A Fuzzy Robust Programming Model for Sustainable Closed-Loop Supply Chain Network Design with Efﬁciency-Oriented Multi-Objective Optimization

: Sustainable closed-loop supply chain (SCLSC) network design and decision-making is a critical problem for enterprises and organizations’ operations because of its excellent economic, environmental, and social performance. This article proposes a multi-objective mixed-integer programming model with targets for minimum total cost, reduction in environmental damage, and maximum social responsibility. In order to deal with the uncertainty caused by the dynamic business environment, a fuzzy robust programming (FRP) approach is applied. Furthermore, an efﬁciency-oriented optimization methodology, hybridizing meta-heuristics and efﬁciency evaluation, is proposed to solve the developed multi-objective model and functions as auxiliary decision-making. Data envelopment analysis is applied to evaluate the sustainability performance of feasible solutions and calculate their efﬁciency. The efﬁciency can comprehensively reﬂect the sustainability performance and guide the evolution process of meta-heuristic algorithms. A numerical case validates the proposed FRP model and efﬁciency-oriented optimization methodology. The results demonstrate that with the proposed methodology, decision-makers not only can obtain a set of efﬁcient schemes but also can determine the optimal scheme with the best sustainability performance.


Introduction
With the increasing awareness of environmental problems, social responsibility, and the competitive market environment, organizations and enterprises focus on taking the sustainability paradigm into supply chain network design [1,2]. The connotation of sustainable development is very extensive. As illustrated in the published report The 2030 Agenda for Sustainable Development by the World Commission on Environment and Development (WCED), energy crisis, employment difficulties, and resource shortage are matters of significant importance in sustainable development. The tendency to design a sustainable supply chain (SSC) network has several motivations including: (1) achieving competitive advantages, following the tide of globalization and maintaining customer loyalty [3]; (2) promoting the conservation and sustainable use of energy and other resources [4]. In the scope of SSC network design, the sustainability paradigm can be considered an integrated paradigm including economic, environmental, and social dimensions [5,6]. Sustainability design in the supply chain contains the concepts of green development and social responsibility, which requires supply chain managers to systematically take environmental and social objectives as strategic considerations [7,8]. Based on the sustainability concept, the SSC network design in this article comprehensively considers an increase in economic benefit and a reduction in environmental damage and social responsibility. The closed-loop supply chain (CLSC) design has drawn more and more attention in dealing with environmental damage and resource shortage [9,10]. The government has published some regulations which require the original equipment manufacturers to recycle, collect, formance of feasible solutions obtained by solving the multi-objective model. The critical issues of performance evaluation are to select appropriate indicators and efficient evaluation methods. The indicators should comprehensively reflect the economic, environmental, and social paradigms and could be measured quantitatively. The evaluation approach should be able to quantitatively calculate and describe the sustainability performance of feasible solutions so that the optimal solution can be selected according to performance sorting. Multi-criteria decision-making (MCDN) is widely used in the comprehensive evaluation of sustainable performance.Şenel et al. apply the concept of octahedron sets to multi-criteria decision-making problems and validate the improved MCDM method [32]. The data envelopment analysis (DEA) method, as a quantitative multi-criteria decision method, is widely applied in evaluating sustainability performance [33][34][35]. Hence, an efficiency-oriented optimization methodology is proposed to solve the multi-objective model and select the optimal solution. In this methodology, multi-objective optimization and performance evaluation are performed in parallel. Meta-heuristic algorithms are utilized to obtain feasible solutions and output them as the decision-making units (DMU) of performance evaluation. The DEA method evaluates DMUs, calculates their efficiency, and selects the optimal solution according to their efficiency sorting.
In general, this study provides a framework for designing a sustainable closed-loop supply chain network under uncertainty, which minimizes net cost and pollution emission and maximizes social performance. The research problem considers both strategic decisions and planning decisions. Strategic decisions involve supplier selection and location of facilities. Planning decisions conclude the quantity of raw material that should be bought from each supplier and the amount of product that exists in each part of the network. Furthermore, for coping with uncertain parameters, we adopt the fuzzy robust programming approach to derive a crisp equivalent model of a certain type. Finally, we propose an efficiency-sorting-based multi-objective optimization algorithm. In this algorithm, the optimization process and the evaluation process are performed in parallel. The individual with the best sustainability performance obtained by the evaluation process can guide the next iteration of the optimization process.
The following research is organized as follows. Section 2 reviews the corresponding literature. The proposed multi-objective mixed-integer programming model and the fuzzy robust counterpart form are developed in Section 3. Section 4 presents the multi-objective intelligent optimization algorithm as the solution method. Section 5 provides a numerical case and corresponding results analyses. Conclusions are represented in Section 6.

