REVISTA INVESTIGACIÓN OPERACIONAL V ol. 29, No. 2,,


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1 REVISTA INVESTIGACIÓN OPERACIONAL V ol. 29, No. 2,, GENETIC OPERATORS FOR THE MULTIOBJECTIVE FLOWSHOW PROBLEM Magdalena Bandala*, María A. OorioLama** School of Computer Science, Univeridad Autónoma de Puebla Ciudad Univeritaria, Puebla, Pue , México * RESUMEN: Uno de lo problema má importante en lo Algoritmo Genético, e la elección correcta de lo operadore de cruza y mutación. Lo operadore genético on má importante para lo cromooma no binario debido a u impacto en lo reultado. Ete trabajo preenta un análii comparativo de diferente operadore de cruza y mutación aplicado a un algoritmo genético para el problema multiobjetivo de calendarización de proceo con tranferencia cero. El algoritmo utilizado etá adaptado de un método de partición propueto por Tagami et al [7] y contruye una frontera de Pareto, minimizando la duración y el tiempo promedio de proceo. ABSTRACT: One of the mut important iue in Genetic Algorithm i the right election of croover and mutation operator. Genetic Operator are even more important for non binary chromoome due to their high impact on the reult. Thi work preent a comparative analyi of different croover and mutation operator applied to a genetic algorithm for the multiobjective flowhop problem. The algorithm ued i adapted from the partition method propoed by Tagami et al[8] and build a Pareto frontier. We minimize the makepan and the mean flowtime. Key Word: mutation operator, partition method, Parteto frontier MSC: 90C29 1. INTRODUCTION In the flowhop problem, we have a et of tak that mut be proceed in everal machine. Not all the tak have to ue all the machine, each tak i proceed in a ubet of tage. It can be aumed tranference with zero time or a continuou flow between the tage in the proce. The flowhop equencing i characterized by unidirectional flow and i NPhard. The flowhop conit of m machine and n different job to be optimally equenced through thee machine. The common aumption ued in modeling the flowhop problem are: All n job are available for proceing at time zero and each job follow identical routing through the machine. Job are independent and available in time 0. Unlimited torage exit between the machine. Each job require m operation and each operation require a different machine. Every machine procee only one job at one time and every job i proceed on one machine at one time. Setup time for the operation are equenceindependent and are included in proceing time. The machine are continuouly available. Individual operation cannot be preempted. * 95
2 In theory, integer programming and the branch and bound technique can be ued to olve the flowhop problem optimally. However, thee method are not viable on large problem. Mot cheduling problem including flowhop problem belong to the NPhard cla for which the computational effort increae exponentially with problem ize. To remedy thi, reearcher have continually focued on developing heuritic and other method. Heuritic method typically do not guarantee optimality of the final olution, but a final olution i reached quickly ad i acceptable for practical ue. Several good heuritic method have appeared in the pat three decade to help minimize makepan for flowhop. Recent advance in metaheuritic earch method that help conduct directed intelligent earch of the olution pace have brought yet new poibilitie to rapidly find good chedule, even if they are not optimal. 2. MIP FORMULATION The Mixed Integer Formulation (MIP) for the flowhop problem conider a et of contraint that avoid colliion in every tage in the proce. The mono objective problem only addree the minimization of total time in the proce, the makepan while the multiobjective may conider the flowtime and the tardine Contraint The model ha two type of contraint. The firt et guarantee that the total time, named makepan (T) exceed the total time of every job in the ytem. The econd type, avoid colliion between each pair of the job in the firt tage, that coincide in the ytem. k T t i + t ij i I k km m J m j j J t + t t + t + My km m J m j i im ( k ) m J ( i ) i m< j t + t t + t + My im ( k ) m J ( i ) m< j ik ik j Cik, i,k I,i k t 0, i I, y = 0,1 i, k I,i k Where t ij i the time that job i tay in tage j and t i i the initial time for job i Objective i The objective i to find a tage equence according to an optimal criteria. The claical three cheduling objective are (1) makepan, (2) mean flowtime and (3) mean tardine. In thi reearch, we only worked with makepan and flowtime Makepan. In cheduling literature, makepan i defined a the imum completion time of all job, or the time taken to complete the lat job on the lat machine in the chedule auming that the proceing of the firt job began at time 0. Makepan i denoted by T and computed a: T = {F j } ik 96
