Supply Chain Management and Genetic Algorithm:introducing a new hybrid genetic crossover operator

Felipe G. S. Teodoro1, Clodoaldo A. M. Lima1, Sarajane M. Peres1

1Escola de Artes, Ciencias e Humanidades – Universidade de Sao Paulo (USP)– 03828.000 – Sao Paulo – SP – Brazil

{fteodoro,c.lima,sarajane}@usp.br

Abstract. The increase in the Brazilian consumer market in recent years has al-lowed the opening of new stores, increasing price competition and diversifyingthe product offering. However, finding the best combination of price, qualityand services, considering the cost of transport between stores is a very hardtask. The complexity of the problem increases when more products and storeswere involved. This is a classical problem in supply chain network design. Theeffective management of the supply chain is recognised as one of the most impor-tant factors in modern business management. In this paper, we propose a newhybrid operator for genetic algorithm used to optimize the purchase of productsin stores geographically separated in order to obtain solutions that combine thepurchase of products with the lowest possible total cost, considering the routebetween each store.

1. IntroductionIn the last two decades, academicians and practitioners has given much attention to thestudy, design and improvement of Supply Chain Management (SCM). The term was firstused by consultants in the early 1980s, and its definition has undergone some very signif-icant developments since then [Oliver and Webber 1992].The concept was used mainlyto discuss the benefits of integrating internal business functions of a company, such aspurchasing, manufacturing, sales and distribution [Harland 1996]. Supply chain (SC)can be viewed as a way to address the customer’s requirement in order to maximize theoverall value generated. The intense competition in global markets, increasingly smallerproduct life cycle, and increasingly customer expectations with respect to the capacityand reliability of the product, delivery, cost, flexibility and service have forced com-panies to implement alternative methods to improve the response capacity of the SC.The ability of a supply chain to respond quickly to market changes and customer de-mands is considered as the carrier of competitive advantage in today’s business world[Christopher and Towill 2001],[Gunasekaran 1999]. Supply chains vary significantly incomplexity and size, but the basic principles apply to all operations whether large orsmall, manufacturing or service, private or public sector.

Considering the Brazilian case, in which has seen an increase in the retail market[Capizzani et al. 2012], it is noticeable that there is an adhesion to developing of studieswhich perform trends analysis, knowledge discovery and processes optimization. In thiscontext is useful to consider the problem of SCM. For the purposes of the present paper,the objective of SCM “is to synchronize the requirements of the customer with the flowof materials from suppliers in order to effect a balance between what are often seen as

conflicting goals of high customer service, low inventory management, and low unit cost”[Stevens 1989]. As an example about a real situation, consider an analysis among prices indifferent stores, the distance among them, and decide about the best cost-benefit relationsfor the consumers.

SCM is a complex problem, due it is a multi-criterion decision making problem,and generally in this context, the decision must satisfy a series of constraints. As statedby [Kumar et al. 2004], such constraints come from internal needs (the needs of the sub-ject that is searching the problem solution) and come from external factors imposed bythe system requirements (for instance, current traffic conditions along the way betweentwo stores involved in the problem). Thus, the analysis related to search a good solutionfor each problem instance. However, the search space of these problems may have morethan one optimal solutions, of which most are undesired locally optimal solutions havinginferior function values [Rezaei and Davoodi 2006]. When solving these problems, if tra-ditional methods get attracted to any of these locally optimal solutions, there is no escape[Deb and Kalyanmoy 2001]. To overcome these difficulties, stochastic search technique,such as Genetic Algorithm, has been successfully used as an optimization technique fordecision-making problems[Davies 1991].

Thus, the main objective of this paper is to describe the development of a strategybased on GA, emphasizing the importance of a new hybrid genetic crossover operator,to provide a set of optimized solutions for a SCM specific problem. The purpose of theproblem addressed is to perform comparative analysis about buying products in multiplestores considering the product prices in each one and the cost of travelling between them,and to provide a solution that comprises the choice of products and stores with minimumcost of shopping and displacement.

This paper is organized as follows: in Section 2 is briefly presented some initia-tives which are related with the strategy proposed in this paper; a formal problem def-inition, considering the application context studied here, is presented in Section 3; thegenetic modelling as well the new hybrid genetic crossover operator is described in Sec-tion 4; the results obtained with the genetic strategy is discussed on Section 5; finally inSection 6 the conclusions are delineated.

