Sustainable Closed-Loop Supply Chain Design Problem: A Hybrid Genetic Algorithm Approach

: In this paper, we propose a solution to the sustainable closed-loop supply chain (SCLSC) design problem. Three factors (economic, environmental, and social) are considered for the problem and the three following requirements are addressed while satisfying associated constraint conditions: (i) minimizing the total cost; (ii) minimizing the total amount of CO 2 emission during production and transportation of products; (iii) maximizing the social influence. Further, to ensure the efficient distribution of products through the SCLSC network, three types of distribution channels (normal delivery, direct delivery, and direct shipment) are considered, enabling a reformulation of the problem as a multi-objective optimization problem that can be solved using Pareto optimal solutions. A mathematical formulation is proposed for the problem, and it is solved using a hybrid genetic algorithm (pro-HGA) approach. The performance of the pro-HGA approach is compared with those of other conventional approaches at varying scales, and the performances of the SCLSC design problems with and without three types of distribution channels are also compared. Finally, we prove that the pro-HGA approach outperforms its competitors, and that the SCLSC design problem with three types of distribution channels is more efficient than that with a single distribution channel.


Introduction
In general, closed-loop supply chain (CLSC) design problems are multi-stage networks endowed with various facilities of forward logistics (FL) and reverse logistics (RL) at each stage. In FL, finished products are provided to customers via manufacturers, distribution centers (DCs), and retailers. On the other hand, in RL, some of the products returned by customers are recovered after performing classification and recovery processes at the collection center and the recovery center, respectively, and the rest are disposed at the disposal center.
Among the major trends in CLSC design problems, the principle of sustainable development has become popular in conventional literatures . According to this principle, economic, environmental, and social factors are usually considered for effectively constructing a sustainable CLSC (SCLSC) design problem.
The economic factors considered include the maximization of the total profit or the minimization of the total cost resulting from construction and operation processes of the CLSC design problem [4][5][6][7][8]16,17,20,22]. Wang and Hsu [5] developed a CLSC design problem to minimize the total cost, which is the aggregate of the total production cost, the total transportation cost, and the total fixed cost. In the CLSC design problem, the supplier, the manufacturer, the DC, and the customer are considered in FL and the dismantler is considered in RL. Following Wang and Hsu [5]. The CLSC design problem by Devika et al. [1] also seeks to minimize the total cost which is the aggregate of the total manufacturing cost, the total transportation cost, the total handling cost, the total purchasing cost, the total opening cost, and the total sale revenue. On the other hand, the CLSC design problem by Chen et al. [6] seeks to maximize the total profit, which is the aggregate of the total revenue (=the total sale revenue of new product) and the total cost (=total processing cost + the total transportation cost + the total fixed cost).
The environmental factors usually considered include the minimization of the total amount or the cost of CO2 or carbon emitted during the production and transportation of materials and products at each stage of the CLSC design problem [3,9,10,18,19,21]. The CLSC design problem by Paksoyet et al. [9] seeks to minimize the total cost of CO2 emissions during the transportation of materials or products at each stage by raw material suppliers, plants, DCs, warehouses, and customers in FL and collection centers, repairing centers, dismantlers, decomposition centers, and final disposals in RL. Talaei et al. [10] showed a simple CLSC design problem to minimize the total amount of CO2 emissions during the establishment of facilities, production and transportation of products, and disposal of waste products at each stage by the manufacturers and markets in FL and disposal centers, collection centers, and inspection centers in RL. Following Talaei et al. [10], Özceylan et al. [3] also showed a CLSC design problem for the automotive industry in Turkey. They minimized the total amount of CO2 emissions during the transportation of finished products and returned products at each stage. The CLSC design problem by Fahimnia et al. [19] seeks to minimize total cost of carbon emitted during the production and transportation of products at each stage by manufacturers, warehouses, and end-users in FL and collection centers, recycling centers and disposal centers in RL.
The relevant social factors usually considered are metrics like the number of job opportunities created by introducing new technology, the number of lost days due to damage to work, and general unemployment [1][2][3]23]. Devika et al. [1] constructed a CLSC design problem to optimize various social influences (e.g., the number of newly created job opportunities, their job types, and damage to work caused either during the establishment of facilities or during the manufacture and handling of products). Özceylan et al. [3] also considered some social influences (e.g., the number of new job opportunities by introducing new technology and the number of lost days due to damage to work) in their CLSC design problem.
As mentioned above, economic, environmental, and social factors have been individually considered in various studies. However, few papers have considered all three factors simultaneously for the SCLSC design problem [1,3,11].
