A Tabu-Search Based Constructive Hyper- heuristics for Scheduling ...

Journal of Industrial and Intelligent Information Vol. 5, No. 2, December 2017

A Tabu-Search Based Constructive Hyperheuristics for Scheduling Problems in Textile Industry Cecilia E. Nugraheni and Luciana Abednego Dept. of Informatics, Parahyangan Catholic University, Bandung, Indonesia Email: [email protected]

Abstract—One type of scheduling problems found in the area of textile industry is the Flow Shop Scheduling Problem (FSSP). The solution of this problem is the sequence of jobs that meet certain requirements. In practice, dispatching rules are often used to determine the order of jobs. This paper proposes a new approach in finding solutions for FSSP. The proposed method applies a tabusearch based hyper-heuristic approach. Hyper-heuristic can be used to combine several basic heuristics to obtain a solution. By using four dispatching rules as the basic heuristics, i.e. FIFO, LIFO, SPT, and LPT, the proposed method, TSHH, has been implemented in a computer program. In order to measure the performance of TSHH, several experiments have been conducted by using Taillard's benchmark datasets. The conclusion obtained is that with a proper heuristics combination and value tenure, TSHH is able to produce better solutions than the solutions of four dispatching rules.

machines and n jobs that will be processed on each machine, an FSSP is a problem to find a sequence of jobs that satisfies some particular criteria. One of the important objectives is minimum makespan, which is the time between the beginning of the execution of the first job on the first machine and the completion of the execution of the last job of the sequence on the last machine. There are many work dedicated to FSSP. The proposed methods, in general, are classified into two techniques: constructive or improvement [1], [2]. The two techniques differ in the way they construct the job sequence. A constructive technique iteratively builds the sequence by adding jobs one by one into the sequence, whereas an improvement technique starts with an initial solution and then by using particular mechanism it iteratively changes the sequence in order to find the best solution [3]. In practice, frequently people apply some heuristics called dispatching rules for constructing the job ordering. A dispatching rule is used to select one of the waiting jobs with the highest priority to be processed. There are many dispatching rules. Four basic dispatching rules considered in this work are FIFO (First in First Out), LIFO (Last In First Out), SPT (Shortest Processing Time), and LPT (Longest Processing Time).

Index Terms—flow shop scheduling problem, dispatching rules, tabu-search, hyper-heuristics

I.

INTRODUCTION

One of the important industries in Indonesia is the textile industry. The textile subsector plays as an important source of foreign exchange. Besides, it is also a labor-intensive industry that absorbs a lot of labor. Nowadays, this industry suffers from some problems. One of them is the condition of production machines. Textile production machines owned by textile factories can be categorized as old machines (around 20 years old). The use of old production machines may cause a large amount of energy consumption and may affect the working speed optimality and the product quality as well. One of the possible solutions of this problem is a good production scheduling. As a manufacturing industry, the optimal production scheduling becomes a very important issue. Scheduling is understood as assigning jobs to machines or operators for specified time period satisfying some constraints. In the context of textile industry, in general, there are two types of scheduling problems: Flow Shop Scheduling Problem (FSSP) and Job Shop Scheduling Problem (JSSP). This work is focused on FSSP. Given m

Figure 1. The General framework of the Hyper-heuristics approach.

The weaknesses of each approach are the lack of generality. It performs well only for a specific problem instance. Hyper-heuristics are proposed to overcome this problem. Hyper-heuristics are search methodologies for choosing or generating (combining, adapting) heuristics (or components of heuristics), in order to solve a range of

Manuscript received July 15, 2017; revised December 2, 2017. © 2017 Journal of Industrial and Intelligent Information doi: 10.18178/jiii.5.2.23-27

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As an illustration, let's consider a small FSSP. There are five jobs that will be processed on three machines. The processing times required for every job by every machine are given in Table I. Schedule resulted by applying dispatching rule FIFO, LIFO, SPT, and LPT is j1, j2, j3, j4,,j5, j5, j4, j3, j2,,j1, j5, j2, j1, j3,,j4, and j4, j3, j1, j2,,j5, respectively.

