Multi-objective robust optimization for multi-stage-multi-product agile closed-loop supply chain under uncertainty in the context of circular economy

Abstract

Purpose

A circular economy (CE) is an economic system that tries to eliminate waste and continually use resources. Due to growing environmental concerns, supply chain (SC) design should be based on the CE considerations. In addition, responding and satisfying customers are the challenges managers constantly encounter. This study aims to improve the design of an agile closed-loop supply chain (CLSC) from the CE point of view.

Design/methodology/approach

In this research, a new multi-stage, multi-product and multi-period design of a CLSC network under uncertainty is proposed that aligns with the goals of CE and SC participants. Recycling of goods is an important part of the CLSC. Therefore, a multi-objective mixed-integer linear programming model (MILP) is proposed to formulate the problem. Besides, a robust counterpart of multi-objective MILP is offered based on robust optimization to cope with the uncertainty of parameters. Finally, the proposed model is solved using the e-constraint method.

Findings

The proposed model aims to provide the strategic choice of economic order to the suppliers and third-party logistic companies. The present study, which is carried out using a numerical example and sensitivity analysis, provides a robust model and solution methodology that are effective and applicable in CE-related problems.

Practical implications

This study shows how all upstream and downstream units of the SC network must work integrated to meet customer needs considering the CE context.

Originality/value

The main goal of the CE is to optimize resources, reduce the use of raw materials, and revitalize waste by recycling. In this study, a comprehensive model that can consider both SC design and CE necessities is developed that considers all SC participants.

1. Introduction

Today, many industries follow a linear model of sourcing materials, using them, and finally disposing of the generated waste. This linear model has negative environmental impacts due to resulting in ecosystem pollution and unsustainable resource usage. In recent years, most companies have focused on sustainable growth, including a switch toward the circular economy (CE) model (Jaeger and Upadhyay, 2020). The CE is a new production and consumption model that endeavors to find solutions to environmental problems. The management of the circular economy supply chain (CESC) aims at revitalizing the flow of different resources within the economy by recycling waste and reducing emissions (Formentini et al., 2021). Due to rising global environmental problems, the transition to a CE should prioritize all industries. However, the transition requires a transparent and controllable supply chain (SC), suggesting the importance of a good SC design (Reimann et al., 2019; Yousefi et al., 2017). Since CE considerations can impose high costs on the SC, it is necessary to establish a comprehensive SC design model that considers CE. Also, integrating sustainable goals into the planning of corporations is of great necessity in completely implementing the CE concept. Firms should be able to cooperate with a network of third parties. In this regard, closed-process chains can be very useful in preventing and reducing the impacts of damaging activities on the environment. This approach is effective in managing the entire SC with the aim of recycling and refurbishing products. Therefore, the implementation of circular SCs, which encompasses the closed-loop design, can provide useful outcomes for the components of the SC (e.g. manufacturer, distributor and supplier) and the environment. In this regard, the closed-loop strategy is more intertwined with the CE concept (Sillanpää and Ncibi, 2019).

Hence, the closed-loop supply chain (CLSC) is highly recommended for dealing with environmental concerns and strict rules on reducing waste in the product lifecycle (Reimann et al., 2019). A CLSC includes both forward and reverses chain movements of the product (Rezaee et al., 2017; Tosarkani et al., 2020). Forward SC comprises product movements from upstream to downstream. On the other hand, reverse logistics (RL) includes movements of either unsold goods or used ones by the client to the upstream side. Indeed, the RL aims to recycle and repair, which is the context of a CE (Sillanpää and Ncibi, 2019; Baptista et al., 2019). An efficient SC is essential to make products available to the customer on time (Yousefi et al., 2021). In the meantime, facing demand and price uncertainty is inevitable (Amaro and Barbosa-Póvoa, 2009). Some shortcomings of RL activities in a production-recycling system complicate the planning and management of SC. These issues include uncertainties, the necessity to adjust the demand for product returns, requirements of RL network, the prohibition of using some substance, random problems in repair and reproduce operations, and the number of variables related to operations (Guide et al., 2003; Tosarkani et al., 2018). The implementation of SC processes is not always precise; usually, the product's demands change with time due to fluctuations in consumption patterns and product life. Hence, companies must deal with demand prediction in dynamic design and modeling (Georgiadis et al., 2011). Supply chain network design (SCND) includes strategic decisions on the number of locations, capacity, efficient product distribution flow, and the demand forecast collaboration between echelons.

Meanwhile, decisions associated with the supplier's selection, contractors and third-party logistics (3PLs) are made aligned with product market development. The real world SCND is nondeterministic, and using robust strategic decision-making would greatly help decision-makers in product return (Klibi et al., 2010). The SCND configuration is a vital strategic issue that impacts practical and tactical activities for optimizing efficiency in a long time (Ramezani et al., 2013a, b). Supplier selection is critical because of purchase costs or market competition, impacting operations and financial aspects (Omurca, 2013). Many SCs do not consider the risk of product returns, which causes customer dissatisfaction and a substantial negative impact on their profits in the market (Yousefi et al., 2020). Concerning product value, time, market demand, and brand protection, manufacturers offer several ways to maximize the value of their returns (Das and Dutta, 2013). Any disruption or risk in SC directly affects the operations and the delivery of products or services to market. In these circumstances, risk management helps SC managers balance the demand and supply by providing appropriate resources and efficient customer delivery. Optimal SCND enables to achieve acceptable results with minimal cost, which is required for success in the market (Goh et al., 2007). Since SCs seek to improve their performance, outsourcing has gained more attention. In this connection, 3PLs are some companies that help network stability of RL services that allow business owners to raise profit margins. In general, a 3PL can be used for outsourcing all or part of the entire RL of many businesses. If returns are not processing wholly and quickly, the costs increase considerably (Efendigil et al., 2008). Here, 3PLs perform some or all of their contracting companies' logistics activities such as warehousing, distribution, inventory and transportation (Sharma and Kumar, 2015). Finally, SC design should not merely be based on profit optimization. Instead, environmental issues must also be considered in the design of SC. In this way, the SC can meet the requirements of the CE.

SCs consist of various components and many factors that play a major part in SCND. Accordingly, an effective model for optimal SCND should take into account all the involving components and elements. Formerly, many researchers have aimed at designing the SC network. However, most of these efforts have failed to consider all the concerning aspects. Furthermore, CLSC is one of the most important driving forces of the CE. Thus, any uncertainty in the design of CLSC can influence the efficiency of the CE. In other words, the uncertainty of the parameters and factors of SC components can affect the goals of the CE. Confronting the uncertainty of parameters, the robust counterpart of the proposed model is also written and solved by a constraint approach. Therefore, the main contribution of this paper is to develop a model for SCND that includes all SC participants and takes into account real-world uncertainty and CE considerations. This study proposes a comprehensive model for SCND to simultaneously optimize SC profits, customer satisfaction, and product returns. Overall, the main aim of this study is to develop a multi-objective mixed-integer linear programming (MILP) model to address the problem of a multi-stage, multi-product and multi-period design CLSC network under uncertainty based on the CE considerations. The proposed multi-objective MILP model developed based on the robust optimization determines the number of produced, disassembled, refurbished, destroyed, purchased and returned products by different components of the CLSC networks. Besides, the decisions on setting up and selecting the refurbishing site, disassembly site, suppliers and 3PLs are made for various periods. Here, the optimal values are obtained by maximizing the total profit of the manufacturer and customer satisfaction and minimizing the failed and reproduced parts through the network. In addition, the multi-objective robust model is solved, and its results are compared with the NSGA-II (Non-dominated Sorting Genetic Algorithm II) method. Notably, ε-constraint and NSGA-II are very popular in modeling and solving multi-objective robust models.

The remainder of this paper is organized as follows. Section 2 reviews the recent studies published in the SCND field and examines the application of CE in SCs. The problem description and formulation are presented in Section 3. Section 4 provides the solution methodology for the developed model. In Section 5, extensive numerical experiments and sensitivity analysis are provided. Finally, the conclusion and future research directions are discussed in Section 6.

2. Literature review

In this section, the recent studies focused on SCND using mathematical programming have been reviewed. Furthermore, Section 2.2 investigates the relationships between CE and SCs.

2.1 Supply chain network design

Focusing on SCND, Hatefi et al. (2015) have designed a multi-stage SC network based on forward and reverse flow, as well as hybrid production-recovery and distribution-collection facilities. Their main objective is to determine the optimal number and location of the facilities based on fuzzy credibility-constrained programming under uncertainty and facilities disruption. Djikanovic and Vujosević (2016) developed a MILP for designing an integrated network of forward and RL for a real case study in Serbia, which provided a mechanism for reusing electrical products and electronic equipment. They applied one-at-a-time methods for sensitivity analysis and minimizing total cost. Taki et al. (2016) proposed the MINLP models to optimize the location number of collecting, processing, and inspection centers in a three-stage RL. The combination of the genetic algorithm (GA) and simulated annealing (SA) is used to solve the proposed model.

