A mixed-integer linear programming approach for circular economy-led closed-loop supply chains in green reverse logistics network design under uncertainty

Abstract

Purpose

To maximize the network total profit by calculating the difference between costs and revenue (first objective function). To maximize the positive impact on the environment by integrating GSCM factors in RL (second objective function). To calculate the efficiency of disassembly centers by SDEA method, which are selected as suppliers and maximize the total efficiency (third objective function). To evaluate the resources and total efficiency of the proposed model to facilitate the allocation resource process, to increase resource efficiency and to improve the efficiency of disassembly centers by Inverse DEA.

Design/methodology/approach

The design of a closed-loop logistics network for after-sales service for mobile phones and digital cameras has been developed by the mixed-integer linear programming method (MILP). Development of MILP method has been performed by simultaneously considering three main objectives including: total network profit, green supply chain factors (environmental sustainability) and maximizing the efficiency of disassembly centers. The proposed model of study is a six-level, multi-objective, single-period and multi-product that focuses on electrical waste. The efficiency of product return centers is calculated by SDEA method and the most efficient centers are selected.

Findings

The results of using the model in a case mining showed that, due to the use of green factors in network design, environmental pollution and undesirable disposal of some electronic waste were reduced. Also, with the reduction of waste disposal, valuable materials entered the market cycle and the network profit increased.

Originality/value

(1) Design a closed-loop reverse logistics network for after-sales services; (2) Introduce a multi-objective multi-echelon mixed integer linear programming model; (3) Sensitivity analysis use Inverse-DEA method to increase the efficiency of inefficient units; (4) Use the GSC factors and DEA method in reverse logistics network.

1. Introduction

The classic Supply Chain (SC), now called the Forward SC, does not play a vital role in End-of-Life (EOL) products. The reverse SC or Reverse Logistics tries to take responsibility for EOL products (Pauliuk et al., 2017). In recent years, reverse logistics (RL) has become one of the most notable issues in different industries. It also has some substantial effects on customer relationships and logistics related to operational capabilities (Bouzon et al., 2016). RL includes a wide range of activities on used products such as recycling, reusing, remanufacturing, collecting, repairing and recycling (Agrawal and Singh, 2019). Generally, RL purpose is to manage the reverse flow, and if efficient management is carried out on reverse operations, it can improve the distribution and collection system of the goods and materials (Safdar et al., 2020). Utilization of RL networks has some advantages, such as minimizing the costs and the number of pollutants, increasing the total profit of the network by recycling materials and components and sending them to markets, etc. (Chan et al., 2020). One of the substantial SC management processes is designing network logistics (Melo et al., 2009). In recent years, some companies such as Dell, Kodak and Xerox decided to design RL networks because of the stringent pressures and regulations regarding environmental and economic profits from remanufacturing and recovery activities (Rubio et al., 2019). A study in China shows that the quality of EOL vehicles plays a vital role when recyclers decide on their recycling behavior (Yu et al., 2020).

RL is the significant dimension of the closed-loop supply chain (CLSC). The CLSC as an integrated approach has been known due to the development of the SC, in which both forward and reverse SC are considered simultaneously. CLSC has two types of duties: firstly, covering the demands of customers and secondly, collecting the EOL products (Krug et al., 2020).

Nowadays, the production of electronic and electrical waste is increasing (Sahu et al., 2018). The management of these wastes plays an essential role in environmental sustainability. This becomes even more important when most of these wastes are reusable or recyclable (Shittu et al., 2020). Traditional waste management was limited to landfilling or recycling and paid less attention to issues such as reuse and repair. Some countries have not adapted to the rules of closing the loops for their industries, such as Electrical and Electronic Equipment (Kazemi et al., 2019). However, some strict rules have been set to adapt to the CLSC strategies for the environment and industries (Ghoushchi and Hushyar, 2020). For instance, the so-called Waste Electrical and Electronic Equipment precept (WEEE), has been set up in 2002 by the European Union and obligated producers of electronic products to collect and recycle their EOL equipment. After setting these rules, dozens of countries inflicted obligations to collect and recycle electronic products (Gu et al., 2016). WEEE management in a CLSC is a complex issue (Islam and Huda, 2018). For this purpose, mixed-integer linear programming approach (MILP) modeling is used to design the RL network (Destyanto et al., 2019).

Considering RL in designing a green and sustainable SC has rarely been reflected in previous researches. Instead, all the focus is on the social and green supply chain (GSC) (Govindan et al., 2016). Therefore, by adding some decision variables, an objective function and a number of constraints into the model, RL can be merged with the green and sustainable SC (Govindan et al., 2015).

Due to the complicated nature of RL design, it is considered as a strategic issue with wide economics ramifications. There are meta-heuristic methods and exact optimization methods in order to solve this type of problems. Despite the short computational time meta-heuristics methods are often not optimal compared to exact techniques and may not be appropriate in this area (Rachih et al., 2019). MILP is one of the common methods for designing a reverse logistics network (Yu and Solvang, 2017; Alshamsi and Diabat, 2018).

Many studies have investigated the importance of decreasing the environmental impacts (Edalatpour et al., 2018; Zaman and Shamsuddin, 2017) and many of them try to explore the relationships between merging sustainability and GSC in RL (Rostamzadeh et al., 2015). In RL networks, some facilities can be selected individually as a supplier due to their tasks in the network. The selected facility has some fundamental tasks in comparison to the other facilities (Newhart et al., 2019). On the one hand, the mentioned facility's efficiency should be calculated, and inefficient components should be identified. One of the methods for evaluating facilities is the data envelopment analysis (DEA) method (Karimi et al., 2020).

Simultaneous data envelopment analysis (SDEA) is used to maximize the profits and efficiency of the centers. Increasing profitability and efficiency and minimizing defects and performance of delivery delay rates is one of the practical applications of this method (Rezaee et al., 2017). Because understanding the results of multi-objective models is crucial for decision-makers, the SDEA method is used to place the optimal Pareto results of the model in order and analyze them (Bal and Satoglu, 2019).

Circular economy (CE) is a concept currently promoted by the European Union (EU), by several national governments including China, Japan, United Kingdom, France, Canada, Netherlands, Sweden, and Finland as well as by several businesses around the world. The European Commission recently estimated that CE-type economic transitions can create 600 billion Euros annual economic gains for the EU manufacturing sector alone (Korhonen et al., 2018). The European Commission recently estimated that CE-type economic transitions can create 600 billion euros annual economic gains for the EU manufacturing sector alone (Commission, 2015). Product reuse, remanufacturing and refurbishment, demand fewer resources and energy and are more economic as well than conventional recycling of materials as low-grade raw materials (Mihelcic et al., 2003). The currently popularized CE concept extends conventional waste and by-product utilization and recycling by emphasizing the utilization of the value embedded in materials as high value applications as possible (Asif et al., 2016). Today, many scientific researches use CEs concepts in e-waste recycling (Bressanelli et al., 2020; Wagner et al., 2019).

This paper aims to develop MILP model to design the RL network, which includes multi product. The network is CLSC and consists of some facilities such as: product return zone, disassembly, remanufacturing, repairing, recycling, disposal and markets. Selecting and developing method of MILP has been accomplished by considering three main objectives simultaneously, including: total network profit, GSC factors and maximizing the efficiency of disassembly centers. Research model is a six-level, multi-objective a single-period multi-product that focuses on WEEE.

Due to limited research in the field of application of MILP in RL and integrating green factors in RL networks, this paper tries to establish a MILP model in RL to show the influence of integrating RL in GSC design. Also, one of the main objectives of this research is to improve the performance of facilities. An overview of this research objectives are as follows:

1. The aim of the first objective function is to maximize the network's total profit by calculating the difference between costs and revenue.

2. The aim of the second objective function is to maximize the positive impact on the environment by considering green factors in the RL network. In fact, the positive impacts on the environment increased by reducing the WEEE.

3. The aim of the third objective function is to calculate the efficiency of disassembly centers by the SDEA method (Klimberg and Ratick, 2008), which are selected as suppliers.

4. To evaluate the total efficiency of the proposed model to facilitate the allocation resource process, to increase resource efficiency, and improve the efficiency of disassembly centers by Inverse DEA.

The rest of the paper is organized as follows: the empirical review of literature is expressed in section 2. Section 3 is allocated to the Problem description. In section 4 the numerical examples are presented, and section 5 focused on conclusion.

2. Literature review

In recent years, organizations have spent a lot of resources on the SC. Managers will soon need to invest just as much in their reverse SC (Kaviani et al., 2020). Rising concerns about environmental warnings have forced manufacturers to try to implement environmental management solutions. Suppliers play an important role in achieving SC goals (Ghoushchi et al., 2018; Kazancoglu et al., 2021). This paper proposes a model for the CLSC-RL network design problem that consists of GSC factors. In addition, we calculate the efficiency of disassembly centers to select the suppliers. Subsequently, to estimate the total input, we changed the total output for disassembly centers.

