Metaheuristics for Identical Parallel Machines Scheduling to Minimize Mean Tardiness

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1 Metaheuristics for Identical Parallel Machines Scheduling to Minimize Mean Tardiness Husam Kaid 1,a, Ibrahim Alharkan 1,b, Atef Ghaleb 1,c, and Mageed A. Ghaleb 1,d 1 College of Engineering, Industrial Engineering Department, King Saud University, Riyadh 11421, Saudi Arabia a yemenhussam@yahoo.com, b imalhark@ksu.edu.sa, c amag16@gmail.com, d mghaleb87@gmail.com Abstract Scheduling of jobs has been a challenging task in manufacturing and the most real life scheduling problems, which involves multi-objectives and multi-machine environments. This paper presents tabu search and simulated annealing approaches for scheduling jobs on identical parallel machines. The identical parallel machine scheduling problem has been considered to minimize the mean tardiness for the jobs. Initially, an initial solution has been obtained using EDD dispatching rule then, simulated annealing and tabu search have been applied to reach a near optimal solution. Computational experiments are performed on problems with up to 10 machines and 150 jobs. The computational results indicate that the two proposed approaches are capable of obtaining better solutions for the given scheduling problem. Moreover, the tabu search approach provides better solution then simulated annealing approach. Keywords Scheduling; Identical parallel machines; Mean tardiness; Tabu search; Simulated annealing. I. INTRODUCTION Scheduling plays a very important role in many modern manufacturing and production systems as well as information-processing environments. The need for scheduling also exists in service وindustries transportation, and in distribution settings. There are two issues commonly considered in scheduling problems: how to allocate jobs on machines and how to sequence jobs on each machine. It is worth noting the distinction between a sequence and a schedule. A sequence usually corresponds to the order in which jobs are to be processed on a given machine. A schedule refers to an allocation of jobs on the machines. Scheduling jobs on parallel machines against their due dates is a very common setting from a practical perspective. It can be found, for example, in the beverage industry where jobs need to be scheduled on parallel bottling machines. Jobs are, for example, beer orders from restaurants, pubs, and wholesalers. Meeting the due dates is important, because it is advantageous for the wholesaler, retailer or local pub not to run out of beer. Another example is the capital-intensive printing industry, where print jobs (like books) are scheduled on parallel presses and finishing lines. Meeting due dates for orders placed by the different book publishers is extremely important, too, as a publisher will lose revenues if a special title (in the worst case the actual bestseller) is not available on the shelves of the bookstores. The pharmaceutical industry usually runs parallel machines to produce drugs. Moreover, they definitely want their blockbusters being available in the drugstores. The real parallel-machine shops, such as the drilling workstation in printed circuit board manufacturing systems. Even auditors can be seen as parallel machines that work on the auditing jobs acquired by their partners. This paper focuses on the problem of scheduling n independent jobs with different due dates on m identical parallel machines. Tabu search and simulated annealing approaches are proposed for scheduling jobs on identical parallel-machines with the objective of minimizing mean tardiness of the jobs. This paper is organized as follows. In Section II, some previous studies are mentioned. In Section III, the scheduling problem introduced and described. Then, a tabu search and simulated annealing algorithms are proposed in Section IV and V, respectively. In Section VI and VII, the instances preparation and parameters setting of the proposed algorithms are introduced, respectively. The proposed algorithms effectiveness are examined in Section VIII by several computational experiments. Finally, Section IX presents the conclusion. II. RELATED WORK The tardiness of jobs in a parallel machine environment has been a very important performance measure for a manufacturing environment. The complexity of the parallel scheduling problems has led to a growing interest for heuristic and Metaheuristics methods, which can produce quality solutions in a reasonable time. Ho and Chang [1] showed that a parallel machine scheduling problem is NPhard. Therefore, the appropriate way to solve these complex parallel scheduling problems is heuristic and Metaheuristics techniques rather than an optimal solution. An identical parallel machine scheduling problem has been reported by Koulamas [2] where SA has been applied to exchange jobs assigned to machines using decomposition. Azizoglu and Kirca [3] solved the problems of parallel machines to minimizing the total tardiness using Branch and bound algorithm and they found that the optimal solutions can be obtained in reasonable times for problem up to 15 jobs. Tabu search application into parallel machine scheduling has been reported by Suresh and Chaudhuri [4] and Armentano and Yamashita [5]. Eom et al. [6] proposed a three-phase heuristic to minimize total weighted tardiness. In the first phase, jobs are listed by the earliest due date and then divided into small job-sets according to a decision parameter. In the second phase, jobs are grouped by the due date within applicable families using apparent tardiness cost with set-up /15/$ IEEE

