Modeling the leader-follower supply chain network under uncertainty and solving by the HGALO algorithm.

The purpose of this article is to develop a competitive supply chain network (SCN) in the face of uncertainty. The objective of the leader chain is to maximize total network profits by strategically locating suppliers, manufacturers, distribution centers, and retailers. Additionally, the follower chain seeks to maximize the network's profit. Both factors, optimal flow allocation to different echelons of the SCN and product pricing, are examined in the leader chain and follower chain. The KKT conditions are used in this article to convert a bi-level model to a one-level model. Additionally, a fuzzy programming technique is used to control the problem's uncertain parameters. According to the results obtained using the fuzzy programming technique, increasing the uncertainty rate increases demand while decreasing the OBFV and average selling price of products. Finally, the problem was untangled using a novel hybrid genetic and ant-lion optimization algorithm (HGALO). The results of problem solving in larger sizes demonstrate HGALO's superior efficiency in comparison with the other algorithm.

Introduction

A supply chain (SC) is a collection of independent companies that encompasses the entire process of acquiring and converting raw materials into finished goods, as well as maintaining, distributing, and transferring them to end customers. Nowadays, researchers emphasize competition both within and across chains (Zhang et al. 2017). Customer competition patterns have altered product and service supply chains, resulting in a shift in market competition between independent businesses and supply chains (Ghomi-Avili et al. 2018). There is a substantial body of literature on the SC design, the majority of which is devoted exclusively to the SC. Until now, the presence of competitors in SC, as well as their possible future emergence, has been inadequately considered. SC, on the other hand, competes for market share (Wang et al. 2016). As a result, even if there is no immediate competitor present, SC must be prepared for future competition (Saberi et al. 2018). Numerous studies have been conducted in the literature to evaluate games between different echelons of the SCN. A non-cooperative game is a type of game in which the buyer and seller compete for the highest possible profit margin (Li and Nagurney 2017). In a non-cooperative Stackelberg game, the strongest player assumes the role of leader and formulates his strategy first, followed by the follower (Yue and You 2017).

To address the aforementioned issue, a new model of the leader–follower network was developed with the goal of optimizing the profit of the leader and follower chains. As a result, this game can be classified as a distinct subgenre of Stackelberg. In the leader and follower SC, four echelons (suppliers, manufacturers, distribution centers, and retailers) are considered, and they compete for product pricing in the market (customers). Due to the uncertainty surrounding the amount of potential market demand, pricing in both leader and follower chains is extremely complicated. As a result, special tools such as fuzzy programming are used to control the potential demand parameters as well as the costs of product and raw material transportation between different echelons of the chain network (Szmelter-Jarosz et al. 2021; Nozari et al. 2022). The presence of strategic decisions in the leader chain is what distinguishes this paper's leader and follower chain networks. The location of facilities at all potential centers, such as suppliers, manufacturers, distribution centers, and retailers, should be made part of the leader chain's strategic planning. The optimal transport of products and raw materials between the two SCN echelons is a tactical decision in the leader and follower chains. Finally, operational decisions include pricing final products in the market (for customers) based on the leader and follower chains in order to make the most money in an uncertain environment.

The leader and follower members use the traditional (forward) SCN. As a result, a mathematical model of SC on two levels will be developed. In the paper presented here, we used the KKT technique to convert a bi-level mathematical model to a one-level mathematical model. Because location and SCN design problems are NP-Hard, this paper solves the problem using a novel combination of genetic algorithms (GA) and ant-lion optimization algorithms (ALO). The operators of the ant-lion optimization algorithm design the chromosome of the problem by locating potential facilities and allocating products optimally between different echelons of the leader and follower chains, whereas the genetic algorithm operators price the products in the market. Finally, statistical comparisons between the developed algorithm and other algorithms such as the genetic algorithm (GA), the ant-lion optimizer (ALO), the gray wolf optimizer (GWO), and the Harris-hawks optimizer (HHO) are made.

The article is structured as follows. The second section discusses the research background and the investigation of the research gap. The third section discusses the bi-level mathematical model of SCN. This section discusses a fuzzy programming technique for controlling uncertain parameters. The fourth section presents a novel combination of GA and ALO algorithms, as well as the initial chromosome design, for the problems of facility location, optimal product flow allocation, and market pricing of products. The fifth section investigates exact and meta-heuristic solution methods, as well as statistical comparisons, in small and large instances. The sixth section will examine the presentation of the research findings and recommendations for future research.

Literature review

Numerous studies have concentrated on the presentation of competitive single- and multi-product games between different SCN echelons as well as product pricing within each firm. Wu et al. (2012) examined five different power scenarios for a one-seller, two-buyer SC: (1) The seller, as the chain's leader, announces its price first, and then the retailers determine their profit margins in competition with one another; (2) The seller, as the chain's first leader, makes the decision, and then retailer No. 1 and then retailer No. 2 announce their decision in response to the seller's price; (3) The retailers determine their profit margins simultaneously, and then the seller makes the decision as a follower. (4) Retailer No. 1 is regarded as the market's initial leader. Then retailer No. 2 and the seller make their decisions sequentially, and 5) Retailer No. 1 serves as the leader, with the other seller and retailer making their decisions concurrently. Leng and Parlar (2010) investigated the Stackelberg game in a chain with multiple suppliers and a single follower manufacturer operating under repurchase agreements and sharing lost sales costs. The manufacturer establishes the contract parameters first, and then the suppliers vote in a concurrent game. Cia et al. (2011) presented a SC with a seller and several buyers confronted with the newsboy problem, in which buyers order during the sales season, causing uncertainty. As a leader, one of the buyers may place an order earlier in the season and receive a discount. Two distinct scenarios are considered: (1) the leader buyer has a single ordering opportunity, and (2) the leader buyer has two ordering opportunities. Zhang and Huang (2010) examined a SC involving a manufacturer and a coalition of suppliers. A common platform is used to create a family of products with interchangeable modules. As the customer and chain leader, the manufacturer initially decides on supplier selection and platform configuration. The suppliers then determine the price and quantity of the order in order to maximize the coalition's overall profit.

Liu et al. (2010) evaluated a SC with a manufacturer who distributes its products to the market via two online retailers. The level of access to demand information varies between the two sales channels, and the game is analyzed using contracts in both centralized and decentralized situations. The findings indicate that when uncertainty is high, the manufacturer prefers a basket of contracts, whereas when uncertainty is low, he selects only one contract. Farahani et al. (2014) analyzed over 200 papers and classified them according to the type of competitive SCN design, solution methods, models, and applications. This article establishes a sound foundation for future research. Pritchard (2007) devised a bi-level algorithm for solving the competing SCN problem, in which two chains compete without the presence of a competitor chain. The proposed algorithm was developed using the Lemke and Howson algorithms as a starting point. They then applied the developed model to a real-world case study and analyzed the data. Saghaeeian and Ramezanian (2018) described a multi-product competitive supply chain network (SCN) organization. They began by expanding a bi-level mathematical model of SC members in this paper. They then transformed the competitive chain model into a one-level mathematical model using the KKT method. Finally, they used a hybrid genetic algorithm to solve the problem.

