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

Transcription

1 Supply Chain Management and Genetic Algorithm: introducing a new hybrid genetic crossover operator Felipe G. S. Teodoro 1, Clodoaldo A. M. Lima 1, Sarajane M. Peres 1 1 Escola de Artes, Ciências e Humanidades Universidade de São Paulo (USP) São Paulo SP Brazil {fteodoro,c.lima,sarajane}@usp.br Abstract. The increase in the Brazilian consumer market in recent years has allowed the opening of new stores, increasing price competition and diversifying the product offering. However, finding the best combination of price, quality and services, considering the cost of transport between stores is a very hard task. The complexity of the problem increases when more products and stores were involved. This is a classical problem in supply chain network design. The effective management of the supply chain is recognised as one of the most important factors in modern business management. In this paper, we propose a new hybrid operator for genetic algorithm used to optimize the purchase of products in stores geographically separated in order to obtain solutions that combine the purchase of products with the lowest possible total cost, considering the route between each store. 1. Introduction In the last two decades, academicians and practitioners has given much attention to the study, design and improvement of Supply Chain Management (SCM). The term was first used by consultants in the early 1980s, and its definition has undergone some very significant developments since then [Oliver and Webber 1992].The concept was used mainly to discuss the benefits of integrating internal business functions of a company, such as purchasing, 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 the overall value generated. The intense competition in global markets, increasingly smaller product life cycle, and increasingly customer expectations with respect to the capacity and reliability of the product, delivery, cost, flexibility and service have forced companies 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 demands is considered as the carrier of competitive advantage in today s business world [Christopher and Towill 2001],[Gunasekaran 1999]. Supply chains vary significantly in complexity and size, but the basic principles apply to all operations whether large or small, 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 studies which perform trends analysis, knowledge discovery and processes optimization. In this context 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 flow of materials from suppliers in order to effect a balance between what are often seen as

2 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 in different stores, the distance among them, and decide about the best cost-benefit relations for 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 stated by [Kumar et al. 2004], such constraints come from internal needs (the needs of the subject that is searching the problem solution) and come from external factors imposed by the system requirements (for instance, current traffic conditions along the way between two stores involved in the problem). Thus, the analysis related to search a good solution for each problem instance. However, the search space of these problems may have more than one optimal solutions, of which most are undesired locally optimal solutions having inferior function values [Rezaei and Davoodi 2006]. When solving these problems, if traditional 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 for decision-making problems[davies 1991]. Thus, the main objective of this paper is to describe the development of a strategy based 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 the problem addressed is to perform comparative analysis about buying products in multiple stores 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 minimum cost of shopping and displacement. This paper is organized as follows: in Section 2 is briefly presented some initiatives which are related with the strategy proposed in this paper; a formal problem definition, considering the application context studied here, is presented in Section 3; the genetic modelling as well the new hybrid genetic crossover operator is described in Section 4; the results obtained with the genetic strategy is discussed on Section 5; finally in Section 6 the conclusions are delineated. 2. Related Works There are several studies about the use of GA in supply chain problems, supplier selection, routes improvement in distribution centers using different encoding and parameters in order to find a good solution. According with the studies of [Ko et al. 2010], GA can solve problems where traditional 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, such as order fulfilment, demand management, supplier relationship management. From the studies revised in [Shen 2007], the author assert that problems related with routing cost estimation are usually NP-Hard and, in this context, GA is one of the more feasible approach to solve them, manly for large-sized problem instances. [Lawrynowicz 2011] also asserts that GAs are efficient tools for solving complex optimization problems, highlighting the problem of minimizing the total cost for a distribution network, which presents some similar features to the problem addressed in this

3 paper. In [Wen and Eberhart 2002] was proposed a GA, using an integer encoding to represent the cargo item sequence to be delivered in order to solve the problem of logistics scheduling problem and optimize the total cost for a location-routing-inventory problem. The proposed approach obtained good performance in terms of quality measure and the computation speed meets requirements. Additionally, [Hsu et al. 2005] uses a similar chromosome encoding scheme, in order to solve the problem of batching orders in warehouses by minimizing travel distance. A similar approach was used by [Lau 2009], in which three different crossover operators were employed, namely, one-point, two-point, and uniform. In order to improve the genetic diversity inside the population, the author uses a uniform mutation with some restrictions related with the problem. The best results were obtained with the uniform crossover. Already [Wen and Eberhart 2002] used four different crossover operators, namely, one-point, two-point, order-based, position-based and two different mutation operators: order-based and position-based. In this case, the best results were obtained with the two-point crossover. The approach discussed in this paper uses a classical GA with an integer encoding to represent the products purchased in various stores and employs different combinations of crossover and mutation operators, in order to build a GA capable to solve the problem addressed. 3. Problem Definition The problem addressed in this paper can be viewed as a graph-theory problem. Let G = (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 v, (u, v) S, S = {S 1, S 2, S 3,, S m } are the set of stores. In the graph, a node is a store and each pair of nodes is connected by an undirected link, i.e, there is a possible path to arrive at all stores. Each store has all products of the set P considered on the problem, where P = {P 1, P 2, P 3,, P n } represent the set of products. The P and S set define an instance of the problem. Each product has a specific price in each store and each product must be purchased only in a single store. Consider the (m, n) matrix W which contain the costs of each product j for each store i, with W ij > 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 that minimizes the total cost, which is calculated based on the prices of the products and the travelling costs. Thus, the problem addressed is mathematically formulated as: Minimize total cost J J = i i C j=1 n W ij + m 1 c=1 dist(s c, S c+1 ) (1) where C V which contains the index set of stores (vertices) that minimizes the cost function J and dist(s c, S c+1 ) represents the path cost between the stores S c and S c+1 subject to