2 (ATCS), and the sequence of jobs within families is improved using the tabu search method. In the third phase, jobs are allocated to machines using a threshold value and a lookahead parameter. The simulation results showed that the proposed heuristic performs better than the ATCS and rolling horizon procedure at a remarkably reduced total weighted tardiness. The work of Mendes et.al. [7] proposes two metaheuristics based on tabu search and memetic approach for solving the identical parallel machine scheduling problem with sequence dependent set up times and minimizing the makespan. An exact method for solving the identical parallel machine scheduling problem has been used by Yalaoui and Chu [8] in order to minimize total tardiness. A problem of considering the scheduling of a given set of independent jobs on unrelated parallel machines in order to minimize the total weighted tardiness has been presented by Liaw et al. [9], a branch-andbound algorithm incorporating various dominance rules has been proposed to find the optimal solution. The scheduling a set of independent jobs with sequence dependent setups on a set of uniform parallel machines is presented by Blidgue et al. [10], the tabu search (TS) approach is proposed to minimize the total tardiness. In this study, to obtain a robust search mechanism, several key components of TS such as candidate list strategies, tabu classifications, tabu tenure and intensification/diver-sification strategies are investigated. In the study of Chen and Wu [11], an effective heuristic based on threshold-accepting methods, tabu lists, and improvement procedures is proposed to minimize total tardiness. Computational results demonstrate that the proposed heuristic is capable of obtaining optimal solutions for small-sized problems, and significantly outperforms an ATCS procedure and a simulated annealing method for problems in larger sizes. Shim and Kim [12] solved the problem of scheduling n independent jobs on identical parallel machines for the objective of minimizing total tardiness of the jobs, branch and bound algorithm is used, the authors found that the proposed algorithm can obtain optimal solutions for problems with up to 30 jobs and 5 machines in a reasonable amount of CPU time. Biskup et al. [13] developed and evaluated efficient heuristic algorithms for finding optimal or near-optimal schedules for the identical parallel-machine problem to minimize total tardiness. The work in Tanakaa and Arakib [14] solves the problems of scheduling n independent jobs on identical parallel machines to minimize total tardiness using branch and bound algorithm, the work highlighted that the proposed algorithm can find an optimal solution efficiently. Indeed, it can handle instances with up to 25 jobs and any number of machines. Moghaddam et al. [15] presented two-level mixed-integer programming model of scheduling n jobs on m unrelated parallel machines that minimizes bi-objectives, namely the number of tardy jobs and the total completion time of all the jobs. The jobs have non-identical due dates and ready times. Furthermore, sequence-dependent setup times, which are included in the proposed model, may be different for each machine depending on their characteristics. Genetic algorithm (GA) is used to the scheduling problem. The related results show the effectiveness of the proposed model and GA for small and large-sized problems. Demirel et al. [16] used genetic algorithm to solve the problems of scheduling n jobs on identical parallel machines to minimize total tardiness. The study highlighted that the proposed genetic algorithm has positive effects on quality of the results when number of iterations increases. Sarıcicek and Celik [17] used two metaheuristics tabu search and simulated annealing to solve parallel machine scheduling with job splitting to minimize total tardiness. The authors found that the SA has a better performance and consumes less time than TS. Valdes et al.[18] considered the problem of scheduling a set of jobs on a set of identical parallel machines where the objective is to minimize the total weighted earliness and tardiness penalties with respect to a common due date. They designed two metaheuristics procedures, Path Relinking and Scatter Search, which combine solutions obtained through a constructive and local search process to obtain high quality solutions for the problem. The computational results show that the algorithms obtain good solutions in moderate computing times. From the previous studies, it can be concluded that parallel machine scheduling problems are NP hard in nature, meaning the problem cannot be solved in polynomial time as n gets large. Quite often the time required to solve such a problem increases exponentially with respect to n. This makes it impractical to apply exhaustive enumeration in almost all cases except where n is relatively small, generally less than 10 jobs. The exact approaches such as branch and bound algorithm are impractical real-life scheduling problems because of the time required. Although the meta-heuristics do not guarantee optimum, they take much less time and give very good solutions. Considering today s competitive environment, producing good solutions in a short time is requirement of today s industry. The appropriate way to solve these complex parallel machine scheduling problem is metaheuristics approaches. Therefore, in this paper the tabu search and simulated annealing approaches will be used for scheduling n jobs on identical parallel machines to minimize mean total tardiness. III. PROBLEM DESCRIPTION The problem addressed here consists of assigning n jobs to m identical parallel machines in which mean tardiness has to be minimized. The problem has following characteristics: There are n jobs each have a processing time and due date. Each job requires only a single operation on one of the m machines. Each machine is capable of processing one item at a time, and all processes are non-pre-emptive. At time zero, all jobs are available for processing and the machines are available all time. The problem could be described as follows: The objective function is to minimize mean tardiness; scheduling

3 environment is identical parallel machines. The following notations will be used to define the given problem: j: index of the job ( j = 1, 2,..., n) n: number of jobs m: Index of the machines ( i=1,2,3...,k) p ij : Processing time of job j on machine i d j : Due date of the j th job C j : Completion time of the j th job T j : Tardiness of the job j, T j =max{0,c j -d j } T : Mean tardiness The above problem can be represented using Graham notations as follows:. The objective function for the given scheduling problem can be given as follows: minimize n T = 1 n T j Assumptions: Each machine is capable of processing one item at a time; All processes are non-pre-emptive; At the starting of the scheduling period all jobs are available; Each machine is assumed to be available from time 0 onwards and can process one job at a time. IV. TABU SEARCH Glover [19] introduced the tabu search approach. Hansen [20] has also developed seminal ideas regarding the method. The philosophy behind TS search derives from and exploits a collection of principles of intelligent problem solving. One of the important elements of TS is the use of flexible memory. The use of flexible memory in TS consists of the dual processes of intensification and diversification for taking advantage of history. Assuming a minimization problem, the general TS algorithm can be outlined as follows: Select an initial solution s 0 ; Initialize memory structures; Repeat Generate a set "A" of non-tabu solutions N s 0 ; s=best solution of A; Update memory structures; if f s <f s 0 then s 0 =s; Until stopping criterion = true. s 0 is the approximation to the optimal solution. According to Laporte et al. [21], TS-based methods have been the most successful metaheuristics for a wide variety of problems, especially for solving the vehicle routing problem. In general, Reinelt [22] stated that the basic TS difficulties include the tabu list design, the mechanism of list management, and the non-prohibited move selection. V. SIMULATED ANNEALING Simulated annealing is an approach of solving combinatorial optimization problems, this approach point to a j=1 straight analogy in which liquids freeze and crystallize or that metals cool and anneal. Simulated annealing approach based on the work of Metropolis et al. [23] for solving combinatorial optimization problems. Basically, the change in energy of a system when it is converges to a steady frozen state by applying a cooling process on that system has been simulated by Metropolis s algorithm. Kirkpatrick et al. [24] and Cerny [25] used these ideas and proposed this approach (Simulated Annealing) to deal with highly nonlinear problems and after that, they proposed it for solving combinatorial optimization problems. Since then, SA has been extensively studied extended to deal with continuous optimization problems also applied to numerous other areas. SA explores the set of all possible solutions, minimizing the chance of being cohesive to local optima by accepting moves that may worsening the value of the objective function to escape from that local optima and move toward a new area in the solution space. A better move is always accepted. Assuming a minimization problem with solution space S, objective function f, and neighborhood structure N, the general SA algorithm can be stated as follows [26]: Select an initial solution s 0 ; Select an initial temperature t 0 >0; Set a temperature change function α; Repeat Repeat Randomly select s N s 0 ; δ=f s -f s 0 ; if δ<0 then s 0 =s else generate random x uniformly in the range (0,1); if x< exp - δ then s= s T 0 Until iteration count = nrep; Set t =α t ; Until stopping criterion = true. s 0 is the approximation to the optimal solution. Reinelt [22] stated that SA provides very good-quality solutions, but it will take a considerable amount of time, as the cooling process has to drop off very slowly and many replications at each cooling process step are necessary. In addition, an important issue should be highly considered is the proper choice of the cooling parameter, which it may need numerous experiments. VI. INSTANCES PREPARATION Tabu search and simulated annealing described above were implemented and tested on a set of 20 instances with number of jobs n = 20, 50, 100, 150, and number of machines m = 2, 3, 5, 7, 10. The proposed algorithms has been coded using Matlab software and the computational experiments for all instances have been conducted on the same computer with following specifications. Processor: Intel (R) Core i7-4702mq; CPU: 2.2 GHz; RAM: 16 GB. The parameters were generated in the following way, processing times are integer values generated from a uniform distribution as U(1, 100). Due date times are generated from a

4 uniform distribution between P(1-τ R and P(1-τ R, as suggested by Potts and Van Wassenhove [27]. Where, P is the sum of processing times of all jobs divided by the number of machines; τ is the tardiness factor and the higher its value the smaller is the distribution mean; R is the due date range factor which controls the distribution dispersion. One value was chosen for these two factors: τ =0.4 and R=0.4 therefore, due date times are from uniform distribution as U (0.4P, 0.8P). VII. PARAMETER TUNING One important issue in designing metaheuristics is the parameter tuning. The values of the parameters of each metaheuristics affects its performance. For this purpose, experiment runs were performed for a wide range of parameters values for both SA and TS algorithms, the best setting are selected and presented in table I. Table I: Parameters setting for TS and SA Metaheuristic Parameter Value Initial Temperature 200 SA Final Temperature No. of Iterations TS Tabu List Size 7 Candidate List Size 50 Number of Iterations 300 TS achieves better performance than SA in this type of scheduling problems. IX. CONCLUSIONS This study dealt with the scheduling problem of identical parallel machines. The problem is NP-hard, meaning the problem cannot be solved in polynomial time as n gets large. Quite often the time required to solve such a problem increases exponentially with respect to n. Two metaheuristics are proposed simulated annealing and tabu search. The proposed algorithms has shown reasonably good results in minimization of mean tardiness. A set of 20 instances have been computed to evaluate the proposed algorithms. The best setting of the parameters values for both SA and TS algorithms are computed using simulation runs for a wide range of the parameters. Computational results indicate that the proposed algorithms has effectively improved the initial solution that obtained using EDD dispatching rule. Moreover, the results show that the TS achieves better mean tardiness than SA in the large size instances (100 and 150 jobs) and performs equally in the instances of 20 and 50 jobs. The SA gives better CPU time in all jobs instances. A very interesting extension of this work would consist of establishing a strategy to evaluate a reduced list of candidates in the neighborhood so as to obtain high quality solutions in a faster computational time. VIII. RESULTS AND ANALYSIS A total of 20 instances have been solved by the proposed metaheuristics to evaluate their performance. All instances are solved initially by EDD rule and the proposed metaheuristics have been used to improve the obtained results of EDD. The percent of improvement (POI) from the initial solution will be used as a measure for finding the best solution method. EDD-Proposed Algorithm SA or TS POI= 100% EDD Results of the TS and SA approaches for all instances are shown in Table II. The results include mean tardiness, POI, and the CPU time in seconds. Fig.1 shows the performance of the proposed approaches according to the value of the objective function mean tardiness. The results indicate that the TS achieves better performance than SA in the large size instances (100 and 150 jobs) and performs equally in the instances of 20 and 50 jobs. The changes of the average CPU times of the two proposed approaches depending on the number of jobs are given in Table II for comparison purpose. Fig. 2 shows the relationship between the instances and average CPU times of the proposed approaches. The figure highlighted that the SA consumes less time than TS and both of them become larger for instances with larger jobs numbers. SA has a better performance and consumes less time than TS. Finally, in Fig. 3 according to the POI, it can be seen that the Fig.1. Mean tardiness for all jobs instances. Fig.2. CPU time for all jobs instances.

5 36 POI from the EDD for SA and TS 95% CI for the Mean POI SA TS Fig.3. POI from the EDD for SA and TS of all jobs instances. Instance No. 1 Job No. Machine No. Table II: Results of TS and SA Initial solution EDD Mean Tardiness POI % CPU Time SA TS SA TS SA TS