2. Related WorksThere are several studies about the use of GA in supply chain problems, supplier selection,routes improvement in distribution centers using different encoding and parameters inorder to find a good solution.

According with the studies of [Ko et al. 2010], GA can solve problems where tra-ditional search and optimization methods are less effective. Second the same authors,there are many well-known SCM problems that can be efficiently solved using GA, suchas order fulfilment, demand management, supplier relationship management. From thestudies revised in [Shen 2007], the author assert that problems related with routing costestimation are usually NP-Hard and, in this context, GA is one of the more feasible ap-proach to solve them, manly for large-sized problem instances.

[Lawrynowicz 2011] also asserts that GAs are efficient tools for solving complexoptimization problems, highlighting the problem of minimizing the total cost for a dis-tribution network, which presents some similar features to the problem addressed in this

paper. In [Wen and Eberhart 2002] was proposed a GA, using an integer encoding to rep-resent the cargo item sequence to be delivered in order to solve the problem of logisticsscheduling problem and optimize the total cost for a location-routing-inventory problem.The proposed approach obtained good performance in terms of quality measure and thecomputation speed meets requirements. Additionally, [Hsu et al. 2005] uses a similarchromosome encoding scheme, in order to solve the problem of batching orders in ware-houses by minimizing travel distance. A similar approach was used by [Lau 2009], inwhich three different crossover operators were employed, namely, one-point, two-point,and uniform. In order to improve the genetic diversity inside the population, the authoruses a uniform mutation with some restrictions related with the problem. The best resultswere obtained with the uniform crossover. Already [Wen and Eberhart 2002] used fourdifferent crossover operators, namely, one-point, two-point, order-based, position-basedand two different mutation operators: order-based and position-based. In this case, thebest results were obtained with the two-point crossover.

The approach discussed in this paper uses a classical GA with an integer encodingto represent the products purchased in various stores and employs different combinationsof crossover and mutation operators, in order to build a GA capable to solve the problemaddressed.

3. Problem Definition

The problem addressed in this paper can be viewed as a graph-theory problem. LetG = (V,A), where V = 1, 2, · · · ,m is the index set of vertices (nodes) and A ={(1, 2), (1, 3), · · · , (u, s)} is the set of undirected arcs (links), u 6= v, ∀(u, v) ∈ S,S = {S1, S2, S3, · · · , Sm} are the set of stores. In the graph, a node is a store and eachpair of nodes is connected by an undirected link, i.e, there is a possible path to arriveat all stores. Each store has all products of the set P considered on the problem, whereP = {P1, P2, P3, · · · , Pn} represent the set of products. The P and S set define an in-stance of the problem. Each product has a specific price in each store and each productmust be purchased only in a single store.

Consider the (m,n) matrix W which contain the costs of each product j for eachstore i, with Wij > 0, i = 1, · · · ,m and j = 1, · · · , n. As mentioned in Section 1,the objective of the proposed approach in this paper is to find the best combination thatminimizes the total cost, which is calculated based on the prices of the products and thetravelling costs. Thus, the problem addressed is mathematically formulated as:

Minimize total cost J

J =∑∀i|i∈C

n∑j=1

Wij +m−1∑c=1

dist(Sc, Sc+1) (1)

where C ⊂ V which contains the index set of stores (vertices) that minimizes thecost function J and dist(Sc, Sc+1) represents the path cost between the stores Sc and Sc+1

subject to

Pj ∈ P|n ≥ j > 0 (2)c ∈ C|m ≥ c > 0, (3)Si ∈ S|m ≥ i > 0. (4)

4. Genetic Algorithm and the Proposed Operator

In order to find a good set of solutions for the SCM problem addressed in this paper,we employ a Genetic Algorithm (GA) with a particular operator especially designed fordeal with the problem described in the Section 3. GAs were first introduced by Holland[Davies 1991] as a kind of stochastic search procedure whose design is based on conceptsof Evolutionary Computation, i.e., the GA’s principles are inspired in the natural selectiontheory and biological reproduction system. It comprises a population of chromosomes(individuals), selection, crossover and mutation. This class of algorithms are feasible tosolve combinatorial problems and its classical design is described in Algorithm 1.