The SCLSC design problem presented by Devika et al. [1] represents a multi-objective optimization problem by considering the three factors to be objective functions. For the economic factor, the problem focused on the minimization of the total cost (=the sum of the costs of manufacture, transportation, handling, purchase and opening − the sum of sale revenues). For the environmental and social factors, the metrics considered were the minimization of the total cost (=the sum of environmental damages − the sum of environmental advantages) and the maximization of the total number (=the sum of the number of the new job opportunity − the sum of the number of lost days due to damage to work), respectively. To solve the multi-objective optimization problem, a metaheuristic approach using an imperialist competitive algorithm and a variable neighborhood search was used. By performing numerical experiments based on various problem scales, it was proven that the meta-heuristic approach outperforms certain conventional hybrid approaches. However, the search speed of the meta-heuristic approach became slower than those of the conventional hybrid approaches as the problem scales were increased. On the other hand, the SCLSC design problem by Özceylan et al. [3] was represented by a single-objective optimization problem, by using the economic factor as the only objective function, and the remaining two factors (environmental and social factors) as constraints. The maximization of total profit (=the sum of total revenues − the sum of total costs) served as the economic factor. The total amount of CO2 emissions as well as the number of job opportunities created and the number of lost days due to damage to work were considered as the environmental and social factors, respectively.
One of the motivations behind the constructions of CLSC design problems is to distribute materials and products efficiently via the CLSC network. Consideration of various distribution channels reinforces the efficiency of production and distribution in a CLSC design problem. In general, three types of distribution channels (normal delivery: NRD, direct delivery: DRD, and direct shipment: DRS) are considered in such a problem [6]. The NRD is generally recognized to be the most basic distribution channel and is used to send materials or products from one stage to an adjoining one. For example, products are sent from the manufacturer to the DC via the NRD in a CLSC design problem that consists of a supplier, a manufacturer, a DC, a retailer, a customer, a collection center and a recovery center. The distribution channel in which a certain quantity of the products can be sent from DC to the customer directly, without routing it through the retailer, is called a DRD. The channel that allows direct shipment from the manufacturer to the customer, without any involvement of the DC and the retailer is called a DRS.
CLSC design problems with the three types of distribution channels mentioned above have been considered in conventional literature [4,8,[13][14][15]. The CLSC design problem by Cardoso et al. [14] consists of a plant, a warehouse, a retailer and a market. Certain products manufactured at the plant are sent to market through the warehouse and the retailer via the NRD, and some others are sent from the plant to the market, without being routed through the warehouse or the retailer, via DRS. The rest of the products at the warehouse are sent to the market via the DRD, without involving the retailer. The CLSC design problem by Son et al. [8] consists of an external supplier, a premanufacturing processor, a parts manufacturer, an end-product manufacturer, and a customer in FL, and a collector, a recycler, and a disposal unit in RL. Certain materials and products are sent from one stage to the next via the NRD, some others are sent from the external supplier to the parts manufacturer via the DRD without involving the pre-manufacturing processors. Varsei and Polyakovskiy [15] designed a CLSC design problem for the wine industry in Australia. The CLSC design problem consisted of a supplier, a winery, a bottling plant, a DC and a demand point at each stage. Certain raw materials (e.g., grapes and bottles) and wine products were sent from one stage to the next via the NRD. Some others were sent from the supplier to the bottling plant via the DRD, without routing them through the winery.
As mentioned above, existing works have considered various distribution channels in their CLSC design problems. Unfortunately, most of them, except for Soleimani and Kannan [13], do not use the three distribution channels simultaneously. Table 1 presents a brief review of conventional works in the literature on SCLSC design problems with regard to three factors of sustainability (economic, environmental, and social factors) and three distribution channels (NRD, DRD, and DRS). As is evident from Table 1, although there are several works that have considered either all three factors for sustainability or all three types of distributions channels, no work has considered both sets simultaneously.
In this paper, we propose a SCLSC design problem considering all three factors for sustainability and all three types of distribution channels simultaneously. The conceptual structure of the SCLSC design problem is described in Section 2.1 As the three factors for sustainability are used as three separate objective functions, the proposed SCLSC design problem is a multi-objective optimization problem and it can be represented by the mathematical formulation discussed in Section 2.2 To effectively solve the multi-objective optimization problem, a hybrid genetic algorithm (pro-HGA) approach is proposed in Section 2.3 that combines the genetic algorithm (GA) with a cuckoo search (CS). Via the numerical experiments presented in Section 3, the SCLSC design problem is explored on five different scales. The performance of the pro-HGA approach is compared with those of other conventional approaches, and the performances of the SCLSC design problems with and without three types of distribution channel are also compared. Finally, some conclusions are summarized and directions for future study are mentioned in Section 4.

