A heuristic approach to logistics network design for end-of-lease computer products recovery

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Transportation Research Part E 44 (2008) 455–474 www.elsevier.com/locate/tre

A heuristic approach to logistics network design for end-of-lease computer products recovery Der-Horng Lee

*,1,

Meng Dong

Department of Civil Engineering, National University of Singapore, Singapore 117576, Singapore Received 8 March 2006; received in revised form 7 September 2006; accepted 5 November 2006

Abstract This paper discusses the logistics network design for end-of-lease computer products recovery by developing a deterministic programming model for systematically managing forward and reverse logistics flows. Due to the complexity of such network design problem, a two-stage heuristic approach is developed to decompose the integrated design of the distribution networks into a location–allocation problem and a revised network flow problem. The applicability of the proposed method is illustrated in a numerical study. Computational experiments demonstrate that high-quality solutions are obtained while modest computational overheads are incurred.  2007 Elsevier Ltd. All rights reserved. Keywords: Reverse logistics; Product recovery; Location–allocation; Tabu search

1. Introduction Recently electronic product recovery is receiving increasing attention. In the United States, the used PC business was estimated between $2–3 billion in 1996. Approximately 25 million obsolete PCs became ready for remanufacture or disposal in 1997. Given a population of approximately 260 million in the United States, that was about under one obsolete computer per 10 persons (Rogers and Tibben-Lembke, 1999). A study completed by Carnegie Mellon University (Carnegie Mellon University, 1997) estimated that approximately 325 million personal computers would have become obsolete in the United States in the 20-year period between 1985 and 2005. Out of that number, it was estimated that 55 million would be placed in landfills and 143 million would be recycled. Nowadays, more and more electronic manufacturers reuse returned products and incorporate product recovery activities into their regular production environments. The motivation of electronic product recovery is twofold. Firstly, the legislation aimed at environmentally conscious production forces electronic manufacturers to take back products which could do harm to or potentially do harm to

*

1

Corresponding author. Tel.: +65 6516 2131; fax: +65 6779 1635. E-mail address: [email protected] (D.-H. Lee). Present address: BLK E1A #07-16, 1 Engineering Drive 2, Singapore 117576, Singapore.

1366-5545/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2006.11.003

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the environment from end-users after the products reach their end of life. For example, Germany has passed a consumer electronics act, which makes the manufacturer responsible for the end-of-life disposition of all consumer electronic products (Rousso and Shah, 1994). The European Commission presented a set of proposed laws on collection and recycling of electrical and electronic wastes in 1998, which set targets for Waste Electrical and Electronics Equipment (WEEE), to increase recycling, to reduce hazardous substances, and to make sure waste left over after recycling is properly disposed (Rogers and Tibben-Lembke, 1999). Secondly, the reuse of the returned products, returned components or reusable materials, can be highly profitable. This is especially the case with high-tech products with a reasonably long product life cycle in the electronic products industry, such as server computers and photocopiers. An example is that IBM, through its IBM global financing organization, presently handles approximately 112 000 units of used IT equipment worldwide each month. This equates to 3 million kg/month of used product being processed by IBM globally, and an annual total amounting to almost 40 000 tons/year. These units are recycled, reused, refurbished and resold into the global marketplace (IBM, 2005). According to Mollenkopf and Closs (2005) and Dong et al. (2006), the major activities of end-of-lease (EOL) computer products recovery at a typical electronic company, referred to as Company A, are described as following. Company A deals in office computer products, such as desktop computer, server computer, laptop computer, CRT display, LCD display, printer and other peripherals, many of which are leased. In the process of forward distribution, the electronic products are shipped from Company’s Original Equipment Manufacturers (OEMs) to customers via forward processing centers. With the focus on EOL asset recovery, Company A has also been involved in returns flow, remanufacturing, remarketing, recycling, and landfill. Because its leased assets are returned, the company highlights the importance of quickly assessing the value of the product in its entirety and deciding on the resale potential of the whole product as distinct from the potential value of individual modules, components, or materials. In the process of reverse distribution, the EOL computer products are taken back from the customers and shipped to the OEMs via collection facilities. The returned products are checked firstly in the process of assets verification to ensure they are in good and ordered conditions when shipped to warehouses. A bill for the cost of replacing missing or non-functional items will be sent to the customer approximately two months after the return of the equipment. Missing or damaged parts on one PC will not impact the lease for the balance of the assets on the lease schedule. The charge for missing or damaged items will be the replacement cost of the component up to the EOL fair market value of the total asset. By functional testing, the returned products are classified into three categories, i.e., products need to be remanufactured, demanufactured, or scrapped. (i) Products that can be readily remanufactured are quickly identified and converted to resalable goods. During remanufacturing process, the returned products can be repaired and repackaged. Thus the products can be either resold to pre-owned market or released market. (ii) For products that cannot be remanufactured. The usable components are disassembled in demanufacturing process. In the electronic production environment, many used parts are considered of equal value to new parts for replacement purposes. As such, the volumes of new components can be significantly reduced by recapturing the components. (iii) Finally, the returned products without market value are sent to the designated location to be dismantled and scrapped. Products are received by weight, and where applicable, by serial number. Products are dismantled and scrapped using certified environmental management systems and processes that meet all governmental regulations. The reusable materials can be resold to the reusable material market. The flow chart of EOL computer products recovery operations at Company A is illustrated in Fig. 1. From a logistics perspective the EOL computer products recovery activities give rise to an additional goods flow from customers back to manufacturers. The management of this flow opposite to the conventional supply chain is concern of the recently evolved field of reverse logistics (Stock, 1998). Implementation of such logistics operations requires setting up additional appropriate logistics infrastructure for the arising flows of used and recovered products. Physical location, facilities and transportation links need to be chosen to transfer forward products from manufacturers to customers and to convey returned products from their former users to manufacturers for the purpose of recovery or safe disposal. Reverse logistics responsibility can be retained within the company or outsourced to the third party logistics providers (Autry et al., 2001). Handling reverse logistics internally allows the company to keep control over the process. On the other hand, when reverse logistics is handled externally, close coordination between