Rezapour et al. (2015) proposed a closed-loop SCN that is price-demand-dependent. Their role in this network was to help the SCN increase its profits. Their model considered strategic objectives such as facility location, tactical objectives such as optimal goods flow allocation, and operational objectives such as product pricing. Finally, they presented a two-step algorithm for resolving the issue. Amiri et al. (2018) developed a bi-level mathematical model for a competitive SC that takes predictive variables into account. They considered using an iterative search method to address the leader's issue. The numerical results indicated that omitting competition discussions results in a loss of market share and profitability for a new competitor's SC. Hassni (2016) proposed a comprehensive model for developing a SCN with multiple periods, multiple products, and multiple stages of inventory control. In the face of static competition, he considered various marketing techniques and methods during the design process. The purpose of the presented model was to achieve an effective customer response in the presence of existing competitors and a lack of demand-meeting prices. For solving his two-objective model, he used the NSGA II algorithm. He indicated that the proposed model and solution algorithm effectively deal with competitive pressure, most notably through the use of an effective marketing strategy. Nobari et al. (2019) presented a two-objective model for developing a competitive closed-loop SCN for new businesses. In the presented paper, they examined competitiveness in both forward and reversed networks concurrently. In the forward chain, competitors compete on selling prices in order to gain a greater share of the market, whereas in reverse logistics, competitors compete on incentive purchase prices. They solved their problem through multi-objective colonial competition.

Sarkar and Bhadouriya (2020) proposed a centralized model for optimizing SC profits and compared it to a Stackelberg game between retailers and manufacturers. Li et al. (2020) presented a four-stage Stackelberg game model with the objective of determining the optimal information investment strategies. This article discusses a SC that is two-tiered, with one manufacturer investing in energy-saving products (ESPs) and another retailer selling the products and incorporating demand forecasting. Noh et al. (2019) proposed a two-tiered SC model that included a manufacturer and a retailer. Three approaches are proposed for resolving the SC's coordination: retailer leader–manufacturer follower, manufacturer leader–retailer follower, and centralized SC. Tang et al. (2020) used a Stackelberg game framework to demonstrate that by establishing a contract between the leader and follower, a CLSC can achieve the same return as a centrally coordinated channel. Ghomi-Avili et al. (2020) presented a robust bi-level mathematical model for a competitive green SCN that takes inventory and pricing decisions into account. By illustrating competition between the leader and follower chains, the bi-level mathematical model method was used, and the model's ability to deal with uncertainties was also used to analyze disruption risks. Nagurney (2021) develops a framework for SCN game theory that incorporates labor constraints through three distinct methods. Equilibrium status was recalculated in response to various imparity-related issues. A case study was then conducted to examine the relationship between COVID-19 and migrant labor.

Sazvar et al. (2021) proposed a bi-objective programming model for designing a CLPSC that examines expired medication reverse flows. Additionally, they controlled the uncertainty parameters via a scenario-based game theory model. A three-party sustainable supply chain network was proposed by Liu et al. (2021). The model was solved using the Nash equilibrium strategy in conjunction with a novel coordination technique. Mozafari et al. (2021) developed a mathematical model of pricing coordination in a two-echelon supply chain in the presence of uncertainty. They used fuzzy numbers to represent demand and production costs. Boronoos et al. (2021) developed a model for a multi-objective closed-loop green supply chain problem that is uncertain. Simultaneously, they used this model to reduce the costs of the entire supply chain network and greenhouse gas emissions in both forward and reverse supply chains. Due to the fact that transportation and operating costs, as well as triangular fuzzy numbers, are considered indefinitely in this demand model, a robust-fuzzy optimization method is used to control these parameters. Liu et al. (2021) proposed a pricing decision model for an uncertain-demand closed-loop supply chain network. They factored in critical decisions such as supplier selection and product flow optimization into their model. Ziari and Sajadieh (2022) proposed a closed-loop supply chain network model based on the pricing structure of the glass industry. Additionally, they investigated network disruptions in the study network. Keshavarz-Ghorbani and Arshadi Khamseh (2022) proposed a multi-period optimization model for a closed-loop supply chain network, taking into account quality control and pricing.

According to the literature review, various researchers have employed a variety of methods and algorithms to address supply chain network problems under a variety of different assumptions. In the majority of cases, well-known algorithms such as the genetic algorithm (GA) and the ant colony algorithm (ACO) have been used. Each algorithm utilizes a unique set of operators to search the solution space. Recent research has shown an increase in the use of population-based algorithms such as GWO, ALO, and others. Numerous studies have demonstrated that the use of population-based algorithms produces superior results. In this paper, taking advantage of the GA operators' strengths (crossover and mutation) and the ALO's high search speed, the operators of the two algorithms were used to search the problem more efficiently and effectively.

Due to the research gap, the manuscript's contributions are divided into two parts: modeling and solution method.

A competitive supply chain network model (leader–follower) based on the Stackelberg game was proposed under uncertainty.

The proposed bi-level model was converted to a one-level model using KKT conditions.

A fuzzy programming approach is used to control the problem's uncertain parameters.

A novel hybrid genetic and ant-lion optimization algorithm is designed for the model's solution.

Problem definition and mathematical modeling

We examine two competing supply chains, one of which is an incumbent whose location of facilities is critical, and the other of which is a new entrant. Each SC is composed of four echelons: suppliers, manufacturers, distribution centers, and retailers. These echelons collaborate to provide final products to a common market. The network of the SC is depicted in Fig. 1. The entrant SC must make several strategic, tactical, and operational decisions, including determining the locations of suppliers, manufacturers, distribution centers, and retailers, to name a few. At the tactical level, each SC determines how the products will be supplied and distributed to the final customers. They determine how much raw material each supplier should provide to each manufacturer, how many products each manufacturer should produce, and how the DCs and retailers should distribute it. Finally, the operational level decision is about product pricing.

Fig. 1

Proposed supply chain network

The supply chains compete for customers by offering substitutable products in a common market. We assume the incumbent SC is currently operating in the market and that its facility locations are known. As a result, the incumbent does not face such strategic choices. The incumbent, on the other hand, would make tactical and operational adjustments in response to the entrant. As a result, there is a foresight-based competition between the entrant and incumbent supply chains, which can be modeled as a leader–follower or Stackelberg game. Since the entrant initiates the game, anticipating and incorporating the incumbent's response, he or she is the leader. And the incumbent is a follower because he or she accommodates the entrant's tactical and operational decisions following entry. A leader–follower game can be modeled as a bi-level programming problem, as is frequently done in the relevant literature. As a result, our objective is to develop and solve a bi-level mathematical model of the considered problem.

