A Game Theory Approach for Supply Chain Coordination Model with Incentive Mechanisms of Discount and Delay in Payments

. Te uncertain nature of supply chains is one of the key challenges managers, and researchers encounter in decision-making. Accordingly, this paper proposes a three-echelon supply chain in which demands are uncertain. Te proposed supply chain has three participants, including supplier, manufacturer, and retailer, while three decentralized, centralized, and coordination models have been formulated to maximize participants’ profts. In the decentralized model, both the manufacturer and retailer independently determine the level of investment and order quantity regarding scenario-based demands. Te centralized model determines the optimal order quantity and investment amounts for the whole network. However, these amounts may be diferent from optimal values for all participants. As such, using game theory, a bilevel adjustable contract based on wholesale price has been proposed as an incentive for players to participate in the co-ordination plan. Results show that the coordination model outperforms others by reducing the network’s costs and increasing profts simultaneously.


Introduction
In today's global economy, supply chain management (SCM) plays a crucial role in any enterprise's major success [1]. In this respect, planning is the foundation of sourcing, production, and logistics, which can afect the supply chain's overall cost, productivity, and quality [2].
Supply chain planning is concerned with coordinating and synchronizing of several activities of diferent functions from the very beginning, e.g., procurement of raw materials for the fnal process and distribution of fnished products, which may require diferent coordination mechanisms due to centralized or decentralized supply chain operations. In a centralized supply chain, a single decision-maker optimizes the entire system's performance by having access to all the required information. On the other hand, in a decentralized system with multiple players, there is no unique planner; instead, the members, who may have conficting objectives, decide according to their own interests [3].
Since supply chain management incorporates all activities related to material fow and product transition from the raw state (extraction) to the fnal state (consumption), as well as the fow of surrounding information, improving the interaction between chain loops will give companies a reliable and sustainable competitive edge [4].
As intermediaries between manufacturers and consumers, retailers are continuously exposed to uncertain risks. Apart from the uncertain demand of consumers, retailers may also face uncertain yields. Uncertain performance in manufacturing and logistics could lead to increased risks caused by uncertainty in the chain. To deal with this, the concept of supply chain coordination has been developed to beneft all chain members, especially in times of uncertainty [5].
With the growing focus on sustainable supply chain management, frms fnd out that outputs across the entire supply chain can be more efciently managed through greater cooperation and better coordination. Coordination in supply chain management is based on centralized and decentralized decision-making. Te supply chain coordination aims to minimize the total supply chain cost. In a decentralized supply chain, each player tries to maximize his own performance.
In centralized decision-making, coordination allows supply chain players to work closely to streamline their decision-making, aiming to maximize the entire supply chain performance. Terefore, supply chain coordination must involve incentive plans to attract customers [6].
In many real-world situations, it is difcult to accurately estimate key parameters, including market demand, supply time, production volume, which are major sources of uncertainty in the supply chain. Te uncertainty in these parameters is an inevitable part of decision-making in the supply chain. In practice, the uncertain demand for many products can be modeled through a scenario-based approach. As a result, the coordination of chain members is an efcient means to improve the overall performance of the supply chain and coordinate various decisions [7].
Studies on supply chain coordination can be classifed into three fows. In the frst fow, a chain mechanism is coordinated only when it leads to the improvement of the whole supply chain. In the second fow, the chain mechanism is coordinated only when it leads to the improvement of the entire supply chain compared to the default solution, which lacks coordination. In the fnal fow, implementing the coordination mechanism could lead to a practical solution across the supply chain system [3].
Tis study investigates a three-echelon supply chain where the supplier, manufacturer, and retailer are each concerned about their own beneft in the game theory, and it seeks to increase the profts for each actor. In this respect, three centralized, decentralized, and coordinated models are employed to formulate the proposed supply chain under uncertain demand. Te following sections discuss the results of applying these models to the petrochemical industry. Te structure of the paper is as follows. Section 2 presents the literature review and specifes the research gap within the present research. Te proposed models and the solution approaches are presented in Section 3. Section 4 is dedicated to the model validation and the fnal results of the models, followed by conclusions in Section 5.