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Collection facility

Original Equipment Manufacturer

Forward processing facility

Customer

Asset verification

Functional testing

Released market

Re-manufacturing

De-manufacturing

Scrapping

Repackaging

Disassembling

Classification of material

Pre-owned market

Reusable componet inventory

Reusable material market

Forward product flow

Logistics network site

Returned product flow

Product recovery operation

Safe disposal

Fig. 1. An illustration of EOL computer products recovery operations.

the parties is required to ensure maximum efficiency (Blumberg, 1999). Autry et al. (2001) reported that in general firms were only somewhat satisfied with the reverse logistics service being provided by their trading partners. As to the EOL computer products recovery with high-tech and high-value products with a reasonably long product life cycle in the electronic product industry, such as server computers and photocopiers, it is the common practice that the electronic company adopts the internal strategy for handling the operations of forward and reverse logistics (Fleischmann and van Nunen, 2003; IBM, 2005). Furthermore, the activities of reverse logistics may have strong influence on the operations of forward logistics such as the occupancy of the storage spaces and transportation capacity. Traditionally, companies use centralized returns center to process returned product. A returns center located at a forward distribution site will operate very differently from one that operates independently. The major perceived disadvantage of the separate network design of centralized returns centers is that the company spends a lot of money to transport product that ultimately may be thrown away (Rogers and Tibben-Lembke, 2001). Recently, a new type of hybrid processing facilities is applied in handling forward and reverse distribution (Ko and Evans, 2005). For example, United Parcel Service Logistic Group processes return activities through one of its warehouses in Louisville, Kentucky. Advantages of building such facilities might include cost savings and pollution reduction as results of sharing material handling equipment and infrastructure (Jayaraman et al., 1999; Ko and Park, 2005). Fleischmann et al. (2001) also showed that logistics network configurations of electronic products remanufacturing involving both forward and reverse flows were different with respect to a sequential solution approach and an integrated solution approach. They found that an integrated approach, optimizing the forward and return network simultaneously, could provide a significant cost benefit against a sequential approach. Thus, an approach for the logistics network design problem for EOL computer products recovery should be based on an integrated point of view.

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Accordingly, formulation of a comprehensive framework with appropriate analytical models for systematically managing forward and reverse logistics flows in an EOL computer products recovery network is urgently needed. Recognizing the inherent complexity of such integrated logistics network design problem, this paper develops a mathematical model and proposes a heuristic solution approach that can near-optimally obtain logistics network linking OEM facilities, hybrid processing facilities and customers.

2. Literature review Reverse logistics is part of a broader supply chain management process called returns management. In a broader sense, reverse logistics refers to the distribution activities involved in product returns, source reduction/conservation, recycling, substitution, reuse, disposal, refurbishment, repair and remanufacturing (Stock, 1992). More precisely, Rogers and Tibben-Lembke (1999) defined reverse logistics as the process of planning, implementing, and controlling the efficient, cost effective flow of raw materials, in-process inventory, finished goods and related information from the point of consumption to the point of origin for the purpose of recapturing value or proper disposal. Reverse logistics encompasses the traditional logistics activities of transportation and inventory management, but its focus shifts to getting product back from customers rather than moving product to customers (Mollenkopf and Closs, 2005). A good returns-handling system can be a source of significant cost saving and even be a profit center (Fleischmann et al., 1997; Stock et al., 2002). It can reduce operating costs and minimize the opportunity costs of writing off defective or out-of-date products. Increased revenues can be realized from ‘‘secondary’’ sales and the goodwill earned from acting in a socially or environmentally responsible manner can produce real value (Mollenkopf and Closs, 2005). In addition to cost benefits, a reverse logistics program can proactively minimize the threat of government regulation and can improve corporate image (Carter and Ellram, 1998). Customers often form their opinions about companies based on their experience with product returns or repairs (Walker, 2000). In the instance of returning products for refurbishing or remanufacturing, customers want handling and processing to be accomplished as fast and smoothly as possible (Daugherty et al., 2005). Reclaiming value from returns has immense service implications, i.e., top-rate reverse logistics handling can impress customers (Daugherty et al., 2002). Thus, customer relations can be impacted positively due to enhanced service quality (Daugherty et al., 2005; Richey et al., 2005a). Because of the opportunity to reclaim value and maintain customer relationships, reverse logistics program design is of strategic importance (Richey et al., 2004; Richey et al., 2005). Rogers and Tibben-Lembke (1999) reported that the issue of how electronic products should be disposed of, and how much recycling should be done of them is a widely discussed issue. The standards will cover a wide variety of electronic products, including cellular phones, games, toys, household appliances, and office machinery. Stimulated by the aforementioned environmental and economical concerns, the efficient logistics network design for EOL computer products recovery has been a challenging issue in the field of reverse logistics, which attempts to minimize the sum of the shipping cost, the operation cost and the disposal cost while at the same time to maximize the sales revenue of reclaimed products. Kooi et al. (1996) developed a mixed integer linear programming (MILP) model for the setup of a multi-echelon logistics network for product recovery with given supply and demand. A linear programming (LP) solver was used for the optimal solution which resulted in a long computation time for the instance of large problem size. Jayaraman et al. (1999) analyzed the logistics network of an electronic equipment remanufacturing company in the USA. The authors present a multi-product capacitated warehouse location MILP that is solved to optimality for different supply and demand scenarios. Krikke et al. (1999) proposed another MILP model for the multi-echelon product recovery network design which focused on the remanufacturing of a certain type of copy photocopier. An LP solver was also used to obtain the optimal solution for the instances of small problem size. Shih (2001) proposed a new MILP model to optimize the infrastructure design and the reverse network flow for the recovery of electrical appliances and computers. Computational results for the scenarios of different product return rates and operation conditions were presented. Fleischmann et al. (2001) developed an MILP model to analyze the impact of product recovery on logistics network design. Due to the complexity of the proposed model, a heuristic algorithm was applied to obtain the solution for cases with large problem size.