Demand function

Demand functions are used to model customer purchasing behavior. There are several different types of demand functions described in the relevant literature, but the linear function is the most frequently used model due to its superior characteristics. Kurata et al. (2007) discuss the advantages of the linear demand function, which include proper tracking, traditional microeconomic analysis, and congruence with actual data. As a result of this advantageous characteristic, we chose to use a linear demand model. Customers are assumed to be price sensitive. As a result, we consider the following price-dependent linear demand function.where the superscripts L and F denote, respectively, the leader and follower. Additionally, the following notations are defined.

The demand for each product from a customer segment m is proportional to its price and the price of its complement products in this demand model. And it is increasing in lockstep with the price of the rival SC's substitutable products.

We permit a product's price to vary across market segments. That is, it is assumed that third-degree discrimination exists in pricing, and chains can charge different prices to different customer segments. The model of supply exceeding demand is prevalent in the pertinent literature.

Mathematical model

The new SC has been designated as the market leader, whereas the incumbent SC has been designated as the follower. Both SCs are divided into four echelons. Suppliers in the first echelon provide raw materials to manufacturers for the creation of new products. The second echelon is made up of numerous manufacturers who ship their new products to the third echelon's DCs. Finally, retailers in the fourth echelon receive products from DCs and resell them to customers in a variety of markets. While the incumbent SC's location of suppliers, manufacturers, DCs, and retailers is critical, the entrant SC should determine the locations (with pre-determined capacity) of its various members. A horizontal competition between two SCs at the retailer level has been considered. Both of them have the ability to influence the other's demand through pricing decisions. Due to capacity constraints, the chain of followers has already been formed, and it is capacity unrestricted. The network structure of the proposed model is depicted in Fig. 1. The problem has been classified as a bi-level mathematical model due to the Stackelberg game. The upper level presented the problem of the first model, while the lower level defined the problem of the follower. Profit maximization is the primary objective of both chains.

The following are the primary assumptions that guided the development of the mathematical model:

At both levels, demand for each product in each market is a linear function of self-price, competitor's price, and also the complement-price of their own chain.

The leader's primary concern has been the need for new facilities.

No shortage of the leader's chain is permitted.

Depending on the price and transportation costs, the probable behavior of customers may result in different demands for retailers.

Each customer can purchase any product in any quantity from any retailer.

In the third stage, the transport price is equal to the distance between each retailer's various markets for both chains.

The chain of the follower does not have a budget constraint and is also incapacitated.

Costs of production and transportation at the upper and lower ends of the spectrum are considered uncertain.

The model is presented using the notations below.

Sets

Parameters

Decision variables

The bi-level mathematical model of the proposed problem is introduced as below.

The upper level (leader’s model)

The lower level (follower’s model)

The objective function (3) seeks to maximize the profit of the leader. Constraint (4) depicts the leader's demand function. Constraints (5)–(8) illustrate the volume of raw materials and products transported between each of the two leadership chain facilities. Constraints (9)–(12) relate to the capacity of network facilities in the leader chain. Constraints (13) and (14) define the range of decision variables in the leader chain.

The objective function (15) seeks to maximize the profit of the follower. Constraint (16) depicts the follower chain's demand function. Constraints (17)–(20) indicate the volume of material or products transported between each of the two follower chain facilities. Constraints (9)–(12) are pertinent to the network's facilities' valence in the leader chain. Constraints (13) and (14) define the range of decision variables in the leader chain. Restriction (21), on the other hand, regulates the demand for leader and follower chains. Therefore, they cannot exceed the potential demand. Restriction (22) defines the leader chain's positive decision variables.

KKT conditions

In this section, we have introduced the Karush–Kuhn–Tucker (KKT) conditions for the bi-level mathematical model. With regard to the KKT conditions for a bi-level mathematical model, the primary necessary condition for the lower-level problem is that its constraints are convex.

Due to KKT conditions, the convexity of its restriction is critical for the lower-level problem and the concavity of its objective function in a bi-level mathematical model. By incorporating the follower's KKT constraints at the upper level, the optimal strategy of the follower is guaranteed if these conditions are met.

The following is a typical bi-level mathematical model (Gümüş and Floudas 2005).

The above bi-level mathematical model is transformed into the following one-level mathematical model using the KKT conditions (Mokhlesian and Zegordi 2014).where denotes the number of dual variables, denotes the Lagrangian multipliers, and denotes the decision variables. Equation (24) satisfies the optimal conditions of the lower-level problem or of the model's followers. For the leader's problem, the Nash equilibrium tactics are obtained by assuming that the follower employs the optimal tactics. Using the KKT conditions, we can convert the bi-level model to a single-level model. As a result, we can be confident that the lower-level optimum solution is global in nature, as defined by the convex constraints and concave objective function. The converted KKT one-level mathematical model is shown below.

Fuzzy programming model

Consider the following linear mathematical programming model with fuzzy parameters (Yildizbaşi et al. 2018):where are the fuzzy parameters used in the problem's objective function, the vector coefficient, and the right-side parameter of the constraints? The probability distribution function of fuzzy parameters is assumed based on the properties of fuzzy numbers. Finally represents the decision vector. For the feasibility and optimization of the problem presented in the above model, it is necessary to control the uncertain parameters given in the objective function and constraints. Hence, assuming parameter as the minimum feasibility of constraints, the controlled model is as follows: is the anticipated value of the fuzzy membership function used in the objective function (OBFV) of the model, which is calculated as follows:

In this paper, the fuzzy parameters are intended as triangular fuzzy, as illustrated in Fig. 2. Figure 2 indicates the possibilistic distribution of the fuzzy parameter . , and represent the optimistic, probable, and pessimistic values of the fuzzy number , respectively, which are characterized by the decision-maker.

Fig. 2

The possibilistic distribution of the fuzzy parameter C

Thus, the expected value (the anticipated value of the fuzzy parameter of the OBFV) can be calculated in the following manner:

As mentioned previously, uncertain parameters include the potential demand, upper and lower-level transport costs, as well as upper and lower-level production costs. As a result, the stated uncertain parameters are controlled using a triangular fuzzy programming method.

As a result of the expressed relationships, we can write the fuzzy programming model as follows:where are dual variables and is the Lagrangian multiplier.

The proposed leader–follower model produces a MINLP. Numerous studies have established the NP-hardness of the SCN design problem (Jayaraman et al. 2003). The developed model addresses two distinct problems: location and allocation. As a result, this model can be reduced to the facility location problem, which has been shown to be NP-hard (Davis and Ray 1969). As a result, the aforementioned leader–follower SCN problem is introduced in this study as NP-hard. Accurately resolving this problem through precise solutions is time-consuming and frequently impractical. Thus, numerous meta-heuristic algorithms with various representations have been proposed to achieve near-optimal solutions, but they are inefficient. The following section describes the HGALO algorithm.