4 P j P n j > 0 (2) c C m c > 0, (3) S i 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 for deal 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 concepts of Evolutionary Computation, i.e., the GA s principles are inspired in the natural selection theory and biological reproduction system. It comprises a population of chromosomes (individuals), selection, crossover and mutation. This class of algorithms are feasible to solve combinatorial problems and its classical design is described in Algorithm 1. Algorithm 1 GA classical algorithm Initialize the initial population; Evaluate each individual (or chromosome) using the fitness function; repeat Select individuals to reproduce or mutate through a selection operator Apply 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 function Prune 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 solving the SCM problem addressed in this paper. This presentation includes the encoding solution to the individuals (the genotype and the phenotype), the strategies applied in order to initializing and pruning the population, the fitness function, and the genetics operators with discussion about the new proposed operator called Directed Crossover. The classical design aforementioned was used to implement the GA algorithm, the changes made will be described in the next section Encoding solution The feasible solution was encoded by a vector C of m integers. Each vector position (allele) represents a product P j P and each element in a vector position represents the store S i S chosen to deliver the product P j in the current solution (chromosome). Figure 1 shows an example of a chromosome C representing a solution for a problem with four products offered by four stores. The adjacency matrix W has the costs of each product in each store. According this solution, the phenotype can be interpreted as: the products P 1 and P 3 must be bought in the store S 1, and the products P 2 and P 3 must be bought in the stores S 3 and S 2, respectively.

5 Figure 1. Graphical example: C is an individual (genotype) and W is the matrix with costs of products in each store Initial Population and Pruning Strategies The initial population is generated by choosing random integer values for the stores in the interval [1, m]. Note that the same store can be selected more than once, due to the random process. The population size is determined by the parameter Φ pop, whose value is defined in the experiments described in Section 5. After the choice of Φ pop, the population size is maintained 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 procedure 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 and elitism. The total exchange consists of replacing the entire population by the new individuals created in each generation. The λ + µ strategy consists of performing the union of the two sets of individual, the current and the new ones, and selecting the best individuals in this unified set in order to compose the new population. Finally, the elitism strategy selects the h best individuals, where h is a parameter to be set Fitness Function The fitness function applied in this work aims to evaluate the individuals in order to allow selecting solutions (individuals) with low global cost (GC) and low travel cost (T L), also excluding solutions composed by a large amount of distinct stores (DE). The global cost for each individual consists in summing all the products costs in the their respective stores determined by the individual s genes. Thus, GC = m j W i,j where i = C[j] and C is the individual under analysis. In order to calculate the value of T L, it was applied the Hoffman-Pavlat algorithm [Hoffman and Pavley 1959]. This algorithm 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, all possible routes related to an individual need to be analysed. Since, in our problem, we have determined the beginning of the route in advanced, it is necessary to perform the Hoffman-Pavlat algorithm n times, where n is the number of distinct stores in the individual.

6 Algorithm 2 Calculate Travel Cost Set Matrix V; Set Chromosome C; Begin TL := 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 > TL TL := Aux; endif endif until SizeOf(E) End Notice that to apply the Hoffman-Pavlat algorithm is required to provide a bidirectional graph containing possible paths among the stores, the initial point in the route and a list with with the distinct stores that need to be visited in the route. This information can be obtained from each individual. Once all possible routes were found, those that do not contain all distinct stores presented in a individual must be discarded. Thus, the cost of each remaining route is evaluated an the minimum cost (T L) is calculated. To calculate DE we apply the equation (5), where E denotes the number of distinct stores and p is a punishment factor, which must be defined with high values in order to 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 function is given by: F itness(c) = DE T C + T L (6) 4.4. Selection Operator In this work, the selection operator is only required to select individuals to the reproduction process. Such operator is implemented using the roulette wheel strategy. In this strategy, the first step is to calculate the cumulative fitness F pop regarding the whole population. After that, the selection probability is calculated for each individual as ρ sel = f C /P pop, considering f C the individual fitness. Thus, the roulette wheel is spinning Φ rou times, i.e., in each spinning, random numbers in the range [0, P pop ] are generated for