Algorithm 1 GA classical algorithmInitialize the initial population;Evaluate each individual (or chromosome) using the fitness function;repeat

Select individuals to reproduce or mutate through a selection operatorApply crossover operators (on pairs of individuals, according to a specific probability)Apply mutation operators (on single individuals, according to a specific probability)Evaluate each new individual using the fitness functionPrune population (typically prune all old individuals; if not, then the worst individuals)

until Stop condition (typically number of generations)

The remaining of this section describes the genetic modelling applied in solvingthe SCM problem addressed in this paper. This presentation includes the encoding solu-tion to the individuals (the genotype and the phenotype), the strategies applied in orderto initializing and pruning the population, the fitness function, and the genetics operatorswith discussion about the new proposed operator called Directed Crossover. The classicaldesign aforementioned was used to implement the GA algorithm, the changes made willbe described in the next section.

4.1. Encoding solution

The feasible solution was encoded by a vector C of m integers. Each vector position(allele) represents a product Pj ∈ P and each element in a vector position representsthe store Si ∈ S chosen to deliver the product Pj in the current solution (chromosome).Figure 1 shows an example of a chromosome C representing a solution for a problemwith four products offered by four stores. The adjacency matrix W has the costs of eachproduct in each store. According this solution, the phenotype can be interpreted as: theproducts P1 and P3 must be bought in the store S1, and the products P2 and P3 must bebought in the stores S3 and S2, respectively.

Figure 1. Graphical example: C is an individual (genotype) and W is the matrixwith costs of products in each store.

4.2. Initial Population and Pruning Strategies

The initial population is generated by choosing random integer values for the stores in theinterval [1,m]. Note that the same store can be selected more than once, due to the randomprocess. The population size is determined by the parameter Φpop, whose value is definedin the experiments described in Section 5. After the choice of Φpop, the population size ismaintained for all generation in the GA optimization process. Once the first population(first generation) is created, all new individuals are evaluated by the fitness function.

In order to maintain the population size, it is required to perform a pruning proce-dure on the population, due the creation of new individuals by the reproduction process.The population pruning strategies adopted in this work were: total exchange, λ + µ andelitism. The total exchange consists of replacing the entire population by the new individ-uals created in each generation. The λ + µ strategy consists of performing the union ofthe two sets of individual, the current and the new ones, and selecting the best individualsin this unified set in order to compose the new population. Finally, the elitism strategyselects the h best individuals, where h is a parameter to be set.

4.3. Fitness Function

The fitness function applied in this work aims to evaluate the individuals in order to allowselecting solutions (individuals) with low global cost (GC) and low travel cost (TL), alsoexcluding solutions composed by a large amount of distinct stores (DE).

The global cost for each individual consists in summing all the products costs inthe their respective stores determined by the individual’s genes. Thus, GC =

∑mj Wi,j

where i = C[j] and C is the individual under analysis. In order to calculate the valueof TL, it was applied the Hoffman-Pavlat algorithm [Hoffman and Pavley 1959]. Thisalgorithm allows calculating the k best possible paths between two vertices in a graph,resulting in a list of weighted connected arcs. In the problem addressed in this paper,the weights correspond to the travel costs between stores.Therefore to calculate TL, allpossible routes related to an individual need to be analysed. Since, in our problem, wehave determined the beginning of the route in advanced, it is necessary to perform theHoffman-Pavlat algorithm n times, where n is the number of distinct stores in the indi-vidual.

Algorithm 2 Calculate Travel CostSet Matrix V;Set Chromosome C;BeginTL := 0;Aux := 0;Start Point := V[0,0];Vector RouteElements := Nil;Vector E := GetDistinctsstores (C)i := 1;repeat

RouteElements := Hoffman–Pavlat(StartPoint, E[i], V);if E ⊆ RouteElements = ∅Aux := GetRouteCost(RouteElements);if Aux > TLTL := Aux;endifendif

until SizeOf(E)End

Notice that to apply the Hoffman-Pavlat algorithm is required to provide a bidi-rectional graph containing possible paths among the stores, the initial point in the routeand a list with with the distinct stores that need to be visited in the route. This informationcan be obtained from each individual. Once all possible routes were found, those that donot contain all distinct stores presented in a individual must be discarded. Thus, the costof each remaining route is evaluated an the minimum cost (TL) is calculated.

To calculate DE we apply the equation (5), where E denotes the number of dis-tinct stores and p is a punishment factor, which must be defined with high values in orderto allow excluding an individual with a lot of distinct stores.