3 i j n Where F j i the flowtime for job j, i.e., the total time taken by job j from the intant of it releae to the hop to the time it i proceed by the lat machine Mean Flowtime. The mean flowtime meaure the average repone of the chedule to individual demand of job for ervice. Mathematically, mean flow time i the average of the flow time of all job. It i uually repreented by F and expreed a: F (x)= n j= 1 2 /JOBS f j 3. MULTIOBJECTIVE FLOWSHOP In equencing job in a flowhop we are likely to confront everal different (and often conflicting) management objective. Conequently, a chedule may have to be evaluated by different type of performance meaure. Some of thee meaure may give importance to completion time (e.g. makepan), ome to due date (e.g. mean tardine, imum tardine), and ome other to peed with which the job flow (e.g. mean flow time). The imultaneou conideration of thee objective i a multiobjective optimization problem. But, even for a ingle objective, flowhop equencing i NPhard. 3.1 Pareto Optimality A common difficulty with multiobjective optimization i the appearance of objective conflict (Han, 1988): none of the feaible olution achieve imultaneou optimization of makepan, mean flow time, mean tardine, machine utilization, etc. in a flowhop, for example. In other word, the individual optimal olution of each objective are uually different. Thu a mot favored olution would be one that would caue minimum objective conflict. Such olution may be viewed a point in a olution pace. The general multiobjective optimization problem contain a number of objective while the olution (2) mut alo atify a number of inequality and equality contraint. Mathematically, the problem may be tated a follow. Minimize F(x) = (F 1(x), F2 (x)) F 1 (x)= T (Makepan) F (x)= 2 n j= 1 f j / JOBS (Flowtime) Here the deciion parameter x i a pdimenional vector made up of p deciion variable. Solution to a multiobjective optimization problem are mathematically expreed in term of nondominated point. The nondomination property of olution may be explained a follow. In a minimization problem, a olution vector x(1) i partially dominated by another vector x(2) (written a x(1) x(2)), when no component value of x(2) i le than x(1) and at leat one component of x(2) i trictly greater than x(1). In a minimization problem if x(1) i partially le than x(2), we ay that the olution x(1) dominate x(2) or the olution x(2) i inferior to x(1). Any ember of uch vector that i not dominated by any other member i aid to be nondominated or noninferior. Thi reearch get the Pareto frontier for olution that minimize the makepan and the mean flowtime. 3.3 Multiobjective Genetic Algorithm 97
4 There are two type of evolutive algorithm for mulitobjective optimization. Algorithm that do not incorporate the Pareto optimality concept for the election proce and algorithm that hierarchize the population according to Pareto domination. For the algorithm that conider Pareto optimization there are two generation. The firt generation of algorithm utilize a ranking for the bet individual. The econd generation ue the concept of elitim for the individual election and for the election of econdary population of individual. Figure. 1 A Pareto Frontier 4. GENETIC OPERATORS 4.1 Croover The genetic algorithm wa implemented in C. The individual ue a non binary repreentation for the chromoome uing the job equence a in Kobayahi et al [1995]. The election of chromoome for croover i made by the creation of an elite ubet. From the original population, the algorithm elect the individual with better qualification ( 8) and then randomly elect the individual that will perform the croover. The elected individual can have different.thi type of election doe not ue the croover probability becaue the element in the elite ubet will perform the croover with the other element in the ubet until the population generated reache the ame ize than the original population. Due to the nonbinary chromoome, we ued two croover: 1) a one point croover with a reparation algorithm propoed by Jayaram [16], that force olution to be feaible, and 2) the ubequence exchange croover preented by Kobayahi et al [1995] One Point Croover It randomly choice a point in the chromoome choen to be the parent and from thi point, it interchange the information from both chromoome to generate new individual. Thi croover method can generate invalid chromoome and require a reparation method that retore feaibility in the chromoome. The algorithm ha the following tep: 1. It randomly elect a cutting point in the parent chromoome. 2. It copie in the new individual, the chain before the cutting point, in the chromoome parent 3. In the firt individual generated, the empty poition will be filled with element from the econd parent that do not exit in the firt individual, in the ame order that they appear in the econd father. f i f i 98
