste throughout the supply chain (Kirilova and Vaklieva-Bancheva, 2017). Greenhouse gas (GHG) emissions directly indicate the supply chain's environmental consequences, such as climate change and global warming. Customer awareness has increased, increasing demand for products that ensure sustainability throughout the supply chain processes, from manufacturing to delivery. Members of the supply chain want to increase demand and reduce costs in addition to considering sustainability issues, so decision-makers must consider all aspects of the sustainable supply chain. Supply chain members add additives to their products for various purposes, such as increasing demand and reducing costs. Excessive use of this type of additive may contaminate food and cause health problems for consumers. Song and Zhuang (2018) have mentioned too much melamine in dairy products by Sanlu Company in 2008, which led to kidney stones and several children's deaths. Consumer health has gained prominence as a result of such tragic events. Health protocols impose restrictions on the use of additives, and the government can also discourage their overuse through the imposition of fines (Song and Zhuang, 2017).
While in a centralized supply chain, a single decisionmaker makes decisions about the entire chain, in a decentralized structure, each member makes decisions based on their objectives, which may conflict with those of other members. Nash equilibrium is achieved when two players with equal power compete in a decentralized structure. The Stackelberg game is used in situations where the player with the most power (the leader) makes a decision based on his or her goal, and the other players (followers) react to that decision. Bilevel programming is a combination of the Stackelberg game and mathematical programming, in which decisions made at one level affect those made at the other. In order to improve their performance, supply chain members cooperate on a contractual or non-contractual basis. Alliances and cost and revenue sharing among supply chain members are examples of cooperative mechanisms.
In this paper, a sustainable food supply chain consisting of one manufacturer and two suppliers is studied. The manufacturer seeks to increase demand by investing in the reduction of GHG emissions. All three economic, environmental, and social dimensions of a sustainable supply chain are considered here. Five scenarios are presented to compare the impacts of competition and cooperation between supply chain members on their goals and decisions in centralized and decentralized structures. A centralized supply chain is considered in the first scenario, while bi-level programming is used to model competition among supply chain members in the other scenarios. We seek to address the following questions by defining and solving this problem: 1) What are the effects of considering different supply chain structures on supply chain members' goals and decisions? 2) What is the impact of competition between supply chain members on their goals and decisions? 3) What are the effects of the three proposed cooperation scenarios between supply chain members on their goals and decisions? 4) What share of the cost of reducing GHG emissions should the second supplier have for a better effect on the supply chain members' economic and environmental goals in the fourth and fifth scenarios? 5) What are the effects of the parameters concerning each of the economic, environmental, and social dimensions of the supply chain on its other dimensions?
The remainder of the article is structured as follows. The second section reviews related research and identifies the research gap. The third section describes the problem and presents the objective functions and constraints for each supply chain member. Section 4 examines five scenarios and presents a mathematical analysis to determine the effects of several critical parameters. Section 5 discusses the solution approach and provides numerical results and sensitivity analyses for several critical parameters. The conclusion, managerial insights, and recommendations for future research are included in the final section.

Literature review
Three subsections comprise the literature review. In the first section, we review articles that examine the competition and cooperation between members of the green, sustainable supply chain. The second section reviews recent publications in the field of food supply chain management. The third section examines articles that employ bi-level programming to simulate supply chain competition. Zhou and Qin (2015) conducted a review of articles on game-theoretical approaches to sustainable supply chain management. The authors conclude that only a few articles have taken into account environmental and social objectives in addition to economic ones. Raj et al. (2018) have investigated centralized and decentralized structures for a sustainable supply chain, including a supplier and a buyer. The Stackelberg game is used to model a decentralized supply chain, in which the supplier (leader) and buyer (follower) compete to maximize their profit by determining the product's green degree and cooperating in the social dimension of the supply chain. Jian et al. (2019) have investigated the price competition between two firms operating in a duopoly market with varying average carbon emissions per unit of homogeneous products. Peng et al. (2019) have examined the primary causes of environmental failure in the context of the behavioral game between government and manufacturing firms. This survey revealed that competition and cooperation in a sustainable supply chain were discussed in a minority of articles, while consumer health was not mentioned as a social dimension in any of them. The following is a review of the same subject as discussed previously for the green supply chain. In these articles, environmental criteria such as GHG emission reduction, product greenness, and energy efficiency have been considered. A few of them considered competition on both price and environmental criteria concurrently. Ji et al. (2017) have investigated carbon emission reduction in a supply chain with a producer as the leader and a retailer as the follower. The consumer is assumed to be interested in purchasing products with lower GHG emissions, and thus the producer invests in GHG reduction. Zhu and He (2017) have examined the effects of competition and cooperation on the greenness of products in a supply chain, which included a manufacturer and a retailer with centralized and decentralized structures. Hafezalkotob (2017) has studied competition and cooperation between two green supply chains, each including a manufacturer and a retailer, along with government intervention. To maximize profit, supply chains determine prices and the level of energy conservation, and the government uses tariff rates to guide them toward economic and environmental goals. Xu et al. (2019) have examined a supply chain, which includes a producer and a retailer, in order to determine the impact of government tax policies on carbon pricing and emissions decisions. Sarkar and Bhadouriya (2020) have presented three alternative models for a bi-level supply chain that includes a retailer and multiple manufacturers. The centralized supply chain is investigated for the first model, and the Stackelberg game is used to address the competition in the second and third models. The manufacturers form a coalition in the third model, and Nash equilibrium is obtained for competition between manufacturers in the second model.