A brief history of the areas required for research is brought in the following. This background is divided into five sub-sections for ease of study. At the end of this section, several references related to the various areas of this manuscript are listed in Table 1 in the order of year of publication.

2.1 RL and CLSC mathematical models for SC

A wide range of researches in recent years has been assigned to RL (Prajapati et al., 2020). In one of researches, a multi-objective optimization model consisting of both cost and the key performance indicators (KPIs), is applied on the CLSC and the asset management process including a pallet provider, a manufacturer and seven retailers have been investigated. For each scenario the simulation of the optimal formation of the asset management process is recognized (Bottani et al., 2015). Another study analyzed a network design problem for CLSC that integrates the collected products with the distribution of the new products; then a mixed-integer nonlinear location-inventory-pricing model was presented to decide on the quantity of inventory, prices of new products, the optimal location for facilities, and motivation values for gathering the exact number of used products (Kaya and Urek, 2016).

Although, a mathematical model developed to design a multi-stage reverse SC. To this aim, a MILP model has been used; the optimum solutions in relation to the number and location of different facilities in the model are presented (John et al., 2018b). In an investigation, a multi-objective programming model is presented for CLSC optimization, efficient supplier selection and allocation according to quantity discount policy; it is modeled based on integrated simultaneous data envelopment Analysis-Nash bargaining game. The objective functions are maximizing the profit and efficiency and minimizing the defective and delivery delay rate (John et al., 2018a, b).

2.2 MILP mathematical models in RL and WEEE

One of the proposed models for solving CLSC problems is MILP. Although, fuzzy MILP is proposed to solve the CLSC problem (Jindal and Sangwan, 2014). MILP model is also suggested while determining the best operation planning strategies (Erdinc et al., 2015). In order to demonstrate the potential of the proposed model, a real-life case study along with several scenarios is studied. Findings of the case study indicate that the model has the potential to enable the decision maker to come with stronger decisions related to both bidding process and operational strategies of the facility (Capraz et al., 2015). Another study investigated RL network design under extended producer responsibility and solved the model using MILP model proposed for computing the optimal configuration of the network (Banguera et al., 2018).

To show the volume of e-waste generated and the importance of proper management, how to deal with this waste in the EU in the period from 2010 to 2017 is shown in Figure 1 (De Laurentiis et al., 2020).

A model proposed for WEEE recovery system and used nonlinear mixed integer programming (Bo et al., 2019). Another MILP model suggested for decision making activities on the WEEE CLSC network. The model assists decision makers to manage product returns with different quality and damage levels (Polat and Gungor, 2021).

2.3 Green and sustainable RL and SC

A mixed-integer nonlinear mathematical SC programming model has been presented to investigate the costs' environmental depreciation and tradeoffs (Fahimnia et al., 2015). Maintaining environmental sustainability is an important issue that had been focused on the SC structure. The environmental pollution reduction, reduction of diseases caused by these contaminants and reduction of waste collection costs need preservation of environmental sustainability in various industries (Yu and Wu, 2018).

The use of green energy resources and green practices can reduce the negative effects on social and environmental sustainability, as it improves financial performance in terms of higher Gross Domestic Product (GDP) per capita, trade openness and greater export opportunities worldwide (Khan et al., 2019). Since the last decade, according to the increase in supply chain sustainability, many experimental and conceptual articles have been published in journals. A systematic review of 362 research articles published in reputable journals in the last six years (2014–2019) shows that this field is influenced by research methods based on multi-criteria decision making and company-level studies. In addition, researchers need to use efficient algorithms and advanced economic modeling and conduct macro-level studies to discover new connections (Khan et al., 2021).

Several studies have focused on the reduction of environmental contamination. Subsequently, several studies connected with green and sustainable RL and SC have been introduced (Qureshi et al., 2019). A reverse logistics network design suggested with WEEE in 2020. This model aims at maximizing economic benefits and minimizing the environmental impacts of wastes (Moslehi et al., 2020).

2.4 Supplier selection in SC

Supplier selection is an essential part of the RL and CLSC (Rezaei et al., 2020). There are some studies in terms of supplier selection in SC management (Rashidi et al., 2020). The effect of visibility in the supplier selection problem has been investigated which presented a multi-objective approach (Yousefi et al., 2017). One of the important issues in the SC is supplier selection in the CLSC. The problem of supplier selection in the CLSC has been applied in almost all fields (Qu et al., 2020; Wu et al., 2019).

2.5 Circular economy

Environmental pressures and climate change have led companies and supply chains to consider new models for environmental protection. The circular economy has emerged as a model of sustainability that can drive economic growth through resource consumption and waste recycling.

The emerging concept of circular economics brings new opportunities and business models to companies such as the common economy. These new concepts are able to promote sustainable practices aimed at optimizing and improving the consumption of basic resources, especially in industries that are concentrated in natural metal resources. However, this issue remains relatively undiscovered, especially given the actual case studies in emerging economies (Jabbour et al., 2020). Two exploratory case studies in Brazil and Scotland show that companies that are less active in the circular economy, face more challenges and tensions due to important uncontrolled success factors (Sehnem et al., 2019).

WEEE industry is very important in the field of the circular economy (Rosa et al., 2019). In terms of the circular economy, the WEEE industry has a significant ability in recycling. The total value of raw materials in WEEE can be estimated at approximately 55 billion Euros (Baldé et al., 2017). In the WEEE industry, there is a need for a systematic review of research in terms of circular economics. A total of 115 studies were reviewed on the sustainability of the WEEE industry from various aspects of the circular economy (Bressanelli et al., 2020).

The circular economy is the opposite of linear economy theory, and its main purpose is to recycle the product and return part of it to the production cycle. Applying the circular economy concept encourages environmental protection and social prosperity (Bradley and Jawahir, 2019). Because the concept of the circular economy is new, its applications are very novel, especially in the field of WEEE. In a study, a way to develop and evaluate innovative recycling strategies for WEEE plastics is presented. Examines the feasibility of recycling strategies. The implementation of the new process represents a significant potential for value recovery based on plastics that would otherwise burn or shrink (Wagner et al., 2019). As well as, the potential for waste collection of electrical and electronic equipment used in information and communication technology to prepare for reuse in the workplace in Ireland was investigated. The results showed that significant amounts of collected items were economically viable and reused (Coughlan and Fitzpatrick, 2020).

Best location for WEEE collection points is proposed and solved through a mathematical model (Ruan Barbosa de Aquino et al., 2021). The circular economy in the WEEE industry has focused mainly on reducing and recycling strategies and paid less attention to reuse and reconstruction (Jaeger and Upadhyay, 2020). Examining the role of blockchain technology for the circular economy to increase organizational performance in the context of the China–Pakistan Economic Corridor shows that the circular economy approach to supply chain management also offers environmental and economic benefits to organizations. It was also found that blockchain technology has a positive role in the rotational economy (Rehman Khan et al., 2021).

Given the broadness of research conducted in the field of current research, in order to facilitate review by readers, the summary of research areas and their authors is given in Table 1.

3. Problem description

Environmental pollution occurs during the returning of products in RL. Some studies focused on this issue and e-waste (Ismail and Hanafiah, 2020). Digital products have been considered to validate the network, and some options such as recycling and disposal centers have been considered (John et al., 2018a, b).

3.1 Structure of the network design

In the proposed network, some issues such as the total profit of the network, environmental pollution problems and the efficiency of some of the centers have been considered and efforts have been made to put up these issues in the most optimal state. Firstly, to maximize profitability, the network tries to send more products to markets which are defined in the first objective function. Secondly, some approaches to solve and decrease environmental pollution utilize green supply chain factors presented, which is defined in the second objective function. Finally, the amount of efficiency of disassembly centers which have been considered as the supplier selection was calculated, has used the SDEA approach; which is defined in the third objective function. The network has been modeled in terms of model 1.

The structure of the network design problem is depicted in Figure 2. Different centers which are considered in this network are product return zones Z, remanufacturing centers R, repairing centers F, disposal and recycling center K, disassembly centers D and markets S. Return products from customers can be inserted into the reverse logistics network and depending on their defects, will be transferred to different centers. First, return products are inserted into the product return zones; in this center, products are divided into two categories: (1) products which require rebuilding are sent directly to the remanufacturing centers, and after rebuilding, they will be sent to the market for sale, and the remaining materials and components will be sent to disposal and recycling center after reconstruction, (2) the remaining products will all be sent to the disassembly centers; in these centers, the products are separated based on their defect, and then are sent to other centers. A group of products in the disassembly centers have been returned because of the incorrect sending of the product according to the customer's order or the variety in the customer's tact and are not defective, so this group of products will be sent to the market for resale. Another group of products has failures and defects and needs repair; these products are sent to repairing centers. There, defective products are repaired and sold as second-hand products again with a discount policy on the market; the remaining components and materials of these products that are hazardous and non-consumable are sent to the disposal and recycling center. Another group of products that remain in the disassembly centers are hazardous, non-consumable and disposable components and materials that are sent directly to the disposal and recycling center. There, harmful components, and materials such as Mercury, Lead (Pb), Cadmium, etc., are directly disposed. On the other hand, some materials and components that are valuable, are recycled and are sold in the market.

