Solving a hybrid flowshop scheduling problem with a decomposition technique and a fuzzy logic based method

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1 Solving a hybrid flowshop scheduling problem with a decomposition technique and a fuzzy logic based method Hicham CHEHADE 1, Farouk YALAOUI 1, Lionel AMODEO 1, Xiaohui LI 2 1 Institut Charles Delaunay, Laboratoire d Optimisation des Systèmes Industriels ICD-LOSI, UMR CNRS 6281 Université de Technologie de Troyes 12 rue Marie Curie, Troyes Cedex, France 2 Chang an University School of electronic and control engineering Nan Er Huan Road, , Xi an, China {hicham.chehade, farouk.yalaoui, lionel.amodeo}@utt.fr, lixiaohui.chd@gmail.com Abstract: - We are interested in this paper in solving a multiobjective hybrid flowshop scheduling problem (HFS). The problem has different parameters and constraints such as release dates, due dates and sequence dependent setup times. Two different objectives should be optimized at once: the makespan and the total tardiness to be minimized. To solve the problem, we have developed two versions of a new decomposition technique based on the NSGA-II algorithm which is called HD-CLAY. This method decomposes a HFS problem with f stages into f sub-problems of parallel machines or single machine. To assess the efficiency of the proposed method, the latter is compared with the classical NSGA-II algorithm and with another metaheuristic coupled to a fuzzy logic controller. The experimental results show the advantages of the decomposition technique. Key-Words: - Scheduling, decomposition technique, fuzzy logic, multiobjective optimization 1 Introduction The studied scheduling problem is a hybrid flowshop scheduling problem. The latter consists of several stages of machines and at least one stage must have two or more machines in parallel. This is the difference with the classical flowshop system. This kind of workshops has been widely appeared in real production industries. The additional parallel machines in some stages can introduce more flexibility and enhance the overall system performances. Multiobjective optimization problems consist of optimizing several criteria simultaneously. The aim is to find a set of non-dominated solutions that fit all the objectives instead of only one optimal solution. None of the non-dominated solutions can be better than the others for all the considered objectives. The two objectives that must be optimized are the makespan and the total tardiness. In scheduling problems, the execution time of the algorithms must be as short as possible. For this reason, we have decided to develop a new resolution technique based on metaheuristics. Different papers have been designated to show the applications of metaheuristics to solve multiobjective optimization problems due to their flexibility [4]. In the feature issue presented by [4], the authors have presented a quick literature review about multiobjective optimization problems and algorithms. Genetic algorithms present some interesting performances in that field [6]. In 1994, Horn et al. [1] have introduced a Niched Pareto Genetic Algorithm (NPGA) which is based on the Pareto dominance relationship. In the same year, Srinivas and Deb [22] have presented the first version of the Non Dominated Sorting Genetic Algorithm. Deb et al. [8] have then developed in 2002 the NSGA-II which is the second version of the previous method and which has given very performing results in different fields. Zitzler and Thiele [26] introduced in 1999 another evolutionary algorithm named the Strength Pareto Evolutionary Algorithm (SPEA) which combines several aspects of the previous developed algorithms in a unique method. This algorithm has been later studied by Zitzler et al. [25] in order to enhance its performances by introducing new techniques leading then to the algorithm SPEA2. In 2010, Khan et al. [13] have proposed a new hybrid ISBN:

2 genetic algorithm for a permutation flow shop scheduling problem with the objective of minimizing the weighted sum of makespan and maximum tardiness, where the maximum tardiness is limited by a given upper bound value. Recently, the researchers have started to develop hybrid metaheuristics which can improve the search ability of the traditional metaheurisitics. For example, [2] have used a method based on the ant colony and the guided local search for solving an assembly line design problem. In a previous work [18], we have proposed a FLC-NSGA-II (Fuzzy logic controlled NSGA-II) to solve a multiobjective parallel machine scheduling problem. Behnamian et al. [1] have proposed a hybrid metaheuristic to solve a machine and sequence