Food supply chain
Researchers have paid more attention in recent years to sustainability in the food supply chain. At most, two of the three sustainability dimensions have been investigated for the sustainable food supply chain. Rohmer et al. (2019) have proposed a multi-objective network design model for the food supply chain, including cost and environmental objective functions. Various environmental dimensions, such as climate change and resource utilization, are considered for the environmental objective function. Li and Zhou (2021) have proposed a multi-objective model for a cold supply chain to improve customer satisfaction, reduce carbon emissions, and minimize costs. Dynamic and static carbon emission of the supply chain is considered to determine environmental effects. Shirzadi et al. (2021) have proposed an inventory-routing model to optimize the total interest of agri-food reverse logistics by considering freshness quality and environmental effects. Collection and reuse of wasted products and controlling GHG emission is considered as the environmental dimension of the green supply chain.
Few articles have studied competition and cooperation for the food supply chain, and some of them are reviewed below. Song and Zhuang (2018) have examined the health effects of additives in a milk supply chain where the manufacturer uses additives to boost demand, and the government regulates the use of these substances through fines. Centralized and decentralized structures are considered for the supply chain, and the Stackelberg game is used to model competition between the government (leader) and the manufacturer (follower) to maximize profit. Song and Zhuang (2017) have proposed three models for the health problem in a milk supply chain involving three players: the supplier, the manufacturer, and the government. In the first model, a centralized supply chain is considered to maximize the total profit. The decentralized structure examines coalitions and competition between the supplier and the manufacturer for the second and third models. Zhu et al. (2018) have investigated the cost-sharing mechanism for the competition on the greenness of products among food supply chain members. It is assumed that the green degree depends on the green degrees of its raw materials. Yu and Cruz (2019) have studied the competition between firms aimed at maximizing profit. A network model is developed for an oligopolistic industry with varying environmental policies, and the effects of various tax policies, such as the flat emission tax rate, on the companies' objectives and decisions are examined. Manteghi et al. (2020) have presented three models for a sustainable food supply chain, including a manufacturer and two suppliers, to investigate the impact of cooperation and competition on sustainability aspects. A centralized structure is investigated for the first model, while competition and cooperation are investigated in the second and third models for a decentralized supply chain. Nematollahi et al. (2021) have examined competition and coordination among agricultural supply chain members, including a conventional farmer, an organic farmer, and an agribusiness enterprise. The novelty of this paper is the examination of organic farming characteristics such as premium, crop yield gap, and organic cost factor. Ma et al. (2021) have used an evolutionary game model to investigate the effect of government punishment and subsidy policies on the behavior of manufacturers and retailers. The social dimension of the food supply chain is examined through the lens of supply chain members' social responsibility to consumers.

Bi-level programming
Because some bi-level models lack the requisite properties for conversion to single-level models, they were solved using heuristic techniques. Yue and You (2017) have investigated competition between members of a biofuel supply chain using mixed-integer bi-level programming. Due to the discrete variable at the follower level, a heuristic algorithm is presented for solving the proposed model. Saranwong and Likasiri (2017) have developed a bi-level model to minimize the cost of transferring products from factories to customers. Four algorithms have been developed for solving the bi-level model, and their results are compared. Wang et al. (2017) have presented a bi-level model for a supply chain with the goal of reducing carbon emissions. A modified evolutionary algorithm is presented to solve the bi-level model because of the discrete variables at the lower level. Tabrizi et al. (2018) have examined competition between members of a fish supply chain using a bi-level model. Pakseresht et al. (2020) have developed a novel multi-objective bi-level model for product family and supply chain reconfiguration. At the upper level, they seek to maximize total profit and customer utility when reconfiguring product families. The goal of the lower level for supply chain reconfiguration is to minimize the total cost of the supply chain. The following is a review of articles that use the KKT method to convert bi-level models to single-level models. Golpîra et al. (2017) have presented a bi-level model to design a green supply chain. The goal of the upper level is to minimize the total cost of the supply chain, and the lower level aims to minimize the risk of uncertainty for retailers. Safaei et al. (2018) have presented a robust bi-level optimization model for the relief supply chain. The transshipment relief points constitute the upper level of the model to minimize costs. Risk minimization is the goal of the supplier relief points, making up the lower level of the model. Sharif et al. (2018) have presented a multi-product bi-level model to maximize the profits of the municipality and buyers. The municipality decides on network design, pricing, outsourcing, bidding, and company selection to maximize its profits at the upper level. The buyers determine price at the lower level to maximize their profits. A heuristic approach is presented to solve the mixed-integer bi-level model, the results of which are compared to those of the KKT method.