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However, all the aforementioned research only focuses on the separate reverse distribution problem in which the interaction between the distribution of forward products and returned products is ignored. In fact, due to the influence of the activities of reverse logistics on forward logistics such as the occupancy of the storage spaces and transportation capacity, the integration of forward and reverse distribution needs to be considered especially at the stage of distribution network design. Due to the inherent complexity of such integrated logistics network design problem, another key issue involved is the efficient solution method. To the best of our knowledge, the algorithm developed in this paper is the first attempt to solve the integrated design problem using meta-heuristics. 3. Problem definition A precise description of the logistics network structure for EOL computer products recovery is illustrated in Fig. 2. An electronic company delivers computer products to a number of geographically dispersed customers from OEM according to their demand. A part of EOL computer products associated with the delivered products and having the same dimensions as the original products are collected from the customers and returned to the OEM for the purpose of recovery or safe disposal. In such integrated logistics network, instead of dealing with separate warehouse or collection centers, a type of hybrid processing facility is considered. Both forward products and EOL returned products are transferred via hybrid processing facilities. As aforementioned, advantages of building such facilities in electronic industry include cost savings and pollution reduction as results of sharing material handling equipment and infrastructure. The objective of such integrated logistics network design is firstly to choose the OEM and select the locations of the hybrid processing facilities, and then to determine the quantities of the forward and EOL returned products shipped among the network. To specify the study scope and facilitate model formulation, four assumptions and simplifications in the proposed model formulations are postulated as follows. (i) Only one OEM is to be chosen in the attempted problem. The number of the hybrid processing facilities to be built up is known in advance.

Fig. 2. A depiction of a logistics network structure for EOL computer products recovery.

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(ii) The capacity for handling each type of EOL returned products at each hybrid processing facility is limited due to the limitation of the certain equipments for the operations of repackaging before the EOL returned products are shipped back to OEM. (iii) The recovered products are viewed as identical to the new products, which is the common practice in the business of leasing high valuable electrical appliances with a reasonably long product life cycle, such as server computers. (iv) Shipment of forward products directly from OEM to customers and collection of EOL returned products directly from customers to OEM are allowed, while the associated transportation cost is much higher than shipping through a hybrid processing facility. The reason for this assumption is that small batch size costs more to ship.

4. Model design Given the aforementioned assumptions, a network flow-based deterministic programming model is formulated to seek the optimal solutions with the goal of minimizing the total cost in the logistics network. The mathematical formulation of the proposed model is detailed below. All the notations for parameters and decision variables determined by the optimization process of the proposed model are summarized in Appendices A and B, respectively. In the attempted logistics network design problem for EOL computer products recovery, the total cost consists of the following three parts: (i) the cost associated with building up hybrid processing facilities, (ii) the cost associated with shipping forward products from OEM to customers via hybrid processing facilities, and (iii) the cost associated with collecting EOL returned products from customers to OEM via hybrid processing facilities. As such, the objective function of the proposed model seeks to minimize the sum of the aforementioned cost, which is given by X XX XX Min C l zl þ C Fij xij þ C Rji y ji : ð1Þ l2RD

i2N

j2N

j2N

i2N

In a large-scale logistics network, both the distances and the shipping quantities have important influence on the shipping cost. Herein, the shipping cost is calculated by the product of distances and the associated shipping quantities between depots and customers. It is noted that in the EOL computer products recovery, the quantities of the EOL returned products also have significant impact on the logistics network structure due to their high-value and reasonably long product life cycle, such as server computers and photocopiers. 4.1. Network flow conservation constraints A depiction of the flow conservation in a logistics network is illustrated in Fig. 3. As to a node i, i 2 N, the quantity of the supply of forward products S Fi represents the quantity of forward products shipped out from node i, while quantity of the demand of forward products DFi means the quantity of forward products shipped F i

R i

R i

F i

Fig. 3. A depiction of the flow conservation in a logistics network.

