1. INTRODUCTION
With the rising awareness about the limited availability of nonrenewable resources, manufacturers are increasingly forced to revise their strategies to ensure the sustainability of their supply chains. One of the strategies that can be adopted for this purpose is to create and use a closedloop supply chain. Indeed, the need to take an environment friendly stance toward the disposal of hazardous waste has forced senior logistics and supply chain managers to pay more attention to reverse logistics processes. Therefore, reverse logistics and closedloop supply chains have become an integral part of many businesses in the areas of manufacturing, distribution, product support, and service delivery (Chaabane et al., 2012). It is widely believed that integrating inventory and routing decisions in supply chain design can reduce strategic, operational, and tactical costs of the resulting supply chain and logistics operations. Given the importance of this issue, the following paragraphs provide a quick review of previous articles on the integration of routing and inventory decisions for perishable goods in closedloop supply chains.
In a study by Huang and Lin (2010), they formulated a model for a multiproduct inventoryrouting problem with a limited fleet, where demand is determined at the time of delivery. If the delivered quantity does not satisfy the demand entirely, the customer must receive another delivery through another tour, which incurs additional costs. If this second delivery does not reach the customer, a stockout cost will be incurred. In this study, a modified ant colony optimization algorithm was used to balance shortage and transportation costs (Huang and Lin, 2010).
Trudeau and Dror (1992) studied the problem of stochastic inventory routing for the distribution of materials from a central warehouse to a large set of customers. They assumed that demands are random and determined when the vehicle arrived at the demand point. It was also assumed a route can be broken down if its actual demand exceeds the capacity of the vehicle. They used a winwin strategy, which, while computationally expensive, could deliver the best system performance and reduce the stockouts in the system to almost zero (Trudeau and Dror, 1992).
A study by Yang and Wee (2002) analyzed an integrated inventory system with one vendor and multiple buyers for perishable products. In this study, production and consumption rates were assumed to be constant. These researchers developed a model for vendor and customer costs and showed that these costs are lower when system decisions are integrated than when supplier and customer decisions are made independently (Yang and Wee, 2002).
In a study by Savelsbergh and Song (2017) they formulated a model for inventory routing with continuous moves in a chain where a product is delivered to a set of customers, which were then divided into two groups, those that are close to the central warehouse and those that are far from that point. To solve this problem, they proposed three heuristic algorithms that determine delivery routes and the amount of product delivered (Savelsbergh and Song, 2007).
Moin et al. (2011) proposed a genetic algorithm for the inventory routing problem with multiple products. They also computed the lower bound and the best solution of the problem with a mixedinteger programming model and compared the results with the results of their proposed genetic algorithm (Moin et al., 2011).
In a study by Aydın (2014) the relationship between inventory management by the vendor and the inventory routing problem was investigated and a genetic algorithm was proposed for solving the vehicle routing part of the problem (Aydın, 2014).
Javid and Azad (2010) presented a model that simultaneously optimizes location, inventory, and routing decisions in a stochastic supply chain. In this study, customer demand was assumed to be uncertain but follow a normal distribution. They used a hybrid algorithm comprised of tabu search and simulated annealing to solve this model (Javid and Azad, 2010).
Mirzapour et al. (2014) studied a multiproduct inventory routing problem, in which heterogeneous vehicles are used to transport products from a set of suppliers to a factory. In this study, it was assumed that the planning horizon is finite and the demand for each product is definite. Green logistics was incorporated into the model by considering the interrelationship between transportation costs and greenhouse gas (GHG) emissions. In this work, GHG emission and fuel consumption were not modeled, and instead, it was assumed that the GHG emission of each vehicle is already known and depends on the distance it travels (Mirzapour Alehashem and Rekik, 2014).
In a study by Cordeau et al. (2015) on the multiproduct routinginventory problem, they proposed a threestep heuristic algorithm based on the decomposition of the supplier decision process (Cordeau et al., 2015).
Xiao and Konak (2017) formulated a green vehicle routing problem with a heterogeneous fleet with the objective of reducing GHG emissions and then used a hybrid method consisting of a genetic algorithm and dynamic programming to solve this problem (Xiao and Konak, 2017).
In a study by Iassinovskaia et al. (2017) they modeled a routinginventory problem in which the manufacturer has a heterogeneous fleet that distributes its products from a central warehouse to customers in given time windows while also collecting the used products for reuse in the next production cycle. They formulated this problem as a mixedinteger linear programming model and solved it with an exact algorithm (Iassinovskaia et al., 2017).
Zhalechian et al. (2016) proposed a new green closedloop locationrouting model with a biobjective nonlinear mathematical formulation aimed at reducing total facility and transportation costs and decreasing the environmental impact in terms of GHG emission and fuel consumption. They used the GAMS software to solve the model (Zhalechian et al., 2016).
