Computers & Operations Research

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1 Coputers & Operations Research 37 (2010) Contents lists available at ScienceDirect Coputers & Operations Research ournal hoepage: A closed-loop logistic odel with a spanning-tree based genetic algorith Hsiao-Fan Wang, Hsin-Wei Hsu Departent of Industrial Engineering and Engineering Manageent, National Tsing Hua University, R721, Engineering Building 1, 101, Section 2, Kuang-Fu Road, Hsinchu 300, Taiwan, ROC A R T I C L E I N F O A B S T R A C T Available online 11 June 2009 Keywords: Integer linear prograing Genetic algorith Closed-loop supply chain Logistics and location decisions Spanning-tree Due to the proble of global waring, the green supply chain anageent, in particular, closed-loop logistics, has drawn the attention of researchers. Although there were logistics odels that were exained in the literatures, ost of the were case based and not in a closed-loop. Therefore, they laced generality and could not serve the purposes of recycling, reuse and recovery required in a green supply chain. In this study, the integration of forward and reverse logistics was investigated, and a generalized closed-loop odel for the logistics planning was proposed by forulating a cyclic logistics networ proble into an integer linear prograing odel. Moreover, the decisions for selecting the places of anufactories, distribution centers, and disantlers with the respective operation units were supported with the iniu cost. A revised spanning-tree based genetic algorith was also developed by using deterinant encoding representation for solving this NP odel. Nuerical experients were presented, and the results showed that the proposed odel and algoriths were able to support the logistic decisions in a closed-loop supply chain efficiently and accurately. Stateent of scope and purposes This study concerns with operations of 3R in the green supply chain logistics and the location selection optiization. Based on `cradle to cradle' principle of a green product, a closed-loop structure of a networ was proposed in order to integrate the environental issues into a traditional logistic syste. Due to NP-hard nature of the odel, a Genetic Algorith, which is based on spanning tree structure was developed. Test probles fro the sall size for accuracy to the large scale for efficiency have been deonstrated with coparison. The proising results have shown the applicability of the proposed odel with the solution procedure Elsevier Ltd. All rights reserved. 1. Introduction Due to the awareness of the environental protection, how to reduce the utilization of the aterials by reusing and reanufacturing the used products has been a critical issue for an enterprise. This induces the concept of the green supply chain anageent and has led to a proble of the closed-loop supply chain anageent. Different fro a conventional supply chain, planning a green supply chain requires an additional function of recycling and thus, a closed-loop chain is a necessary infrastructure for a aterial flow. Then, with well-anaged reverse logistics, 3R of reduce, recovery and reuse for the environental protection can be achieved with cost savings in the procureent, the disposal and the transportation [19]. Technically, the closed-loop logistics coprises two parts: forward logistics and reverse logistics. For the forward logistics, as a conventional logistics, after anufactory, the distributors will Corresponding author. Tel.: ; fax: E-ail address: hfwang@ie.nthu.edu.tw (H.-F. Wang). deliver the final products to the custoers to satisfy their deands and the position of the custoers is typically the end of the process. For the reverse logistics, the flow of used products is processed fro the custoers bac to the disantlers to do the sorting or disassebling for recovery, reuse or disposal [3,15,23,26,29]. The closed-loop logistics anageent is to ensure the least waste of the aterials by following the Conservation Law along the life cycles of the aterials. Generally speaing, for a networ planning proble, there are three issues needed to be considered: the validity of the odel, the efficiency of the solution and the applicability. For a closed-loop logistics proble, the status quo of these issues can be suarized as follows: (1) For odeling: Most of the studies only discussed the reverse logistics odel. Soe studies have proposed the closed-loop odels, but the loops were considered as a prolonged supply chain by lacing of the relation between forward and reverse flows [9,28,31]. Therefore, instead of sharing the sae capacity, the odels often assued the unliited capacities for the reverse logistics or did not state the relation between forward /$ - see front atter 2009 Elsevier Ltd. All rights reserved. doi: /.cor

2 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) and reverse flows, which are not valid for representing the real situations. (2) For applications: Because of the assued unliited capacities for the facilities of distribution centers or disantlers in the reverse logistics, it has resulted in unrealistic route design for the used aterials fro the custoers for recovery, reused or disposal [25]. (3) For solutions: A logistics planning proble with the location selection is an NP-hard proble [8,14,18,30]. Efficient solution procedure reains a challenge for researchers and practitioners. To cope with these issues, in this study, we shall investigate the features and purposes of a green logistics anageent and in particular, the difference of a prolonged supply chain and a closedloop supply chain for green products. Based on the findings, we shall develop a coprehensive green logistics odel to support decisions of facility locations and aterial flows through a closed-loop capacitated supply chain when the realistic applications of 3R green aterials logistics are expected with iniu total cost. Furtherore, an efficient algorith will be developed and evaluated. After the literature review of closed-loop logistics and Genetic Algoriths in Section 2, a atheatical odel for closed-loop logistics will be proposed with the specification of its properties in Section 3. Estiation of its coplexity will be done with nuerical illustrations. In Sections 4 and 5, a revised spanning-tree based genetic algorith is proposed to resolve this odel, which is evaluated using large-scale probles. Finally, in Section 6, the conclusion will be drawn. 