Optimizing a multi-product closed-loop supply chain using NSGA-II, MOSA, and MOPSO meta-heuristic algorithms

Babaveisi, Vahid; Paydar, Mohammad Mahdi; Safaei, Abdul Sattar Article Optimizing a multi-product closed-loop supply chain using NSGA-II, MOSA, and MOPSO meta-heuristic algorithms Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Babaveisi, Vahid; Paydar, Mohammad Mahdi; Safaei, Abdul Sattar (2018) : Optimizing a multi-product closed-loop supply chain using NSGA-II, MOSA, and MOPSO metaheuristic algorithms, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 14, Iss. 2, pp. 305-326, https://doi.org/10.1007/s40092-017-0217-7 This Version is available at: https://hdl.handle.net/10419/195609 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ ORIGINAL RESEARCH Optimizing a multi-product closed-loop supply chain using NSGA-II, MOSA, and MOPSO meta-heuristic algorithms Vahid Babaveisi 1 •Mohammad Mahdi Paydar 1 •Abdul Sattar Safaei 1 Received: 13 March 2017 / Accepted: 16 June 2017 / Published online: 1 July 2017 The Author(s) 2017. This article is an open access publication Abstract This study aims to discuss the solution methodology for a closed-loop supply chain (CLSC) network that includes the collection of used products as well as distribution of the new products. This supply chain is presented on behalf of the problems that can be solved by the proposed meta-heuristic algorithms. A mathematical model is designed for a CLSC that involves three objective functions of maximizing the profit, minimizing the total risk and shortages of products. Since three objective functions are considered, a multi-objective solution methodology can be advantageous. Therefore, several approaches have been studied and an NSGA-II algorithm is first utilized, and then the results are validated using an MOSA and MOPSO algorithms. Priority-based encoding, which is used in all the algorithms, is the core of the solution computations. To compare the performance of the meta-heuristics, random numerical instances are evaluated by four criteria involving mean ideal distance, spread of non-dominance solution, the number of Pareto solutions, and CPU time. In order to enhance the performance of the algorithms, Taguchi method is used for parameter tuning. Finally, sensitivity analyses are performed and the computational results are presented based on the sensitivity analyses in parameter tuning. Keywords Closed-loop logistics NSGA-II MOPSO  MOSA Taguchi Mathematical model Priority-based Introduction The incremental trend of industrialization demands more and more resources forproducing theinnovative and old-fashioned products. However, two main types of resources can be identified. Renewable resources are sources of energy that are not depleted by use, such as wind, or solar power and the like. A non-renewable resource cannot be replaced by the nature such as fossil fuels. In this study, we focus on the non-renewable resources that are noteworthy due to depleting. Considering the increase in demand for resources, these resources will be too expensive as they become less available. Consequently, many companies try to produce products that use resources supplied by the end-user. In this chain, the end-users deliver the used products to collection centers to be used for producing products in manufacturing centers (McWilliams and Siegel 2001). The process of collecting products from the end-users andrecyclingis called ‘‘reverse logistics’’ (Chen et al. 2017). A good instance for reverse logistics is the Japanese Ministry of the Environment for the automotive industry (Sudarto et al. 2016). Recent studies show a good classification for the investigated industries related to CLSC. These studies involve 69 papers between 2003 and 2015 (Govindan and Soleimani 2017). As shown in Table 1, no study is performed in the area of oil and related fields during 12 years. This reason encouraged us to make a research in this area. Oil involves many derivatives such as kerosene, lubricant, gasoline, and tar. In this study, we concentrate on the lubricants. Engine oil that is one of the important products which is used in cars for lubrication of engine parts. Since oil is one of the expensive and non-renewable products, designing an optimized reverse logistics will be advantageous both financially and environmentally (O ¨zceylan and Paksoy 2013). This paper aims to design a closed-loop reverselogistics for collecting used engine oil that will be used in manufacturing &Mohammad Mahdi Paydar [email protected] 1 Department of Industrial Engineering, Babol Noshirvani University of Technology, Babol, Iran 123 J Ind Eng Int (2018) 