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into node i; the quantity of the supply of EOL returned products S Ri represents the quantity of returned EOL products shipped out from node i, while quantity of the demand of EOL returned products DRi means the quantity of returned EOL products shipped into node i. During the shipment of forward products, the quantity of supply of forward P P products at each depot k, k 2 D is equal to the difference between total output m2E xkm and total input i2N xik of forward products, as shown in Eq. (2). X X xkm  xik ¼ S Fk zk 8k 2 D: ð2Þ m2E

i2N

The quantity P of demand of forward Pproducts at each customer n, n 2 C is equal to the difference between total input i2N xin and total output m2E xnm of forward products, as shown in Eq. (3). X X xin  xnm ¼ DFn 8n 2 C: ð3Þ i2N

m2E

During the collection of returned EOL of supply at each customer n, n 2 C is equal Pproducts, the quantity P to the difference between total output i2N y ni and total input m2E y mn of the returned EOL products, as shown in Eq. (4). X X y ni  y mn ¼ S Rn 8n 2 C: ð4Þ i2N

m2E

The quantity P of demand of returned EOL P products at each depot k, k 2 D is equal to the difference between total input m2E y mk to and total output i2N y ki of the returned EOL products, as shown in Eq. (5). X X y mk  y ki ¼ DRk zk 8k 2 D: ð5Þ m2E

i2N

4.2. Capacity constraints Constraints (6) and (7) limit the units of forward and EOL returned products shipped along an arc to its shipping capacity in the network. The realistic meaning of such constraints is that not all products can be shipped via the shortest path in a network due to the limited shipping capacity. xij 6 U Fij

8i 2 N ; j 2 N ;

ð6Þ

U Rji

8j 2 N ; i 2 N :

ð7Þ

y ji 6

Constraint (8) limits the units of EOL returned products transferred through a collection depot to its capacity for dealing with EOL returned products which is due to the limitation of the certain equipments for the operations of repackaging before the EOL returned products are shipped back to OEM. X y nl 6 U l 8l 2 RD: ð8Þ n2C

Constraints (9) and (10) prohibit the units of forward and EOL returned products from being transferred through depots unless the depots are to be built up. X xkm 6 Mzk 8k 2 D; ð9Þ m2E

X

y mk 6 Mzk

8k 2 D:

ð10Þ

m2E

4.3. Other constraints Constraints (11) and (12) guarantee the quantities of forward and EOL returned products shipped along arcs are not less than zero. If the decision variable xij is determined to zero, there is no forward product

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shipped from i, i 2 N node to node j, j 2 N. If the decision variable yij is determined to zero, there is no returned EOL product transferred from i, i 2 N node to node j, j 2 N. xij P 0 8i 2 N ; j 2 N ; y ji P 0 8j 2 N ; i 2 N :

ð11Þ ð12Þ

Constraints (13) and (14) ensure the numbers of OEM and hybrid processing facilities to be built up to specific values, while only one OEM is to be chosen in this paper. Constraint (15) enforces the binary restriction on the location decision variables. X zp ¼ 1; ð13Þ p2CD

X

zl ¼ q;

ð14Þ

l2RD

zk 2 f0; 1g

8k 2 D:

ð15Þ

5. Heuristic solution approach Since the capacity of hybrid processing facilities for handling EOL returned products is taken into consideration, the proposed model reduces to a Capacitated Plant Location Problem (CPLP) by ignoring the shipment of forward products. The CPLP is proved to be NP-complete (Davis and Ray, 1969), and as such, the proposed model is an NP-hard problem. Utilizing conventional Mixed Integer Programming (MIP) tools to solve the proposed model is computationally intractable due to the inherent problem complexity and its large number of variables and constraints. Hence, a heuristic solution approach is a practical method for the problem solution. This paper proposes a two-stage heuristic approach, which decomposes the logistics network design of EOL computer products recovery into a location–allocation problem and a revised network flow problem. A selection strategy is adopted to obtain the locations of the depots at the first stage. A tabu search algorithm is then applied to get the improved shipment solution for EOL returned products at the second stage. The proposed heuristic approach is depicted in Fig. 4, which is discussed in detail in the following sections. 5.1. Finding the locations of depots For given m candidates of hybrid processing facilities, there are C nm possible results of choosing n hybrid 40! processing facilities. For example, if m = 40 andn = 8, there are C 840 ¼ 8!ð408Þ! ¼ 7:7  107 possible results. Enumerating all the possible results is limited due to the requirement of huge computational effort. Hence, random selection strategy is a practical way to choose depots from the candidates. Firstly, the depots are randomly selected from the given candidates. If the rental and subsequent operational costs of the selected depots are larger than the total costs of the best solution obtained thus far, the depots will be randomly chosen again. Secondly, by relaxing the capacity constraint (8), the determination of the quantities of forward and EOL returned products will be reduced to a minimum cost flow problem. The network simplex algorithm is applied to obtain the initial solution of the flows. If the initial solution of the flows can not be obtained by the network simplex algorithm, the depots will be randomly selected again. 5.2. Constructing an initial feasible solution of the shipment of products Since it is assumed that there is no capacity constraint for the hybrid processing facilities in handling forward products, the optimal solution of the shipment of forward products is obtained directly from the results of the network simplex algorithm in the first stage of the proposed heuristics. If the capacity of the hybrid processing facilities for handling the EOL returned products is not exceeded, the EOL returned products

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Fig. 4. An illustration of the process of the two-stage heuristics approach.

can be shipped along the same arcs as the forward products. Otherwise, a repairing strategy will be triggered to make the EOL returned products be shipped directly from the customers to the OEM.