Literature Review
Supply chain coordination has been the subject of many studies: According to Christy et al. [8], the demand of a product is linearly dependent on the retail price and quality of the product. Tey address a closed-loop supply chain where the manufacturer manufactures products according to the demand and sells them through a retailer in the market. A third party collects the used products from costumers and sends to the manufacturer to increase the quality. If the products can retrieve the original quality, thus the process is called remanufacturing. Not every product can retrieve the original quality; thus, manufacturers refurbish these products with lower price. We construct four diferent scenarios-centralized and decentralized led by manufacturer, retailer, and third party. From the comparison of the result obtained in the numerical example, they conclude that the joint proft obtained under centralized, manufacturer-led, and retailerled policies is higher than third party-led policy.
Taleizadeh et al. [9] examined the multiperiod sustainable planning of a closed-loop supply chain using a comprehensive model that considers the social and environmental impacts of supply chain decisions and measures their efects on social and environmental factors through the (global report) guide indices. Te proposed planning model incorporates tactical decisions such as the product's price and logistic decisions. Using a discount for a returned product is an incentive policy designed to increase the tendency of customers to return the products. In addition, two recovery approaches have been considered, including remaking and recycling along with using highquality returned products.
Xiao et al. [10] explored a two-echelon supply chain consisting of a manufacturer and many retailers whose demands lie in unreliability. Tis paper examines two practical issues through a collaborative game approach, where prominent coalition values are demonstrated with an expected favorable proft. We have also examined an advertisement collaboration issue of a manufacturer and retailer where the manufacturer plays the Stackelberg game by forming a coalition with the retailer.
In their research, Malekian [11] considered the increase in price and national advertisements of the manufacturer in a manufacturing supply chain with the efects of the consumers' reference price. Te paper examines a centralized game, followed by two Stackelberg games named "Increasing consumer price" and "Increasing retailer and consumer price." Li et al. [12] presented two novel contract types to achieve supply chain coordination. In this paper, the quality fexibility contract and the capacity reservation contract are proposed as mechanisms for encouraging manufacturers to increase capacity and improve overall supply chain performance. Te proposed model proves that it is possible to coordinate the supply chain by adjusting the precise parameters of the two contracts. Tis will ensure that the total proft in the decentralized supply chain is the same as the one in the centralized system. Ahmad Jauhari et al. [13] develop a closed-loop supply chain model consisting of a single manufacturer, single retailer, and single collector under various coordination scenarios. New products produced from the manufacturing and remanufacturing processes will be sold to the market at the same price. Used products collected by the collector are sorted so that products categorized as recoverable will be sold to the manufacturer. Tere are two recovery processes considered in this paper, namely remanufacturing and refurbishing. Used products below the minimum acceptable quality level of the manufacturer will be categorized as waste and will be disposed of. Tey assume that the manufacturing process is imperfect as it produces rework able defective products. A carbon cap-and-trade policy and investment in green technologies are applied in order to restrict the carbon emissions generated by the production stage of the system. Te demand at the market place depends on the green technology level, the quality of the product, and the selling price. Te proposed model is constructed under fve different scenarios-centralized, decentralized, and three Stackelberg games led, respectively, by the manufacturer, retailer, and collector. A numerical example is provided to illustrate and compare the proposed model under each scenario and investigate the sensitivity of some of the model parameters on the optimal solutions. Te results show that the centralized scenario performs better in maximizing the total proft compared to the decentralized one. However, the retailer-led Stackelberg model tends to give more equitable proft to all players when the selling price is set at the lower level as this will attract more demand.
Yan et al. [14] suggested a method for the supply chain coordination of new crops by considering the strategic behavior of the consumer. Tis paper introduces the optimal consumer performance by considering the characteristics of the supply chain of fresh crops. Afterward, this study focuses on the efect of consumer behavior on decision-making in the supply chain based on a centralized chain approach. Ultimately, it proposes two coordination contracts based on revenue sharing and wholesale price to decentralize decision-making in the supply chain of upcoming crops. Afterward, this study focuses on the efect of consumer behavior on decision-making in the supply chain based on a centralized chain approach. Finally, it suggests two coordination contracts for decentralized decision-making in the supply chain of upcoming crops based on revenue sharing and wholesale pricing.
Heydari et al. [15] analyzed the green channel coordination in a two-echelon supply chain in which demand is a function of pricing and the green quality of the product. In this model, the retailer makes pricing decisions, and the manufacturer adjusts the product's green quality. For channel coordination and a win-win result, the "green division of costs" and the "revenue-sharing" contract have been combined.
Toktaş-Palut [16] introduced a three-echelon forward and reverse green supply chain, where manufacturers invest in green production processes. Tis paper considers the coordination of this integrated green supply chain by introducing a mathematical model associated with fair bilateral contracts between the two sides. Accordingly, a twopart integrated tarif contract is created so that all the supply chain members act based on a rational centralized solution, where both sides can earn a proft.