3.2 The performance of disposal and recycling center

Some factors are applied on the disposal and recycling center. These factors are implemented on the products which are disposable or recyclable: (1) Undesirable disposal, (2) Safe disposal and (3) recycling. These factors are green and help to protect the environment from pollution and destruction and help to develop the economy. The performance of these three factors are as follows:

1. Undesirable disposal, which is mainly released in the deserts or sea and because it is less costly for the network, is considered as the priority and represents the most harm to the environment and is defined by the symbol b₁. This study tries to decrease the number of disposable products and undesirable disposal, and instead of releasing them in the seas or deserts, implement safe disposal or recycle them to produce valuable components; so, they can be sent to markets and be of profit for the RL network.

2. Safe disposal, which mainly involves digging pits and placing special reservoirs in it and burying materials in these reservoirs and then re-covering the pit and reservoir with soil, which would cost more than the first method. It is defined by the symbol b₂.

3. Recycling is another way of reducing the environmental impact of digital waste, because devices such as mobile phones and digital cameras have plastic and useful metals such as gold, copper, silver, etc. They can enter the sales and market cycle after recycling. It is defined by the symbol b₃, which includes two recyclable parts: plastic materials q₁ and precious metals q₂.

3.3 Converting model 1 to the closed-loop model

A percentage of the products sent to the disassembly centers can be returned to the product return zones and kept there. Since disassembly centers have more duties than other centers in the network, while transporting the products from product return zones to these centers, these centers may face a lack of capacity. Therefore, in order to create this functionality in a model 1 where the number of the products are more than the capacity of the disassembly center, they are stored at the center Z and a recurrence link is created from the disassembly center to the product return zones, which converts the model 1 to a closed-loop model. Therefore, the variable Xp_r defines the capacity shortage of the D centers. On the other hand, the variable X'p_dz is the closed-loop variable and is used to provide the ability to return the products from disassembly centers to the product return zones, if necessary.

3.4 Establishing effective relationships between centers

This network has tried to establish effective relationships between the centers such as the connections between disassembly centers and repairing centers, markets, product return zones and disposal and recycling centers. Because of the important role of the disassembly center in terms of connecting with other centers, it was considered as a supplier selection.

4. Model formulation (Model 1)

Since the model is defined for a reverse logistics network that has several centers, there are some complexities and limitations that were tried to overcome the shortcomings and limitations by adding auxiliary variables. The notations related to Model 1 are as follows:

4.1 Objectives and constraints (model 1)

Due to the notations, the linear integer multi-objective programming model is as follows:

The objective function Z₁ is used to maximize the total profit of a network; in a way that the total cost is subtracted from the total income. The different costs considered in this study are processing, collection, disposal, transportation, fixed facility and maintaining costs.(1)$$\begin{array}{l}{Z}_1=Max\left(\begin{array}{l}{\displaystyle \sum _{DC}{\displaystyle \sum _d{\displaystyle \sum _{s}RE{V}_s^{DC}{X}_{ds}^{DC}}}}+{\displaystyle \sum _{RC}{\displaystyle \sum _f{\displaystyle \sum _{s}RE{V}_s^{RC}{X}_{fs}^{RC}}}}+\\ {\displaystyle \sum _{RP}{\displaystyle \sum _r{\displaystyle \sum _{s}RE{V}_s^{RP}{X}_{rs}^{RP}}}}+{\displaystyle \sum _q{\displaystyle \sum _k{\displaystyle \sum _{s}RE{V}_s^q{X}_{ks}^q}}}+{\displaystyle \sum _P{\displaystyle \sum _d{\displaystyle \sum _{z}RE{V}_z^p{X}_{dz}^{\prime p}}}}\end{array}\right)-\\ \left(\begin{array}{l}{\displaystyle \sum _P{\displaystyle \sum _z{\displaystyle \sum _d{X}_{zd}^p\left(t{c}_p{d}_{zd}+P{C}_d^P+C{C}_d^P\right)}}}+{\displaystyle \sum _P{\displaystyle \sum _z{\displaystyle \sum _r{X}_{zr}^p\left(t{c}_p{d}_{zr}+P{C}_r^P+C{C}_r^P\right)}}}+\\ {\displaystyle \sum _P{\displaystyle \sum _d{\displaystyle \sum _z{X}_{dz}^{\prime p}\left(t{c}_p{d}_{zd}\right)}}}+{\displaystyle \sum _{RP}{\displaystyle \sum _r{\displaystyle \sum _s{X}_{rs}^{RP}\left(t{c}_{RP}{d}_{rs}\right)}}}+{\displaystyle \sum _q{\displaystyle \sum _k{\displaystyle \sum _s{X}_{ks}^q\left(t{c}_q{d}_{ks}\right)}}}+\\ {\displaystyle \sum _{RC}{\displaystyle \sum _F{\displaystyle \sum _S{X}_{fs}^{RC}\left(t{c}_{RC}{d}_{fs}\right)}}}+{\displaystyle \sum _{DC}{\displaystyle \sum _d{\displaystyle \sum _s{X}_{ds}^{DC}\left(t{c}_{DC}{d}_{ds}\right)}}}+\\ +{\displaystyle \sum _{FC}{\displaystyle \sum _d{\displaystyle \sum _f{X}_{df}^{FC}\left(t{c}_{FC}{d}_{df}+P{C}_f^{FC}\right)}}}+{\displaystyle \sum _{DI}{\displaystyle \sum _d{\displaystyle \sum _k{\displaystyle \sum _b{X}_{dkb}^{DI}\left(t{c}_{DI}{d}_{dk}+U_B^{DI}\right)}}}}+\\ {\displaystyle \sum _{RK}{\displaystyle \sum _R{\displaystyle \sum _k{\displaystyle \sum _b{X}_{rkb}^{RK}\left(t{c}_{RK}{d}_{rk}+U_B^{RK}\right)}}}}+{\displaystyle \sum _{FK}{\displaystyle \sum _F{\displaystyle \sum _K{\displaystyle \sum _b{X}_{fkb}^{FK}\left(t{c}_{Fk}{d}_{fk}+U_B^{FK}\right)}}}}\\ {\displaystyle \sum _{D}f{f}_{D}Y{d}_D}+{\displaystyle \sum _{R}f{f}_{R}Y{r}_R}+{\displaystyle \sum _{F}f{f}_{f}Y{f}_f+TGC+}{\displaystyle \sum _P{\displaystyle \sum _Z{M}_{big}XP{R}_{PZ}}}\end{array}\right)\end{array}$$

In the objective function Z₂, some indicators of the GSC have been applied to reduce the effects of inappropriate disposal, and if possible, even the possibility of recycling some of the waste and the materials that are recovered are valuable so that these materials can be sent to the market. Three actions can be done at the disposal and recycling center, including undesirable disposal, safe disposal and recycling. Thus, the objective function Z₂ consists of two phases: the first phrase attempts to maximize the quantities of the products that are sent to the market, and the second phrase is used to minimize the quantities of disposed of products (if possible, to recycle more items), and finally, the objective function is to maximize the number of recycled items. And since the implementation and application of this type of objective function in a reverse logistics network are made, the network can be considered as a green RL network.(2)$$Z_2=Max(({\displaystyle \sum _{DC}{\displaystyle \sum _d{\displaystyle \sum _s{X}_{ds}^{DC}}}}+{\displaystyle \sum _{RC}{\displaystyle \sum _f{\displaystyle \sum _s{X}_{fs}^{RC}}}}+{\displaystyle \sum _{RP}{\displaystyle \sum _r{\displaystyle \sum _s{X}_{rs}^{RP}}}}+{\displaystyle \sum _q{\displaystyle \sum _k{\displaystyle \sum _s{X}_{ks}^q}}}+{\displaystyle \sum _P{\displaystyle \sum _d{\displaystyle \sum _z{X}_{dz}^{\prime p}}}}+{\displaystyle \sum _{DI}{\displaystyle \sum _d{\displaystyle \sum _k{\displaystyle \sum _{b\in \{b_3\}}{X}_{dkb}^{DI}}}}})-({\displaystyle \sum _{DI}{\displaystyle \sum _d{\displaystyle \sum _k{\displaystyle \sum _{b\notin \{b_3\}}{\gamma }_b{X}_{dkb}^{DI}}}}}))/({\displaystyle \sum _z{\displaystyle \sum _{p}P{R}_z^p\ast XXP(p)}})$$

The objective function Z₃ is used to calculate the efficiency of the disassembly centers by SDEA, which are selected as suppliers and maximizes efficiency.(3)$$z_3=Max{\displaystyle \sum _d(1-n(d))}$$

Constraint (4) ensures the collection of all the products returned to the product return zones.(4)$${\displaystyle \sum _d{X}_{zd}^p}+{\displaystyle \sum _r{X}_{zr}^p}+Xpr(p,z)=P{R}_z^p\forall p,z$$