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5.3. Obtaining improved shipment solution of returned products Tabu search is a meta-strategy iterative procedure for building extended neighborhood with particular emphasis on avoiding being caught in a local optimum. This global optimization meta-heuristic was initially proposed by Glover (1977), evolved in Glover (1989, 1990) and Glover et al. (1993). It is found through previous research that Tabu search can be an efficient method in solving the network design problem. It is based on introducing flexible memory structure in conjunction with strategic restrictions and aspiration levels to explore search spaces (Barbarosoglu and Ozgur, 1999; Montane´ and Galva˜o, 2005; Shen et al., 2005). A tabu search algorithm is proposed to obtain the improved the solution of the shipment of the EOL returned products in this paper. 5.3.1. Definition of neighborhoods In the context of distribution network, products are shipped along a sequence of nodes from an origin to a destination. Such a sequence of nodes is defined as a channel. Three types of movements are used to define the neighborhoods of a channel, which include Interchange, Insertion and 2-Opt procedures. 5.3.1.1. Interchange procedure. This movement consists of exchanging nodes between two channels, as illustrated in Fig. 5. The movement is in principle attempted for all possible pairs of channels and for all possible combinations of exchange of nodes between the two channels. In the example of Fig. 5, nodes pi and qj are exchanged between the two channels. 5.3.1.2. Insertion procedure. This movement is employed to place a set of nodes (vk:vk 2 u) into a specific channel R, where u is a proper subset of nodes in the distribution network, as illustrated in Fig. 6. In Fig. 6, nodes pi

pi+1

pi

pi-1

q0

p0

qj+1 qj

pi

pi+1

pi-1

q0

p0

qj+1

qj-1

qj

qj-1

Arc between two nodes Removed arc Inserted arc Fig. 5. A sample of neighborhoods: Interchange procedure.

pi+1

pi

pi-1

q0

pi+1

p0

qj+1 qj

qj-1

pi

pi-1

q0

p0

qj+1 qj Arc between two nodes Removed arc Inserted arc

Fig. 6. A sample of neighborhoods: Insertion procedure.

qj-1

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p i+1

q0

pi

pi-1

p i+1

pi-2

p0

pi

q0

p i-1

465

pi-2

p0

Arc between two nodes Removed arc Inserted arc Fig. 7. A sample of neighborhoods: 2-Opt exchange.

are removed from the first channel and inserted into the second channel. The insertion procedure can be elaborated in the following: Step 1: Find the node k 2 u with the minimum transportation cost Cik to any node i already in R. Step 2: Find the arc a(i, j) in channel R which minimizes the value of (Cik + Ckj  Cij) and insert node k between i and j. Step 3: Repeat Step 1 and Step 2 until all nodes in u are placed into channel R. 5.3.1.3. 2-Opt exchange. This movement is applied to intensify the solution improvement in each channel within itself. It consists of replacing two non-adjacent arcs belonging to the channel under consideration by two other arcs not belonging to it, such that the connectivity of the channel is re-established. This movement is illustrated in Fig. 7. Suppose a channel of collecting returned products contains the following set of nodes in the given order S = {v0, v1, . . ., vk}, and let X = {(vi, vi+1); (vj, vj+1)} be a set of two arcs in S which will be replaced by arcs Y = {(vi, vj); (vi+1, vj+1)} if this replacement leads to an improvement. It is necessary to require all nodes in S be different from each other. Once the set X is chosen, the set of Yis directly determined. The steps of the proposed 2-Opt exchange procedure are as follows: Step 1: Calculate the improvement ax by ax = (Ci,i+1 + Cj,j+1)  (Cij + Ci+1,j+1). Step 2: If ax > 0, replace two arcs associated with ax. Step 3: Repeat above steps for all X in S. 5.3.2. Neighborhood search strategy The neighborhood search strategy specifies the movement in the neighborhood to be chosen at each iteration. The neighborhood may be explored in the following ways. 5.3.2.1. The first admissible movement. Sequentially generate the set of solutions in the neighborhood of the current solution. Choose the first feasible movement yielding a solution with the improved objective function value as the next movement. 5.3.2.2. The best admissible movement. Generate and evaluate all solutions in the neighborhood of the current solution. Choose the movement yielding the solution with the best objective function value as the next movement. Since the best admissible movement always produces better results, it is applied in all neighborhoods tested by the proposed tabu search algorithm. Only feasible movements are considered. After completing each step of the neighborhood search, a 2-Opt exchange is executed for each channel, in an attempt to further reduce the total cost of the distribution network. 5.3.3. Short-term memory In the proposed tabu search algorithm, a short-term memory structure is defined to avoid the reassignment of previously moved nodes to the original channels where they were previously moved from. Each movement