DE =m

E∗ p (5)

Finally, since TC, TL and DE are calculated, the value of the individual’s functionis given by:

Fitness(C) =DE

TC + TL(6)

4.4. Selection Operator

In this work, the selection operator is only required to select individuals to the reproduc-tion process. Such operator is implemented using the roulette wheel strategy. In thisstrategy, the first step is to calculate the cumulative fitness Fpop regarding the wholepopulation. After that, the selection probability is calculated for each individual asρsel = fC/Ppop, considering fC the individual fitness. Thus, the roulette wheel is spinningΦrou times, i.e., in each spinning, random numbers in the range [0, Ppop] are generated for

each individual and, if its fC is higher the random number, it is select to reproduce. TheΦrou value is defined in the experiments described in Section 5.

4.5. Crossover Operators

The crossover operators generate new individuals by combining the genetic informationof the individuals parents (the individuals selected to reproduce), so that the children (newindividual) have parts of the parent’s genetic code. In this work, this class of operator isapplied for each GA generation, with a constant probability Φcross. The Φcross value isdefined in the experiments described in Section 5.

Four crossover operators were analysed in the experiments, including the newoperator proposed in this work. The new Directed Crossover is described in a specialsection 4.6 and the other three classical crossover operators are briefly commented below:

• One–Point Crossover: this operator divides the two parents individuals in twoparts, according to a crossover point x | x ∈ [1,m] randomly chosen; thus, eachpart of each parent is combined in order to generate two new individuals;• Two–Point Crossover: this operator is similar the One–Point Crossover; however,

in this case the operator considers two crossover points and divide the each parentin three parts;• Uniform Crossover: in this operator, a number in {0, 1} is randomly sorted for

each gene in the chromosome representation. These sorted numbers are used todecide if the genetic information received by the respective gene in the new indi-vidual (child) comes from the first parent or from the second parent. Thus, twonew individuals are generated with genetic information inherited from the parentsaccording to the random and inverse combination. In Figure 2 is shown a Uni-form Crossover example. This example is contextualized in the SCM problemaddressed in this paper. In this example C1 and C2 are the first and the secondparents respectively, S represents the set of values randomly generated for eachproduct. Both, C ′ and C ′′ are children generated from Uniform Crossover execu-tion.

Figure 2. Uniform Crossover

4.6. Directed Crossover: a new hybrid operator

The Directed Crossover operator development was motivated by the analysis of theclassical crossover operators performance in the SCM problem addressed in this pa-per. The classical crossover operators were not efficient enough for generating a uni-formly optimized population, since it was possible to observe, during the experiments,the difficulties presented by such operators in preserving schema (from Schemata The-ory [Goldberg 1989]).

Besides this, it is important to notice that, in spite of to be possible to reach a goodsolution using the classical crossover operators, it was not possible to reach a set of goodsolutions. Although a set of good solutions is not required in the problem definition, it isdesirable since exogenous variables in the system, like traffic or weather conditions, couldmake impracticable the execution of the best solution proposed in the GA optimizationprocess.

In order to prevent damage in the best schema of each individual and improvethe final set of solutions, we proposed a heuristic adaptation in the Uniform Crossoveroperator. The heuristic has the objective of detecting the most significant schema presentin the individual, aiming to provide conditions to guide de genetic recombination carriedout by the operator.

The Directed Crossover carries out an analysis on the schema driven by the vari-ance in the costs of each product (Pj ∈ P), i.e., products with a higher cost varianceamong different stores indicate alleles that must compose the most relevant schema. Thisanalysis occurs before the first generation of the GA optimization process, as stated inAlgorithmalg:analysis, and the process is exemplified in Figure 3.

Algorithm 3 Analysis of the best schemaCalculate the costs variance (the difference between the maximum cost and the minimum cost)of each product Pj ∈ P using the information in costs matrixW and store the results in a vectorA;Sort the elements of the vector A in decreasing way;Reorder the columns of W following the changes did in the sorting of A;Generate the first generation and proceed with the GA optimization using the new W matrix.

Once the schema analysis is performed, it is applied a heuristic strategy whichdivides the genes of the individual in five group with the same number of alleles. Foreach group is associate a crossover probability as follows: the first group is consideredthe most relevant scheme, thus the crossover probability is set to be 20%; the secondgroup is set to be 40% and so on. Notice that the aim is to avoid breaking good schemawith the usage of crossovers operators. The Figure 4 illustrates the different rates of eachgroup of chromosomes.