Constraint (5) provides maximum allowable flow to all remanufacturing centers from a product return zone.(5)$${\displaystyle \sum _d{X}_{zd}^p{\alpha }_p}\ge {\displaystyle \sum _r{X}_{zr}^p}\forall p,z$$

Constraint (6) defines a number of products that are sent to the disassembly centers D and can be returned to the product return zones Z and stored there; in other words, this constraint ensures the condition of closing-loop of the model.(6)$${\displaystyle \sum _z{X}_{zd}^p{\beta }_p}\ge {\displaystyle \sum _z{X}^{\prime }{P}_{zd}}\forall p,d$$

Constraints (7)–(14) ensure the protection of different components' flow at various centers. That way, constraint (7) guarantees the protection of the flow of the products from disassembly centers D to markets S. Constraint (8) guarantees the protection of the flow of the products from disassembly centers D to repair centers F. Constraint (9) guarantees the protection of the flow of the products from disassembly centers D to the disposal and recycling center K.(7)$${\displaystyle \sum _z{\displaystyle \sum _p\left(X_{zd}^p-X_{dz}^{\prime p}\right)N{C}_{DC}^p}}={\displaystyle \sum _s{X}_{ds}^{DC}}\forall d,DC$$(8)$${\displaystyle \sum _z{\displaystyle \sum _p\left(X_{zd}^p-X_{dz}^{\prime p}\right)N{C}_{FC}^p}}={\displaystyle \sum _f{X}_{df}^{FC}}\forall d,FC$$(9)$${\displaystyle \sum _z{\displaystyle \sum _p\left(X_{zd}^p-X_{dz}^{\prime p}\right)N{C}_{DI}^p}}={\displaystyle \sum _k{\displaystyle \sum _b{X}_{dkb}^{DI}}}\forall d,DI$$

Constraint (10) guarantees the protection of the flow of the products from remanufacturing centers R to markets S. Constraint (11) guarantees the protection of the flow of the products from remanufacturing centers R to disposal and recycling center K.(10)$${\displaystyle \sum _z{\displaystyle \sum _p{X}_{zr}^{p}N{C}_{RP}^p}}={\displaystyle \sum _s{X}_{rs}^{RP}}\forall P,RP$$(11)$${\displaystyle \sum _z{\displaystyle \sum _p{X}_{zr}^{p}N{C}_{RK}^p}}={\displaystyle \sum _k{\displaystyle \sum _b{X}_{rkb}^{RK}}}\forall r,RK$$

Constraint (12) guarantees the protection of the flow of the products from repairing centers F to markets S. Constraint (13) guarantees the protection of the flow of the products from repairing centers F to disposal and recycling center K.(12)$${\displaystyle \sum _d{\displaystyle \sum _{FC}{X}_{df}^{FC}N{C}_{RC}^{FC}}}={\displaystyle \sum _s{X}_{fs}^{RC}}\forall f,RC$$(13)$${\displaystyle \sum _d{\displaystyle \sum _{FC}{X}_{df}^{FC}N{C}_{FK}^{FC}}}={\displaystyle \sum _k{\displaystyle \sum _b{X}_{fkb}^{FK}}}\forall f,FK$$

Constraint (14) guarantees the protection of the flow of recycled materials from disposal and recycling center K to markets S.(14)$${\displaystyle \sum _d{\displaystyle \sum _{DI}{X}_{dk{b}_3}^{DI}{\rho }_q^{DI}}}+{\displaystyle \sum _f{\displaystyle \sum _{FK}{X}_{fk{b}_3}^{FK}{\rho }_q^{FK}}}+{\displaystyle \sum _r{\displaystyle \sum _{rk}{X}_{rk{b}_3}^{RK}{\rho }_q^{Rk}}}={\displaystyle \sum _s{X}_{ks}^q}\forall q,k$$

Constraints (15)–(18) indicate capacity constraints in different centers. Constraint (15) shows capacity constraints in disassembly centers D in the conditions of sending products to these centers from product return zones Z.(15)$${\displaystyle \sum _z{X}_{zd}^p}\le Ca{p}_d^p\times Y{d}_d\forall P,d$$

Constraint (16) shows capacity constraints in remanufacturing centers R in the conditions of sending products to these centers from product return zones Z.(16)$${\displaystyle \sum _z{X}_{zr}^p}\le Ca{p}_r^p\times Y{r}_r\forall r,p$$

Constraint (17) shows capacity constraints in repairing centers F in the conditions of sending products to these centers from disassembly centers D.(17)$${\displaystyle \sum _d{X}_{df}^{FC}}\le Ca{p}_f^{FC}\times Y_f\forall f,FC$$

Constraint (18) shows capacity constraints in disposal and recycling center K in the conditions of sending products to these centers from disassembly centers D, remanufacturing centers R, and repairing centers F.(18)$${\displaystyle \sum _d{\displaystyle \sum _{DI}{X}_{dkb}^{DI}}}+{\displaystyle \sum _f{\displaystyle \sum _{FK}{X}_{fkb}^{FK}}}+{\displaystyle \sum _r{\displaystyle \sum _{FK}{X}_{rkb}^{RK}}}\le Ca{p}_b^{k}Y{b}_b\forall k,b$$

Constraints (19)–(21) ensure that the grading cost is calculated if and only if at least one remanufacturing center is established and indicates the conditions of opening the remanufacturing centers.(19)$${\displaystyle \sum _p{\displaystyle \sum _z{\displaystyle \sum _d{X}_{zd}^p{S}_d^p}}}\le TGD+BM\left(1-Y_{open}\right)$$(20)$$TGC\le BM\times Y_{open}$$(21)$${\displaystyle \sum _r{Y}_r}\le BM\times Y_{open}$$

Constraints (22)–(26) specify limitations related to determining the disassembly centers' efficiency; in other words, the limits of simultaneous data envelopment analysis are expressed. Constraint (22) guarantees that the total input of each decision-making unit (disassembly centers-suppliers) equals variables zero and one. Constraint (23) specifies the inefficiency of the total output of each decision-making unit (disassembly centers-suppliers). Constraint (24) demonstrates that the total output should be less than its relevant total input. Constraint (25)–(26) indicates that input and output weights cannot be a negative value.(22)$${\displaystyle \sum _{i^{\prime }}pi{n}_d^{i^{\prime }}I{n}_d^{i^{\prime }}}=y{d}_d\forall d$$(23)$${\displaystyle \sum _{j^{\prime }}po{u}_d^{j^{\prime }}o{u}_d^{j^{\prime }}}+n_d=y{d}_d\forall d$$(24)$${\displaystyle \sum _{j^{\prime }}po{u}_d^{j^{\prime }}o{u}_{dp}^{j^{\prime }}}-{\displaystyle \sum _{i^{\prime }}pi{n}_d^{i^{\prime }}I{n}_{dp}^{i^{\prime }}}\le 0\forall d,dp$$(25)$$I{n}_d^{i^{\prime }}\ge y{d}_d\times 0.001\forall d,i^{\prime }$$(26)$$o{u}_d^{j^{\prime }}\ge y{d}_d\times 0.001\forall d,j^{\prime }$$

Constraints (27)–(30) express the prioritization problem for the maintenance variable XP_r. Thus, despite these three priority constraints for the maintenance variable XP_r, it is determined that the products in the product return zones Z are first sent to the disassembly D and remanufacturing R centers according to their capacity and then are stored at the product return zones Z.(27)$${\beta }_p=\left({\displaystyle \sum _{r}Cap{R}_r^p}+{\displaystyle \sum _{d}Cap{d}_d^p}\right)-\left({\displaystyle \sum _{z,r}{X}_{pzr}}+{\displaystyle \sum _{z,d}{X}_{pzd}}\right)\forall p$$(28)$$\beta (p)\le g(p)\times BM\forall p$$(29)$$\beta (p)\ge g(p)\forall p$$(30)$${\displaystyle \sum _{z}XP{R}_z^p}\le (1-g(p))\times BM\forall p$$

Since the model is defined for a reverse logistics network that has several centers, there are some complexities and limitations that were tried to overcome the shortcomings and limitations by adding auxiliary variables.

4.1.1 Solution approach

There are three objective functions in model 1, in which they are maximized. To prevent this, an objective function does not decrease the role and the effect of another objective function. To solve the model and optimize it, the global criteria method has been used. This method tries to find a compromise between all three objectives so that it minimizes the whole relevant deviance of all objectives from their main and optimal values Z^*_i. In other words, the global criteria method converts three objectives into a single-objective and then solves it. Finally, by using the global criteria method, the last objective function is as (31):(31)$$\begin{array}{l}{Z}_1=Max\left(\begin{array}{l}{\displaystyle \sum _{DC}{\displaystyle \sum _d{\displaystyle \sum _{x}RE{V}_s^{DC}{X}_{ds}^{DC}}}}+{\displaystyle \sum _{RC}{\displaystyle \sum _f{\displaystyle \sum _{s}RE{V}_s^{RC}{X}_{fs}^{RC}}}}+\\ {\displaystyle \sum _{RP}{\displaystyle \sum _r{\displaystyle \sum _{s}RE{V}_s^{RP}{X}_{rs}^{RP}}}}+{\displaystyle \sum _q{\displaystyle \sum _k{\displaystyle \sum _{s}RE{V}_s^q{X}_{ks}^q}}}+{\displaystyle \sum _P{\displaystyle \sum _d{\displaystyle \sum _{z}RE{V}_z^p{X}_{dz}^{\prime p}}}}\end{array}\right)-\\ \left(\begin{array}{l}{\displaystyle \sum _P{\displaystyle \sum _z{\displaystyle \sum _d{X}_{zd}^p\left(t{c}_p{d}_{zd}+P{C}_d^P+C{C}_d^P\right)}}}+{\displaystyle \sum _P{\displaystyle \sum _z{\displaystyle \sum _r{X}_{zr}^p\left(t{c}_p{d}_{zr}+P{C}_r^P+C{C}_r^P\right)}}}+\\ {\displaystyle \sum _P{\displaystyle \sum _d{\displaystyle \sum _z{X}_{dz}^{\prime p}\left(t{c}_p{d}_{zd}\right)}}}+{\displaystyle \sum _{RP}{\displaystyle \sum _r{\displaystyle \sum _s{X}_{rs}^{RP}\left(t{c}_{RP}{d}_{rs}\right)}}}+{\displaystyle \sum _q{\displaystyle \sum _k{\displaystyle \sum _s{X}_{ks}^q\left(t{c}_q{d}_{ks}\right)}}}+\\ {\displaystyle \sum _{RC}{\displaystyle \sum _f{\displaystyle \sum _s{X}_{fs}^{RC}\left(t{c}_{RC}{d}_{fs}\right)}}}+{\displaystyle \sum _{DC}{\displaystyle \sum _d{\displaystyle \sum _s{X}_{ds}^{DC}\left(t{c}_{DC}{d}_{ds}\right)}}}+\\ +{\displaystyle \sum _{FC}{\displaystyle \sum _d{\displaystyle \sum _f{X}_{df}^{FC}\left(t{c}_{FC}{d}_{df}+P{C}_f^{FC}\right)}}}+{\displaystyle \sum _{DI}{\displaystyle \sum _d{\displaystyle \sum _k{\displaystyle \sum _b{X}_{dkb}^{DI}\left(t{c}_{DI}{d}_{dk}+U_B^{DI}\right)}}}}+\\ {\displaystyle \sum _{RK}{\displaystyle \sum _R{\displaystyle \sum _k{\displaystyle \sum _b{X}_{rkb}^{RK}\left(t{c}_{RK}{d}_{rk}+U_B^{RK}\right)}}}}+{\displaystyle \sum _{FK}{\displaystyle \sum _F{\displaystyle \sum _K{\displaystyle \sum _b{X}_{fkb}^{FK}\left(t{c}_{Fk}{d}_{fk}+U_B^{FK}\right)}}}}\\ {\displaystyle \sum _{D}f{f}_{D}Y{d}_D}+{\displaystyle \sum _{R}f{f}_{R}Y{r}_R}+{\displaystyle \sum _{F}f{f}_{f}Y{f}_f}+TGC+{\displaystyle \sum _P{\displaystyle \sum _Z{M}_{big}XP{R}_{PZ}}}\end{array}\right)\end{array}$$

4.2 Model formulation (model 2)

In this part of the research, providing the possibility of applying important policies to the input and output, resource allocation processes and increasing resource efficiency is done for the decision-makers and managers in the form of an inverse data envelopment analysis. Therefore, an input-oriented model is presented to estimate input values (Hadi-Vencheh and Foroughi, 2006).

In the proposed input-oriented model, the focus is on the disassembly centers. In this model, input values are constant, and the efficiency is either constant or at least better, but the output values change, so the number of inputs to apply to this change of the output values will be determined. The notations of model 2 are presented in the following.

1. The notations related to Model 2 are as follows:

Assumptions

There are four disassembly centers have been considered

There are one input and one output for decision-maker units

Sets

D: Sets of disassembly centers

Parameters

j׳: Sets of output from disassembly centers

i׳: Sets of input for disassembly centers

Variables

β (j): Variable of the initial totals of output

λ (d): Variable of the total of changes in input

α (i): Variable of the initial totals of input

In'po (i): Auxiliary parameter for input to disassembly centers

ou'p (j): Auxiliary parameter for output from disassembly centers

1. Objectives and constraints (Model 2)

Due to the notations, the input-oriented inverse data envelopment analysis is as follows:(32)$$Min({\alpha }_{o1},{\alpha }_{o2},\mathrm{...},{\alpha }_{oi}),i=input\forall DMU$$(33)$${\displaystyle \sum _{d}I{n}_{i^{\prime }d}^{\prime }{\lambda }_d}\le \left(1-n_o\right){\alpha }_{o{i}^{\prime }}\forall i^{\prime }\in I$$(34)$${\displaystyle \sum _{d}o{u}_{j^{\prime }d}^{\prime }{\lambda }_d}\ge {\beta }_{o{j}^{\prime }}\forall j^{\prime }\in J$$(35)$${\alpha }_{i^{\prime }o}\ge I{n}_{i\prime o}^{\prime }\forall i^{\prime }\in I$$

The objective function (32) minimizes the initial value of the input. Constraints specify that the total of input weights cannot be less than their predecessor, and the total output weights cannot be more than their predecessors.

5. Numerical example

The data under investigation is a case study from a company that is one of India's largest 3PL service providers, specializing in reverse logistics, with the simulation and inspiration from related research (John et al., 2017). The data has been achieved and is in line with the subject under review. In this study, two types of products, including mobile phones and digital cameras, have been examined. There are different components and items which make mobile phones and digital cameras. After use, some of these components are hazardous, unusable and are dangerous for the environment, so they should be sent to the disposal and recycling center. Some of them are reusable after recovering and repairing and should be sent to remanufacturing and repairing centers. A few of them are recyclable and can be sent to the disposal and recycling center and be recycled. After recycling, the plastics and some valuable metals are achieved and should be sent to the markets.

The models are solved using GAMS software CPLEX solver, and the results are obtained. It is considered that the unit revenue obtained from different markets for an item is different across the markets; the corresponding value for revenue and the data for product return quantity, as provided by the 3PL company, are given in Table 2.

Also, it is considered that different items have various unit transportation costs per unit distance because of some factors such as the type of the product return, size of the item to be transported and quantities to be transported. Therefore, the unit transportation cost in money unit (in MU) for each product and component, and in terms of three performances, the unit disposal cost has been determined in Table 3.

The capacities of various facilities are chosen so that the total existing capacity of a special type of facility is more than the capacity required to process different items at those facilities. The fixed cost of opening a facility is different depending on the location and capacity. Capacity, unit processing cost, and fixed facility cost of disassembly centers for the two product types are given in Table 4.

The cost of maintaining products in product return zones for both products has a fixed value of 100. The cost of grading products is 10% of the cost of processing the products. The input and output values for disassembly centers have been generated in the form of pin (i, d) = uniform (25,100) and pou(j,d) = uniform (50, 100), respectively.

5.1 Computational results

In this sub-section, in order to illustrate the characteristics of the proposed model (model 1) and to execute it, this model is used to solve a practical example, and its related results are shown. The numerical example is considered for a single-period. First, the multi-objective model results (model 1) are presented as the results of the first objective function, then the results of the second objective function, and finally, the results of the third objective function are reported. After that, the modeling results are presented using the global criteria method in the form of a single-objective function. On the other hand, two types of sensitivity analyses are carried out related to the different parameters, such as α, β and ρ.

The end-of-use products have high-product residual value (high-PRV). Since the parameter α for such products is defined, different values are considered, and changes in the profit function of the proposed model 1 in the network compared with the changes in various values of α.

On the one hand, in the proposed model 1, it is assumed that a number of the products sent from the product return zones to the disassembly centers if there is sufficient capacity, can be returned to the product return zones and stored there; in this case, the parameter β is defined. Therefore, the sensitivity analysis is done on β, and the changes in the profit function of the proposed model 1 in the network are compared with the changes in the values of β. Also, in terms of the products produced during the recycling process, the parameter ρ is defined, in which the changes in the profit function of the proposed model 1 in the network are compared to different values of ρ.

On the other hand, the results of the sensitivity analysis of the output values and the estimation of its input values are reported in the form of an inverse data envelopment analysis (model 2). Input values and the efficiency are kept constant, but the output values are changed to eventually determine how much input is required for this change in output values for disassembly centers.

5.1.1 Computational results of the first objective function

In this case, the objective function aims to maximize the overall profitability of the network. The profit for the first objective function, equation (1), is 4511493.18. First objective function results are shown in Table 5.

5.1.2 Computational results of the second objective function

The second objective function consists of two terms: the first term attempts to maximize the quantities of the products sent to the market, and the second term is used to minimize the quantities of the buried products (if possible, recycle more items) and then the objective function operates to maximize recyclable values. The results are shown in Table 6.