Cao et al. [17] considered a green agricultural supply chain and examined the coordination and optimal decisions of all units within a decentralized and centralized system by using a game model. Te paper proposes a revenue-sharing contract and a repurchase agreement to coordinate the decision-making for greening the whole supply chain and analyzes the efects of green standards at diferent stages.
Zhu et al. [18] introduced a decision-making model in which the conditional value at risk (CVaR) is used as the risk evaluation criteria. To coordinate a dual-channel supply chain, revenue-sharing and repurchase contracts are jointly used to make bilateral contracts for optimal decisionmaking in centralized and decentralized situations. Tis paper proposes joint contacts that could lead to Pareto effciency for a dual-channel supply chain in which a riskaverse consumer is involved in performance and demand uncertainty.
Zhao et al. [19] studied the centralized and decentralized aspects of decision-making and optimal proft of supply chains. Tey illustrated that the uncertainty of manufacturing cost exaggerates the motivation for supply chain formation; however, it might increase the expected proft regarding centralized decision-making. Terefore, an incomplete contract has been designed that determines the wholesale price and order quantity in the frst stage. Once the manufacturing cost is achieved, companies can renegotiate the contract in the second stage. Interestingly, these incomplete contracts can also coordinate the supply chain.
Ganji et al. [20] introduced a new coordination model in the supply chain, which can gain customers' satisfaction by planning a precise logistic system. Tis mathematical model accounts for all coordination among chain members and causes an increase in the members' profts by considering the integrated planning of the supply chain, determining the delivery time, the orders' schedules, determining the vehicle's function based on the freight capacity, and minimizing distribution costs, fxed fuel costs, variable fuel costs, carbon emissions, and the time required for cargo deliveries.
Sarada and Sangeetha [21] examined reverse supply chain coordination with a price-and warranty-dependent stochastic demand under collection uncertainty. Tis paper introduces two theoretical game models of one reverse supply chain (RSC) for the retail of one type of remanufactured product. Te frst model analyzes a two-stage RSC with one manufacturer and one retailer and considers the uncertainties in association with demand, collection quantity, and the system's performance. Te second model studies a three-echelon RSC with one supplier, one manufacturer, and one retailer. Moreover, it considers the stochastic system's demand and performance with price-and warranty period-dependent demand. Te manufacturer provides a warranty for remanufactured products that are produced from returned items retrieved from the customers. Te policies of centralized, decentralized, and revenuesharing contracts are imposed on both models. In the end, the revenue-sharing contract increases the profts of the supply chain members.
Putri Adam et al. [22] develop a coordination mechanism for a closed-loop supply chain, operating under Discrete Dynamics in Nature and Society several policies to control the carbon emission, namely a carbon tax regulation, government incentives policy, green technology investment, and energy-saving investment. Te carbon tax regulation is implemented to lessen the emissions from the manufacturer's activities whereas, to encourage the manufacturer to cut down the emission as well as to increase the product return and energy savings, the government provides incentives based on a target level. Te system operates under a variable market demand which is afected by the retailer's selling price, green technology, and energy-saving levels. Te proposed problem is formulated under two diferent scenarios, which are the centralized model and the decentralized model. To improve the supply chain coordination, they also propose two diferent contracts, namely the green technology revenue-investment-sharing contract (GRIS contract) and the energy saving revenue-investmentsharing contract (ERIS contract). Te system inficted with two types of inspection error in classifying the returned products. Te models are formulated mathematically and optimized using a proposed algorithm. Te result shows that the centralized model performs better in maximizing the total proft compared to the decentralized model. Te results also imply that the government incentives toward product returns, green technology, and energy-saving actually afect the optimal decision of the supply chain system. In addition, the proposed contracts are proven to provide win-win solutions and improve supply chain coordination.
Based on the problem statement and research gap analysis, it can be concluded that the simultaneous use of delays in payment and discounts considering the game theory deserves further study. Additionally, channel coordination is an efcient way to improve the supply chain's overall performance and various decisions. Many past studies on channel coordination have implemented the uncertain nature of market demand with a unique probability distribution, such as a normal distribution. However, in the real world, the market demand could only be estimated with a specifc set of discrete scenarios. When market demand is unclear, determining the optimal order quantity can be challenging. A scenario-based approach must be used to model this issue and optimize decisions in such a situation. Nonetheless, to our knowledge, uncertainty through a scenario-based approach in past studies on supply chain coordination still needs further investigation. Tis research has been conducted through a case study in the real world's petrochemical industry, where a manufacturer sells its products to customers through a distributing channel (retailer). Simultaneously, by providing discounts and delaying payments, the manufacturer seeks to positively afect the customer's perception of its product, thus increasing its market demand. Hence, this study frst investigates the problem under a single-scenario stochastic demand and suggests a basic model. Afterward, it considers the scenario-based stochastic demand through the development of the model and analyzes the problem under decentralized, centralized, and coordinated models (three solutions in the game theory).