2. Literatures review In this section, we probe the literature and categorize studies into two. The first one is the closed-loop logistics, and the second is the Genetic Algoriths used in a logistics networ Closed-loop logistics Closed-loop logistics refers to all those activities associated with the transforation and the flows of goods and services with their inforation fro the sources of the aterials to the end users. Manageent refers to the integration and treatent of these activities, both internal and external of a fir [4]. Therefore, an integrated supply chain anageent ais to close aterial cycles and prevent the leaage of the aterials fro the chain using the inial costs to achieve axial value [6]. Fleischann et al. [10] pointed out that a closed-loop supply chain ay include traditional anufacturers, retailers, with logistics service providers in the forward channel; and specialized parties of secondary aterial dealers and aterial recovery facilities in the bacward channel. One of the ey aspects in the closedloop anageent is the siultaneous iproveent of both econoic and environental perforance throughout the chain by establishing long-ter relationships between buyers and suppliers [35]. Therefore, building a stable closed-loop logistics in a chain is necessary when the obectives of iniizing both the total cost and the involved environental ipacts are desired by the copanies. Therefore, for a closed-loop logistics proble, in addition to the conventional logistics which is described by a networ of suppliers, anufacturing sites, distribution centers (DCs) and custoer locations through logistics, an iportant odule, disantlers or recyclers, is incorporated into a supply chain networ. These disantlers handle the recovered resources into any different types for further use or disposal [3]. That is, if the recycled resources can be used again at disantler sites, the resource should be shipped to a anufacture for reproduction; otherwise, the un-useful resources ust be land filled [29]. This behavior in a closed-loop syste is regarded as reverse logistics. In reality, a distribution center often plays such role as a collector in a recovery syste. Therefore, in this study, depending on the needs, a DC in a closed-loop logistics networ can act as a distributor only or both distributor and collector. The fraewor of closed-loop logistics can be viewed as Fig. 1. The integration of the forward and reverse logistics odel is called the closed-loop logistics odel (CLL). Few studies have considered this issue. Fleischann et al. [9] designed a reversed logistics networ by considering the forward flow together with the reverse flow which has no capacity liit. Extending Fleischann et al.'s odel, Salea et al. [28] proposed a general odel that has been applied to an Iberian copany. However, when suspending the logistics between disantlers and plants, both Fleischann et al. and Salea et al.'s odels did not consider the supplier side and laced the relations between forward and reverse flows. Soeties, the DCs also play the role of the collected centers. Thus, the capacity of DC is used for both distribution and collection. When the aounts of the collection are larger, then the aounts of the distribution ust decrease under the sae capacity. Siilarly, if we consider the supply side, the plants ust allow the aterials flow fro both forward (suppliers) and reverse (disantlers) under the sae capacity. If the aounts of returns are larger in a certain plant, the aounts of orders fro suppliers will decrease. These interactions are the character of a closed-loop supply chain and the odel cannot be separated into two parts independently. Without considering such ind of relations, the odel is siply a prolonged supply chain including forward and reverse chains but not a loop. recycling Suppliers shipping Manufactures shipping DCs Forward : shipping Reverse : recover Custoers shipping Disantlers (recyclers) shipping landfilling Fig. 1. Fraewor of green supply chain logistics.

3 378 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) In addition, Salea et al.'s odel left the unsatisfied deand and did not consider the relations between deands, recovery aounts, and landfilling aounts, which is undesirable and unrealistic in anageent. Üster et al. [31] considered a closed-loop networ design proble, and solved by Benders decoposition. Üster et al. separated anufacturing fro reanufacturing and assued a single source for the custoer supplying. Treating the odel as the proble of the siple assignent has prevented it fro general applications. The realistic logistics solution should provide the best cobination of routings, and each node should be able to connect all nodes in the next stage, but not exclusively assigned. The proble with the location selection of conventional logistics planning has been discussed extensively, and this ind of proble is an NP-hard proble [11,30]. Therefore, to overcoe the shortages of the existing odels and tacle the features of closed-loop logistics, how to develop an efficient solution procedure is another ai of our study Genetic algoriths in logistic networs Genetic Algorith is a coonly used optiizing tool for engineering calculation. It was proposed by Professor John Holland of the University of Michigan in Dengiz et al. [7] offered any exaples on GA, which showed that it can be applied to a wide variety of applicative doains. For the fundaents of GA, one can refer to Gen and Cheng [11]. In the reverse logistics, Min et al. [24] also successfully used GA to develop a ulti-echelon reverse logistics networ for product returns. The concept of applying