Research gap
Members of the sustainable food supply chain strive to produce healthy products while reducing costs, increasing productivity, and offering a variety of products. They also face challenges such as resource scarcity and climate change, necessitating consideration of all aspects of sustainability in decision-making. According to the provided literature review, health is a new issue for the food supply chain. Additives and GHG emissions are harmful to consumer health, and few researchers have investigated them as social and environmental criteria. According to Table 1, the majority of articles have compared centralized and decentralized structures. Members of a decentralized supply chain compete with one another to achieve their objectives through contractual or non-contractual mechanisms. Comparing various scenarios of cooperation, such as alliances and cost-sharing, enables management to make informed decisions. A few researchers have considered cooperation as a mechanism for improving the performance of supply chain members besides the competition. The government and relevant organizations should regulate the use of additives and GHG emissions from manufacturing processes, and these constraints should be incorporated into the model to make it more realistic.

Table 1
Comparison of the major papers mentioned in the literature review Bi-level programming is used to model competition between decision-makers in the presence of constraints, and a few papers have done so. This paper studies the competition and cooperation between members of a food supply chain, including one manufacturer and two suppliers. Reducing GHG emissions has a positive impact on the environment and reduces the risk of endangering consumers' health, but it raises costs. Additionally, while the use of additives is intended to reduce costs and increase the product's attractiveness, it results in an increase in the risk of endangering consumers' health. As a result, reducing GHG emissions and the use of additives play an important role in community health, supply chain economics, and environmental consequences. Profitability for supply chain members, reduction of GHG emissions, and consumer health are considered as economic, environmental, and social criteria for a sustainable supply chain, respectively. Members of the supply chain compete for profit maximization in the proposed problem. Competition between suppliers on a common level is examined in conjunction with competition between suppliers and the manufacturer. Alliances and cost-sharing as cooperation mechanisms are being examined to determine the effect of cooperation on supply chain members' decision-making and goal setting. For the first time, bi-level programming is used to model competition among members of the sustainable food supply chain.

Problem definition
The manufacture of products emits GHG and the negative impacts of GHG emissions on the environment and health of living organisms are evident. The manufacturer increases its costs by investing in the reduction of GHG emissions. Besides, additives are added to products by the manufacturer for purposes such as increasing revenue and reducing costs. Excessive use of additives may be harmful to consumers' health, so supply chain members or external inspectors should control the process. The government can also impose fines to prevent the overuse of substances. The reduction of GHG and the use of additives affect public health, environmental goals, and supply chain member profits, so decisions about them must be made simultaneously. This paper investigates a bi-level dairy supply chain that includes one manufacturer and two suppliers. The manufacturer's objective is to maximize profit by determining the reduction of GHG emissions, the percentage of additives, and the prices. The suppliers compete with each other to maximize their profits by setting the prices of raw materials. The centralized supply chain is considered in the first scenario, and the structures in the other four scenarios are decentralized. In the second scenario, there is competition between members of the supply chain to maximize profit. Cooperation between supply chain members is considered besides the competition in the third to fifth scenarios. In the third scenario, an alliance is formed between the manufacturer and the first supplier. The second supplier cooperates with the manufacturer in the fourth scenario by sharing the cost of reducing GHG emissions with the first supplier. In the fifth scenario, the first supplier allies with the manufacturer, and the second cooperates with it to decrease the cost of reducing GHG emissions. A schematic of the five scenarios is shown in Figure 1. The manufacturer produces two substitutable products with a share of the market demand of i . The value of the customer demand function for each product Figure 1 Centralized (a) and decentralized (b, c, d) proposed scenarios depends on its price, and the product demand decreases by α for each unit of increase in the product price. The products are substitutable, so the demand for each increases by δ, increasing the price of each unit of the other product. The demand for each product is a function of the percentage of additives and reduced GHG emissions besides the price. An increase in the percentage of additives improves the appearance of the product and increases its demand. Customers are environmentally conscious; therefore, the demand for products with lower GHG emissions is higher. The notations are as follows: The demand function is shown in Relation (1). The demand function used in this study can take various forms, but it is simplified by considering it linearly, as in previous research (e.g., Ji et al., 2017;Zhu et al., 2018). Each product requires one unit of raw materials, and Suppliers 1 and 2 are responsible for preparing the raw materials for the first and second products, respectively.
The goal of each supply chain member is to maximize its profit. Manufacturer costs include production costs, the cost of investment in clean technologies to reduce GHG emissions, and the cost of purchasing raw materials. Governments and private organizations adopt strict rules on maximum GHG emissions, regarded as a constraint in the model. The manufacturer also has a budget limit to pay for fines incurred due to the use of additives, GHG emission, and the purchase of raw materials. The suppliers seek to maximize their profits by setting initial prices for raw materials. Each supplier faces capacity constraints to supply raw materials. The mathematical models for the manufacturer and the suppliers are presented in the following sections. (2) defines the objective function of the manufacturer. Product price is the sum of the profit margins considered by the manufacturer and the price of the raw materials, presented in Relation (3). It is assumed that the manufacturer uses additives instead of protein to reduce costs. Each unit of dairy product must contain the minimum amount of nitrogen, presented in Relation (4). Consumption of additives may be detrimental to consumer health, and the health probability function represents the social dimension of the problem, which depends on the percentage of additives used. While the health risk function can be thought of as linearly or exponentially dependent on the percentage of additives, the linear form Zhuang, 2017, 2018) simplifies the analysis, as presented in Relation (5). The amount of additives must not exceed the specified limit, as indicated by Relation (6). Relation (7) shows the limit on manufacturer capacity for both types of products. According to Relation (8), the maximum reduction in GHG emissions is lower than the amount of emissions generated during manufacturing. Relations (9) and (10) determine constraints on budget and maximum GHG emissions. The variables' domains are set in Relation (11).