dependent processing times hybrid flowshop scheduling problem. [9] and [17] have proposed the L-NSGA to solve a reentrant hybrid flowshop scheduling problem and hybrid flowshop scheduling problem respectively, where L-NSGA is the traditional NSGA-II but based on the Lorenz dominance relationship. In this paper, we have proposed a new decomposition heuristic HD-CLAY (A Heuristic based on a Decomposition technique developed by Chehade, Li, Amodeo and Yalaoui). This method decomposes a HFS problem with several stages of machines into certain sub-problems which should be parallel machines or single machine scheduling problem. The sub-problems are solved by NSGA-II algorithm. In a previous work of [17], we have proved that the NSGA-II algorithm is efficient to solve a parallel machines scheduling problem. This decomposition technique can produce better solutions than classical resolution methods. It is based on the decomposition method which was initially developed by Gershwin in 1987 [11]. Two versions of this decomposition technique are proposed in this work. Another multiobjective optimization algorithm is proposed which is an NSGA-II based metaheuristic coupled with a fuzzy logic controller. The reason of applying such a hybrid procedure is that despite the efficiency of metaheuristics to solve complex problems, some difficulties may be encountered while setting their different parameters. In fact, an efficient metaheuristic requires a good parameter setting. Based on that, some studies have recently started to present the use of fuzzy logic controllers to better set the parameters of those metaheuristics. Fuzzy logic was first proposed by Zadeh [24] for system control. In the references [14], [19] and [21], fuzzy logic controllers have been used to guide genetic algorithms to solve single objective problems. Lau et al. [15] have proposed fuzzy logic controllers with the second version of a non-dominated sorting genetic algorithm (FL-NSGA2), then with the second version of a strength Pareto evolutionary algorithm (FL- SPEA2) to solve vehicle routing problems. Yalaoui et al. [23] have solved a reentrant scheduling problem with fuzzy logic controllers coupled to a genetic algorithm (FLC-GA) and to a non-dominated sorting genetic algorithm (FLC-NSGA2). In order to compare the fronts of the non-dominated solutions that are get by the different developed multiobjective algorithms, different measuring criteria may be used. This comparison is important as it will allow to assess the quality of the obtained solutions and to know which algorithm has the best performances. These criteria may be classified in two categories: the first one regroups the criteria that are specific to a single front and the second one to the criteria that are specific to two fronts simultaneously. Among the criteria that are specific to a single front we may cite the number of solutions in the best front n fi, the hypersurface [5], the spacing [5] and the HRS measure [5]. Among the criteria that are specific to two fronts simultaneously, we may cite the distance µ of Riise [20], the Zitzler measure [26], the measure of progression [5] and the measure of Laumanns et al. [16]. In this work, three criteria are used to compare the different multiobjective methods: the number of solutions in the best front, the distance of Riise and the measure of Zitzler. More details about those criteria are presented in the computational experiments section. The rest of this paper is organized as follows. The problem considered in this paper is described in section 2. In section 3, we present the resolution method. Section 4 presents the comparison measures and the numerical results. Conclusion and perspectives are given in section 5. 2 Problem description In a hybrid flowshop scheduling problem, a set of n independent jobs should be executed in a workshop line which has f stages of machines and at least one stage must have more than one parallel machine. Each job j consists of f parts of different operations which should be processed on f stages. The processing times of each operation of job j at stage t is defined as p jt. The sequence dependent setup times s ij t are considered in this paper. They present the transfer time when the production on a machine at stage t is switched from job i to job j (job i is immediately before job j on a machine at t stage t). Without loss of generality, we have set s ij = 0 t and s 0j = 0. The former equals to zero because the job cannot be reprocessed before its finish at any stage, and ISBN:

3 the latter means that if job j is scheduled at the beginning on a machine at stage t, no setup time is required. In addition, if job j has finished its work in the actual stage, no transfer time is required when it enters the next stage. Some assumptions are presented as follows and they must be respected: All the jobs are available at time zero; All the machines in each stage are available at time zero; Each machine can execute only one job at the same time; Each job can be processed only once in each stage; Each job cannot be interrupted during its processing (The preemption is prohibited). 3 Resolution methods In this section, we propose two versions of the method HD-CLAY. First, the solution encoding of this problem and some principles of genetic algorithms are presented. Then, we propose the standard NSGA-II algorithm and the HD-CLAY method. Finally, we present the FLC- NSGA-II which is an NSGA-II based algorithm coupled to a fuzzy logic controller. The computing results are shown in the following section, in order to compare the different proposed methods. 3.1 The basis of a genetic algorithm A multidimensional chromosome is applied here and a matrix with n genes is generated. In each gene, the elements present the index of job j, the index of machine k t at stage t (job j is scheduled on machine k t at stage t) and its position r t at stage s (job j is scheduled at position r t on machine k t at stage t). An example of the chromosome is presented in table 1. If we take for example job 2, it means that, in the first stage, it must be scheduled on machine 3 and in position 2. In stage 2, it must be scheduled on machine 2 and in position 2. TABLE I. EXAMPLE OF THE CHROMOSOME j k r k r In a genetic algorithm, it is very important to choose the suitable operators such as crossover, mutation and selection. In this work, we have chosen a multi-point crossover. The structure of this crossover is presented in a previous work [17]. In each stage, an exchange position is randomly generated. The first part of parent 1 and the second part of parent 2 are copied to generate a new solution (child 1). A procedure similar is applied to generate child 2. In the mutation operator, in each stage, two genes are chosen randomly and exchanged. The tournament selection is applied to decide the choice of best current solutions to crossover. 3.2 NSGA-II The NSGA-II algorithm (non-dominated sorting genetic algorithm) is proposed by Deb et al. [8] as a fast and efficient multiobjective genetic algorithm. This algorithm is based on the Pareto relationship. In the NSGA-II algorithm, an initial population P 0 is first randomly generated. Then, in each generation, the operators of evaluation, selection, crossover and mutation are applied to create the population of children Q t (all the offspring solutions). After that, all the solutions (both from P t and Q t ) are ranked in different non-dominated fronts. Then, according to the front level and the crowding distance, the best solutions are copied in a new population P t+1. The crowding distance of a solution i is calculated based on the perimeter formed by the nearest solutions to i in the same front. The aim is to rank the solutions which are in the same front. The generation is repeated until the stopping criterion is satisfied. 3.3 The decomposition technique HD-CLAY In order to solve the HFS problem, we propose a new method HD-CLAY which decomposes a HFS problem with f stages into f sub-problems. Each sub-problem is solved as a parallel machines or a single machine scheduling problem. The main conception of this method is presented in figure 1. Figure 1. The decomposition method HD-CLAY In the first f-1 sub-problems, the aim is to find a solution which optimizes the objectives. With this solution, the completion times of each job j at stage t (C jt ) are considered as the release date of job j at the next stage (r jt-1 ). At the last stage, we use a multiobjective genetic algorithm. Because each sub-problem is ISBN:

4 considered as a parallel machine or a single machine problem, the problem complexity is reduced and we can find better solutions. In this work, we have proposed two different versions of HD-CLAY HD-CLAY-1 In this method, the first f-1 sub-problems are considered as single objective optimization problems. We use a single objective genetic algorithm to find a solution with the best value of makespan. With this solution, we can consider the completion time of each job at stage t as the release date of job at stage t+1. At the last stage, we apply the NSGA-II algorithm, in order to find the nondominated solutions. The structure of this method is presented in Algorithm 1. Algorithm 1. Structure of the HD-CLAY-1 method j = 1; WHILE (j f-1) DO Generate the job data at