Finally, in Directed Crossover strategy, a number in [0, 1] is randomly sorted foreach gene in the chromosome representation; if this number is greater than the crossoverprobability, the gene receives the genetic information from the parents following the samerules stated for Uniform Crossover.

Figure 3. Analysis of the best schema: the cells in green represent the elementswhich have the minimum costs for each product; the cells in yellow represent theelements with maximum costs for each product.

Figure 4. Probabilities associated to possible schema in the Directed Crossoveroperator.

4.7. Mutation Operator

Mutation is genetic operation which provides diversity for the solutions so as to preventthem from falling into local optima. Not all genes are chosen for performing mutation.The probability of the mutation operation is defined by parameter Pm.

We generate a random number r among [0, 1] for each gene t of all chromosomes.If r is lower than the value defined for the parameter Pm, the value of the gene t is re-placed by one of its possible index values of S randomly chosen. In other words, theprocess of mutation as the crossover, change the stores where the products will be pur-chased. The Figure 5 present an example of the mutation process:

5. Experiments and ResultsIn order to evaluate the genetic modelling some scenarios were created using a realdatabase with products sold by retails stores, combined with a fictitious directed graphto simulate the path among the stores. Thus the cost of 150 different products in 10 storeswere defined, resulting in a matrix W of size 10 × 150. Moreover, it was defined 15possible paths between the stores and the beginning point of the graph. The parametersof the genetic algorithm were set using all combination of values presents in Table 1,andthe punishment factor p was define, empirically, as 0.95.

The combination of these parameters resulted in 1152 scenario for tests. Eachscenario was carried out using 1000 generations. The Table 2 presents the 20 best resultsobtained ordered from higher (better evaluated) to lower (worse evaluated) fitness value.

Note that the 20 best executions have a Pc defined as 100%, λ+µ strategy and Pm

Figure 5. Example of mutation process

Table 1. Parameters used at the simulationCrossover Probability Population Probability Size of

Type of Crossover (Pc) % Mode of Mutation (Pm) % PopulationOne–Point 70 Total Exchange 5 100Two–Point 80 λ+ µ 10 200Uniform 90 Elitism 15 500Directed 100 20 1000

2000

defined as 5%. The best execution reach the fitness of 2313.23 using Directed crossoverand λ+µ strategy. The best average fitness was reached in the execution 9, but analysingeach individual’s phenotype, it was observed that the set of the solutions obtained in theexecution 9 have almost all individuals with the same fitness, and this means that thesolution obtained have only one global maximum. This was verified in all executions thathas a population size smaller than 1000 individuals.

Excluding these results, the best average fitness was reached in the execution 3.This execution has a good solution set, since it has 25 different solutions with the bestfitness value. If we compare this result with the solution set provided by execution 1 thathas the best fitness, we can conclude that the solutions obtained in the execution 1 aremore homogeneous (in terms different solutions) than ones obtained in the execution 3.Analysing the differences between the worst fitness and the best fitness of the execution1, it is lower than the ones provided by the execution 3. This can be observed in figure 6.If we compare the execution 1 with the execution 2, we draw the same conclusions.

6. ConclusionsThe main objective of this paper was to describe the development of a strategy based onGA, emphasizing the proposition of a new hybrid genetic crossover operator, to providea set of optimized solutions for a SCM specific problem.

The proposed operator has proven to be effective in situations where the aim is to

Table 2. 20 best results obtained in the experiments.Sequence Crossover Pc% Population Pm% Size of Best Average

ID Type Mode Population Fitness Fitness1 Directed 100 λ+ µ 5 1000 2313.23 2253.282 Uniform 100 λ+ µ 5 2000 2313.04 2258.353 Uniform 100 λ+ µ 5 1000 2312.92 2284.334 Directed 100 λ+ µ 5 2000 2312.46 2257.315 Two–Points 100 λ+ µ 5 1000 2312.26 2249.816 Two–Points 100 λ+ µ 5 2000 2312.08 2231.027 Uniform 100 λ+ µ 5 400 2311.63 2224.228 Uniform 100 λ+ µ 5 300 2311.57 2276.029 Uniform 100 λ+ µ 5 200 2311.20 2309.1310 Two–Points 100 λ+ µ 5 400 2310.22 2235.5611 Uniform 100 λ+ µ 5 100 2309.51 2307.2012 Two–Points 100 Eletism 5 300 2308.82 2192.3813 Two–Points 100 λ+ µ 5 300 2308.82 2192.3814 One–Point 100 λ+ µ 5 200 2307.76 2289.0415 One–Point 100 λ+ µ 5 300 2307.64 2285.2516 One–Point 100 λ+ µ 5 100 2307.57 2149.2317 One–Point 100 λ+ µ 5 400 2306.92 2265.0018 Directed 100 Eletism 5 2000 2306.72 2088.1819 Directed 100 Eletism 15 1000 2306.62 2074.6220 Two–Points 100 λ+ µ 5 200 2306.50 2281.85