Hybrid genetic ant-lion optimization algorithm

The ant-lion algorithm mimics the hunting mechanism of the ant lion and interacts with the bait, the desired ant, and imitates them all (Mirjalili 2015). As with other population-based algorithms, the ant-lion algorithm approximates optimal solutions to optimization problems by promoting a random set of solutions inspired by the ant-lion interaction. The ant-lion algorithm has two populations: ants and ant lions. This article discusses how to improve the approximate optimal solutions for leader–follower SCN design by utilizing crossover and mutation operators. The ant-lion algorithm's general stages for changing these sets and, finally, for global optimization estimation are as follows:

The ant set is initiated using random values, which is the most critical factor in determining the ant-lion algorithm.

The value of each ant's fit is determined in each replication using the (OBFV).

The ants in the search area are scurrying about the ant lions at random.

The crowd of ant lions is never evaluated. The ant lions are supposed to be positioned in the first reps and will move toward the new position of the ants remaining in the repeats; of course, if they improve, they will move toward the new position of the ants remaining in the repeats.

An ant is assigned to one ant lion, and its position is revealed as the ant gains fitness.

There is an ant lion nearby that, regardless of its distance, has an effect on the ants' movement.

If any ant lion is found to be preferable to the selected ant, it will be replaced.

The stages (B) to (E) are repeated until the final criterion is deemed satisfactory, and

For global optimization, the fitting position and value of the selected ant lion are returned as the most accurate estimate.

Ants' primary role is to inspect the search area. They should proceed randomly through the search area. Ant lions select the best ants and direct them to promising search areas. To solve the problems optimally, the ant-lion algorithm emulates the ants' random steps, immersion in the ant-lion cavity, creation of the cavity, movement of the ant to ant lions, catching the bait, repairing the cavity, and selection of the preferred one. The following par diagrams illustrate the model and programming modules associated with each step. The randomized step used in the ant-lion algorithm to simulate the random steps of an ant is as follows:where cumsum denotes the cumulative sum, n denotes the maximum number of repeats, and t denotes the step size of a random step.t denotes a random step (in this paper, repetition) and rand denotes a random number of births distributed uniformly within the interval [0,1]. To maintain a random step on the search space's boundaries and avoid over-hunting, random steps must be normalized to the following equation.where is the least significant variable in repetition , is the maximum variable in repetition , is the min of random variable , and is the max random step in the variable . Ant-lion algorithm simulates the ants' clogging in the ant-lion hole by changing the ant-lions' random steps. The following equations are presented in this regard:where is the minimum variables in repetition , is a vector containing the maximum of all variables in repetition , is the min variables for variable , is max of all the variables for ant , is the position of the ant lion chosen in repetition .where I is a ratio, is min of all variables in repetition , and shows the vector representing the maximum of all the variables in repetition . In the above equations , in which t is the current repetition, is the max number of repetitions and w's definition is based on the current replication ( when , when , when , when and when ). Parameter in the equation can be used for adjusting the level of operation accuracy. The 2nd stage up to the last stage of the ant-lion algorithm is to catch the ant and rebuild the pit. The following equation stimulates this process:where shows the current repetition, indicates the position of the selected ant lion in repetition , indicates position in repetition . The last operator in the ant-lion algorithm is the elitism in which the best-fitted formed ant lion is stored during the optimization. This is the single ant lion which can affect all ants. This means moving the selected ant lion (randomly selected using a Rolette wheel) to the ant lion. The equation follows both of them:where shows the position of ant on repetition . is random motion in the selected ant lion by the roulette wheel in repetition and is the random motion around the elite in repetition .

In addition to the solution search method described previously, the crossover and mutation operators were used to obtain near-optimal solutions. The operator with two-point crossover is illustrated in Fig. 3.

Fig. 3

Two-point crossover operator

Two crossover points are chosen at random from the parent chromosomes in the two-point crossover. Between these two points on the parent's chromosome, the genes are swapped. Figure 4 illustrates the mutation operator's performance.

Fig. 4

Mutation operator

This operator substitutes a random number for the selected gene. The genetic algorithm's crossover and mutation operators have been used in accordance with the presented contents and in order to move the ant lions toward the near-optimal solution.

According to (Mirjalili et al. 2017), the ALO algorithm has a computational complexity of O(mn), whereas the GA algorithm has a computational complexity of O(2 mn) (Lobo et al. 2000), where n is the population size and m is the individual size. This is equivalent to the HGALO's computational complexity being O (3 mn). GWO, on the other hand, has an O(mn) computational complexity, whereas HHO has an O(m) computational complexity (Mirjalili et al. 2014; Heidari et al. 2019).

Figure 5 illustrates the proposed HGALO algorithm's flowchart and how to use the two algorithms' operators.

Fig. 5

The flowchart of HGALO

The following section discusses a chromosome that solves the leader–follower SCN design problem.

Designing the chromosome for leader–follower SCN design

As it is shown in Fig. 6, consider an echelon of SCN with sources, depots and products. This chromosome's length is and each cell's location represents the priorities of each node (Ghahremani et al. 2019). For example, Fig. 6 shows a chromosome with 3 sources, 4 depots, and 2 products. Also, in this figure, the demand of each product for depots, potential capacities for sources, and transportation costs between nodes are shown.

Fig. 6

A sample of SCN chromosome

The chromosome is decoded in two steps:

Step 1. For the first product, choose the source with the highest priority. If the potential capacity of the chosen sources exceeds the sum of all warehouse demand, the priority of non-selected sources is reduced to zero. Alternatively, assign the highest priority to non-selected sources. Continue in this manner until the capacity of all selected sources exceeds the demand of all depots.

This step specifies the location of the resources to be selected.

Step 2. Determine the highest-priority position among the nodes. If the node is located between two sources, refer to (A); otherwise (B).

Determine the cheapest mode of transport between the selected source and all depots. The optimal flow between two nodes is the smallest of the two (depot demand and source capacity). Reduce the priority of the node to zero if the value of depot demand or source capacity becomes zero.

Determine the least expensive mode of transportation between the selected depot and all sources. The optimal flow between two nodes is the smallest of the two (depot demand and source capacity). Reduce the priority of the node to zero if the value of depot demand or source capacity becomes zero.

Step 3. Repeat this process until all priorities are equal to zero.

Finally, repeat the process for all products.

The decoding of the example presented in Fig. 6 is depicted in Fig. 7.