14:305–326 https://doi.org/10.1007/s40092-017-0217-7 centers. Most of real-world supply chain (SC) problems are complex due to high number of indices which increase the dimension of the problem and it may lead to inefficiency of routine solution approaches (Fahimnia et al. 2013). Regarding increase in the size of problem, exponential growth in complexity makes the model become NP-hard (Park et al. 2007; Jolai et al. 2011). To deal with this problem, meta-heuristic solution methodologies are applied in this study. The algorithms that have been employed in this study are NSGA-II, MOSA, and MOPSO. Inorder to validate the results generated by each algorithm, three meta-heuristics are used. The paper is composed of the following sections. In ‘‘Literature review,’’ a comprehensive study is performed to examine earlier researches and introduce the present research path. The problem is described and the mathematical model is represented in ‘‘Problem statement’’ section. Meta-heuristics are presented in the ‘‘Solution methodology’’. The criteria are introduced in ‘‘Performance measure.’’ Taguchi approach is discussed in ‘‘Parameter setting,’’ and finally, brief results of the study and future research recommendations are provided in ‘‘Conclusion and future research’’ section. Literature review In this section, solution methodologies for solving mathematical models are presented specifically supply chain network models which were proposed by recent researchers. Since the current work focused on the metaheuristic solution approaches the field of the supply chain, the concerned papers are discussed as follows. Bandyopadhyay et al. (2008)proposed a new simulated annealing (SA) called archived multi-objective simulated annealing (AMOSA). The proposed algorithm was compared with two other solution methods: (1) non-dominated sorting genetic algorithm (NSGA-II) (2) Pareto archived evolution strategy (PAES). The results showed better performance of AMOSA than NSGA-II and PAES. Mahdavi et al. (2009) designed a model for a cell formation problem in cellular manufacturing was designed for this problem. Genetic algorithm was used for solving the model. Simple and uniform crossovers as well as a heuristic mutation generate new solutions. A genetic algorithm was implemented for optimization that showed better results than other methods. Pishvaee et al. (2009) discussed a reverse logistics network. This network was formulated using a mixed integer linear programming model. Due to NP- hardness of the problem, a simulated annealing (SA) algorithm was used to solve the proposed model. Prioritybased chromosome representation was used in order to encode solutions. The results of different sizes of problems showed intangible differences between exact and metaheuristic method. Altiparmak et al. (2009) designed a SC network which was solved using a steady-state genetic algorithm (ssGA). The results obtained from ssGA were compared with some exact and heuristic algorithms which outperformed them. Lee et al. (2009) considered a reverse Table 1 Classification of papers related to CLSC Case study No. Articles Without case study 14 Govindan et al. (2013); O ¨zkır and Bas¸lıgil (2013); Nikolaou et al. (2013); Bai and Sarkis (2013); Rashid et al. (2013); Dwivedy and Mittal (2012); Pialot et al. (2012); Chen and Sheu (2013); Gerrard and Kandlikar (2007); Hatcher et al. (2011); Binnemans et al. (2013); Qiang (2015); Corum et al. (2014); Chen et al. (2014) Car parts suppliers 11 Subramoniam et al. (2009); Matsumoto (2010); Subramoniam et al. (2010); Hojas et al. (2011); Subramoniam et al. (2013); Blume and Walther (2013); Amelia et al. (2009); Abdulrahman et al. (2014); Xia et al. (2014); Wang et al. (2014); Tian et al. (2014) Vehicle manufacturer/ remanufacturer 8 Seitz (2007); Saavedra et al. (2013); McKenna et al. (2013); Forslind (2005); Go et al. (2011); Demirel et al. (2014); Kurdve et al. (2015); Ziout et al. (2014) Electronics 8 Georgiadis and Besiou (2008); Mafakheri and Nasiri (2013); Rathore et al. (2011); White et al. (2003); El korchi and Millet (2011); Ravi (2012); Low et al. (2014); Jime ´nez-Parra et al. (2014) Steel 3 Giannetti et al. (2013); Schultmann et al. (2004); Salmi and Wierink (2011) Tannery 2 Hu et al. (2011); Gutterres et al. (2010) Power 2 Ortegon et al. (2013); Jiang et al. (2011) Printing 2 Kerr and Ryan (2001); Cristo ´bal Andrade et al. (2012) Palm oil 2 Ng et al. (2012); Mumtaz et al. (2010) Machine tools 2 Silva et al. (2013); Du et al. (2012) Nutrition 2 Huysveld et al. (2013); Jefferies et al. (2012) Others 13 Fahimnia et al. 2013; Simpson (2012); Zhu and Geng (2013); Matsumoto (2009); Zeng et