outline

Designing a sustainable integrated forward/reverse logistics network

Hamed Farrokhi-Asl Iran University of Science and Technology, Tehran, Iran

Ahmad Makui and Roozbeh Ghousi Department of Industrial Engineering,

Iran University of Science and Technology, Narmak, Iran, and

Masoud Rabbani College of Engineering, University of Tehran, Tehran, Iran

Abstract Purpose – In recent years, governmental regulations and the pressure of non-governmental organizations have convinced corporations to consider sustainable issues in their decisions. A simultaneous design of forward and reverse logistics can keep us away from sub-optimality caused by tackling these two phases (forward and reverse logistics) separately. Design/methodology/approach – Hence, this paper presents a new multi-objective mathematical model for integrated forward and reverse logistics regarding economic, environmental and social issues. A new hybrid multi-objective metaheuristic algorithm is developed to obtain a set of efficient solutions (Pareto solutions). The proposed algorithm hybridizes a well-known, non-dominated genetic algorithm (NSGA-II) with a simulated annealing algorithm. Findings – To validate the algorithm, its results are compared to the obtained solutions from simple NSGA-II with respect to some comparison metrics. The numerical results show the efficiency of the proposed algorithm. Finally, concluding remarks and future research directions are provided. Originality/value – By applying a model presented in this paper, one can reach to sustainable and integrated logistics network which considers forward and reverse flow of commodities simultaneously.

Keywords Logistics, Operations management, Optimization, Supply chain management, Multicriteria programming, Sustainable supply chain, Forward/reverse logistics, Multi-objective optimization, Waste collection problem

Paper type Research paper

1. Introduction In recent years, governmental regulations and the pressure of non-governmental organizations have convinced corporations to consider sustainable issues in their decisions (Rehman et al., 2018). Although there are many definitions of sustainability in the literature, the simplest one is “Utilizing resources to meet the current demands of humans without neglecting the ability of future generations to satisfy their needs” (Beamon, 2008). Early studies of sustainability were intended to concentrate on the environmental issues. But as time goes on, recent articles such as Ahi and Searcy (2013) adopted triple bottom lines approach; researchers consider environmental, economic, and social issues as the main factors of sustainability. Figure 1 shows these triple bottom lines and their relationships in sustainability.

Operations management (OM) is defined as rational management of all operations along the supply chain, from procurement of raw materials to delivery of finished goods to

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Received 7 September 2018 Revised 26 October 2018 Accepted 20 March 2019

Journal of Modelling in Management Vol. 14 No. 4, 2019 pp. 896-921 © EmeraldPublishingLimited 1746-5664 DOI 10.1108/JM2-09-2018-0136

The current issue and full text archive of this journal is available on Emerald Insight at: www.emeraldinsight.com/1746-5664.htm

customers. One of the most important issues of designing a sustainable OM system is the concept of 3 R (i.e. Reduce, Reuse, and Recycle). There is an increasing emphasis on the need to reduce waste, recycle, and reuse products and parts. This is known as Sustainability in Operations (Reid and Sanders, 2011). Reduce is devoted to all activities which are done to decrease the total cost or waste of finished goods. These activities might include the omission of a redundant part of the product or even altering the process of producing an item. Reuse can be collecting the defective products from customers and using effective parts in the production process. Moreover, corporations can fix a faulty component of products and resend it to marketplace with a lower cost. Finally, Recycle consists of collecting waste materials, reducing their risks by treatment facilities, and converting them to reusable raw materials.

Reverse logistics is one of the most essential parts of OM which has attracted researchers’ attention in recent years. There are many different definitions of reverse logistics in the literature. For instance, Dowlatshahi (2000) defines reverse logistics as a process in which a manufacturer systematically accepts previously shipped products or parts from the point of consumption for the possible recycling, remanufacturing, and disposal. Over the past decades, many companies have invested in recycling and reusing activities (Üster et al., 2007). Meade et al. (2007) divided the main elements of the increased attention and reverse logistics investments into two main groups: environmental and business factors. Environmental factors include harmful effects of discarded goods and materials in the environment, regulations and laws on recycling and recovery of the company’s products which have come to end of their lives, and lack of enough accessibility to the resources which are required to produce goods. These environmental factors motivate researchers to address eco-friendly networks toward designing a reverse logistics system.

Business Factors are all aspects of reverse logistics which bring benefits to the company. These factors include benefits which are gained from reusing applicable material or parts in the production processes and decrease in environmental crime by governments. However, the factors are not limited to the above-mentioned benefits.

Configuration of logistics network including the nodes and the arcs between these nodes have several impacts on different aspects of the company such as overall cost of the system and level of customers’ satisfaction (Pishvaee et al., 2010). Designing a network involves various decisions which must be made in different managerial levels including strategic, operational, and tactical levels. Decisions about location, number, volume or capacity of the facilities, established technologies, and so forth are categorized as strategic decisions which have long-term effects on the performance of the company. Some other decisions such as production planning, servicing routes, scheduling, selecting suppliers and et cetera are

Figure 1. Components of

sustainability and their relationships

Forward/ reverse logistics

network

897

examples of operational and tactical levels of managerial decisions which should be made frequently and have short-term influences on the company. In the recent years, researchers such as Lopes et al. (2008), Farrokhi-Asl et al. (2016), and Rabbani et al. (2016) have concluded that separately addressing these decisions, without considering their effects on each other, will lead to sub-optimal designed network with regard to costs, responsiveness of the network, and some other objectives of a system. Additionally, Forward and reverse logistics should be considered integrally since reverse logistics has a great influence on the performance of the forward logistics and vice versa. That is why some common facilities between forward and reverse logistics might be available in the network (Kumar et al., 2016).

In the literature, there are various studies which deal with the problem of designing supply chain and logistics networks. These studies were reviewed by seuring (2013), Eskandarpour et al. (2015), Siddhartha and Sachan (2016). It can be noted that only a few papers have considered investment in environmental issues and to the best of our knowledge, there are not any paperers considering established technologies in facilities of the logistics network. Wang et al. (2011) was the first study which investigate investment in eco-friendly issues in designing phases of the supply chain. Investment in environmental factors provides sustainable competitiveness for corporations and spending a little money on eco-friendly development of corporation, enhances the position of it among customers (Hanss et al., 2016).