The Proposed Method and the Mathematical Model
Tis study investigates a three-echelon competitive supply chain through a mathematical model using the game theory concept. Te proposed model is an extension to the model presented by Hosseini-Motlagh et al. [7] to include the game theory for implementing coordination in the supply chain. Meanwhile, it considers the incentive mechanisms of discounts and delayed payments based on Aljazzar et al. [6].

Te Research Questions
(i) How to coordinate and create efective communication in a three-level supply chain, taking into account incentive mechanisms (discount and delay in payment) in order to equalize benefts using game theory? (ii) According to the proposed mathematical models, what is the proft of the chain members and which of the game theory scenarios do they prefer? (iii) How can a scientifc solution be presented for case study problems in the form of decision-making models? (iv) How can channel coordination be achieved when random demand in the future can only be predicted with a set of discrete scenarios? (v) How can contract parameters be adjusted in a threeechelon supply chain to induce all actors to participate in the coordination program in each demand scenario?

Assumptions
(i) Te proposed supply chain has three echelons, including supplier, manufacturer, and retailer (buyer); (ii) A single product is made of several raw materials; (iii) Te demand rate depends on the discounts; (iv) Te production quantity of the supplier exceeds the manufacturer's demand for raw materials, and the production rate of the fnal products in the manufacturer is faster than the retailer's demand; (v) Shortage is not permissible; (vi) Te manufacturing policy of the manufacturer follows the Hill policy in which equal batches of production and same-size cargoes are made; (vii) Te holding (storing) cost consists of the two physical and monetary components; (viii) Te discounts and delay in payments are considered as decision-making variables; (ix) While planning the permissible delays, the manufacturer and retailer invest in the remaining products collected; (x) Te manufacturer and retailer pay for their remaining products in single payments; (xi) Te maximum discount by the supplier, manufacturer, or retailer cannot exceed the proft margin.

Problem Statement.
Te supply chain system presented in this research constitutes a three-echelon supply chain (supplier, manufacturer-retailer). In this system, the retailer orders a great quantity of manufactured items Q for the annual demand rate D. Te manufacturer manufactures the manufactured items with the annual per rate P, where P > D.
Te manufacturer orders the product quantity αQ from the supplier, where α is the number of raw material units required for the manufacturing of the fnal product. Te manufacturer makes their payments according to the agreed-upon time with the supplier, within a period interestfree. If the manufacturer makes their payment at τ m after the t s , where τ m > t s , the interest rate k s is deducted by the time unit to create equilibrium τ m − t s . Troughout the time period τ m or t s , the manufacturer pays the debt equilibrium to the supplier through the interest rate k m . Consequently, the manufacturer allows the retailer to make their interestfree payment at t m within a period of time. As a result, the retailer can postpone their payment to the manufacturer to after the manufacturer has received the interest rate k m remaining from their account. Hence, the retailer pays their balance to the manufacturer with the interest rate k r . Tis paper frst considers the problem under the single-echelon stochastic demand and proposes basic models. Afterward, the models are developed to create scenario-based stochastic demand, and the problem is dissected under decentralized, centralized, and coordinated models. In the decentralized model under the scenario-based stochastic demand, by considering the scenario-based stochastic CSR-sensitive demand, the retailer and the manufacturer would separately make decisions on the order quantity and investment. Te optimal order quantity and investment are then obtained under the scenario-based stochastic demand according to the whole chain. Te optimal solutions under the centralized model may not be satisfactory to all the three agents (three echelons of the supply chain, i.e., supplier, manufacturer, and retailer). To persuade the three agents to enter the coordination plan, we proposed an adjustable bilevel wholesale price contract. According to the proposed contract, the wholesale price of the supplier and the manufacturer is assigned as the contract parameters, and it is determined in such a way that the proposed results of the contract would have a winning situation for all the agents of the supply chain. Tis research utilizes the study of Hosseini-Motlagh et al. [7] to calculate the uncertainty in the demand rate while estimating the uncertain model within the supply chain. Accordingly, various scenarios are applied to the chain depending on the retailer's demand rate. A demand distribution function based on discounts and uncertain demand in each scenario is made. Based on the probability of each scenario, the total proft rate of the chain, which consists of the proft of all the three suppliers, manufacturers, and retailer echelons, is afected. As a result, the fnal goal is the optimization of the chain's total proft. Figure 1 demonstrates the overall schematic of the problem.

Te Mathematical Model.
In the development of the mathematical model, the paper uses the symbols as follows.

Parameters
i is determined corresponding to the chain members s indicates the supplier w indicates the raw materials of the manufacturer m indicates the manufactured products of the manufacturer r indicates the retailer c indicates the customer A i indicates setup/ordering cost C i indicates production/purchase cost per unit h i indicates fnancial holding cost per unit s i indicates holding (storing) cost per unit n 1 indicates number of transportations done by the supplier to the manufacturer per each period of the manufacturer's raw material n 2 indicates number of transportations done by the manufacturer to the retailer per each retailer period α indicates raw material quantity required for the manufacturing of a fnal product t i indicates permissible delay in payment τ i indicates settlement time k i indicates return on capital P indicates manufacturer's annual production rate d i indicates discount in the monetary unit by factor i to its customer S i indicates demand scenarios index ρ(S i ) indicates probability of a scenario's occurrence S i D s i indicates annual demand rate under scenario S i , D s < P. f(D s i ) indicates probability distribution function of the demand in scenario S i to discount f(d rs ) indicates annual demand of the retailer. In this case, it is assumed that this is a discount and stochastic demand function in various scenarios: D s + f(D s i )d r μ S i indicates average market demand before CSR in scenario S i σ s i indicates maximum increase in medium demand by investing in CSR in scenario S i T indicates length of the time period ψ i indicates annual proft for each factor i.  Figure 2 illustrates the schematics of this model. Using stochastic scenario-based demand in this section, the decentralized basic model is developed. In the real world, in most cases, the market demand can be demonstrated with a discrete stochastic variable with merely three probable scenarios, i.e., (1) high demand, (2) medium demand, and (3) low demand. Terefore, in this problem, three diferent scenarios of S 1 , S 2 , and S 3 have been chosen and demonstrated as such. More precisely, the future demand of the retailer under each probable scenario of