Conceptual Structure of SCLSC Design Problem
The conceptual structure of the SCLSC design problem is depicted in Figure 1. The manufacturer produces products and sends a certain proportion of them (α1%) to the DC using the NRD and the remaining products (α2% = 1 − α1%) are sent to customer directly via the DRS. At the DC, a certain proportion (β1%) of the incoming products are sent to the retailer via the NRD and the rest (β2% = 1 − β1%) are sent to customer directly via the DRD. The retailer sends products to the customer via the NRD. A proportion (γ1%) of the products returned by the customer are transported to the recovery center via the collection center and the rest (γ2% = 1 − γ1%) are sent to the disposal center to be disposed. At the recovery center, a certain proportion (δ1%) of the returned products are recycled into materials and then returned to the manufacturer, and the rest (δ2% = 1 − δ1%) are recovered and then are resold to the customer. Figure 1 also depicts the use of the three factors of sustainability at each stage. The total cost (=the sum of the total fixed costs incurred by the manufacturer, the DC, the retailer, the collection center, the recovery center, and the disposal center + the total transportation costs for all distribution channels + the total handling costs for the manufacturer, the DC, the retailer, the collection center, the recovery center and the disposal center) is considered as the economic factor. The total amount of CO2 emissions during the manufacture of products at the manufacturer, during the recovery of returned products at the recovery center, and during transportation of products between each stage is considered to constitute the environmental factor. The number of job opportunities created, general unemployment, and the number of lost days due to damage to work at the manufacturer are together used as the relevant social factors.

Mathematical Formulation
The following assumptions are made to implement the proposed SCLSC design problem.

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Products of a single type are considered.

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The numbers of facilities considered at each stage at the manufacturer, the DC, the retailer, the customer, the collection center, the recovery center, and the disposal center are fixed and known beforehand.

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Only one facility is opened at each stage at the manufacturer, the DC, the retailer, the collection center, and the recovery center. However, all facilities available at the site of the customer and the disposal center are always considered open.

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The costs to operate the facilities considered at each stage at the manufacturer, the DC, the retailer, the collection center, and the recovery center are all constant, different from each other in value, and known beforehand.

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The unit handling costs of the facilities considered at each stage at the manufacturer, the DC, the retailer, the collection center, and the recovery center are different from each other in value and known beforehand.

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The unit transportation costs among the manufacturer, the DC, the retailer, the customer, the collection center, the recovery center, and the disposal center are different from each other in value and known beforehand.

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The unit amount of CO2 emitted during transportation at each stage, and those emitted during manufacture and recovery at the manufacturer and the recovery center, respectively, are different from each other in value and known beforehand.

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The proposed SCLSC design problem is considered to be in a steady-state situation.
Indices, parameters, and decision variables are defined as follows: Index Set The first objective function (F1), corresponding to the economic factor, is defined as follows to minimize the total cost. The total cost is the aggregate of the total fixed cost, the total handling cost, and the total transportation cost.

Min.
= Total fixed cost (TFC) + Total handling cost (THC) + Total transportation cost (TTC) (1) The term TFC in Equation (2) is calculated as the sum of the fixed costs when the facilities at all manufacturers, DCs, retailers, collection centers and recovery centers are opened. The term THC in Equation (3) is the sum of the handling costs incurred to handle products at the facilities opened at manufacturers, DCs, retailers, collection centers, and recovery centers. The term TTC in Equation (4) is the sum of the transportation costs of products from one stage to another.
The second objective function (F2) corresponding to the environmental factor is defined as follows to minimize the total amount of CO2 emissions during production, recovery and transportation processes.