According to the mentioned facts, this paper presents the designing of multi-objectively integrated forward and reverse logistics in which the network includes production/recovery sites, distribution centers, customer zones, collection centers, recycling facilities, and disposal centers. In addition to decisions about opening a facilities, decision makers should consider the capacity level of each facility, and level of investment in environmental issues of facilities. Despite the fact that many papers in this field have considered a single capacity level of facilities, several levels of capacity are addressed. Moreover, utilizing eco-friendly fleet of vehicles in each facility is considered for the first time. Greenness of the networks is investigated by means of evaluating the volume of CO2 emission in the whole network. In addition, harmful effects of undesirable facilities on the population centers are minimized. The rest of the paper is organized as follows: Section 2 is dedicated to a concise review in the related fields. Section 3 describes multi-objective optimization problems and the dominance concept. Problem definition and mathematical formulation are proposed in Section 4. Methodology is presented in Section 5 to tackle the problem. Some experiments are conducted and obtained results are compared to each other in Section 6. Finally, concluding remarks and future research directions are provided in Section 7.

2. Literature review Chanintrakul et al. (2009) reviewed comprehensively the models of planning a reverse logistics network during 2000 to 2008. They concentrated on the factors such as environmental legislations, economic, and corporate citizenship in their paper. Pishvaee et al. (2010) indicated that most of the proposed mathematical models for logistics network design (forward, backward, or integrated ones) are mixed integer linear programming (MILP) models which are used frequently in the literature and are able to solve a problem by means of commercial optimization software (Sabri and Beamon, 2000; Salema et al., 2007; Demirel and Gökçen, 2008; Rabbani et al., 2015). Most of these researches were aimed to decrease cost or add profit to the company. The models differ in complexity from simply single-product, uncapacitated location models (Cornuéjols et al., 1983) to multi-echelon, multi-product capacitated location models (Pasandideh et al., 2015).

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By drawing more attention to the new concepts such as eco-friendly logistics network, responsiveness to customers’ demand, and robustness, new objective functions have been considered as a multi-objective logistics network design. Ramezani et al. (2013) presented a stochastic multi-objective mathematical model for designing a forward/reverse logistics network. They applied a novel method to determine the systematic supply chain framework to boost profits and increase customer responsiveness as an objective function. They applied epsilon-constraint method to achieve exact Pareto solutions. Furthermore, fluctuations in business environment and demands of customers intensified the importance of considering robust network design. Hatefi and Jolai (2014) presented a robust and reliable model for an integrated forward-reverse logistics network design. They simultaneously took uncertain parameters such as amount of customers’ demand and facility disruption into consideration.

Green logistics relate to all processes which are associated with the production and distribution in a green way; that is, environmental issues are taken into consideration. The importance of green logistics is concluded from the fact that the applied approaches of current logistics in logistics service provider corporations are not sustainable in the long term. Lin et al. (2014) divided the green logistics into three major categories including Green Vehicle Routing Problem (GVRP), pollution routing problem, and vehicle routing problem in reverse logistics. GVRP deals with the problems about which the optimization of energy consumption is investigated (Erdo�gan and Miller-Hooks, 2012; �Cirovi�c et al., 2014). The pollution routing problem introduced by Bektas� and Laporte (2011) deems the reduction of greenhouse gases, particularly CO2 emissions. These gases have bad effects on ecosystems and humans’ health. Efforts to reduce these gases have increased recently (Demir et al., 2014; Kramer et al., 2015). The last type of green logistics are VRP in reverse logistics (VRPRL). Dekker et al. (2013) proposed the definition of reverse logistics as follows: “The process of planning, implementing and controlling backward flows of raw materials in process inventory, packaging, finished goods, and from a manufacturing, distribution or use point to a point of recovery or point of proper disposal”. Considering this categorization, green logistics are divided into three categories; products recycling is also considered as one of these categories. Both recycling management and reduction of CO2 emission, which are the two main features of green logistics issues, are investigated in this paper. In other words, reduction of CO2 emission and applying the proposed model to integrated forward and reverse logistics are green features of the presented model in this paper.

3. Multi-objective optimization The problems of single objective optimization are usually aimed to reach the optimum or rational solution(s). In this regard, making comparison between the available solutions in the feasible area is a simple since we have only one objective function value and we can sort these solutions regarding their objective values. For example, if the problem is maximization problem, f(x) < f(y) shows that solution y is better than solution x. In recent years, researchers have been faced with problems where two or more conflicting objective functions are available. These kinds of the problems are called multi-objective optimization problems. As a result, more attention is paid to eco-friendly and social issues in optimization problems and the new objectives are added to traditional objective functions (Lei et al., 2016; Mortezaei et al., 2015; Gamberi et al., 2015). As such, in multi-objective optimization problems with conflicting objectives, it is almost difficult to find one solution which is the best solution for all goals simultaneously.

A solution which satisfies all constraints is called feasible solution and it is comprised of a number of decision variables. In the multi-objective problems, the term optimization

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means finding the set of solutions satisfying all constraints and giving acceptable values for all objective functions. This term is defined mathematically as follows (Rabbani et al., 2016):

If X = [x1, x2,. . ., xN] T is the vector of decision variables, the aim is to find the vector such

as X* ¼ x*1; x*2; . . . ; x*3 � �T

which satisfies all constraints and optimizes the vector of objective function:

F ¼ f1 xð Þ; f2 xð Þ; . . . ; fm xð Þ � �

(3-1)

s.t.

gi xð Þ � 0 i ¼ 1; 2; . . . ; p (3-2)

hj xð Þ ¼ 0 j ¼ 1; 2; . . . ; q (3-3)

As an aforementioned, finding the solution which optimizes all objectives is usually impossible. Thus, the concept of dominant solutions is defined. Given a general multi- objective maximization problem with n decision variables and m objectives in which (m > 1), the solution x dominates the solution y if we have the following conditions:

fi xð Þ � fi yð Þ 8i 2 1; 2; . . . ; mf g (3-4)

fi xð Þ > fi yð Þ 9i 2 1; 2; . . . ; mf g (3-5)

If there is a solution which is not dominated by any other solution, this solution is named non-dominated solution. All the solutions that are not dominated by any other solutions in the feasible area, are called Pareto optimal solutions. Moreover, their corresponding images on the objective space are called Pareto Frontier.