Daghigh et al. (2016) presented a new multiobjective locationinventory model for designing thirdparty logistics networks with environmental, social, and economic objectives under dynamic conditions. They solved this model with the Epsilon constraint method to reach Pareto optimal solutions (Daghigh et al., 2016).
In a study by Rabbani et al. (2017) they presented a new mathematical formulation for the problem of inventory routing for perishable products. In this study, product production planning was also performed to avoid excess inventory and reduce fuel consumption costs and GHG emissions. These researchers used a genetic algorithm and simulated annealing to solve the formulated model (Rabbani et al., 2017).
Elhedhli and Merrick (2012) developed a green supply chain where the main goal is to reduce carbon dioxide emissions from vehicles. For this purpose, they considered all factors that can affect the carbon dioxide emissions of vehicles and estimated this parameter accordingly (Elhedhli and Merrick, 2012).
Shaw et al. (2012) designed a green supply chain model that considers the role of suppliers in the production of pollutants. In this study, supplier selection was formulated to take into account pollutant generation as well as costeffectiveness. These researchers used a multicriteria decisionmaking technique to solve the model (Shaw et al., 2012;Arabian et al., 2018).
A study by Soysal (2016) presented a closedloop inventory routing model for distribution operations of a beverage company with reverse logistics, demand uncertainty, and multiple products. Ultimately, this model was solved with a linear programming method (Soysal, 2016).
Soleimani et al. (2017) studied a closedloop supply chain with multiple levels, multiple products, and multiple periods, which covers all components and raw materials of products. They optimized and solved this problem by a genetic algorithm (Soleimani et al., 2017).
In a study by Sarkar et al. (2017) they proposed a multilevel closedloop supply chain model with thirdparty logistics. This model takes into account the environmental effects of production and transportation in a production system with returnable products (Sarkar et al., 2017).
RafieMajd et al. (2018) modeled a threelevel supply chain consisting of a supplier, several distribution centers, and multiple retailers in the form of an integrated inventorylocationrouting problem for multiple perishable products with a limited planning horizon. In this study, the planning horizon was divided into multiple periods and the transport fleet was assumed heterogeneous (Rafie Majd et al., 2018).
Panda et al. (2017) studied the impact of considering social responsibility while maximizing the profit of a closedloop supply chain through recycling and retailersupplier coordination (Panda et al., 2017).
In a 2019 study by Waltho et al. (2019) they reviewed the previous works on green supply chain design with a focus on the choice of supply chain policies and environmental objectives. This study stated that better quantitative methods are needed for measuring GHG emissions (Waltho et al., 2019).
Mohtashami et al. (2020) designed a closedloop supply chain model with the objective of reducing the chain’s negative environmental impacts. These researchers considered energy consumption as one of the most important determinants of the said negative impacts. They also considered product recycling and remanufacturing of defective products as efficient approaches for reducing these impacts (Mohtashami et al., 2020).
One of the most recent studies in this field is a 2021 study by Li et al. (2021) where they designed a twolevel green supply chain model, in which pricing policies and profit coordination between chain members are both optimized under uncertain demand. The results of this study showed that the chain members earn more profit when they adopt profit coordination measures than when they do not (Li et al., 2021).
In another 2021 study, Habib et al. (2021) investigated the effect of strategic sustainability orientation on the chain performance. This study presented a conceptual model in which sustainability is measured in terms of market orientation and knowledge management orientation. The results of the implementation of this approach in the textile industry showed that adopting strategies based on environmental indicators serves as a powerful tool for achieving sustainability in supply chains (Habib et al., 2021;Chapnevis et al., 2020).
2. METHODOLOGY
One of the methods that have been successfully used in various fields to handle uncertainly is scenario analysis. This method involves using stochastic variables to represent random quantities. A scenario is a potential future state that represents the outcome of the interaction of different factors under certain conditions. In other words, scenarios are combinations of random parameters that summarize the data in a few simple states. Managers who can predict a wider range of future situations tend to be more prepared and have a greater chance of making better use of opportunities that arise from these situations. Unlike sensitivity analysis, scenario analysis involves changing multiple parameters simultaneously. In this method, an important goal is to avoid common mistakes due to overestimation and underestimation. Naturally, scenarios are created based on possible states of parameters. Expanding a model in this way may turn it into a largescale model. The greater the number of random parameters considered, the larger the model will be. In other words, as we prepare the model to take more uncertainty into account, it becomes more realistic but also larger. The size of the model is proportional to the number of random parameters and the number of possible states considered.
For the mathematical formulation of the green supply chain problem, it is divided into four main parts: factories, warehouses, customers, and disassembly centers. The assumptions of the mathematical model are explained in the following.
Model assumptions are as bellows:

1. Each product must be transferred from the factory to the warehouse and from the warehouse to the customer, (customer demand can only be met through this procedure).

2. For recycling, the products to be disassembled must be collected from customers and transported to a disassembly center.

3. For remanufacturing, disassembled products must be transported from disassembly centers to the factory.

4. Customer demand is definite and known.

5. All product units that enter disassembly centers will be disassembled together.

6. All modes of transport have unlimited capacity.

7. Modes of international transport (e.g., rail, sea, etc.) need not be booked in advance.

8. The distance between the nodes of the network (for the estimation of CO2 emissions) is assumed to be the length of the direct path between facilities (factories, warehouses, customers, and disassembly centers).
The description of model indices is provided in Table 1.
The parameters used in the formulation of the model are described in Table 2.
The proposed formulation has seven variables, which are described in Table 3.
The proposed mathematical model consists of two main parts, one dedicated to objective functions and the other two constraints. In the first part, the model has two objective functions, f_{1} and f_{2}, which minimize the total cost and the CO2 emission of the chain respectively. The total cost is the sum of all fixed costs incurred along the chain from supplier to customer (TFC), all variable costs incurred along the chain (TVC), all costs incurred due to product handling and transport (TTC), and all costs incurred due to spoilage of products (TTP). These costs are formulated in Equations 1 to 5 of the model.
The model also minimizes the total cost of CO2 emission (TE), which consists of CO2 emission due to production activities (EP), CO2 emission due to product handling (EH), CO2 emission due to disassembly activities (ED), CO2 emission due to remanufacturing activities (ER), CO2 emission due to all transportations (ET). These items are formulated in Equations 6 to 11 of the model.
In this model, Equation 12 states that the total number of units of product to be transferred from a factory to a warehouse by any transport option must be equal to or less than the maximum capacity of that factory. According to Equation 13, the total number of units stored in a warehouse must be equal to or less than the maximum capacity of that warehouse. Equation 14 ensures that the total number of units to be transported from a warehouse to a customer by any transport option is equal to or less than the total amount of units that have been transported to that warehouse (from all factories and via all transport options)
According to Equation 15, the total number of units transported from a warehouse to a customer by any transport option must be equal to or greater than the demand of that customer. Equation 16 states that the total number of units to be collected from a customer for transport to a disassembly center must be less than the demand of the customer. According to Equation 17, the total number of units to be collected from a customer and transported to a disassembly center must be less than the capacity of that center.
Equation 18 ensures that the total number of units to be transported from a customer to a disassembly center is equal to or greater than the minimum quantity that can be returned by customers, which is expressed as a percentage of the total demand of the customer.
According to Equation 19 states that the total number of units transported from a disassembly center to a factory must be equal to or greater than the minimum quantity that can be returned by disassembly centers, and must be equal to or greater than the total amount of units entering that center.
According to Equation 20, the total number of units to be sent from a disassembly center to a factory must be equal to or less than the maximum remanufacturing capacity of that factory. Equation 21 computes the quantity of products to be spoiled in each warehouse. Equation 22 obtains the total inventory of product p in period t of scenario s. Equations 23 and 24 specify the types of decision variables.
The general form of a multiobjective problem is as follows:
Where x is the vector of decision variables, ${f}_{1}(x),{f}_{2}(x),\hspace{0.17em}{f}_{3}(x)$ are the objective functions, and S is the solution space. This problem can be solved by the weighting method, which involves normalizing the objective functions and then assigning a weight to each objective. The general form of this weighting method is as follows:
where w_{i} is the weight assigned to each objective function i and ${f}_{i}^{*}$ is the optimal value of this objective function.
The case considered in this study is a dairy company with two factories, 3 warehouses, 2 disassembly centers (separating containers and products, etc.), and six customers for two products and with three scenarios: price stability, price increase, and price reduction. For this case, the goal was to determine the optimal number of factories, warehouses, and disassembly centers.
To illustrate the performance of the proposed model, it was applied to a large number of numerical examples of different sizes that were generated for this purpose. The results of this performance evaluation are presented in this section. But first, we assess the validity of the model by manually checking the chain for a product and testing the effect of constraints on the solution.