Fig. 7

Decoding the sample chromosome

The pseudo-code of decoding the two-echelon SCN

As illustrated in Fig. 7, sources 1 and 2 are located optimally. The following is the pseudo-code for decoding the two-echelon SCN design:

As previously stated, this is a multi-echelon, multi-product, leader–follower SCN design problem, and the initial proposed solution must address these factors. As illustrated in Fig. 9, the priority-based encoding is represented by a matrix, in which M, P, C, I, J, K, and L represent the number of customers, products, raw materials, suppliers, potential DCs, production centers, and potential retailers in the leader SC, respectively. Additionally, I', J', K', and L' denote the number of production centers, suppliers, potential DCs, and potential retailers in the following SC.

Fig. 9

The final solution of the multi-echelon SCN

The designed leader and follower SCN's search area is discrete, which means that no individual from the population components can have an arbitrary value. Allowable values are limited to natural numbers between 1 and N. As a result, the continuous search area in the HGALO algorithm must be changed to a discrete search area. In the solution search area, Fig. 10 illustrates an example of a shipment.

Fig. 10

An example of shipment in the solution search area

Select the largest number in the continuous space of the chromosome in Fig. 10. This number corresponds to the first gene in the new solution. Continue checking the numbers in this manner until all of them have been checked. Figure 11 illustrates the pseudo-code for decoding the leader–follower multi-echelon SCN design.

Fig. 11

The pseudo-code of decoding the leader–follower multi-echelon SCN design

Numerical results

In the preceding section, we examined how to model the leader–follower SCN problem under uncertain conditions and how to control its uncertainty parameters using a fuzzy programming approach (demand, transportation, and purchase costs). The exact method (using Baron Solver) and algorithms are used in this section to evaluate the model and compare the output variables of the problem. The primary objective of this section is to evaluate the HGALO's efficiency in obtaining the near-optimal solution. As a result, a small sample problem is designed first, followed by the solution of the proposed model using Baron Solver. The problem's sensitivity analysis is also discussed below. After solving the model with meta-heuristic algorithms, the performance of the HGALO algorithm is compared to that of the GA, GWO, ALO, and HHO algorithms.

Solving the small size sample problem

A small sample problem is designed in this subsection to evaluate the presented model and compare the output variables. It encompasses two distinct product categories, two distinct raw material categories, four suppliers, four manufacturing facilities, four distribution centers, five retailers, and six customers.

Due to the lack of available benchmarks in the literature, random data from a uniform distribution is generated using the MATLAB software, as illustrated in Table 1.

Table 1

The parameters of the boundaries of production based on uniform distribution

Parameter	Range interval
FixI_i,FixJ_j,FixK_k,FixL_l	∼U(100000,120000)
CapI_ic	∼U(1500,1800)
CapJ_jp,CapK_kp	∼U(1000,2000)
CapL_lp	∼U(500,700)
O_cp	∼U(1,2)
bmppF	∼U(0.3,1.5)
bmppL	∼U(0.4,1.8)
dmpL	∼U(0.5,0.9)
dmpF	∼U(0.3,0.8)
TrIJ~ijcL,TrJK~jkpL,TrKL~klpL,TrLM~lmpL	∼U5,10,U10,15,U15,20
Pc~jpL	∼U2,4,U4,6,U6,8
TrIJ~i^^′j^^′cF,TrJK~j^^′k^^′pF,TrKL~k^^′l^^′pF,TrLM~l^^′mpF	∼U7,12,U12,18,U18,25
Pc~j^^′pF	∼U3,5,U5,8,U8,10
a~mpF	∼U200,300,U300,400,U400,500
a~mpL	∼U350,500,U550,700,U700,900

Due to the model's nonlinearity, Solver Baron is used on the GAMS 24.8.5 software. The numbers and locations of potential facilities, as well as the optimal flow between each node, are shown in Fig. 12 and Tables 2 and 3, respectively, when uncertainty rates () are taken into account.

Fig. 12

The location of the potential facilities and allocation between each node

Table 2

The optimal flow between each node in the leader chain

Decision variable	Amount	Decision variable	Amount	Decision variable	Amount	Decision variable	Amount	Decision variable	Amount
U2,1,1L	421	S2,2,1L	812	Y1,2,1L	1205	X211L	1172	Pr1,1L	526.19
U4,1,2L	405	S2,2,2L	665	Y1,2,2L	1498	X212L	1207	Pr1,2L	424.78
U2,3,1L	391	S2,4,1L	393	Y3,3,1L	1964	X231L	833	Pr2,1L	904.07
U2,3,2L	665	S2,4,2L	833	Y3,3,2L	1321	X232L	1386	Pr2,2L	909.76
U3,2,1L	640	S3,3,1L	1964			X331L	1164	Pr3,1L	517.13
U3,2,2L	497	S3,3,2L	1321			X332L	1527	Pr3,2L	377.2
U3,4,1L	678					X312L	1518	Pr4,1L	797.71
U3,4,2L	444							Pr4,2L	97.76
U3,6,1L	646							Pr5,1L	523.32
U3,6,2L	380							Pr5,2L	830.41
U4,5,1L	393							Pr6,1L	371.22
U4,5,2L	428							Pr6,2L	144.4

Table 3

The optimal flow between each node in the follower chain

Decision variable	Amount	Decision variable	Amount	Decision variable	Amount	Decision variable	Amount	Decision variable	Amount
U2,1,1F	318	S2,2,1F	1014	Y3,2,1F	1014	X231F	1014	Pr1,1L	227.63
U2,1,2F	637	S2,2,2F	1120	Y3,2,2F	1120	X232F	2240	Pr1,2L	972.3
U2,2,1F	696	S3,4,1F	1119	Y4,3,1F	1994	X441F	1994	Pr2,1L	63.23
U2,2,2F	483	S3,4,2F	853	Y4,3,2F	1624	X442F	3248	Pr2,2L	252.57
U4,3,1F	460	S3,5,1F	875					Pr3,1L	624.87
U4,3,2F	367	S3,5,2F	771					Pr3,2L	401.07
U4,5,1F	659							Pr4,1L	676.65
U4,5,2F	486							Pr4,2L	533.49
U5,4,1F	404							Pr5,1L	381.99
U5,4,2F	383							Pr5,2L	440.15
U5,6,1F	471							Pr6,1L	280.67
U5,6,2F	388							Pr6,2L	326.45

According to Fig. 12, the optimal locations are suppliers with numbers 2 and 3, production centers with numbers 1 and 3, distribution centers with numbers 2 and 3, and retailers with numbers 2, 3, and 4. The objective function value (OBFV) of the proposed model is 1989577.94, and the optimal flow between each leader–follower chain is shown in Tables 2 and 3. Additionally, this table shows the prices charged by retailers in the leader–follower chain.

Sensitivity analysis

After verifying and validating the proposed model, it is necessary to analyze its sensitivity. To do so, an uncertainty rate is chosen, and the resulting changes in the OBFV are shown in Table 4.