optimization problems [4]. Rather than search a space of solution directly, hyper-heuristics search a space of heuristics. Fig. 1 shows the general framework for the hyper-heuristics approach. A survey on heuristics for scheduling problem in textile industry is given in [5]. There are nine heuristics that can be used for solving FSSP. In [6] we have proposed a genetic programming based hyper-heuristics approach for combining two of nine heuristics reported in [5]. From AI point of view, FSSP can be regarded as a searching problem, i.e. finding a solution in a solution space. There are many popular searching algorithms. One of them is tabu search. The advantage of tabu search compared to other searching algorithms is its ability to avoid the local optima. This is done by using a tabu list for memorizing which solutions or candidate solutions has already visited. In this work, we investigate the combination application of hyper-heuristics and tabu search for solving FSSP. Many works similar to our work can be found in the literature. Among them, the work that is most closed to our work is the work from Graham and Hussin [7]. In [7], they presented a tabu search hyperheuristics for solving examination timetabling problems. The contribution of our work is the combining four dispatching rules, which are FIFO, LIFO, SPT, and LPT by using tabu search based hyper-heuristics. As far as our knowledge, this topic is not yet reported in the literature. The rest of this paper is structured as follows. Section 2 gives a brief explanation of FSSP. Section 3 explains the proposed algorithm. Section 4 presents the experimental results. Last, conclusion and future work are given in Section 5. II.

TABLE I. PROCESSING TIME m2 1 3 4 5 2

m3 4 2 3 4 3

Total 10 8 13 11 6

Assume we apply LIFO for constructing the schedule, the resulted makespan is 22. III.

PROPOSED ALGORITHM

Given a set of n jobs, a set of k heuristics, and a set of k job ordering associated to each heuristic, the algorithm of the proposed method, TSHH, is as follows. Initially all heuristics are non-tabu heuristics (ti = 0). Starts with initial solution candidate, which is an empty job ordering, TSHH constructs an n-length job ordering by selecting a non-tabu heuristic iteratively. Whenever a non-tabu heuristic h is found, the job with the highest priority according to heuristic h is inserted to the solution candidate. Every time a non-tabu heuristic h is selected, it becomes tabu heuristic and remains tabu until it's tabu tenure value ti equals to 0. In algorithmic notation, TSHH algorithm is given by Fig. 2. let J = {j1, …, jn} be a set of n jobs, H = {h1, …, hk} be a set of k heuristics, and O = {o1, …, ok} be a set of job ordering oi associated to hi on J. for all i  {1, .., n}, ti  0 i1 while i  n do let hm  J be some heuristic in H if tm = 0 then tm  t si  om(1) /* the first job of ordering om */ J  J \ {jm} for all r  {1, .., k}: delete jm from or for all r  {1, .., k} s.t. r  m and tr  0: tr  tr – 1 ii+1 else for all r  {1, .., k}: tr  tr – 1 endif endwhile  s /* the resulted job ordering */

FLOW SHOP SCHEDULING PROBLEMS

Following [3], flow shop scheduling problem can be defined as follows: Given n jobs (items, tasks, …) to be processed in the same sequence on m machines; the processing time of job i on machine j is fixed and given by tij (tij > 0). The flow shop scheduling problem consists of minimizing the time between the beginning of the execution of the first job on the first machine and the completion of the execution of the last job on the last machine; this time is called makespan. There are some assumptions made for this problem:  Every job has to be processed at most once on machine 1, 2, …, m.  Every machine processes only one job at a time  Every job is processed at most on one machine at a time.  The operations are not preemptable.  The set-up times of the operations are included in the processing time and do not depend on the sequence.  The operating sequences of the jobs are the same on every machine and the common sequence has to be determined.

© 2017 Journal of Industrial and Intelligent Information

m1 5 3 6 2 1

j1 j2 j3 j4 j5

Figure 2. TSHH algorithm

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IV.

COMPUTATIONAL EXPERIMENTS

and for all datasets resulted by each dispatching rule and the median, the best, and the worst makespans produced by TSHH. The median, the best, and the worst makespan resulted by TSHH is denoted by M-TH, B-TH, and WTH, respectively. PI stands for Problem Instance and RBM stands for Referenced Benchmark.

