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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.20 No.2 pp.236-247
DOI : https://doi.org/10.7232/iems.2021.20.2.236

# A New Mathematical Model for Redesigning the Reverse Supply Chain with Social Sustainability

Anastasia A. Kurilova*, Maria V. Zelinskaya
Togliatti State University, Russia
Kuban State Agrarian University named after I.T. Trubilin, Russia
*Corresponding Author, E-mail: kurilovaaa@mail.ru; anas.kurilova@mail.ru
February 27, 2021 April 2, 2021 April 5, 2021

## ABSTRACT

The flow of goods in the supply chain is divided into two types of direct and reverse flows. Recently, more attention has been paid to the reverse supply chain because it can create benefit in waste products. In this paper, a nonlinear integer mixed integer multi-objective mathematical programming model for inverse and sustainable supply chain is presented. The objective functions of this model are economic, social and environmental goals. In addition, this model covers four levels including suppliers, collection centers, recycling centers and consumers. The purpose of this article is to assist managers in making decisions at strategic and tactical levels such as choosing a suitable location for the establishment of collection and recycling centers, selecting production technology, determining the amount of production in each recycling center and the use of available vehicles. To solve this mathematical model, a method based on linearization as well as foal planning is developed. Finally, in order to validate and evaluate the performance, the model is examined using a numerical example and sensitivity analysis is performed on its various parameters.

## 1. INTRODUCTION

With the continued emissions of greenhouse gases (GHGs) such as carbon dioxide (CO2) into the atmosphere, global warming has become a major global phenomenon with dire consequences such as rising sea levels (because of the melting of polar ice caps), prolonged droughts, and climate change. One of the main sources of GHG emissions is transportation activities, which account for about 22% of carbon dioxide emission worldwide. Of this amount, 75% comes from vehicles in cities and road transport in general (Hombach et al., 2018;Samimi, 2021).

In recent decades, strict government regulations and rising awareness about global warming have motivated companies to pay more attention to the environmental aspects of their business. With the globalization of business completion and the emergence of highly competitive markets, companies need to maintain an acceptable social image to protect their market share and find new markets for their products while simultaneously addressing the economic aspect of the business, i.e. maximizing revenue at the lowest cost. The concept of sustainable supply chain has been introduced to cover environmental, economic, and social aspects of supply chains in one framework. With due attention to the principles of sustainability in their supply chains, companies can achieve enhanced competitiveness, good social standing, and improved economic performance (reduced cost and increased profit) while also preserving the environment.

One of the core activities in reverse logistics is the transport of return products. In reverse supply chains, products need to be physically transported to certain points for recycling for example. Since this transportation can have significant economic and environmental implications for reverse supply chains, proper design and control are vital for achieving success in such chains (Dat et al., 2012). Thus, reverse logistics can play a notable role in reducing waste generation and raw material consumption, which together have significant environmental benefits in addition to lowering costs and raising revenue.

In the rest of this paper, Section 2 reviews the studies previously conducted on the subject of interest. In Section 3, the problem is defined and its significance is explained. In Section 4, the mathematical model of the problem is formulated. In section 5, the method of solving the model is described. In section 6, the proposed model is evaluated through a numerical example, and in section 7, the results of the example are analyzed and the conclusions are provided.

In a study by Ren et al. (2013), these researchers designed a sustainable supply chain model for analyzing the sustainability of hydrogen supply chains to assist decision- making in this area.

In another study, Govindan et al. (2014) presented a multi-objective optimization model by integrating sustainability in decisions for perishable food supply chains. These researchers developed a two-echelon locationrouting model with time windows for designing sustainable supply chains with economic and environmental objectives, which allows the user to determine the number and location of facilities and the optimal quantity of delivered products for minimum cost and GHG emission. The results of this study showed that this combined approach yields better solutions than other methods.

Boukherroub et al. (2015) introduced an integrated approach to sustainable supply chain planning, in which the mathematical supply chain model is optimized for economic, social, and environmental objectives all together. They solved the model by a weighted goal programming method and implemented it in the wood industry.

Dadhich et al. (2015) used a hybrid life cycle assessment method to analyze a plasterboard supply chain. This study showed how GHG emissions could be reduced by identifying and analyzing the supply chain emission points during the product life cycle.