Wang et al. (2013) proposed a multi-stage CLSC network along with a multi-objective programming model and presented a robust optimization model to determine the stability of the network. Mehrbod et al. (2012), in their study of the SC network, follow two important goals that include reducing costs and increasing customer satisfaction. An interactive fuzzy multi-objective goal programming model is proposed to achieve these goals. Çalık et al. (2017) presented a CLSC network based on multi-product, multi-period and multi-stage designs; indeed, the network is a combination of forward and reverse SC. Then, a multi-objective linear programming (LP) model is introduced. Further, a new solution based on interactive fuzzy programming is provided to solve the model and determine the weights of the objective functions.

Similarly, Vahdani et al. (2012) considered an SC network where nonlinear programming (NLP) is used to improve efficiency and reduce costs. They solved the model with fuzzy stochastic NLP to deal with the uncertainty of the parameters. Garg and Jha (2013) proposed a mathematical model for a single-product multi-period reverse logistic network. The model specifies the variables in each part of the chain, such as returned goods, parts and recovery using certain parameters. Dai (2016) offered a multi-objective LP model in a fuzzy environment for a multi-stage, multi-product CLSC network. The model includes four objectives, which minimizes the total cost, network failure probability, carbon dioxide (CO₂) emission and risk. Özkır and Başlıgıl (2012) designed a CLSC with three types of recycling processes, including recycling products, recycling materials and recycling parts. They proposed a mathematical model to maximize net revenue.

In the same way, Özceylan and Paksoy (2013) designed a multi-product network and provided a MILP model to reduce the costs, and then considered different scenarios (with changes in the model size, returns, demand, plant capacity, product collection rate) to examine the effectiveness of the model. Demirel et al. (2014) provided a multi-stage multi-period CLSC network considering two significant policies: (1) the policy of secondary market and (2) the policy of increasing motivation and inventory in RL flows. A GA based on crisp and fuzzy objective functions is offered to solve the model. Chen et al. (2015) designed a CLSC network to evaluate the optimal use of facilities and analyze transportation and delivery methods with quality-based return classification. Further, a MILP model is introduced and solved by a modified two-stage GA.

Eslamipoor et al. (2015) designed a CLSC under uncertainty. They applied a robust optimization model to cope with uncertainty. The purpose is to minimize the total costs of the SC. Kaya et al. (2014) presented a CLSC network under uncertainty and studied the network by considering two approaches, stochastic optimization and robust optimization. The study is conducted in two phases; the first phase is to determine the capacity of the facility. In the second phase, the optimum amount of production and inventory level is determined. Li et al. (2013) performed a network design and analyzed the behavior of the manufacturer and retailers in backflow based on the evolutionary game and robust strategies. Talaei et al. (2016) provided a multi-stage bi-objective green CLSC model. In this model, location and allocation of facilities are evaluated, and a model in the form of robust fuzzy is proposed to deal with uncertainties in costs and demand rate. The model aimed to minimize costs and also the rate of released CO₂. A robust multi-objective mixed-integer nonlinear programming (MINLP) model is offered to cope with the uncertainty of the network. The model consists of two objective functions: the first one reduces costs, and the second one minimizes environmental impact. The model is solved using an LP-metric method. Heydari et al. (2017) proposed a two-echelon CLSC model by considering one manufacturer and one retailer. Also, they considered quantity discounts and increasing fee contracts to manage the SC. Taleizadeh et al. (2018) studied the pricing strategies as well as the quality level and effort decisions of CLSCs with the manufacturer, retailer, and third party as single-channel forward SC with a dual-recycling channel and dual-channel forward SC with a dual-recycling channel. Taleizadeh et al. (2019) investigated a multi-period multi-echelon sustainable CLSC chain. They considered pricing of the products, logistic decisions, the discount offer on the returned products, remanufacturing, recycling and disposing of very low-quality returned products. As a move towards the CE, Atabaki et al. (2020) redesigned the CLSC network for durable products. They considered different recovery facilities, costs, CO₂ emissions and energy consumption as various objectives. A MILP model was proposed to find optimal supplier selection, location-allocation, transportation mode, assembly technology and recovery level decisions. Also, they utilized possibilistic programming and scenario-based stochastic programming to formulate and solve the robust counterpart of the proposed model. Fu et al. (2021) developed a CLSC network model for products having different market demands. The CLSC consists of suppliers, manufactures and retailers that work to satisfy the demand. They studied the equilibrium of the CLSC network by applying variational inequalities.

2.2 Circular economy and supply chains

CE offers an opportunity to optimize sustainable production and consumption using new models based on continuous growth and limitless resources (Govindan et al., 2018). CE has received considerable attention in political and business circles to overcome the shortcomings of traditional linear operating models, especially in SCM (De Angelis et al., 2018). In this regard, CESCs have been defined based on a shift from product ownership and access in SC relationships and other principles (De Angelis et al., 2018). Thus, a successful transition from the current and linear economic model towards a resource-efficient CE model in SCs requires an understanding of the CE concept and its relationships with various decision factors in supply chain management (SCM). In this regard, Alamerew and Brissaud (2020) investigated this issue using environmental, societal and economic aspects from a reverse SC perspective. Kazancoglu et al. (2018) proposed a holistic three-dimensional hierarchy framework for green SCM performance assessment based on CE to integrate the environmental, economic, logistics, operational, organizational and marketing performance. In the following, Bai et al. (2020) investigated the relationships between sustainable SC flexibility efforts and CE-targeted performance using the DEMATEL method. Del Giudice et al. (2021) analyzed the effect of CE practices on firm performance using multiple regression analysis. This study showed that the three categories of CE practices, including CESC management design, CESC relationship management, and CESC human resource management, affect performance effectively. On the other hand, CE arose from resource regeneration and can optimize resource and environmental sustainability within the CLSC and RL (Tseng et al., 2020). Accordingly, Dev et al. (2020) presented a roadmap to the excellence of operations for sustainable RL by implementing the principles of Industry 4.0 and CE approaches. Also, the main aim of the proposed remanufacturing model in this study is to examine the trade-off between the availability of green transportation and set-up delays. Alamerew and Brissaud (2020) developed a model based on the system dynamics to represent the complex system of RL to recover post-used products at their end-of-life stage. Feng and Gong (2020) developed an analytical model using the linguistic entropy weight method and multi-objective programming to address the green supplier selection and order allocation problems in the CE context.

By reviewing studies on the CLSC and CE, it is observed that few studies have been conducted in uncertain environments. Multi-objective optimization modeling with robust optimization is one of the research areas that is rarely seen in the literature. Therefore, a new robust method is developed for the optimization of the multi-objective model in uncertainty. On the other hand, the SC network consists of many components, and considering all the SC factors in an integrated model is very difficult. Table 1 briefly illustrates the difference between the proposed network and the literature and also shows the comprehensiveness of the proposed CLSC model. Therefore, the contributions of this paper are twofold:

1. A comprehensive model is proposed to take into account the main aspects and components of the CLSC in the context of the CE.

2. The proposed model is extended to consider the uncertainty in CLSC design. In this regard, a new robust optimization approach is utilized to model the uncertainty.

3. Mathematical model

3.1 Problem description

The studied network includes five suppliers: a suppliers' hub, five manufacturers, three distributors in the forward direction, and the opposite direction, five 3PLs, sites of disassembling, refurbishing and disposal (see Figure 1). First, raw materials purchased from suppliers are held by supplier hubs that are a type of third-party company, and then raw materials are sent to manufacturers. In the following, manufacturers produce products according to market demand. Then, products are given to distributors, and consequently to customers. As shown, many products are returned by customers; now, returned products shall be evaluated in the collection center. Due to the capacity limitation of manufacturers for outsourcing, reusable products are given to 3PL companies for reconstruction. After the reconstruction of returned products by 3PL, quality testing is carried out on the product by the manufacturer's representative, and finally, reconstructed products will be sent to the distributors. The rest of the products are given to refurbishing sites to be recovered, sold to suppliers of raw materials and other parts for recycling, transferred to disassembling place, and eventually, as new parts, are delivered to producers. Other useless parts of the products are destroyed at the disposal site. In the real world, many decisions are made in an uncertain environment. Considering the uncertainty in the studied problem is invertible because of the inherent uncertainty of parameters. In the SCND, the long-term strategic decision should be implemented with the least error because the slightest mistake can cause the cost for the manufacturer. In this regard, this study used the analytical research strategy to address the SCND problem under uncertainty in the CE context. In this case, we proposed a multi-objective MILP model based on robust optimization and implemented this model using the available data (a numerical example) to illustrate its performance. According to Figure 2, after developing the proposed model based on robust optimization, this model can be solved using the ε-constraint method to make tactical and strategic decisions, including the selection of optimal supplier and 3PL and determining the optimal amount of production.