4. Problem description The integrated forward/reverse logistics network considered in this paper, is made up several sets of nodes including production/recovery nodes, distribution nodes, customers’ area, collection centers, recycling centers and disposal centers. In addition to decision on locations of facilities, the level of capacity for each established facility in a potential location should be determined. Figure 2 illustrates the relationships between nodes. New products are transported from production centers to the location of opened distribution facilities. There is not any interaction between production centers, and they work independently (i.e. the transportation between production centers are zero). Then, these products are distributed between customers’ area based on their demands. It should be noted that the location of customers’ area are known. At this point, the revers logistics starts and returned products should be collected from the customers’ area to the location of opened collection centers. It is assumed that each customer returns a predefined percentage of demand as a faulty product. Afterwards, the recoverable products are shipped toward production/ recovery centers, then the recovered products are added to the forward logistics. Therefore, we can claim that our presented logistics network is a closed loop. Moreover, recyclable and returned products are shipped to recycling centers and scrapped ones are shipped toward disposal centers. Recycling process also has wastes which should be transported to disposal centers. In other words, the role of collection centers is testing and inspection to make a

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decision on which facility is appropriate to transport returned products. In such an integrated network, combining relevant centers offers cost saving to the whole system (Mousazadeh et al., 2015). A predefined percentage is considered for each type of returned products (i.e. recoverable, recyclable and scrapped). As such, combined distribution, collection and disposal facilities and combined recycling are considered in this paper; distribution, collection, recycling and disposal centers are established at the same location. Therefore, the savings resulted from these combinations should be reflected in the cost objective function.

In addition to the aforementioned issues, some new concepts are addressed in this paper for the first time. One of the main attributes of the study is making a decision about the level of environmental investments in the designing phase. Thus, the environmental decision- making in the network design and taking cautions against environmental pollutants are of interest to us. Plants and companies’ activities are among the main sources of producing pollutants in the environment. Moreover, transportation activities are the significant sources of air pollution and greenhouse gas emissions (Fahimnia et al., 2015). Consequently, we investigate the amount of CO2 emission in the network as the only environmental influence which is a very popular indicator. The volume of CO2 emitted by production, recycling, and disposal centers along with CO2 emitted by fleet of vehicles, are taken into consideration. One new objective function is added to the model which determines the volume of CO2 emitted in the network and the tradeoff between the total costs. Additionally, the environmental influences have been captured.

Furthermore, the paper aims to consider social issues by taking the effects of undesirable facilities such as recycling and disposal centers into account. Because of the undesirable nature of some facilities in the network, we tend to distance these facilities from customers’ area. On the other hand, corporations may decide to increase responsiveness of the network by establishing more facilities near the customers’ area. Hence, the presented network in this study is designed to jointly consider network cost, responsiveness of the network, environmental and social issues.

Figure 2. Proposed network in

this paper

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network

901

It should be noted that all parameters including are assumed to be deterministic and we know an amount of each variable. Prediction a correct value for these parameters are tough challenge, and prediction is always accompanied with error. It is assumed that the demand of customers should be served completely. In addition, only one vehicle is assigned to each node to transport and all load of each established facilities. Flow balance in the location of each facility is checked; that is, the balance between input and output stuffs should be considered for each facility. Finally, capacity constraint is considered for each established facility to inhibit exceedance from the each facility’s capacity. In the meanwhile, each facility has at most one established technology and capacity level.

All in all, questions which should be answered in this network, concerns the location and number of facilities, capacity of each center, technologies established in production, recycling, and disposal centers, the level of investment in environmental issues by purchasing eco-friendly fleet of vehicles for each center, and flow of products among centers. Table I summarizes the problem description.

The following notations are used to formulate the proposed mathematical formulation for a sustainably integrated forward/reverse logistics network:

Sets I = potential locations for production/recovery centers; J = potential locations for distribution centers; K = sets of customers area; M = potential locations for collection centers; E = potential locations for combined distribution and collection centers; E � J; E � M; R = potential locations for recycling centers; D = potential locations for disposal centers; N = potential locations for combined recycling and disposal centers; N � R; N � D; L = set of technologies established in facilities; C = capacity levels; V = set of investment levels in fleet of vehicles; R

0 = set of established recycling centers;

D 0 = set of established disposal centers; and

A = set of all nodes.

Parameters dek = demand of customer k; rek = rate of returned products at customer k; q = recoverable percentage of returned products; qr = recyclable percentage of transmitted products to collection centers; sc = rate of produced waste in recycling centers;

Table I. Attributes of the presented network

Goals Outputs Modeling Characteristics Nodes of the network

Cost Locations MINLP Capacitated facilities Production/recovery Responsiveness Number of each facility MILP Deterministic model Distribution Environmental Technologies Capacitated flow Customers Social Capacity Single product Collection

Fleet of vehicles Undetermined number of each facility

Recycling

Single period Disposal

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fpcil = cost of establishing production/recovery center at potential location i with capacity level c and technology l;

fdcj = cost of establishing distribution center j with capacity level c; fccm = cost of establishing distribution center at location m with capacity c; frcrl = cost of establishing recycling center at location r with capacity level c and technology l; fscdl = cost of establishing disposal center at location d with capacity level c and technology l; DCcc

0 e = saving cost associated with establishing combined facility for distribution center with

capacity level c and collection center with capacity level c0 at the location e; RDcc

0

nll0 = saving cost associated with establishing combined facility for recycling center with capacity level c technology l and disposal center with capacity level c0 technology l0 at the location n;

Trvaa0 = transportation cost of one unit product from node a to node a 0 by vehicles with the environ-

mental investment level v; emvaa0 = CO2 emitted by transportation of one unit product from node a to node a

0 by vehicles with the environmental investment level v;

capci = capacity of production/recovery center i with the capacity level c; cadcj = capacity of distribution center j with the capacity level c; caccm = capacity of collection center m with the capacity level c; carcr = capacity of recycling center r with the capacity level c; cascd = capacity of disposal center d with the capacity level c; vehva = fixed cost of utilizing a fleet of vehicle with environmental investment level v at node a; Resf = expected time for delivering products; Resr = expected time for collecting or returned products; tdcvjk = 1 if delivering time between distribution center j and customer k with fleet of vehicle v is less

than Resf; 0 otherwise; tccvkm = 1 if traveling time between customer k and collection center m with fleet of vehicle v is less

than Resr; 0 otherwise; r = weighting factor for the forward responsiveness; 1 � rð Þ denotes the weight of reverse

logistics; dis1rk = distance between recycling center r and customer area k; dis2dk = distance between disposal center d and customer area k; empli = CO2 emitted by producing one unit at production center i with technology l; emplr = CO2 emitted by processing one unit at recycling center r with technology l; and empld = CO2 emitted by processing one unit at disposal center i with technology l.