Table 4

Changing of the OBFV through adjustment of the uncertainty rates

Uncertainty rates (α=β)	OBFV	Average selling price of products (leader chain)	Average selling price of products (follower chain)
0.9	1927883.3	500.71	397.56
0.8	1934265.5	507.87	410.97
0.7	1943648.4	516.90	418.74
0.6	1964793.3	525.17	423.94
0.5	1989577.9	535.32	431.75
0.4	2027543.2	545.67	440.34
0.3	2063654.2	550.94	453.17
0.2	2094678.5	555.47	462.84
0.1	2124876.8	556.24	473.67

According to Table 4, as the uncertainty rate increases, the demand for products increases, while the OBFV and average selling price of products decrease. Figure 13 depicts the trend of changes in OBFV at various levels of uncertainty.

Fig. 13

Changes in the OBFV and average selling price of products by changing the uncertainty rates

Tuning the meta-heuristics algorithms by Taguchi method

Prior to solving the problem with meta-heuristic algorithms, the Taguchi method was used to tune the parameters of the GA, HGALO, ALO, HHO, and GWO algorithms. Prior to beginning the Taguchi process, it is necessary to identify the relevant factors. Then, the appropriate levels for each factor should be determined, followed by the appropriate test design for these control factors. After establishing the design of the experiment, the experiments are conducted and the results are analyzed to determine the optimal parameter combination. Three levels are considered in this paper for each factor in accordance with Table 5. We can determine the design and implementation of experiments for each algorithm by considering the number of factors and the numbers of their levels. Given that the proposed model is a single objective function, we use the RPD index to tune the parameters during the data analysis.

Table 5

The value of proposed and optimized parameters of meta-heuristic algorithms

Algorithm	Parameter	Level one	Level two	Level three	Optimum level
GA	Maxit	50	100	200	200
	N-pop	50	100	200	100
	P_c	0.7	0.8	0.9	0.7
	P_m	0.03	0.05	0.07	0.05
HGALO	Maxit	50	100	200	200
	N-ant	50	100	200	100
	b_i	500	1000	1500	1500
	a_i	200	300	400	400
	P_c	0.7	0.8	0.9	0.8
	P_m	0.03	0.05	0.07	0.05
ALO	Maxit	50	100	200	200
	N-ant	50	100	200	200
	b_i	500	1000	1500	1000
	a_i	200	300	400	300
GWO	Maxit	50	100	200	200
	N-Wolf	50	100	200	200
	A	1	2	3	3
	C	1	2	3	2
HHO	Maxit	50	100	200	200
	N-pop	50	100	200	100
	Strategyrate	0.05	0.1	0.15	0.05

Solving sample problems in small size

This subsection evaluates the meta-heuristic algorithms used to solve the previous section's small size example. The purpose of this analysis is to determine how algorithms converge toward a near-optimal solution and its output variables. Thus, a sample problem is considered that includes two types of products, two types of raw materials, four suppliers, four manufacturing centers, four distribution centers, five retailers, and six customers. After solving the numerical example using meta-heuristic algorithms, the algorithms' convergence toward achieving the OBFV is determined, as illustrated in Fig. 14.

Fig. 14

The convergence of the algorithms in achieving the OBFV

As illustrated in Fig. 14, the HGALO algorithm converges faster and has a greater OBFV than other algorithms. Table 6 shows the OBFV obtained by solving a numerical example and comparing it to Baron's solver.

Table 6

The OBFV obtained from different solution approaches

Algorithm	OBFV	Gap between baron solver and algorithms (%)	CPU-time
Baron solver	1989577.94	–	84.25
HGALO	1989398.88	0.009	33.14
GA	1989100.44	0.024	27.26
ALO	1989458.57	0.006	25.94
GWO	1989299.40	0.014	20.34
HHO	1989219.82	0.018	26.47

According to the results of Table 6, the maximum gap is equal to 0.024 percent, indicating the algorithms' high efficiency in achieving the OBFV in less time than the Baron Solver. Additionally, the results indicate that meta-heuristic algorithms solve problems 60% faster than Baron Solver.

The optimal location and number of facilities for solving small numerical examples are shown in Table 7.

Table 7

The optimal location and number of facilities obtained from different solution methods

Algorithm	Suppliers	Production centers	DCs	Retailers
Baron solver	2–3	1–3	2–3	2–3–4
HGALO	2–3	1–3	2–3	2–3–4
GA	1–3	1–3	2–3	2–3–4
ALO	2–3	1–3	1–3	1–3–4
GWO	1–3	2–3	2–3	2–3–4
HHO	1–2	1–3	2–3	2–3–4

According to Table 7, all solution methods produce the same number of optimal locations. Finally, Table 8 illustrates the average selling price of products (Leader and Follower chains) when the uncertainty rate is set to 0.5.

Table 8

The Average selling price of products (leader and follower chain) obtained from different solution methods

Algorithm	Average selling price of products (leader chain)	Average selling price of products (follower chain)
Baron solver	535.32	431.75
HGALO	535.72	430.17
GA	535.24	429.37
ALO	536.14	431.20
GWO	534.94	430.25
HHO	534.86	430.86

Additionally, the results of Table 8 demonstrate the close proximity of the average selling prices of products (leader and follower chains) obtained using various solution methods in comparison with the Baron Solver.

Solving sample problems in large size

After evaluating the meta-heuristic algorithms, this subsection solves the leader–follower model using the HGALO, GA, ALO, GWO, and HHO algorithms. As a result, ten sample problems of varying sizes are designed to evaluate the performance of the proposed algorithms. Tables 9, 10, and 11 illustrate the size of the exemplified problems and the OBFV and CPU-time of the proposed algorithms (11).

Table 9

The sample problems size

Sample problem no	I	J	K	L	M	P	C
1	5	5	5	7	8	3	2
2	7	6	6	8	10	3	2
3	8	7	7	10	12	3	2
4	9	8	8	12	15	4	2
5	10	10	10	14	18	4	3
6	12	12	12	16	21	4	3
7	15	14	14	18	23	5	3
8	18	16	16	20	25	5	4
9	20	18	18	22	28	5	4
10	25	20	20	25	30	6	4

Table 10

OBFV obtained from solving sample problems, using different solution methods

Sample problem	OBFV
	HGALO	GA	ALO	GWO	HHO	(Baron solver)
1	2416876.1	2415789.6	2416556.7	2415824.2	2416090.5	2427541.6
2	2826497.4	2824613.6	2824700.6	2824796.6	2826164.8	2846237.1
3	3124685.5	3124259.7	3124555.6	3131394.7	3124664.3	3147820.3
4	3264975.6	3261974.3	3262077.7	3264291.1	3263119.5	–
5	3478945.7	3471165.1	3477121.3	3477352.2	3472619.1	–
6	3864472.3	3854687.7	3859479.8	3859047.6	3861011.6	–
7	3947651.2	3924686.4	3950976.8	3942017.6	3931025.3	–
8	4157666.4	4111794.6	4142973.8	4141845.1	4159253.9	–
9	4268794.3	4213428.6	4220017.0	4241020.9	4266565.5	–
10	4378261.4	4304789.8	4325278.2	4350018.1	4364981.6	–
Mean	3571554.6	3550718.9	3560373.7	3564760.8	3569877.5