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is characterized by three sets of indices which define its attributes: (i) the index of the node involved in the movement, (ii) the index of the channel where the node is removed, and (iii) the index of the channel where the node is inserted. The short-term memory approach maintains a list of the sets of indices that was used just recently form the tabu list. A movement is considered as tabu when all the involved indices are in the current tabu list. The tabu duration represents the number of iterations that a movement remains tabu. If the tabu duration is short, the tabu movements are allowed to be recovered after a few iterations which make the search emphasize intensification. On the other hand, if the tabu duration is long, more movements are considered as tabu and the search will be extended to the areas that have not been visited yet which makes the search focus on diversification. It was suggested by Glover and Laguna (1997) that the size of tabu duration should depend on the problem size and characteristics. Thus, due to the tradeoff between intensification and diversification, variable tabu duration is applied in the proposed tabu search algorithm. The variable duration is randomly chosen from a uniform distribution with the minimum value to be five and the maximum value to be ten. An aspiration criterion is designed to overrule the tabu status of a movement if it leads to a better objective function value than the best found so far. The algorithm is terminated by either one of the following two stopping criteria: (i) no feasible movement exists; or (ii) the maximum number of iterations is reached. For example, if at iteration k, nodes v1 2 R1 and v2 2 R2 are exchanged. It is forbidden to put v1 in R1 and v2 in R2 again during iterations (k + 1, k + 2, . . ., k + C) where C is the tabu duration randomly chosen from (Cmin, Cmax). If all improved solutions are within the tabu duration, the algorithm chooses the oldest tabu movement and proceeds accordingly.

5.3.4. General structure of the proposed tabu search algorithm The proposed tabu search algorithm is characterized by a set of parameters: (Nmax, Amax, Bmax, N1, N2, P, Cmax, Cmin), where Nmax maximum number of iterations without any improvement Amax maximum number of neighborhoods to be searched Bmax maximum number of 2-Opt exchanges (N1, N2) maximum number of nodes to be interchanged between two channels Cmin lower bound in tabu duration Cmax upper bound in tabu duration The general structure of the proposed tabu search algorithm is illustrated as following: Step 0: Generate an initial shipment solution of the returned products based on the result of Network Simplex algorithm. Step 1: The following steps are executed to search the neighborhoods of the current solution. Step 1.1: Randomly choose a channel of collecting returned products from the current solution, R1. Step 1.2: Randomly choose another channel of collecting returned products from the current solution, R2. Step 1.3: Randomly choose u nodes out of R1, where u 6 min(N1, jR1j) and jR1j represents the number of nodes in R1. Step 1.4: Randomly choose v nodes out of R2, where v 6 min(N2, jR2j) and jR2j represents the number of nodes in R2. Step 1.5: If the mutual interchange between above u nodes of R1 and v nodes of R2 exceeds the capacity of any hybrid processing facility, repeat Step 1.3 and Step 1.4. Step 1.6: Insert the chosen v nodes into R1 by using the insertion procedure. Step 1.7: Insert the chosen u nodes into R2 by using the insertion procedure. Step 1.8: Apply the 2-Opt exchange to R1 for Bmax times. Step 1.9: Apply the 2-Opt exchange to R2 for Bmax times. Step 2: Repeat Step 1 for Amax times to obtain the improved solutions which result in smaller value of the objective function.

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Step 3: If more than one improved solutions are found, implement the non-tabu solution with the minimum objective function value. If all improved solutions are within the tabu duration, implement the oldest tabu movement. Step 4: Update the tabu list: place the implemented movement into the tabu list and remove the oldest movement from the tabu list. Step 5: Check the stopping criterion: If there is no improved solution found, repeat all above steps until the number of non-improved iterations reaches Nmax. 5.4. Updating the best solution If the total cost resulted from the newly obtained solution is less than the best solution obtained thus far, the best solution will be replaced by the newly obtained solution. The algorithm terminates if the maximum number of iteration Rmax is reached; otherwise, all above steps will be repeated. 6. Model application and numerical results To illustrate the applicability of the proposed method, a numerical study was conducted. Company A is an international electrical company in Asia Pacific (AP) region, which is shipping forward computer products from OEM to a number of globally dispersed customers via hybrid processing facilities. Meanwhile, Company A presently is also handling a large number of units of EOL computer products. These units are remanufactured and resold into the global marketplace. The remanufacturing is carried out in the OEM plants using the same equipment. The objective of this study is to make a recommendation to Company A on which cities in the AP region should be selected as the hybrid processing facilities, which OEM should be chosen and how the forward and returned products should be transferred in such logistics network so as to minimize the total cost. Using the proposed method, the numerical results of near-optimal solutions were determined and then compared to the corresponding lower bound of the optimal solutions. 6.1. Experiments design As shown in Table 1, five sets of testing problems are created to represent the different logistics network structure of Company A in AP. In each problem set, five problem samples are randomly generated. In total, twenty-five samples with different sample size are tested. The proposed heuristic algorithm is coded in C++ and tested on a Pentium IV 1.60 GHz PC with 256 MB of RAM. Other problem related parameters are generated by the following formulae, where the square brackets denote the random number generation from a uniform distribution in the indicated range. • • • • •

Demand for forward products of each customer: DF = [10, 30]; Supply of returned products of each customer: SR = DF; Shipping capacity of arcs connecting the OEM and hybrid processing facilities: U1 = [150, 400]; Shipping capacity of arcs connecting the hybrid processing facilities and customers: U2 = [30, 80]; Shipping capacity of arcs connecting the OEM and customers: U3 = [5, 10];