a spanning tree to supply chain networ probles was first proposed by Syarif et al. [30]. To utilize the characteristic of a spanning tree to set up the code of a genetic algorith, Syarif et al.'s ethod was able to deterine the locations ofanufacturingsitesanddcs.thisindofprobleisaindof fixed charge transportation proble (FCTP), and both Jo et al. [18] and Gottlieb and Paulann [14] successfully adopted spanning treebased GA with Prüfer encoding to solve FCTP. In the literature, various encoding ethods have been used to represent trees, and they can be classified broadly into three categories: edge, node, and edge-node encodings. Edge encoding has been found to be a poor representation, while in node- or vertexbased encoding, the nodes rather than the edges are represented in the encoding [5]. A popular encoding ethod for trees called Prüfer encoding is based on the Prüfer nuber [16], which represents a tree of n nodes with n 2 digits, where each digit is an integer between 1 and n. However, epirical investigations have shown that Prüfer encoding is a poor ethod in evolutionary algoriths and thus should be avoided [13,20,34]. Abuali et al. [1] based on nodes to offer Deterinant Encoding, and they proved that the code is better than Prüfer's. Both the wors of Chou et al. [5] and Yao and Hsu [33] for a logistic networ further confired this conclusion. There are two issues that are needed to be addressed for the initialization of a GA: the population size and the procedure to initialize the population. For the population size, Goldberg [12] has shown the need of increasing population exponentially with the length of the chroosoe string in order to generate good solutions. Regarding the generation of the initial population, there were two ways proposed in the literature: rando initialization and heuristic type initialization. Doris et al. [27] used alternative approach to generating initial solutions in place of rando ethod and got a better result. Baer and Ayechew [2] copared rando, heuristic type, and ixed population containing both rando and heuristic initial solutions, and concluded that an initial population of heuristic solutions will lead to high-quality solutions in a relatively sall nuber of generations of the GA. However, a possible drawbac is that such a population will lac of diversity needed to obtain near-optial solutions Conclusion Reviewing the above-entioned literature on closed-loop logistics, it is noted that ost of the existing odels lac of the relations between forward and reverse logistics ebedded in a closed-loop syste. Since 3R of recovery, recycle and reuse are activities to fulfill the basic principle of cradle to cradle in closed-loop anageent, therefore, Conservation Law should be obeyed at each node or state and will be satisfied in our proposed closed-loop logistics odel. In addition, the closed-loop logistics is an NP-hard proble. Thus an efficient algorith is essential for enterprises. Fro the literatures, we can note that GAs have been successfully applied to a wide range of doains including reverse logistics. Aong varieties of versions, spanning-tree based GAs with Deterinant Encoding technique has shown their potential in solving forward logistics with good outcoes. Therefore, we shall adopt such technique to develop a GA to solve the proposed closed-loop logistic odel. 3. A atheatical prograing odel for closed-loop supply chain logistics Fro the concepts we described above, we now that the closedloop supply chain is different fro a conventional supply chain. The probles involved are ore coplex, and need ore than double efforts to analyze both forward and reverse logistics siultaneously. To easure the effectiveness of the logistics in a closed-loop networ, the cost is norally considered by a copany. Besides, in a ultistage supply chain networ proble, the following conditions should be satisfied in odeling [17,30,33,34]. (a) The deand of each custoer ust be satisfied. (b) The flow is only allowed to be transferred between two consecutive stages. (c) The nuber of facilities that can be opened and their capacities are both liited. Because they are also the basic conditions for closed-loop logistics, we shall consider the as our assuptions in odeling. Note that there are essentially five stages along a green logistic networ: suppliers, anufacturers, DCs, custoers, and disantlers. Apart fro the coon conditions of the satisfied deand in (a), and liited capacities in (c); fro Assuption (b), it can be noted that there are no flows between the facilities at the sae stage. One special issue of closed-loop logistics is the recycling rate, including the recovery and landfilling rates. Laan et al. [21] pointed out that in the recovery systes; a coon assuption is that the aounts of the returned products depend on the deand of the products. To adopt this assuption, the recovery aount is assued to be a percentage of the custoer deand in our odel. Then, this leads to the fourth assuption as (d) The recovery and landfilling rates are given. The goal of this study is to design a closed-loop supply chain logistics syste that can iniize the total transportation and the operation costs by deterining locations of the facilities and the flows of the operation units along each capacity-constrained stage when the deand of custoers and the recycling rates are given. This closed-loop syste is eant to support long-ter steady-state

4 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) logistics decisions. Therefore, fro the econoic point of view, we can suggest the inial cost flows and opening facilities in the syste The proposed closed-loop logistics odel Consider the integer-valued basic logistics units in our syste, in this section, based on four assuptions and the networ structure; we shall propose a atheatical odel to describe such logistic syste. Before odeling, we define the related paraeters and notations as below: Indices I J K L M Paraeters a i b Sc pd pc l pl d l e s i t u l v w Ru l f g h φ Variables the nuber of suppliers with i = 1, 2,..., I the nuber of anufactories with = 1, 2,..., J the nuber of DCs with = 1, 2,..., K the nuber of custoers with l = 1, 2,..., L the nuber of disantlers with = 1, 2,..., M capacity of supplier i capacity of anufactory total capacity of forward and reverse logistics in the DC the percentage of total capacity for reverse logistics in DC recovery percentage of custoer l the landfilling rate of disantler deand of the custoer l capacity of disantler unit cost of production in anufactory using aterials fro supplier i unit cost of transportation fro each anufactory to each DC unit cost of transportation fro DC to custoer l Unit cost of transportation fro DC to disantler unit cost of transportation fro disantler to anufactory unit cost of recovery in DC fro custoer l fixed cost for operating anufactory fixed cost for operating DC fixed cost for operating disantler fixed cost for landfilling per unit x i quantity produced at anufactory using raw aterials fro supply i y aount shipped fro anufactory to DC z l aount shipped fro DC to custoer l o aount shipped fro DC to disantler Rd aount shipped fro disantler to anufactory Rz l quantity recovered at DC fro custoer l { 1 if production taes place at anufactory α = 0 otherwise { 1 if DC is opened β = 0 otherwise { 1 if disantler is opened δ = 0 otherwise Because the recovery and landfilling rates are the estiated proportional values of the deand and the recovery aount, they are non-integral. Siilarly, the percentage of the capacity for the reverse logistics in DC is also estiated proportionally with real nubers. Therefore, in order to aintain integral properties, Gauss sybol is used in our atheatical odel and the odel is shown below with TC as the total cost: Obect function: in TC = s i x i + t y + u l z l + v o i l + w Rd + Ru l Rz l + f i α l + g β + h δ + φ pl o (1) Subect to x i a i, i (2) y b α, (3) x i + Rd = y, i (4) z l + o Sc β, l (5) y = z l, l (6) o pd Sc β, : floor for Gauss sybol (7) Rz l = o, l (8) Rz l pc l z l, l : ceiling for Gauss sybol (9) z l d l, l (10) Rd + pl o e δ, : floor for Gauss sybol (11) o = Rd + pl o, : floor for Gauss sybol (12) α, β, δ {0, 1},,, (13) x i, y, z l, o, Rd, Rz l N {0} i,,, l, (14) The obective is to iniize the total cost of the transportation and the operations, and the obective function (1) represents this goal. The constraints ainly contain two types: one is for liited capacities and the other is for the law of the flow conservation. Constraints (2) and (3) represent the liit of the capacity for suppliers and anufactories in forward logistics. Constraint (5) shows that the total flows of forward and bacward cannot exceed the total capacity of DC. Constraints (7) and (11) ean the reverse liit of the

5 380 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) capacity for DCs and disantlers. Constraint (9) describes the custoer recovery relationship with the recovery rate. Constraints (4), (6), (8) and (12) satisfy the law of the flow conservation by in-flow equal to out-flow. Constraint (10) is to satisfy the custoer deand. Constraint (13) denotes the binary variables, and Constraint (14) is the non-negative, integral condition in our odel. The paraeter of pd is used to describe the role of DC.Ifpd =0, DC has a sole duty for distribution in the forward logistics; and when pd = 1, DC ay play a singular role for Collection Center. If pd falls in between zero and one, it eans DC not only can be a distribution center, but also a collection center. The concept is siilar for Manufactories. If a Manufactory accepts the resources fro the disantlers for reuses, it acts as both Manufacture and Reanufacture; otherwise it only uses raw aterials for anufactory. These concepts have been presented in our general closed-loop logistics odel. Because the variables denote the basic units of logistics, apart fro 0 1 decision variables, all other variables are all integers and thus Gauss' sybols are introduced in the odel. The floor and ceiling of Gaussian are defined below: Definition 3.1. The floor or the ceiling of a real nuber x is an integer denoted by x and x, respectively, and defined, respectively, as below: x ={x x x, x y, y, x I} x ={x x x, x y, y, x I} Because of Gaussian, the odel will be further transfored into a linear odel The transfored integer linear prograing odel To transfor into a coputable odel, we propose the following process to transfor two rates into linear fors as below: Let us consider the Constraints (7) and (9) first. Since pd Sc β and pc l z l are the oint given input values of all paraeters: pd, pc l,andsc ;and z l equals to the deand of custoer l, because custoer deand ust be satisfied by Constraint (10) and the optial solution exists if and only if Constraint (10) equals to the lower bound, therefore, by rewriting SP = pd Sc and ZP l = pc l z l, Constraints (7), (9), and (10) can be transfored, respectively, into (7a), (9a) and (10a) as below: o SP β, (7a) Rz l ZP l, l (9a) z l = d l, l (10a) As regards Constraints (11) and (12) which are different fro the situation above with decision value, o are not nown in advance, and also pl are the paraeters related to the estiated landfilling aounts. To transfor these two constraints with Gauss's sybol, three additional inequalities are needed as defined below. OP pl o, (15) OP pl o ε, where ε 1 (16) OP N {0} (17) Rd + OP e δ, (11a) o = Rd + OP, (12a) where a very sall real nuber ε > 0 is given to ensure inequality holds for integer solutions. Then, the original odel with Gauss's sybol can be transfored into an integer linear progra (ILP) which is suarized as below: Closed-loop logistics odel (CLL odel): in TC = s i x i + t y + u l z l i l + v o + w Rd + Ru l Rz l + f i α + g β l + h δ + φ OP (1) Subect to x i a i, i (2) y b α, (3) x i + Rd = y, i (4) z l + o Sc β, l (5) y = z l, l (6) o SP β, (7a) Rz l = o, l (8) Rz l ZP l, l (9a) z l = d l, l (10a) OP pl o, (15) OP pl o ε (16) Rd + OP e δ, (11a) o = Rd + OP, (12a) α, β, δ {0, 1},,, (13) x i, y, z l, o, Rd, Rz l, OP N {0}, (14) ε 1 i,,, l, (17)

6 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) Table 1 The size and estiated constants of the exaple. Suppliers Manufactories DCs Custoers Disantlers pd (%) pc l (%) pl (%) φ Table 2 Capacity, deand (in unit) and fixed cost (US$). Supplier Manufactory DC Custoer Disantler Capacity Capacity Fixed cost Capacity Fixed cost Deand Capacity Fixed cost Table 3 Unit shipping cost for each stage (US$). Supplier Manufactory Manufactory DC Table 4 The optial solution of nuerical exaple. Obective value x i x 13 = 40 x 15 = 460 x 22 = 415 x 23 = 60 x 33 = 390 y y 22 = 550 y 31 = 490 y 51 = 319 y 52 = 141 z l z 12 = 109 z 13 = 400 z 14 = 300 z 21 = z 22 = 191 o o 11 = 61 o 21 = 89 Rd Rd 12 = 135 Rz l Rz 11 = 50 Rz 22 = 30 Rz 32 = 40 Rz 41 = 11 Rz 42 = 19 α α 2 = 1 α 3 = 1 α 5 = 1 β β 1 = 1 β 2 = 1 δ δ 1 = 1 OP OP 1 = 15 DC Custoer An illustrative exaple Custoer DC DC Disantler Disantler Manufactory In this closed-loop logistics odel, there are (I+2J+4K+2L+4M) constraints, and (I J+J K+K L+L K+K M+M J+J+K+2M) variables including (J+K+M) binary variables, in which additional 2M variables and M constraints are derived fro transforation. With this structure, the nuber of variables is always ore than that of constraints and thus the odel is always feasible. In this paragraph, we shall use a sall exaple to illustrate the properties of the proble and the odel. Tables 1 3 are the given data. The exaple contains 3 suppliers, 5 anufactories, 3 distribution centers, 4 custoers and 2 disantlers. Five types of roles are involved with the respective nubers (recovery, landfilling, and percentage of capacity for reverse in DC) as shown in Table 1, and three rates are assued to be equal with respect to each custoer l, disantler, and DC, respectively. Tables 2 and 3 list all the unit costs of operation and transportation, respectively. In this exaple, with I = 3, J = 5, K = 3, L = 4andM = 2, there are 41 constraints, and 82 variables. Using both LINGO 8.0 and ILOG- CPLEX 7.0 with at ost 1(s) elapsed tie, we obtained the optial solution as shown in Table 4 and Fig. 2: Fro this nuerical exaple, it can be seen that with the conservation law, all logistic units were reserved in the syste of the forward and bacward flows. Also, fro this solution, it can be seen that only three anufactory sites out of five, two distribution centers out of three, and one disantler out of two are needed to eet the overall deands of four custoers and recycling. Thus, the odel is able to serve optial green supply chain anageent fro econoic viewpoint. In the closed-loop logistics planning proble, the recovery and landfilling rates are the ost critical yet uncertain factors. In this nuerical exaple, they are assued to be 10%. Hsu and Wang [17] have carried out paraeter analysis to obtain the ranges of landfilling and recovery rates with the sae optial logistics pattern. Also, they pointed out that because of the recovery rate is ore sensitive than

7 382 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) Forward flows Reverse flows 50 C1 M1 400 DC C S1 S M2 550 M DC C3 400 S M DC3 600 C4 300 M5 d1 540 d2 380 Fig. 2. Optial distribution pattern of the illustrative exaple. the landfilling rate, therefore, with a saller tolerance range, any change of recovery rate should be given ore attention in control and anageent. In the following illustrations, reference to the superscript nubers in Fig. 3 (lie a,b ) will be followed Revised deterinant encoding 4. Revised spanning-based genetic algorith Since the CLL odel for the green logistics proble is a capacitated location-allocation proble; and also can be viewed as a ultiple-choice Knapsac proble, it is nown to be NP-hard [8,11,14,18]. Furtherore, the odel is neither total uniodular [32], nor decoposable, therefore, an efficient algorith should be developed to solve this odel, which is the ai of this section. Fro an enterprise' viewpoint, the logistics anageent should ai at iniizing cost or axiizing profit. To achieve this purpose, the strategy is to choose the right nuber and right locations of the facilities (e.g., anufactories, DCs, and disantlers) to open. Although this ind of proble can be forulated into an integer linear progra, it cannot provide a good solution for large-scaled probles within a short tie. This becoes severe when the software progras use the Siplex-based algorith with exponential-tie coplexity. Based on the reviewed literature, we can conclude that two aor shortcoings of the existing spanning-tree-based genetic algorith needed to be iproved and revised for our purposes of solution. The first is related to spanning-tree based encoding, and the second is genetic operations. Fig. 3 shows the flow diagra of our algorith, and each state in the figure will be discussed in details below. The spanning-tree based encoding norally is used in an acyclic proble. Although our proble lies in a cyclic logistics networ, it is divided into several spanning-trees based on encodings between two consecutive stages which are not cyclic and the encoding is used to present the networ structure. Since the property of our proble is not a real spanning-tree proble (we ust use the encoding of spanning tree), we can relax the rules of spanning-tree properties and at the sae tie, iprove the solution. The details of our encoding process will be explained as follow: In a ultistage supply chain networ proble, it can be easily presented by a spanning tree. Fig. 4 shows the first two consecutive stages which have been presented by the spanning tree between suppliers and anufactories. Fro the literatures, we have learned that both Prüfer and deterinant encoding can be used for the encodings of the spanning tree proble in the fraewor of a GA. However, because the deterinant encoding is a siple node-based indirect encoding strategy to overcoe the bottlenecs of Prüfer encoding [1], we shall adopt deterinant encoding in our study The initial chroosoe and deterinate encoding (Fig. 3 a ) The deterinate encoding contains two parts: encoding and decoding.

8 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) Start Load the proble data, population size (pop_size), crossover (cr) and utation rate (r) and ax generation ties (ax_gen) a Use deterinate encoding to deterine the initial population b Deterine the flows and all total cost Decoding: The decoding algorith treats each allele of the gene to correspond to its position in the chroosoe, and the position represents its direct connecting node. The first gene is decoded as fixed-position 2, the second as fixed-position 3, and so on. The procedure of deterinate decoding process is as follows: Step 1. Let C be the given deterination string and l be its length. If C() is the th allele in chroosoe and 1 l, the nuber of the nodes in the given graph G is l+1, where a node is denoted as node (x) and1 x l + 1. Step 2. Set = 1, if 0 < < l+1,gotostep3,elsestop. Step 3. Connect node (+1) with node C(), Set = +1, go bac to Step 2. N Do chroosoe(n) Rando nuber < cr N Rando nuber < r n:=n+1 N If n >pop_size e μ parents and λoffsprings copete for survival and the μ best solutions are selected Y Y Y Do n:=1; c Two-points crossover d Exchange utation b Deterine the flows and all total cost N For exaple, let C = [ ] represent a chroosoe coded by the deterinant encoding. It iplies that there are nine nodesinthenetworcorrespondingtoeachfixedposition[ ]withthelins(2,3), (3,4), (4,2), (5,5), (6,8), (7,5), (8,3) and (9,9) in the tree. Each of the fixed corresponding position is equal to the order of the gene plus one as shown in Fig. 5. The generated tree ay not be legal and need to be repaired by reallocating genes at appropriate positions to generate a legal tree [5,33]. There are three `illegal' situations in the deterinant encoding, and we can find all cases in Fig. 5. The first concerns cycling, of which nodes 2, 3, and 4 are connected with a cycle. This eans that a route starting fro node 2 or 3 or 4 returns to itself and it will ae the spanning tree illegal. The second is related to reflexivity. It eans that the node connects to itself as nodes 5 and 9, and it also aes the tree illegal. The last one is called issing node 1, and it taes place when the chroosoe does not contain node 1. Based on our proposed deterinate encoding, only the last proble of the issing node 1 ay happen in our proble, but it is coparatively easy to solve and will be discussed later. After choosing the encoding type, we can start GA by evaluating the initial chroosoe. Fig. 6 is an exaple of our chroosoe presentation. There are two parts in a chroosoe that we ust consider. Part 1 refers to the places that should be opened, and Part 2 is the evaluation of the deterinant encoding for setting up the flows. Part I: Are the facilities open or not? If ax_gen>750 or ten iterations have the sae best solution Step 1. Randoly ae the 0 1 values for the first three digits (for anufactories, distributions and disantlers). Step 2. Feasibility Test 1: If the total opened capacity satisfies the total custoer deand, then the code of part I is finished and go to part II. Otherwise, go bac to Step 1. Encoding: Y End Fig. 3. The flow diagra of revised spanning-tree based GA. Assue there are N nodes, Step 1. Generate an N 1 length of deterinate encoding; Step 2. Use the rando or heuristic ethod to set the codes. (The rando or heuristic ethod will be explained at the initialization procedure below.) Part II: Deterinant encoding for setting up the flows We used a heuristic of revised deterinant encoding to set up the flows in our proble. Based on the assuption (b), the units are restricted to flow only between different stages of the networ. In order to ensure feasibility of a substring in deterinant encoding, we should restrict the range of the encoding value. Using the flows between I suppliers and J anufactories as an exaple, the deterinant encoding has N 1 chroosoe length, with N nodes, and there are I+J 1 genes in a chroosoe between suppliers and anufactories. For the first I 1 genes, let it be the nuber in between I+1 and I+J, and the last J genes be the nuber between 1 and I. The cases of reflexivity and cycling will never occur provided that the range of the encoding value is restricted to certain values. Therefore, there is only one case which needs to deal with and is the proble of issing node 1. Missing node 1 in deterinant

9 384 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) S1 S M1 400 M2 550 M3 490 S1 40 S M M M1 400 S M4 300 M5 460 M5 390 S M4 300 Fig. 4. The solution presented by a spanning tree. setting can effectively help us to find a good solution, and the rando setting is used to avoid optiu generated fro local population. In conclusion, although the deterinant encoding is an indirect encoding strategy, the decoding algorith is very siple, and the only thing needed to be aware is to repair the issing node 1 [33]. This step described above not only repairs the illegal node; but by connecting the node in the fixed position with a iniu cost, also provides an opportunity to iprove the solution. This is ipossible for Prüfer encoding. Fig. 5. Deterinant encoding and the decoding process. encoding will cause `illegal' case of the first node absent fro our code, so we ust proceed validity test for it. Validity test: In our proble, only without node 1 in the deterinant encoding is counted as illegal and needed to be repaired. As the nodes correspond to the gene in the chroosoe, the repairing process is to test the nodes at the fixed positions with the costs connecting to node 1, and the one with the iniu cost will be replaced by node 1. Thus, issing node 1 proble is resolved. After encoding, the initial chroosoe can be obtained by rando and heuristic generation with the restrictions on the range of the encoding value as below. The initialization procedure: We use two inds of algoriths for setting deterinate encoding: one is to randoly set the genes by this process, and the other is a heuristic ethod. Let ρ% of the population be randoly generated, and (100 ρ)% of the population be heuristically generated. 1. By rando generation: For the first ρ% chroosoes, we randoly generate the value of the positions in the chroosoe but tae into accounts of the restrictions on the range of the encoding value for each chroosoe. 2. By heuristic generation: For the first (100 ρ)% percent chroosoes, we do the following steps. Let q = 2. (a) The heuristic ethod begins fro the qth fixed position in the deterinant encoding, of which the gene with the iniu cost is allocated to this fixed position. If ore than one of the iniu points exists, any one will be arbitrarily chosen. (b) Set q = q+1. If q I+J (If it is in the first stage of I suppliers and J anufacturers), then go to step 2(a); otherwise, stop. In our research, the ratio for the use of heuristic setting and rando setting is 9:1 (ρ% = 10%) in the initial population. The heuristic The flows and cost of deterinate encoding (Fig. 3 b ) We generate a rando nuber strea to deterine the order of flows and the details will be described by using the exaple between suppliers and anufactories with a strea set up by the rando nuber between 1 and i+ 1. Step 1. Use the rando strea to deterine which of the fixed positions will be chosen, then select the saller capacity to be the flow between the fixed positions and the corresponding gene in the chroosoe. This eans that there is a need to assign the available aount of units to x i = in{a i, b }. Step 2. Update the availability a i = a i x i and b = b x i. Step 3. If there is no available aount of units to assign, then stop; otherwise, there is a reaining supply of node r and deand of node s, then add edge (r, s) to the tree and assign the available aount of units x rs = in{a r, b s } to the edge. Deterinate encoding with n nodes will have (n 1) routings. In other words, I suppliers and J anufactories have at the ost (I+J 1) connection lines. Although the total possible routing nuber is I J, the obective of the optial cost saving will result in inial routing nuber. Based on the result, a spanning tree can be used to present the logistics networ and avoid unnecessary routings and thus coputation efficiency can be iproved. In a closed-loop supply chain, there are two inds of systes, push for reverse and pull for forward. In the beginning of our GA, we ust start fro the push syste to deterine the reverse aounts, and then we can now how uch the suppliers should assign to the anufactories in the pull syste. Push syste: the reverse supply chain The reverse logistics is a push syste, and the flows are deterined by the custoers' recovered rates. Then starting fro the DCs and disantlers stages, the location and allocation will be ended at the anufactories and disantlers stages. Pull syste: the forward supply chain

10 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) Manufactors 0-1 variables Disantlers 0-1 variables Distribution 0-1 variables deterinant encoding Fig. 6. The chroosoe in revised spanning-tree-based GA. The forward logistics is a pull syste, and the flows are deterined by the custoer deand. Then through DCs and custoers, the flows are allocated to the stages of anufactories and DCs exchange. The stages of suppliers and anufactories have to be done after the pull and push processes are ipleented. The flows will be evaluated at all stages, and the cost can be calculated by Eq. (1) Genetic operations Genetic operations are very iportant in genetic algoriths, and we focused on the crossover and utation ethod, crossover and utation rate, population size, fitness ethod or selection ethod, and terination condition for discussion. Research on these issues is enorous, and we adopt the one ostly coonly used in spanning tree probles in the following illustrations: BasedontheworsofYaoandHsu[33], Chouetal.[5], and Booer [22], the proper ethods of crossover and utation types for spanning tree probles are two-point crossover and exchange utation. The two-point crossover (Fig. 3 c ): (1) Generates two rando positions; head and tail. (2) The alleles of the first chroosoe fro the head position to the tail are exchanged with the second chroosoe in the sae range. The ethod of exchange utation (Fig. 3 d ): (1) Randoly selects two positions in a given chroosoe. (2) Exchanges both genes fro these two rando positions. In order to ae sure the feasibility after applying a crossover or utation, we also used a saple heuristic here which can be referred to Appendix A. Fro the experients of Syarif et al. [30], the crossover rate is suggested to be 0.4, the utation rate is equal to 0.2, and the population size is set to 100. For the population size, we will especially discuss the influence when the proble size increases in our nuerical exaples in the next section. The selection ethod in Fig. 3 e adopts the (μ + λ) ethod suggested by Chou et al. [5], of which μ parents and λ off springs copete for survival, and the μ best solutions are selected for the next generation. Several terination conditions are established fro nuber of generations, coputing tie, and fitness convergence. Fitness convergence occurs when all the chroosoes in the population have the sae fitness value. In this study, fitness convergence is selected as the terination criterion. Naely, we stop the evolutionary process in GA when the best chroosoe on hand was not iproved in the last 10 generations. Siultaneously, if the nuber of generations is greater than 750, we also stop the algoriths Suary of the proposed algorith Based on the above description, we use revised deterinate encoding with heuristic initial population to overcoe the bottlenecs of Prüfer encoding in a spanning-tree-based GA, and deterine the cost and flows fro reverse to forward logistics. By applying twopoint crossover and exchange utation with the (μ + λ) selection ethod, we can achieve better generation. This process repeats until the terination condition is satisfied. This revised algorith can obtain better feasible solution in ters of high efficiency and reasonable accuracy. This evidence can be supported fro the following exaples when the coparison is done with optiization algoriths of LINGO 8.0 and CPLEX Evaluation of the algorith To test the accuracy and efficiency of the proposed algorith, previous exaple was adopted as a base for coparison. To test the efficiency, different sizes of the test probles were used through doubling the nubers of the nodes at each stage as shown in Table 5; and running 30 ties for each proble. A total of 150 experients were executed by our algorith. The results were copared with ILOG-CPLEX. These experients were all done by a PC with Intel Pentiu M processor 1.86 GHz, 1.0G RAM. The test proble 1 is the illustrative exaple above of which I = 3, J = 5, K = 3, L = 4andM = 2. In test proble 2, I = 6, J = 10, K = 6, L = 8andM = 4, there are 82 constraints, and 304 variables (including 20 binary variables), and optial solution is by LINGO. In test proble 3, I = 12, J = 20, K = 12, L = 16 and M = 8, there are 164 constraints, and 1168 variables (including 40 binary variables). The proble size increased to 328 constraints and 4576 variables in proble 4, and proble 5 reaches to 656 constraints and variables. We can observe that the size of the proble increased iensely Preliinary evaluation To evaluate the efficiency and accuracy of our algorith, LINGO was first adopted for these purposes as it is the ost coonly used software. The results are shown in Table 6. For Test Proble 3, after 10 9 iterations and 20 in(s) of elapsed runtie, LINGO failed to obtain the final solution, and so did test proble 4. For the proble 5, LINGO failed to find a feasible solution before 35 in, and after 40 in, a feasible solution was found. With our revised spanningtree-based GA, two cases were considered: one was done with sae population size of 100 for five test probles regardless of the proble size, and the other was done by increasing the population size with the proble size in order to obtain ore accurate results. Table 6 suarizes the test results. Fro Table 6, it can be observed that LINGO fails to solve such ind of large-scale probles, whereas our algorith is capable of doing so. Besides, with our algorith, increasing the population size with proble size only iproved slight accuracy of the proble, yet

11 386 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) Table 5 Thesizeoftestprobles. Test probles Suppliers Manufactories DCs Custoers Disantlers pd (%) pc l (%) pl (%) φ Table 6 Proble size with the sae and different population sizes. 30 Ties each probles Test proble LINGO Optial (US$) (feasible (feasible (feasible solution) solution) solution) Tie (s) 6 13 > 1200 > 1200 > 2400 Revised ST-GA Min_cost (US$) (population size = 100) [deviation] [0] [62] [1061] [4173] [4716] Ave_cost (US$) Ave_tie (s) [percentage of tie] [34%] [48.85%] [ < 3.74%] Revised ST-GA (population size increase with proble size) Pop_size Min_cost (US$) [deviation] [0] [62] [121] [2410] [ 417] Ave_cost (US$) Ave_tie (s) [percentage of tie] [18.67%] [52.62%] [ < 10.52%] Deviation = ST-GA Min_cost LINGO optial. Percentage of tie = ST-GA Ave_tie/LINGO tie. requires large coputation tie. Therefore, we do not have to use large population size to ipleent our algorith as the proble size increases Advanced evaluation While LINGO cannot solve large-scale probles with its branchand-bound ethod; CPLEX is considered to be ore efficient by branch-and-cut ethod, and thus was adopted for further experients. The coparisons of our algorith and CPLEX with error rates are shown in Table 7, of which the values in boldface are the best solutions. In Table 7, we can see that although the CPLEX software is ore efficient, it still cannot find the optial solution in large probles. To evaluate accuracy, the error rates of our revised ST-GA with respect to the optial solutions or out-of-eory feasible solutions obtained fro CPLEX are less than about 1%. In test proble 5, although the operation tie is higher, but we can even get a better result than CPLEX. With regard to efficiency, because tests 4 and 5 ran out of eory earlier than test 3, we noralized run tie with respect to the CPLEX tie of test 3 in percentage. Fig. 7 shows the results of the evaluation: the left diagra shows the error rate of the revised ST-GA and the transfored tie percentage of the revised ST-GA and CPLEX; the right diagra is an estiated situation for test 4 copared with tests 1, 2 and 3. Fro this evaluation, it is evident that even though the proble sizes were large, our algorith showed a saller error rate and less operation tie than CPLEX. Therefore, the revised ST-GA can provide sufficiently accurate solutions with the efficient coputation tie for our closed-loop logistics proble. In the previous experients, we increased the population size in order to obtain a solution that is near-optial or a feasible solution when out of eory incurred by the adopted software. However, increasing the population size rapidly increases operation tie. The error rates of tests 3, 4 and 5 without increasing population size are shown in Table 8. The results showed that the error rates of population size increased a little, but the average operation tie(s) decreased substantially without increasing population size. We can ae the process ore efficient and ust sacrifice accuracy slightly when a population size equal to 100 is used. This facilitates real proble applications. For exaple, if an error is 2% is acceptable by a anager, then we ay choose the population size of 100 for finding solution Discussion and suary In order to investigate the influence of the proble structures on the solution perforance, we further generated two test probles. That is, (1), 20 suppliers, 15 anufactories, 12 DCs, 50 custoers and 5 disantlers, and (2) 10 suppliers, 6 anufactories, 8 DCs, 100 custoers and 5 disantlers, respectively. These two test probles have 1852 variables and 1802 variables, respectively. As LINGO cannot solve the, coparison with CPLEX is done and shown in Table 9. It can be seen that at the first test proble of 1852 variables, our algorith obtains optial solution. The error rate of our algorith with the second test proble is less than 0.1%. However, the run ties of both probles are less than that of CPLEX. Fro these additional experients, we ay confir the accuracy and efficiency of the proposed algorith.

12 H.-F. Wang, H.-W. Hsu / Coputers & Operations Research 37 (2010) Table 7 Coparisons of CPLEX and the revised ST-GA. Proble ILOG-CPLEX Revised ST-GA Min_cost (Ave) Ave_tie (s) Ave_tie (s) Min_cost (Ave) [error rate] Test (82 variables) ( ) (pop_size = 50) [0%] Test (304 variables) ( ) (pop_size = 100) [0.106%] Test > (1168 variables) (feasible solution) (Out of eory) ( ) (pop_size = 200) [0.107%] Test > (4576 variables) (feasible solution) (Out of eory) ( ) (pop_size = 400) [1.05%] Test > (18112 variables) (feasible solution) (Out of eory) ( ) (pop_size = 400) [ -] Error rate = (GA (CPLEX optial or feasible solution when out of eory))/(cplex optial or feasible solution when out of eory). percentage 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% 82 GA tie percentage GA tie percentage GA's solution error rate GA's solution error rate ILOG-CPLEX tie percentage ILOG-CPLEX tie percentage 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Variables percentage Variables Fig. 7. Coparisons of error rate and tie by percentage. Table 8 The error rates of equal and different population sizes in tests 3 and 4. Proble Revised ST-GA (population size = 100) Revised ST-GA (population size increase with proble size) Error rate (%) Ave_tie Error rate (%) Ave_tie Test Test Test Furtherore, fro our experients, we have observed that although a larger population size can iprove the solution, it consues huge coputation tie. The trade-off between these is to find a suitable population size in the consideration of the error rate and tie. Therefore, if we set the acceptable error rate in advance, the respective population size can be deterined. In our experients, 2% of the acceptable error rate was assued, and thus the population size of 100 was used. In reality, to control the error and ae