Literature Review
Due to the importance of excellent designing of sustainable closed-loop supply chain networks, many papers are conducted in this area and the research on the uncertainty of the supply chain has become deeper. This section is dedicated to conducting a review of papers involving SCLSC network design with uncertainty. Multi-objective optimization has been an efficient method for solving the problem of supply chain network design in the case of seeking multi-objective performance. Wang et al. considered economic performance and environmental influence in designing a green supply chain network. The multi-objective model is solved by the normalized normal constraint approach [18].
On the other hand, uncertainty is another important aspect of the supply chain network. Govindan et al. reviewed the studies relating to supply chain network design under uncertainty [36]. They pointed out that designing a reliable supply chain network can efficiently deal with high uncertainty. Pishvaee et al. developed a robust optimization model for coping with a closed-loop supply chain including customers in the first and second markets [37]. Recently, most of the research takes into consideration multi-objective optimization and uncertainty optimization. Talaei et al. firstly proposed fuzzy robust optimization approach for dealing with uncertain cost and demand rates in a carbon-efficient green closed-loop supply chain network [21]. The fuzzy robust model can obtain the optimal solutions which incur a cost increase named "robustness price" compared with the determined model. On the basis of the fuzzy robust optimization method, Nayeri et al. solved a multi-objective FRO model for designing a sustainable closed-loop tire supply chain network by the goal programming approach [38]. Soleimani et al. proposed a fuzzy multi-objective model for designing a sustainable green supply chain with social consideration [39]. The mathematical model was solved by the ε-constraint method and genetic algorithm. Safaei et al. adopted robust optimization for coping with uncertain scenarios in the cardboard closed-loop supply chain [40]. Setiawan et al. optimized a multi-objective mask closed-loop supply chain considering environmental impact [41]. The objective functions were defined by the fuzzy membership degree function and converted into a single-objective model by ε-constraint. Ghahremani-Nahr et al. applied robust fuzzy programming to address the effect of uncertainty in the closed-loop supply chain [42]. The multi-objective whale optimization algorithm is applied for solving the developed equivalent model. Zhang et al. developed a multi-objective robust fuzzy optimization model for coping with the inherent uncertainty in the closed-loop supply chain [5]. The uncertainty was divided into two categories: one was uncertainty parameters solved by fuzzy membership theory and the other was managed with the robust optimization method. Pourmehdi et al. adopted a scenario-based stochastic programming method for dealing with the uncertainty in the steel sustainable closed-loop supply chain [43]. The preemptive fuzzy goal programming, as a particle multi-objective method, was utilized to solve the mathematical model. Yu et al. designed and optimized a multi-objective hazardous waste network [44]. Stochastic programming was used to define uncertain parameters and a sample average approximation-based goal programming method was applied for solving the model. Goodarzian et al. developed a multi-objective fuzzy robust optimization model for designing a pharmaceutical supply chain network [26]. They applied several metaheuristic algorithms named MOSEO, MOSAM, MOKA, and MOFFA, for obtaining the optimal solution. Isaloo et al. developed a scenario-based multi-objective model with the targets of minimum cost and pollution emission for configuring the plastic injection supply chain network [45]. Fathollahi-Fard et al. proposed a multi-objective two-stage stochastic programming model for maximizing economic and social performance [46]. Furthermore, there are many uncertain factors in the sustainable closed-loop supply chain network, such as the quantification of social impact. Many studies focus on modeling these factors. Soleimani et al. adopted a fuzzy set to model the lost working days due to work damages for described social factors [39]. Lee et al. extended fuzzy sets and proposed Cubic sets [47]. Ma et al. adopted interval uncertainty sets, including box sets and polyhedron sets, to model uncertainty parameters in the shared cycle recycling supply chain [6].
Their results demonstrated that the proposed hybrid meta-heuristic algorithms had better solution efficiency than current methods.
From the aforementioned review: • Most recent researches pay attention to multi-objective optimization for seeking an excellent trade-off between economic and environmental performance. Few take into consideration social performance.

