Computers & Operations Research ( ) –
www.elsevier.com/locate/cor
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 supplychain. Accordingly, third partylogistics providers (3PLs) must operate supplychains 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 independentlyconducted 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 byan 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 supplychain. The success of a companywill depend on its abilityto achieve
effective integration of worldwide organizational relationships within a supplychain[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
Beyond the current interest in supply chain management, recent attention has been given to extend-ing the traditional forward supplychain to incorporate a reverse logistic element owextend-ing to liberalized return policies, environmental concern, and a growing emphasis on customer service and parts reuse. Implementation of reverse logistics especiallyin product returns would allow not onlyfor savings in inventorycarrying 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 efficientlydesigned 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 uncertaintyin terms of the timing and quantityof products. Thus, managing return flow usuallyrequires a specialized infrastructure and relativelyhigh handling cost and time. For that reason, demand for reverse logistics services from third partylogistics providers (3PLs) is increasing [7]. The market for 3PLs was estimated at more than $45 billion in 1999 and is growing bynearly 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 supplychain, 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 greatlyreduced. Some 3PLs offer complete supplychain solutions on warehousing, order fulfillment, and especiallyvalue-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, inventorylevels, and return processes, leading to reduced logistics costs in a global market. A prerequisite for meeting these requirements is that the 3PLs have a properlyintegrated logistics system for both forward and reverse flows, and must operate supplychains 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 inter-related 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 certaintywho 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 mayhave to be made among the qualitymeasures related to service for the various clients. For example, improving service for one client mayresult in degradation of service for other clients. In fact, it is extremelydifficult to take into account those factors simultaneouslyin a mathematical model.
To handle these problems, we thus employa 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
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
config-urations involving both forward and reverse flows were different with respect to a sequential solution approach and an integrated solution approach. Theyfound 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 algorithm-based heuristic will be presented in order to handle a realisticallysized problem. Finally, we will apply the proposed model to an example problem and show the numerical results.
2. Literature review
Facilitylocation decisions represent an important aspect of strategic planning for supplychain man-agement. 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 bysingle company. The primarypurpose of the company-owned net-work is to take care of its own products and customers. However, 3PLs’ netnet-works 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 facilitylocation problems. In the first category, the facilitycapacities in each year are given as parameters in terms of a capacityconstraint, and then the problem is concerned with optimal facility locations[12]. The second categoryallows the facilitycapacityto be a variable. The capacities are modeled bycontinuous variables and the optimal value of a capacityin 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 facilitytypes with finite capacities are considered. In his model, the capacities of the plants over the planning horizon are determined bythe 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-commoditycapacitated
location problem. Theypresented 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 relativelyfew analytical models for the design/operation of reverse flows. However, several authors have published related works, mostlywith 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 manysources to a few demand locations. A review of these models is given below. Most research has onlyaddressed the static situation.
In reuse logistics models, Kroon et al.[24]reported a case studyconcerning 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 studyaddressing 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 recoveryof discarded copiers. Theypresented 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 recoveryfacilities 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. Theyproposed 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 supplychain management by3PLs 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 inFig. 1. The network consists of the client’s facilities,
Mar Markets
Co
Collellectioion cn center
Cli Clients Wa Warehoususeses Cli Clients Mar Markets 3PL 3PLs
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 by3PLs from which theyare 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 facilityin 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 facilityopening, facilityclosing, 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 facilityopenings, 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
stayopen 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 capacityon 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 capacityof client’s plant i; i ∈ I, i ∈ T , Mjt= the maximum capacityof warehouse j; j ∈ J, t ∈ T ,