ensure solutions with with high genetic diversity, but with the same best fitness value. Toachieve this goal, it also was necessary to use large populations of individuals. This is alimitation of the proposed approach. However, the proposed approach can be adapted toaddress other optimization problems in SCM.

For problems where the goal is to achieve the best solution, the two-point and/oruniform crossover can be used. In the experiments performed, the use of these operatorsproduced solutions with low diversity, but with fitness value equivalent to those obtainedwith the proposed operator.

As future work, we intend to apply the proposed approach in other optimizationproblems. Furthermore, we intend to compare the proposed approach with other evolu-tionary strategies.

Figure 6. Comparison between executions 1 and 3.

ReferencesCapizzani, M., Huerta, F. J. R., and Oliveira, P. R. (2012). Retail in latin america: Trends,

challenges and opportunities. IESE Business School - STUDY 170, pages 8–12.

Christopher, M. and Towill, D. (2001). An integrated model for the design of agile supplychains. Int. Journal of Physical Distribution and Logistics Management, 31(4).

Davies, L. (1991). Handbook of Genetic Algorithms. Van Nostrand Reinhold.

Deb, K. and Kalyanmoy, D. (2001). Multi-Objective Optimization Using EvolutionaryAlgorithms. Wiley, 1 edition.

Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learn-ing. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition.

Gunasekaran, A. (1999). Agile manufacturing: A framework for research and develop-ment. International Journal of Production Economics, 62(1-2):87–105.

Harland, C. M. (1996). Supply chain management: relationships, chains and networks.British Journal of Management, 7(s1):S63–S80.

Hoffman, W. and Pavley, R. (1959). A method for the solution of the nth best pathproblem. Journal of the ACM, 6.

Hsu, C.-M., Chen, K.-Y., and Chen, M.-C. (2005). Batching orders in warehouses byminimizing travel distance with genetic algorithms. Comput. Ind., 56(2):169–178.

Ko, M., Tiwari, A., and Mehnen, J. (2010). Review article: A review of soft computingapplications in supply chain management. Applied Soft Computing, 10(3):661–674.

Kumar, M., Vrat, P., and Shankar, R. (2004). A fuzzy goal programming approach forvendor selection problem in a supply chain. Computer & Industrial Eng., 46:69–85.

Lau, H. C. W. (2009). Cost optimization of the supply chain network using genetic algo-rithms. IEEE Transactions on Knowledge and Data Engineering, 8(8):1–36.

Lawrynowicz, A. (2011). A survey of evolutionary algorithms for production and logisticsoptimization. Research in Logistics and Production, 1(2):57–91.

Oliver, R. K. and Webber, M. D. (1992). Supply-chain management: logistics catches upwith strategy. Logistics: The Strategic Issues.

Rezaei, J. and Davoodi, M. (2006). Genetic algorithm for inventory lot-sizing with sup-plier selection under fuzzy demand and costs. In Ali, M. and Dapoigny, R., editors,Advances in Applied Artificial Intelligence, volume 4031 of Lecture Notes in ComputerScience, pages 1100–1110. Springer Berlin Heidelberg.

Shen, Z.-J. (2007). Integrated supply chain design models: A survey and future researchdirections. Journal of Industrial and Management Optimization, 3(1):1–27.

Stevens, G. C. (1989). Integrating the supply chains. International Journal of PhysicalDistribution and Materials Management, 8(8):3–8.

Wen, C. and Eberhart, R. (2002). Genetic algorithm for logistics scheduling problem.In Evolutionary Computation, 2002. CEC ’02. Proc. of the 2002 Cong. on, volume 1,pages 512–516.