•
For different kinds of uncertainty in the supply chain network, there are targeted uncertainty technologies for coping with them. • For solving the multi-objective mathematical model, firstly, some studies convert the multi-objective optimization problem into a single-objective problem by ε-constraint and other methods. Secondly, some exact solution methods are applied, such as goal programming. Thirdly, metaheuristic approaches are utilized which adopt the concept of the Pareto optimal solution. However, when solving multi-objective SCLSC network optimization problems, metaheuristic algorithms are rarely applied. Environmental pollution will happen in the process of transportation and operation involving production centers, repair centers, recycling centers, and disposal centers. In the real world, production, repair, recycling, disposal, and transportation activity lead to carbon emissions or solid waste emissions.

Social factor modeling
In this article, lost working days due to occupational accidents are adopted to evaluate social performance. This evaluation indicator is related to the facilities opening involving production centers, repair centers, and recycling centers. However, in the real world, prior information about lost working days is difficult to obtain due to different production technologies, staff equality, and so on. Hence, fuzzy programming is utilized to measure lost working days. Figure 2 shows the fuzzy membership level of lost working days due to work damage. In Figure 2, LDideal represents the ideal number of lost working days, and max LD is the maximum number. Then, the membership degree functions of lost working days involving the aforementioned facilities are calculated as Formula (1). Environmental pollution will happen in the process of transportation and operation involving production centers, repair centers, recycling centers, and disposal centers. In the real world, production, repair, recycling, disposal, and transportation activity lead to carbon emissions or solid waste emissions.

Social Factor Modeling
In this article, lost working days due to occupational accidents are adopted to evaluate social performance. This evaluation indicator is related to the facilities opening involving production centers, repair centers, and recycling centers. However, in the real world, prior information about lost working days is difficult to obtain due to different production technologies, staff equality, and so on. Hence, fuzzy programming is utilized to measure lost working days. Figure 2 shows the fuzzy membership level of lost working days due to work damage. In Figure 2, LDideal represents the ideal number of lost working days, and LDmax is the maximum number. Then, the membership degree functions of lost working days involving the aforementioned facilities are calculated as Formula (1). where G max e represents the maximum mean of lost working days of production centers, repair centers, and recycle centers. G e indicates the number of lost working days of the abovementioned three facilities.

{ }
e e e max e e max e ee e where max e G represents the maximum mean of lost working days of production centers, repair centers, and recycle centers. e G indicates the number of lost working days of the abovementioned three facilities.

Notations
Firstly, we present the indices, parameters, and decision variables involved to form the mathematical model. Demands of repair center r for raw material n  ij TC Transport cost of raw material per kg from raw material supplier i to production center j  jk TC Transport cost of product per kg from production center j to distribution Transport cost of product per kg from distribution center k to customer c

Notations
Firstly, we present the indices, parameters, and decision variables involved to form the mathematical model. Demands of repair center r for raw material n TC ij Transport cost of raw material per kg from raw material supplier i to production center j TC jk Transport cost of product per kg from production center j to distribution center k TC kc Transport cost of product per kg from distribution center k to customer c TC ir Transport cost of raw material per kg from raw material supplier i to repair center r TC cb Transport cost of product per kg from customer c to recycling center b TC br Transport cost of product per kg from recycling center b to repair center r TC rk Transport cost of product per kg from repair center r to distribution center k TC bd Transport cost of raw material per kg from recycling center b to disposal center d Membership degree for the number of lost working days because of work damage for each worker in recycling center b The first objective function seeks to minimize the total net cost, which indicates the economic aspect of the SRCLSC network. The net cost consists of transportation costs between facilities, facility opening costs, procurement costs of components, production cost, repair cost, testing cost, disassembly cost, disposing cost, and revenue.