A. Experiment Setup For measuring the performance of the proposed method, some experiments have been conducted. The objective of the experiments is to answer the following questions: 1. How is the performance of the proposed method, TSHH, compared to the original dispatching rules? 2. Which combination of dispatching rules performs the best? 3. Does the tabu tenure have an impact on the performance of TSHH? In this work, we ran our experiments with the objective of minimizing the makespan of Taillard's benchmark problem datasets [9]. Taillard's benchmark consists of 8120 instances, 10 each of one particular size. Taillard's datasets range from 20 to 500 jobs and 5 to 20 machines. For comparison purposes, following [9], we use the outputs of NEH algorithm as reference solutions. For every problem instance, we ran each dispatching rule (FIFO, LIFO, SPT, and LPT) and the proposed method, TSHH, with eleven combination of dispatching rules (FIFO-LIFO-SPT-LPT, FIFO-LIFO-SPT, FIFOLIFO-LPT, FIFO-SPT-LPT, LIFO-SPT-LPT, FIFOLIFO, FIFO-SPT, FIFO-LPT, LIFO-SPT, LIFO-LPT, and SPT-LPT) and seven different tabu tenure values (0,1,2,3,4,5,and 10). Thus, for each problem instance, we have produced 81 makespans. For each problem instance, we calculate the minimum, maximum, and median values based on 81 resulted makespans. The minimum and maximum values will be used to know the best and the worst performance of TSHH, whereas the median value will be used as indicator of TSHH performance in general. The used of median value instead of mean (average) in stochastic algorithmic performance is inspired by Ivkovic et.al. in [10]. For comparing the solutions of each dispatching rule and the proposed algorithm, following [8], we used Relative Percent Deviation (RPD) and Average Relative Percent Deviation (ARPD). The definition of Relative Percent Deviation and Average Percentage Relative Deviation is respectively given by: 𝑅𝑃𝐷𝑖 =

𝐻𝑆𝑖 − 𝑅𝑆𝑖 100% 𝑅𝑆𝑖

TABLE II. RESULTED MAKESPANS Dataset 20 jobs 5 machines PI RBM FIFO LIFO 1 1299 1448 1473 2 1365 1545 1533 3 1132 1597 1354 4 1329 1754 1556 5 1305 1431 1434 6 1251 1616 1583 7 1251 1528 1502 8 1215 1428 1504 9 1284 1468 1609 10 1127 1404 1513 Dataset 20 jobs 10 machines 1 1681 2004 2026 2 1766 2104 2068 3 1562 1812 1947 4 1416 1726 1866 5 1502 1944 1828 6 1456 1877 1643 7 1531 1935 1984 8 1626 2044 2048 9 1639 1978 1913 10 1656 2051 2103 Dataset 20 jobs 20 machines 1 2443 2770 2788 2 2134 2543 2648 3 2414 2625 2866 4 2257 2800 2793 5 2370 2829 2804 6 2349 2597 2715 7 2383 2723 2737 8 2249 2697 2749 9 2306 2713 2771 10 2257 2830 2696 Dataset 50 jobs 5 machines 1 2979 3095 3196 2 2882 3515 3052 3 2650 2900 3159 4 2782 3073 3309 5 2868 3071 3413 6 2835 3195 3312 7 2806 3450 3108 8 2700 3140 3426 9 2606 2930 2912 10 2801 3188 3397 Dataset 50 jobs 10 machines 1 3175 3754 3742 2 3073 3685 3505 3 2994 3612 3835 4 3218 3669 3989 5 3186 3741 3980 6 3148 3738 3746 7 3277 3678 4040 8 3170 3773 3887 9 3025 3792 3816 10 3267 3845 3945 Dataset 50 jobs 20 machines 1 4006 5094 4877 2 3958 4730 4635 3 3866 4592 4639 4 3953 4797 4896 5 3872 4748 4459

(1)

𝐼

1 𝐴𝑅𝑃𝐷 = ∑ 𝑅𝑃𝐷𝑖 𝐼

(2)

𝑖=1

where:    

I number of problem instances, HSi heuristic solution of problem instance i, RSi reference solution of problem instance i, RPDi percentage relative deviation of problem instance i.

B. Experimental Results Table II shows the makespans for all problem instances © 2017 Journal of Industrial and Intelligent Information

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SPT 1472 1526 1499 1640 1437 1457 1504 1422 1585 1365

LPT 1556 1594 1408 1723 1592 1604 1479 1531 1538 1387

M-TH B-TH 1538 1394 1516 1467 1423 1341 1649 1568 1501 1387 1639 1527 1499 1403 1505 1378 1513 1374 1405 1322

W-TH 1596 1611 1624 1807 1621 1762 1593 1677 1585 1608

2026 2190 1830 1944 1805 1867 1665 1974 1874 1929

1943 2046 1902 1746 1944 1885 1910 1999 2010 2034

2028 2067 1875 1861 1927 1825 1881 1954 1970 2010

1877 1936 1790 1741 1823 1681 1719 1804 1860 1925

2134 2200 1984 2031 2088 1920 2055 2062 2143 2191

2937 2785 2829 2712 2827 2615 2693 2658 2871 2618

2727 2577 2668 2784 2755 2740 2822 2646 2800 2592

2828 2632 2743 2758 2746 2749 2782 2722 2736 2710

2668 2518 2634 2639 2601 2598 2693 2539 2621 2529

2966 2785 2866 2878 2905 2888 2926 2834 2975 2954

3240 3329 3116 3400 3281 3197 3151 3160 3012 3468

3119 3224 3086 3134 3222 3489 3300 3364 2830 3125

3245 3202 3050 3159 3204 3263 3352 3240 2927 3254

3024 3069 2881 3037 3129 3073 2998 3040 2788 3116

3415 3504 3283 3405 3375 3564 3519 3506 3147 3467

3781 3688 3635 3995 3769 3803 3946 3946 3910 3768

3969 3748 3783 4018 4021 3699 3849 3769 3632 3927

3812 3693 3700 3843 3649 3780 3822 3865 3760 3836

3717 3586 3456 3653 3496 3593 3694 3636 3563 3708

4123 3808 3885 4115 3980 4079 4088 4080 3976 4138

4906 4819 4411 4710 4824

5044 4692 4957 4883 4739

4929 4608 4752 4802 4642

4662 4447 4438 4648 4432

5122 4767 5048 5094 4983

Journal of Industrial and Intelligent Information Vol. 5, No. 2, December 2017

6 3861 4946 4593 4752 7 3927 4742 4664 4926 8 3914 4763 5028 4831 9 3970 4823 4814 4608 10 4036 4901 4851 4857 Dataset 100 jobs 5 machines 1 5514 5943 6209 5232 2 5284 5878 5998 5834 3 5222 5880 5819 6068 4 5023 5675 5656 5731 5 5261 6095 6067 5868 6 5154 5753 5789 5832 7 5282 5935 6029 6082 8 5140 6068 5738 6032 9 5489 6193 6082 6211 10 5336 6157 5897 6140 Dataset 100 jobs 10 machines 1 5897 6982 6742 6805 2 5466 6558 6504 6214 3 5747 6667 6635 6886 4 5924 7300 7052 6774 5 5672 6844 6417 6788 6 5395 6591 6319 6394 7 5717 6765 6545 6651 8 5752 6517 6761 6735 9 6016 6859 6941 6962 10 5937 6930 6915 6462 Dataset 100 jobs 20 machines 1 6520 7840 7846 7700 2 6550 7591 7690 7783 3 6621 7755 7915 7719 4 6589 7885 7777 7698 5 6697 7729 7808 8021 6 6813 8072 7849 7770 7 6578 8033 7992 7851 8 6791 8138 8204 8039 9 6679 7907 7880 7711 10 6680 8099 7936 8206 Dataset 200 jobs 10 machines 1 10949 12193 11960 12012 2 10677 12796 12356 12373 3 11080 12556 12371 12357 4 11057 12198 12325 12066 5 10615 12110 12523 12574 6 10495 12113 12197 12013 7 10950 12848 12544 12720 8 10834 12294 12518 12098 9 10565 12010 12297 12085 10 10808 12274 12379 12177 Dataset 200 jobs 20 machines 1 11683 13576 13872 13736 2 11678 13628 13615 13571 3 11724 14152 13310 13817 4 11796 13479 13699 13600 5 11570 13686 13837 13602 6 11805 13917 16411 13759 7 11876 13836 13935 13738 8 11824 13855 13774 13796 9 11801 13409 13728 13775 10 11890 14101 13461 13714 Dataset 500 jobs 20 machines 1 26744 30121 29956 30756 2 27215 31202 30301 31409 3 26941 30447 31167 30770 4 26928 30355 31122 30597 5 26928 30099 30563 30578 6 27047 30946 31058 30103 7 26820 30792 30496 30728 8 27230 31034 30710 30873 9 26541 30634 29773 30270 10 27103 30148 30664 30109