3.2 Problem formulation

According to the given description in the previous sections and the complexity of a CLSC network, a robust model is presented for the fluctuations and uncertainty factors. At first, the deterministic MILP model is proposed, and then this model is developed based on robust optimization. The proposed model has the following assumptions (see Table 2):

1. Locations of suppliers, 3PL companies and customers are known and fixed;

2. Model is developed to address the multi-product and multi-period problems;

3. The capacity of disassembly, refurbishing, collection and disposal sites are known;

4. Costs, demand, and rate of returns are considered in an uncertain environment;

5. After reproducing by 3PL, returned products are sold at a lower price, but with high quality.

3.3 The proposed deterministic model

In this section, the deterministic multi-objective MILP model is proposed to address the SCND problem in the CE context. The developed model includes three objectives. The first objective function, presented in Equation (1), maximizes total profit, including the total income minus the total cost. The first and second parts show the manufacturer's benefits gained from producing and selling the products. Profit of reproducing products by 3PL, provided for the manufacturer, is specified in parts three and four. A benefit from the sale of returned and useless products, which is offered to suppliers, is shown in the fifth part. Other costs are as follows: payment amount for the purchase of parts from suppliers, disassembling cost, refurbishing components cost, cost of setting up refurbishing sites, cost of disposing of parts, cost of setting up disassembling sites, the return assessment cost, evaluating the cost of products reproduced by 3PL, cost of inventory maintained by the supplier hub and total cost of an inventory shortage. The second objective function, introduced in Equation (2), represents the customers' satisfaction rate and, in fact, SC agility. The third objective function minimizes the failure of parts purchased from suppliers as well as reproduced products by the 3PL (see Equation 3). One of the objectives of CE is to reduce waste. Reducing the amount of purchased and reproduced parts can indirectly decrease the potential waste. Therefore, the third objective function is defined in line with the goals of CE.(1)$\text{Max}{Z}_1 ={\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^{N}S{P}_j^t{P}_{jn}^t}}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^{N}P{C}_j^t{P}_{jn}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^{M}TS{P}_j^{t}T{P}_{jm}^t}}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^{M}TP{C}_j^{t}T{P}_{jm}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^{T}S{A}_{jk}^t{y}_{jk}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^T{r}_{ik}^t{Q}_{ik}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^{T}D{C}_i^{t}N{P}_i^t}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{O}_{il}^{t}D{X}_{il}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{p}_{il}^t{U}_{il}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T{h}_i^t{V}_i^t}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^{T}SD{C}_j^t{F}_j^t}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^{T}E{C}_j^{t}R{P}_j^t}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^M{N}_j^{t}T{P}_{jm}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^{T}in{v}_i^{t}hc}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^{T}inv{s}_j^{t}S{C}_j^t}}$(2)$\text{Max}{Z}_2=\frac{{\displaystyle {\sum }_{j=1}^J{\displaystyle {\sum }_{w=1}^W{\displaystyle {\sum }_{f=1}^F{\displaystyle {\sum }_{t=1}^{T}SD{C}_{jwf}^t}}}}}{{\displaystyle {\sum }_{j=1}^J{\displaystyle {\sum }_{f=1}^F{\displaystyle {\sum }_{t=1}^T{D}_{jf}^t}}}}$(3)$\text{Min}{Z}_3={\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^{T}SR{F}_{ik}^t{Q}_{ik}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{t=1}^{T}TR{F}_{jm}^{t}T{P}_{jm}^t}}}$

Regarding the constraints of the proposed model, 29 constraints can be defined for the studied problem (see Equations 4–32). Products sent to the 3PL, suppliers, and disassembling sites are equal to the total returns of the collection center to the evaluation site; this issue is shown in Equation (4). Equation (5) suggests that all products, manufactured and remanufactured, are given to a distributor. Equation (6) shows the capacity of distributors to receive the products. In addition to stating the shortage probability, Equation (7) indicates each customer's demand to be met for each product in each period and also states the shortage probability. Equation (8) guarantees that the amounts of incoming and outgoing products to distributors are equal. Equation (9) shows that the shortage amount is lower than the total demand. The minimum and maximum amount of returns is shown in Equation (10). Equation (11) states that the products produced by the manufacturer are equal to the number of refurbished and bought parts. Equation (12) guarantees that the number of returned parts is equal to those of disassembled ones. Equation (13) shows that some of the parts from the disassembling site are destroyed, and the others are re-used. Based on Equation (14), returned products greater than the capacity of suppliers will not be given to them. Equation (15) shows the minimum and the maximum number of purchases from external suppliers. Equations (16) and (17) illustrate the capacity of the disassembly and refurbish sites.(4)${\displaystyle \sum _{m=1}^M{A}_{jm}^t}+{\displaystyle \sum _{k=1}^{K}S{A}_{jk}^t}+R{D}_j^t=R{P}_j^t\forall j,t$(5)${\displaystyle \sum _{n=1}^N{P}_{jn}^t}+{\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}=FD{C}_{jw}^t\forall w,j,t$(6)$FD{C}_{jw}^t\le DC{C}_{jw}^t.DC{B}_w^t\forall j,w,t$(7)${\displaystyle \sum _{w=1}^{W}SD{C}_{jwf}^t}+inv{s}_j^t=D_{fj}^t\forall j,f,t$(8)$FD{C}_{jw}^t={\displaystyle \sum _{f=1}^{F}SD{C}_{jwf}^t}\forall j,w,t$(9)$inv{s}_j^t\le {\displaystyle \sum _{f=1}^F{D}_{fj}^{t^t}}\forall j,t$(10)$MI{R}_j\left({\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}+P_j^t\right)\le R{P}_j^t\le MA{R}_j\left({\displaystyle \sum _{m=1}^{M}T{P}_j^t}+p_j^t\right)\forall j,t$(11)${\displaystyle \sum _{j=1}^J{q}_{ij}}{\displaystyle \sum _{n=1}^N{P}_{jn}^t}={\displaystyle \sum _{l=1}^{L}D{X}_{il}^t}+H{Q}_i^t\forall i,t$(12)$T_i^t ={\displaystyle \sum _{j=1}^J{q}_{ij}^{t}R{D}_j^t}\forall i,t$(13)${\displaystyle \sum _{l=1}^L{X}_{il}^t}+V_i^t=N{P}_i^t\forall i,t$(14)${\displaystyle \sum _{k=1}^{K}S{A}_{jk}^t}\le S{R}_{jk}^t{U}_k^t\forall j,k,t$(15)$u_k^t{V}_k^t\le {\displaystyle \sum _{i=1}^I{b}_{ik}{Q}_{ik}^t}\le u_k^t{B}_k^t\forall k,t$(16)$e_i^t{T}_i^t\le E_i^t\forall i,t$(17)$g_{il}{x}_{il}^t\le G_{il}^t{U}_{il}^t\forall i,l,t$

Equation (18) shows the inventory amount of the suppliers' hub in each period. Equations (19) and (20) illustrate the maximum percentage of useable and corrupted parts. Equation (21) indicates that the products repaired in each period are lower than the produced products by manufacturers. Equation (22) indicates the maximum allowable number of setups of refurbishing sites in each period. In Equation (23), the limited capacity of the collection center is guaranteed. Equation (24) shows the equality of the number of incoming and outgoing products to the 3PL. Equation (25) limits the capacity of 3PL. Equation (26) describes the minimum and the maximum production capacity of the manufacturer. The percentage of products in the collection center to be delivered to the 3PL is shown in Equation (27). Equation (28) limits the capacity of the supplier hub to purchase parts. Finally, Equations (29) and (30) guarantee the return of products to the disassembling sites and the 3PL.(18)$in{v}_i^t=in{v}_i^{t-1}+{\displaystyle \sum _{k=1}^K{Q}_{ik}^t}-H{Q}_i^t\forall i,t$(19)${\displaystyle \sum _{l=1}^L{x}_{il}^t}\le O_{i}N{P}_i^t\forall i,t$(20)$V_i^t\le (1-O_i)N{P}_i^t\forall i,t$(21)${\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}\le {\displaystyle \sum _{n=1}^N{P}_{jn}^t}\forall j,t$(22)${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{U}_{il}^t}}\le C\forall t$(23)${\displaystyle \sum _{j=1}^{J}R{P}_j^t}\le L\forall t$(24)$T{P}_{jm}^t=A_{jm}^t\forall j,m,t$(25)$TM{C}_m^t{u}_m^t\le {\displaystyle \sum _{j=1}^{J}T{P}_{jm}^t}\le T{C}_m^t{u}_m^t\forall m,t$(26)$MP{C}_j^t\le {\displaystyle \sum _{n=1}^N{P}_{jn}^t}\le M{P}_j^t\forall j,t$(27)${\displaystyle \sum _{m=1}^M{A}_{jm}^t}=rot.R{P}_j^t\forall j,t$(28)$in{v}_i^t\le M{C}_i^t\forall i,t$(29)$R{D}_j^t\le M.F_j^t\forall j,t$(30)$A_{jm}^t\le M.U_m^t\forall j,m,t$(31)$U_{il}^t,F_j^t,u_k^t,u_m^t\in \{\mathrm{0,1}\}\forall i,j,k,l,m,t$(32)$P_{jn}^t,R_j^t,Q_{ik}^t,T_i^t,X_{il}^t,V_i^t,A_{jk}^t,A_{jm}^t,B_{jm}^t,R_j^{\prime t}\ge 0\forall i,j,k,l,n,m,t$