5.1.3 Computational results of the third objective function

The third objective function, equation, serves to maximize the efficiency of the various centers of the disassembly centers. The results are shown in Table 7.

5.1.4 Computational results of the single-objective function

The single-objective function minimizes the difference between the first, second and third objective goals with their ideal values by the global criteria method. The results are shown in Table 8.

According to the results shown in Table 8, the effect of the second and the third objective functions on the first objective function in relation to the profit is greater than the effect of the first and the third objective function. On the second objective function, which is related to the green factors and it is also greater than the effect of the first and the second objective function. On the third objective function which is related to the value of efficiency.

5.1.5 Sensitivity analysis on parameters α, β and ρ

A sensitivity analysis on α, β and ρ parameters is performed with these parameters' different values in this section. In different experiments on these parameters, the values of the changes for the first, second and third objective functions are considered.

5.1.5.1 Sensitivity analysis on parameter α

The experiment set 1: α₁ is set to a specific value starting from 0.4, and α₂ is 0.5 and varied up to a point where the network is more profitable as compared to the base case. In this experiment, α₁ = 0.4, and all the other cases are obtained by a 0.5% change in the value of α₂. The values of other parameters are: β₁ = 0.25, β₂ = 0.25, ρ₁ = 0.2, ρ₂ = 0.1. These experiments were performed in three cases 1, 2 and 3 and the results are shown in Table 8. Also, the changes of the first to third objective functions are shown in Figure 3.

According to Figure 3, by an increase of 0.5% in α₂, under a constant α₁, the value of the first objective function (profit) increases and the values of the second and the third objective functions are not varied. Similarly, in set 2, α₂ is fixed and α₁ is changed in three cases 4, 5 and 6. The results are according to Table 8. In this case, the changes to the objective functions are like set 1.

5.1.5.2 Sensitivity analysis on parameter β

The experiment set 3: β₁ is set to a specific value starting from 0.25, and β₂ is 0.25 and varied up to a point where the network is more profitable as compared to the base case. In this experiment, β_{1 =} 0.25, and all the other cases are obtained by a 5% change in the value of β₂. The values of other parameters are: α₁ = 0.4, α₂ = 0.5, ρ₁ = 0.2, ρ₂ = 0.1. These experiments were performed in two cases, 7 and 8, and the results are shown in Table 8. Also, the changes of the first to third objective functions are shown in Figure 4.

According to Figure 4 by an increase of 0.5% in β₂, under a constant β₁, the values of the first objective function (profit) and the second objective function increase and the values of the third objective function are not varied. On the other hand, it is observed that by increasing the first objective function in the sensitivity analysis, the β values have increased significantly in comparison to the sensitivity analysis of the α values. Also, the values of the second objective function have increased, which increases the positive environmental effects. It can be concluded that the parameter β, in comparison to the parameter α, is more sensitive to changes in values. Similarly, in set 4, β₂ is fixed and β₁ is changed in two cases, 9 and 10. The results are according to Table 8. In this case, the changes to the objective functions are like set 3.

5.1.5.3 Sensitivity analysis on parameter ρ

The experiment set 5: ρ₁ is set to a specific value starting from 0, and ρ₂ is 0 and varied up to a point where the network is more profitable as compared to the base case. In this experiment, ρ₁ = 0 and all the other cases are obtained by a 10% change in the value of ρ₂. The values of other parameters are: α₁ = 0.4, α₂ = 0.5, β₁ = 0.25, β₂ = 0.25. These experiments were performed in 4 cases 11 to 14 and the results are shown in Table 8. Also, the changes of the first to third objective functions are shown in Figure 5.

According to Table 8 and Figure 5, by keeping the value of ρ₁ constant and increasing the value of ρ₂ from 0 to 1 by 10%, the first objective function (profit) is almost constant in the range of 0–0.6 and is increased in the range of 0.6–1. It can be concluded that the first objective function for ρ₂, in the range of 0.6–1, shows more sensitivity, and has an increasing trend. In this sensitivity analysis, only the first objective function value has been changed, and the second and the third objective functions have not received significant changes. Similarly, in set 6, ρ₂ is fixed and ρ₁ is changed in 4 cases 15 to 18, the results are according to Table 8. In this case, the changes to the target functions are like set 6.

5.1.5.4 Crossover-sensitivity analysis of parameters α, β and ρ

In this section, the sensitivity analysis of the parameters is examined simultaneously in such a way that the changes in the values of these parameters are not separated, and all three parameters are analyzed together. The effects of these changes on all three objective functions will also be examined.

5.1.5.5 First scenario

At this part of the analysis, the information from the previous experiments is used. Changes in all three parameters, and, in the case where the profit has the highest value, are checked.

1. The experiment set 7: in this situation, changes to the parameters are considered, and the profit function has the highest value; therefore, case 3 from experiment set 1, case 8 from experiment set 3 and case 14 from experiment set 5 are considered, and these values of the parameters are crossover-analyzed simultaneously, in which the first parameters α₁, β₁ and ρ₁ are constant. By combining the changes in the values of the parameters for the first objective function, the profit is 5872760.2796, which is greater than the values of all three earning functions in Table 8. Also, as for the second objective function, it yields a value of 0.1261, which is somewhat improved compared to its initial value of 0.1044 and contributes to maintaining environmental sustainability. The third objective function, which shows the performance values, has not changed the original value. Finally, in experiment set 7, the increase in the value of the profit function is 23.17%, and the increase in the second objective function is 17.2%. The results are shown in Figure 6a.

2. The experiment set 8: in this situation, changes to the parameters are considered, and the profit function has the highest value; therefore, case 6 from experiment set 2, case 10 from experiment set 4, and case 18 from experiment set 6 are considered, and these values of the parameters are crossover-analyzed simultaneously, in which the second parameters α₁, β₂ and ρ₂ are constant. It can be seen in Table 8 that by combining the changes of the values of the parameters, for the first objective function, the profit is 5830135.1537, which is smaller than the profit of case 6 and is greater than the profit of cases 10 and 18. Also, for the second objective function, it has a value of 0.1261, which has improved somewhat compared to its initial value of 0.1044 and has a greater contribution to maintaining environmental sustainability. The third objective function, which shows the performance values, has not changed the original value. Finally, in experiment set 8, the increase in the profit function is 22.61%, and the increase in the second objective function is 17.2%. The results are shown in Figure 6b.

5.1.5.6 Second scenario

At this part of the analysis, the information from the previous experiments is used. The changes in all three parameters are examined in such a way that the value is constant before these changes and begins to increase after these changes. According to the results obtained from Table 8, there are two modes in relation to the changes of the values of α and ρ, and for these changes, the first objective function has begun to change its value.

1. The experiment set 9: in this situation, changes to the parameters are considered, and the profit function begins to change; therefore, case 2 from experiment 1 and case 12 from experiment 5 are considered, and these values of the parameters are crossover-analyzed simultaneously, in which the first parameters α₁ and ρ₁ are constant. In this part, β is β₁ = 0.25, β₂ = 0.25. The obtained profit function is 4511555.5114, which according to the values of the profit function of Table 8, equals the profit of case 2 and is lower than the profit of case 1. There are no changes in the second and the third objective functions compared to cases 2 and 12. Finally, in experiment set 9, the increase in the profit function is 0.001%. The results are shown in Figure 7a.

2. The experiment set 10: in this situation, changes to the parameters are considered, and the profit function begins to change; therefore, case 5 from experiment 2 and case 17 from experiment 6 are considered, and these values of the parameters are crossover-analyzed simultaneously, in which the first parameters α₂ and1₂ are constant. In this part β is β₁ = 0.25, β₂ = 0.25. The obtained profit function is 4513116.9228, which, according to the values of the profit function of Table 9, is greater than the profit of cases 5 and 17. There no changes in the second and the third objective functions compared to cases 2 and 12. Finally, in experiment 10, the increase in the profit function is 0.03%. The results are shown in Figure 7b.

According to the results of Table 9 and considering several scenarios, for the reverse logistics network for both the digital camera and mobile phone products, in the real world, the network's profit can be increased and the destructive environmental effects can be reduced. Choosing the right supplier can also increase efficiency.

5.1.6 Sensitivity analysis of input/output values and the estimation of input values (scenario 1)

According to equation (32), an input-axis model is presented, which, under constant efficiency or its improvement, examines changes in input values with respect to the increase in output values. The current situation for input and output values for the disassembly centers and their efficiency are shown in columns 2, 3 and 4 in Table 9. This section aims to determine how much of the input is required to increase the output, assuming that the values of efficiency are maintained. By solving the inverse DEA model, the desirable situation 1 in columns 5 and 6 in Table 9 show the changes in output values in which each output value changes compared to its initial value, and after that, the new value of input has been obtained.

