A genetic algorithm-based heuristic for the dynamic ... - CiteSeerX

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Computers & Operations Research

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A genetic algorithm-based heuristic for the dynamic integrated forward/reverse logistics network for 3PLs Hyun Jeung Ko, Gerald W. Evans∗ Department of Industrial Engineering, University of Louisville, Louisville, Kentucky 40292, USA

Abstract Today’s competitive business environment has resulted in increasing cooperation among individual companies as members of a supply chain. Accordingly, third party logistics providers (3PLs) must operate supply chains for a number of different clients who want to improve their logistics operations for both forward and reverse flows. As a result of the dynamic environment in which these supply chains must operate, 3PLs must make a sequence of inter-related decisions over time. However, in the past, the design of distribution networks has been independently conducted with respect to forward and reverse flows. Thus, this paper presents a mixed integer nonlinear programming model for the design of a dynamic integrated distribution network to account for the integrated aspect of optimizing the forward and return network simultaneously. Since such network design problems belong to a class of NP hard problems, a genetic algorithm-based heuristic with associated numerical results is presented and tested in a set of problems by an exact algorithm. Finally, a solution of a network plan would help in the determination of various resource plans for capacities of material handling equipments and human resources. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: 3PLs; Distribution networks; Reverse logistics; Forward logistics; Genetic algorithms

1. Introduction Today’s competitive business environment has resulted in increasing cooperation among individual companies as members of a supply chain. The success of a company will depend on its ability to achieve effective integration of worldwide organizational relationships within a supply chain [1]. ∗ Corresponding author. Tel.: +1 502 852 0143; fax: +1 502 852 5633.

E-mail address: [email protected] (G.W. Evans). 0305-0548/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2005.03.004

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Beyond the current interest in supply chain management, recent attention has been given to extending the traditional forward supply chain to incorporate a reverse logistic element owing to liberalized return policies, environmental concern, and a growing emphasis on customer service and parts reuse. Implementation of reverse logistics especially in product returns would allow not only for savings in inventory carrying cost, transportation cost, and waste disposal cost due to returned products, but also for the improvement of customer loyalty and futures sales [2–5]. Business processes in most companies are efficiently designed for forward flows only, the reason for this is that reverse logistics has been, often mistakenly, recognized as an unprofitable activity [6]. More specifically, one of the main difficulties associated with implementing reverse logistics activities is the degree of uncertainty in terms of the timing and quantity of products. Thus, managing return flow usually requires a specialized infrastructure and relatively high handling cost and time. For that reason, demand for reverse logistics services from third party logistics providers (3PLs) is increasing [7]. The market for 3PLs was estimated at more than $45 billion in 1999 and is growing by nearly 18 percent annually [8]. In addition, 74% of Fortune 500 companies used 3PLs’ services during 2000. These services involved transportation management, freight payment, warehouse management, shipment tracking, and reverse logistics. Virtually, all of the companies reported positive cost reduction results due to the avoidance of insurance and employee costs and material handling equipment and technology purchases [8]. To account for the integrated aspect of a supply chain, 3PLs such as UPS, FeDex, Genco, etc. thus are playing an increasing role in logistics elements. The main advantage of outsourcing services to 3PLs is that these 3PLs allow companies to get into a new business, a new market, or a reverse logistics program without interrupting forward flows; in addition, logistics costs can be greatly reduced. Some 3PLs offer complete supply chain solutions on warehousing, order fulfillment, and especially value-added services such as repackaging, re-labeling, assembly, light manufacturing, and repair. In addition, 3PLs have also become important players in reverse logistics since the implementation of return operations requires a specialized infrastructure needing special information systems for tracking/capturing data, dedicated equipment for the processing of returns, and specialist trained nonstandard manufacturing processes. As such, this paper deals with warehousing and transportation operations since these are the key operations in a 3PL market. In such operations, clients expect 3PLs to improve lead times, fill rates, back-orders, inventory levels, and return processes, leading to reduced logistics costs in a global market. A prerequisite for meeting these requirements is that the 3PLs have a properly integrated logistics system for both forward and reverse flows, and must operate supply chains for a number of different clients. The requirements for individual clients as well as clients’ markets change over time. As a result of the dynamic environment in which these supply chains must operate, 3PLs must make a sequence of interrelated decisions over time. In order to make these decisions successfully, 3PLs are faced with several complicating factors. For example, a 3PL cannot forecast with much certainty who its clients will be, and hence the location of the clients’ manufacturing facilities or the clients’ markets, the volume of the products to be handled, or even the products themselves. A second complicating factor is the fact that trade-offs may have to be made among the quality measures related to service for the various clients. For example, improving service for one client may result in degradation of service for other clients. In fact, it is extremely difficult to take into account those factors simultaneously in a mathematical model. To handle these problems, we thus employ a dynamic modeling approach. In this approach, a decision maker decides on an appropriate time interval such as monthly, quarterly, or yearly. In each time interval, the parameters are assumed to be deterministic. Accordingly, an appropriate time interval could depend

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on the particular industry. For this integrated network, instead of dealing with separate warehouse or collection centers, we also considered a type of a hybrid warehouse-repair facility. For example, United Parcel Service Logistic Group processes return activities through one of its warehouses in Louisville, Kentucky. An advantage of installing a hybrid facility might include savings as a result of sharing material handling equipment, infrastructure, and so on [9]. Fleischmann et al. [10] showed that network configurations 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 network design problem for 3PLs should be based on an integrated point of view. With consideration of the factors noted above, this paper thus will present a mixed integer nonlinear programming model for the design of a dynamic integrated distribution network for 3PLs. In fact, this type of network design problem belongs to the class of NP-hard problems [11], so that a genetic algorithmbased heuristic will be presented in order to handle a realistically sized problem. Finally, we will apply the proposed model to an example problem and show the numerical results.