Supplier model
Relation (12) defines the objective function for each supplier. The amount of raw materials supplied by each supplier is limited, and Relation (13) specifies the maximum capacity of each supplier. Relation (14) identifies the types of variables.

Competition and cooperation
In this section, the relationships between the supply chain members are studied in five scenarios. In the first scenario, the supply chain is assumed to be centralized, and a decision-maker decides about the entire supply chain to maximize total profit. It is assumed that each supply chain member decides to maximize its profit in the second scenario. In the third scenario, the first supplier forms an alliance with the manufacturer to improve the supply chain goals in manufacturing the first product. The second supplier's share of reducing GHG emissions is considered a cooperation mechanism in the fourth model besides competition between the supply chain members. In the fifth scenario, both suppliers cooperate with the manufacturer. The first supplier forms an alliance with the manufacturer, and the second one shares the cost of reducing GHG emissions. The above scenarios are considered to investigate the effect of the supply chain structure and the type of relationship between the chain members on the goals and decisions.

Cooperation
In the first scenario, it is assumed that a single decisionmaker decides about the entire supply chain. This scenario is considered an ideal case compared to the others and is less common in practice. The decision-maker aims to maximize the total profit of the supply chain by determining product prices, the percentage of melamine, the percentage of protein, and the reduction of GHG emissions. The objective function in this scenario is the sum of those of the supply chain members, as presented in Relation (15).
It is appropriate to consider a centralized supply chain for economic purposes (supply chain profit). Still, it is necessary to study the effect of a centralized structure on the environmental and social dimensions of a sustainable supply chain.

Decentralized competition
In this scenario, there is competition (vertical) between the manufacturer and suppliers to maximize profit. The manufacturer seeks to maximize its profit by determining the profit margin, percentage of additives, percentage of protein, and reduction of GHG emissions. According to Relation (7), production capacity is limited, so there is (4,5,6,7,8,9,10,11,13,14) competition (horizontal) between the suppliers to maximize profit. Nash equilibrium is obtained for the competition between the suppliers. Bi-level programming is used for considering the competition between the manufacturer (leader) and the suppliers (followers), as presented below.
Subject to (3,4,5,6,7,8,9,10,11) Subject to (13,14) The existence and uniqueness of the Nash equilibrium obtained for the suppliers must be proven. First, lemma (1) is used to demonstrate that the Nash equilibrium between the two suppliers obtains at least one solution, and the uniqueness of the Nash equilibrium is then proven in lemma (2).

Lemma 1:
The game between the suppliers in the second scenario to determine wholesale price involves at least one Nash equilibrium. Lemma 2: The game between the suppliers in the second scenario to determine wholesale price involves exactly one Nash equilibrium.