Table 11

CPU-time obtained from solving sample problems, using different solution methods

Sample problem	HGALO	GA	ALO	GWO	HHO	(Baron solver)
1	67.1	54.4	55.5	46.4	52.4	126.47
2	71.2	58.6	65.4	56.6	62.3	347.68
3	80.3	69.8	80.4	67.3	73.5	846.82
4	91.7	80.6	94.7	85.2	88.4	> 1000
5	112.3	99.3	120.4	110.3	109.2	> 1000
6	140.4	122.3	148.5	134.1	134.5	> 1000
7	170.6	152.3	187.0	181.3	167.7	> 1000
8	210.3	190.0	237.6	229.8	208.0	> 1000
9	261.7	235.4	292.7	281.3	258.4	> 1000
10	328.7	290.7	364.0	353.1	329.4	> 1000
Mean	153.4	135.3	164.6	154.5	148.3	> 1000

According to the results of Tables 9, 10, and 11, the exact solution cannot be found by increasing the size of the problem. It is obvious that the CPU-time required to solve the model using the exact method is quite high, as the third sample problem requires 846.82 s. By comparing the meta-heuristic algorithms for solving large-scale sample problems, it is possible to deduce that the OBFV obtained from the HGALO algorithm was superior to the other algorithms in all sample problems. Additionally, when the RPD index of the algorithms is compared, it is observed that as the size of the problem increases, the GA algorithm loses its ability to obtain a near-optimal solution. Simultaneously, it consumes less CPU-time than the other algorithms.

Figure 15 illustrates the changes in the OBFV and CPU-time averages as a result of the proposed meta-heuristics algorithms in all sample problems.

Fig. 15

Changes in averages of the RPD % and CPU-time in large sample sizes

The T Test was used at a 95% confidence level to compare the significant difference between the OBFV and CPU-time obtained from the proposed algorithms. If the P value is less than 0.05, the hypothesis test indicates that there is a significant difference in the averages of that computational index between the algorithms used. T Test output results for large sample size problems using these algorithms are shown in Table 12.

Table 12

T Test outputs at 95% confidence level

Algorithm	Index	Means different	Confidence interval 95%		T value	P value
			Lower bound	Upper bound
GA-HGALO	OBFV	20,836	3725	37,946	2.75	0.022
GA-ALO		9655	1212	18,097	2.59	0.029
GA-GWO		14,042	2980	25,104	2.87	0.018
GA-HHO		19,159	− 634	38,951	2.19	0.056
ALO-HGALO		11,181	− 1819	24,181	1.95	0.084
ALO-GWO		4387	− 3134	11,908	1.32	0.220
ALO-HHO		9504	− 6948	25,955	1.31	0.224
GWO-HGALO		6794	− 349	13,936	2.15	0.060
GWO-HHO		5117	− 4678	14,911	1.18	0.268
HHO-HGALO		1677	− 3592	6946	0.72	0.490
GA-HGALO	CPU-time	18.09	11.96	24.21	6.679	0.000
GA-ALO		29.28	12.35	46.21	3.91	0.004
GA-GWO		19.20	2.18	36.22	2.55	0.031
GA-HHO		13.04	4.69	21.39	3.53	0.006
ALO-HGALO		11.19	− 0.19	22.57	2.22	0.053
ALO-GWO		10.08	8.272	11.88	12.61	0.000
ALO-HHO		16.24	7.28	25.20	4.10	0.003
GWO-HGALO		1.11	− 10.47	12.69	0.22	0.833
GWO-HHO		6.16	− 3.03	15.35	1.52	0.164
HHO-HGALO		5.05	1.97	8.13	3.71	0.005

The P value for the OBFV index is less than 0.05, indicating a significant difference between the averages of the OBFV index for the GA–HGALO, GA–ALO, and GA–GWO algorithms. Additionally, there is no discernible difference in the averages of this index between the other algorithms. Due to a lack of decision-making regarding the most efficient algorithm for solving the proposed mathematical model, the TOPSIS method for ranking algorithms has been discussed using two indicators of OBFV and CPU-time. The results of these comparisons are shown in Table 13.

Table 13

Algorithm performance ranking results

Algorithm	Mean of OBFV	Mean of CPU-time	Utility weight	Rank
HGALO	3,571,554.6	153.43	0.6435	1
GA	3,550,718.9	135.34	0.4618	5
ALO	3,560,373.7	164.62	0.5218	4
GWO	3,564,760.8	154.54	0.5517	3
HHO	3,569,877.5	148.38	0.5611	2

According to the results in Table 13, the HGALO algorithm obtained the optimal value of the objective function when compared to other algorithms. While the GA algorithm took less time to solve the model than other algorithms. However, based on the TOPSIS ranking results, it is observed that the HGALO algorithm, with a utility weight of 0.6435, is more efficient than other algorithms when both OBFV and CPU-time are considered. Apart from the scientific findings, the proposed model has the strongest correlation with the activities of companies manufacturing electronic and automotive components in the global SC. Within these networks, manufacturing firms compete fiercely for market share and product pricing. As a result, large and small businesses compete against one another under the leader and follower chains, respectively. Under conditions of uncertainty, the model presented in this paper can provide beneficial management results for managers of companies that manufacture electronic and automotive components, enabling managers to make appropriate decisions about product pricing, volume of production, distribution, and facility location.

Conclusion

The purpose of this paper was to develop a new leader–follower SCN model that maximizes both the leader and follower chains' profits. In the leader and follower SC, four echelons (suppliers, manufacturers, distribution centers, and retailers) are considered, and they compete for product pricing in the market (customers). The high degree of complexity associated with pricing the two leader and follower chains is a result of the uncertainty surrounding the amount of potential market demand. Special tools such as fuzzy programming are used to control the potential demand parameters and the transportation costs of products and raw materials between different echelons of the chain network. The difference between the leader and follower chain networks in this paper is due to strategic decisions made in the leader chain. The location of facilities at all potential centers, such as suppliers, manufacturers, distribution centers, and retailers, should be made part of the leader chain's strategic planning. The tactical decisions made in the leader and follower chains concern the optimal flow of products and raw materials between the two SCN echelons. Finally, operational decisions include pricing final products in the market (for customers) by leader and follower chains in order to maximize profit in the face of uncertainty.