Table 1 Generated problem sets of integrated distribution Problem set

No. of potential OEM

No. of potential hybrid processing facility

No. of hybrid processing facility to be rented

Capacity of hybrid processing facility

No. of customers

1 2 3 4 5

12 15 20 20 30

14 20 30 30 40

4 6 6 6 8

550 700 900 1200 1700

30 40 50 70 100

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• Transportation cost per item along arcs connecting the OEM and hybrid processing facilities: C1 = [5, 20]; • Transportation cost per item along arcs connecting the hybrid processing facilities and customers: C2 = [10, 30]; • Transportation cost per item along arcs connecting the OEM and customers: C3 = [50, 60]; 6.2. Heuristic parameters setting In order to implement the proposed algorithm efficiently, a sensitivity analysis is performed. As aforementioned, the tabu duration is set to a range with uniform distribution from five to ten. It is also found through preliminary experiments that one is the proper value of the maximum number of nodes to be interchanged between two channels, and three 2-Opt exchanges are sufficient in order to obtain the satisfied solution. The other three parameters, Rmax, Nmax, and Amax are tested in the following sensitivity analysis. The results with different maximum numbers of iteration Rmax are reported in Fig. 8. It is observed that the maximum number of iteration does not make a significant impact to the performance of the proposed heuristic in all the five Problem Sets. It is also noted that the CPU time increases dramatically with the growing of the maximum number of iteration. Fig. 9 illustrates the index of iteration where the final solution is obtained with different maximum number of iteration. It can be seen that about 80% of the final solutions are obtained within 20 iterations. Thus, the maximum number of iteration Rmax = 20 is applied in the subsequent experiments. Respectively, Figs. 10 and 11 show the total costs and average CPU time with different non-improved iteration number Nmax. It can be found that non-improved iteration number Nmax = 5 yields better solutions (lower total costs) in Problem Sets 1, 2, 3 and 4 and spends less average CPU time. Nmax = 15 leads to the best solution and spends less average CPU time in Problem Set 5. Taking the solution quality and computational efficiency into consideration simultaneously, the parameter Nmax = 5 is chosen for Problem Set 1, 2, 3 and 4 while Nmax = 15 is chosen for Problem Set 5. 160000

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4 Problem Set 5

Total Cost ($)

140000 120000 100000 80000 60000 40000 20000 0 10

20

30

40

50

Maximum Number of Iteration

Average Index of Iteration

Fig. 8. Results with different maximum numbers of iteration.

45 40 35 30 25 20 15 10 5 0

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4 Problem Set 5

10

20 30 40 Maximum Number of Iteration

50

Fig. 9. Average index of iteration where the final solution is obtained with different maximum numbers of iteration.

Total Cost ($)

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160000 140000 120000 100000 80000 60000 40000 20000 0

469

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4 Problem Set 5

5

10

15

20

25

Non-improved Iteration Number

Average CPU Time (ms)

Fig. 10. Results with different non-improved iteration number.

5000000

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4 Problem Set 5

4000000 3000000 2000000 1000000 0 5

10

15

20

25

Non-improved Iteration Number Fig. 11. Average CPU times with different non-improved iteration number.

The results with different numbers of iteration of neighborhood search Amax are illustrated in Fig. 12. It is observed that Amax = 15 leads to the best solutions for Problem Sets 1, 2 and 3 while Amax = 25 yield the best results for Problem Sets with larger sample size. Thus, Amax = 15 is applied in the three problem sets with smaller sample size and Amax = 25 is chosen for Problem Sets 4 and 5. 6.3. Results comparison with estimated lower bounds

Total Cost ($)

The lower bound of total cost in the distribution network is obtained by relaxing Capacity Constraint (8) of the hybrid processing facilities for handling the returned products. A linear programming solver, CPLEX of 160000 140000 120000 100000 80000 60000 40000 20000 0

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4 Problem Set 5

5 10 15 20 25 Iteration Number of Neighborhood Search Fig. 12. Results with different iteration number of neighborhood search.

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version 9.1, is adopted for the purpose of solution comparison. The comparison of results between the objective function values obtained from the proposed heuristic and the estimated lower bounds obtained by CPLEX are reported in Table 2. The computational times for the heuristic presented in this paper are fairly consistent within a given problem size, and increase reasonably with increasing problem size. On the other Table 2 Computational results for different problem sets Problem set

Objective function value ($)

CPU time (s)

Lower bound value ($)

CPU time (s)

N = 12 · 14 · 30 1–1 1–2 1–3 1–4 1–5

36 187 37 234 33 366 30 728 28 638

116 108 102 96 112

35 260 36 064 32 838 29 264 27 216

21 18 16 15 20

N = 15 · 20 · 40 2–1 2–2 2–3 2–4 2–5

38 984 36 969 40 292 35 014 44 727

240 220 242 251 252

35 662 34 154 35 662 30 472 39 606

103 95 117 85 132

N = 20 · 30 · 50 3–1 3–2 3–3 3–4 3–5

55 629 48 021 54 899 54 405 56 926

445 530 522 440 452

47 440 45 610 48 136 48 470 48 394

702 801 720 1098 952

N = 20 · 30 · 70 4–1 4–2 4–3 4–4 4–5

89 016 80 303 79 404 88 134 84 395

1148 859 970 1016 951

76 080 68 278 70 388 79 942 822 68

3718 5871 4783 6226 4845

N = 30 · 40 · 100 5–1 5–2 5–3 5–4 5–5

120 962 112 735 108 590 100 731 90 391

2818 2560 2762 3278 2939

102 886 96 836 96 778 89 852 82 202

18 372 20 346 15 983 17 459 15 676

14 12 10 8 6 4 2

Problem Set Fig. 13. Gap between the final solution and lower bound versus problem set.