2. Literature review Facility location decisions represent an important aspect of strategic planning for supply chain management. These decisions are instrumental in the construction of a distribution network and involve the determination of the sets of locations for facilities (e.g., warehouse, consolidation facilities, repair centers etc.), the capacities of the facilities, and the types of facilities. 3PLs’ logistics networks typically differ from the logistics networks owned by single company. The primary purpose of the company-owned network is to take care of its own products and customers. However, 3PLs’ networks must consider a number of various clients over time. The network design issues can be divided into two categories with respect to the material flows: forward flow and reverse flow. Current 3PLs tend to provide logistics services for both flows. However, most studies of the network problem have involved the separation of the two types of flows. There are three categories of problems addressed based on modeling assumptions in the dynamic capacitated facility location problems. In the first category, the facility capacities in each year are given as parameters in terms of a capacity constraint, and then the problem is concerned with optimal facility locations [12]. The second category allows the facility capacity to be a variable. The capacities are modeled by continuous variables and the optimal value of a capacity in each location is selected in the solution process. Several heuristic procedures for solving continuous expansion sizes have been developed in the works of Jacobsen [13], Rao and Rutenberg [14], and Fong and Srinivasan [15]. The last category deals with the problem where the number of possible expansion sizes is small. For example, this type of problems occurs in telecommunication facilities that produce a small number of telecommunication products in a limited set of sizes [16,17]. Shulman [18] proposed a mixed integer linear programming model in which multiple facility types with finite capacities are considered. In his model, the capacities of the plants over the planning horizon are determined by the placement of multiple facilities at opened plants. He introduced a Lagrangean relaxation method to solve the capacitated dynamic plant location problem. Hinojosa et al. [19] addressed the use of mixed integer programming for solving a multi-period two-echelon multi-commodity capacitated

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location problem. They presented a Lagrangean relaxation method together with a heuristic procedure that constructed feasible solutions for the original problem from the solutions at the lower bounds obtained by the relaxed problems. Canel et al. [20] developed an algorithm to solve the capacitated, multi-commodity, dynamic, multi-stage facility location problem. Their algorithm consisted of two parts: in the first part a branch and bound procedure is used to generate several candidate solutions for each period, and then dynamic programming is used to find an optimal sequence of configurations over the multi-period planning horizon. For a reverse logistics area, there have been relatively few analytical models for the design/operation of reverse flows. However, several authors have published related works, mostly with respect to product recall [21] or end-of-use returns [22–25]. In location models with such reverse logistics, one special characteristic of a reverse network is the existence of a convergence structure from many sources to a few demand locations. A review of these models is given below. Most research has only addressed the static situation. In reuse logistics models, Kroon et al. [24] reported a case study concerning the design of a logistics system for reusable transportation packages. They proposed the use of an MILP in which the major decisions were the determination of the number of containers, the number of container depots and their locations, and the service, distribution, and collection fees. Spengler et al. [25] dealt with the recycling of industrial by-products in the German steel industry. They proposed an MILP model based on the modified multi-level warehouse location problem. The model was solved using a modified Benders decomposition. In recycling logistics models, Barros et al. [26] reported a case study addressing the design of a logistics network for the recycling of sand coming from construction waste in the Netherlands. They presented an MILP model based on a multi-level capacitated warehouse location problem. The model determined the optimal number, capacities, and locations of the depots and cleaning facilities. Louwers et al. [27] considered the design of a recycling network for carpet waste. They proposed a continuous location model that used a linear approximation to the more accurate nonlinear model. In remanufacturing logistics models, Kirkke et al. [23] described a case study, dealing with a reverse logistics network for the returns, processing, and recovery of discarded copiers. They presented an MILP model based on a multi-level uncapacitated warehouse location model. The products taken back from the customers were stored at pre-determined locations (sources) and from there were routed via recovery processing facilities to the demand locations (sinks). The model was used to determine the locations and capacities of the recovery facilities as well as the transportation links connecting various locations. Jayaraman et al. [9] analyzed the logistics network of an electronic equipment remanufacturing company in the USA. They proposed a single period MILP model based on a multi-product capacitated warehouse location model. The model aimed to determine the location of distribution/remanufacturing facilities, the transshipment, production, and stocking of the optimal quantities of the remanufactured products and cores.

3. Modeling a network for 3PLs In this paper, the model for dynamic supply chain management by 3PLs belongs to a class of the multi-period, two-echelon, multi-commodity, capacitated location models. The main differences of this model as compared to existing location models lie in handing forward and reverse flows simultaneously. The network structure of this model is illustrated in Fig. 1. The network consists of the client’s facilities,


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Clients Cli

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Clients Cli

Collectio Colle ion n ccenter

Warehous Wa uses es

3PLs 3PL

Markets Mar

Markets Mar

Fig. 1. An integrated network structure.