To solve the problem of sustainable supply chain network design, Pop et al. (2015) developed an efficient reverse distribution system consisting of a plant, a set of potential points for distribution centers, each with its own capacity, and a set of consumers with different demands.

Zhalechian et al. (2016) presented a model for sustainable location-routing-inventory management in closedloop supply chains under uncertainty. In this study, uncertainty was investigated using the fuzzy-probabilistic programming approach. In addition, a hybrid metaheuristic algorithm was developed for solving the large-scale instances of the problem.

In a study by Feitó-Cespón et al. (2017), they presented a model for redesigning sustainable reverse supply chains under uncertainty. The model presented in this study is a multi-objective probabilistic mixed integer nonlinear programming model that integrates economic, social, and environmental objectives. A multi-criteria programming approach was also developed to consider the uncertainty in this model.

Petridis et al. (2018) introduced a mixed-integer linear programming model for sustainable biomass supply chains. The results of this study showed that concentrating on environmental aspects significantly reduces the CO2 emission from biomass transport. It was also reported that environmental and economic criteria work in the same direction, but social criteria work in the opposite direction.

Mota et al. (2018) introduced a sustainable closedloop supply chain model with uncertainty. These researchers provided a decision management tool for designing and planning a sustainable supply chain. The multi- objective mixed-integer linear model of this study considers all three aspects of sustainability in the objective function.

A study by Tsao et al. (2018) presented a model for sustainable supply chain design under uncertainty that aims to maximize social benefits and minimize costs and environmental damage. In this study, uncertainty was integrated into the model by fuzzy programming.

Das (2018) designed a sustainable supply chain model based on the lean system principles under deterministic environmental conditions. The purpose of this study was to integrate the lean system into the design of supply chain models to improve the overall system performance.

Afra and Behnamian (2021) developed a heuristic method for reverse supply chain planning in multiproduct and multi-period conditions. In this study, the goal was to optimize the volume of transport in the network as well as the routes by which transport should be performed. To solve this problem, a heuristic method based on the Lagrange technique was formulated.

Kargar et al. (2020) proposed a mathematical model for a reverse supply chain with the purpose of municipal waste collection. The objectives of this mathematical model were to reduce all costs, maximize the use of existing technologies, and reduce all residual medical waste. This mathematical model was solved with the fuzzy goal programming method

After this review of the relevant literature, the innovations of this article that distinguish it from previous works on the subject can be summarized as follows:

• •Social factors are integrated into the design process by incorporating the number of working days lost due to risks and problems arising from the use of production technologies and the safety risks and health problems arising from the consumption of recycled products into the third objective function of the model.

• •A multi-objective mixed integer nonlinear model is formulated for redesigning the sustainable inverse supply chain network and is linearized by exact mathematical methods so that it is turned into a convex programming model.

## 2. METHODOLOGY

The model presented in this paper is a sustainable reverse supply chain design model that incorporates location, allocation, and vehicle selection problems. In this model, it is assumed that used products are collected from consumers by a set of collection centers and then transferred to recycling centers, where they are processed (recycled or remanufactured) into a product with some market value and then transferred to the target markets for sale. It is also assumed that several different types of products can be collected and transportation across the network can be done by several different vehicles. The schematic diagram of the considered model is illustrated in Figure 1.

In this study, the goal is to formulate the following policies:

• •Location (selecting a suitable location for establishing collection and recycling centers from among a set of available candidate options)

• •Production planning (determining the production quantity of each recycling center)

• •Production technology selection

• •Vehicle planning (determining how many of each type of vehicle to be used)

These goals must be achieved such that customer demand is fully met, supply chain costs are minimized, the positive environmental effects of supply chain activities are maximized, and its social harms are minimized.

This section introduces indices, parameters, variables, constraints, and objective functions and describes how they are modeled.

All of the sets, indices, parameters, and variables used in the mathematical model are listed in Table 1.

Objective function 1 minimizes the supply chain costs given in Equation (1), which include fixed transport cost (FTC), variable transport cost (VCT), cost of recycling (COR), and cost of operating the collection centers (COG).

$m i n f 1 = F C T + V C T + C O R + C O G$
(1)

FTC consists of cost items of Equation (2), which include the fixed cost of transport between suppliers and collection centers (the first term), the fixed cost of transport between collection centers and recycling centers (the second term), and the fixed cost of transport between recycling centers and end consumers (the third term).