The Second Objective Function: Environmental Factor
The second objective function seeks to minimize pollution emissions for maximizing environmental performance. Two aspects make up the environment: carbon emission and solid waste emission. The former is produced by vehicle emission during transportation Processes 2022, 10, 1963 9 of 24 between facilities. The latter is produced during the operation of the facilities, involving the process of production, repair, disassembly, testing, and disposal.

The Third Objective Function: Social Factor
The third objective function seeks to minimize the lost working days due to work damage by maximizing the membership degree of all corresponding facilities, which represents the social aspect.

Flow Balance Constraints
Constraint (9) guarantees that the number of raw materials should be transited from raw materials suppliers and that all of the raw materials are used to produce new products. Relation (10) ensures the quality of the inflow and outflow of new products for each distribution center. Equations (11) and (12) confirm the recycled products from customers to recycling centers are dismantled into various raw materials, which are transported to repair centers and disposal centers. Constraint (13) shows that the repair centers make all the raw materials for the new products from raw materials suppliers and recycling centers.

Demand and Recycling Constraints
Equation (5) calculates the number of new products transported from distribution centers to customers and ensures that the demand for new products in customers is satisfied. Equation (6) calculates the number of recycled EOL products transited from customers to recycling centers. Expression (7) calculates the number of raw materials sent to the repair center from raw materials suppliers and calculates that raw material suppliers will satisfy the demand for components in repair centers.

Carbon Cap Constraints
Formula (8) represents that the carbon emission of the SCLSC network should be less than or equal to the carbon emission capacity.

Capacity Constraints
Constraint sets (14)- (17) guarantee that the inflow and outflow cannot be more than the capacity of opened production centers and distribution centers. Constraint (18) shows that the recycled products flowing into recycling centers are less than or equal to the capacity of recycling centers for products. Formula (19) shows that the recycled products flowing out of recycling centers should not exceed the capacity of recycling centers for each raw material. Similarly, relation (20) and (21) separately indicate the capacity of repair centers for incoming raw materials and outgoing products. Constraint (22) ensures the products flowing into disposal centers cannot be more than the capacity for raw materials.

Lost Working Days Constraints
Constraint sets (23)- (25) show the limitations of the fuzzy membership degree relating to the lost working days because of occupational involving production centers, repair centers and recycling centers.
3.5.6. Binary and Non-Negative Constraint (26) and (27) indicate the characteristics of decision variables.

Fuzzy Robust Optimization Model
To deal with the uncertainty parameters in the SCLSC network, the FRP approach proposed by Talaei et al., is employed in this article. FRP has an excellent performance in the problem including uncertainty parameters with epistemic characteristics [21]. The FRP approach is an extended form of the chance constraint fuzzy programming. For better understanding, the proposed model can be abstracted to the compacted form of the possibility linear programming problem as follows: Bx ≤ e where vector f and coefficient matrix B and N are crisp parameters, when vector c 1 , d and e are uncertainty parameters. Formula (28) represents the membership function of the trapezoidal fuzzy member r by four sensitive spots (r 1 , r 2 , r 3 , r 4 ).
Knowing that constraints (29) and (30) with uncertain parameters should be formulated with a satisfaction level of α k means decision makers will be satisfied as to the necessity for each constraint. Therefore, the deterministic model can be formed as follows: According to the fuzzy robust optimization proposed by Talaei et al. [21], the FRP counterpart of Formula (40) can be presented as follows: Z max in Formula (41) is defined as follows: In Formula (41), the first expression indicates minimizing the expected value of the first objective function. The second expression measures the difference between the most pessimistic value and the expected value. η represents the weight of the second part of Formula (41). Moreover, π k is the unit penalty for the possible deviation from each constraint with uncertain parameters. Equation (42) calculates the worst case of the first objective function.
Hence, the developed multi-objective FRP model for SCLSC network design can be written as follows.
The second and third objective functions s.t.
Constraint (5)- (27) where φ, ϕ, ς, ξ are satisfaction levels for constraints with uncertain parameters and η is the weight of the deviation of excepted value and maximum value of the first objective function. Furthermore, π 1 , π 2 , π 3 , π 4 are the unit penalty of deviation of constraints with uncertain parameters.