warehouses, repair facilities, and market places. In forward flow, manufacturers produce products at factories and store them at warehouses operated by 3PLs from which they are shipped to the customers over time. In reverse flow, we consider repair centers where the inspection and separation are carried out, and then the collected products are shipped to manufacturers over time. A hybrid warehouse-repair facility in this paper is defined as installing a warehouse and a repair center at the same location. In addition, the clients of the 3PL have specific terms for their contracts such as a month, half a year, one year, and so on. Thus, some customers break off a contract, and others start to make a contract at the same time in a certain time period. Thus, 3PLs must handle facility opening, facility closing, and expansion decisions over time in order to manage their networks based on the trade-offs for the various customers. As such, this paper assumes that the locations of clients’ plants and the clients’ markets, together with products to be shipped, are known, and demands of the products and product returns are known over the planning time horizon. We denote index sets by: P = {1, . . . , NP }, set of clients’ product types, I = {1 . . . , NI }, set of clients’ plant locations, J = {1, . . . , NJ }, set of existing warehouses and new potential sites, L = {1, . . . , NL}, set of existing repair facilities and new potential sites, K = {1, . . . , NK}, set of fixed customer locations, T = {1, . . . , NT }, set of time periods. At the beginning of the period, there exists a network structure which includes the entire set of locations for the customers’ plants as well as warehouse/repair facilities where operating facilities exist. There may be a need to redesign a distribution network in case of changes in the number of customers’ plants and the structure of demand patterns in each product over time. As a result, the appropriate timings for warehouse and repair facility openings, expansions, and closings need to be considered. To account for the model complexity, we assume that (1) if there is an expansion decision at any opened facility, this facility would

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stay open over the planned time horizon, (2) there is the maximum number of times over the planning horizon for which expansion can occur, (3) warehouses and repair facilities have limited capacity on an expansion decision, and (4) there are savings associated with opening a hybrid warehouse-repair facility. We denote the parameters by: Mit = the maximum production capacity of client’s plant i; i ∈ I, i ∈ T , Mj t = the maximum capacity of warehouse j ; j ∈ J, t ∈ T , Mlt = the maximum capacity of repair facility l; l ∈ L, t ∈ T , mwj t = the modular expansion size of warehouse j ; j ∈ J, t ∈ T , mr lt = the modular expansion size of repair facility l; l ∈ L, t ∈ T , ubj = the maximum for modular expansion size of warehouse j ; j ∈ J , ubl = the maximum of modular expansion size repair facility l; l ∈ L, p = per unit storage capacity by product p, p ∈ P , dpkt = demand of product p at customer k in period t; p ∈ P , k ∈ K, t ∈ T , rpkt = the amount of returns of product p from customer k in period t; p ∈ P , k ∈ K, t ∈ T , Next, we assume a cost structure that includes transportation costs of products and maintenance costs. f wj t = the fixed operating cost for warehouse j in period t; j ∈ J, t ∈ T , swj t = the setup cost for installing warehouse j in period t; j ∈ J, t ∈ T , ewj t = the fixed cost of expanding modular size on warehouse j in period t; j ∈ J, t ∈ T , vwj t = the operating cost for modular expansion of warehouse j in period t; j ∈ J, t ∈ T , f r lt = the fixed operating cost for repair center l in period t; l ∈ L, t ∈ T , sr lt = the setup cost for installing repair center l in period t; l ∈ L, t ∈ T , er lt = the fixed cost of expanding repair center l in period t; l ∈ L, t ∈ T , vr lt = the variable cost associated with the expansion of repair center l in period t; l ∈ L, t ∈ T , wr t = the costs of savings associated with opening an integrated warehouse-repair facility in period, t ∈ T , f cpijkt = the unit variable cost of serving demand of product p at customer k from plant i and warehouse j in period t, including transportation and handling cost; p ∈ P , i ∈ I , j ∈ J , k ∈ K, t ∈ T , r = unit variable cost of returns of product p from customer k via repair center l to plant i in cpklit period t, including transportation and handling cost; p ∈ P , k ∈ K,l ∈ L, i ∈ I , t ∈ T , The decision variables of the problem are: f

Xpijkt = forward flow: amount of demand of product p at customer k served from plant i and warehouse j in period t; p ∈ P , i ∈ I , j ∈ J , k ∈ K, t ∈ T , r = reverse flow: amount of returns of product p from customer k to be returned via repair Xpklit center l to plant i in period t; p ∈ P , k ∈ K, l ∈ L, i ∈ I0 , t ∈ T , Vj t = the integer value of modularized expansion for warehouse j in period t if warehouse j is installed; j ∈ J , t ∈ T ,


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Wlt = the integer value of modularized expansion for repair center l in period t if repair center l is installed; l ∈ L, t ∈ T , 1, if warehouse j is open in period t, j ∈ J, Zj t = 0, otherwise, Aj t =

Glt =

Blt =

if warehouse j is expanded in period t, j ∈ J, otherwise,

1, 0,

1, if repair center l is open in period t, l ∈ L, 0, otherwise,

if repair center l is expeaded in period t, l ∈ L, otherwise,

1, 0,

(P) Minimize   f wj t Zj t + t∈T

+

j ∈J

j ∈J

and

f r lt Glt +

l∈L

+

l∈L

and

t=1

(ew j t Aj t + vw j t Vj t ) +

and

and

t 2

l∈L

f

f

cpijkt Xpijkt +

p∈P i∈I j ∈J k∈K

f

Xpijkt Mit ,

swj t Zj t (1 − Zj t−1 ) t 2

sr lt Glt (1 − Glt−1 )

(er lt Blt + vr lt Wlt ) −

l∈L

Subject to

j ∈J

st l1 Gl1 +

t=1

j ∈J

+

swj 1 Zj 1 +

=j =1

p∈P k∈K l∈L i∈I0

wr t Zt Gt 

r r , cpklit Xpklit

∀i ∈ I, p ∈ P , t ∈ T ,

(1)