$F C T = ∑ i ∑ j ∑ m C F T m V S R i j m + ∑ j ∑ k ∑ m C F T m V R P j k n + ∑ k ∑ l ∑ m C F T m V P C k l m$
(2)

VCT is comprised of cost items given in Equation (3), which include the variable cost of transport between suppliers and collection centers (the first term), the variable cost of transport between collection centers and recycling centers (the second term), and variable fixed cost of transport between recycling centers and end consumers (the third term).

$V C T = ∑ i ∑ j ∑ p ∑ m C U T m Q S R i j m p V S R i j m d i j + ∑ j ∑ k ∑ w ∑ p C U T m Q R P j k u p V R P j t w d j k + ∑ k ∑ 1 ∑ w ∑ p ∑ , C U T m Q P C u k n V P C k k d k l$
(3)

The costs of using recycling centers are formulated in Equation (4), where the first term is fixed cost and the second term is the variable costs of using recycling centers.

$C O R = ∑ k ∑ t C F P k t P k t + ∑ k ∑ p C U P k p ∑ ​ ∑ m ∑ t Q P C k l m p t$
(4)

The cost of using collection centers is obtained using Equation (5). The first term of this equation is the fixed cost and its second term is the variable cost of using collection centers.

$C O G = ∑ j C F R j R j + ∑ j ∑ p C U R j p ∑ k ∑ m Q P R j k m p$
(5)

Objective function 2 tries to maximize the positive environmental impacts of the supply chain as formulated in Equation (6). The terms of this equation are the positive environmental impact of products (EIP), the negative environmental impact of transportation along the supply chain (EIT), the negative environmental impact of electricity consumption in the entire system (EIE), the negative environmental impact of infrastructure and water consumption (EIB) and the positive environmental impact of waste disposal (EID).

$max f 2 = E I P − E I T − E I E − E I B + E I D$
(6)

The positive environmental impact of products is quantified by Equation (7).

$E I P = ∑ k ∑ l ∑ m ∑ p ∑ t I p Q P C k l m p t$
(7)

The negative environmental impact of transportation is given in relation (8). The first term of this equation is the negative environmental impact of transportation between suppliers and collection centers, the second term is the negative environmental impact of transportation between collection centers and recycling centers, and the third term is the negative environmental impact of transportation between recycling centers and end consumers.

$E I T = ∑ i ∑ j ∑ m I T m V S R i j m d i j ∑ p Q S R i j m p + ∑ j ∑ k ∑ m I T m V R P j k m d j k ∑ p Q R P j k m p + ∑ k ∑ l ∑ m I T m V P C k l m d k l ∑ p ∑ t Q P C k l m p t$
(8)

The negative environmental impact of electricity consumption in the entire system is determined using Equation (9), where the first, second, and thirds terms are related to the electricity consumption of suppliers, collection centers, and recycling centers, respectively.

$E I E = I E ∑ k ∑ t c f e k t P k t + I E ∑ j c f e j R j + I E ∑ p c v e p ∑ k ∑ l ∑ m ∑ t Q P C k l m p t$
(9)

The negative impact of infrastructure and water consumption are described in Equation (10). The first term of this equation is the impact of infrastructure and the second term is related to water consumption.

$E I B = I P ( ∑ k ∑ t α k P k t + ∑ j β j R j ) + I A ∑ p ∑ t c v a p t ∑ k ∑ ​ ∑ m Q P C k l m p t$
(10)

The positive environmental impact of waste disposal is formulated as Equation (11).

$E I D = ∑ p I V p ∑ i ∑ j ∑ m Q S R i j m p$
(11)

Finally, the third objective function, which is dedicated to the social impact of business, is formulated as Equation (12). This objective function tries to minimize the risks posed by the products and the number of working days lost because of safety problems related to production technology.