Min.
= Total amount of CO2 emitted during production and recovery processes (TCP) + Total amount of CO2 emitted during transportation process (TCT) The term TCP in Equation (6) denotes the total amount of CO2 emitted during production and recovery processes at manufacturers and recovery centers. The term TCT denotes the total amount of CO2 emitted during transportation between each pair of stages. In Equation (7), the parameters values of vc and CO2V are randomly generated according to the methods described in [3,9].
The third objective function (F3) corresponding to the social factor is defined as follows to maximize the social influence, which consists of (i) the number of job opportunities created, (ii) the number of lost days due to damage to work, and (iii) the number of unemployed workers at the manufacturers. The three components are already mentioned and studied by many conventional literatures [24][25][26][27]. Equation (8) The three aforementioned objective functions should be optimized with respect to the following constraints.
= 0,1 , ∀ ∈ , , , , , , ≥ 0, ∀ ∈ , ∀ ∈ , ∀ ∈ , ∀ ∈ , ∀ ∈ , ∀ ∈ , ∀ ∈ Equation (9) indicates that the amount of transported goods from a manufacturer to a DC is less than or equal to the capacity of the DC. In a similar vein, inequalities (10) to (18) represent the quantity limitation for transportation between each pair of stages. Equations (19)-(23) impose the restriction that only one facility may be opened at each stage at each manufacturer, DC, retailer, collection center, and recovery center. Relations (24)-(28) represent the fact that each decision variable should take the values of 0 or 1. Inequality (29) represents non-negativity of the variables concerned.

Proposed HGA Approach
Most complicated network problems are known to be NP-complete [16,28,29], and the SCLSC design problem proposed in this paper is no exception. Existing works in the literature [4,[29][30][31] have shown that meta-heuristics-based approaches such as GA, CS, and Tabu search (TS) have been adapted to solve complicated network problems efficiently. However, there exist many situations in which most conventional single-based meta-heuristics approaches do not perform particularly well [28,31,32]. To address this gap, various hybrid approaches have been developed that use GA or other similar approaches [29,[31][32][33][34][35][36][37].
Zhang et al. [35] and Xinyu and Liang [36] developed the hybrid GA (HGA) approaches that combine GA with TS to reinforce the hybrid search ability by utilizing GA for global searches and TS for local searches. However, their HGA approaches suffer from weaknesses in applications to complicated network problems. First, the search schemes used in their HGA approaches require considerable computation time, as all individuals in a current population are adapted to produce the TS list and this process is time-consuming. Second, the search quality for locating global optimal solutions via GA sometimes deteriorates, because the Tabu list in TS is produced based on the initial population. Lin et al. [33] developed an HGA approach that utilizes GA for global searches and the iterative hill climbing method for local searches. The approach was adapted to solve complicated network problems of several sizes. Although the authors proved the superior efficiency of the approach compared to that of a conventional GA, its search speed is significantly slower than that of GA, especially for problems of large size. Therefore, this paper proposes an improved HGA approach, called the pro-HGA approach, that reinforces search speed and search quality to solve the SCLSC design problem efficiently. The pro-HGA approach combines GA for global searches with CS for local searches. First, a new population is produced by using three operators (crossover, mutation, and selection) of GA. Then, CS is applied to the new population produced by GA to locate more respective solutions. To enhance the search speed and search quality of the pro-HGA approach, the respective solutions are continuously produced, and they are used to carry out continuous improvement during GA and CS search processes. For this strategy, the Lévy flight scheme [38] of CS is adapted to all solutions of the population produced during the GA search process. The implementation procedure of the pro-HGA approach is depicted in detail as follows.

Results and Discussions
Via numerical experiments, the mathematical formulation of the SCLSC design problem suggested in Section 2.2 is implemented at five different scales as presented in Table 2. Table 2. The SCLSC design problem at five different scales.
The parameter settings to implement the GA, HGA, and pro-HGA approaches are as follows: Total number of generations is 1000, population size is 20, crossover rate is 0.5, and mutation rate is 0.3. The number of host nests is 20,  [20,30], sum = U [15,20]. To eliminate the randomness in the implementation of each approach, each method is evaluated over 30 independent runs. Various measures of performance are used to compare the performances of the approaches, as recorded in Table 3.