Mlt= the maximum capacityof repair facilityl; l ∈ L, t ∈ T , mwjt= the modular expansion size of warehouse j; j ∈ J, t ∈ T , mrlt= 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 capacitybyproduct 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 wjt= the fixed operating cost for warehouse j in period t; j ∈ J, t ∈ T ,
swjt= the setup cost for installing warehouse j in period t; j ∈ J, t ∈ T ,
ewjt= the fixed cost of expanding modular size on warehouse j in period t; j ∈ J, t ∈ T , vwjt= the operating cost for modular expansion of warehouse j in period t; j ∈ J, t ∈ T , f rlt= the fixed operating cost for repair center l in period t; l ∈ L, t ∈ T ,
srlt= the setup cost for installing repair center l in period t; l ∈ L, t ∈ T , erlt= the fixed cost of expanding repair center l in period t; l ∈ L, t ∈ T ,
vrlt= the variable cost associated with the expansion of repair center l in period t; l ∈ L, t ∈ T , wrt= the costs of savings associated with opening an integrated warehouse-repair facility
in period,t ∈ T ,
cfpijkt= 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 ,
crpklit= unit variable cost of returns of product p from customer k via repair center l to plant i in
period t, including transportation and handling cost;p ∈ P , k ∈ K,l ∈ L, i ∈ I, t ∈ T ,
The decision variables of the problem are:
Xfpijkt= forward flow: amount of demand of product p at customer k served from plant i and
warehouse j in periodt; p ∈ P , i ∈ I, j ∈ J , k ∈ K, t ∈ T ,
Xr
pklit= reverse flow: amount of returns of product p from customer k to be returned via repair
center l to planti in period t; p ∈ P , k ∈ K, l ∈ L, i ∈ I0,t ∈ T ,
Vjt= the integer value of modularized expansion for warehouse j in period t if
Wlt= the integer value of modularized expansion for repair center l in period t if repair center l is installed; l ∈ L, t ∈ T ,
Zjt =
1, if warehouse j is open in period t, j ∈ J, 0, otherwise,
Ajt =
1, if warehouse j is expanded in period t, j ∈ J, 0, otherwise,
Glt=
1, if repair center l is open in period t, l ∈ L, 0, otherwise,
Blt=
1, if repair center l is expeaded in period t, l ∈ L, 0, otherwise, (P) Minimize t∈T j∈J f wjtZjt + j∈J andt=1 swj1Zj1+ j∈J and t2 swjtZjt(1 − Zjt−1) + l∈L f rltGlt+ l∈Landt=1 stl1Gl1+ l∈Landt2 srltGlt(1 − Glt−1) + j∈J (ewjtAjt + vwjtVjt) + l∈L (erltBlt+ vrltWlt) − =j=1 wrtZtGt + p∈P i∈I j∈J k∈K cfpijktX f pijkt+ p∈P k∈K l∈L i∈I0 crpklitXpklitr , (1) Subject to j∈J k∈K XfpijktMit, ∀i ∈ I, p ∈ P, t ∈ T , (2) i∈I j∈J Xfpijktdpkt, ∀p ∈ P, k ∈ K, t ∈ T , (3) p∈P l∈L k∈K pXfpijkMjtZjt + t =1 mwjVj, ∀j ∈ J, t ∈ T , (4)
VjtubjAjt, ∀j ∈ J, t ∈ T , (5) (NT − t + 1)Ajt NT =t Zj, ∀j ∈ J, t ∈ T , (6) l∈L k∈K Xpklitr Mit, ∀i ∈ I, p ∈ P, t ∈ T , (7) l∈L i∈I Xrpklitrpkt, ∀k ∈ K, p ∈ P, t ∈ T , (8) p∈P i∈I k∈K Xpklitr MltGlt+ t =1 mrlWl, ∀k ∈ K, ∀l ∈ L, t ∈ T , (9) WltublBlt, ∀l ∈ L, t ∈ T , (10) (NT − t − 1)Blt NT =t Gl, ∀l ∈ L, t ∈ T , (11)
0Xfpijkt, Xpklitr , p ∈ P, ∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T , (12) Vjt ∈ {0, 1, 2, .., ubj}, Wlt ∈ {0, 1, 2, .., ubl}, ∀j ∈ J, ∀l ∈ L, t ∈ T , (13)
Zjt, Glt, Ajt, Blt ∈ {0, 1} ∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T . (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 capacitylimitations on warehouses including expansion size across the time period. Constraint (5) ensures that expansion is onlypossible if a warehouse has alreadybeen opened. Constraint (6) also assures that if there is an expansion decision at anyfacility, this facilitywould 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 capacitylimitations on repair centers including expansion size across the time period. Constraint (10) ensures that expansion is only possible if a repair center has alreadytaken place. Constraint (11) also assures that if there is an expansion decision at anyrepair center, this facilitywould 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
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 byJaramillo 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 complexityof the constraints, we first divide the original problem into two sub-problems based on forward and reverse flows. Then, we use onlythe binaryand integer decision variables to represent a chromosome for both forward and reverse flows simultaneously. Thus, each chromosome developed in this study is based on anN ×M dimensional array, whereN 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 capacitylimitations 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 t∈T p∈P i∈I j∈J k∈K cpijktf Xpijktf , (15) Subject to i∈I j∈J Xpijktf dpkt, ∀p ∈ P, k ∈ K, t ∈ T , (16) p∈P i∈I XfpijkMjtZjt+ t m=1 Vjm, ∀j ∈ J, k ∈ K, t ∈ T , (17) 0Xpijktf , 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 cpklitr Xpklitr , (19)
Warehouses 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 Xpklitr rpkt, ∀p ∈ P, ∀k ∈ K, t ∈ T , (20) p∈P i∈I0 XrpklitMltGlt+ t m=1 Wlm, ∀k ∈ K, ∀l ∈ L, t ∈ T , (21) 0Xpklitr , p ∈ P, ∀i ∈ I, ∀j ∈ J, ∀l ∈ L, t ∈ T . (22)
As a result, the complex constraints in the overall GA procedure reduce to onlycapacityconstraints, 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 keyissue for a successful GA implementation. For example, the representation
of a chromosome in this paper is illustrated inFig. 2. The solution (chromosome) has two warehouses and
two repair centers and two time periods. Thus, each chromosome is represented bya 2× 12 array, where each row represents a time period and the number of column is given bynumber of warehouses∗3+ number of repair center∗3. Each facilityhas three genes; the first gene represents opening (= 1)/closing (= 0) decisions with binarystrings; the second gene represents the expansion ( = 1)/no expansion ( = 0) decisions using binarystrings; 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 timeT = 1 is obtained bymultiplying 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 waythat it copies the top 20% of the chromosomes of a population to a new population.