stage j Initialize the solutions at stage j Evaluate the initial solutions at stage j Sort the initial solutions at stage j WHILE (Stopping criteria is not satisfied) DO Create the offspring population Q t (with the operations of selection, crossover and mutation), evaluate all the solutions Compose the populations of parents and children in R t = P t and Q t Sort the solutions of R t based on the makespan Choose the best solution based on the value of the makespan. The completion time of jobs of this solution are considered as the release date of jobs at the stage j+1. Generate the job data at stage f (the last stage) Initialize the solutions at stage f Evaluate the initial solutions at stage f Sort the initial solutions at stage f WHILE (Stopping criteria is not satisfied) DO Create the offspring population Qt (with the operations of selection, crossover and mutation), evaluate all the solutions Compose the populations of parents and children in R t =P t and Q t Sort the solutions of Rt in different non dominated fronts Fi based on the Pareto dominance P t+1 =0; i=1 WHILE (P t+1 + F i < N) DO P t+1 P t+1 and F i i = i + 1 Rank the solutions of F i based on the crowding distance, add N- P t+1 solutions in P t+1 by descending order of the crowding distance HD-CLAY-2 In the HD-CLAY-1 algorithm, the first f-1 problem is solved as a single objective problem. In fact, in a scheduling problem, several solutions can give the same value for one objective. For example, the minimization of the makespan cannot guarantee a solution which is also competitive on another criterion (they may have the same value of makespan, but the value of total tardiness is greater than another solution). This case cannot guarantee to find global optimal solutions. To solve this problem, we have proposed the second version of our decomposition method (HD-CLAY-2). In this method, all the sub-problems are treated as multiobjective problems. For the first f-1 sub-problems, we have applied NSGA-II based on the two objectives: the makespan and the total completion times. Then we have chosen the extreme solution in the non-dominated front which has the best value of makespan. The completion time of each job at stage t are considered as the release date of job at the next stage t+1. At the last stage, the NSGA-II algorithm is applied. The aim is to minimize the makespan and the total tardiness. 3.4 The NSGA-II algorithm with the fuzzy logic controller Usually, a genetic algorithm uses many parameters (the probability of crossover, the probability of mutation, the number of generations, the initial population size etc.). Since the setting of those parameters is very difficult, a fuzzy logic controller (FLC) is proposed. It is used to better set the values of two parameters which are: the probability of crossover Pc and the probability of mutation Pm. In this study, the average objective values of the solutions in the non-dominated front and their diversity are considered as the inputs of the fuzzy logic controller. The modifications in the values of the two parameters are guided by the output of the FLC. Appropriate values of these parameters can produce better results, by avoiding premature convergence and falling into local optimum. This enhancement is carried out every ten consecutive generations [15] in order to provide sufficient required time for the modification in the algorithm. Fuzzy logic was firstly presented by Zadeh [24], as a powerful and useful tool applied in many areas. A FLC consists of 3 parts: fuzzification, decision making, and defuzzification. We have used here a similar FLC of Lau et al. [15] to guide the NSGA-II ISBN:

5 algorithm, with a difference in the parameters of the membership functions. The details are described below as well as in a previous work [18] Fuzzification The membership functions are in order to associate the system input and output values with the fuzzy input and output membership values. In this study, the values of f a (t) = f(t-1) - f(t-2) and d(t-1) are considered as the system input values where f(t-1) and f(t-2) are the average values of the objective function and d(t-1) is the sum of the hamming distance between the individuals of the entire generation. Then ΔP c (t) and ΔP m (t) are considered as the system output values. The membership functions are presented in figure 2. They are usually chosen based on the expert knowledge and experience. The triangular membership functions are used in this study, and the parameters are defined according to several tests. The meanings of each linguistic term are presented in table Decision making When the system input values are transferred into