(2)

j ∈J k∈K

f

Xpijkt dpkt ,

∀p ∈ P , k ∈ K, t ∈ T ,

(3)

i∈I j ∈J

p∈P l∈L k∈K

f

p Xpij k Mj t Zj t +

t =1

mwj Vj ,

∀j ∈ J, t ∈ T ,

(4)

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Vj t ubj Aj t ,

l∈L k∈K

l∈L i∈I

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∀j ∈ J, t ∈ T , NT

Zj ,

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(5) ∀j ∈ J, t ∈ T ,

(6)

r Xpklit Mit ,

∀i ∈ I, p ∈ P , t ∈ T ,

(7)

r Xpklit rpkt ,

∀k ∈ K, p ∈ P , t ∈ T ,

(8)

(NT − t + 1)Aj t

(

=t

r Xpklit Mlt Glt +

p∈P i∈I k∈K

Wlt ubl Blt ,

t

mr l Wl ,

∀k ∈ K, ∀l ∈ L, t ∈ T ,

∀l ∈ L, t ∈ T ,

(NT − t − 1)Blt

NT

Gl ,

(9)

=1

(10) ∀l ∈ L, t ∈ T ,

(11)

=t

f

0 Xpijkt ,

r Xpklit ,

p ∈ P , ∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T ,

Vj t ∈ {0, 1, 2, .., ubj }, Wlt ∈ {0, 1, 2, .., ubl }, Zj t , Glt , Aj t , Blt ∈ {0, 1}

∀j ∈ J, ∀l ∈ L, t ∈ T ,

∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T .

(12) (13) (14)

This model has the objective of minimizing the total cost that consists of the costs of fixed operating, opening, and expansion of facilities, transportation costs, the savings from integrated facilities, and expansion costs in forward and reverse flows. Constraint (2) assures that the plants of clients have limited capacities during contract terms. Constraint (3) guarantees that clients’ market demands are satisfied. Constraint (4) is the capacity limitations on warehouses including expansion size across the time period. Constraint (5) ensures that expansion is only possible if a warehouse has already been opened. Constraint (6) also assures that if there is an expansion decision at any facility, this facility would not be closed. Constraint (7) ensures that there are no return flows at unopened facilities. Constraint (8) ensures that the returned products should be sent to clients. Constraint (9) impose the capacity limitations on repair centers including expansion size across the time period. Constraint (10) ensures that expansion is only possible if a repair center has already taken place. Constraint (11) also assures that if there is an expansion decision at any repair center, this facility would not be closed. Constraint (12) preserves the nonnegativity restrictions on the decision variables while constraints (13) and (14) ensures the integer and binary variables, respectively. 4. Solution methodology The decisions to be made in dynamic location problems involve the timing of facility installations on the network while considering various performance measures. However, obtaining optimal solutions for


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dynamic facility location problems in polynomial time is not possible since location problems belong to the class of NP-hard problems [11]. Furthermore, the proposed mathematical model in this paper deals with both forward and reverse flows simultaneously. Also, it includes nonlinear components in the objective function (1) and a large number of constraints. Thus, we propose a genetic algorithm-based heuristic in order to obtain good solutions. Several authors have showed the effectiveness of using a genetic algorithm (GA) for location problems. Gen et al. [28] proposed the spanning tree-based GA for the capacitated plant location problem. Jaramillo et al. [29] suggested the use of a GA as an alternative procedure for generating optimal or near-optimal solutions for location problems. The solution procedure in this paper is a more extended version than that suggested by Jaramillo et al. [29]. In detail, the genetic algorithm-based heuristic is coded in C++. It consists of genetic operations and a simplex method for a transshipment problem. In order to avoid complexity of the constraints, we first divide the original problem into two sub-problems based on forward and reverse flows. Then, we use only the binary and integer decision variables to represent a chromosome for both forward and reverse flows simultaneously. Thus, each chromosome developed in this study is based on an N × M dimensional array, where N is the total number of time periods and M is the number of decision variables related to the candidate facilities. The decision variables represent the decisions of opening, expanding, and amounts of expansion for each possible candidate warehouse and repair center; therefore, M is computed as the product of the number 3, the total number of warehouses, and the total number of repair centers. Next, given the set of these variables using a GA procedure, the decisions of allocating customers to open facilities within capacity limitations should be made, but it is difficult to determine these continuous values of optimal flow decisions using a general GA procedure only. In order to overcome this difficulty, we add a sub-procedure, involving a simplex transshipment algorithm, within an overall GA procedure. Mathematically, the transshipment problem for each flow is shown below: (1) Forward flows Minimize

f

f

cpijkt Xpijkt ,

(15)

t∈T p∈P i∈I j ∈J k∈K

Subject to

f

Xpijkt dpkt ,

∀p ∈ P , k ∈ K, t ∈ T ,

(16)

i∈I j ∈J

p∈P i∈I

f Xpij k Mjt Zjt

f

0 Xpijkt ,

+

t

Vj m ,

∀j ∈ J, k ∈ K, t ∈ T ,

(17)

m=1

p ∈ P , ∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T .

(18)

(2) Reverse flows Minimize

t∈T p∈P k∈K l∈L i∈I

r r cpklit Xpklit ,

(19)

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H.J. Ko, G.W. Evans / Computers & Operations Research Warehouses

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Repair centers

1

2

1

2

T=1

1

1

5

0

0

0

1

0

0

0

0

0

T=2

1

1

6

1

0

0

0

0

0

1

1

4

Fig. 2. A genetic representation scheme.