5 4. In the econd individual generated, the empty poition will be filled with element from the firt parent that do not exit in the econd individual, in the ame order that they appear in the firt father. 5. FIRST Author Characteritic Year GENERATION VEGA Schaffer Divide population according to the number of objective, give priority to only one MOGA Foneca and Fleming, Hierarchize the population according to Pareto dominance with niche NSGA Sriniva y Deb Hierarchize the population by lide. Keep diverity NPGA SECOND GENERATION Elitim + Niche Horn, Nafplioti and Goldberg Make tournament between individual. Hierarchize the population according to the199 tournament. 4 Author Characteritic Year SPEA Zitzler and Thiele Ue a population to help nondominated individual. Hierarchize the external population PAES Knowleand and Ue an auxiliar population for the Corne nondominated individual. 199 Divide the objective pace in grid. Make9 election by elitim and Pareto. NSGA II Deb, Agrawal, Pratap Hierarchize population according to NSGA:. and Meyarivan PESA Corne, Knowle and Ue auxiliary population for the nondominated Oate individual Divide the objective pace in grid and ue a0 election criterion baed on elitim and Pareto. NSGAII Ded and Goel Control elitim SPEA2 Zitzler, Laumann and Thiele Figure. 2 Multiobjective algorithm evolution Subequence Exchange Croover Baed on Pareto dominance. Keep diverity Thi method interchange ubequence that have the ame element in each chromoome parent an interchange the chain. It originate valid chromoome and doe not need a reparation algorithm. 99
6 4.2 Mutation Figure. 3 Subequence Exchange Croover Mutation i made by interchanging adjacent job in the chromoome. The job ued for mutation are choen randomly. During all experiment, the population ize i et i et a 25, the mutation rate a 0.1 with only one interchange in the chromoome. In the Pareto partitioning method, the number of partitioning region i et a the GENETIC ALGORITHM UTILIZED The genetic algorithm propoed i baed in the Pareto partitioning algorithm propoed by Takanori Tagami and Tohru Kawabe [1998]. The Pareto partitioning algorithm conit in dividing the olution pace in pecific region, a a grid where every objective function ha a certain number of pecific region and the objective i to put olution in the Pareto frontier. The evaluation function (fitne) for each individual p i i defined a f i = 1/n i. The value of n i denote the number of olution nondominated in the region p i. There are algorithm that work with a very imilar method and divide the olution pace in cubic ubpace that contitute a grid where the objective i to locate individual that are nondominated in an uniform and ditributed way in the Pareto frontier. Thoe algorithm are: PAES (Pareto Archived Evolution Strategy), Knowle and Corne, (1999), PESA (Pareto Envelopebaed Selection Algorithm), Knowle and Corne, (2000), PESA II Corne, Jerram, Knowle and Oate (2001). The Pareto partitioning algorithm partition can be tated a follow: Algorithm Step 1 Generate a random initial population with M individual. Step 2 Calculate the makepan and the flowtime for each individual in the initial population. Step 3 Calculate the fitne of each olution in the current population according to the multobjective ranking. i) Aign a rank R i to every individual in the population. The rank will correpond to the number of member in the population that dominate the individual. 100
7 Step 4 ii) Aign the fitne f i to all the nondominated individual. The evaluation function f i of every individual p i i defined a follow: f = (f f (R )) i i 1 f 1 = (f M 1 f min where M denote the population ize, f and f min denote the imun and the minimum fitne value. Generate a new population from the initial population, uing the croover operator Step 5 Apply mutation to the new population. Step 6 Apply the evaluation function (Fitne) to both population. Step 7 Select M individual from all population on the bai of their fitne. Step 8 6. COMPUTATIONAL RESULTS If a terminal condition i atified, top and return the bet individual. Otherwie go to Step 2. ) We teted everal ize for the numerical example preented by Bagchi in [1999]. The algorithm i implemented in C (gcc Linux/3.2.2) V 5.1 under Linux. We teted problem that range from 15 to 25 job and from 10 to 15 machine. The parameter conidered are: population ize ( p ), mutation probability 2 ( p m ), number of region in the Pareto partition ( p ) and imum number of generation ( t ). The value for the parameter are reported in Table 1. Parameter Value p 50, 70, ( p ) ( 50), (70), (100) p 0.1 m t 100, 300,400,500 Table 1 Figure 4 how the graph obtained with the etting that yield the bet reult for an example with 20 job and 10 machine. Figure 5 correpond to an example with 25 job and 15 machine. The Pareto frontier can be een in both graph. 101
8 Figure. 4 Example with 20 job and 10 machine Execution time in econd of CPU, were obtained with the clock function in language C for unix. We teted intance with 25 job and 15 machine and with 20 job and 10 machine and reported the execution time. Figure. 6 Example with 25 job and 15 machine Table 2 and 3 how the execution average time in econd for different population ize ( ), and with generation ranging from 100 to Table 4 how that with a bigger of job and machine, the average execution time increae with the ize of the population and the number of generation, although the time i le for p =50 and t =100. For our tet the bet reult were obtained with a ize of p =100 and t =500 with the ame execution time for both example. Figure 7 how the average computational time and the population ize for different number of generation. It can be een that the computational time increae for bigger intance p 102