Decision variables xvaa0 = amount of products which are shipped from node a to node a

0 by vehicle v; ypcil = 1 if the production center with capacity level c and technology l is established at potential loca-

tion i; 0 otherwise; ydcj = 1 if the distribution center with capacity level c is established at potential location j; 0

otherwise; yccm = 1 if the collection center with capacity level c is established at potential location m; 0 otherwise; yrcrl = 1 if the recycling center with capacity level c and technology l is established at potential location

r; 0 otherwise; yscdl = 1 if the disposal center with capacity level c and technology l is established at potential location

d; 0 otherwise; and zva = 1 if the fleet of vehicle with environmental investment level v is applied at node a.

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According to the aforementioned notations, the multi-objective, mathematical model for designing a sustainably integrated logistics network can be formulated as follows:

Min Z1 ¼X i

X l

X c fpxil yp

c il þ

X j

X c fdci yd

c j

þ X m

X c fccmyc

c m þ

X r

X c

X l

frcrl yr c rl þ

X d

X c

X l

fscdl ys c dl

� X e

X c

X c0 DCcc

0 e yd

c eyc

c0 e �

X n

X l

X l0

X c

X c0 RDcc

0 nll0yr

c nlys

c0 nl0

þ X v

X a vehvaz

v a þ

X v

X a

X a0

trvaa0x v aa0 (4-1)

Max Z2 ¼ r X k

X j

X v xvjktdc

v jk

� �� X k

dek � �

þ 1 � rð Þ X k

X m

X v xvkmtcc

v km

� �� X k

dekrek � �

þ (4-2)

Max Z3 ¼ Minr0 ;k;l;c;d0 dis1r0 kyrcr0 l; dis2d0kys c d0l

� � (4-3)

Min Z4 ¼X a

X a0

X v emvaa0x

v aa0 þ

X c

X i

X l

empliyp c il

X j

X v xvij

� � þ X

c

X r

X l

emrlryr c rl

X j

X v xvmr

� � þ X

c

X d

X l

empldyp c dl

X r

X v xvrd þ

X m

X d

xvmd � �

(4-4)

s.t.X v

X j

xvjk ¼ dek 8k 2 K (4-5)

X v zva # 1 8a 2 A (4-6)

X a0

xv aa0 #Mz

v a 8a 2 A; v 2 V (4-7)

X v

X m

xvkm ¼ dekrek 8k 2 K (4-8)

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X i

X v xvij ¼

X k

X v xvjk 8j 2 J (4-9)

X k

X v xvkm ¼

X i

X v xvmi þ

X r

X v xvmr þ

X d

X v xvmd 8m 2 M (4-10)

X i

X v xvmi ¼ q

X k

X v xvkm 8m 2 M (4-11)

X r

X v xvmr ¼ qr

X k

X v xvkm 8m 2 M (4-12)

X d

X v xvrd ¼ sc

X m

X v xvmr 8r 2 R (4-13)

X j

X v xvij #

X l

X c capci yp

c il 8i 2 I (4-14)

X k

X v xvjk #

X c cadcj yd

c j 8j 2 J (4-15)

X k

X v xvkm #

X c caccmyc

c m 8m 2 M (4-16)

X m

X v xvmr #

X c

X l

carcryr c rl 8r 2 R (4-17)

X m

X v xvmd þ

X r

X v xvrd #

X c

X l

cascdys c dl 8d 2 D (4-18)

X c

X l

ycil # 1 8i 2 I (4-19)

X c ycj # 1 8j 2 J (4-20)

X c ycm #1 8m 2 M (4-21)

X c

X l

ycrl # 1 8r 2 R (4-22)

X c

X l

ycdl # 1 8d 2 D (4-23)

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xv aa0 � 0 8a; a

0 2 A; v 2 V (4-24)

ypcil; yd c j ; yc

c m; yr

c rl; ys

c dl; z

v a 2 0; 1f g 8i 2 I; j 2 J; m

2 M; c 2 C; r 2 R; d 2 D; l 2 L a 2 A; v 2 V (4-25)

Objective function (4-1) minimizes total cost of designing a network including establishment cost of facilities with respect to utilized technology and capacity level, transportation cost between nodes, applying a fleet of vehicles, and saving cost associated with combining distribution, collection, combining recycling, and disposal centers. Objective function (4-2) maximizes the forward and reverse responsiveness of the network. We modify the objective function presented by Altiparmak et al. (2006). Objective function (4-3) is used to increase the distance between the undesirable facilities (i.e. recycling and disposal centers) and customers’ area by maximizing the minimum distance between them. Farrokhi-Asl et al. (2016) applied this objective function to consider the social impact of undesirable facilities. Objective function (4-4) calculates the amount of the CO2 emitted in the network by considering CO2 emission in production, recycling and disposal centers, as well as, CO2 emission which is caused by the fleet of vehicles.

Constraint (4-5) ensures that the demand of customers is met. Constraint (4-6) and (4-7) guarantee that only one type of fleet of vehicles is assigned to each node and all load of each established facilities are shipped by solely one fleet type. The amount of returned products which initializes the reverse logistics is determined in Constraint (4-8). Constraints (4-9) through (4-13) consider the flow between nodes and ensure that the flow is balanced between nodes. Constraints (4-14) through (4-18) are capacity constraints which also ban the units of products, returned products, recoverable products, recyclable products, and scrapped products from being transferred to facilities which are not opened. Constraints (4-19) to (4-23) ensure that each facility has at most one established technology and capacity level. Constraints (4-24) and (4-25) specify the range of decision variables.