Subject to l∈L i∈I

p∈P i∈I0

r Xpklit rpkt ,

∀p ∈ P , ∀k ∈ K, t ∈ T ,

r Xpklit Mlt Glt +

r 0 Xpklit ,

t

Wlm ,

∀k ∈ K, ∀l ∈ L, t ∈ T ,

(20)

(21)

m=1

p ∈ P , ∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T .

(22)

As a result, the complex constraints in the overall GA procedure reduce to only capacity constraints, checking total capacities of facilities in each time period. 4.1. Chromosome representation Prior to the application of GA, we need to design suitable chromosomes representing the candidate solutions since this step is a key issue for a successful GA implementation. For example, the representation of a chromosome in this paper is illustrated in Fig. 2. The solution (chromosome) has two warehouses and two repair centers and two time periods. Thus, each chromosome is represented by a 2 × 12 array, where each row represents a time period and the number of column is given by number of warehouses ∗ 3+ number of repair center ∗ 3. Each facility has three genes; the first gene represents opening ( = 1)/closing ( = 0) decisions with binary strings; the second gene represents the expansion ( = 1)/no expansion ( = 0) decisions using binary strings; the third gene represents the amounts of expansion which is obtained by multiplying the value of the third gene (integer values) and a predetermined modular expansion size. Instead of handling continuous variables for the amount of expansion, we assume that there is a modular expansion size since expanding a space for just a few units does not occur in reality. In detail, if the modular expansion size is 100, the total expansion of warehouse 1 in time T = 1 is obtained by multiplying the third gene of warehouse 1 ( = 5) and 100. Consequently, the total expansion becomes 500 ( = 5*100). 4.2. Genetic operators 4.2.1. Cloning operator The cloning operator involves keeping the best solutions. In our algorithm, the procedure works in such a way that it copies the top 20% of the chromosomes of a population to a new population.


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4.2.2. Parent selection operator The parent selection operator is an important process that directs a GA search toward promising regions in a search space. Two parents are selected from the solutions of a particular generation by selection methods that assign reproductive opportunities to each individual parent in the population. There are several selection methods, such as roulette wheel selection, tournament selection, rank selection, elitism selection, random selection and so on [28]. For this study, we used a binary tournament selection that works by forming two teams of chromosomes [30]. Each team consists of two chromosomes randomly drawn from the current population. The two best chromosomes that are taken from one of the two teams are chosen for crossover operation. In this way, two offsprings are generated and entered into a new population. 4.2.3. Crossover operator The crossover operator generates new children by combining the information contained in the chromosomes of the parents so that new chromosomes will have the good parts of the parents’ chromosomes. A crossover probability indicates how often crossover will be performed. There are several types of crossovers, including single-point crossover, multi-point crossover, and uniform crossover [28]. Herein, we applied the two-point crossover in which one is used for warehouses and the other one for repair centers. The two locations of the crossover points are randomly selected in only opening/closing decisions of facilities in the initial time period (T = 1) since expansion decisions are dependent upon installation decisions. Then, the blocks of the two parents’ strings are swapped to produce two children. Fig. 3 shows the detailed procedure. 4.2.4. Mutation operator After recombination, some children undergo mutation. Mutation operates by inverting each bit in the solution with some small probability, usually from zero percent to 10 percent. The rationale is to provide a small amount of randomness, and to prevent solutions from being trapped at a local optimum. The type of mutation varies depending on the encoding as well as the crossover. In the GA used for this work, the mutation operator first randomly selects a time period and a bit value of only opening/closing decision variables on a chromosome. Then, a bit value is flipped from 0 to 1, or from 1 to 0. If the changed bit value is 0, the corresponding two bits for expansion and amount of expansion are changed to zero; otherwise they are randomly generated. Hence, a good level of diversity in each generation is achieved. 4.3. Fitness function Decoding the chromosome generates a candidate solution and its fitness value based on the fitness function. The fitness value is the measure of goodness of a solution with respect to the original objective function and the amount of infeasibility. The fitness function is formed by adding a penalty to the original objective function. In detail, the components of the original objective function are the opening costs of facilities, the operating costs of facilities, the expansion costs of facilities, the cost savings from integrated facilities, and transportation costs in forward and reverse flows. In particular, we first calculate the cost components of the objective function, except for the transportation costs. Then, based on the set of the variables derived from the chromosomes, a fitness value of each chromosome is obtained by applying a simplex transshipment algorithm for optimal customer allocation to the opened facilities. Finally, the penalty function is needed when some candidate solutions in a

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Fig. 3. The description of crossover operation.

population turn out to be infeasible, exceeding the capacity limit of some warehouses or repair centers. Whenever each facility in any time period exceeds the capacity limit, the penalty value is assessed and is subsequently added to the original objective function. A penalty value is considerably larger than any possible objective value corresponding to the current population of individuals. The penalty function is mathematically expressed as follows: f Penalty function = pv × f Xpijkt , Mjt , Zjt , Vjt j ∈J t∈T

+

l∈L t∈T

r pv × f (Xpklit , Mlt , Glt , Wlt ),

(23)

where pv is the penalty value. t f f f Xpij k , Mjt , Zjt , Vjt = 1 if Xpij k > Mjt Xjt + Vj m , otherwise 0,

(24)

r r r f Xpklit , Mlt , Glt , Wlt = 1 if Xpklit > Mlt Glt + Wlm , otherwise 0.