Efficiency-Oriented Multi-Objective Optimization
Because the proposed model is a multi-objective mixed-integer linear programming problem, objective functions conflict with each other and it is difficult to find the optimal solution. In general, a set of feasible solutions can be obtained by solving the multi-objective model. In this article, an efficiency-oriented optimization methodology is proposed to select the optimal solution according to the efficiency sorting of feasible solutions. This optimization methodology hybridizes multi-objective meta-heuristic algorithms and the DEA model. Meta-heuristic algorithms can obtain feasible solutions which are considered the input DMUs of the DEA method. The traditional CCR model can distinguish efficient DMUs and inefficient DMUs, while the second goals-based DEA model can further sort the efficient DMUs according to their cross-efficiency values. In this methodology, the evaluation results of the DEA methods can guide the evolution process of the population in meta-heuristic algorithms. The steps of the efficiency-oriented optimization methodology are presented in Figure 3.

Efficiency-Oriented Multi-Objective Optimization
Because the proposed model is a multi-objective mixed-integer linear programming problem, objective functions conflict with each other and it is difficult to find the optimal solution. In general, a set of feasible solutions can be obtained by solving the multi-objective model. In this article, an efficiency-oriented optimization methodology is proposed to select the optimal solution according to the efficiency sorting of feasible solutions. This optimization methodology hybridizes multi-objective meta-heuristic algorithms and the DEA model. Meta-heuristic algorithms can obtain feasible solutions which are considered the input DMUs of the DEA method. The traditional CCR model can distinguish efficient DMUs and inefficient DMUs, while the second goals-based DEA model can further sort the efficient DMUs according to their cross-efficiency values. In this methodology, the evaluation results of the DEA methods can guide the evolution process of the population in meta-heuristic algorithms. The steps of the efficiency-oriented optimization methodology are presented in Figure 3.
Step 1: Initialize algorithm parameters and generate the initial population.
Step 2: Perform the evolution process on the parent population based on the special evolution mechanism and generate offspring population. The offspring population will converge to the optimal solution with the best cross-efficiency.
Step 3: Update the archive according to the dominant relationship of individuals and select the superior Pareto optimal solutions. The individuals in the archive are considered DMUs of DEA methods.
Step 4: On one hand, divide the Pareto optimal individuals into efficient individuals and inefficient individuals based on the CCR model. On the other hand, furtherly evaluate efficient DMUs, calculate their cross-efficiency values and sort them. The individual with the best cross-efficiency is selected as the optimal solution.
Step 5: Cycle from step 2 to step 4 until the maximum iteration number is met. Finally, the Pareto optimal solutions in the final archive are considered feasible solutions and the solution with the best cross-efficiency is regarded as the optimal one.  Step 1: Initialize algorithm parameters and generate the initial population.
Step 2: Perform the evolution process on the parent population based on the special evolution mechanism and generate offspring population. The offspring population will converge to the optimal solution with the best cross-efficiency.
Step 3: Update the archive according to the dominant relationship of individuals and select the superior Pareto optimal solutions. The individuals in the archive are considered DMUs of DEA methods.
Step 4: On one hand, divide the Pareto optimal individuals into efficient individuals and inefficient individuals based on the CCR model. On the other hand, furtherly evaluate efficient DMUs, calculate their cross-efficiency values and sort them. The individual with the best cross-efficiency is selected as the optimal solution.
Step 5: Cycle from step 2 to step 4 until the maximum iteration number is met. Finally, the Pareto optimal solutions in the final archive are considered feasible solutions and the solution with the best cross-efficiency is regarded as the optimal one.