4. Solution method

4.1 Robust optimization approach

In this section, the proposed deterministic model is developed based on robust optimization to consider the uncertainty concept to address the studied problem. The robust counterpart of the first objective function (Z₁) with the uncertainty of parameters can be written as follows:(33)$\text{Max}{Z}_1$(34)$Z_1+\psi \left({\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^N{\widehat{SP}}_j^t{P}_{jn}^t}}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^N{\widehat{PC}}_j^t{P}_{jn}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^M{\widehat{TSP}}_j^{t}T{P}_{jm}^t}}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^M{\widehat{TPC}}_j^{t}T{P}_{jm}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^T{\widehat{SA}}_{jk}^t{y}_{jk}^t}}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^T{\widehat{r}}_{ik}^t{Q}_{ik}^t}}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T{\widehat{DC}}_j^{t}N{P}_i^t}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{\widehat{O}}_{il}^{t}D{X}_{il}^t}}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{\widehat{p}}_{il}^t{U}_{il}^t}}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T{\widehat{h}}_i^t{V}_i^t}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\widehat{SDC}}_j^t{F}_j^t}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\widehat{EC}}_j^{t}R{P}_j^t}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{M=1}^M{\widehat{N}}_j^{t}T{P}_{jm}^t}}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^{T}in{v}_i^t\widehat{hc}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\widehat{SC}}_j^{t}inv{s}_j^t}}\right)\le {\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^{N}S{P}_j^t{P}_{jn}^t}}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^{N}P{C}_j^t{P}_{jn}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^{M}TS{P}_j^{t}T{P}_{jm}^t}}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{m=1}^{M}TP{C}_j^{t}T{P}_{jm}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^{T}S{A}_{jk}^t{y}_{jk}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^T{r}_{ik}^t{Q}_{ik}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^{T}D{C}_j^{t}N{P}_i^t}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{O}_{il}^{t}D{X}_{il}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{l=1}^L{\displaystyle \sum _{t=1}^T{p}_{il}^t{U}_{il}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^T{h}_i^t{V}_i^t}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^{T}SD{C}_j^t{F}_j^t}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^{T}E{C}_j^{t}R{P}_j^t}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{M=1}^M{N}_j^{t}T{P}_{jm}^t}}}-{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{t=1}^{T}in{v}_i^{t}hc}}-{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{t=1}^{T}S{C}_j^{t}inv{s}_j^t}}$

The robust counterpart of the second objective function (Z₂) is presented in Equations (35) and (36) as follows:(35)$\text{Max}{Z}_2$(36)$Z_2+\mathrm{\Psi }\left(\frac{{\displaystyle {\sum }_{j=1}^J{\displaystyle {\sum }_{w=1}^W{\displaystyle {\sum }_{f=1}^F{\displaystyle {\sum }_{t=1}^{T}SD{C}_{jwf}^t}}}}}{{\displaystyle {\sum }_{j=1}^J{\displaystyle {\sum }_{f=1}^F{\displaystyle {\sum }_{t=1}^T{\widehat{D}}_{jf}^t}}}}\right)\le \frac{{\displaystyle {\sum }_{j=1}^J{\displaystyle {\sum }_{w=1}^W{\displaystyle {\sum }_{f=1}^F{\displaystyle {\sum }_{t=1}^{T}SD{C}_{jwf}^t}}}}}{{\displaystyle {\sum }_{j=1}^J{\displaystyle {\sum }_{f=1}^F{\displaystyle {\sum }_{t=1}^T{D}_{jf}^t}}}}$

The robust counterpart of the third objective function (Z₃) is written as follows:(37)$\text{Min}{Z}_3$(38)$Z_3\ge {\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^{T}SR{F}_{ik}^t{Q}_{ik}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{t=1}^{T}TR{F}_{jm}^{t}T{P}_{jm}^t}}}+\mathrm{\Psi }\left({\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{t=1}^T{\widehat{SRF}}_{ik}^t{Q}_{ik}^t}}}+{\displaystyle \sum _{j=1}^J{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{t=1}^T{\widehat{TRF}}_{jm}^{t}T{P}_{jm}^t}}}\right)$

Regarding the introduced constraints in the previous section, Equation (7) is transformed into Equation (39). Also, Equation (9) is converted into Equation (40) based on robust optimization. Besides, the robust counterpart of Equation (10) is converted into two Equations (41) and (42). Thus, the new robust models are solved by solving Equations (33)–(42) as well as Equations (4)–(32), except for Equations (4)–(6).(39)${\displaystyle \sum _{w=1}^{W}SD{C}_{jwf}^t}+inv{s}_j^t=D_{fj}^t+\mathrm{\Psi }({\widehat{D}}_{fj}^t)\forall j,f,t$(40)$inv{s}_j^t+\mathrm{\Psi }(0.05{\widehat{D}}_{fj}^t)\le D_{fj}^t\forall j,f,t$(41)$MI{R}_j\left({\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}+p_j^t\right)+\mathrm{\Psi }{\widehat{MIR}}_j\left({\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}+p_j^t\right)\le R{P}_j^t\forall j,t$(42)$R{P}_j^t+\mathrm{\Psi }{\widehat{MAR}}_j\left({\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}+p_j^t\right)\le MA{R}_j\left({\displaystyle \sum _{m=1}^{M}T{P}_{jm}^t}+p_j^t\right)\forall j,t$

In the proposed convex mathematical programming models by Soyster (1973), different strategies are evaluated to demonstrate the sustainability and feasibility of the model. Ben-Tal and Nemirovski (1999) provided a robust model counterpart for LP under uncertainty. Then, sustainable solutions for the robust counterpart of LP are obtained using the tools of analysis and calculation. The result showed that LP robust counterparts based on ellipsoidal sets become quadratic cone programming. Mulvey et al. (1995) proposed a robust optimization approach, due to data turbulence and disruption, whereby the data of the problem are defined based on a set of scenarios. The proposed approach showed that, by using volatility data, the answer is less sensitive as compared to an optimal model. Bertsimas (2003) provided an approach for suboptimal solutions to make the solution be either feasible or stay close to optimal. Indeed, the approach aimed to reduce the robust cost, and indicate the level of conservatism and its flexibility against changes in demand. Also, a robust optimization approach is developed by Bertsimas and Sim (2004) and Bertsimas and Thiele (2004) to control SC efficiency under uncertain demand. In the following, a robust optimization approach presented by Li et al. (2011) is introduced. In a robust optimization set-based approach under uncertainty, the goal is to pick the best possible solution from a safe answer against uncertainty. Consider the following mixed-integer optimization problem:$\max {\displaystyle \sum _m{c}_m{x}_m}+{\displaystyle \sum _k{d}_k{y}_k},$(43)$s.t.{\displaystyle \sum _m{\tilde{a}}_{im}{x}_m}+{\displaystyle \sum _k{\tilde{b}}_{ik}{y}_k}\le {\tilde{p}}_i\forall i$where x and $y$ are integers and continuous variables, respectively. ${\tilde{a}}_{im}$, ${\tilde{b}}_{ik}$ and ${\tilde{p}}_i$ are the actual values of the uncertain parameters. The values of these parameters are as follows:(44)${\tilde{a}}_{im}=a_{im}+{\xi }_{im}{\widehat{a}}_{im}\forall m\in M_i$(45)${\tilde{b}}_{ik}=b_{ik}+{\xi }_{ik}{\widehat{b}}_{ik}\forall k\in K_i$(46)${\tilde{p}}_i=p_i+{\xi }_{i0}{\widehat{p}}_i$where $M_i$ and $K_i$ are the subsets of continuous and discrete variables index, and all coefficients indexed by them are uncertain. $a_{im}$, $b_{ik}$ and $p_i$ represent the nominal values of the parameters. ${\widehat{a}}_{im}$, ${\widehat{b}}_{ik}$ and ${\widehat{p}}_i$ show positive values deviation (nominal values). ${\xi }_{im}$, ${\xi }_{ik}$ and ${\xi }_{i0}$ are variables related to the set U, and the goal is to find a feasible solution for each $\xi$ in the set to be safe against infeasibility. Based on the shape of uncertainty interval $U$, the robust counterpart formulation of the problem is different. Li et al. (2011) defined the uncertainty box as:(47)$U_{\infty }=\{\xi |\mathrm{\Vert }\xi {\mathrm{\Vert }}_{\infty }\le \psi \}=\{\xi |\left|\xi \right|\le \psi ,\forall j\in J\}$where $\psi$ is an adjustment parameter, which controls the size of the uncertainty set. Therefore, the box robust counterpart model in Equation (1) is as follows:$\text{max} {\displaystyle \sum _m{c}_m{x}_m}+{\displaystyle \sum _k{d}_k{y}_k}$(48)$s.t.{\displaystyle \sum _m{a}_{im}{x}_m}+{\displaystyle \sum _k{b}_{ik}{y}_k}+\mathrm{\Psi }\{{\displaystyle \sum _{m\in M_i}{\xi }_{im}{â}_{im}{x}_m}+{\displaystyle \sum _{k\in K_i}{\xi }_{ik}{\widehat{b}}_{ik}{y}_k}-{\xi }_{i0}{\widehat{p}}_{is}\}\le p_i$

This model is used to provide a robust SC network and also deal with uncertainties in the parameters.