Some terms in this mathematical formulation are nonlinear so the initial proposed model is Mixed Integer Non-linear Programming (MINLP). To avoid the complexity of such a model, the above-mentioned model is linearized by introducing some new variables and new constraints. The terms

X e

X c

X c0 DCcc

0 e yd

c eyc

c0 e and

X n

X l

X l0 X

c

X c0 RDcc

0 nll0yr

c nlys

c0 nl0 in the

first objective function are non-linear since they comprise the multiplication of two binary variables. We reformulate the objective function as follows:

Var1cc 0

e ¼ ydce ycc 0 e (4-26)

Var2cc 0

nll0 ¼ yrcnl ysc 0 nl0 (4-27)

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Var1cc 0

e ; Var2 cc0 nll0 2 0; 1f g 8e 2 E; n 2 N; c; c

0 2 C; l; l0 2 L (4-28)

Min Z1 ¼ X i

X l

X c fpxil yp

c il þ

X j

X c fdci yd

c j

þ X m

X c fccmyc

c m þ

X r

X c

X l

frcrl yr c rl þ

X d

X c

X l

fscdl ys c dl

� X e

X c

X c0 DCcc

0 e Var1

cc0 e �

X n

X l

X l0

X c

X c0 RDcc

0 nll0Var2

cc0 nll0

þ X v

X a vehvaz

v a þ

X v

X a

X a0

trvaa0x v aa0 (4-29)

As the objective function is minimization, it intends to put the value of 1 for the both auxiliary variables. Consequently, we should not allow these variables to take only value 1 by considering following conditions: when both ydce and yc

c0 e are 1, Var1

cc0 e should be 1 and

the same conditions for Var2cc 0

nll0 should be applied. To consider the above-mentioned conditions we add the following constraints to the model:

2Var1cc 0

e #yd c e þ ycc

0 e 8e 2 E; c; c

0 2 C (4-30)

2Var2cc 0

nll0 #yr c nl þ ysc

0 nl0 8n 2 N; c; c

0 2 C; l; l0 2 L (4-31)

Moreover, the objective function (4-3) has non-linear term which can be linearized by introducing one new variable named Var3 to eliminate the term minimization in this objective functions. We reformulate the objective function as follows by adding two new constraints:

Max Z3 ¼ Var3 (4-32)

Var3# dis1r0 kyr c r0 l 8r

0 2 R0; k 2 K; l 2 L; c 2 C (4-33)

Var3# dis2d0kys c d0l 8d

0 2 D0; k 2 K; l 2 L; c 2 C (4-34)

Lastly, the fourth objective function includes the non-linear terms, namely multiplication of decision variables. For example, in this objective function, the term ypcil

X j

X v xvij

is non-

linear. To eliminate this complexity, we introduce Var4cil as follows:

Var4cil ¼ ypcil X j

X v xvij

� � (4-35)

X j

X v xvij # Var4

c il #Myp

c il 8i 2 I; l 2 L; c 2 C (4-36)

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In the above formulas, M denotes the big number. If ypcil is zero, X

j

X v xvij is zero, so the

value of Var4cil Will be zero. Conversely, if yp c il is 1, Var4

c il will takes the minimum number

which it can take; as a result, Var4cil will be X

j

X v xvij. Similarly, other non-linear terms are

converted to linear ones as follows:

Var5crl ¼ yrcrl X m

X v xvmr

� � (4-35)

X m

X v xvmr #Var5

c rl # Myr

c rl 8r 2 R; l 2 L; c 2 C (4-36)

Var6cdl ¼ ypcdl X r

X v xvrd

� � (4-37)

X r

X v xvrd #Var6

c dl # Myp

c dl 8d 2 D; l 2 L; c 2 C (4-38)

Var7cdl ¼ ypcdl X m

X v xvmd

� � (4-39)

X m

X v xvmd # Var7

c dl #Myp

c dl 8d 2 D; l 2 L; c 2 C (4-40)

Finally, the forth objective function, namely equation (4-4), is transformed into the following equation by applying the above-mentioned transformations.

Min Z4 ¼ X a

X a0

X v emvaa0x

v aa0 þ

X c

X i

X l

empliVar4 c il

þ X c

X r

X l

emrlrVar5 c rl þ

X c

X d

X l

empld Var6 c dl þ Var7cdl

� � (4-41)

5. Methodology GA has been the most popular meta-heuristic approach exploited for multi-objective design and optimization’s problems since most multi-objective GAs do not require the user to prioritize, scale, or weigh objectives (Konak et al., 2006). From the earlier years of the nineteenth century, interest in multi objective problems (MOPs) with Pareto approaches has always grown rapidly. Population-based metaheuristics such as non- dominated sorting genetic algorithm (NSGA-II) seem appropriate to solve MOPs, since they deal simultaneously with a set of solutions that allow us to find several members of the Pareto optimal set in a single run of the algorithm. Moreover, Pareto population-based metaheuristics approaches such as NSGA-II are less sensitive to the convexity of the Pareto front (Talbi, 2009). For reason of NP-hard nature of the problem (Pishvaee et al., 2010), exact methods which try to find Pareto optimal set (i.e. goal attainment, goal programming, weighted sum, epsilon constraint, and so forth) are unable to tackle the

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large sized problems. In this regard, we present a hybrid metaheuristic algorithm, namely, multi-objective hybrid genetic algorithm and simulated annealing algorithm (MOHGASA) to solve the problem. NSGA-II (modified version of GA to solve MOPs) shows the acceptable performance in the supply chain management (Garg et al., 2015; Zohal and Soleimani, 2016); interestingly, that is what has motivated us to use this algorithm.

In recent years, interest in applying hybrid metaheuristic approaches has risen rapidly. In comparison to simple metaheuristics, hybrid ones usually gain better results in the field of optimization problems. The combination of population-based metaheuristics and single-based metaheuristics provides a very powerful search scheme. There are different approaches to hybridize algorithms. The two competing purposes governing the design of metaheuristic algorithms are diversification and intensification. Diversification ensures that all parts of the search space is explored. On the other hand, intensification is an absolutely essential factor since the refinement of the current solution will usually generate better solutions. The competency of P-metaheuristics is their ability to explore more spaces, however, they are weak in intensification. To overcome this weakness we couple genetic algorithm (GA) with SA which is capable of intensifying better solutions. In this paper, MOHGASA is developed to solve the sustainably integrated forward/reverse logistics network design as depicted in Figure 3.

5.1 Solution representation Firstly, we illustrate two concepts which are used in this paper to encode and decode the solutions, namely genotype and phenotype space. Genotype denotes the space in which a solution is codded by a number of matrices filled with numbers. Phenotype is the meaningful and feasible solution space. These two spaces are often not the same. Each solution is usually represented through combination of numbers and matrices. Afterwards, the representation of this solution is transformed into a meaningful solution by applying one mapping function. The role of mapping function is producing a meaningful and often feasible solution from genotype space (Rabbani et al., 2018). In this paper, we have utilized the approaches of solution representation introduced by Gen et al. (2006) for the first time. We modified this approaches to make it appropriate for applying to the proposed problem. Structure of the first string in genotype space is demonstrated in Table II. Each solution is represented by four distinct strings. The first string is representative of the flow between nodes. The second one is related to capacity level. The third one denotes the established technology at each location and finally the last one relates to fleet of vehicles used for each node. Each string has a different length. The first string comprises I þ J þ J þ K þ K þ M þ M þ I þ M þ Rþ M þ D þ R þ D numbers. In the other words, the string includes seven segments corresponding to one echelon of the network (Table II). In this representation, each segment consists of random numbers generated in the range of [0-1]. Then, this numbers are sorted in a descending order. Furthermore, the priority of facilities for each segment of the problem is determined by this method.

The length of second, third and fourth strings is I þ J þ Mþ R þ D, I þ R þ D and I þ J þ K þ M þ R þ D, respectively. Each string consists of a set of randomly real numbers generated in the range of [0-1]. Afterwards, this random numbers are transformed into the meaningful numbers. For example, if we have c levels for capacity,

the numbers in the range of 0 � 1c h i

are allocated to the first level of capacity. Numbers

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in the range of 1c � 2c h i

are associated with the second level of capacity and so on.

Similarly, this method are applicable to two other strings. Table III demonstrates an illustrative example with three levels of technology to describe how we can specify each established facility’s technology. Finally, mapping function should be applied to determine the flow between nodes. Mapping function is used for each segment of the string. Pseudo code for mapping function shows that how mapping function works and how solutions are generated. It should be noted that we initially specify the flows in the forward logistics and then we go for the reverse logistics. In the forward logistics, the second segment’s flow should be determined before than the first segment. Moreover, in the reverse logistics, decoding the fourth and fifth segments is impossible before the third segment.

Figure 3. Flowchart of the proposed algorithm

Generate ini�al random popula�on (with size nPop)

Sort popula�on using non- dominated criterion

Assign a rank to each solu�on based on its non-domina�on level

Calculate crowding distance to dis�nguish between same rank

solu�ons

Binary tournament selec�on

Crossover

Muta�on

Generate offspring popula�on

Mixing popula�on and offsprings

Sor�ng a mixed popula�on considering non-domina�on

concept

Using eli�sm criterion to select solu�ons from mixed popula�on

Generate new popula�on

Sending all rank 1 solu�ons to archive

Upda�ng archive Stopping criterion

Output the final archive

Yes

No

Simulated annealing as a local search

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F ir st

se gm

en t

I- J

Se co nd

se gm

en t

J- K

T hi rd

se gm

en t

K -M

F or th

se gm

en t

M -I

F if th

se gm

en t

M -R

Si xt h

se gm

en t

M -D

Se ve nt h

se gm

en t

R -D

N od e

1 2

1 2

1 2

1 2

3 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 P ri or it y nu

m be r

0. 2

0. 01

0. 4

0. 68

0. 11

02 0. 15

0. 5

0. 03

0. 9

0. 77

0. 1

0. 11

0. 1

0. 22

0. 02

0. 89

0. 7

0. 6

0. 15

0. 13

0. 3

0. 1

0. 6

0. 55

0. 13

0. 8

0. 9

0. 99

O rd er ed

nu m be r 3

4 2

1 4

2 3

1 5

1 2

4 3

3 2

4 1

1 2

3 4

3 4

1 2

4 3

2 1

Table II. An illustration

example for the first string of solution

representation

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Inputs: Q: set of source nodes P: set of destination nodes demp: demand of destination p capq: capacity of source q v = vehicle type is used regarding the fourth string Trvqp: Cost of transshipment between node q and p by vehicle v emvqp: CO2 emission between q and p by vehicle v S: related segment in the first string

Outputs: xqp: amount of shipment between nodes ya: binary variable shows opened facilities

while Rp demp > 0 Step 1: xqp = 0; Vq, p Step 2: select a node based on corresponding segment in the first

string l = ar gmax S (t), t [ |Q| þ |P|Vp, q

Step 3: if l [ Q then a source is selected q * = l p * = argmin{Fq*p|p [ P} else p * = l and a demand node is selected q * = argmin {Fp*q|q [ Q}

Step 4: xq*p* ¼ min demp*; capq* � �

Update demand and capacities Step 5: if capq = 0 then S(t) = 0 | t is corresponding to q

if demp = then S(t) = 0 | t is corresponding to p Step 6: for 1 to |Q|

if X p xqp > 0, yq = 1

End

In the above-mentioned mapping function, Fpq denotes the utility function which is defined for each pair of source and destination nodes. We calculate Fpq by the following equation. In addition, we apply this function to specify flows between each pair of nodes in each echelon of the network.

Fpq ¼ 1

max eqp j8q; p � �

� min eqp j8q; p � � epq

þ 1 max Trqp j8q; p

� � � min Trqp j8q; p

� � Trpq (5-1)

Table III. An illustration example for third string in the solution representation

Production centers

Recycle centers

Disposal centers

Random number 0.05 0.87 0.66 0.31 0.12 0.55 0.9 Established technology 1 3 2 1 1 2 3

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5.2 Crossover and mutation operators The performance of NSGA-II algorithm is considerably dependent on its operators, namely Crossover and Mutation. By applying these operators, we search through the solution space, explore, and exploit the good solutions. In the literature of GA, there are many Crossover and Mutation Operators that could be applied regarding the type of a problem.

Firstly, we select parents to perform crossover operation by tournament selection procedure (Talbi, 2009). After selection of parents, we specify the characteristics of solutions with four components (i.e. flow between nodes, capacity level, established technology, and fleet of vehicles). In this stage, we decide which component must be selected for crossover operator. For this reason, we generate an integer random number in a range [1-15] and specify which policy has to be chosen to apply in this stage. For the numbers in the range 1-4, only one of the solution’s strings is selected to perform the crossover operator. For the range 5–10, we select two strings from four available strings to perform crossover operator. For range 11–14, we choose three chromosomes from four chromosomes. If the generated random number is 15, the crossover operator is implemented for all chromosomes of solution. Because of continuous nature of the proposed algorithm, we use a crossover operator that are compatible with continuous problems. The mathematical formulation for applying crossover operator is as follows:

y1i ¼ ax1i þ 1 � að Þx2i y2i ¼ ax2i þ 1 � að Þx1i

0 < a < 1

( (5-2)

Where y1i and y 2 i are the first offspring and the second offspring’s i-th component in the

string. x1i and x 2 i are first parents and second parent’s i-th component. In addition, a is the

randomly real number between 0 and 1. Furthermore, we use a mutation operator which is applicable to real numbers which is

normally distributed mutation. In this regard, mutation is applied to solutions obtained from the previous operator. Probability of applying mutation operator to each gene in chromosomes are proportional to Pm which is the parameter of mutation operator. A Gaussian or normal distribution N (0, 1) is used, where N (0, 1) is a number of independently random Gaussian numbers with a mean of 0 and standard deviation 1. The mathematical formulation for mutation operator is as follows:

yi ¼ xiþijN 0; 1ð Þj (5-3)

5.3 Simulated annealing After applying NSGA-II operators, it is time to apply the SA which is obtained from of the previous solutions, to carry out local search. In this regard, SA is an algorithm to solve single objective optimization problems. It’s easy to transform a multi-objective optimization problem into single one by allocating weights to each objective, but this method has some drawbacks such as inability to extract non-supported solutions (Farrokhi-Asl et al., 2016). To tackle this problem, we have used a modified version of SA algorithm to solve multi- objective problems. SA imitates the process of cooling alloys to solve optimization problems (Aarts and korst, 1988). At each temperature (T), a solution x0 is accepted to be substituted for the current solution with the probability P which is calculated by the following equation. Specific attributes of SA increase the chance of replacing the worse solutions with the current solutions to escape from local optima:

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A ¼ min 1; exp � d E x 0 ; x

� � T

�� � (5-4)

d E x 0 ; x

� � ¼ E x0ð Þ � E xð Þ (5-5)

where E(x) denotes the level of energy solution, x. goes on as the algorithm and the temperature decreases by multiplying a coefficient at the end of each iteration. Consequently, the chance of worse solutions to be accepted decreases. Pseudo Code of the SA shows the pseudo code of the algorithm.

Inputs: K ‘’iteration numbers‘’ Lk ‘’ Sequence of epoch durations’’ Tk ‘’Sequence temperatures, Tk þ1 < Tk’’ x ‘’ Initial feasible solution’’ main loop:

1: for k = 1,. . ..,K 2: for i = 1,. . .,Lk 3: x0= perturb(x) 4: d E = E(x0) – E(x) 5: u = rand(0,1) 6: if u< min (1,exp(–d E/Tk)) 7: x = x0 8: end 9: end 10: end

Concerning multi-objective problems, we use the Pareto concept to define an energy function for each solution. Energy level in this approach is defined as the percentage of Pareto optimal solutions which dominates the solution. Therefore, we define the rx as the percentage of solutions which belongs to Pareto optimal solutions dominating the solution x:

rx ¼ y 2 rj y � xf g (5-6)

E xð Þ ¼ m rxð Þ (5-7)

Where m is defined as a function of r . If one solution belongs to Pareto set, the level of energy will be zero. Figure 4 demonstrates two solutions. The solid circle has less energy in comparison to empty circle with respect to Pareto optimal set showed by dash lines.

In this problem, the Pareto optimal set is not available. Thus we use approximated Pareto solutions obtained from each iteration of the algorithm (F). ~F is the union of set F, current solution x, and proposed solution x0. So, ~Fxis the solution which belongs to set ~F and dominates solution x:

eFx ¼ fy 2 eFjy � xg (5-8) Finally, the difference between the level of current solution and proposed solution is determined by the following equation. The obtained number will be less than one. If two solutions are available in the approximated Pareto solutions set, this difference will be zero.

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d E x; x 0� � ¼ 1

jeFj j eFxj � jfFx0 j

(5-9)

6. Experimental results The performance of the proposed hybrid algorithm is compared with NSGA-II and the associated results are analyzed. The algorithms are coded in MATLAB R2013a software and run on Intel Core i5 2.27 GHz personal computer with 4 GB RAM. Parameters of the algorithm (number of iterations, population size, number of inner iteration for SA, crossover parameter, mutation parameter, a in crossover equation, b in SA, cooling rate) are tuned by Taguchi method in Minitab 16 software. 10 test problems are generated among the test problems in the literature (Pishvaee et al., 2010) and in addition to the literature, characteristics of other parameters used in this paper are summarized in Table IV. Attributes of each size of the test problems are shown in Table V. Each instance runs 5 times and the results are collected. In this section MOHGASA refers to the algorithm which has been presented in this paper. To evaluate the performance of algorithms, we utilize six performance metrics including Number of Pareto Solutions (NPS), Computational Time, Spacing Metric (SM), Space Covered (SC), Diversification Metric (DM) and Coverage Metric (CM).

Figure 4. The obtained energy

level regarding Pareto optimal set

Table IV. Characteristics of test

problems

Problem Number of each facility Attribute of facilities Production Distribution Customers Collection Recycling Disposal Capacity Technology Vehicle

1 2 4 10 3 2 1 4 4 4 2 2 5 10 4 3 2 2 2 2 3 3 6 12 6 5 3 3 3 3 4 5 10 12 10 5 3 3 3 3 5 10 20 20 10 6 5 2 2 2 6 12 24 30 15 7 5 2 2 2 7 15 30 30 20 15 10 2 2 2 8 20 30 35 20 15 10 3 3 3 9 25 40 40 25 20 15 2 2 2 10 50 80 80 40 30 20 2 2 2

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Number of Pareto solutions (NPS) evaluates the ability of algorithms to find efficient solutions. Table VI summarizes the results for test problems. Additionally, computational time which has been spent to achieve the approximated Pareto solutions are shown in this Table. As shown in this table, the average performance of the MOHGASA is better than simple NSGA-II in generating Pareto solutions .In almost all test problems except problem 3 and 9, the proposed algorithm outperforms NSGA-II with respect to NPS factor. However, in some runs of these two problems, MOHGASA has found more Pareto solutions than NSGA- II. It should be noted that metaheuristic algorithms have a possibilistic nature and the performance of algorithms may vary in different runs. Conversely, computational time for MOHGACA is absolutely higher than NSGA-II in all test problems. However, because of the better performance of this algorithm and strategic nature of our decision about this problem, the increase in effort to achieve better solutions is rational and acceptable.