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 byselection 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 byforming 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 bycombining the information contained in the chro-mosomes of the parents so that new chrochro-mosomes will have the good parts of the parents’ chrochro-mosomes. A crossover probabilityindicates 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 cen-ters. The two locations of the crossover points are randomlyselected in onlyopening/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. 3shows the detailed procedure.
4.2.4. Mutation operator
After recombination, some children undergo mutation. Mutation operates byinverting 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 randomlyselects a time period and a bit value of onlyopening/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 theyare randomlygenerated. Hence, a good level of diversityin 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 byadding a penaltyto 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 transporta-tion costs. Then, based on the set of the variables derived from the chromosomes, a fitness value of each chromosome is obtained byapplying a simplex transshipment algorithm for optimal customer alloca-tion to the opened facilities. Finally, the penalty funcalloca-tion is needed when some candidate solualloca-tions in a
Fig. 3. The description of crossover operation.
population turn out to be infeasible, exceeding the capacitylimit of some warehouses or repair centers. Whenever each facilityin anytime period exceeds the capacitylimit, the penaltyvalue is assessed and is subsequentlyadded to the original objective function. A penaltyvalue is considerablylarger than any possible objective value corresponding to the current population of individuals. The penaltyfunction is mathematicallyexpressed as follows: Penaltyfunction= j∈J t∈T pv × f Xpijktf , Mjt, Zjt, Vjt + l∈L t∈T pv × f (Xrpklit, Mlt, Glt, Wlt), (23)
wherepv is the penaltyvalue.
fXfpijk, Mjt, Zjt, Vjt = 1 if p∈P i∈I Xpijkf > MjtXjt+ t m=1 Vjm, otherwise 0, (24) fXrpklit, Mlt, Glt, Wlt = 1 if p∈P i∈I Xpklitr > MltGlt+ r m=1 Wlm, otherwise 0. (25)
4.4. Aoverall 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
anyfacilityin each period is randomlymade. Second, if a facilityis open, an expansion and amounts of expansion decisions are randomlyassigned; if a facilityis 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 penaltyfunction. The objective function is calculated from a chromosome itself and the sub-algorithm used for the transshipment problem. The penaltyfunction is obtained bychecking the violation of capacity limits of the facilities.
(3) Create a new population byrepeating 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, respectivelyon 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 inTable 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 bythe 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. 4shows the best fitness values at each generation as a function of the number of generations.
Tables 5and6show 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
Table 1
Facilitydata in the base-line model
Index Warehouse Repair center Plant of client
x y x y x y Mit 1 122.67 34.76 122.67 34.76 38.73 100.55 3000 2 36.56 94.63 36.56 94.63 117.05 60.94 6000 3 116.01 64.09 116.01 64.09 4 8.16 140.26 8.16 140.26 5 97.32 1.06 97.32 1.06 6 130.63 65.96 130.63 65.96 7 137.39 110.19 137.39 110.19 8 98.50 95.59 98.50 95.59 9 41.64 95.97 41.64 95.97 10 82.41 146.71 82.41 146.71 Table 2
Customer data of client 1 in the base-line model Index Client 1
Coordinate t = 1 t = 2 T = 3
x y Demand Return Demand Return Demand Return
1 148.06 31.36 104 11 102 11 101 10 2 24.8 16.81 94 11 105 11 94 10 3 35.48 12.38 96 10 95 11 99 10 4 138.98 127.8 108 11 95 10 100 11 5 98.6 138.7 110 10 107 11 108 10 6 28.22 14.07 94 11 103 11 94 10 7 91.95 17.89 0 0 98 11 109 11 8 94.79 48.55 0 0 99 11 95 10 9 22.03 136.17 0 0 106 11 100 10 10 84.21 97.08 0 0 99 10 101 10 11 60.34 89.21 0 0 0 0 93 10 12 117.39 0.72 0 0 0 0 101 10 13 34.96 14.13 0 0 0 0 96 10 14 10.03 52.38 0 0 0 0 97 10 15 118.57 115.08 0 0 0 0 108 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.