4.2 ε–constraint method

The multi-objective mathematical programming methods are used to solve the proposed model in this study. We use $\epsilon$-constraint as an effective method to solve the developed multi-objective model. The form of a multi-objective optimization problem based on $\epsilon$-constraint presented in the study of Haimes et al. (1971) has been provided. Based on the procedure of this method, the multi-objective function is transformed into a single objective function. Accordingly, a multi-objective model is developed as follow:(49)$\begin{array}{l}\text{Min} \left(f_1\left(x\right),f_2\left(x\right),\mathrm{...},f_p\left(x\right)\right)\\ s.t\text{.}\\ g_1\left(x\right)\le 0,\mathrm{...},g_q\left(x\right)\le 0\\ x\in F,\end{array}$where x is the vector of decision variables and $\{f_1\left(x\right),f_2\left(x\right),\mathrm{...},f_p\left(x\right)\}$, and $\left\{g_1\left(x\right),g_2\left(x\right),\mathrm{...},g_q\left(x\right)\right\}$ are p objective and q constraint functions, respectively. F is the feasible solution space. According to Equation (49), the model is solved with respect to one of the objective functions, while the other objective functions are put in constraints. For example, the output model may be as follows:(50)$\begin{array}{l}\text{Min} f_1\left(x\right)\\ s.t.\\ f_2\left(x\right)\le {\epsilon }_2\\ f_3\left(x\right)\le {\epsilon }_3\\ ⋮\\ f_p\left(x\right)\le {\epsilon }_p\\ g_1\left(x\right)\le 0,\mathrm{...},g_q\left(x\right)\le 0\\ x\in F,\end{array}$

A setting for parameters ${\epsilon }_i$ should be considered by which an efficient solution is found. The Model (50) is solved for all of the objective functions. By parametric variation in the right-hand side of the constrained objective functions, the efficient solutions are obtained.

5. Analysis of the results

In this section, a numerical example is developed to implement the proposed model. Then, the outputs of implementing this model are discussed and sensitivity analysis on these results is performed.

5.1 Numerical example

In this study, a numerical example has been used to demonstrate the efficiency of the proposed model for designing a robust CLSC network in the CE context. This example is designed based on the study by Amin and Zhang (2012a, b). The parameters considered in their research are definite, which in this study have been defined in an uncertain environment due to the nature of the studied problem. To put it precisely, these parameters are simulated according to the theoretical contributions of this research, i.e. developing the multi-period model, considering the uncertainty in costs, demand and rate of returns, re-designing the CLSC network by adding 3PLs, supplier hub, and other members. In this example, it is assumed that five manufacturers produce five product types for four customers. Each product is composed of five different parts, which must be purchased from external suppliers. Choosing the right suppliers and the economic value of purchases are those issues that manufacturers should always consider as a vital decision. The matter of returned products by the customer is another critical issue that plays a vital role in the CLSC. Thus, the manufacturer gives the returned parts to the 3PL company to organize, disassemble, and refurbish the rest of the products. The maximum number of sites for refurbishing is 6. By solving the proposed model, the optimum number of sites is found. Some parameters of the proposed model, as shown in Tables 3 and 4, are generated uniformly. Variability of uncertainty is considered 0.1; also, calculations have been conducted for ten periods. Notably, all calculations are implemented using CPLEX solver in an environment of Gams 24.1.2 software on an HP, 8 GB RAM, Core i7 computer.

5.2 Discussion

In this section, we analyzed three objectives defined in our model independently under different levels of uncertainty illustrated in Figures 3–5. We also do a sensitivity analysis on the values of the first objective function (maximizing the total profit in the CLSC) for different levels of uncertainty and values of the other two objectives (see Figures 6 and 7). Furthermore, to validate the results obtained from the implementation of the proposed solution approach and its performance, we compared these results with the NSGA-II algorithm (see Table 10).

5.2.1 Theoretical implications

Regarding the theoretical contributions, this study tried to confront the uncertainty in CLSC design by proposing a multi-objective MILP model based on robust optimization. To put it precisely, developing the multi-period model, applying the uncertainty concept in considering costs, demand and rate of returns, re-designing the CLSC network by adding 3PLs, supplier hub and other members in the CE context can be considered as theoretical contributions of this study. Also, this study considers reducing waste as one of the goals of CE in the proposed model. In other words, reducing the amount of purchased and reproduced parts can indirectly decrease the potential waste. Therefore, one of the objective functions of this model is defined in line with the goals of CE as a contribution compared to previous studies to illustrate the impact of CESC management design on network performance. Customer satisfaction is one of the main goals of various industries, which is investigated in the proposed model considering the uncertainty in the amount of demand. On the other hand, due to limited production capacity, 3PLs have been used to compensate for the lack of capacity in the CLSC network and to increase SC agility in responding to customers. The proposed model consists of three objective functions; the first one is maximizing the profit, the second function is seeking to maximize customer satisfaction (responsiveness to customers) and the third one minimizes the failure rate of parts and products. In the developed robust counterpart model, uncertainty is considered in all functions, as well as the demand and product returns. The level of uncertainty (ψ) is variable and is considered as in interval [0–0.08].

5.2.2 Practical and managerial implications

In this subsection, the proposed model is numerically analyzed and managerial implications of the results are provided. In Figures 3–5, it can be seen how much the amount of each objective function will be if functions are solved separately. With increasing the uncertainty level of a robust counterpart, the outputs of objective functions differ in comparison to the optimized and deterministic mode. Since the proposed model is sensitive to customer's responsiveness, small changes in the uncertainty level lead to considerable changes in the results. Based on Figures 3 and 4, the trend of the first and second objective functions until the uncertainty level of 0.01 is almost constant. After that, these two functions experience a relatively sharp decline in their values. So that the values of the first and second objective functions at the uncertainty level of 0.08 reach slightly higher than 198,000,000 and lower than 0.2, respectively. In the CE context, increasing the uncertainty results in the augmentation of the third objective function which can impact the CE goals. According to Figure 5, when the variation of SC parameters increases the number of returned and reproduced products may have a positive jump. This can increase the profitability of the whole SC; however, this issue is against the CE concept. An increase in the number of produced and reproduced products can implicitly lead to the accumulation of waste. In this case, the waste cannot be reprocessed and reused by the customers, and accordingly, the main objective of CE is not satisfied.

For further investigation, we investigate the behavior of the first objective function with respect to the second and third functions. To do this, we focus on analyzing the total cost or obtained total profit considered in the first developed model. According to analyzing the value of Z₁ with respect to Z₂, Figure 6 demonstrates that costs should be paid to keep agility as well as network response to customer demands at a high level. Based on Figure 6, at any level of uncertainty, the cost paid for the optimal amount of profit is diminished. By fixing the minimum amount of response ($Z_2$) and solving the first objective function, we can reach the solutions of $Z_1$. To put it precisely, the obtained total profit can be greater than the values mentioned in Figure 3 by changing the value of $Z_2$. Accordingly, it can be seen that models tend to be more responsive and profitable compared to the previous case. Comparing the profit values with failure rates shown in Figure 7 indicates that when the failure rate is increased, the total profit is reduced. The amount of this reduction is proportional to the level of uncertainty. Focusing on Figure 7, it can be said that the reduction rate of the value of the first objective function is decreased by increasing the value of Z₃.