Alliance between the manufacturer and the first supplier
In this scenario, it is assumed that the manufacturer allies with the first supplier. The alliance seeks to maximize the profits of the coalition members by determining the price of the first product, profit margin of the second product, percentage of additives, percentage of protein, and reduction of GHG emissions. The second supplier competes with the coalition by setting the wholesale price of the raw materials of the second product to maximize its profit. Bi-level programming is used to model the competition between the alliance (leader) and the second supplier (follower). Relation (18) presents the alliance's profit, and Relation (3) is considered only for the second product in the third scenario. The bi-level programming model of the third scenario is presented below. Subject to (3,4,5,6,7,8,9,10,11,13,14) Subject to (13,14) (16)

Decentralized competition scenario with the cost-sharing mechanism
There is competition in this scenario between the manufacturer (leader) and the suppliers (followers) to maximize profit, as in the second scenario. The manufacturer seeks to maximize its profit by determining the product price, percentage of additives, percentage of protein, and reduction of GHG emission. The suppliers compete with each other by setting raw material prices to maximize profit. In this scenario, it is assumed that the second supplier contributes to reducing GHG emissions with a share of , and the objective functions of the first and second suppliers are presented in Relations (21) and (22). The manufacturer's objective function is presented in Relation (20). The second supplier's share of the cost of reducing GHG emission is very important in this model in the provision of a balance between economic and environmental goals. The bi-level programming model of the fourth scenario is presented below.

Lemma 3:
The game between the suppliers in the fourth scenario has at least one Nash equilibrium. Lemma 4: The game between the suppliers in the fourth scenario has exactly one Nash equilibrium.

Alliance with the first supplier and cooperation mechanism for the second
In this scenario, the first supplier forms an alliance with the manufacturer, and the second supplier shares the cost of reducing GHG emissions. The coalition members compete with the second supplier, and their goal is to maximize the profit of the coalition by determining the price of the (20) first product, profit margin of the second product, percentage of melamine, percentage of protein, and reduction of GHG emission. The second supplier's goal is to maximize its profit by setting the wholesale price of the raw materials of the second product. There are two types of mechanisms for cooperation in this scenario. The manufacturer seeks to reduce the price and increase the first product demand by allying with the first supplier, as in the third scenario. In the second mechanism, the second supplier shares the cost of reducing GHG emissions by the manufacturer to increase the demand for the products and reduce GHG emissions, as in the fourth scenario. The coalition's profits and the second supplier are presented in Relations (23) and (24), respectively. The bi-level programming model of the scenario is presented below. Subject to (3,4,5,6,7,8,9,10,11,13,14) Subject to (13,14)

Mathematical analysis of the scenarios
Because the price of products has a direct effect on customers' desire to purchase them, the sensitivity of demand to the product's own price as well as the price of the competing product is effective in determining the price. The impact of these two parameters on raw material prices and supplier profits is examined in this section. The suppliers are assumed to have sufficient capacity to supply raw materials for the products, and Relationship (13) is omitted.

Proposition 1: In the second and fourth scenarios,
A proof for proposition (1) is presented in Appendix (5).
The manufacturer seeks to reduce the product price by increasing the dependence of demand on the price to prevent a sharp decline in the demand for the products. The suppliers try to avoid a sharp drop in profit due to the decrease in demand by lowering raw material prices. The suppliers' profits also decrease as the prices of their raw materials decrease. Competition over price increases with the increasing dependence of product demand on the price of the other product, so the manufacturer and suppliers can increase the (23) prices of its product and raw materials by increasing competitiveness. Their profits also increase with an increase in the price of raw materials.

Proposition 2: In the third and fifth scenarios,
A proof for proposition (2) is presented in Appendix (6). As in proposition (1), the coalition members seek to reduce the product prices to prevent a sharp drop in demand by increasing its dependence on price. The second supplier also reduces the price of its raw materials to avoid a sharp decline in profit, increasing the dependence of demand on the product prices. The second supplier's profit decreases with a reduction in the price of raw materials. Product price competitiveness increases with an increase in the dependence of product demand on the price of the other product. The second supplier seeks to increase the product price by increasing competitiveness, leading to an increase in the price of raw materials and the profit of the second supplier.

Solution method and numerical results
In the second to fourth scenarios, supply chain members compete with one another to maximize their profits. Bilevel programming is used to model competition in these scenarios due to the existence of constraints. The manufacturer is the upper level of the bi-level model in the second and fourth scenarios, while suppliers compete to maximize their profit at the lower level. Sections 4.2 and 4.4 established that the Nash game between suppliers has a unique solution, implying that bi-level models can be solved in these scenarios. In the third and fifth scenarios, the alliance members comprise the model's upper level, while the second supplier constitutes the model's lower level. According to a literature review in Section 2.3 (Bi-level programming), researchers have solved bi-level models using two general approaches. Certain bi-level models exhibit characteristics that enable their conversion to a single-level model via the KKT approach. To solve other models that lack these characteristics, approximate approaches such as heuristic and metaheuristic algorithms are used. Lemma 5 establishes that the Slater condition is satisfied for suppliers, allowing for the conversion of bi-level models in these scenarios to singlelevel models via the KKT method. The solution space of the suppliers in the second to fifth scenarios is convex, and the objective functions are concave, so sufficient KKT conditions are met. The optimal values for the variables can be obtained through the KKT conditions if one of the constraint qualifications is met (Kuhn, 2006).