It can be analyzed that in the first and the second centers, which performance values are 0.92 and 1 and have better performances than the third and the fourth centers, which performances are 0.69 and 0.43, respectively, the final input values in the fifth column of Table 9 have changed due to the changes in the output values in the fourth column of each center and the need to increase of the input is lower; but for the third and the fourth disassembly centers with a weaker performance level, which is respectively 0.69 for the third center and 0.43 for the fourth center, it can be analyzed for the changes in the output values in the fourth column of Table 9 for each center in which the increase of the final input in the fifth column for the third and the fourth centers is more than the first and the second disassembly centers. Finally, to analyze the changes in the output values for the third and the fourth centers with weak performances level, it is necessary to increase the input for allocating resources. It can be concluded that by increasing efficiency to achieve the desired output level, the less input use to allocate resources must be considered and economical for the network.

5.1.7 Sensitivity analysis of input/output values (scenario 2)

In this section, as in the previous one, there were changes in the output values, but the efficiency of the total disassembly centers was upgraded to 1. In this situation, model 2 estimates the values of the input changes again. This section aims to determine how much of the input is required to increase the output, assuming that the values of efficiency are upgraded to 1. By solving the inverse DEA model, the desirable situation 1 in columns 5 and 6 in Table 10 show the changes in output values in which each output value changes compared to its initial value, and after that, the new value of input has been obtained.

By increasing the efficiency to 1, the final input changes in the desirable situation two decreases in comparison to the changes of the final input in the desirable situation 1. For example, the final input in the desirable situation one in column 6 has decreased from 54.864 to 53.093 in column 9 at the desirable situation two for the first center. This has been unchanged for the second center. For the third center, the number 52.376 in column 6 has been decreased to 45.785 in column 9, and for the fourth center, the number 103.141 in column 6 has been decreased to 74.626 in column 9.

6. Conclusion

Today, designing a reverse logistics network to provide after-sales service has become one of the most important competitive advantages for many companies. In this regard, companies are trying to have a reverse logistics network with appropriate and profitable service.

In this research, a mixed-integer linear programming model for closed loop reverse logistics is proposed in order to provide after-sales service for products. This MILP network is multi-product and includes six levels of product return zones, remanufacturing centers, repairing centers, disposal and recycling center, disassembly centers and markets. The proposed model is multi-objective. This model focuses on electrical products. In this network, an attempt has been made to establish effective relations between the centers. In addition to maximize profits, this network tries to preserve and sustain the environment by mathematically integrating the concepts of GSC management and by increasing the number of useable items, optimal disposal, and recycling. This model also analyzes the efficiency of disassembly centers with the help of SDEA.

The network has three objective functions aimed at maximizing profits, minimizing damage to the environment and increasing the efficiency of the centers. Solving approach of the model is as follows: First, triple objective functions are solved separately and the results are reported. First step was performed with the aim of determining total profit of the model, environmental impact and efficiency of disassembly centers separately. Then, the objective functions become a single-objective function and is solved by the comprehensive criterion method. Sensitivity analysis performed by changing the parameters separately and cross in different scenarios.

The proposed model was used in a case mining of after-sales service for mobile phone and digital camera products. The results compared to the results of previous research showed that: The first objective function has created a significant increase in the profit of the total network. In the objective function 2, a significant number of products are properly disposed and the rest are recycled and valuable materials are obtained from this recycling. This objective function has played an important role in the sustainability of the environment and the reduction of pollution from improper disposals, as well as in the profitability of the network. In the objective function 3, the efficiency values of disassembly centers were determined.

As well as, another innovation of the present study is improving low-efficiency disassembly centers. In order to increase the performance of low-efficiency disassembly centers, suitable inputs were proposed using inverse DEA. Since a number of products are stored in the product return centers, the condition of being closed-loop is also satisfied. This model can be used in the design of after-sales service networks for digital goods. Other applications of the model are to help maintain environmental sustainability and to estimate network resources with respect to rational inputs and outputs and to determine suppliers in network components.

Combining GSC indicators in the reverse logistics model, developing the reverse logistics model to select the supplier, and combining the reverse logistics model and Inverse DEA to estimate financial resources are the most important innovations of the proposed model.

Global Criteria is a Multi-criteria Decision-making (MCDM) method used to solve and optimize multi-objective models. The advantages of the method include considering several goals at the same time and presenting the results. Another advantage of this method is the shorter solution time than other methods for large-scale models, which makes it easier to use this method.

Finally, considering possible uncertainties for the inputs of the disassembly centers, converting single-cycle model to multi-cycle model, and combining the proposed model with the robust stochastic model are the suggestions for research development.

Figure 1

EEE put on the market and WEEE collected and treated, EU-27, 2010–2017

[Figure omitted. See PDF]

Figure 2

Structure of the network design

[Figure omitted. See PDF]

Figure 3

Experiment results of sets 1 and 2

[Figure omitted. See PDF]

Figure 4

Experiment results of the sets 3 and 4

[Figure omitted. See PDF]

Figure 5

Experiment results of the sets 5 and 6

[Figure omitted. See PDF]

Figure 6

Results of the experiment sets 7 and 8

[Figure omitted. See PDF]

Figure 7

Results of the experiment sets 9 and 10

[Figure omitted. See PDF]

Table 1

Some researches related to current research areas

Year	Authors	CE	GSCM	RL	CLSC	WEEE	MILP	SDEA
2016	Kaya and Urek				√		√
2016	Govindan, Paam and Abtahi		√		√	√
2015	Govindan, Jafarian and Nourbakhsh		√	√
2017	H. Yu and Solvang			√			√
2017	Pauliuk et al.			√
2017	Rezaee, Yousefi and Hayati							√
2018	Alshamsi and Diabat			√			√
2019	Kazemi, Modak and Govindan				√	√
2019	Qureshi et al.		√	√
2019	Bo, Wang and Wan					√	√
2019	Ahmadi and Amin				√	√
2019	Wagner et al.	√				√
2020	Chan et al.			√
2020	Krug, Guillaume and Battaïa				√
2020	Fan, Pintarič and Klemeš			√			√
2020	Moslehi, Sahebi and Teymouri			√		√
2021	Polat and Gungor			√	√	√	√
2020	Qu et al.		√		√
2020	Bressanelli et al.	√				√
2021	Ruan Barbosa de Aquino et al.	√				√
2021	Khan et al.	√	√	√

Assumptions

The logistics network in this study consists of six levels, five product return zones, four disassembly centers, three remanufacturing center, two markets, three repairing centers, one disposal and recycling centers

This network is considered under certain conditions of return of products

A single mode of transportation is considered

The network designed for single-period

This network considered input and output for disassembly centers. There is one input i and one output j for disassembly centers

Sets

Z: Set of product return zones

D: Set of disassembly centers

F: Set of repairing/reconditioning centers

R: Set of remanufacturing centers

S: Set of markets

K: Set of disposal and recycling centers

P: Set of all product types

A: Set of all combinations of any two types of facilities across which flow of different

N: Set of all entities in the network whose locations have to be fixed

I: Set of all types of products/components/materials in the network

DI: Set of disposable items

DC: Set of directly reusable components

FC: Set of repairable components

RC: Set of repaired/reconditioned components

RP: Set of remanufactured products

RK: Set of disposal items which delivered from the remanufacturing center

FK: Set of disposal items which delivered from repairing center

B: Set of disposal operations (b1, b2), and recycling operations (b3)

q: Set of recycled items, plastic, and Metal, (q1, q2)

i, j: Index set for different types of items in the network

i': Set of inputs to disassembly centers

J': Set of outputs from disassembly centers

m, n: Index set for different types of facilities in the network

Parameters

PRim: Number of products returned of type i at product return zone m, where i ∈ P, m ∈ Z

PCin: Unit processing cost of product/component/material i, at the facility n, where n∈ N, i∈ I

CCin: Unit cost of the product returned i, i ∈ p

Ui: Unit cost of disposal of the material i

dmn: Distance between facilities m and n, where (m, n) ∈ A

tci: Unit transportation cost of product/component/material i per unit distance, where i ∈ I

Sin: Unit cost of grading for product i

fn: Fixed cost of the facility n, where n ∈ N

hi: Unit maintaining cost of product in product return zones Z, per each product i

Pin: Input values for disassembly centers D

$Pout$: Output values for disassembly centers D

BM: A big number

REVij: Revenue obtained from market n for the item i where i ∈ {DC, RC, RP}

NCij: Number of units of an item of type j produced from product/component i

αi: Percentage of high-pressure reducing valve (high-PRV) items in a product type i, where i ∈ P

βi: Percentage of returnable product i from D to Z, i ∈ P

ρq: Percentage of product q which can be produced during recycling operation in K, q ∈ q1, q2

γb: Importance ratio for disposal operation b1 and b2

W: Weight of the objectives function

xpp: The number of product p1 and product p2

MU: Monetary units

TP: Total profit

TGC: Total grading cost

Capin: Capacity of the facility n, for product/component/material i where n ∈ N, i ∈ I

Variables

Ximn: Quantity of product/component/material i shipped from facility m to facility n, where (m, n) ∈A, i ∈ I