Multi-Objective Particle Swarm Optimization (MOPSO) Algorithm
MOPSO is a commonly used swarm intelligent optimization algorithm. Based on traditional particle swarm optimization, the non-dominated sorting mechanism and elite archive are introduced into MOPSO. The main characteristic is the updating process of position and velocity of particles which is calculated in Equations (50) and (51): x i (k + 1)= x i (k)+v i (k + 1) (51)

Efficiency Sorting Strategy
The DEA approach is applied to evaluate the feasible solutions obtained in each iteration and calculate their efficiency values. In the DEA method, the normalization and standardization of indicator values are not required. The traditional DEA model can divide the decision-making units (DMUs) into efficient units and inefficient units. For further evaluating and distinguishing the efficiency of efficient DMUs, some studies focus on the extended DEA model for quantitatively sorting objective data. In this article, based on the efficiency sorting multi-objective optimization framework proposed by Wang et al. (2020), the secondary goals-based DEA model is utilized to evaluate and sort the DMUs in an efficiency-oriented solution methodology. The principles of the traditional DEA model and secondary goals-based DEA model are presented as follows.

CCR Model
The traditional DEA model (i.e., CCR model) can calculate the self-evaluation efficiency values of DMUs. The CCR model with constant returns to scale is presented as follows. where E dd is the self-evaluation efficiency value of dth DMU, y rd is the value of rth output indicator of dth DMU and u rd is the corresponding weight, x id is the value of ith input indicator of dth DMU and v is the corresponding weight. Relation (53) represents that the efficiency value of v DMU should be smaller than or equal to 1. Expression (54) indicates the weights of output and input should be more than 0. ε states a non-Archimedes number that is smaller than any positive number. If the E dd is equal to 1, the DMU is DEA efficient; otherwise, the DMU is inefficient.

Secondary Goals-Based DEA Model
The traditional CCR model can distinguish DMU sets by dividing them into DEA efficient units and inefficient units. However, when the self-evaluation efficiency values of multiple DMUs are equal to 1, the CCR model cannot distinguish and further sort them. In this article, the secondary goals-based DEA model proposed by Doyle et al. is utilized to further evaluate the cross-evaluation efficiency value of DMUs and sort them. Differently from the self-evaluation based CCR model, the secondary goals-based DEA model is based on cross-evaluation, which is shown as follows.
where objective function (36A) aims to minimize ∑ n j=1 z d j (i.e., ensure z d j to be 0) for obtaining a set of optimal weights. Expressions (56)-(58) specify the calculation and definition of self-evaluation efficiency. The constraints (59) and (60) specify the comparison of cross-evaluation efficiency and self-evaluation efficiency of dth DMU. If z d j = 1, h d j ≤ 0 according to constraint (59) and the cross-evaluation efficiency of dth DMU is greater than its self-evaluation efficiency according to constraint (60). Otherwise, h d j ≥ 0 and the cross-evaluation efficiency of dth DMU is smaller than its self-evaluation efficiency. Expression (63) specifies that the weights of input and output indicators should be more than 0. According to the cross-evaluation efficiency value obtained by the secondary goals-based DEA model, the efficient DMUs can be distinguished and sorted.

Indicator Selection
In order to evaluate the feasible solutions obtained in each iteration and calculate the efficiency, it is critical to select the appropriate input and output indicators. In the DEA model, the indicator selection should follow the principles below: (1) The indicators should be quantitative to avoid the influence of subjective preference. According to the abovementioned principles, the evaluation indicators are presented as follows: The input indicators contain: (1) Cost indicators, including transportation cost, facility opening cost, ordering cost, and facility processing cost:

Results and Discussion
Two efficiency-oriented solution methodologies are applied to solve the mathematical model developed in this article, which is secondary goals-based NSGA-II (SG-NSGA-II) and secondary goals-based MOPSO (SG-MOPSO), respectively. In an attempt to ensure the fairness of the experiments, the corresponding control parameters of NSGA-II and MOPSO are set as follows: maximum number of iterations = 100; population size = 200; variable dimension = 336; in NSGA-II, crossover percentage = 0.7; mutation possibility = 0.02; in MOPSO, inertia weight = 0.7298; personal learning coefficient = 1.4962; global learning coefficient = 1.4962.
Both efficiency-oriented solution methodologies were run 10 times. The Pareto front obtained by SG-NSGA-II and SG-MOPSO is shown in Figure 4. The x-axis indicates the total economic performance (the first objective function), the y-axis represents the environmental pollution emission (the second objective function), and the z-axis represents the social impact (the third function). In Figure 4 it can be seen that the distribution degree of feasible solutions obtained by SG-NSGA-II is higher than that obtained by SG-MOPSO. The aforementioned phenomenon indicates that the solution diversity is similar in the two efficiency-oriented solution methodologies. To further distinguish the quality and performance of obtained feasible solutions, Tables 4 and 5 present the self-evaluation values and cross-efficiency values of the feasible solutions obtained by SG-NSGA-2 and SG-MOPSO. In Tables 4 and 5, "self" indicates self-evaluation efficiency calculated by the traditional CCR model and "cross" represents cross-evaluation efficiency obtained by the secondary goals-based DEA model. Comparing the self-evaluation efficiency, the DMUs can be divided into DEA efficient units and inefficient units. As presented in Tables 4 and 5, the values of input and output indicators are presented. According to self-evaluation efficiency, it can conclude that all obtained schemes are DEA relatively efficient. Furthermore, all feasible solutions can be distinguished and sorted based on cross-evaluation efficiency. As shown in Tables 4 and 5, the optimal schemes with the optimal cross-evaluation efficiency are in bold.
The important decisions relating to network configuration solved by SG-NSGA-II and SG-MOPSO are presented in Table 6. Table 6 presents the network configuration decision of the optimal scheme obtained by both efficiency-oriented solution methodologies. For example, according to the obtained schemes obtained by SG-NSGA-II, the raw material supplier 4 is selected. As can be seen in Table 6, production center 1 and 2 are opened, distribution center 2, 4, 5, and 6 are opened, repair center 1 and 3 are opened, recycling center 5 is opened, and disposal centers are opened in potential location 1 and 2.

Comparison of Algorithms
This section performs a comparison of the solution performance of two efficiencyoriented solution methodologies and the sustainability performance of their solutions. Figure 5 shows the comparison of the values of objective functions obtained by SG-NSGA-II and SG-MOPSO. It can be seen that the net cost and the pollution emission solved by SG-NSGA-II are lower than that of SG-MOPSO. Furthermore, there are fewer lost working days in the case of SG-NSGA-II, which represents that the social performance of the scheme obtained by SG-MOPSO is better than SG-NSGA-II. Furthermore, Figure 6 presents the comparison of cross-efficiency values of feasible schemes obtained by two efficiencyoriented solution methodologies. It can be concluded that in total, the cross-efficiency values of all feasible solutions obtained by SG-MOPSO are better than that of SG-NSGA-II. On the other hand, the sustainability performance of the optimal scheme of SG-MOPSO is greater than that of SG-NSGA-II. Hence, SG-MOPSO has better solution performance than SG-NSGA-II.

Comparison of Algorithms
This section performs a comparison of the solution performance of two efficiencyoriented solution methodologies and the sustainability performance of their solutions. Figure 5 shows the comparison of the values of objective functions obtained by SG-NSGA-II and SG-MOPSO. It can be seen that the net cost and the pollution emission solved by SG-NSGA-II are lower than that of SG-MOPSO. Furthermore, there are fewer lost working days in the case of SG-NSGA-II, which represents that the social performance of the scheme obtained by SG-MOPSO is better than SG-NSGA-II. Furthermore, Figure 6 presents the comparison of cross-efficiency values of feasible schemes obtained by two efficiency-oriented solution methodologies. It can be concluded that in total, the cross-efficiency values of all feasible solutions obtained by SG-MOPSO are better than that of SG-NSGA-II. On the other hand, the sustainability performance of the optimal scheme of SG-MOPSO is greater than that of SG-NSGA-II. Hence, SG-MOPSO has better solution performance than SG-NSGA-II.