Spacing Metrics (SM) provides some information about how Pareto solutions, which have been obtained by each algorithm, are distributed uniformly. This metrics are calculated as follows:

SM ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 N � 1 �

XN i¼1

di � d � �2s

(6-1)

Where di is the Euclidean distance between solution i and the nearest solution which belongs to Pareto solution sets solutions. d is the average value of all di and N denotes the number of Pareto solutions.

Table V. Range of additional data used in this paper

Parameters Range Parameter Range

qr 0.6 cascd U(300,1000) sc 0.3 vehva U(10000,30000) frcrl U(250000,350000) emp

l i U(10,20)

fscdl U(250000,350000) emp l r U(15,30)

RDcc 0

nll0 U(50000,100000) emp l d U(20,40)

emvaa0 U(1,10) car c r U(400,1000)

Table VI. Average number of Pareto solutions and computational time for test problems

No. of Pareto solution Computational time(s) Problem no. NSGA-II MOHGACA NSGA-II MOHCG

1 8.4 10.2 60 85 2 12.2 13.4 129 226 3 10.4 10.2 200 312 4 8.4 10 250 318 5 15.4 14.2 431 533 6 9 11.6 422 604 7 8.2 9 504 732 8 7.8 10.8 620 878 9 10.4 9.2 789 1025 10 8.2 10.4 1012 1726 Average 9.84 10.9 441.7 643.9

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Diversification Metric (DM) determines the dispersion of Pareto solution sets and is specified as follows: max xti � yti

� � is the Euclidean distance between the non-dominated solutions xti and y

t i :

DM ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn i¼1

max kxti � ytik � �s

(6-2)

Based on Table VII, the performance of these two algorithms are almost close to each other regarding spacing metrics; that is to say, the distribution of Pareto solution in both algorithm is similar to each other. However, the MOHGASA is slightly better than NSGA-II in this metric. Likewise, MOHGASA excels in comparison to NSGA-II in diversification metric. It should be mentioned that fewer amounts are more desirable for spacing metric; conversely, more amounts are better for diversification metric.

Zitzler and Thiele (1999) presented a comparison metric called Space Covered (SC). This metric evaluates the size of the space which is covered by the obtained Pareto solutions. For.

bi-objective problems, each Pareto solution has two values for objective functions (f1(xi), f2(xi)), therefore each dominated solution represents a rectangle in the phenotype space. This could be extendible for multi-objective problems by multiplying the value of objective functions. It should be mentioned that firstly we make the numbers without scale and then calculate this metric for each set of data. Obtained results in Table VIII show in almost all test problems MOHGASA outperforms NSGA-II with respect to space covered measure. In other words, in most problems the proposed algorithm operates better than NSGA-II.

Table VII. SM Measure for test

problems

Spacing metric Diversification metric Problem no. NSGA-II MOHGASA NSGA-II MOHGASA

1 22396 19875 242389 56852 2 15625 14923 54698 112254 3 19782 22525 68952 115007 4 20005 21121 77925 89156 5 15625 9123 80230 99125 6 15813 17127 100251 121689 7 16236 17001 201119 215623 8 28751 28644 87322 228963 9 33985 34089 90178 156102 10 98888 99005 97146 196601 Average 28710 28343 110021 139137

Table VIII. SC Measure for test

problems

Problem characters NSGA-II MOHCG

1 0.44704794 0.99616126 2 0.59951867 0.82326392 3 0.04364909 0.84413564 4 0.76659407 0.32837142 5 0.11898274 0.73186372 6 0.17894898 0.55376153 7 0.21079523 0.49145762 8 0.91141854 0.62225788 9 0.02321573 0.30560697 10 0.61871324 0.26880212 Average 0.3918884 0.5965684

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In the Coverage Metric (CM), two non-dominated sets of solutions, which have been obtained by the distinct algorithms, are compared to each other. The given x and x0 are two sets of Pareto solutions obtained through each algorithm. The CM function maps the pair (x,x0) to interval [0,1] by applying following equation (Zitzler and Thiele, 1999):

CM x; x 0� � ¼ j a0 2 x0; a 2 x : a � a0� �jjx0 j (6-3)

The results of this measure are summarized in Table IX. MOHGASA is capable of providing Pareto solutions dominating 39 per cent of the solutions generated by NSGA-II. When MOHGASA was compared to NSGA-II, the generated solutions by NSGA-II dominated more than 17 per cent of MOHGASA’s, which is less than 39 per cent of MOHCG’s.

7. Conclusion Because of the increasing growth in governmental or non-governmental regulations, organizations’ pressure motivated us to address sustainable issues in our new proposed logistics network. This paper aimed to escape from sub-optimality through tackling forward and reverse logistics separately. Likewise, this paper presented a new multi- objective mathematical model for integrated forward and reverse logistics with simultaneous consideration of economic, environmental, and social issues in design of forward and reverse logistics. Because of NP-hard nature of the problem, precise methods are unable to find the exact Pareto optimal set. Therefore, one new hybrid multi-objective metaheuristic algorithm-called multi objective hybrid genetic algorithm- and SA were developed to provide a set of efficient solutions (approximated Pareto solutions). To evaluate the performance of the proposed algorithm some instances are generated with respect to instances which are available in the literature. In addition, the performance of the algorithm was compared to NSGA-II which is one of the well-known metaheuristic algorithm to solve MOPs. We utilized six comparison metrics including number of Pareto solutions (NPS), computational time, spacing metric (SM), space covered (SC), diversification metric (DM) and coverage metric (CM). MOHGASA outperformed NSGA- II in all metrics except Spacing Metric and Computational Time. Considering the results obtained by MOHGASA and the better solutions, the higher computational time seems rational since the proposed algorithm discovers more space in the feasible area by applying local search.

Regarding the direction of future research, it is appropriate to incorporate stochastic parameters in the network to change the parameters of the business environment . For example, demand of customer areas and rate of returned products can be uncertain. However, our proposed algorithm proved to be competitive for solving a presented problem. Moreover, other multi-objective metaheuristics algorithms such as multi objective particle swarm optimization (MOPSO) or NSGA-II combined with other local searches, can offer promising avenues for developing richer and integrated logistics networks.

Table IX. Average value for coverage of two sets in test problems

x/x0 NSGA-II MOHCG

NSGA-II 0 0.178 MOHCG 0.389 0

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Corresponding author Hamed Farrokhi-Asl can be contacted at: hamed.farrokhi@ut.ac.ir

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