Yn: {0,1} variable, Yn = 1f the facility n is opened else Yn = 0, ∀ n ∈ N

Xpr: Auxiliary variable, capacity deficiency variable for R and D (Maintenance variable for the Z center)

X'pdz: Closed-loop variable to create functionality for returning goods from the center of disassembly centers D to product return zones Z if necessary

ρp: The integer auxiliary variable for the maintenance variable, in the direction to prioritize the maintenance variable in order to send the products to the D and R centers, and then store them in Z

gp: {0,1} Integer auxiliary variable for the maintenance variable, in the direction to prioritize the maintenance variable in order to send the products to the D and R centers, and then store them in Z

n(d): Inefficiency variable of disassembly centers D

in: Input variable for disassembly centers D

ou: Output variable for disassembly centers D

Table 2

Product return quantity and unit revenue generated (in MU) from markets for different items

Product return quantity
Product return zone	Z1		Z2		Z3		Z4		Z5
Product type	P1	P2	P1	P2	P1	P2	P1	P2	P1	P2
Quantity of product return	72	105	63	76	54	64	90	114	80	93

Unit revenue
Component/material	DC1	DC2	RC1	RP1	RP2	q1	q2
S1	380	310	970	3,650	2,600	3,000	2,700
S2	390	320	990	3,800	2,700	4,000	3,700

Table 3

Unit transportation cost and unit disposal cost (in MU)

Product/component		P1	P2	RP1	RP2	FC1	DC1	DC2	DI1	DI2	RC1	q1	q2	Rk1	Fk1
Unit transportation cost		7	7	5.5	5	2	3	2.5	3	2.5	2	6	8	2	2
Unit disposal cost	b1								70	60				100	50
	b2								210	180				300	150
	b3								400	400				500	350

Table 4

Capacity, unit processing cost and fixed facility cost of all centers

Disassembly centers									Disposal and recycling center
	1		2		3		4
Product	P1	P2	P1	P2	P1	P2	P1	P2	b1	b2	b3
Capacity	130	160	180	210	80	120	110	140	100,000	500	1,000
Unit processing cost (MU)	166	138	175	142	170	135	162	147
Fixed facility cost (MU)	30,000		28,000		24,000		27,000

	Repairing centers			Remanufacturing centers
				R1		R2		R3
Product	F1	F2	F3	P1	P2	P1	P2	P1	P2
Capacity	360	470	410	50	70	80	80	60	80
Unit processing cost (MU)	110	115	104	1,040	980	1,120	1,090	1,190	1,030
Fixed facility cost (MU)	27,500	33,500	22,000	47,000		54,000		61,500

Table 5

Results of the first objective function

	Z1	Z2	Z3	Z4	Z5
$PR$	1770	1,390	1,180	2040	1730
$Xpr$	1,493	1,390	1,180	1,334	1,163

	S1	S2		S1	S2
XDC			R1		240
D1		436	R2		320
D2	293	293	R3		280
D3		300	Xq
D4		376	q1		400
XRP			q2		200

Table 6

Results of the second objective function

S				K
		S1	S2			b1	b2	b3
XRP	R1	240		XDI	D1	9	218	209
	R2	160	160		D2	133		453
	R3	280			D3	56	94	150
XDC	D1	218	218		D4		188	188
	D2	586		XRK	R1	120
	D3	300			R2	160
	D4		376		R3	140
XRC	F1		293	XFK	F1	293
	F2	188			F2	188
	F3	368			F3	368
Xq	q1		384
	q2	200			Sum	1,467	1,500

Table 7

Results of the third objective function

	D1	D2	D3	D4
n	0.071	0	0.304	0.575
Input	0.019	0.024	0.021	0.013
Output	0.011	0.014	0.012	0.008

Table 8

Results of the single-objective function

	Solving separately	Solving with global criteria method	Difference (%)
First objective function	4511493.1851	4347879.495	3.62
Second objective function	0.1044	0.104	0.38
Third objective function	3.0491	3.049	0.003

Table 9

Results of sensitivity analysis on parameters α, β and ρ

Experiment	Case	Parameters	First objective function (total profit)	Percentage of change (%)	Second objective function	Percentage of change (%)	Third objective function	Percentage of change (%)
Experiment set 1	Case 1 (base case)	α1 = 0.4, α2 = 0	4511493.18	–	0.1044	–	3.0491	–
	Case 2	α1 = 0.4, α2 = 2	4511555.51	0.001	0.1044	0	3.0491	0
	Case 3	α1 = 0.4, α2 = 10	4512621.60	0.02	0.1044	0	3.0491	0
Experiment set 2	Case 4 (base case)	α1 = 0.4, α2 = 0.5	4511493.18	–	0.1044	–	3.0491	–
	Case 5	α1 = 2.5, α2 = 0.5	4511545.96	0.001	0.1044	0	3.0491	0
	Case 6	α1 = 10, α2 = 0.5	4512324.39	0.018	0.1044	0	3.0491	0
Experiment set 3	Case 7 (base case)	β1 = 0.25, β2 = 0.2	4511493.18	–	0.1044	–	3.0491	–
	Case 8	β1 = 0.25, β2 = 1	5276045.94	14.49	0.1261	17.2	3.0491	0
Experiment set 4	Case 9 (base case)	β1 = 0.25, β2 = 0.2	4511493.18	–	0.1044	–	3.0491	–
	Case 10	β1 = 1, β2 = 0.25	5157321.39	12.52	0.1261	17.2	3.0491	0
Experiment set 5	Case 11 (base case)	ρ1 = 0, ρ2 = 0	4511493.18	–	0.1044	–	3.0491	–
	Case 12	ρ1 = 0, ρ2 = 0.6	4511493.18	0	0.1044	0	3.0491	0
	Case 13	ρ1 = 0, ρ2 = 0.7	4562938.50	1.12	0.1044	0	3.0491	0
	Case 14	ρ1 = 0, ρ2 = 1	5002746.48	9.81	0.1044	0	3.0491	0
Experiment set 6	Case 15 (base case)	ρ1 = 0, ρ2 = 0	4511493.18	–	0.1044	–	3.0491	–
	Case 16	ρ1 = 0.5, ρ2 = 0	4511493.18	0	0.1044	0	3.0491	0
	Case 17	ρ1 = 0.6, ρ2 = 0	4513064.13	0.03	0.1044	0	3.0491	0
	Case 18	ρ1 = 1, ρ2 = 0	5150739.83	12.41	0.1044	0	3.0491	0
Experiment set 7	Case 3	α1 = 0.4, α2 = 0	4512621.60	0.02	0.1044	0	3.0491	0
	Case 8	β1 = 0.25, β2 = 1	5276045.94	14.49	0.1261	17.2	3.0491	0
	Case 14	ρ1 = 0, ρ2 = 1	5002746.48	9.81	0.1044	0	3.0491	0
	Total result	α1 = 0.4, α2 = 0.25	5872760.27	23.17	0.1261	17.2	3.0491	0
		β1 = 0.25, β2 = 1
		ρ1 = 0, ρ2 = 1
Experiment set 8	Case 6	α1 = 10, α2 = 0.5	4512324.39	0.018	0.1044	0	3.0491	0
	Case 10	β1 = 1, β2 = 0.25	5157321.39	12.52	0.1261	17.2	3.0491	0
	Case 18	ρ1 = 1, ρ2 = 0	5150739.83	12.41	0.1044	0	3.0491	0
	Total result	α1 = 10, α2 = 0.5	5830135.15	22.61	0.1261	17.2	3.0491
		β1 = 1, β2 = 0.25
		ρ1 = 1, ρ2 = 0
Experiment set 9	Case 2	α1 = 0.4, α2 = 0.25	4511555.51	0.001	0.1044	0	3.0491	0
	Case 12	ρ1 = 0, ρ2 = 0.6	4511493.18	0	0.1044	0	3.0491	0
	Total result	α1 = 0.4, α2 = 0.25	4511555.51	0.001	0.1044	0	3.0491	0
		ρ1 = 0, ρ2 = 0.6
Experiment 10	Case 5	α1 = 0.25, α2 = 0.5	4511545.96	0.001	0.1044	0	3.0491	0
	Case 17	ρ1 = 0.6, ρ2 = 0	4513064.13	0.03	0.1044	0	3.0491	0
	Total result	α1 = 0.25, α2 = 0.5 ρ1 = 0.6, ρ2 = 0	4513116.92	0.0.3	0.1044	0	3.0491	0

Table 10

Result of sensitivity analysis of input/output values (inverse data envelopment analysis)

	Current situation			Desirable situation 1			Desirable situation 2
1	2	3	4	5	6	7	8	9
Disassembly centers	Initial input	Initial output	Efficiency	Final output	Final input	Desirable efficiency	Final output	Final input
D1	53.093	87.927	0.92	90	54.864	1	90	53.093
D2	41.216	73.491	1	80	44.866	1	80	44.866
D3	46.785	58.027	0.696	65	52.376	1	65	46.785
D4	74.626	56.516	0.435	80	103.141	1	80	74.626