al. (2013); O ¨stlin et al. (2009); Pigosso et al. (2010); Queiruga et al. (2012); Shaharudin et al. (2014); Murakami et al. (2014); Goodall et al. (2014); Gelbmann and Hammerl (2014); Song et al. (2014) 306 J Ind Eng Int (2018) 14:305–326 123 SC. The proposed model was solved using a hybrid genetic algorithm. A novel method is used for crossover to generate elite offsprings called weight mapping crossover (WMX). Pishvaee et al. (2010) designed a bi-objective model for a SC which involves two objective functions of maximizing responsiveness and minimizing costs. Due to insufficient search capability of genetic algorithm, a memetic algorithm is used for local search improvement. Since two objective functions are defined in their model, a multi-objective algorithm is implemented for optimizing the proposed model. Umar et al. (2012) discussed a vehicle routing problem (VRP) which was formulated as a singleobjective mathematical model. Since GA was implemented for solving this problem, the priority-based method was used for genetic representation. Subramanian et al. (2013) considered a CLSC network, and then the simulated annealing algorithm was conducted in the proposed CLSC using a priority-based encoding for chromosome representation. Yang et al. (2013) considered a mixed-model assembly line which contains a multi-objective function. To solve this model, an extended version of the genetic algorithm is used, called multi-objective genetic algorithm (MOGA). Partial representation is implemented for chromosome representation. Lotfi and Tavakkoli-Moghaddam (2013) discussed a transportation problem which was solved using a genetic algorithm. New operators of crossover and mutation were proposed in their paper. The priority-based algorithm was tested against spanning-tree- based algorithm by different instances of small, medium and large-sized problems that outperformed for medium and large-sized problems. Ghasimi et al. (2014) proposed a model for a defective goods SC network with an objective function of minimizing costs. This model determines the appropriate length of each cycle based on the importance of customer lead time using just-in-time logistics. Finally, the results of CPLEX solver and the GA are compared that CPLEX showed better results. The chromosomes are displayed in a matrix representing quantity variable. Yadegari Table 2 A brief of reviewed meta-heuristic algorithm Author(s) Implemented meta-heuristic algorithm Field of the research Memetic GA NSGAII NRGA ABC Hybrid GA SA MOPSO MOGA MOSA Bandyopadhyay et al. (2008) * Theoretical discussion Pishvaee et al. (2009) * Reverse logistics Lee et al. (2009)*Reverse logistics Mahdavi et al. (2009)*Cellular manufacturing Altiparmak et al. 2009 *Forward logistics Pishvaee et al. 2010 *Forward/reverse logistics Umar et al. (2012)*Vehicle routing Lotfi and Tavakkoli- Moghaddam (2013) *Transportation problem Yang et al. (2013)*Line balancing model Subramanian et al. (2013)*Closed-loop supply chain Ghasimi et al. (2014)*Forward logistics Yadegari et al. (2015)*Forward/reverse logistics Pasandideh et al. (2015a,b)*Forward logistics Santosa and Kresna (2015)*Facility location problem Liu et al. (2015)*Oil–gas production Sarrafha et al. (2015)*Forward logistics Kumar et al. (2015)*Reverse logistics Mousavi et al. (2016)** * Inventory model Current study ***Closed-loop supply chain J Ind Eng Int (2018) 14:305–326 307 123 et al. (2015) implemented a memetic algorithm for an NP- hard problem of an integrated forward-reverse logistics network. Random path-based direct decoding procedure is used for the delivery and recovery path generation. Pasandideh et al. (2015) implemented two bi-objective optimization methods of NRGA and NSGA-II to solve the model for a SC network. Priority-based was used for chromosome representation. NSGA-II for outperformed NRGA in this problem. Santosa and Kresna (2015) designed a mathematical model for a warehouse location problem. Simulated annealing was used for solving the model. Quantity of variables construct the chromosome matrix. An 11% gap exists between exact and metaheuristic methods. Since computation time is more important, SA outperforms the exact method. Liu et al. (2015) designed a multi-objective model. Two maximizing and one minimizing objective functions are used. To solve the model, a modified version of NSGA-II called I-NSGA- II is implemented which enhances diversity and convergence of the solutions. Sarrafha et al. (2015) designed a SC that was formulated as an MILP model. A multi-objective biogeography-based optimization (MOBBO) was utilized as a solution methodology. Habitat structure was used to represent