Decision Variables
Moreover, the occurrence probability of each scenario is illustrated with ρ(S i ). It is noteworthy that the number of demand scenarios can easily be renewed with a few adjustments in the proposed models.
Based on the decentralized model, the present problem is modeled as a manufacturer-Stackelberg game, where the manufacturer optimizes their decision by considering the most appropriate response of the retailer. Te optimal solutions of the Stackelberg game can be determined through the backward induction method. In this method, the retailer's problem is frst optimized, and the retailer's best response to any investment in CSR done by the manufacturer can be obtained. To maximize their proft performance in future demand scenarios, retailers must select their order quantity under scenario-based stochastic demand. Furthermore, the order quantity selected must ensure that the retailer does not incur losses under any demand scenario. Afterward, the manufacturer's problem is optimized by considering the best retailer's response to the manufacturer's proft performance, and the optimal investment in CSR is achieved. Ultimately, the retailer's optimal order quantity is obtained based on the manufacturer's investment in CSR. The cost of ordering raw materials, the cost of selling raw materials, the cost of holding raw materials in the storage rooms, interest rate, the cost of setup manufactured products, the cost of manufacturing the products, the financial cost of holding and storing (physical) products, in the storage room, the cost of financial holding caused by delays in sending to the retailer, and lost opportunities.
The income from selling manufactured products to the retailer, the interested rate paid by the retailer to the manufacturer, and the interest obtained throughout the permissible delay.
The cost of ordering, the cost of selling the manufactured products, the cost of financial holding of the manufactured products, the cost of storing the manufactured products, and the interested rate paid to the manufacturer.
Annual income of the retailer including selling the manufactured products, the interested obtain from the permissible delay in the delivery by the seller, and the income from investing in the sold products.
The cost of setup, the cost of producing raw materials, the financial costs of holding and storing (physical), the cost of financial holding caused by delays in sending to the manufacturer, lost opportunities.
Selling the raw material to the supplier minus the payment interest rate. Annual Income-Annual cost Annual Income-Annual cost Annual Income-Annual cost Terefore, the retailer's proft function can be written as the function (1) as follows: where Since the order quantity has been adjusted before the occurrence of each demand scenario, the proftability of the supplier and manufacturer is independent of the probable scenarios. Hence, the average proft of the supplier and the manufacturer in all the scenarios is illustrated in function (2) for the supplier and function (5) where ψ s denotes the supplier's annual income, and c s denotes the supplier's annual cost, illustrated in functions (3) and (4) In the previous equation, for all the small amounts of the interest rate k s , e k s τ m,w − t s − 1 has been approximated as k s (τ m,w − t s ).
where in this equation, when k s ≪ 0, we have: In equation (6), ψ m denotes the manufacturer's annual income, and c m denotes the manufacturer's annual cost, which are illustrated in equations (7) and (9), respectively.
At k s ≪ 0, we have: Discrete Dynamics in Nature and Society At k s , k m ≪ 0, we have: In the scenario-based decentralized decision-making model, the manufacturer frst optimizes their investment unit in CSR i.e. (η). Afterward, the retailer chooses the optimal order quantity. Since the optimal order quantity and the investment unit in CSR are not directly calculated through the equations (2), (5), and (1), a solution to obtain the optimal order quantity and investment in CSR in the scenario-based decentralized model is illustrated below. In other words, since equations (2), (5), and (1) are associated, the iterative search method suggested below can be used to fnd the desirable values of decision-making variables.
Step 3.1: , and move to step 2 to calculate the optimal order quantity, otherwise, we move to step 4.
Step 4: (Optimal Values) We obtain the maximum value for φ m (Q * K 1 , t s , t m , τ m,w , τ r , n 1 , n 2 , d m , d s , d r , η k 1 ) by citing the parameters Q * K 1 , η k 1 , and putting Q * dc � Q k 1 and η * dc � η k 1 . Afterward, we obtain the optimal values for Q * dc , η * dc and end the iterative method. Figure 3 demonstrates the schematics of the centralized model. Supply chain agents operate as an integrated vertical supply chain with a unique decision-maker under the centralized structure. Te unique decision-maker of the supply chain, which is facing the scenario-based stochastic demand, must adjust Q and η before the start of the selling season to give the chain's maximum total proft expected. Similar to the methodology implemented in the decentralized model, the scenario-based centralized model can be described as follows:

Te Centralized Model.
In the scenario-based stochastic model in the centralized system, the mathematical model is able to maximize the expected total proft of the chain by considering various demand scenarios.
Te restrictions of the model above guarantee that the total proft of the supply chain in the centralized adjustments will exceed that of decentralized adjustments in all the scenarios. If the expected proft for the whole supply chain under the centralized mode in a scenario (S i ) is lower than that of the decentralized model, the obtained optimal results will not be desirable. In other words, equations [φ c SC,S i (Q, η) ≥ φ * dc SC,S i ∀S i ] must be satisfed. In the equations above, Te optimal values for the two decision-making variables (for instance, the order quantity and the investment unit (CSR)) cannot be directly calculated through equation (11). Below, an iterative approach has been devised to calculate the optimal values for (Q * c , η * c ) in the adjustment of the centralized scenario:
Otherwise, we move to step 4.
Step 5: We determine the maximum value of φ c SC (Q K 2 , t s , t m , τ m,w , τ r , n 1 , n 2 , d s , d m , d r , η K 1 ) by citing the parameters Q K 2 , η K 1 , and putting Q * c � Q k 2 and η * c � η k 1 . Afterward, we obtain the optimal values for Q * c , η * c , and end the iterative method.
Although the centralized model increases the total proft of the supply chain compared to the decentralized one, the centralized solution does not necessarily increase the proft of all the agents of each echelon in the supply chain. Since the independent agents of the supply chain intend to maximize their proft, the chain agents who incur losses will not be interested in choosing centralized decisions. We will propose a new wholesale price-based coordinated contract to resolve the channel confict and achieve a coordinated system under scenario-based stochastic demand in the next section. Figure 4 demonstrates the schematics of the coordinated model.

Te Coordinated Model.
Te optimal decision-making variables derived from a centralized model may reduce the retailer or manufacturer's proft compared to one derived from a decentralized model. Te centralized method, however, is far more benefcial than its decentralized counterpart. In supply chain management, coordinated contracts have been used to encourage the decentralized SC agents for decentralized decision-making. Various coordination plans have been provided for supply chain networks, such as wholesale prices, delayed payments, repayments, and quantitative degrees of fexibility. Previous contracts for coordinated models are implemented when future market demands can be forecast using a single scenario. However, in most cases, an uncertain demand can be determined through a set of Discrete Dynamics in Nature and Society separate scenarios with relevant probabilities. In this section, we propose a new coordinated contract known as the adjustable bilevel wholesale price contract to achieve channel coordination among the next supply chain's three agents i.e. the manufacturer, supplier, and retailer, under a scenariobased stochastic demand. Te proposed contract has been designed in a way that will beneft all the chain's agents regardless of the demand scenario in the future. Tis can be acknowledged as the unique quality of the proposed contract. Literature on past coordinated contracts can only apply if the future market demand is forecast through a scenario. In the problem under question, the proft of the agents directly depends on the demand scenario, which will occur in the future. Consequently, a new coordinated contract is needed to provide a win-win situation regardless of the scenario. To solve the problem, in the proposed contract of this study, the contract parameters have been designed by considering probable demand scenarios in the future.
In the proposed contract, when the real demand in scenario S i is determined, the manufacturer presents the wholesale price W b M,S i to the retailer. As a result, the retailer presents a certain wholesale price W b R,S i to the customer. Hence, the parameters of the proposed contract of each scenario can be states as W b M,S i , W b R,S i . Tese two parameters must be determined in a way that the proft of all the agents will rise after participation in the coordinated model. It is noteworthy that the contract parameters W b M,S i , W b R,S i in each scenario S i are independent from another scenario.
Te proposed coordinated contract must be acceptable to all three agents i.e. the supplier, manufacturer, and retailer. Otherwise, they will refuse to participate in the coordinated model. Hence, at minimum, the necessities of all three agents must be taken into account for participation. Te term "bilevel" in the name of the proposed contract refers to the fact that not only will the proposed contract determine the wholesale price of the manufacturer, but it will also determine the wholesale price of the retailer at the same time so that under various demand scenarios all three parties proft from the coordinated model. Tese two contract parameters must be simultaneously adjusted so that all three agents participate in the coordinated model. As a result, we consider the steps below as the steps to the coordinated model in the supply chain: (i) Step 1: Before the start of selling season (i.e. before demand fulfllment), the retailer introduces a centralized-based order. (ii) Step 2: While investing in the CSR of each product according to the occurrence probability of each demand scenario, the manufacturer produces the retailer's order quantity based on the manufacturing restrictions and investment in centralized CSR. (iii) Step 3: Depending on the needs of the manufacturer, the retailer provides the manufacturer with the raw materials needed to produce the product. (iv) Step 4: After the onset of the selling season (i.e. after determining the demand quantity), depending on the agreement between diferent levels of the chain and contract parameters W b M,S i , W b R,S i , a certain time is considered for the delivery and settlement of the services between two sides, based on which various scenarios are considered for delays and late payments.
(v) Step 5: Te customer pays W b R,S i to the retailer and the retailer pays the W b M,S i to the manufacturer. According to the coordinated model, before achieving the demand scenario, the members of the supply chain (i.e. the supplier, manufacturer, and retailer) work together to discuss the order quantity, CSR investment, and contract parameters. As a result, the order quantity of the manufacturer must be commensurate with the optimal order quantity of the retailer under the centralized model, meaning equation (11) is applicable in this case. As can be seen, under the coordinated model, the retailer and manufacturer choose centralized solutions to order quantity and CSR investment, respectively. Moreover, the wholesale price is reevaluated by the manufacturer and supplier. Te contract parameters must be adjusted in a way that the proft of all the SC agents increases by choosing the centralized solution under any scenario compared to the decentralized model. If not, they will refuse to cooperate in the structure of coordinated decision-making. Hence, the contract parameters By applying mathematics, we will achieve the equation below for the model: Manufacturer Supplier CSR Q Figure 4: Confguration of the coordinated model. 10 Discrete Dynamics in Nature and Society Both contract parameters (W b M,S i , W b R,S i ), which satisfy equation (15), will also increase the proft of all the members (agents) of the chain under the coordinated model compared to the decentralized model. Terefore, the new wholesale prices (contract parameters) under the proposed contract will be satisfactory for all chain agents and will enable them to beneft from the coordinated model simultaneously.
In the proposed game model, the objective functions equal the multiplication of all the agents' proft functions after participation in the coordinated model. Te proposed game model can be adjusted as below: Max For this model, the restrictions are as follows: By solving the model above, the precise contract parameter acceptable to all three agents is determined. By achieving channel coordination in all the probable demand scenarios, the proposed Nash equilibrium model will ofer a win-win situation for all three agents.