RNDS(Sj)
Rates of Pareto optimal solutions in the S* [40] DIR(Sj) Average distance between Pareto optimal solutions and the set S* [40] CPU time (sec.) Average CPU time required for each run The values of |Sj|, RNDS(Sj), and DIR(Sj) presented in Table 3 are the averages obtained over 30 independent runs. The three sub-problems generated from the original problem which had been suggested in Section 2.2 are used to aid the analyses of computation results as follows [31]. The experimental results are summarized in Tables 4-8. The boldfaces in Tables 4-8 Table 4. Computation results for each approach in Scale 1.  Table 5. Computation results for each approach in Scale 2.  Table 6. Computation results for each approach in Scale 3. In Table 4, the number of Pareto optimal solutions of Problem 1 obtained via the GA approach is greater than that obtained via the HGA and pro-HGA approaches for the same problem in terms of the values |Sj|. However, in Problem 2, the number of the Pareto optimal solutions obtained via the pro-HGA approach is the same as that obtained via the HGA approach, and they are both greater than the number of solutions obtained via the GA approach. In Problem 3, the pro-HGA approach locates two Pareto optimal solutions, but the GA and HGA approaches fail to locate any. The values of |Sj| also influence the rate of Pareto optimal solutions in S*. In other words, in terms of RNDS(Sj), the rate corresponding to the GA approach is higher than those corresponding to the HGA and pro-HGA approaches in Problem 1. In contrast, the rates corresponding to the HGA and pro-HGA approaches are higher than that corresponding to the GA approach in Problems 2 and 3. In terms of DIR(Sj), the pro-HGA approach exhibits significantly better results than the GA and HGA approaches in Problems 1 and 3, which can be inferred from the fact that the average distance between the Pareto optimal solutions and elements of S* is inversely proportional to the quality of the approach.
According to the data presented in Table 5, the pro-HGA approach performs significantly better in terms of |Sj| compared to the GA and HGA approaches in Problems 1, 2, and 3. Similar results are observed in terms of RNDS(Sj) and DIR(Sj). The computation results based on |Sj|, RNDS(Sj), and DIR(Sj) indicate that the search scheme used in the pro-HGA approach is more efficient than those used in the GA and HGA approaches.
According to the data presented in Table 6, the pro-HGA approach outperforms the GA and HGA approaches in terms of |Sj| and RNDS(Sj) in Problems 1 and 3, but not for Problem 2. On the other hand, in terms of DIR(Sj), the performance of the GA approach is slightly better than that of the HGA and pro-HGA approaches in Problem 1. However, for Problems 2 and 3, the performance of the pro-HGA approach is superior to those of the others.
In terms of the average CPU times required, as recorded in Tables 4-6, no significant differences among the approaches are observed, even though the computation speeds of the GA approach were slightly quicker than those of the HGA and pro-HGA approaches. In the data corresponding to Problems 1, 2, and 3 presented in Table 7, the pro-HGA approach performs the best in terms of |Sj|, RNDS(Sj), and DIR(Sj) compared to the GA and HGA approaches. A similar observation also follows from the results tabulated in Table 8 in all the Problems, the performances of the pro-HGA approach are significantly superior to those of the GA and HGA approaches in terms of |Sj|, RNDS(Sj), and DIR(Sj). The search speeds of the GA approach were observed to be slightly quicker than those of the HGA and pro-HGA approaches, but no significant difference was noticed.    From Figure 2, it is evident that five Pareto optimal solutions obtained via the pro-HGA approach coincide with the S*, and that the GA and HGA approaches do not show any advantages compared to it in this regard. Similar results are observed regarding the number of Pareto optimal solutions in terms of |Sj| for Problem 1 as presented in Table 8. Similar situations are also depicted in Figures 3 and 4. The pro-HGA approach exhibits two Pareto optimal solutions in Figure 3 and three solutions in Figure 4 that coincide with those of S*, but none of the solutions obtained via the GA and HGA approaches coincide in position with elements of S*.
Based on the experimental results presented in Tables 4-8 and Figures 2-4, we can reach the following conclusion.