$m i n f 3 = θ w ( ∑ k ∑ l ∑ m ∑ p ∑ t e t Q P C k l m p t ) + θ d ( ∑ k ∑ l ∑ m ∑ p ∑ t r t Q P C k l m p t )$
(12)

The constraints that are considered in this paper are as follows:

$∑ j ∑ m Q S R i j m p ≤ G i p$
(13)

$∑ k ∑ m Q R P j a m p = ∑ m Q S R j m p ∀ j . p$
(14)

$∑ I ∑ m ∑ p Q P C k m p t = ∑ j ∑ m Q R P j m p ∀ k . p$
(15)

Equation (13) states that the total quantity of product p sent from supplier i to collection centers should exceed the capacity of the supplier for that product. Equation (14) ensures that the total quantity of product p that is transported from a collection center (j) to recycling centers does not exceed the input of that collection center. Equation (15) states that the total quantity of product p that is transported from a recycling center (k) to consumers should not exceed the input of that recycling center.

$∑ l ∑ m ∑ t Q P C k i m p t ≤ ∑ t C a k p t$
(16)

(17)

Constraint (16) states that the quantity of product p that is produced by technology t in the recycling center k and transported to the customer l by vehicle m should not exceed the capacity of that recycling center. Constraint (17) ensures that the quantity of product p produced by technology t and transported to customer l from the recycling center k does not exceed the demand of that customer for that product.

$∑ p Q R P j k n p C T m p + H R P j k n = V R P j k n ∀ j . k . m$
(18)

$∑ p Q R P j k m p C T m p + H S R i j m = V S R i j m ∀ i . j . m$
(19)

$∑ p ∑ t Q P C j k m p t C T m p + H P C k l m = V P C j k n ∀ k . l . m$
(20)

Constraints (18), (19), and (20) state that the quantity of product transported between two points by a particular vehicle divided by the capacity of that vehicle must be equal to the number of trips between these two points by that vehicle.

$V S R i j m + H S R i j m ≥ 0 ∀ i . j . m$
(21)

$V R P j k m + H R P j k m ≥ 0 ∀ j . k . m$
(22)

$V P C k l m + H P C k l m ≥ 0 ∀ k . l . m$
(23)

Constraints (21), (22), and (23) ensure that slack variables are concentrated on the difference between the numbers of shipments.

$∑ i ∑ j V S R i j m + ∑ j ∑ k V R P j k n + ∑ k ∑ l V P C k l m ≤ N v m ∀ m$
(24)

$∑ k ∑ m ∑ p Q R P j k m p = R j ∑ k ∑ m ∑ p Q R P j k n p ∀ j$
(25)

$∑ l ∑ m ∑ p Q P C k l m p t = P k t ∑ l ∑ m ∑ p Q P C k l m p t ∀ k$
(26)

$∑ t P k t ≤ 1 ∀ k$
(27)

(28)

$Q S R i j m p . Q R P j k n p . Q P C k l m p t ≥ 0 ∀ i . j . k . l . m . p . t$
(29)

$R j , P k t ∈ { 0 , 1 } ∀ j . k . t$
(30)

Constraint (24) states that the total number of trips made along the supply chain by vehicle m should not exceed the total number of available trips for that vehicle. Constraint (25) ensures that materials can flow from the collection center j to the recycling center k only when that recycling center is established. Constraint (26) states that materials can flow from the recycling center k to customer l only when that recycling center is established. Constraint (27) guarantees that if the recycling center k is established, it will be assigned exactly one production technology. Constraints (28), (29), and (30) specify the bounds of decision variables.

## 3. RESULTS

The model presented in this paper is a multiobjective mixed integer nonlinear programming model (MOMINLP). The nonlinearity of this model originates from the multiplication of integer variables by continuous variables in the first and second objective functions and the multiplication of binary variables by continuous variables in some of the constraints. These nonlinearities make the solution space non-convex, causing the result to be a general optimal solution (Bazaraa et al., 2013). In this paper, exact mathematical linearization methods are used to solve this problem. Thus, the model is first linearized and then solved by the goal programming method. A diagram of the solution process is shown in Figure 2.

As mentioned, the presence of nonlinear terms in the first and second objective functions and several constraints makes the model nonlinear. This section describes how this nonlinear model is linearized to facilitate the solution process.

In the economic and social objective functions, nonlinear terms are the multiplication of integer variables by continuous variables. To linearize these terms, first, these integer variables should be converted to binary variables. To do so, it is necessary to determine the upper bound of these variables and then linearize the multiplication of binary variables by continuous variables.

The first integer variable is VSRijm, the bound limit of which is denoted by VSRUijm. This upper bound is the maximum number of trips between the supplier the collection and I center j by vehicle m, which depends on the quantity of product supplied by this supplier and the capacity of the collection center. However, since collection centers are assumed to have an unlimited capacity, this upper bound only depends on the supplier. However, this upper bound should also not exceed the maximum number of trips available for the vehicle. Hence, this bound is calculated as follows.

$V S R U i j m = m i n { ∑ p G i p C T m p , N v m }$
(31)

The next integer variable whose upper limit needs to be determined is VRPjkm. the upper limit of this variable, denoted by VRPUjkm, is the maximum number of trips made between the collection center j and the recycling center k by vehicle m, which value depends on the production capacity of the recycling center k with technology t for product p. This upper bound should also not exceed the maximum number of trips available for the vehicle.

$V R P U j k m = Min { ∑ p C a k p t C T m p , N v m }$
(32)

Similarly, the upper bound of the integer variable VPCklm is denoted by VPCUklm and is the maximum number of trips made between the recycling center k and the customer l by vehicle m. This bound depends on the production capacity of the recycling center and the customer demand. VPCUklm should also not exceed the maximum number of trips available for the vehicle. Thus, this upper bound is calculated by the following equation.

$V P C U k l m = m i n { m a x { ∑ p C a k p t C T m p , ∑ p D l p C T m p } , N v m }$
(33)

Now that the upper limits of integer variables are determined, they can be used to convert the integer variables into a set of binary variables as shown below.

$V S R i j m = ∑ a = 0 d 2 a x i j m a d = [ log 2 r S R U α r ] x i j m a ∈ { 0.1 }$
(34)

$V R P j k n = ∑ b = 0 d ' 2 b y j k m b d ' = [ log 2 V R P U μ ] y i j m b ∈ { 0 , 1 }$
(35)

$V P C k l m = ∑ c = 0 d σ 2 c z k l m c ∀ k . l . m d ' ' = [ log 2 V P C U u n ] z k / m c ∈ { 0 , 1 }$
(36)

After the above steps, the multiplication of integer variables and continuous variables, which is a nonlinear term, must be linearized. For this purpose, we define the parameter M as a very large number and add the following constraints to the model.

$A K I j m p a ≤ M x i j m a$
(37)

(38)

(39)

(40)

(41)

(42)

$K j k m p b ≥ Q R P j k m p − M ( 1 − y j k m b ) ∀ j . k . m . p . b$
(43)

$B j k m b = ∑ b 2 b B K j k n p b ∀ j . k . m . p$
(44)

$C K k l m p t c ≤ M z k l m c ∀ k . l . m . p . t . c$
(45)

$C K k l m p t c ≤ Q P C k l m p t ∀ k . l . m . p . t . c$
(46)

$C K k l m p t c ≥ Q P C k l m p t − M ( 1 − z k l m c ) ∀ k . l . m . p . t . c$
(47)

$C k l m p t = ∑ c 2 c C K k l m p t c ∀ k . l . m . p . t . c$
(48)

Constraints (25) and (26) contain nonlinear terms in the form of the multiplication of binary variables and integer variables. These terms are linearized in the same way as described above using the following constraints.

$Q R P R j k m p ≤ M R j ∀ j . k . m . p$
(49)

$Q R P R j k m p ≤ Q R P j k m p ∀ j . k . m . p$
(50)

$Q R P R j k m p ≥ Q R P j k m p − M ( 1 − R j ) ∀ j . k . m . p$
(51)

$Q P C P k l m p t ≤ M P k t ∀ k . l . m . p . t$
(52)

$Q P C P k l m p t ≤ Q P C k l m p t ∀ k . l . m . p . t$
(53)

$Q P C P k l m p t ≥ Q P C k l m p t − M ( 1 − P k t ) ∀ k . l . m . p . t$
(54)

The model presented in this study is a multiobjective nonlinear model. To solve such a multiobjective model, it is necessary to use multi-objective solution methods. One of the methods most commonly used for this purpose is the fuzzy goal programming method. The main advantage of the fuzzy approach is the ability to measure how much each objective function is satisfied explicitly. This feature helps decision-makers to adopt an effective method based on the degree of satisfaction and priority of each objective (Torabi and Hassini, 2008). The fuzzy goal programming method starts with defining a positive ideal solution and a negative ideal solution for each objective, the value of which can be calculated as shown below.

$f 1 P I S = m i n f 1 , f 1 N I S = m a x f 1 f 2 P I S = m a x f 2 , f 2 N I S = m i n f 2 f 3 P I S = m i n f 3 , f 3 N I S = m a x f 3$
(55)

Based on these positive and negative ideal solutions, one can define a linear membership function for each objective function as given below.

$μ 1 = { 1 f 1 < f 1 P I S f 1 N I S − f 1 f 1 N I S − f 1 P I S f 1 P I S < f 1 < f 1 N I S 0 f 1 N I S < f$
(56)

$μ 2 = { 1 f 2 > f 2 P I S f 2 − f 2 N I S f 2 P I S − f 2 N I S f 2 N I S < f 2 < f 2 P I S 0 f 2 N I S > f 2$
(57)

$μ 3 = { 1 f 3 < f 3 P I S f 3 N I S − f 3 f 3 N I S − f 3 P I S f 3 P I S < f 3 < f 3 N I S 0 f 3 N I S < f 3$
(58)

Here, μi is the level of satisfaction of the i-th objective. After the above steps, the following equation is used to turn the multi-objective programming model into its equivalent single-objective model.

$max Z G = ∑ n w n μ n$
(59)

## 4. DISCUSSION

This section examines the results of the model for a numerical example. For this example, the results were produced by coding the model in GAMS. In this example, there are five suppliers, three collection centers, two recycling centers, three types of production technology, and seven consumers. Also, there are 3 types of transport vehicles with low (4 tons), medium (10 tons) and high capacity (15 tons), for which the number of available trips is respectively 100, 80 and 60 (shipments per year). The amount of demand for recycled products is definite and known and all demands need to be fully met. The capacity of suppliers and recycling centers is known, but collection centers have an unlimited capacity. The quantities of raw materials (used products), the production capacity of recycling centers, and the demand of customers are given in Tables 2 to 4.

The model was solved using GAMS and CPLEX server. The optimal values of objective functions in the solution are given in Table 5.

This section presents the results of a sensitivity analysis performed to examine the effect of changes in demand on the values of objective functions. In this sensitivity analysis, first, the changes in the results related to the social aspect of the model were examined. This process was then repeated for other aspects of the model. In each step of this sensitivity analysis, all model parameters except demand were given a constant value and the changes resulting from demand fluctuations were recorded. The changes applied to demand ranged from 80% to 120%. The results of this sensitivity analysis are presented in Table (6) and plot-ted in Figures 3, 4, and 5.

In the first diagram, it can be seen that with the ex-ception of point 1, the value of the social objective function increases with rising demand, which means there is a positive relationship between these two parameters. The trend of change in the economic objective function after applying demands of different magnitudes is shown in the second diagram. As can be seen, increasing the demand does not cause a regular change in the value of the economic objective function. Nevertheless, the lowest costs have occurred at the 100% and the 120% demand levels. The changes in the environmental objective function following the changes in demand are plotted in the third diagram. It can be seen that increasing demand has not made a regular change in the value of the environmental objective function. However, the lowest values of this function have also occurred at the 100% and the 120% demand levels.

## 5. CONCLUSION

This paper presented a three-objective nonlinear model for a four-level multi-product reverse supply chain consisting of suppliers, collection centers, recycling centers, and customers with limited production and transport capacity. The objectives of the model were to minimize the costs of the supply chain and maximize its positive social and environmental impacts. To solve the proposed model, it was linearized so that it can be solved with the goal programming method. The solution process was also coded in GAMS software. The study also analyzed the effects of working days lost because of safety problems related to production technologies and the dangers posed by the products. It was found that increasing these parameters greatly increases the social objective function. However, no clear relationship was observed between them and economic and environmental aspects. In future studies, it is recommended to define some parameters of the model as uncertain variables. It may also be beneficial to try solving the model with other multi-objective solution methods.

## Figure

Proposed reverse supply chain network.

Flowchart of the solution process.

Effect of demand on the first objective function.

Effect of demand on the second objective function.

Effect of demand on the third objective function.

## Table

Description of model indices, parameters, and variables

Quantity of raw materials (used products) provided by suppliers

Production capacity of recycling centers in terms of production technology

Demand for recycled products

Optimal values of objective functions

Results of sensitivity analysis for demand

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