Robustness Analyses
This section is dedicated to performing sensitivity analysis on FRO and the deterministic model. The results obtained by solving FRO and the deterministic model are compared. The uncertain parameters are defined as the expected value of the trapezoidal fuzzy member in the deterministic model. Figures 7 and 8 present the comparison of objective function values obtained by solving two models. In Figures 7 and 8, the "FRO" in the x-axis indicates the fuzzy robust optimization model and "Deterministic" represents the deterministic model. The "net cost" indicates the economic performance (the first objective function), the "pollution emission" represents the environmental (the second objective function), and "lost working days" is the social impact (the third objective function). The difference in the net cost between the fuzzy robust programming model and the deterministic model is defined as "robustness cost" which is the cost that incorporates into the supply chain to face the uncertain environment. Furthermore, the environmental and social performance obtained by the fuzzy robust programming model is better than that of the deterministic model. Based on the mentioned point, considering the fuzzy robust model in the real world may lead to better sustainable performance and economic performance in the long term.

Robustness Analyses
This section is dedicated to performing sensitivity analysis on FRO and the deterministic model. The results obtained by solving FRO and the deterministic model are compared. The uncertain parameters are defined as the expected value of the trapezoidal fuzzy member in the deterministic model. Figures 7 and 8 present the comparison of objective function values obtained by solving two models. In Figures 7 and 8, the "FRO" in the x-axis indicates the fuzzy robust optimization model and "Deterministic" represents the deterministic model. The "net cost" indicates the economic performance (the first objective function), the "pollution emission" represents the environmental (the second objective function), and "lost working days" is the social impact (the third objective function). The difference in the net cost between the fuzzy robust programming model and the deterministic model is defined as "robustness cost" which is the cost that incorporates into the supply chain to face the uncertain environment. Furthermore, the environmental and social performance obtained by the fuzzy robust programming model is better than that of the deterministic model. Based on the mentioned point, considering the fuzzy robust model in the real world may lead to better sustainable performance and economic performance in the long term.

Conclusions
This article addresses a sustainable closed-loop supply chain network design and design problem under high uncertainty. A multi-objective mixed-integer programming model is developed for the purpose of the minimum total cost, reduction in environmental damage, and maximum social responsibility. The social issue is measured by lost working days caused by occupational accidents which are defined quantitatively by fuzzy programming. Furthermore, in order to cope with the uncertainty, a fuzzy robust program-

Conclusions
This article addresses a sustainable closed-loop supply chain network design and design problem under high uncertainty. A multi-objective mixed-integer programming model is developed for the purpose of the minimum total cost, reduction in environmental damage, and maximum social responsibility. The social issue is measured by lost working days caused by occupational accidents which are defined quantitatively by fuzzy programming. Furthermore, in order to cope with the uncertainty, a fuzzy robust programming approach is applied to convert the developed model. An efficiency-oriented optimization methodology is proposed to solve the multi-objective FRP model. Based on this methodology, decision-makers can obtain a set of feasible solutions, evaluate comprehensively their sustainability performance, and select the optimal solution according to cross-evaluation efficiency values. Meanwhile, decision-makers can distinguish efficient units and inefficient units of feasible solutions according to self-evaluation efficiency. In this efficiency-oriented optimization methodology, the cross-evaluation efficiency-sorting strategy can guide the evolution process of meta-heuristic algorithms. In summary, the numerical case validates the proposed FRP model and the efficiency-oriented optimization methodology. Finally, we perform sensitivity analyses on the robustness of FRP and the deterministic model. The results illustrate that the FRP model can efficiently deal with the uncertainty in the SCLSC network.
The main contributions are presented as follows: