A Memetic Algorithm for Parallel Machine Scheduling

Transcription

1 A Memetic Algorithm for Parallel Machine Scheduling Serafettin Alpay Eskişehir Osmangazi University, Industrial Engineering Department, Eskisehir, Turkiye Abstract - This paper focuses on the problem of scheduling a set of independent jobs with setup times and job splitting, on a set of identical parallel machines such that total tardiness is minimized. In this study, it is assumed that a job can be split into sub-jobs and these sub-jobs can be processed independently on parallel machines. Because the problem is NP-hard, a memetic algorithm (MA) is proposed. Several test problems are solved via MA and its performance is compared to solutions obtained via simulated annealing (SA) and tabu search (TS) approaches from the literature. Experimental results reveal that proposed MA produces better total tardiness performance than SA and TS. Keywords- Memetic Algorithms, Parallel Machines, Scheduling, Total Tardiness, Job Splitting 1 Introduction The scheduling of jobs on parallel machines provides for interesting and challenging combinatorial problems that continue to interest researchers [1]. Parallel machine scheduling comes down to assigning each operation to one of the machines and sequencing the operations assigned to the same machine. We may have identical, uniform or unrelated parallel machines. If the machines are identical, then the processing time of each job is the same on all machines. Uniform machines work at different speeds, i.e., the processing time of each job differs by a constant factor for the individual machines. If the machines are unrelated, then there is no relation between the processing times of the jobs and the machines. [2] Generally, there are two decisions to be made in parallel-machine scheduling problems. One is to assign jobs to the machines, and the other is to determine the sequence of the jobs on each machine [3]. Therefore, parallel machine problems generally seem to be harder than single machine problems [4]. There are various published papers in parallel machine scheduling problems. The common objectives studied in this area include the minimization of completion time, tardiness, and make-span [5]. As the single machine total tardiness problem is strongly NP-hard, it follows that the parallel-machine total tardiness problem is strongly NPhard, too [4]. It was showed that even the minimization of total tardiness in two identical machine scheduling problem was NP-hard [6]. In this paper, an identical parallel machine scheduling problem with the objective of minimizing of total tardiness is addressed while considering a job splitting property of the jobs. The jobs can be split into a number of sub-jobs that can be processed independently on two or more parallel machines at the same time. Such a problem is called the parallel machine scheduling problem with splitting jobs [7]. Also in the paper, a job is assumed to be composed of a number of unit-jobs and unit-jobs from a job are identical. So, the processing times and the due dates of unit-jobs are the same. A set of unit-jobs from a job is defined as a sub-job that is processed on a machine consecutively and the sub-jobs from a job are processed on the parallel machines independently. Although many studies have been published on parallel machine problems [8], there are very few research results on identical parallel machine scheduling problems with job splitting properties [7]. Serafini [19] studied identical parallel machine problem with a job-splitting property. He considered the objective of minimizing the maximum weighted tardiness for both uniform and unrelated parallel machine environments. He showed that minimizing maximum weighted tardiness can be done in polynomial time. Xing and Zhang [9] also consider the same problem with the objective of minimizing makespan and they proposed a heuristic algorithm to solve it. Kim et al. [10] proposed a two-phase heuristic algorithm for identical parallel machine scheduling problems with the objective of minimizing total tardiness. Logendran and Surbur [11] reported a methodology for minimizing the total weighted tardiness of all jobs intended to be processed on unrelated parallel machines while each job can only be split into two portions. Tahar et al. [12] studied the problem of scheduling a set of independent jobs with sequence-dependent setup times and job splitting, on a set of identical parallel machines such that maximum completion time (makespan) is minimized. They suggested a heuristic algorithm using a linear programming modeling with setup times and job splitting considerations. They tested the performance of their algorithm on large number of randomly generated instances. Shim and Kim [7] proposed a branch and bound

2 (B&B) algorithm for the identical parallel machine scheduling problem with the objective of minimizing total tardiness considering the job splitting property. They developed several dominance properties and lower bounds for the problem and incorporated them with their B&B algorithm. They concluded that the suggested algorithm solves problems of moderate sizes in a reasonable amount of computation time. Sarıçiçek and Çelik [13] dealt with the scheduling problem of identical parallel machines with splitting jobs. They formulated a mixed integer programming model for the problem and proposed two meta-heuristics: simulated annealing (SA) and tabu search (TS). Their computational results showed that SA has a better performance and consumes less time than TS, so they suggested SA as a better heuristic method than TS for considered problem. In this paper, a memetic algorithm (MA) is proposed for parallel machine scheduling problem addressed above. Several test problems provided by Sarıçiçek and Çelik [13] are solved by MA and its performance is compared to their results obtained via SA and TS heuristics. 2 Problem Description Some definitions used in this paper are given below. Job : a production order unit to be composed of a number of unit-jobs Unit-Job: A unit of a minimum process batch for a job. Unit-jobs from a job are identical so their processing times and their due dates are the same. Sub-Jobs: A set of unit-jobs from a job that is processed on a machine consecutively. Sub-jobs from a job are processed on the parallel machines independently. Tardiness of a job: T j = max{0, C j d j }, where C j and d j are the completion time and due date of job j, respectively Completion time of a job: The time when all sub-jobs from the job are completed. of the sub-job just processed and setup times are independent of sequences of the sub-jobs. Other notations: m: number of parallel machines n: number of jobs i: index for machines, i=1,,m j: index for jobs, j=1,,n k: index for position of sub-job of job j on machine, k=1,,n u j : number of unit-jobs of job j p j : processing time of each unit-job of job j s j : sequence-independent setup time for job j d j : due date of job j M: a large constant number which is at least as large as the sum of the processing times and setup times of all jobs x ijk : 1; if the sub-job of j is processed on machine i in the kth position, 0; otherwise: y ijk : The number of unit-jobs of the sub-job of job j which is processed on machine i in the kth position; T ijk : Tardiness of the sub-job of a job j processed on machine i in the kth position. G j : a decision variable that can have a value greater than T ijk and is constrained to be non-negative. Based on the definitions, notations and assumptions above, Sarıçiçek and Çelik [13] presented a mathematical model which is also considered in this study. The objective function of the model is minimizing the total tardiness and expressed as: (1) For the problem addressed here, it is assumed that: As the machines are identical, processing times of a unit-job on all the machines are equal, All jobs are available at time zero. Each machine can process only one sub-job at time. Each sub-job can be processed on only one machine. A setup operation is required before a sub-job is processed on a machine, if the job type of a subjob to be processed is different from the job type

3 Mathematical model: Start Generation of Initial Population Local Search Evaluation Stopping Criteria Satisfied? N Selection Crossover Mutation Y Stop Fig. 1 Flowchart of the Proposed MA As the problem is NP-hard, finding an exact solution to the model in polynomial time is possible for only small sized problems in which n is relatively small, i.e. n<10. If n gets large, the time required to solve such problems increases exponentially with respect to n [13]. On the other hand, heuristic methods can produce good solutions (possibly even an optimal solution) quickly. In this paper, a memetic algorithm is proposed for the problem. 3 Proposed Memetic Algorithm Memetic Algorithm also called hybrid genetic algorithm, represents a new meta-heuristic for combinatorial optimization problems [14]. The proposed MA integrates Variable Neighborhood Decent (VND) with the Genetic Algorithm (GA) to improve the solutions. During the processing of MA, VND is executed on each individual in the population to find the better solution. If it is found, it is replaced with the original individual. The flowchart of the proposed MA is illustrated in Figure 1. Step 1: The initial population is generated by randomly. Each chromosome is encoded by using Earliest Due Date (EDD) dispatching rule on each initial solution. What that means is that randomly assigned unit jobs on each machine are sequenced by using EDD rule. Step 2: To improve the solution quality, the local search based on VND is executed on each individual. Step 3: Each individual is evaluated and obtained its fitness function value, that is, total tardiness. Then, all total tardiness values are compared with each other to find the local best solution. If the local best solution is better than the global best solution found so far in whole process, the global best solution is updated. Step 4: If the stopping criteria is satisfied, the iterative process is stopped and the global best solution is returned. Otherwise, the process is continued with step 5. Step 5: New population is generated by executing selection, crossover and mutation operators respectively. Then the process is continued with step Solution representation and selection mechanism The most commonly used solution representation for the parallel machine scheduling problem is an array of jobs for each machine that represents the processing order of the jobs assigned to that machine [15]. As the jobs are considered to be consisted of unit jobs in this study, the arrays are based on unit jobs represented by a unique job numbers so the unit jobs of a job have the same job number. An example solution representation for 2 jobs, 2

4 machines, 3 unit jobs is given in Figure 2. The figure shows a single array composed by two groups of elements for two machines. The number of elements related to each machine is counted by multiplying the number of jobs and the number of unit jobs. For the example, the array consists of 6 (2x3) elements for each machine so there are total of 12 elements in the array. Fig. 2. Example solution encoding The jobs given in the example are represented by the digits of 1 and 2 respectively in Figure 2. Each 1 denotes a unit job of job 1 assigned to the machines and each 2 shows a unit job of a job 2 assigned to the machines. 0 denotes no unit-jobs assigned to that position. Consequently, according to the figure 2, the processing order of the unit jobs is 1,1,2 for machine 1 and 2,2,1 for machine 2. The selection is performed using a roulette wheel selection mechanism. The idea behind the roulette wheel selection is that each individual is given a chance to become a parent in proportion to its fitness. 3.2 Genetic operators In genetic algorithms, genetic operators are used to combine existing solutions into new better solutions (crossover operator) and to generate diversity (mutation operator). Crossover: In general, the goal of the crossover operator is to generate two good individuals, called offspring, from the two selected progenitors. One of the most used crossover operators to the parallel machine case is the one-pointcrossover [15]. So one-point-crossover is performed in this study. This type of crossover includes one point which is randomly selected for dividing first parent [16]. Once the parents have been selected, the crossover operator is applied according to the crossover probability that is 0.55 in the study. To prevent the generation of infeasible solutions after crossover operation, a repair operator is applied on the new offsprings. Repair operator can both repair infeasible total number of unit-jobs for each job by considering unchanged elements beyond the crossover point and infeasible solutions contains more than one sub-jobs of the same job on the same machine. Mutation: Once the offspring is obtained, the mutation operator is applied according to the mutation probability that is 0.05 in the study. The mutation operator is used to reduce the convergence rate [17]. In other words, the application of the mutation operator can increase the diversity of the population and avoid local optimization [16]. Two genes in the chromosome are randomly selected and exchanged their values. If a chromosome after mutation becomes infeasible according to the constraints 4 in the model then the repair operator is applied to maintain feasibility. 3.3 Local search The local search procedure is employed to enrich and diversify the population produced by the genetic crossover operator without visiting other regions of the search space [16]. In this study, VND is used as local search procedure. VND is a variant of Variable Neighborhood Search (VNS) where the change of neighborhood is realized in a deterministic way during the local search phase [14]. Two different search neighborhoods [18] incorporated into the VND are defined below. Forward insertion neighborhood By starting from the leftmost unit job in an individual, all unit jobs are shifted forward to other positions respectively. After each shifting operation, a new solution is generated. From the generated solutions, non-feasible ones violate the constraints 4 in the model are eliminated. If an improvement occurs during the generating process, the original individual is updated. The process is stopped when no more improvement is found and the current solution is accepted as locally optimal. Backward insertion neighborhood This type of neighborhood is very similar to forward insertion neighborhood except all shifting operations are performed in reverse order by starting from the rightmost unit job. The process continues until no shifting operation gives a better result and the current solution is accepted as locally optimal. All shifting operations for both search neighborhoods are performed without considering 0 values in the chromosomes to reduce the CPU times to complete local searches. Based on the above neighborhood structures, the local search phase is described below:

5 Local search 0 Select first individual from the population. 1 Take the individual as the current solution and execute backward insertion neighborhood search. The local optimal solution the current solution. 2 Take the current solution and execute forward insertion neighborhood search. The local optimal solution the current solution. 3 If the local search was performed for all individuals then Stop, otherwise select the next individual on which local search has not been performed yet from the population and go to step 1. 4 Computational Results The proposed MA was coded in Borland Delphi 7 and run 10 times at each problem on Intel Core 2 Duo T Ghz computer. The size of population varies according to the problem size. All other required parameter values for the problem are generated in a same way as Sarıçiçek and Çelik do: the processing time of a unit-job is integer and randomly generated from uniform distribution with range [5, 60]. Setup time for a job may be short, middle or long and it is generated from the discrete uniform distribution with range [5, 60] for short, [60, 120] for middle and [120, 180] for long. Due date of job j, d j, is generated from the discrete uniform distribution with range [α (s j + u j p j )/m, β (s j + u j p j )/m], where s j, p j and u j denote the setup time of job j, processing time of a unit-job of job j and the number of unit-jobs associated with job j respectively, α and β are parameters used to control tightness (and range) of due dates. A pair of values for (α, β) is used as (0, 1.2) in this study. The performance related results of the proposed MA for average of 100 generations were compared with the results from the search heuristic approaches of SA and TS for the test problems and detailed in the Tables 1, 2, 3 respectively. The results for SA and TS on the test problems are reported by Sarıçiçek and Çelik [13]. The reported performance measures are based on Total Tardiness (TT) and CPU time. A comparison of the results of total tardiness of SA, TS and MA is reasonable but, because of the difference of the test platforms, the CPU time results of MA and other heuristics are not comparable. Nevertheless, the CPU time results of MA are given in this study for information purposes only. Also in the study, the number of tardy jobs (n.of.tj) results obtained from MA runs are reported as another performance measure. Table 1 Test problems 1 and obtained results Table 2 Test problems 2 and obtained results Table 3 Test problems 3 and obtained results Tables show means of the three performance measures of interest: number of tardy jobs, total tardiness and CPU time. The tables are organized by the type of search procedure used: MA, TS and SA. 5 Conclusions The results indicate that total tardiness performance of MA is superior to that of other heuristics for all test problems. As the problem size grows, MA gives much better performance in terms of total tardiness when

6 compared to TS and SA. Only for 40 jobs, 20 machines, 12 unit jobs and medium setup problem, the TT performance of MA is nearly same or similar to the performances of TS and SA. It should be noted that the local search presented here will probably need more CPU time to end up searching for the feasible search space than those of TS and SA heuristics as the problem size grows. So, one further investigation would be to enhance the computational efficiency of this approach especially for really large size scheduling problems. 6 References [1] A. Cossari, J.C. Ho and G. Paletta, A new heuristic for workload balancing on identical parallel machines and a statistical perspective on the workload balancing criteria, Computers and Operations Research, 39, 2012, pp [2] I. A. Chaudhry, S. Mahmood and R. Ahmad, Minimizing Makespan for Machine Scheduling and Worker Assignment Problem in Identical Parallel Machine Models Using GA, Proceedings of the World Congress on Engineering 2010, Vol III, WCE 2010, June 30 - July 2, 2010, London, U.K. [3] S. Shim and Y. Kim, Scheduling on parallel identical machines to minimize total tardiness, European Journal of Operation Research, 177, 2007, pp [4] D. Biskup, J. Herrmann and J.N.D. Gupta, Scheduling identical parallel machines to minimize total tardiness, Int. J. Production Economics, 115, 2008, pp [5] D. Kim, K. Kim, W. Jang and F.F. Chen, Unrelated parallel machine scheduling with setup times using simulated annealing, Robotics and Computer Integrated Manufacturing, 18, 2002, pp [6] R.M. Karp, Reducibility among combinatorial problems: complexity of computer computations, NewYork: Plenum Press, 1972, p [11] R. Logendran, F. Subur, Unrelated parallel machine scheduling with job splitting, IIE Transactions, 36, 2004, pp [12] D.N. Tahar, F. Yalaoui, C. Chu and L. Amodeo, A linear programming approach for identical parallel machine scheduling with job splitting and sequence dependent setup times, Int. J. Prod. Econ., 99, 2006, pp [13] İ. Sarıçiçek and C. Çelik, Two Meta-heuristics for Parallel Machine Scheduling with Job Splitting to Minimize Total Tardiness, Applied Mathematical Modelling, 35, 2011, pp [14] B. Wang and G. Zhang, Hybrid VNS and Memetic Algorithm for Solving the Job Shop Scheduling Problem, 18th International Conference on Industrial Engineering and Engineering Management (IE&EM 2011), 2011, China, pp [15] E. Vallada and R. Ruiz, A Genetic Algorithm for the Unrelated Parallel Machine Scheduling Problem with Sequence Dependent Setup Times, European Journal of Operational Research, 211, 2011, pp [16] M. Souki, S.B. Youssef and A. Rebai, Memetic Algorithm for Operating Room Admissions, International Conference on Computers & Industrial Engineering, CIE 2009, , 2009 [17] C.R.A., Reeves, Genetic Algorithm for Flowshop Sequencing, Computers and Operations Research, 22, 1, 1995, pp [18] Ş. Alpay, GRASP with path relinking for a multiple objective sequencing problem for a mixed-model assembly line, International Journal of Production Research, Vol 47, No. 21, 2009, pp [19] P. Serafini, Scheduling jobs on several machines with job splitting property, Oper. Res., 44, 1996, pp [7] S. Shim and Y. Kim, A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property, Computers & Operations Research, 35, 2008, [8] M. Ranjbar, M. Davari and R. Leus, Two branch-andbound algorithms for the robust parallel machine scheduling problem, Computers & Operations Research, 39, 2012, pp [9] W. Xing, J. Zhang, Parallel machine scheduling with splitting jobs, Discr. Appl. Math., 103, 2000, pp [10] Y.D. Kim, S.O. Shim, S.B. Kim, Y.C. Choi, H.M. Yoon, Parallel machine scheduling considering a job splitting property, Int. J. Prod. Res,. 42, 2004, pp