Table 3
Customer data of client 2 in the base-line model Index Client 2
Coordinate t = 1 t = 2 T = 3
x y Demand Return Demand Return Demand Return
1 85.06 74.91 191 20 0 0 207 21 2 142.28 53.39 207 20 0 0 198 20 3 146.86 126.00 205 21 0 0 194 20 4 72.97 86.95 199 20 0 0 201 21 5 89.88 141.82 202 21 207 20 203 20 6 13.93 52.39 197 21 210 20 205 20 7 9.30 37.89 202 20 195 21 205 20 8 35.05 44.99 200 21 191 21 196 20 9 121.52 2.90 196 20 208 21 194 21 10 94.80 125.85 0 0 204 20 206 20 11 132.83 29.85 0 0 201 21 200 20 12 26.38 46.43 0 0 197 21 204 20 13 95.08 139.79 0 0 201 20 210 20 14 108.91 98.62 0 0 194 20 201 21 15 141.36 37.98 0 0 196 21 195 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 dummyvariables 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 t∈T j∈J f wjtZjt+ j∈J andt=1 swj1Zj1+ j∈J andt1 swjtZjt + l∈L f rltGlt+ l∈Landt=1 srt1Gl1+ l∈Landt2 srltGlt + j∈J (ewjtAjt+ vwjtVjt) + l∈L (erltBlt+ vrltWlt) − =j=l wrtHt + p∈P i∈I j∈J k∈K cpijktf Xpijktf + p∈P k∈K l∈L i∈I0 crpklitXrpklit . (26)
Table 4
The parameters of the base-line model
Fixed operating cost per warehouse f wjt $18,000
Setup cost for installing a warehouse swjt $120,000
Maximum capacityof warehouses Mjt 3000 units
Modular expansion size of warehouse mwjt 600 units
Fixed cost of expanding warehouse ewjt $2,100
Variable cost of modular expansion for a warehouse vwjt $15,000
Fixed operating cost per repair center f rlt $3,000
Setup cost for installing a repair center srlt $30,000
Maximum capacityof repair center Mlt 300 units
Modular expansion size of repair center mrlt 100 units
Fixed cost of expanding repair center erlt $200
Variable cost of modular expansion for repair center vrlt $4,500 Savings from opening an hybrid warehouse-repair facility wrt $6,000
Maximum of modular expansion of warehouse ubjt 4
Maximum of modular expansion of repair facility ublt 4
Unit transportation cost of client’s plant-warehouse cfijt 0.05
Unit transportation cost of warehouse-customer cfjkt 0.5
Euclidean distance between locationsx and y in time t kxyt
Unit forward sipping cost cfpijkt cfijtkijt+ cfjktkjkt Unit transportation cost of customer-collection center crkl 0.05 Unit transportation cost of repair center-client’s plant crli 1.5
Unit backward shipping cost crpklit cklrkkl+ clirkli
To elaborate,Zjt ,G jt, andHtwere added as dummyvariables. First, through the use ofZjt , the following constraints were added into the set of original constraints:
Zjt+ Zjt−1− Z jt0, ∀j ∈ J, t(2) ∈ T , (27)
Zjt+ Zjt−1+ Z jt2, ∀j ∈ J, t(2) ∈ T , (28)
−2Zjt+ Zjt−1+ Zjt 1, ∀j ∈ J, t(2) ∈ T , (29)
2Zjt− Zjt−1− Zjt 1, ∀j ∈ J, t(2) ∈ T . (30)
Constraint (27) assures that ifZjt = 0 and Zjt−1= 0, Zjt should be zero; constraint (28) ensures that if
Zjt= 1 and Zjt−1= 1, Z jtshould be zero; constraint (29) assures thatZjt= 0 and Zjt−1= 1, Zjt should
be zero; constraint (30) ensures thatZjt= 1 and Zjt−1= 0, Z jtshould be one.