9 p t (100,1000) Execution Average Time 50 (0.01, ) (0.02,0.04) 100 (0.04,0.05) Table 2 Execution Time (ec) for n =20 and m =10 5. CONCLUSIONS The implemented election produce diverity in the population, becaue it doe not reproduce children equal to their parent. In addition, 65% to 70% of the individual are apt for croover. From both croover method, the ubequence interchange ha more computational work, becaue the number of comparion neceary to find the ubequence with equal element. When the chain contain greater number of element (20 to 25), the reultant chain do not preent a great change among them and it correponding parent. Thi ituation limit the exploration, ince it generate very imilar or equal chain ometime. p t (100, 1000) Execution Average Time 50 (0.02, ) 70 (0.03, 0.04) (0.04, 0.06) Table 3 Execution Time (ec) for n =25 and m =15 It i difficult to find ubequence of more than 5 character with ignificant change. If the chain correpond to mall ubequence, they may be different between them but very imilar to their parent, and a mutation procedure i neceary. For thi reaon, the method of croover with chromoome reparation i ued. For more than 10 work, the method of croover with chromoome reparation offer a good exploration in a maller computational time. N m p =50 p =0.1 m t (100,1000) p =70 p =0.1 m t (100,1000) Table 4 Execution Time (ec) In the mutation cae, tet were done interchanging adjacent and random work. Better chain are obtained with a randomly mutation. Bet reult are obtained with a population of 100 individual and a probability of 0.1 in 500 generation, a hown in table
10 Parámeter Bet Value p ( p ) ( 100) p m 0.1 t 500 Table 5 Bet Parameter 0.06 Tiempo en egundo n=20,m=10 n=25,m= Tamaño de población Figure 7 Time v. Population ize Received April 2006 Revied March 2007 REFERENCES BAGCHI, T. (1999) Multiobjetive Scheduling by Genetic Algorithm, Kluwer Academic Publiher, Boton/Dordrecht/London. COELLO, C.A.; D. A. VELDHUIZEN, and G.B. LAMONT, (2002) Evolutionary Algorithm for Solving Multiobjective Problem. Kluwer Academic Publiher, Boton/Dordrecht/London. 104
11 CORNE, D.W; J. D. KNOWLES, and M. J. OATE(2000), The ParetoEnvelope baed Selection Algorithm for Multiobjetive Optimization. Parallel Problem Solving from NaturePPSN VI, Springer Lecture Note in Computer Science, Springer, Berlin. CORNE, D.W. N. R. JERRAM, J. D. KNOWLES and M. J. OATES (2001) PESA11: Regionbaed Selection in Evolutionary Multiobjetive Optimization. Proceeding of the Genetic and Evolutionary Computation Conference (GECCO2001), Morgan Kaufmann Publiher, Illinoi FINK, A. and V.B.STEFAN, (2001) Solving the Continuo FlowShop Scheduling Problem by Metaheuritic. Technical Univerity of Braunchweig, Germany. FONSECA C. m. AND P. J. FLEMING. (1993) Genetic Algorithm for Multiobjective Optimization: Formulation, Dicuion and Generalization. In Stephanie Forret, editor, Proceeding of the Fifth International Conference on Genetic Algorithm, , San Mateo, California, Univerity of Illinoi at Urbana Champaign, Morgan Kauffman Publiher, Illinoi. GREINER, D. (2000) A ummarized overview of evolutionary multiobjetive algorithm and new approache CEANI, USA,. ISHIBUCHI, H., T. YOSHIDA, and T. MURATA, (2000) Balance between Genetic Search and Local Search in Memetic Algorithm for Multiobjective Permutation Flowhop Scheduling. Deparment of Indutrial Engineering, Oaka Prefecture Univerity, Japan; Deparment of Informatic, Faculty of Informatic, Kanai Univerity, Japan. JAYARAM,K. (1998) Multiobjetive Production Scheduling, M Tech Thei, Deparment of indutrial and Management Engineering, Indian Intitute of Technology Kanpur. KNOWLES, J. D. and D. W. CORNE Approximating the Nondominated Front uing the Pareto Archived Evolution Strategy. Evolutionary Computation, 8(2) pp Maachuett Intitute of Technology, KOBAYASHI, S; I. ONO, and M. YAMAMURA, (1995), An Efficient Genetic Algorithm for Job Shop Scheduling Problem, Proceeding of the Sixth International Conference on Genetic Algorithm Univerity of Pittburgh, KURI A and, J. GALAVIZ (2002), Algoritmo Genético, Intituto Politécnico Nacional, Univeridad Nacional Autónoma de México, Fondo de Cultura Económica, México PÉREZ, B y M.A. OSORIO,(2003) Análii Comparativo de Heurítica para Problema de Calendarización de Trabajo con Tranferencia Cero Memoria del Primer Congreo Mexicano de Computación Evolutiva, CIMAT, RÍOS, R. Z. y J. F. BARD (1999) Heurítica para Secuenciamiento de Tarea en Línea de Flujo. Reporte Técnico, Univeridad Autónoma de Nuevo León. RUIZ, R. y C. MORATO (2003) Evaluación de Heurítica para el problema del taller de flujo. Reporte Técnico, Departamento de etadítica e Invetigación Operativa Aplicada y Calidad. Univeridad Politécnica de Valencia, Epaña/. TAGAMI, T AND T. KAWABE (1998) Genetic Algorithm with a Pareto Partitioning Method for Multiobjetive Flowhop Scheduling, Technical Report, Kagawa Junior College, Univerity of Tukuba, Japan,. 105
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