the fuzzy input by the membership functions, a number of IF-THEN rules are applied to get the output membership values. These rules are based on the expert knowledge and experience. We have chosen the decision tables of Lau et al. [15] in our work. They are presented in tables 3 and Defuzzification The aim of the defuzzification is to calculate the deviation values of the parameters (P c, P m ) by the fuzzy output membership values and the membership functions of ΔP c (t) and ΔP m (t). There are many defuzzification methods such as AI (adaptive integration), COA (center of area), CDD (constraint decision defuzzification) and COG (center of gravity). The COG defuzzification method is adopted here, as it is the best known defuzzification operator. This method computes the gravity center of the area under the membership function FLC-NSGA-II In order to improve the performances of the NSGA-II, a fuzzy logic controller (FLC) is proposed here to change the crossover and mutation probabilities each ten consecutive generations. The aim is to provide sufficient times for the NSGA-II to respond the changes. The average fitness value of the population and the population diversity are taken as the input of the fuzzy controller. In fact, appropriate crossover and mutation probabilities can provide good convergence and diversity of the obtained results [18]. Since the studied problem is a multiobjective optimization problem, the average fitness value f(t) is the average value of the makespan and the total tardiness of the population at iteration t. The value of d(t) is the degree of population diversity at iteration t. It is the average of the difference of all pairs of chromosomes of all the solutions in the obtained population and it can be computed as shown in equation (1). N N n 1 1 ( gil, g jl ) d( t) (1) N( N 1) i 1 j 1 k 1 n 2 Where N is the population size, the value of n is the chromosome length and g ik is the values of three elements of the k th gene of the i th chromosome, and δ(g ik, g jk ) = 1 if the two genes are not the same, 0 otherwise. TABLE II. MEANINGS OF THE LINGUISTIC TERMS Linguistic terms for Meaning Linguistic terms for d(t-1) Meaning fa(t), ΔP c (t), ΔP m (t) NLR Negative VS Very small larger NL Negative large S Small NM Negative SS Slightly small medium NS Negative small LM Lower medium Z Zero M Medium PS Positive small UM Upper medium PM Positive SL Slightly large medium PL Positive large L Large PLR Positive larger VL Very large ISBN:

6 f(t-1) - f(t-2) d(t-1) Δp α (t) Δp β (t) Figure 2. Membership functions The same parameters of the NSGA-II are used for setting the FLC-NSGA-II. The population size and the number of generations are fixed at 100. The initial crossover probability is 0.9, and the initial mutation probability is 0.1. The structure of the FLC-NSGA-II is shown in Algorithm 2. 4 Computational experiments Here we show first the protocol applied to generate the job data and the multiobjective comparison measures. Then, we compare the two versions of the decomposition method. After that, we compare the best version of the decomposition technique with NSGA-II. At last, the best version of the decomposition technique is compared with the FLC-NSGA-II algorithm. ISBN:

7 TABLE III. DECISION TABLE FOR PARAMETER P C d(t-1) \ f a (t) NLR NL NM NS Z PS PM PL PLR VL PLR PLR PL PL PM PM PS PS Z L PLR PL PL PM PM PS PS Z NS SL PL PL PM PM PS PS Z NS NS UM PL PM PM PS PS Z NS NS NM M PM PM PS PS Z NS NS NM NM LM PM PS PS Z NS NS NM NM NL SS PS PS Z NS NS NM NM NL NL S PS Z NS NS NM NM NL NL NLR VS Z NS NS NM NM NL NL NLR NLR TABLE IV. DECISION TABLE FOR PARAMETER P m d(t-1) \ f a (t) NLR NL NM NS Z PS PM PL PLR VL NLR NLR NL NL NM NM NS NS Z L NLR NL NL NM NM NS NS Z PS SL NL NL NM NM NS NS Z PS PS UM NL NM NM NS NS Z PS PS PM M NM NM NS NS Z PS PS PM PM LM NM NS NS Z PS PS PM PM PL SS NS NS Z PS PS PM PM PL PL S NS Z PS PS PM PM PL PL PLR VS Z PS PS PM PM PL PL PLR PLR Algorithm 2: Structure of the FLC-NSGA-II Generate the initial population P0 of size N Evaluate these solutions Sort these solutions by non-dominated front and crowding distance WHILE stopping criterion is not satisfied DO Create the offspring population Q t (with the operations of selection, crossover and mutation), evaluate all the solutions Compose the populations of parents and the children in R t = P t + Q t Sort the solutions of R t in different non dominated fronts Fi by the Pareto dominance P t+1 = 0 i=1 WHILE P t+1 + F i < N DO P t+1 P t+1 + F i i=i+1 Rank the solutions of F i by the crowding distance, add N- P t+1 solutions in P t+1 by descending order of the crowding distance IF t mod 10=0 THEN The values of f(t-1)-f(t-2) and d(t-1) are considered as the input of FLC, update the Pc and Pm with the FLC END IF In this work, we have used the same method used in [17] to generate all the jobs data: the processing times, the release date, the due date and the sequence dependent setup times are randomly generated between lower and upper values. Three measures are applied in this work to compare the obtained results, since they do not require computing the absolute optimal front: The number of nondominated solutions in the front, the measure C of Zitzler [26] and µ d (DYA distance) of [9]. The latter is a modified measure based on the µ d distance of Riise [20]. ISBN:

8 First, the HD-CLAY-2 method is compared with the HD-CLAY-1. For each configuration, we have tested several instances. The results are given in table 5. The elements in each line show the index of configuration, the mean value of µ d, the best value and the worst value of µd, the mean values of C 1 and C 2, the mean number of non-dominated solutions : n s1 shows the number of nondominated solutions in the HD-CLAY-1 front, n s2 for the HD-CLAY-2 front. The mean values of µd distance are always negative, and the mean values of C1 are always smaller than C2. That means that the front of HD-CLAY-2 is below the front of HD-CLAY-1. The HD-CLAY-1 uses the single objective genetic algorithm to solve the first f-1 problems. It cannot guarantee to find an appropriate solution which is competitive on both considered objectives. The HD-CLAY-2 method can avoid this problem. It uses NSGA-II algorithm for solving each sub-problem. The results show that the HD-CLAY-2 dominates the HD-CLAY-1. Secondly, the best decomposition method HD- CLAY-2 is compared with the original NSGA-II algorithm, the experimental results are shown in table 6. In this table, we can observe that the mean value of µd distance are always negative (the best case and the worst case are also negatives) for all the tested instances. The value of C 1 is zero means that none of solution in the front of HD-CLAY-2 is dominated by the solution in NSGA-II front. On the contrary, C 2 is equal to 1 means that all the solutions in NSGA-II front are dominated by those of HD-CLAY-2 front. The results show the advantage of our proposed method. Finally, we compare the HD-CLAY-2 decomposition technique with the FLC-NSGA-II algorithm. This comparison (table 7) shows the advantages of HD- CLAY-2. In fact, the fronts obtained with HD-CLAY-2 are better than those obtained by FLC-NSGA-II taking in consideration the two objectives. The mean values of µ d are negatives for all the tested instances. This means that the HD-CLAY-2 fronts are under the fronts of FLC- NSGA-II which suits the two objectives. C 1 is smaller than C 2 means that the solutions obtained by HD-CLAY- 2 dominate those obtained by FLC-NSGA-II. 5 Conclusion In this paper, we have proposed a new decomposition technique to solve a hybrid flowshop scheduling problem with sequence dependent setup times in a crisis environment. The aim is to minimize two different objectives simultaneously: the makespan and the total tardiness. Our main contribution is to propose a new resolution technique based on the decomposition method. This method decomposes a hybrid flowshop scheduling problem into several sub-problems which are single machine problems or parallel machine problems. Another method FLC-NSGA-II based on coupling a multiobjective genetic algorithm NSGA-II with a fuzzy logic controller is also proposed. The experimental results show the advantage of the decomposition method HD-CLAY especially in its second version. In our further works, more computational experiments could be realized in order to confirm the efficiency of the HD- CLAY-2 method. It would be also interesting to test this decomposition technique with other metaheuristics such as multiobjective ant colony algorithms or multiobjective particle swarm optimization. References: [1] Behnamian, J., Fatemi Ghomi, S., and Zandieh, M A multi-phase covering pareto-optimal front method to multi-objective scheduling in a realistic hybrid flowshop using a hybrid metaheuristic. Expert System with Applications 36: [2] Chehade, H., Yalaoui, F., Amodeo, L., and De Guglielmo, P Ant colony optimization for assembly lines design problem. In Proceedings of the 8th international FLINS'08 conference on computational intelligence in decision and control, Madrid, Spain. [3] Chehade, H., Yalaoui, F., Amodeo, L., and De Guglielmo, P Multi-objective optimization problem for the dimensioning of buffers. Journal of Decision Systems, 18(2), [4] Coello, C., A. 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