6 REFERENCES [1] J. C. Ho and Y.-L. Chang, "Minimizing the number of tardy jobs for< i> m</i> parallel machines," European Journal of Operational Research, vol. 84, pp , [2] C. Koulamas, "Decomposition and hybrid simulated annealing heuristics for the parallel machine total tardiness problem," Naval Research Logistics (NRL), vol. 44, pp , [3] M. Azizoglu and O. Kirca, "Tardiness minimization on parallel machines," International Journal of Production Economics, vol. 55, pp , [4] V. Suresh and D. Chaudhuri, "Bicriteria scheduling problem for unrelated parallel machines," Computers & industrial engineering, vol. 30, pp , [5] V. A. A. a. D. S. Yamashita, "Tabu search for scheduling on identical parallel machines to minimize mean tardiness," Journal of intelligent manufacturing, vol. 11, pp , [6] D.-H. Eom, H.-J. Shin, I.-H. Kwun, J.-K. Shim, and S.-S. Kim, "Scheduling jobs on parallel machines with sequence-dependent family set-up times," The International Journal of Advanced Manufacturing Technology, vol. 19, pp , [7] A. S. Mendes, F. M. Müller, P. M. França, and P. Moscato, "Comparing meta-heuristic approaches for parallel machine scheduling problems," Production Planning & Control, vol. 13, pp , [8] F. Yalaoui and C. Chu, "Parallel machine scheduling to minimize total tardiness," International Journal of Production Economics, vol. 76, pp , [9] C.-F. Liaw, Y.-K. Lin, C.-Y. Cheng, and M. Chen, "Scheduling unrelated parallel machines to minimize total weighted tardiness," Computers & Operations Research, vol. 30, pp , [10] Ü. Bilge, F. Kıraç, M. Kurtulan, and P. Pekgün, "A tabu search algorithm for parallel machine total tardiness problem," Computers & Operations Research, vol. 31, pp , [11] J.-F. Chen and T.-H. Wu, "Total tardiness minimization on unrelated parallel machine scheduling with auxiliary equipment constraints," Omega, vol. 34, pp , [12] S.-O. Shim and Y.-D. Kim, "Scheduling on parallel identical machines to minimize total tardiness," European Journal of Operational Research, vol. 177, pp , [13] D. Biskup, J. Herrmann, and J. N. Gupta, "Scheduling identical parallel machines to minimize total tardiness," International Journal of Production Economics, vol. 115, pp , [14] S. Tanaka and M. Araki, "A branch-and-bound algorithm with Lagrangian relaxation to minimize total tardiness on identical parallel machines," International Journal of Production Economics, vol. 113, pp , [15] R. Tavakkoli-Moghaddam, F. Taheri, M. Bazzazi, M. Izadi, and F. Sassani, "Design of a genetic algorithm for bi-objective unrelated parallel machines scheduling with sequence-dependent setup times and precedence constraints," Computers & Operations Research, vol. 36, pp , [16] T. Demirel, V. Ozkir, N. Demirel, and B. Taşdelen, "A genetic algorithm approach for minimizing total tardiness in parallel machine scheduling problems," in Proceedings of the World Congress on Engineering, [17] İ. Sarıçiçek and C. Çelik, "Two meta-heuristics for parallel machine scheduling with job splitting to minimize total tardiness," Applied Mathematical Modelling, vol. 35, pp , [18] R. Alvarez-Valdes, J. Tamarit, and F. Villa, "Minimizing weighted earliness-tardiness on parallel machines using hybrid metaheuristics," Computers & Operations Research, [19] F. Glover and G. A. Kochenberger, "Critical event tabu search for multidimensional knapsack problems," in Meta-Heuristics, ed: Springer, 1996, pp [20] P. Hansen, "The steepest ascent mildest descent heuristic for combinatorial programming," in Congress on numerical methods in combinatorial optimization, Capri, Italy, 1986, pp [21] G. Laporte, M. Gendreau, J. Y. Potvin, and F. Semet, "Classical and modern heuristics for the vehicle routing problem," International transactions in operational research, vol. 7, pp , [22] G. Reinelt, The traveling salesman: computational solutions for TSP applications: Springer-Verlag, [23] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, "Equation of state calculations by fast computing machines," The journal of chemical physics, vol. 21, pp , [24] S. Kirkpatrick, "Optimization by simulated annealing: Quantitative studies," Journal of statistical physics, vol. 34, pp , [25] V. Černý, "Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm," Journal of optimization theory and applications, vol. 45, pp , [26] C. R. Reeves, Modern heuristic techniques for combinatorial problems: John Wiley & Sons, Inc., [27] C. N. Potts and L. Van Wassenhove, "A decomposition algorithm for the single machine total tardiness problem," Operations Research Letters, vol. 1, pp , Husam Kaid is a Researcher and Master Student in Industrial Engineering Department, College of Engineering, King Saud University, Saudi Arabia. His area of expertise is manufacturing systems. He received the BSc in Industrial Engineering from University of Taiz, Taiz, Yemen. Ibrahim Alharkan is an Associate Professor and Chairman of deanship of admission at King Saud University, Research interests are in the areas of production planning and control, modeling and simulation of industrial and service systems, and applied operations research. These areas of interest include production planning and control, inventory control, production sequencing, scheduling and lot sizing; expert system; simulated annealing algorithms; genetic algorithms; Tabu search; scatter search algorithms, and total quality management, quality control, maintenance planning and scheduling, project scheduling Atef Ghaleb is a Researcher and Master Student in Industrial Engineering Department, College of Engineering, King Saud University, Saudi Arabia. His area of expertise is manufacturing systems. He received the BSc in Industrial Engineering from University of Taiz, Taiz, Yemen. Mageed A. Ghaleb is a Researcher and Master Student in Industrial Engineering Department, College of Engineering, King Saud University, Saudi Arabia. His area of expertise is Industrial systems. He received the BSc in Industrial Engineering from University of Taiz, Taiz, Yemen.

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