Now, we are currently seeking to optimize the three introduced objective functions in the developed mathematical model simultaneously. Optimizing several objective functions simultaneously in the developed models based on robust optimization is challenging. To address this issue, the proposed model is solved using $\epsilon$-constraint method. Table 5 shows the results of solving the proposed model by taking all objective functions into account to maximize profit simultaneously. The results indicate that in optimizing functions, all functions cannot reach their ideal solutions simultaneously. Indeed, for obtaining optimal and acceptable results, the number of functions should be ignored. For example, the third objective function should go away from the ideal solution to have responsiveness and profit at a high level. The model has a feasible solution for modes $Z_2\ge 0.9$ and $Z_3\le 140000$, and the value of the third objective function should be increased because the proposed model in $Z_2\ge 0.9$ and $Z_3\le 130000$ does not have any solution. Similarly, in the case of $Z_2\ge 0.98$ and $Z_3\le 160000,$ it can be seen that by changing and ignoring the amount of the objective functions, the obtained profit is different. Due to the needs of any organization, the value of each objective function can be calculated with respect to other functions concerning the network approach. The remarkable point in the obtained outputs is that profit increases by adding any amount to responsibility as a positive rating and level of network failure as a negative rating. In these circumstances, manufacturers should decide how to compete with their rivals, and what amounts of profit, responsibility, and network failure to have. It also should not be forgotten that the decision should be purposefully for a long time, and the survival of its production should not be sacrificed for short-term gains. The outputs of some variables which are obtained from solving the proposed model are shown in Tables 6–9. The calculations have been done for $Z_2\ge 0.9$, $Z_3\le 155000$, and Z₁ = 0.08. By solving the model, the manufacturer can understand that in every part of the network, how many products and components are available.

On the other hand, the important issue and challenge, discussed in the CLSC network that increases the amount of network complexity, is the rate of returns. Manufacturers are continually looking to estimate the amount of their profit by determining the number of returns. In contrast to the viewpoint of traditional manufacturers, the number of returns in the SC has a significant impact on profit in the network. Figure 8 indicates that when the rate of returns is increased, the objective function related to profit experienced a considerable increase in different uncertainty levels; $Z_2\ge 0.7$ and $Z_3\le 145000$. Based on Figure 8, the obtained profit when the return rate is equal to 0.95 is much more than the profit in the case in which the return rate is 0.65.

As stated, the main goal of the proposed model is to specify the number of products and parts available in the whole CLSC network. Purchase amount from a supplier, outsourced amount of returns to third party companies, and most importantly, the supply-demand balance are the objectives of CLSC network. Accordingly, we compare the outputs of the proposed approach with the NSGA-II algorithm to illustrate its superiority by defining three different problems based on the above-mentioned objectives. In fact, the NSGA-II is employed to solve the developed multi-objective model to compare its results with the proposed approach. Table 10 shows the run time, diversity and Pareto front for three problems with different dimensions. Based on this table, it can be said that the run time and diversity of the proposed solution approach is less than the NSGA-II algorithm according to defined problems with different dimensions. This demonstrates that the proposed solution approach provides a more accurate solution in less time in comparison with another algorithm. In other words, since the parameters of the NSGA-II are tuned by trial and error, this algorithm does not necessarily give optimal solutions. On the other hand, Table 10 represents that the proposed solution approach has better near-optimal solutions (NOS) than the NSGA-II algorithm. Also, the results provided in Table 10 show that the outputs of the proposed solution approach and the NSGA-II are consistent. Furthermore, it can be said that the proposed solution approach outperforms the NSGA-II algorithm considerably in addressing a more complex problem (the first problem).

6. Conclusion, limitations and scope for future research

Today, managers and manufacturers use innovative ways to stay in a competitive market regarding the high quality and the access costs in a network. SC design is a critical and strategic issue that provides an effective platform for efficient SC management. This study seeks to provide an integrated structure of the SC to reduce its overall costs and product degradation and increase the agility throughout the SC. To achieve these goals, using the combined problem-solving models and introducing a new and comprehensive network is of cardinal importance. Therefore, this study proposed a model to consider both forward and backward processes, which can be applied to many industries such as electronics, digital equipment and transportation industries. Thus, one of the main differences between this study and other previous researches is that a new network of CLSC is designed to include all members of the chain, while other researches have not covered it in general. In this case, adding each member to the CLSC network would complicate the network model and increase the uncertainty. Accordingly, this study proposed a mathematical model to address such complexity and uncertainty. In the proposed model and designed network, customer satisfaction is also considered; this important challenge has always affected the performance of managers and manufacturers. Therefore, our proposed model and designed network can support managers to confront the changes and sudden events of the market, so that increases the competitiveness and credibility in the market. Furthermore, companies not only are responsible for the environment but also must respond to the products returned by or after the end of their lifetime due to strict government regulations, environmental concerns, social responsibility and customer awareness. Thus, one of the main goals of the proposed model is to help preserve the environment by utilizing, reusing and recycling the products and parts. This issue is very important from the viewpoint of CE. Recycling products and reducing waste is one of the main objectives of CE. On the other hand, making decisions on returning products can increase the profit and employment rate in many countries in addition to meeting legal and customer requirements. Finally, it can be said that this model provides a comprehensive insight into the CLSC for managers to enable them to make critical and right decisions under uncertain conditions to reduce the damage on the CLSC network and gain the most revenue simultaneously. In this case, managers boost the SC flexibility to respond to the external environment.

Considering the theoretical contributions, the main aim of this study was developing a multi-objective MILP model for re-designing a CLSC network under uncertainty in the CE context. This network includes supplier, supplier hub, and manufacturer, distributor in forwarding movement as well as 3PLs, disposing site, refurbishing and disassembling locations in RL. To address the network complexity, uncertainty and also information in the real world, the proposed deterministic model has been developed based on robust optimization. The proposed robust counterpart model is based on the box set. Solving mathematical models in a situation of uncertainty is difficult by itself; when the model is multi-objective, the complexity and difficulty increases then. The parameters related to the three defined objective functions, including maximizing profit, maximizing responsiveness to customers (increasing the agility), and minimizing the failure rate of parts and products, are considered uncertain, as well as demand and rates of returns. One of the major challenges in manufacturing is making strategic decisions on supplier and contractor selection and determining the optimal order quantity from them. By solving the proposed model using $\epsilon$-constraint method, the manufacturer can make important decisions regarding the supplier and 3PL selection, economic order quantity to the suppliers, the optimal value of outsourcing to 3PLs, and specifying the parts and products available in each part of the chain. This, in turn, allow the manufacturer to determine the production capacity. Furthermore, the proposed robust mathematical model is evaluated in different levels of uncertainty and different return rate, and the results showed that the model is robust and feasible for different scenarios.

To mention some of the limitations of the current study, we know that CE includes many aspects and details for a successful implementation in supply chains. It is clear that considering all these aspects in a single model can increase the complexity of the model and leads to less practical results. In this research, CE was studied from the viewpoint of reducing waste by recycling products. However, supply chain is a comprehensive concept which contains many elements and details, and CE can be applied to all the elements of the supply chain. Also, CE concepts can be considered in the supply chain based on strategic, tactical and operational levels. Therefore, proposing a single model which can take into account all these aspects can be very difficult or even impossible. On the other hand, separate analysis of supply chain elements with regard to CE may result in non-optimal overall performance of the supply chain. Data collection from the supply chain elements and analyzing the data in an integrated manner is another limitation of CE and supply chain studies.

In this study, the location of suppliers and 3PLs are considered fixed and specific, which can be considered uncertain for future research. In this regard, transportation between facilities, as well as customer service time, taxes and tariffs can be applied in the mathematical model. Future researchers could also define an objective function to consider CO₂ emissions in designing a CLSC to highlight the concept of sustainability. In this study, the robust counterpart model was developed based on the box set that could be modeled with polyhedral, ellipsoidal or a combination of interval-polyhedral-ellipsoidal sets and solved using other metaheuristic methods like particle swarm optimization (PSO). Furthermore, other CE aspects such as minimum use of raw materials, maximum reuse of products, renewable energy and systems thinking can also be considered in modeling a CLSC.