In this paper, the KKT method is used to convert a bi-level mathematical model to a one-level mathematical model. Due to the NP-Hard nature of location and SCN design problems, this paper employs a novel hybrid of genetic algorithm (GA) and ant-lion optimization algorithm (ALO) to solve the problem. The ant-lion optimization algorithm operators design the problem's chromosome, locate potential facilities, and optimally allocate products between the different echelons of the leader and follower chains, while the genetic algorithm operators price the products in the market. Finally, statistical comparisons between the developed algorithm and the genetic algorithm are performed. According to the findings, as the uncertainty rate increases, demand also increases, while the OBFV and average selling price of products decrease. By comparing the meta-heuristic algorithms for solving large-size sample problems, it is possible to deduce that the OBFV obtained from the HGALO algorithm was superior to the GA algorithm in all sample problems. Additionally, the P value for both the OBFV and the OBFV is less than 0.05, indicating a significant difference between the averages of this index in the GA–HGALO, GA–ALO, and GA–GWO algorithms. Finally, the high OBFV of the HGALO algorithm in solving sample problems demonstrates the algorithm's superior efficiency in comparison with other algorithms. In addition to the proposed algorithm, other novel algorithms such as the whale optimization algorithm, the slap swarm algorithm, and others may be used to solve the introduced model. Additionally, it is suggested that researchers consider the facility location problem in the follower's model and control the uncertainty parameters using a robust possibilistic optimization method. Additionally, researchers can use a permutation-based encoding to encode the solution.

a~mp.	The uncertain potential demand of customer segment m for product p
bmhp.	The price sensitivity coefficient of customer segment m. If h=p, it is self-price sensitivity, and if h≠p, it is cross-price sensitivity and shows the sensitivity of customer’s demand from segment m for product p to the price of product h. The products p and h are complementary and produced by the same SC
dmp.	The cross-price sensitivity to the price of the other SC. That is, dmpF shows the sensitivity of customer’s demand from segment m for product p of the leader SC to the price of product p of the follower SC. And, dmpL represents a similar concept in the followers’ demand model
Prmp.	The price of product p for customer segment m

I	Sets of suppliers in the leader chain i={1,2,⋯,I}
J	Sets of potential manufactures in the leader chain j={1,2,⋯,J}
K	Sets of potential DCs in the leader chain k={1,2,⋯,K}
L	Sets of potential retailers in the leader chain l={1,2,⋯,L}
M	Sets of fixed markets (Customers) m={1,2,⋯,M}
I^^′	Sets of suppliers in the follower chain i^^′={1,2,⋯,I^^′}
J^^′	Sets of manufactures in the follower chain j^^′={1,2,⋯,J^^′}
K^^′	Sets of potential DCs in the follower chain k^^′={1,2,⋯,K^^′}
L^^′	Sets of potential retailers in the follower chain l^^′={1,2,⋯,L^^′}
P	Set of Products p,h={1,2,⋯,P}
C	Set of raw materials c={1,2,⋯,C}

FixI_i	Fixed cost of supplier i in the leader chain
FixJ_j	Fixed cost of manufacture j in the leader chain
FixK_k	Fixed cost of DC k in the leader chain
FixL_l	Fixed cost of retailer l in the leader chain
TrIJ~ijcL	The transportation cost of material c between supplier i and manufacture j in the leader chain
TrJK~jkpL	The transportation cost of product p between manufacture j and DC k in the leader chain
TrKL~klpL	The transportation cost of product p between DC k and retailer l in the leader chain
TrLM~lmpL	The transportation cost of product p between retailer l and customer m in the leader chain
Pc~jpL	The cost of producing, product p in manufacture j in the leader chain
O_cp	The raw material c needed to produce a unit of product p
CapI_ic	Maximum supplier i capacity of raw material c
CapJ_jp	Maximum manufacture j capacity of product p
CapK_kp	Maximum DC k capacity of product p
CapL_lp	Maximum retailer l capacity of product p
TrIJ~i^^′j^^′cF	The transportation cost of material c between supplier i^^′ and manufacture j^^′ in the follower chain
TrJK~j^^′k^^′pF	The transportation cost of product p between manufacture j^^′ and DC k^^′ in the follower chain
TrKL~k^^′l^^′pF	The transportation cost of product p between DC k^^′ and retailer l^^′ in the follower chain
TrLM~l^^′mpF	The transportation cost of product p between retailer l^^′ and customer m in the follower chain
Pc~j^^′pF	The cost of producing, product p in manufacture j^^′ in the follower chain

ZI_i	1 if supplier i is chosen; 0 otherwise
ZJ_j	1 if manufacture j is chosen; 0 otherwise
ZK_k	1 if DC k is chosen; 0 otherwise
ZL_l	1 if retailer l is selected; 0 otherwise
XijcL	Amount of raw material c transported between supplier i and manufacture j in the leader chain
YjkpL	Amount of product p transported between manufacture j and DC k in the leader chain
SklpL	Amount of product p transported between DC k and retailer l in the leader chain
UlmpL	Amount of product p transported between retailer l and customer m in the leader chain
Xi^^′j^^′cF	Amount of raw material c transported between supplier i^^′ and manufacture j^^′ in the follower chain
Yj^^′k^^′pF	Amount of product p transported between manufacture j^^′ and DC k^^′ in the follower chain
Sk^^′l^^′pF	Amount of product p transported between DC k^^′ and retailer l^^′ in the follower chain
Ul^^′mpF	Amount of product p transported between retailer l^^′ and customer m in the follower chain
PrmpL	The price of product p for customer m in the leader chain
PrmpF	The price of product p for customer m in the follower chain

Parameter	Optimistic	Probable	Pessimistic
a~mpF	a1mpF	a2mpF	a3mpF
a~mpL	a1mpL	a2mpL	a3mpL
TrIJ~ijcL	TrIJ1ijcL	TrIJ2ijcL	TrIJ3ijcL
TrJK~jkpL	TrJK1jkpL	TrJK2jkpL	TrJk3jkpL
TrKL~klpL	TrKL1klpL	TrKL2klpL	TrKL3klpL
TrLM~lmpL	TrLM1lmpL	TrLM2lmpL	TrLM3lmpL
Pc~jpL	Pc1jpL	Pc2jpL	Pc3jpL
TrIJ~i^^′j^^′cF	TrIJ1i^^′j^^′cF	TrIJ2i^^′j^^′cF	TrIJ3i^^′j^^′cF
TrJK~j^^′k^^′pF	TrJK1j^^′k^^′pF	TrJK2j^^′k^^′pF	TrJK3j^^′k^^′pF
TrKL~k^^′l^^′pF	TtKL1k^^′l^^′pF	TtKL2k^^′l^^′pF	TtKL3k^^′l^^′pF
TrLM~l^^′mpF	TrLM1l^^′mpF	TrLM2l^^′mpF	TrLM3l^^′mpF
Pc~j^^′pF	Pc1j^^′pF	Pc2j^^′pF	Pc3j^^′pF