Model Verifcation.
To verify the proposed model of this study, the model presented by Aljazzar et al. [6] is used to analyze the total chain proft under various circumstances, and compare them with the results of this study. For this purpose, the initial values used by Aljazzar et al. [6] are presented in Table 1.
Now, based on the modeling proposed in Aljazzar et al. and by determining the annual income ψ, annual cost c, and annual proft ϕ of each agent of the chain i.e. the supplier, manufacturer, and retailer, we will determined the fnal proft of the chain, which is equal to the total proft of each agent. Subsequently, we will compare the results in diferent modes with the paper in question. Table 2 demonstrates this matter.
Based on Table 2, the diference between the chain's total proft obtained here and obtained by Aljazzar et al. is less than 5% in diferent cases, indicating the high precision of the modeling and confrming the verifcation. Meanwhile, the paired samples t-test is used for comparison of two population means (Table 3).
where there are n pairs, d is the mean and S d is the standard deviation of their diferences. the test statistic has Student's tdistribution with n − 1 degrees of freedom.
Terefore, since 0.82 < 2.365, the null hypothesis is not rejected and the model is valid. By verifying the model, we will later analyze the results obtained from the case study for the three coordinated, centralized, and decentralized models. Elasticity of demand to a discount 70 -

Case Study (Shahid Tondgooyan Petrochemical Co.).
Founded on 1998-04-26 at Imam Khomeini port, in site IV of the petrochemical Special Economic Zone, Shahid Tondgooyan Petrochemical Co. is located in the northeast of the Persian Gulf Coast in Khuzestan province. It is a 34hectare area for the production of PET (Polyethylene terephthalate) and PTA (Pure terephthalic acid). In order to meet part of its obligations under the second fve-year plan of the Islamic Republic of Iran and in light of the expanding market for the consumption of these products both domestically and abroad, the National Petrochemical Company implements this plan according to these objectives: (i) Providing raw materials to downstream units of the country's textile and packaging industries, and saving currency. One of the most important raw materials for the production of PET, whose production and consumption process has been of signifcant importance all over the world, is PTA. PET is also one of the most important polyesters used in the production of synthetic fber and cotton in the textile industry, all types of bottles, cans, and packages used as food, medicine, and health packaging, as well as in the production of plastic flms, which is increasing in production and consumption. Moreover, this company produces two signifcant products: polyester yarn (POY) and polyester fber (STAPLE). Table 4 demonstrates the data collected from the company to determine the results of the paper's main model. Moreover, Tables 5-8 illustrate the costs of the setup of related units: Table 9 demonstrates the annual production rate for 4 types of products i.e. PTA, BG, TG, and POU. Table 10 illustrates the volume of raw materials needed to manufacture products.
Following the presentation of values and the calculation of other parameters in Sensor Cplex in GAMS, Table 11 below shows the results derived from solving the model based on the time spent on solving, as well as the objective function value for the supplier.