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The pro-HGA approach has performed better than the GA and HGA approaches in terms of the metrics |Sj|, RNDS(Sj), and DIR(Sj), as the problem sizes were increased from scales 1 to scale 5. However, the former does not exhibit any advantage in terms of the CPU times required compared to the latter pair. With respect to the Pareto optimal solutions obtained via each approach in comparison with the S*, the performance of the pro-HGA approach is superior to those of the GA and HGA approaches, which proves that the search scheme used in the pro-HGA approach is more efficient than those used in the GA and HGA approaches.
As recorded in Table 1, the SCLSC design problem proposed in this paper utilizes various distribution channels (i.e., NRD, DRD, and DRS) and this feature distinguishes it from methods presented in other works in the literature. Therefore, we can compare the performances of the following SCLSC design problems.
. SCLSC-S: the SCLSC design problem with a single distribution channel, i.e., NRD . SCLSC-V: the SCLSC design problem with various distribution channels, i.e., NRD, DRD, and DRS Tables 9 and 10 present the computation results for SCLSC-S and SCLSC-V at scales 4 and 5. The boldfaces in Tables 9 and 10 indicate the best solutions for each measure.  Table 10. Computation results for SCLSC-S and SCLSC-V at Scale 5. From the data presented in case of Problem 1 in Table 9, the performances of SCLSC-V are observed to be significantly better than those of the SCLSC-S in terms of the metrics |Sj|, RNDS(Sj), and DIR(Sj). In Problem 2, SCLSC-V again exhibits slightly better performances than SCLSC-S in terms of |Sj| and RNDS(Sj), but not for DIR(Sj). However, in Problem 3, SCLSC-V no longer exhibits any comparative merits in terms of |Sj| and RNDS(Sj). Similar results are gleaned by analyzing the results presented in Table 10. In Problems 1 and 3, SCLSC-V significantly outperforms SCLSC-S in terms of |Sj|, RNDS(Sj), and DIR(Sj), even though the latter performs slightly better in terms of |Sj| and RNDS (Sj) in Problem 2. In Problems 1, 2 and 3 of Tables 9 and 10, the search speeds of SCLSC-S are slightly quicker than those of SCLSC-V as is apparent from the average CPU times required. Figures 5-7 depict the Pareto optimal solutions for SCLSC-S and SCLSC-V with respect to Problems 1, 2, and 3 at Scale 5.   From Figure 5, it is evident that four Pareto optimal solutions out of five coincide with S* for SCLSC-V, while the same is true for two out of seven solutions for SCLSC-S. These results are the same as the distributions of Pareto optimal solutions in terms of |Sj| for Problem 1 as presented in Table 10. From Figure 6, we can observe that all Pareto optimal solutions for SCLSC-V and SCLSC-S coincide with S*. However, in Figure 7, all Pareto optimal solutions for SCLSC-V coincide with S*, but none of the solutions for the SCLSC-S do.
Based on the experimental results presented in Tables 9 and 10 and Figures 5-7, the following conclusion can be reached.
• SCLSC-V outperforms SCLSC-S with respect to most performance measures, although the former has slightly lower search speeds than the latter. This indicates that the SCLSC design problem with various distribution channels, i.e., NRD, DRD, and DRS is more efficient in locating optimal solutions than that with a single distribution channel, i.e., NRD.

Conclusions
In this paper, we have proposed an SCLSC design problem. Three factors (economic, environmental, and social factors) and three distribution channels (NRD, DRD, and DRS) have been considered for the construction of the problem. The economic factor is concerned with minimizing the total cost, which is the aggregate of the total fixed cost, the total handling cost, and the total transportation cost. The environmental factor also is concerned with minimizing the total amount of CO2 emissions during the transportation and manufacturing processes. On the other hand, the social factor is concerned with maximizing the social influence of the process, which consists of the number of job opportunities created, the number of unemployed workers and the number of lost days due to damage to work. As these three factors have been considered in the proposed SCLSC design problem simultaneously, the problem has been formulated as a multi-objective optimization problem. A pro-HGA approach that combines GA with CS has been proposed to solve the SCLSC design problem.
The SCLSC design problem has been presented in five different scales with the help of numerical experiments and the performance of the pro-HGA approach has been compared with those of two conventional approaches (GA and HGA). Various measures of performance, such as Pareto optimal solutions, have been used for the comparison. The experimental results have conclusively shown that the pro-HGA approach is more efficient than the GA and HGA approaches with respect to various measures of performance, although the former shows no advantage over the latter with respect to the average CPU time required to complete the process.
SCLSC design problems with and without various distribution channels have also been compared in the paper. SCLSC-S (the SCLSC design problem with single distribution channel, i.e., NRD) and SCLSC-V (the SCLSC design problem with various distribution channels, i.e., NRD. DRD, and DRS) have been compared at two separate scales. Experimental results demonstrated that SCLSC-V is more efficient in locating optimal solutions than SCLSC-S with respect to most performance measures, except the average CPU time required.
With regard to the direction of our future studies, we wish to present SCLSC design problems of a larger scale using real field data to compare the performances of SCLSC-V and SCLSC-S. We also wish to consider other meta-heuristic approaches, such as particle swam optimization, ant colony optimization, etc. to construct a more efficient pro-HGA approach.