Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A memetic algorithm with extended random path encoding for a closed-loop supply chain model with flexible delivery

A memetic algorithm with extended random path encoding for a closed-loop supply chain model with... Logist. Res. (2016) 9:22 DOI 10.1007/s12159-016-0150-y ORIGINAL PAPER A memetic algorithm with extended random path encoding for a closed-loop supply chain model with flexible delivery 1 2,3 Elham Behmanesh Ju¨rgen Pannek Received: 22 January 2016 / Accepted: 21 October 2016 / Published online: 7 November 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Logistics network design is a major strategic 1 Introduction issue in supply chain management of both forward and reverse flow, which industrial players are forced but not Supply chain management (SCM) describes the discipline equipped to handle. To avoid sub-optimal solution derived of optimizing the delivery of goods, services and infor- by separated design, we consider an integrated forward mation from supplier to customer. Logistics network reverse logistics network design, which is enriched by design is known as one of the comprehensive strategic using a complete delivery graph. We formulate the cyclic decision problems due to its impact on the efficiency and seven-stage logistics network problem as a NP hard mixed responsiveness of the supply chain including reducing cost integer linear programming model. To find the near opti- and improving service quality. To this end, an optimal mal solution, we apply a memetic algorithm with a choice regarding number, location, capacity and type of neighborhood search mechanism and a novel chromosome plants, warehouses, and distribution centers as well as the representation including two segments. The power of amount of shipped materials needs to be obtained. extended random path-based direct encoding method is Within the full material cycle, we distinguish between shown by a comparison to commercial package in terms of the forward supply chain from the upstream supplier to the both quality of solution and computational time. We show downstream customer, and the reverse one for possible that the proposed algorithm is able to efficiently find a good recycling, reusage and disposal. solution for the flexible integrated logistics network. Due to economic benefits and environmental protection, industrial players are under a lot of pressure to take back Keywords Memetic algorithm  Closed-loop supply the product after its use. Therefore, the reverse supply design  Random path  Flexible delivery chain is becoming more relevant in both theory and prac- tice. Although some firms such as General Motors, Kodak, and Xerox concentrate on reverse logistics and have obtained significant success in this area [1, 2], most logistics networks are not equipped to handle the reverse flow. To avoid sub-optimality of a solution derived by the This article is part of a focus collection on ‘‘Dynamics in Logistics: separated design, we consider forward and reverse channel Digital Technologies and Related Management Methods’’. as an integrated model and analyse closed-loop supply & Elham Behmanesh chain as a system in total. To manage logistics system beh@biba.uni-bremen.de efficiently in terms of cost and delivery time as well as increase customer satisfaction, flexible and productive International Graduate School for Dynamics in Logistics, University of Bremen, Bremen, Germany network design models are of particular interest. Many research works have been published in the fields of opti- Faculty of Production Engineering, University of Bremen, Bremen, Germany mizing of supply chain network (SCN) [3, 4], while flex- ible delivery and multistage planning conditions are rarely BIBA Bremer Institut fur Produktion und Logistik GmbH, found in the literature. Bremen, Germany 123 22 Page 2 of 12 Logist. Res. (2016) 9:22 In this study, we investigate the integration of forward 2 Literature review and problem definition and reverse logistics network design, cf. Fig. 1. In the forward flow, new products are shipped from plants to Focusing on an efficient solution methodology, we split our customers via distribution centers and retailers to satisfy literature review to the research areas network design with their demands. In the reverse flow, returned products are forward, reverse and integrated flows, and to flexible collected in collection-inspection centers for sorting dis- delivery paths. assembling for recovery, reuse or disposal, cf. [5, 6] for related frameworks. To enhance the logistic network effi- 2.1 Forward, reverse and integrated logistics network ciency and flexibility, we consider a full delivery graph in the forward flow with normal delivery (products are delivered from one echelon to another), direct delivery In previous studies, the design of forward and reverse (products are transported from distribution centers to cus- logistics network was typically separated. In the field of tomers or via plants to retailers directly) and direct ship- forward logistics many models were developed as part of a ment (products are transported from plants to customers traditional logistics network design. The common MILP directly). model involves the choice of facilities to be open and the distribution network design to satisfy the demand with The objective of this paper is to develop and optimize a seven-stage closed-loop supply chain network with respect minimum cost. Yeh [7] developed a MILP model for a supplier-pro- to number and capacity of plants, distribution centers (Dcs), retailers, collection/inspection centers (Cos) and duction-distribution network. He revised existing mathe- disposal centers (Dis) as well as the product flow between matical models presented by other researchers and the facilities. The aim of this study is to minimize the total developed an efficient hybrid heuristic, which was enriched cost including the transportation cost as well as operation by applying three different local search techniques. The cost. efficiency of the proposed methodology was demonstrated The remainder of this paper is structured as follows. via computational results. Syarif et al. [8] proposed a MILP After systematically reviewing related literature in Sect. 2, formulation for a fixed charge and multistage transporta- we present our cyclic logistics network problem as a mixed tion problem for a single commodity supply chain model. They considered a spanning tree-based GA using Pru¨fer integer linear programming (MILP) model in Sect. 3.As the problem is NP hard and therefore difficult to solve number representation to solve this problem. Some com- parison between results obtained by this method and using classical methods such as branch-and-bound, Sect. 4 presents a random path, flexible delivery-based memetic LINDO showed the efficiency of proposed method. A two- echelon facility location problem was studied by Tragan- algorithm (MA) with a neighborhood search mechanisms using a new chromosome representation. To assess the talerngsak et al. [9]. In this work, each facility in the sec- quality of the approach, we compare respective results for ond echelon was limited in capacity and could only be test cases to solutions obtained by LINGO in Sect. 5. The supplied by one facility in the first echelon. Also, each final Sect. 6 concludes the paper and points out directions customer is serviced by only one facility of the second of future research. echelon. The authors presented a mathematical model for the problem and developed a Lagrangian relaxation-based branch-and-bound algorithm to solve it. A different two- stage distribution-planning problem was addressed by Gen et al. [10] to minimize the total cost including transporta- tion and opening costs. They presented a priority-based genetic algorithm (pb-GA) with a new decoding and encoding method. Also they introduced a new crossover operator called Weight Mapping Crossover and analyzed the effect of the latter on the computational performance. They showed the efficiency of proposed method with regards to solution quality and computing time in com- parison to two different GA approaches. Due to legislative changes regarding end-of-life (EOL) [11] issues as well as economic factors [12], considering the forward logistic network and omitting any reverse flow is impractical. A general review of the current develop- ments in reverse logistics was reported by Pokharel and Fig. 1 Framework of the flexible IFRLN 123 Logist. Res. (2016) 9:22 Page 3 of 12 22 Mutha [13]. They identified three factors, which differ for minimize the total cost including fixed opening costs, reverse logistics and a traditional supply chain: (1) Most transportation cost, processing costs and penalties for non- logistics networks are not equipped to handle returned utilized capacities. Pishvaee et al. [32] proposed a linear product movement; (2) Reverse distribution costs may be multi-objective programming model to improve the total higher than moving the original product from the plant to cost as well as responsiveness of the integrated forward/ customer; (3) Returned products may not be transported, reverse logistics network. To find the set of non-dominated stored, or handled in the same manner as in the regular solutions, the authors proposed a solution algorithm based channel [14]. In a study by Jayaraman et al. [15], an ana- on a new dynamic search strategy using three different lytical model to minimize reverse distribution costs was local searches. Within the model, they allowed for a hybrid developed. This MILP model limited supply of customer distribution-collection facility. The authors compared their demand from a single distribution center. In addition, there pareto-optimal solutions to recent GA results. was a tight bound on the number of collection and refur- bishing sites. Apart from the formulation of a reverse dis- 2.2 Flexible delivery path tribution problem, the authors also presented a new heuristic solution method. The algorithm has three com- To increase market share, companies try to increase cus- ponents: random selection of potential collection and tomer satisfaction via fast delivery, which requires supply refurbishing sites, heuristic concentration and heuristic chain responsiveness [32, 33]. To deal with the issues of expansion. Min et al. [16] designed a MINLP model to cost efficiency and network responsiveness simultaneously, minimize the overall reverse logistics costs including spa- researchers have proposed models to optimize the supply tial and temporal consolidation of returned products. The chain network, respectively [4, 32]. However, results are authors presented a mathematical model for the problem typically limited to shipments between consecutive stages and solved it using GA. There are other studies on reverse or just indirect shipment mechanisms [4, 32, 34]. Lin et al. logistics, which are limited to specific applications, such as [35], formulated a MILP model by including direct ship- carper recycling by Louwers et al. [17] and Realff et al. ment and direct delivery as well as inventory control for a [18], battery recycling by Schultmann et al. [19] as well as three stages forward logistics network. To solve the prob- Kannan et al. [20], tire recycling by Figueiredo and May- lem, they proposed a hybrid evolutionary algorithm com- erele [21], paper recycling by Pati et al. [22], plastic posed of Wanger within algorithm, GA, and fuzzy logic recycling by Huysman et al. [23], bottle recycling by Shen controller. Pishvaee and Rabbani [36] studied the respon- et al. [24], sand recycling by Listes and Dekker [25]. siveness of a three-stage forward logistics network when Notable work with a remanufacturing focus was presented (1) direct shipment between plant to customer is allowed by Krikke et al. [26] on copiers and Srivastava [27]on and (2) direct shipment is forbidden. They proposed two appliances and personal computers. Currently, no general mixed integer programming models for these conditions model for reverse logistics exits. and proved that both of these problems can be modeled by In recent years, some researches started to integrate a bipartite graph. To tackle these NP hard problems, a forward and reverse networks to close products cycles. The novel heuristic solution method was considered based on a aim is to avoid sub-optimality of a solution due to sepa- new chromosome representation derived from graph the- rated design [28]. Lee and Dong [29] proposed a MILP ory. Based on the above review, extending the following model, which is capable to manage the forward and reverse restrictions can be considered a potential field of research: flows at the same time for end-of-lease computer products. • Flexibility in delivery paths as a measure to shorten the They presented the first attempt of solving the integrated delivery time is typically ignored for simple and design problem using a Tabu search-based MA. Lu and completely ignored for integrated forward/reverse Bostel [30] designed a two-level location problem as a logistics networks. MILP model with three types of facility (producers, • The total number of echelons in most of the developed remanufacturing centers and intermediate centers). This IFRLN models is not more than five echelons. model considers forward and reverse flows and their • It is still a critical need to develop an efficient solution interactions simultaneously. The focus of this research was to cope with NP hard problems as well as a general on remanufacturing to reduce costs of production and raw model to be applicable to a wide range of industries. materials. The model was solved using Lagrangian heuristics, which requires lower and upper bound of the Within the literature, various facility location models based objective function. Pishvaee et al. [31] focused on a MILP on mixed integer programming (MIP) were considered to model to integrate reverse logistics activities into the for- determine maximal profit, optimal number and capacity of ward supply chain. To deal with uncertainty, they pre- facilities as well as the optimal flow between them. sented a scenario-based stochastic optimization model to Although typically larger models are required to represent 123 22 Page 4 of 12 Logist. Res. (2016) 9:22 real supply chains, researchers developed many heuristics customers C in forward flow, as well as collection/inspection [7, 15, 34, 37] and metaheuristics such as genetic algorithm centers Co and disposal centers Di in reverse flow, cf. Fig. 2 [4, 6, 20, 38–42], simulated annealing [43–45], tabu search for a schematic sketch.We like to point out that in accordance [29, 46], memetic algorithm [32, 47] and scatter search to Fig. 1, we consider a hybrid manufacturing-recovery-re- [34] to solve these NP hard problems. However, there is cycling facility as well as a hybrid collection-inspection still a critical need in this area to increase the efficiency of facility. Establishing several facilities at the same location solution approaches, especially, when the complexity of can decrease the price in comparison with separating design. the model increases (Melo et al. [48]). To adapt problem (1), we impose the following The problem addressed in this work includes integrated assumptions: design of forward and reverse logistics as well as flexibility • The set of nodes is given by N ¼ S [ P [ Dc [ in delivery paths for a seven-stage closed-loop supply chain R [ C [ Co [ Di. network. The proposed model as a complete and general • There are no edges between facilities of the same stage, network covers the previously described cases in the litera- the delivery graph is complete and the return graph is ture with less complexity. Additionally, the full delivery simple, i.e., E ¼ðS  PÞ[ðP  DcÞ[ðP  RÞ[ðP graph allows us to solve the conflicting goals profit and CÞ[ðDc  RÞ[ðDc  CÞ[ðR  CÞ[ðC  CoÞ[ responsiveness, which otherwise may lead to greater cost ðCo  DiÞ[ðCo  PÞ. [32]. From the computational point of view, we incorporate • The demands of each customer are deterministic and the graph structure in the chromosome representation, must be satisfied. thereby avoiding different model and solution methodolo- • The number of facilities per stage and respective gies as, e.g., considered by Pishvaee and Rabbani [36]. capacities are limited. • All cost parameters (fixed and variables) are known in advance. 3 Description for integrated forward/reverse • The transportation rates are perfect and there are no logistics network return storages. Moreover, the return rate p as well as the disposal disposal recovery and disposal rates p and ð1  p Þ To support the presentation of the proposed mathematical j j model, we consider the general model area of our problem. are fixed. All returned products from each customer must be collected. To this end, we consider G ¼ðN; EÞ to be a digraph where N denotes the set of all nodes and E the set of all edges in • The inspection cost per item for the returned products are included in the collection cost. the closed-loop network. The cost for node i 2 N are denoted by c , and the unit transportation cost on edge • The un-recyclable returned products will be sent to the disposal center. The remaining products are returned to ði; jÞ2 E are given by c . The respective decision variables ij the same plant. y 2f0; 1g and x 2 N represent whether a stage i 2 N is i ij 0 • The required recycled materials are assumed to be of used and which quantity is transported between node i and the same quality as the raw materials bought from j. To determine the optimal distribution network and suppliers and any plant chooses the raw material from capacity of each node, we minimize the transportation and the collection/inspection center over suppliers. operation cost of the proposed network, which reveals the • Customers have no special preference. It means, price following mixed integer minimization problem: X X is the same in all facilities. min c x þ c y ij ij i i x ;y ij i The objective of this model is to minimize the total cost of i2N ði;jÞ2E the proposed supply chain, which is composed of fixed ð1Þ s:t: a x 6 b y i ij i i costs for facilities and variable costs for transportation. In ði;jÞ2E terms of the above notation, the cost function, the sign and x  0; y 2f0; 1g ij i the integer conditions remain identical. The constraints in Next, we specialize this model to reflect the IFRLN properties. 3.1 Mathematical formulation and assumptions The previously described IFRLN setting represents an inte- grated supply chain with seven echelons consisting of sup- pliers S, plants P, distribution centers Dc, retailers R and Fig. 2 Underlying structure of MILP 123 Logist. Res. (2016) 9:22 Page 5 of 12 22 (1) are specialized and we have that the capacities in each to a specific problem is to decide how to design a chromo- node induce the inequalities some. The tree-based representation is known to be one way for representing network problems. Different methods have b 8i 2 S ij been developed to encode trees. One of them is matrix-en- ð2Þ b y 8i 2 N nfS [ Cg: i i ði;jÞ2E coding, was is developed by Michalewicz [51]. In this method, the solution is presented by a jKj jJj matrix where return Additionally, by assumption only a fraction p is |K| and |J| are the number of sources and depots, respectively. disposal returned by customers and a fraction p of the returned Although this solution approach has a simple representation, products has to be disposed off. Apart from these excep- applying this method requires the development of a special tions, the supply chain network is subject to the law of the crossover and mutation operator for obtaining a feasible flow conservation, i.e., in-flow and out-flow in each node solution as well as huge amount of memory. Another tree- must be identical for these nodes. These conditions reveal based representation is the Pru¨fer number. The use of the x 8j 2 N nfC [ Cog ij Pru¨fer number representation for solving various network ði;jÞ2E problems was introduced by Gen and Cheng [52]. It requires > P return p x 8j 2 C > ij j an array of the length jKjþjJj 2 with |K| sources and |J| X ði;jÞ2E depots. Since this method may compute infeasible solutions x ¼ P ð3Þ jk disposal > p x 8j 2 Co > ij [39], a repair mechanism has been developed. In this regard, ðj;kÞ2E ði;jÞ2CoDi Jo et al. [39] presented the procedure for repairing infeasible disposal > ð1  p Þ x 8j 2 Co ij chromosomes. Later, Gen et al. [53] introduced determinant ði;jÞ2CoP encoding using priority which does not need any repair mechanism to guarantee feasibility of solutions. In this Last, the demands of customers must be satisfied. approach, solutions are encoded as arrays of size jKjþjJj,in x ¼ b 8j 2 C ij j which the position of each cell represents the sources and ð4Þ ði;jÞ2E depots and the value in cells represent the priorities. From the literature [41], we have found that both Pru¨fer and determinant encoding are efficient for the encoding of the spanning tree problem. However, as the determinant 4 Solution approach encoding overcomes the bottlenecks of Pru¨fer encoding [54], we utilize determinant encoding in our study. In the Because of our IFRLN model is a capacitated allocation and following encoding and decoding are discussed. multi-choice problem, it is known as a NP hard problem [6, 38, 39, 49]. Hence, although the problem can be refor- 4.1.1 Random path-based direct encoding method mulated into an integer linear programming, we cannot compute a suitable solution for large-scaled problems within The delivery and recovery path can be conventionally a short time. There are three main options to tackle NP hard determined by applying the random path direct encoding problems: probabilistic algorithms, approximation algo- method introduced by Lin et al. [55]. Using this method rithms and metaheuristic algorithms. To reduce the search computation time can be greatly cut down. One gene in a space and increase the solution quality, we consider the class chromosome is characterized by two factors: locus, the of metaheuristic algorithm to solve this model. According to position of the gene within the structure of chromosome, [50], memetic algorithms are appropriate for the proposed and allele, the value the gene takes. In this method, each model. The basic feature of MA is a multi-directional and gene is initialised with a random value from its domain and global search by generating a population of solutions as well it contains M groups where M is the total number of cus- as local search to improve intensification of the search. tomers. Each group represents a delivery path in forward According to the reviewed literature, two major issues affect flow as well as recovery path in reverse flow. Due to the performance of memetic algorithm [41], i.e., the chro- existence of three different delivery paths in the proposed mosome representation and the memetic operators. problem, we extend the random path-based direct encoding method by adding a second segment into the chromosome. 4.1 Chromosome representation 4.1.2 Extended random path-based direct encoding A chromosome must have the necessary gene information for solving the problem. Selecting a proper chromosome Although applying the new delivery paths improves the representation highly affects the performance of meta- flexibility and efficiency of the supply chain network, it heuristic algorithm. Therefore, the first step of applying MA makes the problem more complex. In Fig. 3 the 123 22 Page 6 of 12 Logist. Res. (2016) 9:22 Fig. 3 Representation of extended random path-based direct encoding method representation of the extended random path-based direct lowest. The procedure of encoding by extended random encoding method in two segments is shown. The first seg- path-based direct encoding is shown in Algorithm 1 below. ment is encoded by using random path-based direct encoding method which shows the delivery path for each customer. Algorithm 1 The pseudocode procedure of initializa- The second segment of a chromosome contains two parts: the tion by extended random path-based direct encoding first part with J locus including the guide information Input: Number of customers M regarding plant assignments in the network, and the second Number of collection/inspection centers N part of length K containing the information of the distribution Number of disposal centers O centers. As shown in Fig. 3, the length of chromosome is Number of plants J ð7  MÞþ J þ K where M, J and K are the total number of Number of retailers L customers, plants and distribution centers, respectively. Number of distribution centers K Each sequence of seven subsequent genes forms a group. Number of suppliers I Each group encodes four potential delivery paths through Step 1:  (first segment) plant, distribution center and retailer to customer as well as a 1: for i =0: M − 1 do recovery path from customer through collection/inspection 2: ch [7 ∗ i +1] ← random(1,N ) to disposal center or plant. The first three alleles of a group 3: ch [7 ∗ i +2] ← random(1,O) represent the reverse flow of the network, while the next four 4: ch [7 ∗ i +3] ← random(1,J ) alleles of that group show the forward flow from supplier to 5: ch [7 ∗ i +4] ← random(1,L) customers. As an illustration, a randomly assigned ID to 6: ch [7 ∗ i +5] ← random(1,K) these facilities in the reverse and forward flow is shown in 7: ch [7 ∗ i +6] ← ch [7 ∗ i +3] k k Fig. 3. Each locus in the second part is assigned an integer in 8: ch [7 ∗ i +7] ← random(1,I) the set f0; 2g for plants due to existence of three delivery 9: end for options for each plant in the network. Regarding distribution Step 2:  (second segment, plant delivery path) center, an integer from f0; 1g is chosen to represent the two 10: for i =0: J − 1 do respective delivery options. The second segment is involved 11: ch [7 ∗ M + i] ← random(0, 2) by determining the sort of delivery path for the selected plant 12: end for as well as distribution center in first segment. Step 3:  (second segment, Dc delivery path) It should be noted that applying this encoding approach 13: for i =0: K − 1 do 14: ch [7 ∗ M + J + i] ← random(0, 1) might generate infeasible solutions, which violate the 15: end for facility capacity constraint; hence, a repairing procedure is Output: Chromosome ch [·] needed. If the total demand of a depot from a source k exceeds its capacity, the depot will be assigned to another Remark 1 According to the assumption presented in source with sufficient product supply so that the trans- Sect. 3, returned products have to be directed to the portation cost between that source and the depot is the 123 Logist. Res. (2016) 9:22 Page 7 of 12 22 original plant. To follow this limitation, the third and sixth skipping distribution centers, path number 2 is selected. position of first segment of the chromosome representation Similarly, path number 3 is chosen if retailers are skipped. for any customer should be identical. Last, if direct shipment is selected, the delivery path number 4 will be implemented. Remark 2 Since, the third and sixth position of first An important difference between the traditional random segment are identical, 6 has been considered as the number path-based direct decoding method and the method adopted of each unit, instead of 7, to apply the chromosome in this paper is that we include the delivery path informa- representation. tion of the second segment. The detailed decoding proce- dure is shown in Fig. 5. Each locus in this segment is 4.1.3 Extended random path-based direct decoding assigned to an integer in the range of f0; 1; 2g for plants and f0; 1g for distribution centers. Here, we encode normal Decoding is the mapping from chromosomes to candidate delivery for plants and distribution centers by P ¼ 0 and solution to the problem. As an example, Fig. 4 represents Dc ¼ 0 respectively, where j and k denote the ID of the an instance of a delivery and recovery path in our model. plant and of the distribution center. Moreover, P ¼ 0 and In each gene unit, four delivery paths can be designed by Dc ¼ 1 as well as P ¼ 1 represent direct delivery and k j applying normal delivery, direct shipment and direct P ¼ 2 direct shipment. The paths displayed in Figure 5 delivery. All of them are from a neighborhood. For correspond to respective choices, i.e., we have instance, we can obtain the neighborhood given in Algo- Path1 () P ¼ 0; Dc ¼ 0 j k rithm 1 from the sample of gene unit shown in Fig. 4 that shows the delivery path to customer 2. Considering the Path2 () P ¼ 1; Dc 2f0; 1g j k second chromosome (customer 2) in Fig. 4 as an example, Path3 () P ¼ 0; Dc ¼ 1 j k we start by supplier 2 and continue via plant 4, distribution Path4 () P ¼ 2; Dc 2f0; 1g: j k center 1 and retailer 3 in forward flow as well as collection/ inspection center 3, disposal center 1 and plant 4 in the It should be noted that because the amount of returned reverse flow. Due to construction, four different delivery products shipped to each one should be known for paths are possible, cf. Figure 4. The delivery and recovery decoding the forward flow, decoding of the forward flow is path 1 occurs if normal delivery is chosen for all stages. By impossible until the reverse flow is decoded. Fig. 4 Delivery path for a sample of gene unit Fig. 5 Presentation of the second segment of the extended random path-based direct encoding 123 22 Page 8 of 12 Logist. Res. (2016) 9:22 4.2 Evaluation Algorithm 3 Pseudocode of the local search for the proposed model Fitter individuals have higher chances of survival. The Input: One parent A evaluation assigns a fitness value to each individual, Number of customers M thereby inducing a measure. In our study, we apply the cost Number of collection/inspection centers N function as the fitness value. This fitness value is computed Number of disposal centers O for the decoded chromosome to analyze the accuracy and Number of plants J efficiency of the proposed MA. Number of retailers L Number of distribution centers K Number of suppliers I 4.3 Crossover and Local Search 1: Randomly select position a in chromosome of A 2: b ← mod(a, 7) Crossover is known as the most important recombination I if b =0 of both parents’ feature to explore new solution within the N if b =1 search space. There are many variants of crossover oper- O if b =2 ations developed in the literature, cf. [10]. Based on the 3: X ← characteristics of the chromosome, we chose the two-cut ⎪ J if b ∈{3, 6} point crossover, which applies the steps shown in Algo- L if b =4 rithm 2. K if b =5 4: n ← round(random(30/100 ∗ X, 70/100 ∗ X)) 5: for i =1: n do Algorithm 2 Pseudocode of the two point 6: A ← A crossover for the proposed model 7: c ← random(1,X) Input: Two parents 8: A (a) ← c 1: Generate two random positions 9: Evaluate fitness function for A 2: Swap data beyond the two points between parents 10: end for Output: Two offsprings 11: Select best chromosome among n new instances Output: One offspring After crossover, the population is merged and sorted according to its fitness value. In the next step, a local 4.4 Selection method search technique is applied, i.e., if the fitness value of a new solution is better than the old one, the latter is We apply the well-known roulette wheel selection for replaced. The detailed procedure is shown in Algo- generating the next generation of chromosomes. The rithm 3. The chromosome showing the best fitness is strategy of roulette wheel is a probabilistic selection based selected. on fitness. 123 Logist. Res. (2016) 9:22 Page 9 of 12 22 4.5 Procedure of proposed memetic algorithm 5 Test problems and computational results Combing the aforementioned components, we obtain the To assess the accuracy and efficiency of the developed procedure displayed in Algorithm 4 for solving our prob- MA, we generated various test problems of different sizes lem IFRLN. to compare the results obtained by our MA from Algorithm 4 and a branch-and-bound algorithm from LINGO15. Since the logistics network framework in this study differs from Algorithm 4 Pseudocode of the proposed memetic previous studies, the size of test problems considered in algorithm this work is selected randomly as shown in Table 1. The Input: Number of population n first six test problems relatively small and the number of Number of crossover population m decisions variables are 128, 209, 234, 468, 1006 and 1780, 1: for k =1: n do respectively, and the remaining problems are large sized. 2: Encode ch [·] by Algorithm 1 Other parameters are generated randomly using uniform 3: Evaluate ch [·] according to fitness function distributions shown in Table 2. Each test problem has been 4: end for 5: Set i ← 0 solved 20 times to test the robustness of the method. 6: while termination condition not satisfied do Our implementation was written in MATLAB R2015b r TM 7: for k = n +1 : n + m do and run on the PC with Intel Core i5 2.40 GHz with 12 8: Select two parents via roulette wheel GB RAM. For our testing, we considered population size 9: Generate ch [·] by Crossover Algorithm 2 n ¼ 60 and crossover rate c ¼ 0:5. As a stopping criterion 10: end for for Algorithm 4, we imposed a maximum iteration number 11: Merge ch[·]= ch [·] k=1:n+m of 100 as well as a maximum number of iteration without 12: Evaluate and sort ch[·] by fitness value improvement 8, 10, 12, 20, 25 and 30 for our small size 13: Select first n elements of ch[·] problems, respectively. For the large size problems, we 14: Obtain ch[·] via Algorithm 3 with ch [·] increased the latter bound by 5 for each test problem. Also, 15: if fitness value of ch[·] is better than of ch [·] we set the number of local search iterations to be equal to then the number of retailers L for each test problem. 16: ch [·] ← ch[·] To evaluate the performance of proposed MA, firstly, 17: end if we employed LINGO15 to solve the optimization problem 18: i ← i +1 and obtained the results displayed in Table 3. Although 19: end while LINGO provides results for small size problems quickly, Output: Chromosome of optimal solution ch [·] Table 3 indicates that LINGO is unsuitable for solving the large size problems and it is run out of memory. According to Table 4, the proposed MA provides good Note that as we apply only one crossover and search step solutions for our small size problems, which allows us to before selecting the next generation, our method belongs to trust the method also for large size problems. To compare the class of steady state MA. Table 1 Settings of test Problem Supplier Plants Distribution Retailers Customers Col/Ins Disposal problems centers centers centers 12 2 5 8 2 2 1 22 3 8 9 3 3 2 3 2 4 6 10 2 2 1 42 4 10 16 4 4 2 53 6 15 24 6 6 2 64 8 20 32 8 8 4 7 6 12 30 48 12 12 6 8 8 16 40 64 16 16 8 912 24 40 96 24 24 12 10 16 32 40 128 32 32 16 11 24 48 60 192 48 48 24 12 32 64 80 256 64 64 32 123 22 Page 10 of 12 Logist. Res. (2016) 9:22 Table 2 Parameters values used in the test problems the optimal solutions obtained by LINGO with the results of our MA Algorithm 4, the percentage error is calculated Parameters Range using formula (5). b ; j 2 S Uniform (200, 1100) MA  LINGO answer answer Error percent ¼  100 ð5Þ b ; j 2 P Uniform (100, 1000) LINGO answer b ; j 2 Dc Uniform (50, 900) Based on Table 5, we observe that the error percentages b ; j 2 R Uniform (50, 850) for the small size problems are zero, which indicate the b ; j 2 D Uniform (100, 500) high accuracy of proposed MA. Although the operation b ; j 2 Co Uniform (20, 100) time is higher compared to LINGO, our implementation b ; j 2 Di Uniform (20, 100) allows us to derive results for the large size problems. return p 10% Hence, the proposed MA demonstrated that it is capable to disposal 50% prepare sufficiently accurate solution with the efficient c Uniform (1,3) ij computation time for our integrated forward/reverse c ; j 2 P Uniform (100, 2500) logistics problem with flexible delivery. c ; j 2 Dc Uniform (100, 2100) c ; j 2 R Uniform (100, 400) 6 Conclusion c ; j 2 Co Uniform (100, 500) c ; j 2 Di Uniform (50, 400) In this paper, we focused on a comprehensive mixed integer linear programming formulation for a seven-stage closed-loop network design problem. We applied the Table 3 Results obtained by LINGO extended direct delivery path representation-based meme- Problem Problem size Solution tic algorithm, which was developed for a full delivery graph and a combined forward/reverse logistics design to 12  2  5  8  2  2  1 2905 decrease delivery time and avoid sub-optimal solutions, 22  3  8  9  3  3  2 2335 respectively. The aim of this work is to minimize total cost, 32  4  6  10  2  2  1 2345 which we addressed as allocation problem to find the 42  4  10  16  4  4  2 1160 optimal number and capacity for any facility as well as the 53  6  15  24  6  6  2 4100 optimal transportation flow between facilities. Since the 64  8  20  32  8  8  4 11365 basic problem is NP hard, the combination with flexibility 76  12  30  48  12  12 6– in delivery path makes the search space of the problem 88  16  40  64  16  16 8– much larger and more complex and NP hard as well. 912  24  40  96  24  24  12 – Because existing methods are unable to solve this problem, 10 16  32  40  128  32  32  16 – we proposed a MA approach to compute a near optimal 11 24  48  60  192  48  48  24 – solution for large size problems. In this study, we intro- 12 32  64  80  256  64  64  32 – duced a new chromosome representation for MA to Table 4 Results for Algorithm Test problem Min-cost Max cost Ave cost Min time (s) Max time (s) Ave-time (s) 4 with n ¼ 60 and m ¼ 30 over 20 runs 1 2905 2905 2905 2.3 4.3 3.05 2 2335 2735 2402 4.3 10 6.7 3 2345 2885 2381 6.7 13 9.3 4 1160 1560 1225.5 18 47 32.5 5 4100 4920 4576 19 57 38.45 6 11,365 12415 11821 175 410 275.75 7 17,268 21205 19324 260 430 310.6 8 24,933 30446 26995 570 630 600.25 9 33,555 40043 36571.4 1780 2010 1903 10 51,343 60251 52692.95 3740 4060 3935 11 11,986 15600 13132.2 4100 5600 4700 12 13,400 15804 14227.7 6200 7500 6680 123 Logist. Res. (2016) 9:22 Page 11 of 12 22 Table 5 Comparison of results Problem LINGO MA Error percent from LINGO and proposed MA Min-cost Ave-time (s) Min-cost Ave-time (s) 1 2905 0.1 2905 3.05 0 2 2335 0.12 2335 6.7 0 3 2345 0.12 2345 9.3 0 4 1160 0.14 1160 32.5 0 5 4100 0.16 4100 38.45 0 6 11,365 0.17 11,365 275.75 0 7. Yeh W (2005) A hybrid heuristic algorithm for the multistage enhance its search ability for the proposed flexible model. supply chain network problem. Int J Adv Manuf Technol We verified correctness of the proposed method using 26:675–685 numerical experiments and LINGO15. Additionally, we 8. Syarif A, Yun YS, Gen M (2002) study on multi-stage logistic showed that the method is capable to solve larger size chain network: a apanning tree-based genetic algorithm approach. Comput Ind Eng 43(1):299–314 problems which cannot be solved by LINGO. 9. Tragantalerngsak S, Holt S, Ronnqvist J (2000) An exact method Apart from costs aspect considered here, other aims such for two-echelon, single-source, capacitated facility location as responsiveness and robustness can be considered in problem. Eur J Oper Res 123:473–489 designing integrated forward/reverse logistics network that 10. Gen M, Altiparmak F, Lin L (2006) A genetic algorithm for two- stage transportation problem using priority-based encoding. Oper needs updating the algorithm to be capable to solve multi- Res-Spektrum 28:337–354 objective models. Moreover, to be close to the real-world 11. Neto JQF, Walther G, Bloemhof J, Nunen AEEV, Spengler T application, multi-product multi-capacity and multi-period (2010) From closed-loop to sustainable supply cchain: the weee networks with uncertainty as well as considering inventory case. Int J Prod Res 48(15):4463–4481 12. Alumur SA, Nickel S, da Gamad FS, Verter V (2012) Multi- can be employed. Last, the effect of different parameters on period reverse logistics network design. Eur J Oper Res the behavior of the proposed metaheuristic algorithm can 220(1):67–78 be analyzed in depth. 13. Mutha SPA (2009) Perspectives in reverse logistics: a review. Conserv Recycl 53(4):175–182 Acknowledgements The authors would like to appreciate the support 14. Sarkis J, Darnal N, Nehman G, Priest J (1995) The role of supply International Graduate School (IGS) of Bremen University to support, chain management within the industrial ecosystem. In: The 1995 help and advice as well as the Deutscher Akademischer Austausch IEEE international symposium on electronics and the environ- Dienst (DAAD) for financial support of this research under the GSSP ment, Orlando, pp 229–234 programme of the IGS. 15. Jayaraman V, Patterson R, Rolland E (2003) The design of reverse distribution networks: model and solution procedure. Eur Open Access This article is distributed under the terms of the Creative J Oper Res 150:128–149 Commons Attribution 4.0 International License (http://creative 16. Min H, Ko CS, Ko HJ (2006) The spatial and temporal consol- commons.org/licenses/by/4.0/), which permits unrestricted use, distri- idation of returned pproduct in a closed-loop supply chain net- bution, and reproduction in any medium, provided you give appropriate work. Comput Ind Eng 51(2):309–320 credit to the original author(s) and the source, provide a link to the 17. Louwers D, Kip B, Peters E, Souren F, Flapper S (1999) A Creative Commons license, and indicate if changes were made. facility location allocation model for reusing carpet materials. Comput Ind Eng 36:855–869 18. Realff M, Ammons J, Newton D (2004) Robust reverse propro- References duct system design for carpet recycling. IIE Trans 36:767–776 19. Schultmann F, Engels B, Rentz O (2003) Closed-loop supply 1. Meade L, Sarkis J, Presley A (2007) The theory and practice of cchain for spent batteries. Interfaces 33:57–71 reverse logistics. Int J Logist Syst Manag 3(1):56–84 20. Kannan G, Pokharel S, Kumar P (2009) A hybrid approach using 2. Uster H, Easwaran G, Akali E, etinkaya S (2007) Benders ism and fuzzy tosis for the selection of reverse logistics provider. decomposition with alternative multiple cuts for a multi-product Resour Conser Recycl 54(1):28–36 closed-loop supply chain network design model. Naval Res 21. Figueiredo J, Mayerle S (2008) Designing minimum-cost recy- Logist (NRL) 54(8):890–907 cling collection networks with required throughput. Transp Res 3. Georgiadis M, Tsiakis P, Longinidis P, Sofioglou M (2011) Part E 44:731–752 Optimal design of supply chain networks under uncertain tran- 22. Pati R, Vrat P, Kumar P (2008) A goal programming model for sient demand variations. Omega 39(3):254–272 paper recycling system. Omega 36:405–417 4. Altiparmak F, Gen M, Lin L, Paksoy T (2006) A genetic algo- 23. Huysman S, Debaveya S, Schaubroeck T, Meester SD, Ardente F, rithm approach for multi-objective potimization of supply chain Mathieux F, Dewulf J (2015) The recyclability benefit rate of networks. Comput Ind Eng 51:197–216 closed-loop and open-loop systems: a case study on plastic 5. Beamon B (1999) Designing the green supply chain. Logistics recycling in flanders. Resour Conserv Recycl 101:53–60 12(4):332–342 24. Shen L, Worrel E, Patel M (2010) Open-loop recycling: a lca case 6. Govindan K, Soleimani H, Kannan D (2015) Reverse logistics study of pet bottle-to-fibre recycling. Resour Conser Recycl and closed supply chain: a comprehensive review to explore the 55(1):34–52 future. Eur J Oper Res 240(3):603–626 123 22 Page 12 of 12 Logist. Res. (2016) 9:22 25. Listes O, Dekker R (2005) A stochastic approach to a case study 41. Wang H, Hsu H (2010) A closed-loop logistic model with a for product recovery network design. Eur J Oper Res spanning- tree based genetic algorithm. Comput Oper Res 160:268–287 37:376–389 26. Krikke H, Harten V, Schuur A (1999) Reverse logistic network 42. Dengiz B, Altiparmak F, Smith AE (1997) Local search genetic redesign for copiers. OR Spektrum 21:381–409 algorithm for optimal design of reliable networks. IEEE Trans 27. Srivastava S (2008) Network design for reverse logistics. Omega Evol Comput 1(3):179–188 36:535–548 43. Krikpatrick S (1984) Optimizing by simulated annealing: quan- 28. Fleischmann M, Beullens P, Bloemhof-ruwaard JM, Wassenhove titative studies. J Stat Phys 34(5):975–986 L (2001) The impact of product recovery on logistics network 44. Pishvaee MS, Kianfar K, Karimi B (2010) Reverse logistics design. Prod Oper Manag 10:156–173 network design simulated annealing. Int J Adv Manuf Technol 29. Lee D, Dong M (2007) A heuristic approach to logistics network 47(1):269–281 design for end-of lease computer pproduct recovery. Transp Res 45. Jayaraman V, Ross A (2003) A simulated annealing mathodology Part E 44:455–474 to distribution network design and management. Eur J Oper Res 30. Lu Z, Bostel N (2007) A facility location model for logistics 144:629–645 systems including reverse flows: the case of remanufacturing 46. Sung C, Song S (2003) Integrated service network design for a activities. Comput Oper Res 34:299–323 cross-docking supply chain network. J Oper Res Soc 31. Pishvaee MS, Jolai F, Razmi J (2009) A stochastic optimization 54:1283–1295 model for integrated forward/reverse logistics network design. 47. Moscato P, Cotta C (2003) An introduction to memetic algo- J Manuf Syst 28:107–114 rithms. Inteligencia Artificial, Revista Iberoamericana de 32. Pishvaee MS, Farahani RZ, Dullaert W (2010) A memetic Inteligencia Artificial 19:131–148 algorithm for bi-objective integrated forward/reverse logistics 48. Melo M, Nickel S, da Gama FS (2009) Facility location and network design. Comput Oper Res 37:1100–1112 supply chain management: a review. Eur J Oper Res 33. Kilibi W, Martel A, Guitouni A (2010) The design of robust 196:401–412 value-creating supply chain networks: a critical review. Eur J 49. Eckert C, Gottlieb J (2002) Direct representation and variation Oper Res 203:283–293 operators for the fixed charge transportation problem, in parallel 34. Du F, Evans G (2008) A bi-objective reverse logistics network problem solving from nature. Springer, Berlin analysis for post-sale service. Comput Oper Res 35:2617–2634 50. Behmanesh E, Pannek J (2016) Modeling and random path-based 35. Lin L, Gen M, Wang X (2009) Integrated multistage logistics direct encoding for a closed loop supply chain model with flex- network design by using hybrid evolutionary algorithm. Com- ible delivery paths. In: The 7th IFAC conference on Management puters and Industrial Engineering 56:854–873 and Control of Production and Logistics, (Accepted) 36. Pishvaee MS, Rabbani M (2011) A graph theoretic-based heu- 51. Michalewicz Z, Vignaux G, Hobbs M (1991) A non-standard heuris algorithm for responsive supply chain network design with genetic algorithm for the transportation problem. ORSA J Com- direct and indirect shipment. Adv Eng Softw 42:57–63 put 3:307–316 37. Amiri A (2006) Designing a distribution network in a supply 52. Gen M, Cheng R (1997) Genetic algorithms and engineering chain system: formulated and efficient solution procedure. Eur J design. Wiley, New york Oper Res 171:567–576 53. Gen M, Cheng R, Lin L (2008) Network model and optimization: 38. Gottlieb J, Paulmann L (1998) Genetic algorithms for the fixed multiobjective genetic algorithm approach. Springer, Berlin charge transportation problem. In: Inn IC(Ed), IEEE World 54. Abuali F, Wainwright R, Schoenefeld D (1995) Determinant Congress on evolutionary computation (I. C. INN, ed.), factorization: a new encoding scheme for spanning trees applied pp 330–335 to the probabilistics minimum spanning tree problem. In Eshel- 39. Jo JB, Yinzhen L, Gen M (2007) Nonlinear fixed charge trans- man L (ed), Proceedings of the 6th international conference on portation problem by spanning tree-based genetic algorithm. genetic algorithms, pp 470–477 Comput Ind Eng 53:290–298 55. Gen M, Cheng R (2000) Genetic algorithms and engineering 40. Min H, Ko H (2008) The dynamic design of a reverse logistics optimization. Wiley, New York network from the perspective of third-party logistics service providers. Int J Prod Econ 113:176–192 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Logistics Research Springer Journals

A memetic algorithm with extended random path encoding for a closed-loop supply chain model with flexible delivery

Logistics Research , Volume 9 (1) – Nov 7, 2016

Loading next page...
 
/lp/springer-journals/a-memetic-algorithm-with-extended-random-path-encoding-for-a-closed-dknVxl5I86
Publisher
Springer Journals
Copyright
Copyright © 2016 by The Author(s)
Subject
Engineering; Engineering Economics, Organization, Logistics, Marketing; Logistics; Industrial and Production Engineering; Simulation and Modeling; Operation Research/Decision Theory
ISSN
1865-035X
eISSN
1865-0368
DOI
10.1007/s12159-016-0150-y
Publisher site
See Article on Publisher Site

Abstract

Logist. Res. (2016) 9:22 DOI 10.1007/s12159-016-0150-y ORIGINAL PAPER A memetic algorithm with extended random path encoding for a closed-loop supply chain model with flexible delivery 1 2,3 Elham Behmanesh Ju¨rgen Pannek Received: 22 January 2016 / Accepted: 21 October 2016 / Published online: 7 November 2016 The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Logistics network design is a major strategic 1 Introduction issue in supply chain management of both forward and reverse flow, which industrial players are forced but not Supply chain management (SCM) describes the discipline equipped to handle. To avoid sub-optimal solution derived of optimizing the delivery of goods, services and infor- by separated design, we consider an integrated forward mation from supplier to customer. Logistics network reverse logistics network design, which is enriched by design is known as one of the comprehensive strategic using a complete delivery graph. We formulate the cyclic decision problems due to its impact on the efficiency and seven-stage logistics network problem as a NP hard mixed responsiveness of the supply chain including reducing cost integer linear programming model. To find the near opti- and improving service quality. To this end, an optimal mal solution, we apply a memetic algorithm with a choice regarding number, location, capacity and type of neighborhood search mechanism and a novel chromosome plants, warehouses, and distribution centers as well as the representation including two segments. The power of amount of shipped materials needs to be obtained. extended random path-based direct encoding method is Within the full material cycle, we distinguish between shown by a comparison to commercial package in terms of the forward supply chain from the upstream supplier to the both quality of solution and computational time. We show downstream customer, and the reverse one for possible that the proposed algorithm is able to efficiently find a good recycling, reusage and disposal. solution for the flexible integrated logistics network. Due to economic benefits and environmental protection, industrial players are under a lot of pressure to take back Keywords Memetic algorithm  Closed-loop supply the product after its use. Therefore, the reverse supply design  Random path  Flexible delivery chain is becoming more relevant in both theory and prac- tice. Although some firms such as General Motors, Kodak, and Xerox concentrate on reverse logistics and have obtained significant success in this area [1, 2], most logistics networks are not equipped to handle the reverse flow. To avoid sub-optimality of a solution derived by the This article is part of a focus collection on ‘‘Dynamics in Logistics: separated design, we consider forward and reverse channel Digital Technologies and Related Management Methods’’. as an integrated model and analyse closed-loop supply & Elham Behmanesh chain as a system in total. To manage logistics system beh@biba.uni-bremen.de efficiently in terms of cost and delivery time as well as increase customer satisfaction, flexible and productive International Graduate School for Dynamics in Logistics, University of Bremen, Bremen, Germany network design models are of particular interest. Many research works have been published in the fields of opti- Faculty of Production Engineering, University of Bremen, Bremen, Germany mizing of supply chain network (SCN) [3, 4], while flex- ible delivery and multistage planning conditions are rarely BIBA Bremer Institut fur Produktion und Logistik GmbH, found in the literature. Bremen, Germany 123 22 Page 2 of 12 Logist. Res. (2016) 9:22 In this study, we investigate the integration of forward 2 Literature review and problem definition and reverse logistics network design, cf. Fig. 1. In the forward flow, new products are shipped from plants to Focusing on an efficient solution methodology, we split our customers via distribution centers and retailers to satisfy literature review to the research areas network design with their demands. In the reverse flow, returned products are forward, reverse and integrated flows, and to flexible collected in collection-inspection centers for sorting dis- delivery paths. assembling for recovery, reuse or disposal, cf. [5, 6] for related frameworks. To enhance the logistic network effi- 2.1 Forward, reverse and integrated logistics network ciency and flexibility, we consider a full delivery graph in the forward flow with normal delivery (products are delivered from one echelon to another), direct delivery In previous studies, the design of forward and reverse (products are transported from distribution centers to cus- logistics network was typically separated. In the field of tomers or via plants to retailers directly) and direct ship- forward logistics many models were developed as part of a ment (products are transported from plants to customers traditional logistics network design. The common MILP directly). model involves the choice of facilities to be open and the distribution network design to satisfy the demand with The objective of this paper is to develop and optimize a seven-stage closed-loop supply chain network with respect minimum cost. Yeh [7] developed a MILP model for a supplier-pro- to number and capacity of plants, distribution centers (Dcs), retailers, collection/inspection centers (Cos) and duction-distribution network. He revised existing mathe- disposal centers (Dis) as well as the product flow between matical models presented by other researchers and the facilities. The aim of this study is to minimize the total developed an efficient hybrid heuristic, which was enriched cost including the transportation cost as well as operation by applying three different local search techniques. The cost. efficiency of the proposed methodology was demonstrated The remainder of this paper is structured as follows. via computational results. Syarif et al. [8] proposed a MILP After systematically reviewing related literature in Sect. 2, formulation for a fixed charge and multistage transporta- we present our cyclic logistics network problem as a mixed tion problem for a single commodity supply chain model. They considered a spanning tree-based GA using Pru¨fer integer linear programming (MILP) model in Sect. 3.As the problem is NP hard and therefore difficult to solve number representation to solve this problem. Some com- parison between results obtained by this method and using classical methods such as branch-and-bound, Sect. 4 presents a random path, flexible delivery-based memetic LINDO showed the efficiency of proposed method. A two- echelon facility location problem was studied by Tragan- algorithm (MA) with a neighborhood search mechanisms using a new chromosome representation. To assess the talerngsak et al. [9]. In this work, each facility in the sec- quality of the approach, we compare respective results for ond echelon was limited in capacity and could only be test cases to solutions obtained by LINGO in Sect. 5. The supplied by one facility in the first echelon. Also, each final Sect. 6 concludes the paper and points out directions customer is serviced by only one facility of the second of future research. echelon. The authors presented a mathematical model for the problem and developed a Lagrangian relaxation-based branch-and-bound algorithm to solve it. A different two- stage distribution-planning problem was addressed by Gen et al. [10] to minimize the total cost including transporta- tion and opening costs. They presented a priority-based genetic algorithm (pb-GA) with a new decoding and encoding method. Also they introduced a new crossover operator called Weight Mapping Crossover and analyzed the effect of the latter on the computational performance. They showed the efficiency of proposed method with regards to solution quality and computing time in com- parison to two different GA approaches. Due to legislative changes regarding end-of-life (EOL) [11] issues as well as economic factors [12], considering the forward logistic network and omitting any reverse flow is impractical. A general review of the current develop- ments in reverse logistics was reported by Pokharel and Fig. 1 Framework of the flexible IFRLN 123 Logist. Res. (2016) 9:22 Page 3 of 12 22 Mutha [13]. They identified three factors, which differ for minimize the total cost including fixed opening costs, reverse logistics and a traditional supply chain: (1) Most transportation cost, processing costs and penalties for non- logistics networks are not equipped to handle returned utilized capacities. Pishvaee et al. [32] proposed a linear product movement; (2) Reverse distribution costs may be multi-objective programming model to improve the total higher than moving the original product from the plant to cost as well as responsiveness of the integrated forward/ customer; (3) Returned products may not be transported, reverse logistics network. To find the set of non-dominated stored, or handled in the same manner as in the regular solutions, the authors proposed a solution algorithm based channel [14]. In a study by Jayaraman et al. [15], an ana- on a new dynamic search strategy using three different lytical model to minimize reverse distribution costs was local searches. Within the model, they allowed for a hybrid developed. This MILP model limited supply of customer distribution-collection facility. The authors compared their demand from a single distribution center. In addition, there pareto-optimal solutions to recent GA results. was a tight bound on the number of collection and refur- bishing sites. Apart from the formulation of a reverse dis- 2.2 Flexible delivery path tribution problem, the authors also presented a new heuristic solution method. The algorithm has three com- To increase market share, companies try to increase cus- ponents: random selection of potential collection and tomer satisfaction via fast delivery, which requires supply refurbishing sites, heuristic concentration and heuristic chain responsiveness [32, 33]. To deal with the issues of expansion. Min et al. [16] designed a MINLP model to cost efficiency and network responsiveness simultaneously, minimize the overall reverse logistics costs including spa- researchers have proposed models to optimize the supply tial and temporal consolidation of returned products. The chain network, respectively [4, 32]. However, results are authors presented a mathematical model for the problem typically limited to shipments between consecutive stages and solved it using GA. There are other studies on reverse or just indirect shipment mechanisms [4, 32, 34]. Lin et al. logistics, which are limited to specific applications, such as [35], formulated a MILP model by including direct ship- carper recycling by Louwers et al. [17] and Realff et al. ment and direct delivery as well as inventory control for a [18], battery recycling by Schultmann et al. [19] as well as three stages forward logistics network. To solve the prob- Kannan et al. [20], tire recycling by Figueiredo and May- lem, they proposed a hybrid evolutionary algorithm com- erele [21], paper recycling by Pati et al. [22], plastic posed of Wanger within algorithm, GA, and fuzzy logic recycling by Huysman et al. [23], bottle recycling by Shen controller. Pishvaee and Rabbani [36] studied the respon- et al. [24], sand recycling by Listes and Dekker [25]. siveness of a three-stage forward logistics network when Notable work with a remanufacturing focus was presented (1) direct shipment between plant to customer is allowed by Krikke et al. [26] on copiers and Srivastava [27]on and (2) direct shipment is forbidden. They proposed two appliances and personal computers. Currently, no general mixed integer programming models for these conditions model for reverse logistics exits. and proved that both of these problems can be modeled by In recent years, some researches started to integrate a bipartite graph. To tackle these NP hard problems, a forward and reverse networks to close products cycles. The novel heuristic solution method was considered based on a aim is to avoid sub-optimality of a solution due to sepa- new chromosome representation derived from graph the- rated design [28]. Lee and Dong [29] proposed a MILP ory. Based on the above review, extending the following model, which is capable to manage the forward and reverse restrictions can be considered a potential field of research: flows at the same time for end-of-lease computer products. • Flexibility in delivery paths as a measure to shorten the They presented the first attempt of solving the integrated delivery time is typically ignored for simple and design problem using a Tabu search-based MA. Lu and completely ignored for integrated forward/reverse Bostel [30] designed a two-level location problem as a logistics networks. MILP model with three types of facility (producers, • The total number of echelons in most of the developed remanufacturing centers and intermediate centers). This IFRLN models is not more than five echelons. model considers forward and reverse flows and their • It is still a critical need to develop an efficient solution interactions simultaneously. The focus of this research was to cope with NP hard problems as well as a general on remanufacturing to reduce costs of production and raw model to be applicable to a wide range of industries. materials. The model was solved using Lagrangian heuristics, which requires lower and upper bound of the Within the literature, various facility location models based objective function. Pishvaee et al. [31] focused on a MILP on mixed integer programming (MIP) were considered to model to integrate reverse logistics activities into the for- determine maximal profit, optimal number and capacity of ward supply chain. To deal with uncertainty, they pre- facilities as well as the optimal flow between them. sented a scenario-based stochastic optimization model to Although typically larger models are required to represent 123 22 Page 4 of 12 Logist. Res. (2016) 9:22 real supply chains, researchers developed many heuristics customers C in forward flow, as well as collection/inspection [7, 15, 34, 37] and metaheuristics such as genetic algorithm centers Co and disposal centers Di in reverse flow, cf. Fig. 2 [4, 6, 20, 38–42], simulated annealing [43–45], tabu search for a schematic sketch.We like to point out that in accordance [29, 46], memetic algorithm [32, 47] and scatter search to Fig. 1, we consider a hybrid manufacturing-recovery-re- [34] to solve these NP hard problems. However, there is cycling facility as well as a hybrid collection-inspection still a critical need in this area to increase the efficiency of facility. Establishing several facilities at the same location solution approaches, especially, when the complexity of can decrease the price in comparison with separating design. the model increases (Melo et al. [48]). To adapt problem (1), we impose the following The problem addressed in this work includes integrated assumptions: design of forward and reverse logistics as well as flexibility • The set of nodes is given by N ¼ S [ P [ Dc [ in delivery paths for a seven-stage closed-loop supply chain R [ C [ Co [ Di. network. The proposed model as a complete and general • There are no edges between facilities of the same stage, network covers the previously described cases in the litera- the delivery graph is complete and the return graph is ture with less complexity. Additionally, the full delivery simple, i.e., E ¼ðS  PÞ[ðP  DcÞ[ðP  RÞ[ðP graph allows us to solve the conflicting goals profit and CÞ[ðDc  RÞ[ðDc  CÞ[ðR  CÞ[ðC  CoÞ[ responsiveness, which otherwise may lead to greater cost ðCo  DiÞ[ðCo  PÞ. [32]. From the computational point of view, we incorporate • The demands of each customer are deterministic and the graph structure in the chromosome representation, must be satisfied. thereby avoiding different model and solution methodolo- • The number of facilities per stage and respective gies as, e.g., considered by Pishvaee and Rabbani [36]. capacities are limited. • All cost parameters (fixed and variables) are known in advance. 3 Description for integrated forward/reverse • The transportation rates are perfect and there are no logistics network return storages. Moreover, the return rate p as well as the disposal disposal recovery and disposal rates p and ð1  p Þ To support the presentation of the proposed mathematical j j model, we consider the general model area of our problem. are fixed. All returned products from each customer must be collected. To this end, we consider G ¼ðN; EÞ to be a digraph where N denotes the set of all nodes and E the set of all edges in • The inspection cost per item for the returned products are included in the collection cost. the closed-loop network. The cost for node i 2 N are denoted by c , and the unit transportation cost on edge • The un-recyclable returned products will be sent to the disposal center. The remaining products are returned to ði; jÞ2 E are given by c . The respective decision variables ij the same plant. y 2f0; 1g and x 2 N represent whether a stage i 2 N is i ij 0 • The required recycled materials are assumed to be of used and which quantity is transported between node i and the same quality as the raw materials bought from j. To determine the optimal distribution network and suppliers and any plant chooses the raw material from capacity of each node, we minimize the transportation and the collection/inspection center over suppliers. operation cost of the proposed network, which reveals the • Customers have no special preference. It means, price following mixed integer minimization problem: X X is the same in all facilities. min c x þ c y ij ij i i x ;y ij i The objective of this model is to minimize the total cost of i2N ði;jÞ2E the proposed supply chain, which is composed of fixed ð1Þ s:t: a x 6 b y i ij i i costs for facilities and variable costs for transportation. In ði;jÞ2E terms of the above notation, the cost function, the sign and x  0; y 2f0; 1g ij i the integer conditions remain identical. The constraints in Next, we specialize this model to reflect the IFRLN properties. 3.1 Mathematical formulation and assumptions The previously described IFRLN setting represents an inte- grated supply chain with seven echelons consisting of sup- pliers S, plants P, distribution centers Dc, retailers R and Fig. 2 Underlying structure of MILP 123 Logist. Res. (2016) 9:22 Page 5 of 12 22 (1) are specialized and we have that the capacities in each to a specific problem is to decide how to design a chromo- node induce the inequalities some. The tree-based representation is known to be one way for representing network problems. Different methods have b 8i 2 S ij been developed to encode trees. One of them is matrix-en- ð2Þ b y 8i 2 N nfS [ Cg: i i ði;jÞ2E coding, was is developed by Michalewicz [51]. In this method, the solution is presented by a jKj jJj matrix where return Additionally, by assumption only a fraction p is |K| and |J| are the number of sources and depots, respectively. disposal returned by customers and a fraction p of the returned Although this solution approach has a simple representation, products has to be disposed off. Apart from these excep- applying this method requires the development of a special tions, the supply chain network is subject to the law of the crossover and mutation operator for obtaining a feasible flow conservation, i.e., in-flow and out-flow in each node solution as well as huge amount of memory. Another tree- must be identical for these nodes. These conditions reveal based representation is the Pru¨fer number. The use of the x 8j 2 N nfC [ Cog ij Pru¨fer number representation for solving various network ði;jÞ2E problems was introduced by Gen and Cheng [52]. It requires > P return p x 8j 2 C > ij j an array of the length jKjþjJj 2 with |K| sources and |J| X ði;jÞ2E depots. Since this method may compute infeasible solutions x ¼ P ð3Þ jk disposal > p x 8j 2 Co > ij [39], a repair mechanism has been developed. In this regard, ðj;kÞ2E ði;jÞ2CoDi Jo et al. [39] presented the procedure for repairing infeasible disposal > ð1  p Þ x 8j 2 Co ij chromosomes. Later, Gen et al. [53] introduced determinant ði;jÞ2CoP encoding using priority which does not need any repair mechanism to guarantee feasibility of solutions. In this Last, the demands of customers must be satisfied. approach, solutions are encoded as arrays of size jKjþjJj,in x ¼ b 8j 2 C ij j which the position of each cell represents the sources and ð4Þ ði;jÞ2E depots and the value in cells represent the priorities. From the literature [41], we have found that both Pru¨fer and determinant encoding are efficient for the encoding of the spanning tree problem. However, as the determinant 4 Solution approach encoding overcomes the bottlenecks of Pru¨fer encoding [54], we utilize determinant encoding in our study. In the Because of our IFRLN model is a capacitated allocation and following encoding and decoding are discussed. multi-choice problem, it is known as a NP hard problem [6, 38, 39, 49]. Hence, although the problem can be refor- 4.1.1 Random path-based direct encoding method mulated into an integer linear programming, we cannot compute a suitable solution for large-scaled problems within The delivery and recovery path can be conventionally a short time. There are three main options to tackle NP hard determined by applying the random path direct encoding problems: probabilistic algorithms, approximation algo- method introduced by Lin et al. [55]. Using this method rithms and metaheuristic algorithms. To reduce the search computation time can be greatly cut down. One gene in a space and increase the solution quality, we consider the class chromosome is characterized by two factors: locus, the of metaheuristic algorithm to solve this model. According to position of the gene within the structure of chromosome, [50], memetic algorithms are appropriate for the proposed and allele, the value the gene takes. In this method, each model. The basic feature of MA is a multi-directional and gene is initialised with a random value from its domain and global search by generating a population of solutions as well it contains M groups where M is the total number of cus- as local search to improve intensification of the search. tomers. Each group represents a delivery path in forward According to the reviewed literature, two major issues affect flow as well as recovery path in reverse flow. Due to the performance of memetic algorithm [41], i.e., the chro- existence of three different delivery paths in the proposed mosome representation and the memetic operators. problem, we extend the random path-based direct encoding method by adding a second segment into the chromosome. 4.1 Chromosome representation 4.1.2 Extended random path-based direct encoding A chromosome must have the necessary gene information for solving the problem. Selecting a proper chromosome Although applying the new delivery paths improves the representation highly affects the performance of meta- flexibility and efficiency of the supply chain network, it heuristic algorithm. Therefore, the first step of applying MA makes the problem more complex. In Fig. 3 the 123 22 Page 6 of 12 Logist. Res. (2016) 9:22 Fig. 3 Representation of extended random path-based direct encoding method representation of the extended random path-based direct lowest. The procedure of encoding by extended random encoding method in two segments is shown. The first seg- path-based direct encoding is shown in Algorithm 1 below. ment is encoded by using random path-based direct encoding method which shows the delivery path for each customer. Algorithm 1 The pseudocode procedure of initializa- The second segment of a chromosome contains two parts: the tion by extended random path-based direct encoding first part with J locus including the guide information Input: Number of customers M regarding plant assignments in the network, and the second Number of collection/inspection centers N part of length K containing the information of the distribution Number of disposal centers O centers. As shown in Fig. 3, the length of chromosome is Number of plants J ð7  MÞþ J þ K where M, J and K are the total number of Number of retailers L customers, plants and distribution centers, respectively. Number of distribution centers K Each sequence of seven subsequent genes forms a group. Number of suppliers I Each group encodes four potential delivery paths through Step 1:  (first segment) plant, distribution center and retailer to customer as well as a 1: for i =0: M − 1 do recovery path from customer through collection/inspection 2: ch [7 ∗ i +1] ← random(1,N ) to disposal center or plant. The first three alleles of a group 3: ch [7 ∗ i +2] ← random(1,O) represent the reverse flow of the network, while the next four 4: ch [7 ∗ i +3] ← random(1,J ) alleles of that group show the forward flow from supplier to 5: ch [7 ∗ i +4] ← random(1,L) customers. As an illustration, a randomly assigned ID to 6: ch [7 ∗ i +5] ← random(1,K) these facilities in the reverse and forward flow is shown in 7: ch [7 ∗ i +6] ← ch [7 ∗ i +3] k k Fig. 3. Each locus in the second part is assigned an integer in 8: ch [7 ∗ i +7] ← random(1,I) the set f0; 2g for plants due to existence of three delivery 9: end for options for each plant in the network. Regarding distribution Step 2:  (second segment, plant delivery path) center, an integer from f0; 1g is chosen to represent the two 10: for i =0: J − 1 do respective delivery options. The second segment is involved 11: ch [7 ∗ M + i] ← random(0, 2) by determining the sort of delivery path for the selected plant 12: end for as well as distribution center in first segment. Step 3:  (second segment, Dc delivery path) It should be noted that applying this encoding approach 13: for i =0: K − 1 do 14: ch [7 ∗ M + J + i] ← random(0, 1) might generate infeasible solutions, which violate the 15: end for facility capacity constraint; hence, a repairing procedure is Output: Chromosome ch [·] needed. If the total demand of a depot from a source k exceeds its capacity, the depot will be assigned to another Remark 1 According to the assumption presented in source with sufficient product supply so that the trans- Sect. 3, returned products have to be directed to the portation cost between that source and the depot is the 123 Logist. Res. (2016) 9:22 Page 7 of 12 22 original plant. To follow this limitation, the third and sixth skipping distribution centers, path number 2 is selected. position of first segment of the chromosome representation Similarly, path number 3 is chosen if retailers are skipped. for any customer should be identical. Last, if direct shipment is selected, the delivery path number 4 will be implemented. Remark 2 Since, the third and sixth position of first An important difference between the traditional random segment are identical, 6 has been considered as the number path-based direct decoding method and the method adopted of each unit, instead of 7, to apply the chromosome in this paper is that we include the delivery path informa- representation. tion of the second segment. The detailed decoding proce- dure is shown in Fig. 5. Each locus in this segment is 4.1.3 Extended random path-based direct decoding assigned to an integer in the range of f0; 1; 2g for plants and f0; 1g for distribution centers. Here, we encode normal Decoding is the mapping from chromosomes to candidate delivery for plants and distribution centers by P ¼ 0 and solution to the problem. As an example, Fig. 4 represents Dc ¼ 0 respectively, where j and k denote the ID of the an instance of a delivery and recovery path in our model. plant and of the distribution center. Moreover, P ¼ 0 and In each gene unit, four delivery paths can be designed by Dc ¼ 1 as well as P ¼ 1 represent direct delivery and k j applying normal delivery, direct shipment and direct P ¼ 2 direct shipment. The paths displayed in Figure 5 delivery. All of them are from a neighborhood. For correspond to respective choices, i.e., we have instance, we can obtain the neighborhood given in Algo- Path1 () P ¼ 0; Dc ¼ 0 j k rithm 1 from the sample of gene unit shown in Fig. 4 that shows the delivery path to customer 2. Considering the Path2 () P ¼ 1; Dc 2f0; 1g j k second chromosome (customer 2) in Fig. 4 as an example, Path3 () P ¼ 0; Dc ¼ 1 j k we start by supplier 2 and continue via plant 4, distribution Path4 () P ¼ 2; Dc 2f0; 1g: j k center 1 and retailer 3 in forward flow as well as collection/ inspection center 3, disposal center 1 and plant 4 in the It should be noted that because the amount of returned reverse flow. Due to construction, four different delivery products shipped to each one should be known for paths are possible, cf. Figure 4. The delivery and recovery decoding the forward flow, decoding of the forward flow is path 1 occurs if normal delivery is chosen for all stages. By impossible until the reverse flow is decoded. Fig. 4 Delivery path for a sample of gene unit Fig. 5 Presentation of the second segment of the extended random path-based direct encoding 123 22 Page 8 of 12 Logist. Res. (2016) 9:22 4.2 Evaluation Algorithm 3 Pseudocode of the local search for the proposed model Fitter individuals have higher chances of survival. The Input: One parent A evaluation assigns a fitness value to each individual, Number of customers M thereby inducing a measure. In our study, we apply the cost Number of collection/inspection centers N function as the fitness value. This fitness value is computed Number of disposal centers O for the decoded chromosome to analyze the accuracy and Number of plants J efficiency of the proposed MA. Number of retailers L Number of distribution centers K Number of suppliers I 4.3 Crossover and Local Search 1: Randomly select position a in chromosome of A 2: b ← mod(a, 7) Crossover is known as the most important recombination I if b =0 of both parents’ feature to explore new solution within the N if b =1 search space. There are many variants of crossover oper- O if b =2 ations developed in the literature, cf. [10]. Based on the 3: X ← characteristics of the chromosome, we chose the two-cut ⎪ J if b ∈{3, 6} point crossover, which applies the steps shown in Algo- L if b =4 rithm 2. K if b =5 4: n ← round(random(30/100 ∗ X, 70/100 ∗ X)) 5: for i =1: n do Algorithm 2 Pseudocode of the two point 6: A ← A crossover for the proposed model 7: c ← random(1,X) Input: Two parents 8: A (a) ← c 1: Generate two random positions 9: Evaluate fitness function for A 2: Swap data beyond the two points between parents 10: end for Output: Two offsprings 11: Select best chromosome among n new instances Output: One offspring After crossover, the population is merged and sorted according to its fitness value. In the next step, a local 4.4 Selection method search technique is applied, i.e., if the fitness value of a new solution is better than the old one, the latter is We apply the well-known roulette wheel selection for replaced. The detailed procedure is shown in Algo- generating the next generation of chromosomes. The rithm 3. The chromosome showing the best fitness is strategy of roulette wheel is a probabilistic selection based selected. on fitness. 123 Logist. Res. (2016) 9:22 Page 9 of 12 22 4.5 Procedure of proposed memetic algorithm 5 Test problems and computational results Combing the aforementioned components, we obtain the To assess the accuracy and efficiency of the developed procedure displayed in Algorithm 4 for solving our prob- MA, we generated various test problems of different sizes lem IFRLN. to compare the results obtained by our MA from Algorithm 4 and a branch-and-bound algorithm from LINGO15. Since the logistics network framework in this study differs from Algorithm 4 Pseudocode of the proposed memetic previous studies, the size of test problems considered in algorithm this work is selected randomly as shown in Table 1. The Input: Number of population n first six test problems relatively small and the number of Number of crossover population m decisions variables are 128, 209, 234, 468, 1006 and 1780, 1: for k =1: n do respectively, and the remaining problems are large sized. 2: Encode ch [·] by Algorithm 1 Other parameters are generated randomly using uniform 3: Evaluate ch [·] according to fitness function distributions shown in Table 2. Each test problem has been 4: end for 5: Set i ← 0 solved 20 times to test the robustness of the method. 6: while termination condition not satisfied do Our implementation was written in MATLAB R2015b r TM 7: for k = n +1 : n + m do and run on the PC with Intel Core i5 2.40 GHz with 12 8: Select two parents via roulette wheel GB RAM. For our testing, we considered population size 9: Generate ch [·] by Crossover Algorithm 2 n ¼ 60 and crossover rate c ¼ 0:5. As a stopping criterion 10: end for for Algorithm 4, we imposed a maximum iteration number 11: Merge ch[·]= ch [·] k=1:n+m of 100 as well as a maximum number of iteration without 12: Evaluate and sort ch[·] by fitness value improvement 8, 10, 12, 20, 25 and 30 for our small size 13: Select first n elements of ch[·] problems, respectively. For the large size problems, we 14: Obtain ch[·] via Algorithm 3 with ch [·] increased the latter bound by 5 for each test problem. Also, 15: if fitness value of ch[·] is better than of ch [·] we set the number of local search iterations to be equal to then the number of retailers L for each test problem. 16: ch [·] ← ch[·] To evaluate the performance of proposed MA, firstly, 17: end if we employed LINGO15 to solve the optimization problem 18: i ← i +1 and obtained the results displayed in Table 3. Although 19: end while LINGO provides results for small size problems quickly, Output: Chromosome of optimal solution ch [·] Table 3 indicates that LINGO is unsuitable for solving the large size problems and it is run out of memory. According to Table 4, the proposed MA provides good Note that as we apply only one crossover and search step solutions for our small size problems, which allows us to before selecting the next generation, our method belongs to trust the method also for large size problems. To compare the class of steady state MA. Table 1 Settings of test Problem Supplier Plants Distribution Retailers Customers Col/Ins Disposal problems centers centers centers 12 2 5 8 2 2 1 22 3 8 9 3 3 2 3 2 4 6 10 2 2 1 42 4 10 16 4 4 2 53 6 15 24 6 6 2 64 8 20 32 8 8 4 7 6 12 30 48 12 12 6 8 8 16 40 64 16 16 8 912 24 40 96 24 24 12 10 16 32 40 128 32 32 16 11 24 48 60 192 48 48 24 12 32 64 80 256 64 64 32 123 22 Page 10 of 12 Logist. Res. (2016) 9:22 Table 2 Parameters values used in the test problems the optimal solutions obtained by LINGO with the results of our MA Algorithm 4, the percentage error is calculated Parameters Range using formula (5). b ; j 2 S Uniform (200, 1100) MA  LINGO answer answer Error percent ¼  100 ð5Þ b ; j 2 P Uniform (100, 1000) LINGO answer b ; j 2 Dc Uniform (50, 900) Based on Table 5, we observe that the error percentages b ; j 2 R Uniform (50, 850) for the small size problems are zero, which indicate the b ; j 2 D Uniform (100, 500) high accuracy of proposed MA. Although the operation b ; j 2 Co Uniform (20, 100) time is higher compared to LINGO, our implementation b ; j 2 Di Uniform (20, 100) allows us to derive results for the large size problems. return p 10% Hence, the proposed MA demonstrated that it is capable to disposal 50% prepare sufficiently accurate solution with the efficient c Uniform (1,3) ij computation time for our integrated forward/reverse c ; j 2 P Uniform (100, 2500) logistics problem with flexible delivery. c ; j 2 Dc Uniform (100, 2100) c ; j 2 R Uniform (100, 400) 6 Conclusion c ; j 2 Co Uniform (100, 500) c ; j 2 Di Uniform (50, 400) In this paper, we focused on a comprehensive mixed integer linear programming formulation for a seven-stage closed-loop network design problem. We applied the Table 3 Results obtained by LINGO extended direct delivery path representation-based meme- Problem Problem size Solution tic algorithm, which was developed for a full delivery graph and a combined forward/reverse logistics design to 12  2  5  8  2  2  1 2905 decrease delivery time and avoid sub-optimal solutions, 22  3  8  9  3  3  2 2335 respectively. The aim of this work is to minimize total cost, 32  4  6  10  2  2  1 2345 which we addressed as allocation problem to find the 42  4  10  16  4  4  2 1160 optimal number and capacity for any facility as well as the 53  6  15  24  6  6  2 4100 optimal transportation flow between facilities. Since the 64  8  20  32  8  8  4 11365 basic problem is NP hard, the combination with flexibility 76  12  30  48  12  12 6– in delivery path makes the search space of the problem 88  16  40  64  16  16 8– much larger and more complex and NP hard as well. 912  24  40  96  24  24  12 – Because existing methods are unable to solve this problem, 10 16  32  40  128  32  32  16 – we proposed a MA approach to compute a near optimal 11 24  48  60  192  48  48  24 – solution for large size problems. In this study, we intro- 12 32  64  80  256  64  64  32 – duced a new chromosome representation for MA to Table 4 Results for Algorithm Test problem Min-cost Max cost Ave cost Min time (s) Max time (s) Ave-time (s) 4 with n ¼ 60 and m ¼ 30 over 20 runs 1 2905 2905 2905 2.3 4.3 3.05 2 2335 2735 2402 4.3 10 6.7 3 2345 2885 2381 6.7 13 9.3 4 1160 1560 1225.5 18 47 32.5 5 4100 4920 4576 19 57 38.45 6 11,365 12415 11821 175 410 275.75 7 17,268 21205 19324 260 430 310.6 8 24,933 30446 26995 570 630 600.25 9 33,555 40043 36571.4 1780 2010 1903 10 51,343 60251 52692.95 3740 4060 3935 11 11,986 15600 13132.2 4100 5600 4700 12 13,400 15804 14227.7 6200 7500 6680 123 Logist. Res. (2016) 9:22 Page 11 of 12 22 Table 5 Comparison of results Problem LINGO MA Error percent from LINGO and proposed MA Min-cost Ave-time (s) Min-cost Ave-time (s) 1 2905 0.1 2905 3.05 0 2 2335 0.12 2335 6.7 0 3 2345 0.12 2345 9.3 0 4 1160 0.14 1160 32.5 0 5 4100 0.16 4100 38.45 0 6 11,365 0.17 11,365 275.75 0 7. Yeh W (2005) A hybrid heuristic algorithm for the multistage enhance its search ability for the proposed flexible model. supply chain network problem. Int J Adv Manuf Technol We verified correctness of the proposed method using 26:675–685 numerical experiments and LINGO15. Additionally, we 8. Syarif A, Yun YS, Gen M (2002) study on multi-stage logistic showed that the method is capable to solve larger size chain network: a apanning tree-based genetic algorithm approach. Comput Ind Eng 43(1):299–314 problems which cannot be solved by LINGO. 9. Tragantalerngsak S, Holt S, Ronnqvist J (2000) An exact method Apart from costs aspect considered here, other aims such for two-echelon, single-source, capacitated facility location as responsiveness and robustness can be considered in problem. Eur J Oper Res 123:473–489 designing integrated forward/reverse logistics network that 10. Gen M, Altiparmak F, Lin L (2006) A genetic algorithm for two- stage transportation problem using priority-based encoding. Oper needs updating the algorithm to be capable to solve multi- Res-Spektrum 28:337–354 objective models. Moreover, to be close to the real-world 11. Neto JQF, Walther G, Bloemhof J, Nunen AEEV, Spengler T application, multi-product multi-capacity and multi-period (2010) From closed-loop to sustainable supply cchain: the weee networks with uncertainty as well as considering inventory case. Int J Prod Res 48(15):4463–4481 12. Alumur SA, Nickel S, da Gamad FS, Verter V (2012) Multi- can be employed. Last, the effect of different parameters on period reverse logistics network design. Eur J Oper Res the behavior of the proposed metaheuristic algorithm can 220(1):67–78 be analyzed in depth. 13. Mutha SPA (2009) Perspectives in reverse logistics: a review. Conserv Recycl 53(4):175–182 Acknowledgements The authors would like to appreciate the support 14. Sarkis J, Darnal N, Nehman G, Priest J (1995) The role of supply International Graduate School (IGS) of Bremen University to support, chain management within the industrial ecosystem. In: The 1995 help and advice as well as the Deutscher Akademischer Austausch IEEE international symposium on electronics and the environ- Dienst (DAAD) for financial support of this research under the GSSP ment, Orlando, pp 229–234 programme of the IGS. 15. Jayaraman V, Patterson R, Rolland E (2003) The design of reverse distribution networks: model and solution procedure. Eur Open Access This article is distributed under the terms of the Creative J Oper Res 150:128–149 Commons Attribution 4.0 International License (http://creative 16. Min H, Ko CS, Ko HJ (2006) The spatial and temporal consol- commons.org/licenses/by/4.0/), which permits unrestricted use, distri- idation of returned pproduct in a closed-loop supply chain net- bution, and reproduction in any medium, provided you give appropriate work. Comput Ind Eng 51(2):309–320 credit to the original author(s) and the source, provide a link to the 17. Louwers D, Kip B, Peters E, Souren F, Flapper S (1999) A Creative Commons license, and indicate if changes were made. facility location allocation model for reusing carpet materials. Comput Ind Eng 36:855–869 18. Realff M, Ammons J, Newton D (2004) Robust reverse propro- References duct system design for carpet recycling. IIE Trans 36:767–776 19. Schultmann F, Engels B, Rentz O (2003) Closed-loop supply 1. Meade L, Sarkis J, Presley A (2007) The theory and practice of cchain for spent batteries. Interfaces 33:57–71 reverse logistics. Int J Logist Syst Manag 3(1):56–84 20. Kannan G, Pokharel S, Kumar P (2009) A hybrid approach using 2. Uster H, Easwaran G, Akali E, etinkaya S (2007) Benders ism and fuzzy tosis for the selection of reverse logistics provider. decomposition with alternative multiple cuts for a multi-product Resour Conser Recycl 54(1):28–36 closed-loop supply chain network design model. Naval Res 21. Figueiredo J, Mayerle S (2008) Designing minimum-cost recy- Logist (NRL) 54(8):890–907 cling collection networks with required throughput. Transp Res 3. Georgiadis M, Tsiakis P, Longinidis P, Sofioglou M (2011) Part E 44:731–752 Optimal design of supply chain networks under uncertain tran- 22. Pati R, Vrat P, Kumar P (2008) A goal programming model for sient demand variations. Omega 39(3):254–272 paper recycling system. Omega 36:405–417 4. Altiparmak F, Gen M, Lin L, Paksoy T (2006) A genetic algo- 23. Huysman S, Debaveya S, Schaubroeck T, Meester SD, Ardente F, rithm approach for multi-objective potimization of supply chain Mathieux F, Dewulf J (2015) The recyclability benefit rate of networks. Comput Ind Eng 51:197–216 closed-loop and open-loop systems: a case study on plastic 5. Beamon B (1999) Designing the green supply chain. Logistics recycling in flanders. Resour Conserv Recycl 101:53–60 12(4):332–342 24. Shen L, Worrel E, Patel M (2010) Open-loop recycling: a lca case 6. Govindan K, Soleimani H, Kannan D (2015) Reverse logistics study of pet bottle-to-fibre recycling. Resour Conser Recycl and closed supply chain: a comprehensive review to explore the 55(1):34–52 future. Eur J Oper Res 240(3):603–626 123 22 Page 12 of 12 Logist. Res. (2016) 9:22 25. Listes O, Dekker R (2005) A stochastic approach to a case study 41. Wang H, Hsu H (2010) A closed-loop logistic model with a for product recovery network design. Eur J Oper Res spanning- tree based genetic algorithm. Comput Oper Res 160:268–287 37:376–389 26. Krikke H, Harten V, Schuur A (1999) Reverse logistic network 42. Dengiz B, Altiparmak F, Smith AE (1997) Local search genetic redesign for copiers. OR Spektrum 21:381–409 algorithm for optimal design of reliable networks. IEEE Trans 27. Srivastava S (2008) Network design for reverse logistics. Omega Evol Comput 1(3):179–188 36:535–548 43. Krikpatrick S (1984) Optimizing by simulated annealing: quan- 28. Fleischmann M, Beullens P, Bloemhof-ruwaard JM, Wassenhove titative studies. J Stat Phys 34(5):975–986 L (2001) The impact of product recovery on logistics network 44. Pishvaee MS, Kianfar K, Karimi B (2010) Reverse logistics design. Prod Oper Manag 10:156–173 network design simulated annealing. Int J Adv Manuf Technol 29. Lee D, Dong M (2007) A heuristic approach to logistics network 47(1):269–281 design for end-of lease computer pproduct recovery. Transp Res 45. Jayaraman V, Ross A (2003) A simulated annealing mathodology Part E 44:455–474 to distribution network design and management. Eur J Oper Res 30. Lu Z, Bostel N (2007) A facility location model for logistics 144:629–645 systems including reverse flows: the case of remanufacturing 46. Sung C, Song S (2003) Integrated service network design for a activities. Comput Oper Res 34:299–323 cross-docking supply chain network. J Oper Res Soc 31. Pishvaee MS, Jolai F, Razmi J (2009) A stochastic optimization 54:1283–1295 model for integrated forward/reverse logistics network design. 47. Moscato P, Cotta C (2003) An introduction to memetic algo- J Manuf Syst 28:107–114 rithms. Inteligencia Artificial, Revista Iberoamericana de 32. Pishvaee MS, Farahani RZ, Dullaert W (2010) A memetic Inteligencia Artificial 19:131–148 algorithm for bi-objective integrated forward/reverse logistics 48. Melo M, Nickel S, da Gama FS (2009) Facility location and network design. Comput Oper Res 37:1100–1112 supply chain management: a review. Eur J Oper Res 33. Kilibi W, Martel A, Guitouni A (2010) The design of robust 196:401–412 value-creating supply chain networks: a critical review. Eur J 49. Eckert C, Gottlieb J (2002) Direct representation and variation Oper Res 203:283–293 operators for the fixed charge transportation problem, in parallel 34. Du F, Evans G (2008) A bi-objective reverse logistics network problem solving from nature. Springer, Berlin analysis for post-sale service. Comput Oper Res 35:2617–2634 50. Behmanesh E, Pannek J (2016) Modeling and random path-based 35. Lin L, Gen M, Wang X (2009) Integrated multistage logistics direct encoding for a closed loop supply chain model with flex- network design by using hybrid evolutionary algorithm. Com- ible delivery paths. In: The 7th IFAC conference on Management puters and Industrial Engineering 56:854–873 and Control of Production and Logistics, (Accepted) 36. Pishvaee MS, Rabbani M (2011) A graph theoretic-based heu- 51. Michalewicz Z, Vignaux G, Hobbs M (1991) A non-standard heuris algorithm for responsive supply chain network design with genetic algorithm for the transportation problem. ORSA J Com- direct and indirect shipment. Adv Eng Softw 42:57–63 put 3:307–316 37. Amiri A (2006) Designing a distribution network in a supply 52. Gen M, Cheng R (1997) Genetic algorithms and engineering chain system: formulated and efficient solution procedure. Eur J design. Wiley, New york Oper Res 171:567–576 53. Gen M, Cheng R, Lin L (2008) Network model and optimization: 38. Gottlieb J, Paulmann L (1998) Genetic algorithms for the fixed multiobjective genetic algorithm approach. Springer, Berlin charge transportation problem. In: Inn IC(Ed), IEEE World 54. Abuali F, Wainwright R, Schoenefeld D (1995) Determinant Congress on evolutionary computation (I. C. INN, ed.), factorization: a new encoding scheme for spanning trees applied pp 330–335 to the probabilistics minimum spanning tree problem. In Eshel- 39. Jo JB, Yinzhen L, Gen M (2007) Nonlinear fixed charge trans- man L (ed), Proceedings of the 6th international conference on portation problem by spanning tree-based genetic algorithm. genetic algorithms, pp 470–477 Comput Ind Eng 53:290–298 55. Gen M, Cheng R (2000) Genetic algorithms and engineering 40. Min H, Ko H (2008) The dynamic design of a reverse logistics optimization. Wiley, New York network from the perspective of third-party logistics service providers. Int J Prod Econ 113:176–192

Journal

Logistics ResearchSpringer Journals

Published: Nov 7, 2016

References