4617 4749 4743 4646 4805

4772 4736 4832 4806 4823

4550 4557 4645 4560 4598

5011 4958 5045 5028 5078

5980 5759 5968 5506 5934 5743 6059 5748 6152 5886

6182 5871 5939 5667 6020 5717 6037 5931 6152 6115

5976 5712 5647 5412 5867 5539 5779 5612 5924 5838

6419 6044 6082 5863 6313 5923 6351 6148 6363 6556

6823 6677 6831 6919 6505 6470 6730 6611 6712 6697

6848 6523 6828 6968 6699 6495 6639 6698 6930 6752

6608 6182 6596 6692 6428 6282 6407 6407 6767 6491

7767 7727 8072 7981 7766 7848 7905 8223 7817 8282

7586 7646 7667 7467 7828 8085 7598 8113 7938 7866

7766 7744 7958 7855 7765 7947 7906 8220 7830 8073

7562 7514 7633 7537 7562 7734 7715 7894 7701 7661

8016 7925 8370 8172 8036 8184 8129 8368 8155 8512

12347 12604 12582 12053 12257 12249 12307 12360 12448 12147

12183 12342 12376 12250 12401 12033 12432 12273 12072 12279

11945 11973 12143 11957 12020 11626 12217 11857 11852 12052

12571 12711 12678 12543 12828 12666 13001 12565 12407 12520

13473 13537 13717 13915 13358 13866 13837 13928 13297 13731

13775 13817 13757 13774 13540 13771 13957 13807 13553 13753

13479 16504 13210 13401 13303 13296 13715 13480 13304 13526

14103 14238 14148 14267 13986 14137 14240 14086 13923 14163

30345 31174 30247 31121 30374 30806 30769 30611 30384 30956

30379 30940 30434 30559 30612 30499 30651 30751 30361 30215

29718 30501 30013 30156 30082 29700 30094 30225 29688 29824

31504 31890 31227 31452 31133 31262 31550 31666 31004 30800

From the resulted makespans in Table II, we have calculated the RPD and the ARPD of each heuristic. The ARPD of FIFO, LIFO, SPT, LPT, M-TH, B-TH, and WTH, are shown in Fig. 3. From Fig. 3, it can be concluded that TSHH tends to produce results that are not much different from the results from original dispatching rules. This is based on the value of M-TH. However, for some cases, TSHH can yield a much better, and even worse, solution. To find out which combination of dispatching rules is the best, we have calculated the ARPD relatively to each possible combination. Since there are four dispatching rules, there are eleven possible combinations to consider. The ARPDs of all dispatching rule combinations are given by Fig. 4. We refer FIFO, LIFO, SPT, and LPT by number 1, 2, 3, and 4, respectively. For examples, H-134 represents the combinations of FIFO, SPT, and LPT, and H-23 represents the combinations of LIFO and SPT.

Figure 3. Resulted ARPD of each heuristic.

Figure 4. Resulted ARPD of each dispatching rules combination.

Figure 5. Resulted ARPD of each tabu tenure value.