(25)

p∈P i∈I

p∈P i∈I

m=1

m=1


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4.4. A overall genetic algorithm-based heuristic procedure After choosing an appropriate representation scheme, the overall algorithm of the proposed genetic based heuristic can be described as follows: (1) Read the required data and generate an initial population based on population size, in which each chromosome is a two dimensional array, representing decision values for warehouses and repair centers according to each period of time. In each chromosome, first, the opening( = 1)/closing( = 0) decision of any facility in each period is randomly made. Second, if a facility is open, an expansion and amounts of expansion decisions are randomly assigned; if a facility is closed, the values of decision variables are zero. (2) Set the generation to be zero and evaluate the fitness function of each chromosome in a population. The fitness function is the sum of the objective function of the original optimization problem and the penalty function. The objective function is calculated from a chromosome itself and the sub-algorithm used for the transshipment problem. The penalty function is obtained by checking the violation of capacity limits of the facilities. (3) Create a new population by repeating generation operations (cloning, parent selection, crossover, and mutation) until the new population is complete. The combined roulette wheel and elitism method is used for the parent selection method. Two-point crossover and random mutation are used on positions in a chromosome. (4) Replace new offsprings in a new population. (5) Stop the iterations if the end condition is satisfied; otherwise, go to the next generation.

5. A base-line case The GA described in the previous chapter has been applied to a base-line model for a 3PL. There were two clients, ten possible warehouses and repair centers, and a three-period planning horizon. The potential locations for market clusters, warehouses, repair centers, and plants of clients were generated from a uniform distribution with minimum and maximum distances of 0 and 150, respectively on the x and y coordinate system. Also, demands of the customer zones were assumed to be known and then generated from a uniform distribution with minimum of 90 units and a maximum of 120 units. The amount of returns was assumed to be 10% of the customer demands. The details of the data are summarized in Tables 1–3. For simplification due to extensive data requirements, we assume that storage space per product (p ) is taken as one, and cost parameters are constant over the planning horizon. Additional parameters of the base-line model are shown in Table 4. The mixed integer program associated with the base-line case has 1800 continuous variables, 180 integer variables, and 372 constraints. This problem is solved by the proposed GA-based heuristic where the parameter values were set through extensive experiments. These values are as follows: population size = 300, maximum number of generations = 50, cloning = 20%, crossover rate = 80%, and mutation rate = 0.2–0.5%. The problem is executed on a PC with Pentium IV CPU 3.00 GHz processor. The solution required about 8.77 min. Fig. 4 shows the best fitness values at each generation as a function of the number of generations. Tables 5 and 6 show the best solution with an objective function value of $1,274,860. There are two opening facilities for forward flows where warehouse (3) is open at the beginning of the period and is

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Table 1 Facility data in the base-line model Index

Warehouse

1 2 3 4 5 6 7 8 9 10

Repair center

Plant of client

x

y

x

y

x

y

Mit

122.67 36.56 116.01 8.16 97.32 130.63 137.39 98.50 41.64 82.41

34.76 94.63 64.09 140.26 1.06 65.96 110.19 95.59 95.97 146.71

122.67 36.56 116.01 8.16 97.32 130.63 137.39 98.50 41.64 82.41

34.76 94.63 64.09 140.26 1.06 65.96 110.19 95.59 95.97 146.71

38.73 117.05

100.55 60.94

3000 6000

Table 2 Customer data of client 1 in the base-line model Index

Client 1 t =1

Coordinate

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

t =2

T =3

x

y

Demand

Return

Demand

Return

Demand

Return

148.06 24.8 35.48 138.98 98.6 28.22 91.95 94.79 22.03 84.21 60.34 117.39 34.96 10.03 118.57

31.36 16.81 12.38 127.8 138.7 14.07 17.89 48.55 136.17 97.08 89.21 0.72 14.13 52.38 115.08

104 94 96 108 110 94 0 0 0 0 0 0 0 0 0

11 11 10 11 10 11 0 0 0 0 0 0 0 0 0

102 105 95 95 107 103 98 99 106 99 0 0 0 0 0

11 11 11 10 11 11 11 11 11 10 0 0 0 0 0

101 94 99 100 108 94 109 95 100 101 93 101 96 97 108

10 10 10 11 10 10 11 10 10 10 10 10 10 10 10

expanded; warehouse (2) is open at the beginning of the second period and expanded. In reverse flows, there also two opening facilities where repair centers (2) and (3) are open at the beginning of the first period; repair center (2) is expanded at the third time period. The locations of these opened repair centers are the same as the warehouses so that opening hybrid warehouse-repair facilities are recommended for achieving possible cost savings.


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Table 3 Customer data of client 2 in the base-line model Index

Client 2 t =1

Coordinate

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

t =2

T =3

x

y

Demand

Return

Demand

Return

Demand

Return

85.06 142.28 146.86 72.97 89.88 13.93 9.30 35.05 121.52 94.80 132.83 26.38 95.08 108.91 141.36

74.91 53.39 126.00 86.95 141.82 52.39 37.89 44.99 2.90 125.85 29.85 46.43 139.79 98.62 37.98

191 207 205 199 202 197 202 200 196 0 0 0 0 0 0

20 20 21 20 21 21 20 21 20 0 0 0 0 0 0

0 0 0 0 207 210 195 191 208 204 201 197 201 194 196

0 0 0 0 20 20 21 21 21 20 21 21 20 20 21

207 198 194 201 203 205 205 196 194 206 200 204 210 201 195

21 20 20 21 20 20 20 20 21 20 20 20 20 21 21

6. An equivalent linear model with experimentation In order to assess the computational effectiveness of the GA, the original mathematical model was converted into a linear model through the use of dummy variables and additional constraints owing to the nonlinear components in the objective function. There are three nonlinear terms to be considered, dealing with the costs of opening warehouses, the costs of opening repair centers, and the costs of savings over the planning horizon. The transformed objective function is as follows:

Minimize +

l∈L

+

t∈T





j ∈J

j ∈J

f r lt Glt + l∈L

and

t=1

(ew jt Ajt + vw jt Vjt ) +

p∈P i∈I j ∈J k∈K

and

t=1

l∈L

j ∈J

and

t 2

l∈L f

f

and

swjt Zjt t 1

sr lt Glt

(er lt Blt + vr lt Wlt ) −

cpijkt Xpijkt +

swj 1 Zj 1 +

sr t1 Gl1 +

j ∈J

+

f wjt Zjt +

=j =l

p∈P k∈K l∈L i∈I0

wr t Ht 

r r . cpklit Xpklit

(26)

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Table 4 The parameters of the base-line model Fixed operating cost per warehouse Setup cost for installing a warehouse Maximum capacity of warehouses Modular expansion size of warehouse Fixed cost of expanding warehouse Variable cost of modular expansion for a warehouse Fixed operating cost per repair center Setup cost for installing a repair center Maximum capacity of repair center Modular expansion size of repair center Fixed cost of expanding repair center Variable cost of modular expansion for repair center Savings from opening an hybrid warehouse-repair facility Maximum of modular expansion of warehouse Maximum of modular expansion of repair facility Unit transportation cost of client’s plant-warehouse

f wjt swjt Mjt mwjt ew jt vwjt f r lt sr lt Mlt mr lt er lt vr lt wr t ubjt ublt f cij t

$18,000 $120,000 3000 units 600 units $2,100 $15,000 $3,000 $30,000 300 units 100 units $200 $4,500 $6,000 4 4 0.05

Unit transportation cost of warehouse-customer Euclidean distance between locations x and y in time t Unit forward sipping cost Unit transportation cost of customer-collection center Unit transportation cost of repair center-client’s plant Unit backward shipping cost

cj kt kxyt f cpijkt r ckl r cli r cpklit

0.5

f

f

f

cij t kij t + cj kt kj kt 0.05 1.5 r k + cr k ckl kl li li

To elaborate, Zjt , Gjt , and Ht were added as dummy variables. First, through the use of Zjt , the following constraints were added into the set of original constraints: Zjt + Zj t−1 − Zjt 0,

∀j ∈ J, t ( 2) ∈ T ,

(27)

Zjt + Zj t−1 + Zjt 2,

∀j ∈ J, t ( 2) ∈ T ,

(28)

−2Zjt + Zj t−1 + Zjt 1, 2Zjt − Zj t−1 − Zjt 1,

∀j ∈ J, t ( 2) ∈ T , ∀j ∈ J, t ( 2) ∈ T .

(29) (30)

Constraint (27) assures that if Zjt = 0 and Zj t−1 = 0, Zjt should be zero; constraint (28) ensures that if Zjt = 1 and Zj t−1 = 1, Zjt should be zero; constraint (29) assures that Zjt = 0 and Zj t−1 = 1, Zjt should be zero; constraint (30) ensures that Zjt = 1 and Zj t−1 = 0, Zjt should be one. Second, by the use of Gjt , the same logic was applied and then the following constraints should be also added as follows: Gjt + Gj t−1 − Gjt 0,

∀l ∈ L, t ( 2) ∈ T ,

(31)

Gjt + Gj t−1 − Gjt 2,

∀l ∈ L, t ( 2) ∈ T ,

(32)


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2.50E+06

fitness value

2.00E+06

1.50E+06

1.00E+06

5.00E+05

0.00E+00 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 generation

Fig. 4. Convergence of fitness values.

Table 5 The summary of the solutions in the base-line model Index

Warehouse

Repair centers

2

T =1 T =2 T =3

3

2

3

Z

A

V

Z

A

V

G

B

W

G

B

W

0 1 1

0 1 0

0 1 0

1 1 1

1 0 0

1 0 0

1 1 1

0 0 1

0 0 2

1 1 1

0 0 0

0 0 0

Table 6 The cost summary of the base-line model Cost components Cost of operating warehouses Cost of forward transportation Cost of operating repair centers Cost of reverse transportation Cost savings

$382,200 $724,765 $87,600 $116,298 $3,600

Total cost

$1,274,860

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Table 7 The results of the test problems No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

P

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3

W

5 5 5 5 5 5 10 10 10 10 10 10 10 15 20 10 15 20

R

5 5 5 5 5 5 10 10 10 10 10 10 10 15 20 10 15 20

C

T

30 50 70 30 50 70 30 50 70 30 50 70 60 120 180 60 120 180

3 3 3 4 4 4 3 3 3 4 4 4 3 4 5 3 4 5

Optimal solution

Genetic heuristic

Gap

OBJ* ($)

Time (m)

OBJ ($)

Time (m)

(%)

86,597 116,091 174,244 93,731 124,805 191,206 87,127 145,842 174,064 94,289 432,537 508,945 1,244,980 N/A N/A 2,243,190 N/A N/A

0.11 0.08 8.33 0.21 0.11 128.11 2.00 4.17 358.13 3.00 316.00 27.28 4.48 N/A N/A 4.02 N/A N/A

87,272 117,018 175,899 97,223 129,239 199,682 88,726 148,804 180,463 100,889 462,518 550,588 1,274,860 3,162,030 4,818,570 2,340,650 3,927,340 7,209,830

2.06 2.45 5.40 2.08 2.75 5.25 2.10 6.15 10.63 2.33 6.53 11.68 8.77 86.70 207.98 12.81 74.33 219.95

0.78 0.80 0.95 3.73 3.55 4.43 1.84 2.03 3.68 7.00 6.93 8.18 2.40 N/A N/A 4.34 N/A N/A

P: the total number of clients in each time period; W: the total number of warehouses in each time period; R: the total number of repair centers in each time period; C: the total number of customer zones in each time period; T: the total number of time periods.