7 each individual and, if its f C is higher the random number, it is select to reproduce. The Φ rou value is defined in the experiments described in Section Crossover Operators The crossover operators generate new individuals by combining the genetic information of the individuals parents (the individuals selected to reproduce), so that the children (new individual) have parts of the parent s genetic code. In this work, this class of operator is applied for each GA generation, with a constant probability Φ cross. The Φ cross value is defined in the experiments described in Section 5. Four crossover operators were analysed in the experiments, including the new operator proposed in this work. The new Directed Crossover is described in a special section 4.6 and the other three classical crossover operators are briefly commented below: One Point Crossover: this operator divides the two parents individuals in two parts, according to a crossover point x x [1, m] randomly chosen; thus, each part 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 parent in 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 to decide if the genetic information received by the respective gene in the new individual (child) comes from the first parent or from the second parent. Thus, two new individuals are generated with genetic information inherited from the parents according to the random and inverse combination. In Figure 2 is shown a Uniform Crossover example. This example is contextualized in the SCM problem addressed in this paper. In this example C1 and C2 are the first and the second parents respectively, S represents the set of values randomly generated for each product. Both, C and C are children generated from Uniform Crossover execution. Figure 2. Uniform Crossover

8 4.6. Directed Crossover: a new hybrid operator The Directed Crossover operator development was motivated by the analysis of the classical crossover operators performance in the SCM problem addressed in this paper. The classical crossover operators were not efficient enough for generating a uniformly optimized population, since it was possible to observe, during the experiments, the difficulties presented by such operators in preserving schema (from Schemata Theory [Goldberg 1989]). Besides this, it is important to notice that, in spite of to be possible to reach a good solution using the classical crossover operators, it was not possible to reach a set of good solutions. Although a set of good solutions is not required in the problem definition, it is desirable since exogenous variables in the system, like traffic or weather conditions, could make impracticable the execution of the best solution proposed in the GA optimization process. In order to prevent damage in the best schema of each individual and improve the final set of solutions, we proposed a heuristic adaptation in the Uniform Crossover operator. The heuristic has the objective of detecting the most significant schema present in the individual, aiming to provide conditions to guide de genetic recombination carried out by the operator. The Directed Crossover carries out an analysis on the schema driven by the variance in the costs of each product (P j P), i.e., products with a higher cost variance among different stores indicate alleles that must compose the most relevant schema. This analysis occurs before the first generation of the GA optimization process, as stated in Algorithmalg:analysis, and the process is exemplified in Figure 3. Algorithm 3 Analysis of the best schema Calculate the costs variance (the difference between the maximum cost and the minimum cost) of each product P j P using the information in costs matrix W and store the results in a vector A; 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 which divides the genes of the individual in five group with the same number of alleles. For each group is associate a crossover probability as follows: the first group is considered the most relevant scheme, thus the crossover probability is set to be 20%; the second group is set to be 40% and so on. Notice that the aim is to avoid breaking good schema with the usage of crossovers operators. The Figure 4 illustrates the different rates of each group of chromosomes. Finally, in Directed Crossover strategy, a number in [0, 1] is randomly sorted for each gene in the chromosome representation; if this number is greater than the crossover probability, the gene receives the genetic information from the parents following the same rules stated for Uniform Crossover.

9 Figure 3. Analysis of the best schema: the cells in green represent the elements which have the minimum costs for each product; the cells in yellow represent the elements with maximum costs for each product. Figure 4. Probabilities associated to possible schema in the Directed Crossover operator Mutation Operator Mutation is genetic operation which provides diversity for the solutions so as to prevent them from falling into local optima. Not all genes are chosen for performing mutation. The probability of the mutation operation is defined by parameter P m. 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 P m, the value of the gene t is replaced by one of its possible index values of S randomly chosen. In other words, the process of mutation as the crossover, change the stores where the products will be purchased. The Figure 5 present an example of the mutation process: 5. Experiments and Results In order to evaluate the genetic modelling some scenarios were created using a real database with products sold by retails stores, combined with a fictitious directed graph to simulate the path among the stores. Thus the cost of 150 different products in 10 stores were defined, resulting in a matrix W of size Moreover, it was defined 15 possible paths between the stores and the beginning point of the graph. The parameters of the genetic algorithm were set using all combination of values presents in Table 1,and the punishment factor p was define, empirically, as The combination of these parameters resulted in 1152 scenario for tests. Each scenario was carried out using 1000 generations. The Table 2 presents the 20 best results obtained ordered from higher (better evaluated) to lower (worse evaluated) fitness value. Note that the 20 best executions have a P c defined as 100%, λ+µ strategy and P m