Lemma 5:
Slater's quality condition is met for the second to fifth scenarios.
A proof for lemma (5) appears in Appendix (7). The KKT conditions for each supplier are set as follows. First, the L i function is formed for each supplier, as presented in Relation (25). The first KKT condition is obtained as the first derivative of the function concerning the variable for each supplier ( L i w i ) and is set to zero in Relation (26). The complementary slackness condition is also considered for each objective function, as shown in Relation (27). Relations (22) and (23) are considered for each supplier and added to the upper-level constraints to convert the bilevel models to single-level ones in the second to fifth scenarios. The objective functions for the scenarios are nonlinear and so are some of their constraints.
In the following, a numerical example of a real case is considered to investigate the practical applications of the scenarios presented in the previous section. The information provided in this section is taken from Song and Zhuang (2018). The dairy supply chain is considered in this paper, where quality control is very important for the products. The percentage of protein in each product is measured for that purpose. The density of nitrogen in the protein is 16%, and the amount of nitrogen in 1 kg of the dairy product is measured to determine its protein amount, which must be at least .0228. The cost of using protein is .84 RMB kg . The density of nitrogen in melamine is .666, which is higher than that in protein, and the cost of using melamine is .7 RMB kg , which is lower than the cost of using protein. The manufacturer uses melamine instead of protein to reduce costs and increase demand, but excessive melamine use causes health problems. In this paper, it is assumed that the probability of endangering health is linearly dependent on the percentage of melamine, and is estimated to be 106.3. The government has imposed a fine of 1500 to prevent the manufacturer from using too much melamine; the maximum allowable amount to be used in the products is .0001. The carbon dioxide emission from the production of each unit of a dairy product is 1214 dc 3 Co2 kg (Food and Organization, 2010). The social cost of GHG emissions is .0007 RMB dc 3 Co2 (Gillingham and Stock, 2018), and the manufacturer seeks to reduce GHG emissions to increase demand. The values of the other parameters are estimated according to experts' opinions and the values obtained from the previous articles. The first scenario involves a single-level model, and the others turn into singlelevel models using KKT conditions. The objective function is nonlinear in all the scenarios, as are some of the constraints. All the scenarios are solved in the GAMS software with the BARON solver.
The supply chain's structure, as well as competition and cooperation among members, all have an effect on decisions and sustainable goals, which will be examined through the solution of proposed models for the scenarios defined in Section 5.1. One of the objectives is to examine and quantify the second supplier's contribution to the cost of reducing GHG emissions in order to achieve a balance between the various dimensions of a sustainable supply chain. The effect of various values of this parameter on decisions and goals is examined in Section 5.2, and its appropriate value is determined. Budget is a critical parameter associated with model constraints that influences members' decisions on the demand, pricing, and investment to reduce GHG emissions. Sections 5.3 examine the impact of the budget. The cost of GHG emission reduction and the government penalty for endangering consumer health are two critical parameters relating to the environmental and social dimensions, respectively, that are effective in other sustainable dimensions of the food supply chain. As a result, these two parameters were chosen for numerical analysis in Sections 5.4 and 5.5.

Comparison of the scenarios
The optimal values of the variables for all the five scenarios and those of the objective functions of the chain members and the total profit are presented in Tables 2 and 3, respectively. On that basis, the total profit is greater in the centralized scenario than in the others, and GHG emission is the least there. According to Song and Zhuang (2017), the collaboration between supply chain members is more economically advantageous than the competition, and our findings corroborate their findings. Furthermore, among the decentralized scenarios, the alliance between the manufacturer and the first supplier in the third scenario provides the highest total supply chain profit. The coalition members tend to reduce GHG emission and increase demand in the fifth scenario by having the second supplier share the costs, but the increase in revenue due to increased demand is less than the increase in the cost of reducing GHG emissions; therefore, total profit is less in this scenario than in the third. This feature is also evident in comparing total chain profit in the second and fourth scenarios. Zhu et al. (2018) demonstrated that sharing the cost of greening (reducing negative environmental effects) among supply chain members improves their environmental performance, and this conclusion was confirmed in this article. Additionally, the fifth scenario has the lowest GHG emissions of the decentralized scenarios, and the fourth scenario has lower GHG emissions than the second. The difference in total profit between the third and fifth scenarios is less than the difference between them in GHG emissions, so the fifth scenario is the most suitable among the decentralized structures to balance the economic and environmental goals. The supply chain members must decide how to cooperate according to the priority of economic and environmental goals. In all five scenarios, the percentage of additives used is the same and equal to zero. According to Table 2, product price is higher in the second and fourth scenarios than in the others because of the competition, so the products' demand is less in these scenarios than in the others. The price of raw materials required for the first product is higher than that for the second product in all five scenarios because the cost of raw materials and supply capacity for the first product are higher than those for the second product.