Figure 1

CLSC network configuration

[Figure omitted. See PDF]

Figure 2

The proposed approach framework of this study

[Figure omitted. See PDF]

Figure 3

The value of Z₁ for different levels of uncertainty regardless of other functions

[Figure omitted. See PDF]

Figure 4

The value of Z₂ for different levels of uncertainty regardless of other functions

[Figure omitted. See PDF]

Figure 5

The value of Z₃ for different levels of uncertainty regardless of other functions

[Figure omitted. See PDF]

Figure 6

The value of Z₁ for different levels of uncertainty and different values of Z₂

[Figure omitted. See PDF]

Figure 7

The amount of $Z_1$ for different levels of uncertainty and different values of $Z_3$

[Figure omitted. See PDF]

Figure 8

The amount of $Z_1$ for different return rates and various levels of uncertainty

[Figure omitted. See PDF]

Table 1

A brief review of studies on the CLSC

Authors	Single period	Multi-period	Suppliers	Manufacturing	Distribution centers	Retailers	Collection center	Refurbishing centers	Disassembling site	Recycling center	Disposal center	Third-party logistics	Supplier hub	Minimizing cost	Solution method
Gupta and Evans (2009)		✓	✓	✓	✓	✓		✓	✓	✓	✓			✓	Goal programming
Lee and Dong (2009)		✓		✓	✓		✓	✓						✓	Stochastic MINLP
Francas and Minner (2009)	✓			✓				✓							SP
Easwaran and Üster (2010)	✓			✓	✓	✓	✓			✓					MILP
Wang and Hsu (2010)	✓		✓	✓	✓	✓				✓				✓	NLP, ILP
Kannan et al. (2010)	✓					✓	✓			✓	✓			✓	MILP
Pishvaee and Torabi (2010)	✓			✓	✓		✓	✓		✓				✓	Fuzzy programming
Pishvaee et al. (2011)	✓						✓	✓			✓			✓	Robust MILP
Zhang et al. (2011)		✓		✓				✓						✓	ILP
Paksoy et al. (2011)	✓		✓		✓	✓	✓	✓	✓	✓	✓			✓	LP
Zarandi et al. (2011)	✓			✓	✓	✓								✓	LP
Krikke (2011)	✓		✓	✓			✓	✓	✓	✓	✓				MILP
Vahdani et al. (2012)	✓			✓	✓		✓			✓				✓	Fuzzy MINLP
Abdallah et al. (2012)		✓			✓									✓	MINLP
Amin and Zhang (2012a)	✓			✓		✓	✓	✓	✓	✓	✓				MILP
Hasani et al. (2012)		✓	✓	✓	✓	✓								✓	LP, NLP
Amin and Zhang (2012b)	✓		✓	✓		✓			✓	✓					MILP
Özceylan and Paksoy (2013)		✓	✓	✓		✓	✓	✓	✓		✓			✓	MILP
Ramezani et al. (2013a)	✓		✓	✓	✓	✓	✓	✓			✓				Stochastic MILP
Ramezani et al. (2013b)	✓		✓	✓	✓	✓	✓				✓				Robust MILP
Amin and Zhang (2013)	✓			✓			✓	✓			✓			✓	Stochastic MILP
Demirel et al. (2014)		✓	✓	✓	✓		✓	✓	✓		✓				Fuzzy MIP
Accorsi et al. (2015)	✓		✓	✓	✓	✓	✓			✓	✓			✓	MILP
Talaei et al. (2016)	✓			✓			✓	✓			✓				Robust fuzzy MILP
Heydari et al. (2017)	✓		✓	✓		✓				✓	✓			✓	NLP
Taleizadeh et al. (2018)	✓		✓	✓		✓	✓			✓		✓		✓	NLP
Taleizadeh et al. (2019)		✓	✓	✓		✓	✓			✓	✓	✓		✓	NLP
Atabaki et al. (2020)	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓			✓	Robust MILP
Current research		✓	✓	✓	✓		✓	✓	✓		✓	✓	✓	✓	Robust MILP

Note(s): LP: Linear Programming; ILP: Integer Linear Programming; NLP: Nonlinear Programming; MILP: Mixed Integer Linear Programming; MINLP: Mixed Integer Nonlinear Programming; SP: Stochastic Programming

Table 2

Definition of indices, parameters and variables

Sets and indices
$I$	The set of parts	$l$	The set of refurbishing sites
$J$	The set of products	$m$	The set of 3PLs
$K$	The set of suppliers	$n$	The set of manufacturers
$T$	The set of periods	$w$	The set of distributors
$F$	The set of customers
Decision variables
$p_{jn}^t$	The number of the product $j$ produced by the manufacturer $n$ in period $t$	$N{P}_i^t$	The number of part $i$ obtained from the disassembly site in period $t$
$R{D}_j^t$	The number of the returned product $j$ that must be disassembled in period $t$	$D{X}_{il}^t$	The number of part $i$ that is refurbished in the refurbishing site $l$ in period $t$
$Q_{ik}^t$	The number of part $i$ that must be purchased from the external supplier $k$ in period $t$	$V_i^t$	The number of part $i$ that must be destroyed in period $t$
$U_{il}^t$	The binary variable for setting up the refurbishing site $l$ for the part $i$ in period $t$	$A_{jm}^t$	The number of the returned products $j$ that must be given to 3PL of $m$ in period $t$
$F_j^t$	The binary variable for setting up the disassembly site for the product $j$ in period $t$	$T{P}_{jm}^t$	The number of the product $j$ that is refurbished by 3PL $m$ in period $t$
$U_k^t$	The binary variable for the supplier $k$ in period $t$	$u_m^t$	The binary variable for 3PL $m$ in period $t$
$S{A}_{jk}^t$	The number of the returned product $j$ that must be given to the supplier $k$ in period $t$	$R{P}_j^t$	The number of the returned product $j$ that goes to the evaluation site in period $t$
$in{v}_i^t$	The amount of the inventory for the part $i$ in period $t$	$H{Q}_i^t$	The number of part i sent from the supplier hub in period $t$
$inv{s}_j^t$	The shortage of product $j$, in period $t$	$DC{B}_w^t$	A binary variable for selecting distributor $w$ in period $t$
$SD{C}_{jwf}^t$	The amount of product $j$ that distributor $w$, must give to customer $f$ in period $t$	$DC{C}_{jw}^t$	The capacity of distributer $w$ for product $j$ in period $t$
$FD{C}_{jw}^t$	The amount of product $j$ that must be given to distributor $w$, in period t
Parameters
$S{P}_j^t$	The unit of the sale price for the product $j$ in period $t$	$MA{R}_j$	The maximum returned percentage from the product $j$
$TS{P}_j^t$	The unit of the sale price for the reproduced product $j$ in period $t$	$MI{R}_j$	The minimum returned percentage from the product $j$
$P{C}_{jt}$	The unit of direct production price for the product $j$ in period $t$	$O_i$	The maximum useable percentage from part $i$
$D_{jf}^t$	The demand for the product $j$ in period $t$	$C$	The maximum number of times for setting up refurbishing sites
$SD{C}_j^t$	The cost of setting up a disassembly site for product $j$ in period $t$	$R{F}_{ik}^t$	The failure rate of part $i$ that is produced by supplier $k$ in period $t$
$E_i^t$	The maximum capacity of the disassembly site for arranging part $i$in period $t$	$E{C}_j^t$	The evaluation cost of the returned product $j$ in period $t$
$D{C}_i^t$	The unit cost of the disassembly for part $i$ in period $t$	$T{C}_m^t$	The capacity of 3PL $m$ in period $t$
$h_i^t$	The unit cost of disposing of part $i$ in period $t$	$TM{C}_m^t$	The minimum production amount of 3PL of $m$ in the period $t$
$e_i^t$	The resource used for the disassembly of one unit of part $i$ in period $t$	$S{R}_{jk}^t$	The maximum capacity of the supplier $k$ for taking the returned product in period $t$
$o_{il}^t$	The unit cost of refurbishing for part $i$ in refurbishing place $l$ in period $t$	$V_k^t$	The minimum number of purchases from supplier $k$ in period $t$
$p_{il}^t$	The setup cost of refurbishing site $l$ for part $i$ in period $t$	$TP{C}_j$	The cost paid to 3PL for reproducing each unit of the product $j$ in period $t$
$G_{il}^t$	The maximum capacity of the refurbishing site $l$ for the refurbishing of part $i$ in period $t$	$y_{jk}^t$	The unit of the sale price for the product$j$ to external supplier $k$ in period $t$
Hc	The fixed cost of maintaining inventory	$L$	The capacity of the evaluation site
$r_{ik}^t$	The cost for purchasing part $i$ from the external supplier $k$ in period $t$	$MP{C}_j^t$	The minimum production size of product $j$ by the producer in period $t$
$B_K^t$	The maximum reserved capacity for the external supplier $k$ in period $t$	$M{P}_j^t$	The maximum production size of product $j$ by the producer in period $t$
$M{C}_i^t$	The maximum capacity for the supplier hub of part $i$ in period $t$	$TR{F}_{jm}^t$	The failure rate for the manufactured product $j$ by the 3PL $m$ in period $t$
$N_j^t$	The cost of testing manufactured products by 3PL in period $t$	Rot	The percentage of products that must be delivered to 3PL
$S{C}_j^t$	Shortage cost of product $j$ in period $t$