10 Figure 5. Example of mutation process Table 1. Parameters used at the simulation Crossover Probability Population Probability Size of Type of Crossover (Pc) % Mode of Mutation (Pm) % Population One Point 70 Total Exchange Two Point 80 λ + µ Uniform 90 Elitism Directed defined as 5%. The best execution reach the fitness of using Directed crossover and λ + µ strategy. The best average fitness was reached in the execution 9, but analysing each individual s phenotype, it was observed that the set of the solutions obtained in the execution 9 have almost all individuals with the same fitness, and this means that the solution obtained have only one global maximum. This was verified in all executions that has 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 best fitness value. If we compare this result with the solution set provided by execution 1 that has the best fitness, we can conclude that the solutions obtained in the execution 1 are more 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 execution 1, 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. Conclusions The main objective of this paper was to describe the development of a strategy based on GA, emphasizing the proposition of a new hybrid genetic crossover operator, to provide a set of optimized solutions for a SCM specific problem. The proposed operator has proven to be effective in situations where the aim is to

11 Table best results obtained in the experiments. Sequence Crossover Pc% Population Pm% Size of Best Average ID Type Mode Population Fitness Fitness 1 Directed 100 λ + µ Uniform 100 λ + µ Uniform 100 λ + µ Directed 100 λ + µ Two Points 100 λ + µ Two Points 100 λ + µ Uniform 100 λ + µ Uniform 100 λ + µ Uniform 100 λ + µ Two Points 100 λ + µ Uniform 100 λ + µ Two Points 100 Eletism Two Points 100 λ + µ One Point 100 λ + µ One Point 100 λ + µ One Point 100 λ + µ One Point 100 λ + µ Directed 100 Eletism Directed 100 Eletism Two Points 100 λ + µ ensure solutions with with high genetic diversity, but with the same best fitness value. To achieve this goal, it also was necessary to use large populations of individuals. This is a limitation of the proposed approach. However, the proposed approach can be adapted to address other optimization problems in SCM. For problems where the goal is to achieve the best solution, the two-point and/or uniform crossover can be used. In the experiments performed, the use of these operators produced solutions with low diversity, but with fitness value equivalent to those obtained with the proposed operator. As future work, we intend to apply the proposed approach in other optimization problems. Furthermore, we intend to compare the proposed approach with other evolutionary strategies. Figure 6. Comparison between executions 1 and 3.

12 References Capizzani, M., Huerta, F. J. R., and Oliveira, P. R. (2012). Retail in latin america: Trends, challenges and opportunities. IESE Business School - STUDY 170, pages Christopher, M. and Towill, D. (2001). An integrated model for the design of agile supply chains. 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 Evolutionary Algorithms. Wiley, 1 edition. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition. Gunasekaran, A. (1999). Agile manufacturing: A framework for research and development. International Journal of Production Economics, 62(1-2): 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). problem. Journal of the ACM, 6. A method for the solution of the nth best path Hsu, C.-M., Chen, K.-Y., and Chen, M.-C. (2005). Batching orders in warehouses by minimizing travel distance with genetic algorithms. Comput. Ind., 56(2): Ko, M., Tiwari, A., and Mehnen, J. (2010). Review article: A review of soft computing applications in supply chain management. Applied Soft Computing, 10(3): Kumar, M., Vrat, P., and Shankar, R. (2004). A fuzzy goal programming approach for vendor selection problem in a supply chain. Computer & Industrial Eng., 46: Lau, H. C. W. (2009). Cost optimization of the supply chain network using genetic algorithms. IEEE Transactions on Knowledge and Data Engineering, 8(8):1 36. Lawrynowicz, A. (2011). A survey of evolutionary algorithms for production and logistics optimization. Research in Logistics and Production, 1(2): Oliver, R. K. and Webber, M. D. (1992). Supply-chain management: logistics catches up with strategy. Logistics: The Strategic Issues. Rezaei, J. and Davoodi, M. (2006). Genetic algorithm for inventory lot-sizing with supplier selection under fuzzy demand and costs. In Ali, M. and Dapoigny, R., editors, Advances in Applied Artificial Intelligence, volume 4031 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg. Shen, Z.-J. (2007). Integrated supply chain design models: A survey and future research directions. Journal of Industrial and Management Optimization, 3(1):1 27. Stevens, G. C. (1989). Integrating the supply chains. International Journal of Physical Distribution and Materials Management, 8(8):3 8. Wen, C. and Eberhart, R. (2002). Genetic algorithm for logistics scheduling problem. In Evolutionary Computation, CEC 02. Proc. of the 2002 Cong. on, volume 1, pages