the chromosome. The results of their method are compared with NSGA-II and MOSA algorithms. Kumar et al. (2015) designed a mathematical model for a reverse logistics. To solve the model, an artificial bee colony approach was utilized. Mousavi et al. (2016) discussed a bi-objective mathematical model for an inventory optimization problem. The reviewed papers are classified in Table 2, according to the implemented algorithms. Among reviewed papers, the papers that utilized priority-based representation are classified according to the type of the proposed supply chain and its specifications in Table 3. Based on the reviewed papers, none of the studies have been performed on the closed-loop supply chains considering multi-objective, multi-product, and multi-period models. Current paper considers a CLSC for collecting used engine oil that is multi-product, multi-period, and multi-objective. In this study, the NSGA-II, MOSA, and MOPSO multi-objective solution methods are utilized to get the optimal solution for the proposed multi-objective mathematical model. Problem statement Used engine oil collection involves several layers that have been studied practically to obtain information for designing the supply chain network. The flow of collected used oil starts from vendors and continues to collection centers and then to the manufacturing centers. Two types of manufacturing centers are considered. Type one is manufacturing centers using the original oil to produce product type one; the second type is manufacturing centers that purify used engine oil and use it in the type two product. In the forward supply chain, products are distributed from Table 3 The main classification of supply chain papers considering priority-based representation Author(s) Objective function Product (s) Period(s) Type of supply chain Single objective Multiobjective Single product Multiple products Single period Multiperiod Open Closedloop Forward Reverse Forward & reverse Pishvaee et al. (2009) ** * * Lee et al. (2009)* * * * Altiparmak et al. (2009) **** Pishvaee et al. (2010) ** * * Subramanian et al. (2013) ** * * Yadegari et al. (2015) ** * * Pasandideh et al. (2015a,b) ** ** Sarrafha et al. (2015) ** ** Current study ** * * 308 J Ind Eng Int (2018) 14:305–326 123 manufacturing centers to distribution centers and finally to vendors. Described CLSC is presented in Fig. 1. Assumptions, notations, and parameters •Manufacturing centers are two types. The first one is manufacturing type one that produces products type one using original oil and the second one is manufacturing type two which produces products by used engine oil. •Some hybrid centers are defined which function as distribution and collection centers. •Used engine oil is supplied by vendors and original oil is provided by the suppliers. •The capacity of containers that is used for transferring the used engine oil from vendors to collection centers is PO1. •Vehicles which transfer oil from collection centers to manufacturing centers have a capacity of PO2. Considering the defined problem and its assumptions, a model is designed for the proposed SC. The notations, parameters, and decision variables are as follows: Indices cIndex of the first type of products; c=1,…,C dIndex of the second type of products; d=1,…,D rIndex of products; r=c[d iIndex of the first type of manufacturing plants (MPs); i=1,…,I jIndex of the second type of MPs; j=1,…,J pIndex of MPs; p=i[j eIndex of distribution centers (DCs); e=1,…,E fIndex of collection centers (CCs); f=1,…,F hIndex of hybrid (multi-purpose) centers; h=1,…,H kIndex of distribution and multi-purpose centers; k=d[h k0Index of collection and multi-purpose centers; k0=c[h vIndex of vendors; v=1,…,V tIndex of periods; t=1,…,T Parameters C rpt Production cost related to product rwhich is produced in MP pin period t WEI r Weight of product r TC1 rpkt Cost of shipping product rfrom MP pto DC kin period t TC2 rkvt Cost of shipping product rfrom DC kto vendor vin period t TC3vk0tCost of shipping one unit of used engine oil from vendor vto CC k 0 in period t TC4k0pt Cost of shipping one unit of used engine oil from CC k 0 to MP p in period t Cap rpt Capacity of MP rfor product pin period t Cap0 kt Capacity of DC kin period t Cap00 k0tNumber of available PO2 containers in CC k 0 in period t Cap vt 000 Capacity of vendor vto supply used oil in period t BE vt Purchase cost for each PO1 container of used oil from vendor vin period t PP rpt Price of product rin MP pin period t Brt Shortage cost of product rin period t s r Amount of oil to produce one unit of product r PO1 Weight of small container to ship used oil from vendors to collection