Effect of the second supplier's share of the cost of reduction of GHG emission
This section investigates the effects of on the supply chain members' variables and objective functions in the fourth and fifth scenarios. As shown in Table 4, the manufacturer's incentive to reduce GHG emissions increases as the second supplier's share of GHG reduction costs increases in the fourth scenario. The reduction of GHG emissions increases the manufacturer's non-production costs, and the manufacturer seeks to compensate for the loss by increasing the prices of its products. The suppliers' profits decrease as raw material prices and product demand are reduced, according to Table 5. The total profit decreases with an increase in the second supplier's share of the cost of reducing GHG emissions. The above share is not economical in the fourth scenario but leads to a drastic decrease in GHG emission and is environmentally friendly. According to Table 6, the demand for the second product decreases in the fifth scenario with an increase in the second supplier's share of the cost of reducing GHG emission, as in the fourth scenario, but the demand for the first product remains constant. Table 7 shows that the   Table 5 Optimal values of the supply chain members' objective functions in the fourth scenario 164,223.419 740,150.874 9,060,027.416 10,964,400 0.2 1,159,200 727,052.089 9,076,883.911 10,963,140 0.3 1,152,989.679 707,582.635 9,098,291.28 10,958,860 0.4 1,145,125.046 677,282.252 9,126,463.43 10,948,870 coalition members increase their profits by increasing the prices of the products, and the profit of the second supplier decreases as the demand for the second product decreases. A sharp decline in the profit of the second supplier causes a decrease in the total profit, with an increase in the parameter. The reduction of GHG emissions in the fifth scenario is close to that in the centralized scenario, and the second supplier's profit decline is not severe where it has a 0.2 share of the coalition members' costs of reducing GHG emissions.

Effect of the manufacturer's budget
To investigate the effect of the budget parameter, it is assumed to vary from 1,000,000 to 3,000,000. The cost of reducing GHG emissions is higher than the other production costs, so the manufacturer and coalition members seek to reduce GHG emissions by increasing the budget to increase demand and profit. The pricing strategy of the chain members varies as the budgets for different products and scenarios increase. According to Figures (2a) to (2d), the manufacturer and coalition members seek to increase demand by reducing the price of the second product in all the scenarios and the first product in the second and fourth scenarios. In the first, third, and fifth scenarios, the manufacturer cannot increase the demand for the first product due to the limited supply capacity for the raw materials of the first product and seeks to maximize profit by increasing price. According to Figure (2f), the total profit also increases in all the scenarios with an increase in the budget. The reduction of GHG emissions as the budget increases is greater in the third and fifth scenarios than in the second and fourth because price competition is less severe in the former scenarios. The rate of increase in total profit is higher in the second and fourth scenarios than in the others because customer demand is less in these scenarios due to price competition and supply capacity limitations do not prevent the increase in the chain members' profits.

Effect of the cost of reducing GHG emission
We study the effects of the cost of reducing GHG emissions by having the value of the parameter range from 0.8 to 1.3. According to Figure (3e), GHG emission increases in all the scenarios with the increasing cost of reducing GHG emissions. The increase in GHG emission is greater in the first, third, and fifth scenarios than in the other two. Figures (3a) to (d) show that the supply chain members cannot invest in reducing GHG emissions for all the scenarios due to increased cost. They determine a lower price to prevent a sharp decline in customer demand. Manteghi et al. (2020) demonstrated that as the cost of reducing GHG emissions increases, the total profit of the supply chain decreases, and the validity of this proposition is also demonstrated in this article. Apart from that, according to Figure (3f), total profit decreases in all scenarios as the cost of reducing GHG emissions increases. The total profit is higher in the fifth scenario than in the third, where r is less than one, and it is higher in the fourth scenario than in the second.

Effect of government penalty on the endangerment of consumer health
According to Table 8, the coalition faces a higher penalty for using melamine in the third and fifth scenarios than in the second and fourth. In the second and fourth scenarios, there is greater price competition between members, and manufacturer costs increase as the penalty increases, so the manufacturer avoids using melamine to avoid a decrease in demand. In this case, the manufacturer prefers to avoid using melamine rather than reduce the price and increase GHG emissions in order to maintain demand for the first product while increasing demand for the second, as illustrated in Figures (4-a) to (c) (e). The percentages of melamine used in the third and fifth scenarios for different values of parameter g are higher than those for the second and fourth scenarios. Figures (4-a) to (e) show that the coalition members prefer to use melamine and reduce product price and GHG emissions to maintain the demand for the first product and increase that of the second product. The optimal values of the variables remain constant with an increase in the parameter in the second and fourth scenarios, so the total profit is constant in these scenarios, as shown in Figure (4-f). In the centralized scenario, the decision-maker reduces the optimal value of melamine by increasing parameter g , seeking to prevent a decline in demand by reducing price and GHG emissions according to Figures (4-a) to (e), and these changes reduce the total profit of the supply chain as shown in Figure (4f). An interesting result is obtained for the third and fifth scenarios, where total supply chain profit increases with an increase in the parameter. The demand for the second product increases with lower prices and less GHG emission and melamine use. The increase in demand leads to an increase in the total profit of the supply chain, according to Figure  (4-f). Total supply chain profit is higher in the fifth scenario than in the third, where g is less than 230.