The results extracted from the software are given in Tables 4 to 7. According to the initial data, customer 1 had needed 37 units of product 1, which have been supplied through warehouse 2. Product 1 has been produced by factories 1 and 2, which have produced respectively 72 and 58 units of this product. The total number of units of product 1 transferred out of warehouses is 130, of which 58 have come out of warehouse 1 and 72 from warehouse 2. The number of items returned by customers and sent to centers i1 and i2 is 24.6 and 14.4, respectively. Also, 9.84 disassembled items have been sent to factory f1 and 5.76 to f2. All of these results are correct and consistent with the constraints of the model, confirming the validity of the model.
3. RESULTS AND DISCUSSION
Industry has long been aware of the importance of quality as an undeniable determinant of success. Quality has always been a key part of any business strategy to increase market share and many quality experiments have been designed to achieve globally competitive quality levels.
One of the most powerful techniques for improving quality and productivity is the design of experiments. This method involves conducting a series of experiments by making certain purposeful changes in a system or process to examine their impacts on the performance characteristics or response of that system or process.
In other words, this method is the systematic manipulation of a number of variables so that some conclusions can be drawn from their effects. Here, we first compare the results of the proposed model in its deterministic and robust modes for the nominal data Table 8.
After solving the model for the three scenarios of price stability, price increase, and price reduction, the mean and variance of customer demand in these scenarios were obtained as shown in Table 9.
The breakdown of customer demand in each scenario is as shown in Table 10.
The stochastic twostep planning considers all three scenarios and produces the best solution based on the conditions of the scenarios and their probabilities. Normally, we would decide to open or close a facility based on the characteristics of each scenario (price stability, price increase, or price reduction), but in the twostep planning, we first make a decision about the facilities and then respond to customer demands based on the available facilities.
In this paper, we analyzed and compared the ratio of changes in the objective function to changes in demand in different scenarios. An overview of these changes is illustrated in Figure 2.
Compared to the first scenario, with a 19% decrease in demand for the first product and a 13% decrease in demand for the second product in the second scenario, the objective function will decrease by 15%. In the third scenario, with a 34% decrease in demand for the first product a 35% decrease in demand for the second product, there will be a 33% reduction in the objective function value compared to the first scenario Figure 3.
As shown in Figure 3, the cost of CO2 emission (in the second objective function) is about 10 times the cost in the first objective function, and this gap grows larger in other scenarios.
The fixed cost is incurred only once during the construction of the facility, and the transportation costs in the studied case are lower than the costs due to CO2 production. Therefore, the conditions that satisfy the estimated customer demand while reducing fixed and variable costs as well as environmental costs are the points specified in the above figure. Here, the costs in the objective functions are calculated separately for scenarios and cost items. The results show that the highest costs are related to CO2 emission due to internal transportation and after that the direct costs of transportation inside and outside the system. Therefore, for the case under study, it is very important for the facilities to be located as closest as possible to each other even if this means they are located farther away from some customers.
4. CONCLUSION
In this study, we developed a model for a green multi level supply chain. For this purpose, we first defined a chain consisting of multiple components including factories, warehouses, customers, and disassembly centers. Then, we developed a biobjective mathematical model for facility locationallocation and product distribution in this supply chain. The first objective of this model is to minimize the total sum of fixed and variable costs of the supply chain and its second objective is to minimize the chain’s total CO2 emission.
Since increased demand increases the circulation of materials outside the system, including the transport of products from factories to customers and transport of returned products from customers to factories, as well as the circulation of materials inside the system, including the transport of products from factories to warehouses and transport of returned and disassembled products to factories, it results in increased transportation and therefore increased CO2 emission.
It should be mentioned that although international relations and green supply chain are considered in this research, but the study have not focused on digital area.