Table 3

The values of parameters used in implementing the proposed model

Parameter	Value	Parameter	Value
$S{P}_j^t$	U(500,1000)	$v_k^t$	U(60000,80000)
$TS{P}_j^t$	U(400,950)	$MA{R}_j$	0.8
$P{C}_j^t$	U(100,150)	$MI{R}_j$	0.2
$D_{jf}^t$	U(1000,2500)	$o_i^t$	0.7
$SD{C}_j^t$	U(5,10)	C	6
$E_i^t$	U(1500,2000)	$SR{F}_{ik}^t$	U(0.05,0.11)
$D{C}_i^t$	U(2,5)	$E{C}_j^t$	U(1,3)
$h_i^t$	U(5,10)	$T{C}_m^t$	U(2500,6500)
$e_i^t$	U(1,2)	$TM{C}_m^t$	U(1500,1850)
$o_{il}^t$	U(2,4)	$S{R}_{jk}^t$	U(350,600)
$p_{il}^t$	U(4,5)	$DC{C}_{jw}^t$	U(6000,8000)
$G_{il}^t$	U(10000,40000)	$TP{C}_j$	U(50,100)
$hc$	2	$y_{jk}^t$	U(35,100)
$r_{ik}^t$	U(10,20)	$L$	100,000
$B_k^t$	U(200000,300000)	$MP{C}_j^t$	U(3500,4500)
$M{C}_i^t$	U(10000,50000)	$M{P}_j^t$	U(5000,8500)
$N_j^t$	U(2,4)	$TR{F}_{jm}^t$	U(0.02,0.1)
$S{C}_j^t$	U(50,200)	Rot	0.8

Table 4

Parameters related to parts and refurbishing sites

The amount of using part $i$ for each unit of product $j\left(q_{ij}\right)$
j/i	1	2	3	4	5
1	2	1	3	1	3
2	1	3	2	1	2
3	3	2	1	4	1
4	2	1	2	3	4
5	1	3	2	2	2

The resource used for refurbishing one unit of part $i$ in the refurbishing site $l$ in period $t\left(g_{ij}\right)$
l/i	1	2	3	4	5	6
1	1	1	1	1	1	1
2	1	1	1	1	1	1
3	1	1	1	1	1	1
4	1	1	1	1	1	1
5	1	1	1	1	1	1

The resources used by the supplier $k$ to manufacture each unit of product $i$ $\left(b_{ik}\right)$
k/i	1	2	3	4	5
1	1.5	2	3	1	3
2	2	1	1	3	1
3	2	1.5	1	3	2.5
4	1.5	3	2.5	2	3
5	3	2	3	2	1.5

Table 5

The amount of Z₁ for different levels of uncertainty and different values of Z₂ and Z₃

O.F	Z2 ≥ 0.7	Z2 ≥ 0.8	Z2 ≥ 0.85	Z2 ≥ 0.85	Z2 ≥ 0.9	Z2 ≥ 0.9	Z2 ≥ 0.95	Z2 ≥ 0.95	Z2 ≥ 0.98
ψ	Z3 ≤ 125,000	Z3 ≤ 125,000	Z3 ≤ 130,000	Z3 ≤ 140,000	Z3 ≤ 140,000	Z3 ≤ 155,000	Z3 ≤ 155,000	Z3 ≤ 160,000	Z3 ≤ 160,000
0	158,621,766	158,621,766	166,802,043	175,541,011	175,139,272	185,693,504	185,629,550	188,678,526	188,141,711
0.001	158,553,688	158,553,688	166,375,033	175,503,264	173,924,183	185,655,704	185,153,350	188,641,626	186,506,928
0.004	158,348,967	156,371,026	166,375,033	175,390,054	173,924,183	185,542,335	185,153,350	186,468,934	186,506,928
0.006	158,209,108	156,371,026	166,375,033	174,498,546	173,924,183	185,084,881	185,153,350	186,468,934	186,506,928
0.008	158,069,329	156,371,026	166,375,033	174,498,546	173,924,183	185,084,881	185,153,350	186,468,934	186,506,928
0.01	157,929,632	156,371,026	166,375,033	174,498,546	173,924,183	185,084,881	185,153,350	186,468,934	186,506,928
0.04	157,929,632	156,371,026	166,375,033	174,498,546	173,924,183	185,084,881	185,153,350	186,468,934	186,506,928
0.06	157,929,632	156,371,026	166,375,033	174,498,546	173,924,183	185,084,881	185,153,350	186,468,934	186,506,928
0.08	157,929,632	156,371,026	166,375,033	174,498,546	173,924,183	185,084,881	185,153,350	186,468,934	186,506,928

Table 6

The number of product $j$ manufactured by the producer $n$ in period $t$ $\left(p_{jn}^t\right)$

j.n/t	1	2	3	4	5	6	7	8	9	10
1.1			2,117
1.2	1,879		1,875
1.3		1,196			1,765	908
1.4				6,105	3,780		3,556	2,514		1,372
1.5	3,767	4,463		143	436	3,904	1,317	4,079	3,393	4,392
2.1	1,259					909
2.2	3,697	2,639	166	4,173		4,032		524
2.3				254			4,973	5,003	1,048	907
2.4		3,503
2.5			4,276		4,023
3.1		3,502				3,187		2,040
3.2		247			4,406		4,442	2,853	5,520
3.3			2,221	3,559						4,209
3.4										1,286
4.1				6,315	1,785		470
4.2		1,507	2,016				1,010	2,452
4.3	2,066		1,884		3,028	3,790			5,335
4.4	4,159									3,434
5.1					4,398		3,049	2,620	5,373
5.2					1,205					4,509
5.3	4,441	2,494				5,007
5.4								1,631
5.5				6,652			2,962		171

Table 7

The number of part $i$ that must be purchased from the external supplier $k$ in period $t\left(Q_{ik}^t\right)$

i.k/t	1	2	3	4	5	6	7	8	9	10
1.1	32,100	40,921
1.2			17,895	14,598		36,903	46,274
1.3	10,453				50,627	4,735				50,550
1.4			38,705	20,825				61,992	35,667
2.1								8,629		31,868
2.2				45,524	16,881
2.3	39,695		27,467						42,538
2.4			9,165		25,411		30,184			15,572
2.5		40,264				50,195		38,011
3.1		33,619			25,285
3.2			25,666
3.3						37,152
3.4	28,378	26,072		62,696					20,300	5,731
3.5	29,581				29,571	9,564	57,597	31,605	31,485	46,939
4.1	20,636			82,165		49,896		31,978	48,944
4.2		24,679	373		45,541			20,027		73,589
4.3			16,033				51,160		11,507
4.4	46,074
4.5		10,191	25,962	3,993
5.1			42,148		68,616		25,092
5.2	58,070								73,997
5.3		38,039		20,645		4,305		22,752
5.4						33,766	17,595	27,180
5.5				37,832						37,877

Table 8

The number of the returned product $j$ that must be given to 3PL m in period t for reproduction $\left(T{P}_{jm}^t\right)$

j.m/t	1	2	3	4	5	6	7	8	9	10
1.1	1,335		3,294			173			538
1.2	637	2,141				2,390		1,402
1.3			672		1,591	228	2,129			1,471
1.4				1,190					2,855
2.1					1,089	1,406				40
2.2		761			1,723		1,597	1,539	2,030	1,694
2.3	2,683	869		1,418
2.4			1,548
3.1		1,921		2,169			824	2,215		2,680
3.2	641				1,030	453	1,403
3.3		918	1,426	309	401		1,301	307	1,229
3.4	1,892	910		1,081		2,735			363
3.5
4.1		1,507			1,633
4.2	1,416			1,467					1,008
4.3						1,761		2,452	516	1,449
4.4			1,752		40		1,480			1,045
4.5
5.1				716		1,710	1,737		2,031
5.2			1,413	1,056
5.3	144
5.4	1,450	1,451			1,434			1,781		1,561

Table 9

The number of returned product $j$ that goes to the evaluation site in period t ($R{P}_j^t$)

j/t	1	2	3	4	5	6	7	8	9	10
1	2,466	2,676	4,957	1,487	1,989	3,489	2,662	1,753	4,242	1,839
2	3,354	2,037	1,935	1,772	3,515	1,758	1,996	1,923	2,537	2,168
3	3,166	4,686	1,782	4,449	1,788	3,984	4,410	3,152	1,990	3,350
4	1,770	1,883	2,191	1,833	2,091	2,201	1,850	3,064	1,905	3,118
5	1,992	1,814	1,767	2,215	1,793	2,137	2,171	2,226	2,539	1,951

Table 10

Comparing the performance of the proposed model and the NSGA-II algorithm (iterations = 10)

Problem	Number of companies	Number of customers	Number of products	Number of parts	NSGA-II algorithm			Proposed solution approach
					NOS	Diversity	Time (min)	NOS	Diversity	Time (min)
1	5	5	5	5	4	79,845.35	73.61	3	63,251.21	63.24
2	5	4	5	3	5	63,251.32	56.32	4	56,325.25	53.25
3	5	5	4	5	3	45,326.35	76.39	3	58,365.25	70.32