Conclusion and managerial insights
Consumer demand is constantly increasing in the food industry, so firms have to meet the new needs of customers to stay competitive. Nowadays, customers prioritize quality and sustainability when purchasing food products. In this paper, it is assumed that the manufacturer Figure 2 Effects of B on (a) demand for the first product, (b) demand for the second product, (c) price of the first product, (d) price of the second product, (e) reduction of GHG emission, and (f) supply chain profit reduces GHG emissions from the production process to increase customer demand. The reduction of GHG emissions increases manufacturer costs. Manufacturers use additives to cut costs and boost demand, which may have a negative effect on consumer health. The government imposes fines on the manufacturer for the use of additives  to protect consumer health. The amount of additives used and the amount of GHG emissions reduced impact other dimensions of a sustainable supply chain. Five scenarios were used in this paper to examine the impact of supply chain structure and relationships among its members. The following summarizes the article's contribution: (1) Examining the impact of competition and cooperation in the sustainable food supply chain using five scenarios (2) Examination of supplier competition as well as producer-supplier competition (3) Utilizing both contractual and non-contractual mechanisms of cooperation (4) Considering constraints such as maximum emissions, the use of additives, and the available budget (5) Proposing a bi-level model for supply chain member competition and resolving it.
The above five scenarios have been solved for a numerical example, and the results demonstrate that the total profit of the supply chain in the third scenario is the greatest among those in the decentralized scenarios, and GHG emissions are least in the fifth scenario among the Figure 4 Effects of g on (a) price of the first product, (b) price of the second product, (c) demand for the first product, (d) demand for the second product, (e) reduction of GHG emissions, and (f) supply chain profit decentralized scenarios. It is economical to increase the budget in the scenarios where the supply chain members compete. Moreover, it is environment-friendly to increase the budget for the supply chain members that have formed coalitions. Using both alliance and cost-sharing cooperation methods is appropriate for the achievement of a balance between the economic and environmental goals of the sustainable supply chain. The total profits of the supply chain in the fourth and fifth scenarios decrease as the second supplier's share of the cost of reducing GHG emissions increases. The amount of reduction of GHG emissions in the fifth scenario is close to the value of the variable in the centralized scenario, and the second supplier's profit decline is not severe, where the second supplier has a 0.2 share of the cost of reducing GHG emissions. Consumer health is at higher risk in the first, third, and fifth scenarios than others.
The supply chain's structure and the relationships between its members have a significant impact on economic, environmental, and social objectives. Competitive pricing is economically advantageous for other supply chain members, and the manufacturer contributes to their economic objectives by sharing product market information. Different scenarios have varying effects on economic objectives. The numerical results section determined that cooperation between supply chain members is economically more advantageous than the competition. Furthermore, the supplier's share of the manufacturer's cost is not economically appropriate. Increasing budgets is economically advantageous for supply chain competitors. In contrast, the effect of structures on environmental goals is nearly the polar opposite of the effect on economic goals. Contributing to the cost of GHG reduction is environmentally friendly, however, and goes against the economic objective. When supply chain members cooperate, increasing the budget is environmentally responsible. Increased costs associated with GHG reduction have a detrimental effect on the environmental goals of the supply chain members that comprise the alliance. The social impact of alliances between supply chain members is worse than that of competition between them because consumers are more likely to be at risk in these scenarios. Decision-makers must consider all aspects of a sustainable supply chain, and it is appropriate to use both coalition cooperation and cost-sharing mechanisms to achieve a balance between economic and environmental dimensions. Uncertainty and risk management have gained plenty of significance in the food supply chain, so these issues should be considered in future research on the food supply chain. Furthermore, different criteria can be considered for the environmental and social dimensions of the food supply chain. Additionally, future research should examine the effect of alternative demand and the probability of health risk functions on the decisions and objectives of supply chain members.
The matrix determinant is calculated in Relation (33), and we know that > holds for the model. It can be concluded from Relation (34) that the determinant of the matrix is always positive, and the matrix is negative definite if parameters r 1 and r 2 are set to 1. According to these relations, the matrix J w i , r + J T w i , r is negative definite, and the game between the two suppliers is strictly concave diagonal, so it has a unique solution.

Appendix (3)
There is no difference between the solution spaces for the suppliers in the second and fourth scenarios, so the solution space is non-empty, convex, and closed for the suppliers in both scenarios. It can be seen from Relations (21) and (22) that the suppliers' objective functions are continuous. The second derivative of the objective function of each supplier is negative with respect to the wholesale price of its raw materials ( 2 Z S i w it