5-5

5-3

5-1

4-5

4-3

4-1

3-5

3-3

3-1

2-5

2-3

2-1

1-5

1-3

0 1-1

Gap between Final Solution and Lower Bound (%)

16

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11.5914

12

11.9567

10.4097

10 (%)

Solution and Lower Bound

Average Gap between Final

14

10.5353

8 6 4

3.4032

2 0

1

2

3 Problem Set

4

5

Fig. 14. Average gap between the final solution and lower bound versus problem set.

hand, the average computational times for CPLEX are 18, 106.4, 854.6, 5088.6 and 17567.2 s for Problem Sets 1, 2, 3, 4 and 5, respectively, demonstrating that computational times using CPLEX even for the lower bounds deteriorate substantially as problem sizes become large. Furthermore, as the problem size increases to Problem Set 3, 4 and 5, computational times for the proposed heuristic for the objective function values are better on average than using CPLEX for the lower bounds. Thus, the proposed heuristic algorithm is efficient in solving the problems. The gap between the final solutions obtained by the proposed heuristic approach and the lower bound obtained by CPLEX for each Problem Set is illustrated in Fig. 13. Fig. 14 shows the trend of the average gap when the problem size grows. As it can be seen, in Problem Set 1 where there is the smallest sample size, the average gap is less than 4%. In other problem sets, the average gap ranges from 10% to 12%. Since the lower bound of total cost is obtained by relaxing the capacity constraint of the hybrid processing facilities, there exists inherent difference between the optimal solution and the lower bound. Therefore, the gap between the final solution and the lower bound is acceptable, which further demonstrates the good solution quality brought by the proposed heuristic algorithm. 7. Conclusions and limitations This paper has explored the logistics network design for EOL computer products recovery. The deterministic programming model has been proposed by considering the hybrid processing facilities for systematically managing forward and EOL returned product flows. Due to the problem complexity and the large number of variables and constraints, a two-stage heuristic approach has been developed to decompose the integrated design of the multi-echelon forward and reverse logistics distribution networks into a location–allocation problem and a revised network flow problem. The locations of the depots are randomly chosen at the first stage. The optimal solution of the shipment of the forward products and the initial solution of the shipment of the returned products are obtained by applying the network simplex algorithm. According to the capacity constraints of the hybrid processing facilities for handling the returned products, a repair strategy is performed to construct the initial feasible solution of the shipment of returned products. A tabu search algorithm is applied to obtain the improved solution of shipping the returned products. Computational results are generated from a set of twenty-five test problems. The average gap between the final solution obtained by the proposed heuristic approach and the lower bound obtained by CPLEX is less than 4% in Problem Set 1. In other problem sets, the average gap ranges from 10% to 12%. The numerical experiments have suggested that the proposed heuristic solution algorithm performs well in terms of solution quality and computational time consumed. This paper contributes to the research of reverse logistics firstly by developing a mathematical model for the integration of forward and reverse distribution network design, and the locations of facilities jointly used by forward and reverse logistics operations. Secondly, this paper has developed a two-stage heuristic algorithm as the first attempt in solving the integrated forward and reverse logistics network design problem using metaheuristics.

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The fact that this research centered solely on the EOL computer products recovery is a limitation. It should be acknowledged that this particular industry may have greater incentives for investing in reverse logistics because they realize profits from the recovery and resale of returned products. In the proposed model, both the quantities of the returned EOL products and the ordered forward products are assumed to be known in advance. For future research, the problem of the network design for heterogeneous products recovery with uncertain return rate is to be studied. Acknowledgements The authors would like to express their heartfelt appreciation to the two anonymous referees for the valuable comments that improve the content and the presentation of this paper. The valuable suggestions of Professor Wayne K. Talley to improve this paper are also gratefully acknowledged. Appendix A. Definitions of model parameters Definitions of the parameters shown in the proposed model are summarized below.

Notation

Definition

M q CD = {1, . . ., a} RD = {1, . . ., b} D = CD [ RD C = {1, . . ., c} E = RD [ C N=D[C A = a(i, j) S Fk DFn S Rn DRk U Fij U Rji Ul C Fij C Rji Cl

a sufficiently large constant number of hybrid processing facilities to be built up set of potential OEMs set of potential hybrid processing facilities set of potential depots in the logistics network set of customers set of potential hybrid processing facilities and customers set of nodes in the logistics network set of arcs connecting node i and node j in the logistics network, "i,j 2 N quantity of the supply of forward products at node k, "k 2 D quantity of the demand of forward products at node n, "n 2 C quantity of the supply of returned products at node n, "n 2 C quantity of the demand of returned products at node k, "k 2 D shipping capacity of arc a(i, j) in the unit of forward products, "i, j 2 N shipping capacity of arc a(j, i) in the unit of returned products, "j, i 2 N capacity for handling returned products at hybrid processing facility l, "l 2 RD shipping cost per unit of forward products shipped along arc a(j, i), "i, j 2 N shipping cost per unit of returned products shipped along arc a(j, i), "j, i 2 N fixed cost associated with building up hybrid processing facility l, "l 2 RD

Appendix B. Definitions of model variables Definitions of the decision variables shown in the proposed model are summarized below. Notation

Definition

xij yij

quantity of forward products shipped along arc a(i, j), "i, j 2 N quantity of returned products shipped along arc a(j, i), "j, i 2 N  1 if the potential depot i is to be chosed; 8k 2 D ¼ 0 otherwise

zk

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