Te Results of the Models.
After verifying the model, we use game theory-based models to determine the optimal results of the three centralized, decentralized, and coordinated models. Figures 5-7 demonstrate the convergence diagram of the proft rate obtained from each model, respectively: As can be seen in Figures 4-7, the uncoordinated model quickly achieves convergence, and it dwindles steadily. Accordingly, the total proft of the chain for this model equals 560935.05 in fnancial units. Te centralized model also reaches convergence after 152 repetitions and remains consistent. Te coordinated model reaches convergence after 60 times of repetition, the optimal value for the highest proft rate of all the members of the chain. Additionally, its downward trend stops. Table 12 illustrates the results of the values obtained from each model and the time spent on solving the model using them. Te table can provide adequate comparison of the performance of each model to determine the chain's total proft.

Sensitivity Analysis
Sensitivity analysis determines how much the dependent variable will change if the value of an independent variable is changed in a specifc and defned situation and assumes that other variables are constant. Te values presented in Table 4 are changed here to determine the efect each increase or decrease will have on the fnal chain's proft chain's proft. Te sensitivity analysis is investigated here using all three models.
Based on Table 13, with the increase in the number of suppliers, the chain's proft has increased in all three models. In general, by doubling the number of suppliers, about 139.25% has been added to the overall proft of the chain (based on the analysis of the coordinated model, which performed best among other models). Tis is because, for example, by increasing the number of suppliers, the number of raw materials required for production increases, and the number of exchanges between the supplier and the manufacturer decreases, contributing to reducing the costs of the entire chain and increasing profts.
From Table 14, it can be observed that the average proft of the entire chain has increased with the increase in the number of raw materials. In other words, by doubling the number of raw materials, the proft of the entire chain has doubled by about 231.31. Because with the increase of the raw amount, in addition to the transfer costs, construction costs, and side costs have decreased, all of which indicate a reduction in the overall costs of the chain for all actors and an increase in the chain's proft by doubling the number of w.
Based on Table 15, with the increase in the number of manufactured products, the chain's proft also increases. In other words, by doubling the number of products (m), the chain's proft has increased by 88.40%. In this case, the chain costs and the actors' profts have increased. In addition to the reasons mentioned concerning the increase in the value of w, this is due to the fact that the costs related to the decrease in the movement and the number of customers, and consequently, the agreement between the actors have increased, which results to more proft to the chain.
From Table 16 for the sensitivity analysis of the number of retailers, it can be observed that with the increase in the number of retailers, the total cost of the network has decreased, and its proft has increased. In other words, by doubling the number of retailers, about 38.03% has been added to the overall proft of the chain.

Discrete Dynamics in Nature and Society
Based on Table 17, the increase in the allowed delay has caused an increase in the overall chain's proft because this delay increases the amount of production, and as a result, the income from the sale of more products increases. Tis increase is smaller than the other factors mentioned above. Specifcally, Annual proft for each factor i 3-5%           by doubling the allowed delay time from 3 to 6 months, the proft of the entire chain has increased by 19.32%. Table 18 investigates the sensitivity analysis of the settlement time. It can be observed that with the increase in the settlement time, the total proft of the network has increased by 4.42%.
Based on the analysis done in Tables 19-22, by doubling the setup/ordering cost, production/purchasing cost, fnancial holding cost per unit, and holding (storing) cost per unit, the total chain's proft increase by 4.07%, 0.89%, 0.048%, and 0.27%, respectively.          Discrete Dynamics in Nature and Society As shown in Figure 8, the number of raw materials has the greatest impact on the overall costs of the supply chain and the proft of actors.

Conclusion
Nowadays, the gas and oil industry has caused fundamental changes in regional growth and development. Moreover, today's need for oil products and their various functions in diferent felds, and existing potentials have led to signifcant investments in this area. Regarding this, the frst step is to provide raw materials for the production of oil derivatives by conducting the necessary operations. If these materials can be provided in a shorter time and with adequate quality, the supply chain will be more efective and the organization's objectives can be met. Te main objective of this paper was to increase the efectiveness of the supply chain plan and create coordination between the components. A three-level mathematical model was used to decrease the chain's total costs and increase its total proft. After verifying the model through GAMS to examine the results of each chain echelon's cost, income, and proft, three centralized, decentralized, and coordinated models based on game theory were used to solve the model. Comparing the results, it was found that the coordinated model has better performance in reducing the chain's costs and boosting profts than the other models. In particular, it can be stated that the coordinated model outperforms the decentralized model by 136.38% and the centralized model by 17.63%, while it increases the chain proft too. Meanwhile, it was observed that the decentralized model spent more time solving the model, than the other two. Te coordinated model has the advantage of assessing the computation solution space for the agents with greater precision, and it can present a higher proft for each of the echelons in the chain. Tese models can be extended in several possible directions. First, consider a specifc system of transportation or distribution to move the products from the centers to the fnal destination (customer). Second, create a backup for bottle distribution centers that, if the manufactured product is unavailable, receive the requested product with compensation from other distribution centers. Tird add other supply chain echelons to the model.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors certify that there is no confict of interests (considering both fnancial and nonfnancial gains) with any organization regarding the material discussed in the paper. Tis study did not receive any funding in any form.