Second, bythe use ofG jt, the same logic was applied and then the following constraints should be also
added as follows:
Gjt+ Gjt−1− G jt0, ∀l ∈ L, t(2) ∈ T , (31)
0.00E+00 5.00E+05 1.00E+06 1.50E+06 2.00E+06 2.50E+06 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 fitness value generation
Fig. 4. Convergence of fitness values.
Table 5
The summaryof the solutions in the base-line model
Index Warehouse Repair centers
2 3 2 3 Z A V Z A V G B W G B W T = 1 0 0 0 1 1 1 1 0 0 1 0 0 T = 2 1 1 1 1 0 0 1 0 0 1 0 0 T = 3 1 0 0 1 0 0 1 1 2 1 0 0 Table 6
The cost summaryof the base-line model Cost components
Cost of operating warehouses $382,200
Cost of forward transportation $724,765 Cost of operating repair centers $87,600 Cost of reverse transportation $116,298
Cost savings $3,600
Table 7
The results of the test problems
No. P W R C T Optimal solution Genetic heuristic Gap
OBJ* ($) Time (m) OBJ ($) Time (m) (%)
1 1 5 5 30 3 86,597 0.11 87,272 2.06 0.78 2 1 5 5 50 3 116,091 0.08 117,018 2.45 0.80 3 1 5 5 70 3 174,244 8.33 175,899 5.40 0.95 4 1 5 5 30 4 93,731 0.21 97,223 2.08 3.73 5 1 5 5 50 4 124,805 0.11 129,239 2.75 3.55 6 1 5 5 70 4 191,206 128.11 199,682 5.25 4.43 7 1 10 10 30 3 87,127 2.00 88,726 2.10 1.84 8 1 10 10 50 3 145,842 4.17 148,804 6.15 2.03 9 1 10 10 70 3 174,064 358.13 180,463 10.63 3.68 10 1 10 10 30 4 94,289 3.00 100,889 2.33 7.00 11 1 10 10 50 4 432,537 316.00 462,518 6.53 6.93 12 1 10 10 70 4 508,945 27.28 550,588 11.68 8.18 13 2 10 10 60 3 1,244,980 4.48 1,274,860 8.77 2.40
14 2 15 15 120 4 N/A N/A 3,162,030 86.70 N/A
15 2 20 20 180 5 N/A N/A 4,818,570 207.98 N/A
16 3 10 10 60 3 2,243,190 4.02 2,340,650 12.81 4.34
17 3 15 15 120 4 N/A N/A 3,927,340 74.33 N/A
18 3 20 20 180 5 N/A N/A 7,209,830 219.95 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+ Gjt−1+ G jt1, ∀l ∈ L, t(2) ∈ T , (33)
2Gjt− Gjt−1− G jt1, ∀l ∈ L, t(2) ∈ T . (34)
Finally, forHt calculating cost savings from opening hybrid facilities, additional constraints were added
into the set of original constraints as follows:
Zjt+ Glt− 2H(=j=l)t0, ∀j ∈ J, ∀l ∈ L, t ∈ T , (35)
Zjt+ Glt− H(=j=l)t1, ∀j ∈ J, ∀l ∈ L, t ∈ T . (36)
Constraint (35) assures that if either Zjt orGlt is 0 where j = l, Ht should be zero. Constraint (36)
ensures that if bothZjt= 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 uniformlydistributed 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 byapplying 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 7because of the
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 inTable 7. The first column of the table indicates
the index for test problems. The second through sixth columns specifythe 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 by100(OBJ − OBJ∗)/OBJ∗.
On the basis of the results fromTable 7, the range of gaps with respect to solution qualityis 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 significantlyvaryalthough the increases of the number of warehouses, repair centers, and customers are allowed. Also, the differences in the gap performances is almost unaffected byadding 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 maycontinuouslydeviate 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%, respectivelywhen increasing onlyone 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 inTable 5was detected. In fact, the results indicate that the performance stronglydepends 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) apparentlywere 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 keypossible factors of affecting the computing complexityin the proposed mathematical model mayresult from a total number of time periods and demand pattern across the planning horizon. Especiallyfor problems (14), (15), (17) and (18), theyall have over four time periods and fail to solve byan exact solution approach. However, the proposed heuristic solved them less than 4 h. Thus, this maybe explained bythe fact whya 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 supplychain management since theyare under pressure for filling customers’ orders on time as well as for efficientlytaking returned products back from customers after selling products. In terms of product flows, there are both forward flows and reverse flows in an integrated supplychain. 3PLs are playing an increasing role in supporting such integrated supplychain 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
realisticallysized 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 transship-ment 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 complexityof 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.
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