−2Gjt + Gj t−1 + Gjt 1, 2Gjt − Gj t−1 − Gjt 1,

∀l ∈ L, t ( 2) ∈ T , ∀l ∈ L, t ( 2) ∈ T .

(33) (34)

Finally, for Ht calculating cost savings from opening hybrid facilities, additional constraints were added into the set of original constraints as follows: Zjt + Glt − 2H(=j =l)t 0, Zjt + Glt − H(=j =l)t 1,

∀j ∈ J, ∀l ∈ L, t ∈ T , ∀j ∈ J, ∀l ∈ L, t ∈ T .

(35) (36)

Constraint (35) assures that if either Zjt or Glt is 0 where j = l, Ht should be zero. Constraint (36) ensures that if both Zjt = 1 and Glt = 1 where j = 1, Ht should be one. As such, a total 18 test problems of varying size were constructed in consideration of computation time and the data requirements. The coordinates of locations were generated as uniformly distributed random numbers on the intervals [0, 150]. Also demands of customers for three clients were generated uniformly on the intervals [90, 110], [190, 210], and [290, 310], respectively. Optimal solutions were obtained by applying the LINGO mathematical programming software [31] to solve the test problems on a PC with Pentium IV CPU 3.00 GHz processor. In fact, the optimal solutions were not obtained for problems (14), (15), (17) and (18) shown in Table 7 because of the


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increased computation time requiring more than 24 h. The parameter values for the proposed GA were set through extensive experiments. These values are as follows: population size = 300, maximum number of generation = 50, cloning = 20%, crossover rate = 80%, and mutation rate = 0.2–0.5%. The summary of the results from all of the test problems is as shown in Table 7. The first column of the table indicates the index for test problems. The second through sixth columns specify the problem dimensions. The next two columns provide information on the performances of objective values ($) and computing time (in minutes). The last column indicates the gaps measured by 100 (OBJ − OBJ∗ )/OBJ∗ . On the basis of the results from Table 7, the range of gaps with respect to solution quality is from 0.78% of problem (1) to 8.18% of problem (12). On average, the gap performances within the same time period such as those of problems (1)–(3), problems (4)–(6), problems (7)–(9), and problems (10)–(12) does not significantly vary although the increases of the number of warehouses, repair centers, and customers are allowed. Also, the differences in the gap performances is almost unaffected by adding more clients, showing the values of 3.68% of problem (9), 2.40% of problem (13), and 4.34% of problem (16). However, the results suggest that the trend of the gaps may continuously deviate from optimal solution as the total number of time periods increases. To facilitate this, the two group of test problems (1)–(3) and (4)–(6) show that the average gap shifts from 0.84% to 3.9%, respectively when increasing only one more time period; for the groups of problems (7)–(9) and (10)–(12), the average gap shifts from 2.51% to 7.37%. As for computation time considerations, it is difficult to conclude that the exact approach outperforms the GA since no consistent pattern reported in Table 5 was detected. In fact, the results indicate that the performance strongly depends on the problem structures to be solved. To elaborate, problem (11) required more time to solve than problem (12) although problem dimension was small. Furthermore, problems (8), (10), and (13) apparently were larger in problem dimension than problem (6), but the computing time of problem (6) was almost about 30 times as long as those of them. The key possible factors of affecting the computing complexity in the proposed mathematical model may result from a total number of time periods and demand pattern across the planning horizon. Especially for problems (14), (15), (17) and (18), they all have over four time periods and fail to solve by an exact solution approach. However, the proposed heuristic solved them less than 4 h. Thus, this may be explained by the fact why a heuristic approach is needed for the proposed problem setting. This lends hope that implementation of the proposed heuristic will enable much larger problems to be solved.

7. Conclusions and future works A growing number of companies have begun to realize the importance of implementing integrated supply chain management since they are under pressure for filling customers’ orders on time as well as for efficiently taking returned products back from customers after selling products. In terms of product flows, there are both forward flows and reverse flows in an integrated supply chain. 3PLs are playing an increasing role in supporting such integrated supply chain management using sophisticated information systems and dedicated equipments. Thus, the objective of this study is to develop an optimization model and associated algorithm to design an integrated logistics network for 3PLs. In order to formulate the problem to be solved for 3PLs, we presented a mixed integer nonlinear programming model that is a multi-period, two-echelon, multi-commodity, capacitated network design problem, considering forward and reverse flows simultaneously. Since such network design problems belong to a class of NP hard problems, various heuristics have been developed in order to solve a

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realistically sized problem. However, no dominant approach has been reported for solving network design problems in terms of computation time and a degree of optimality. Therefore, we proposed a GA-based heuristic that consists of genetic operations and simplex transshipment algorithm. The solutions obtained from the proposed method were compared to the optimal solutions using the 18 test problems. The results indicated that the rage of gaps with respect to solution quality was from 0.78% to 8.18%. In addition, the proposed heuristic solved all the test problems in reasonable amount of computation time; the exact solution approach did not solve some of the test problems owing to complexity of the problem structure. Finally, an interesting extension of this work is to compare the proposed solution method to other heuristics involving, for example, Lagrangean relaxation, tabu search, and scatter search. Acknowledgements We acknowledge the helpful comments and suggestions of the editor and two anonymous referees. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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