and hybrid centers PO2 Weight of each tanker of used engine oil shipped from CC to MP D rvt Demand of vendor vfor product rin period t RI k 0 v Rate of risk of supplying used oil by vendor vto CC k0 EC e Establishment cost of DC of e Hybrid center Collection center Vendor Manfacturing center type two Manufacturing center Type one Distribution center Fig. 1 Illustration of the proposed closed-loop logistics J Ind Eng Int (2018) 14:305–326 309 123 EC f 0 Establishment cost CC of f EC h 00 Establishment cost of h MA big number Decision variables Q rpt Quantity of production of product rin MP pin period t X rpkt Quantity of product rshipped from MP pto DC kin period t Y rkvt Quantity of product rshipped from DC kto vendor vin period t Y0 vk0tAmount of used oil transferred from vendor vto CC k 0 in period t X0 k0pt Amount of used oil transferred from CC k0to MP pin period t L rvt Shortage of product rfor vendor vin period t DES e If DC eis open 1, otherwise 0 CES f If CC fis open 1, otherwise 0 HES h If HC his open 1, otherwise 0 Mathematical model The first objective function (1) is maximizing total profit. It is obtained by SC costs from total incomes. Max Z1¼INCOMES COSTS ð1Þ Term (2) shows incomes obtained from selling products in the forward direction of the SC network. INCOMES ¼X rX pX tX kX v PPrptYrkvt ð2Þ Terms (3) and (4) show facility establishment and production costs, respectively. COSTS ¼X e ECeDESeþX f EC0fCESf þX h EC00hHEShð3Þ þX rX pX t CrptQrpt ð4Þ Transportation costs between facilities are obtained from term (5). þX rX pX kX t TC1rpktWEIrXrpkt þX rX kX vX t TC2rkvtWEIrYrkvt þX vX k0X t TC3vk0tPO1Y0 vk0t þX k0X pX t TC4k0pt PO1X0 k0pt ð5Þ Terms (6) and (7) represent used engine oil purchase cost and forward products shortage costs, respectively. þX vX k0X t BEvt Y0 vk0t  ð6Þ þX rX vX t BrtLrvt ð7Þ Minimization of total risks of purchasing used oil from vendors is represented as the second objective function in Eq. (8). Min Z2¼X vX k0X t RIk0vY0 vk0tð8Þ Finally, minimizing the shortage of products is obtained from the third objective function in term (9). Min Z3¼X rX tPvLrvt PvDrvt  ð9Þ Constraints The number of products transferred from manufacturing centers to distribution centers should equal the quantity of production in manufacturing centers (10). Qrpt X k Xrpkt ¼08r;p;tð10Þ Constraints (11) present the balance of flow in distribution centers. To generate a feasible solution, it is mandatory that output from DCs be less than or equal to input to DCs. X p Xrpkt X v Yrkvt ¼08r;k;tð11Þ Constraint (12) presents shortages of forward products. If the number of transferred products to vendors are less than their demands, there will be shortages of products. X k Yrkvt þLrvt ¼Drvt 8r;v;tð12Þ The balance of flow in collection centers is specified using constraint (13). PO1X v Y0 vk0tPO2X r X0 k0rt ! ¼08k0 ;tð13Þ Maximum allowed production in manufacturing centers is bounded by their capacities in term (14). Capacity is defined for each product rin each manufacturing center pin period t. Qrpt caprpt 8r;p;tð14Þ Each vendor has a certain capacity to supply used engine oil that is shown in constraint (15). This capacity is the amount of used engine oil that can be provided by each vendor in each period as the resources of producing the second type of products. 310 J Ind Eng Int (2018) 14:305–326 123 PO1X k0 Y0 vk0tcap000 vt 8v;tð15Þ Collection centers have a capacity of processing used engine oil due to the number of the vehicle for shipment that is shown in Eq. (16). X v Y0 vk0tPO2cap00 k0t8k0 ;tð16Þ Since products type two are produced by used engine oil, the quantity of production is specified using constraint (17). X d WEIdsdQdjt ¼X k0 PO2X0 k0jt 8j;tð17Þ Constraints (18)–(25) are defined to control variables to be activated when a center is open. If a center is open, the right side of the constraint can take amount. Xrpet MDESe8r;p;e;tð18Þ Xrpht MHESh8r;p;h;tð19Þ Yrevt MDESe8r;e;v;tð20Þ Yrhvt MHESh8r;h;v;tð21Þ Y0 vct MCESf8v;c;fð22Þ Y0 vht MHESh8v;h;tð23Þ X0 fjt MCESf8f;j;tð24Þ X0 hjt MHESh8h;j;tð25Þ The domain of decision variables is specified in Eq. (26). DESe20;1 fg 8e2E CESf20;1 fg 8f2F HESh20;1fg 8h2H Qrpt 0;integer 8r;p;t Xrpkt 0;integer 8r;p;k;t Yrkvt 0;integer 8r;k;v;t Lrvt 0;integer 8r;v;t Y0 vk0t;X0 k0vt 8v,k0 ;t 8k0 ;v;t ð26Þ Solution methodology Different solution methods are utilized in order to solve problems. In case of simple problems, routine methods can be used. However, by increasing the problem size, the routine solution methods would not work efficiently (Leu and Yang 1999). In these cases, using optimization techniques that are called evolutionary techniques are advantageous. Genetic algorithm (GA), particle swarm optimization (PSO), and simulated annealing (SA) are of these methods. In this paper, three meta-heuristic algorithms are utilized for optimization. Using different algorithms can validate the accuracy of results. Since the model involves three objective functions, using multi-objective solution methodology is necessary. Non-dominated sorting genetic algorithm (NSGAII), multi-objective simulated annealing (MOSA), and multiobjective particle swarm optimization (MOPSO) are used for this purpose. Two encoding schemes are proposed which are priority-based and permutation encoding. In the prioritybased representation, the priorities of the nodes are shown by the values in a list. According to this method, the nodes with higher values of priorities are selected earlier (Peng and Wei 2008; Taleizadeh et al. 2010). In this paper, priority-based encoding is utilized due to its compatibility with the problem. Chromosome representation Chromosomes are encoded using three methods: (1) encoding based on edge, (2) encoding based on vertex, and (3) encoding based on edge-and-vertex (Gen and Cheng 2000). Using the last method, sources and depots are used. When |K| is the number of sources and |J| is the number of depots, the dimension of the proposed matrix for GA representation will be |K||J| and the length of this matrix is |K|?|J|. The values of genes for this chromosome are between 1 and |K|?|J|. Each gene presents two kinds of values that is the locus and the allele. The locus shows the position of the value that is representative of the source or depot and the allele which is the value of the priority. Since transportation trees are generated for a set of sources and depots, the highest priority should be considered in each chromosome to select a source or depot (Altiparmak et al. 2009). These transportation problems should be solved to obtain the flow between a source and a depot. As the constrained optimization problems require constraints to be met, feasibility check is necessary. To prevent from generating infeasible solutions, using the repair mechanism is mandatory. Priority-based representation method is used to generate feasible solutions without using excessive repair mechanism (Gen and Cheng 2000). To apply priority-based encoding, stages should be clarified in the SC. The proposed SC is divided into four stages which is shown in Fig. 2. These stages are used in all the proposed algorithms such as NSGA-II, MOPSO and MOSA. NSGA-II NSGA-II is a multi-objective optimization algorithm which is used to generate Pareto solutions for multi-objective models. The main procedure of NSGA-II is based on the GA that is shown in Fig. 13. Generated populations are J Ind Eng Int (2018) 14:305–326 311 123 ranked using non-dominated sorting procedure (Pasandideh et al. 2015; Safaei et al. 2017). Ranked solutions create the Pareto fronts. The crowding distance and the rank are calculated for each solution. New solutions generated by crossover and mutation operators are added to the current populations. New population is sorted and the population size is selected from these solutions (Pasandideh et al. 2015; Taleizadeh et al. 2011). Populations consist of socalled chromosomes that are the strings of values called genes. According to this definition, a chromosome is an initial solution. A chromosome is designed for each stage. The representation for each stage is shown in Fig. 3. The encoding procedure that is specified for each stage is presented in the ‘‘Appendix’’. Crossover and mutation operators Due to defining priorities in these chromosomes, mutation and crossover should be specified in a way that duplication will be prevented. To perform crossover, order crossover (OX) was implemented. Two cut points are determined in each chromosome. The first cut point is determined in each row and the second one is determined in each column. As shown in Fig. 4, the darker parts are selected from the first parent and the white part is selected from the second one. The rest of priorities in each row are obtained from the second parent (Pasandideh et al. 2015). In case of mutation, two positions are changed randomly for all the periods. The mutation is shown in Fig. 5. Hybrid center Manufacturing center Type one Collection center Vendor Manfacturing center type two Stage 1 Stage 2 Stage 3 Stage 4 Fig. 2 Representation of stages in the supply chain V+C+H period 1 period t period 2 p eriod t-1 Fig. 3 Chromosome representation for stage 1 6 8 9 5 3 4 2 7 1 p eriod 1 period t Order crossover 8 9 5 1 2 4 6 7 3 7 8 6 2 3 4 1 9 5 9 3 8 5 6 4 2 7 1 period 1 period t 9 1 3 4 5 8 2 7 6 7 9 4 2 3 5 1 8 6 6 8 9 5 3 4 2 7 1 period 1 period t 8 9 5 1 2 4 3 7 6 7 8 6 2 3 4 1 9 5 Fig. 4 Crossover representation 312 J Ind Eng Int (2018) 14:305–326 123 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Appendix See Figs. 13,14,15,16,17,18, and 19. 321 0.4 0.3 0.2 321 321 0.4 0.3 0.2 321 population Mean of Means crossover rate mutation rate iteration 321 16 14 12 10 8 321 321 16 14 12 10 8 321 population Mean of SN ratios crossover rate mutation rate iteration 321 0.8 0.7 0.6 0.5 321 321 0.8 0.7 0.6 0.5 321 population Mean of Means crossover rate mutation rate iteration 321 6 4 2 321 321 6 4 2 321 population Mean of SN ratios crossover rate mutation rate iteration 321 0.18 0.16 0.14 0.12 0.10 321 321 0.18 0.16 0.14 0.12 0.10 321 population Mean of Means crossover rate mutation rate iteration 321 22 20 18 16 14 321 321 22 20 18 16 14 321 population Mean of SN ratios crossover rate mutation rate iteration Fig. 12 Mean of means and S/N ratio of instances 1, 5 and 10 J Ind Eng Int (2018) 14:305–326 319 123 Non-domination sorting First front Second front Third front Direct transfer Selection according to crowding distance NSGA-II loop Operator s No selection Fig. 13 Procedure of the NSGA-II C+H+M+R period 1 period t period 2 p eriod t-1 Fig. 14 Chromosome representation for stage 2 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 period t-1 Fig. 15 Chromosome representation for stage 3 k=1 D+H+V k=K D+H+V period 1 period t period 2 p eriod t-1 Fig. 16 Chromosome representation for stage 4 320 J Ind Eng Int (2018) 14:305–326 123 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 p eriod t-1 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 period t-1 Fig. 17 Swap 1 3 1 5 5 4 2 4 7 1 7 3 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 period t-1 7 1 7 3 5 4 2 4 1 3 1 5 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 period t-1 Fig. 18 Reversion 2 8 9 1 1 3 1 5 5 4 2 4 7 1 7 3 8 2 3 9 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 period t-1 2 8 9 1 5 4 2 4 7 1 7 3 1 3 1 5 8 2 3 9 k=1 period 1 period t M+R+D+H k=K M+R+D+H period 2 period t-1 12 Fig. 19 Insertion J Ind Eng Int (2018) 14:305–326 321 123 Procedure of priority-based decoding for stage 1 Inputs: {} ,: vct V: Set of vendors, C: Set of collection centers, H: Set of hybrid centers, T:set of periods, O C H set of collection and hybrid centers, tvc : Transportation cost from vendor v to collection center c = :{ , } : : vht vot ot vt for period t, v, c, t, tvh : Transportation cost from vendor v to hybrid center h for period t, v, h, t, tvo tvc tvh , cap Capacity of distribution center o for period t, o, t, cap ∀∀∀ ∀∀∀ ′∀∀ ′′′ Capacity of vendor v for period t, v, t, ch(t (v+c+h)): chromosome, v V, c C, h H, t T ∀∀ ×∀∈∈∈∈ Outputs: : vot Y C The amount of used oil tranported from vendor v to collection center o in period t For t=1 to T While () () 0:all ch ≠ Step1.Chromosome generation {} argmax (), ||||||;ran ch u u v c h ←∈++ Step2.determining sources and depots {} * * * * * arg min | , , arg min{ | ( oct vot If ran v, v ran , a vendor is selected o tvo ch(t,o) 0, v V selecting a collection center with minimum transportation cost else o ran a collection center is selected vtvoch ≤ ← =≠∀∈ ← =,) 0, },t v o O selecting a vendor with minimum transportation cost≠∀∈ Step3. ** * * min{ , }, vot ot vt Y C cap cap assigning from capacity to shipment between vendor and collection center ′′′ = Step4. Updating capacities **** ot ot vot cap cap Y C ′′ =− , **** vt vt vot cap cap Y C ′′ ′′ =− End End Step5. Determining which distribution center(s) to be opened ** * ** * ** ** (:, ,:)) 0 1, (:, ,:)) 0 1 vot o vot o If sum(Y C o and o c CC If sum(Y C o and o c HC ≥≤ = ≥> = 322 J Ind Eng Int (2018) 14:305–326 123 Procedure of priority-based decoding for stage 2 Inputs: {} {} , ,: : owt C: Set of collection centers H: Set of hybrid centers, r: set of manufacturing type two, T: set of periods,O C H set of collection and hybrid centers,W= r Set of manufacturing centers, tow : Transpo = : : kwt vot rtation cost from collection center to manufacturing center for period t, o, w, t, cap Capacity of producing product k in manufacturing center w for period t, k, w, t, Y C amount of used oil ∀∀ ∀ ∀∀∀ transported from vendor v to collection center o for period t, v, o, t, ch(t (c+h+r)): chromosome, c C, h H, r W, t T ∀∀∀ ×∈∈∈∈ Outputs: : owt XC The amount of used oil tranported from collection center o to manufacturing w in period t For t=1 to T While () () 0:all ch ≠ Step1.Chromosome generation {} arg max ( ), | | | | | | ;ran ch u u c h r ←∈++ Step2.determining sources and depots {} * * * * arg min | , , owt If ran o , o ran , a collection center is selected r tow ch(t,r) 0, o O selecting a manufacturing type two with minimum transportation cos t else rran a manufacturing type two is selecte ≤ ← =≠∀∈ ← * *arg min{ | ( , ) 0, }, or t d o tow ch t o r W selecting a collection center with minimum transportation cost=≠∀∈ Step3. * vo t v AYC= ∑ , * kkkr t k B weight teta cap=×× ∑ ** min{ , }, ort XC A B assigning from capacity to shipment between collection center and manufacturing type two = Step4. Updating capacities : ** ort AAXC=− , ** ort BBXC=− End End J Ind Eng Int (2018) 14:305–326 323 123 Procedure of priority-based decoding for stage 3 Inputs: {} {} , ,: ,: D: Set of distribution centers H: Set of hybrid centers, m: set of manufacturing type one, r: set of manufacturing type two, T: set of periods,O D H set of distribution and hybrid centers,W= m r Set o= : kwot kwt f manufacturing centers, two : Transportation cost of product k from manufacturing center to distribution center for period t, k, w, o, t, cap Capacity of producing product k in manufacturin ∀∀ ∀∀ : : kovt ort g center w for period t, k, w, t, Y D amount of product k transported from distribution o to vendor v for period t, k, o, v, t, XC amount of used oil transported from collection center o to ∀∀∀ ∀∀ ∀ ∀ , :() : kk manufacturing center r for period t, o, r t, weight weight kg of product k, teta amount of oil needed for producing each product ch(t k (m+r+d+h)): chromosome, k K, m W, r W, d D, h H, t T ∀∀∀ ×× ∈ ∈ ∈ ∈ ∈ ∈ Outputs: : kwot XD The amount of product k tranported from manufacturing w to distribution o in period t For t=1 to T While () () 0:all ch ≠ Step1.Chromosome generation: {} arg max ( ), | | (| | | | | | | |) ;ran ch u u k m r d h ←∈×+++ Step2. Selecting product: *, |||||||| ran kmrdh ⎡⎤ =⎢⎥ +++ ⎢⎥ Step3.determining sources and depots {} ** * * * * ((||||||||), ||||, arg min | , , kwot aran k-1) m r d h If a m r w a , a manufacturing center is selected ot wo ch(t,o) 0, w W selecting a distribution center with minimum transportation cost else oa a =− × +++ ≤+ ← =≠∀∈ ← ** *arg min{ | ( , ) 0, }, kwot distribution center is selected w two ch t w o O selecting a manufacturing center with minimum transportation cost=≠∀∈ If * ki≤ go to step 4 otherwise go to step 6 ** kovt v AYD= ∑ , ** kwt Bcap= Step4. Determining amount of shipment from manufacturing center to distribution center *** min{ , }, kwot XD A B= Step5. Updating capacities: *** *** ,, kwot kwot A A XD B B XD=− =− Else Step6. Determining amount of shipment from manufacturing center to distribution center owt o A XC ′=∑ ** kovt v AYD= ∑ *** ** min{ , , }, kwot kwt XD A cap A ′ = Step7. Updating capacities *** * * *** kwot k k kwot A A XD weight teta XD ′′ =− − × × *** ** ** *** kwot kwt kwt kwot B B XD , cap cap XD=− = − End End 324 J Ind Eng Int (2018) 14:305–326 123 Procedure of priority-based decoding for stage 4 Inputs: {} , ,: kovt D: Set of distribution centers H: Set of hybrid centers, V:set of vendors,T: set of periods, O D H set of distribution and hybrid centers, tov : Transportation cost of product k from distribution c = : : ot kvt enter to vendor for period t, k, o, v, t, cap Capacity of producing product k in manufacturing center w for period t, k, w, t, D Demand of vendor v for product k for period t, k, v, t, ∀∀∀∀ ′∀∀∀ ∀∀∀ :() , k weight weight kg of product k, ch(t k (d+h+v)): chromosome, k K, d D, h H, v V t T×× ∈ ∈ ∈ ∈ ∈ Outputs: : kovt Y D The amount of product k transported from distribution o to vendor v in period t For t=1 to T While () () 0:all ch ≠ Step1.Chromosome generation: {} arg max ( ), | | (| | | | | |) ;ran ch u u k d h v ←∈×++ Step2. Selecting product: *, |||||| ran k dhv ⎡⎤ =⎢⎥ ++ ⎢⎥ Step3.determining sources and depots {} ** * * * * ((||||||||), ||||, arg min | , , kovt aran k-1) m r d h If a d h o a , a distribution center is selected v tov ch(t,v) 0, o O selecting a vendor with minimum transportation cost else v a a vendor is sel =− × +++ ≤+ ← =≠∀∈ ← ** *arg min{ | ( , ) 0, }, kovt ected o tov ch t o v V selecting a distribution with minimum transportation cost=≠∀∈ Step4. Determining amount of shipment from distribution center to vendor *** * ** min{ , }, kovt ot kvt YD cap D ′ = Step5. Updating capacities *** *** ,, ot ot kvt kvt k o v t k o v t cap cap Y D D D Y D ′′ =− =− End End References Altiparmak F, Gen M, Lin L, Karaoglan I (2009) A steady-state genetic algorithm for multi-product supply chain network design. Comput Ind Eng 56:521–537 Bandyopadhyay S, Saha S, Maulik U, Deb K (2008) A simulated annealing-based multiobjective optimization algorithm: AMOSA. 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