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C. E. Nugraheni and L. Abednego, “On the development of hyper heuristics based framework for scheduling problems in textile industry,” IJMO, vol. 6, no. 5. 2016. [4] C. E. Nugraheni and L. Abednego, “Collaboration of multi-agent and hyper-heuristics systems for production scheduling problem,” International Journal of Computer, Electrical, Automation, Control and Information Engineering, vol. 7, no. 8, pp. 1136-1141, 2013. [5] C. E. Nugraheni and L. Abednego, “A survey on heuristics for scheduling problem in textile industry,” in Proc. ICEAI, 2015. [6] C. E. Nugraheni and L. Abednego, “A comparison of heuristics for scheduling problems in textile industry,” Journal Teknologi, vol. 78, no. 6-6. 2016. [7] G. Kendall and N. M. Hussin, “A tabu search hyper-heuristic approach to the examination timetabling problem at the MARA University of Technology,” PATAT 2004, pp. 270-293, 2004. [8] E. Taillard, “Some efficient heuristic methods for the flow shop sequencing problem,” European Journal of Operational Research, vol. 47, pp. 65-74, 1990. [9] P. Semanco and V. Modrak, “A comparison of constructive heuristics with the objective of minimizing makespan in the flow shop scheduling problem,” Acta Polytechnica Hungarica, vol. 9, no. 5, 2012. [10] N. Ivkovic, D. Jakobovic, and M. Golub, “Measuring performance of optimization algorithms in evolutionary computation,” Int. Journal of Machine Learning and Computing, vol. 6, no. 3, pp. 167-171, June 2016.

[3]

Fig. 4 shows the resulted ARPD of each dispatching rules combination. It is clear that FIFO-LPT produced the best result. Last, we have calculated the ARPD of each tabu tenure value. The resulted computation is given by Fig. 5. From Fig. 5, in average, we can see that tabu tenure 5 yields the best result. V.

CONCLUSIONS AND FUTURE WORK

We have presented an algorithm for finding the solution of Flow Shop Scheduling Problem using tabu search based hyper-heuristics approach. From the experimental results, we have shown that this approach is capable of producing a better solution than the solutions resulted in the original dispatching rules. It is also shown that there is no guarantee that this approach will always give the best solution. However, with a proper selection of dispatching rules combination and tabu tenure settings, the proposed method will perform well. The best dispatching rules combination is FIFO-LPT, whereas the best tabu tenure value is 5. As future work, we will continue our work on the other type of scheduling problem found in textile industry which is Job Shop Scheduling Problem. We will investigate whether this approach can be applied to this scheduling problem and measure the performance. Besides makespan, some other optimization criteria will also be considered.

Cecilia E. Nugraheni received the bachelor and master degrees in Informatics Engineering from Institut Teknologi Bandung, Indonesia. She received the Ph.D. degree in Informatics from Institut für Informatik, Ludwig-Maximilians Universität, Munich, Germany. Currently, she is a lecturer at Informatics Dept. Parahyangan Catholic University, Bandung, Indonesia. Her research interests are formal specification and verification of reactive systems, multi agent systems, soft computing, production scheduling optimization, meta- and hyper-heuristics.

ACKNOWLEDGMENT This work was supported in by the Ministries of research, technology, and higher education of Republic Indonesia under Grant DIPA-042.06.1.401516/2017.

Luciana Abednego received the bachelor degree in Computer Science from Parahyangan Catholic University. She received the master degree in Informatics Engineering from Institut Teknologi Bandung. Currently, she is a lecturer at Informatics Dept. Parahyangan Catholic University, Bandung, Indonesia. Her research interests include programming and algorithms, computer graphics, machine learning, production scheduling optimization, meta- and hyper-heuristics.

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[2]

V. Modrák and R. S. Pandian, “Flow shop scheduling algorithm to minimize completion time for n-Jobs m-Machines problem,” Technical Gazette, vol. 17, no. 3, pp. 273-278. R. Ruiz and C. Maroto, “A comprehensive review and evaluation of permutation flowshops heuristics,” European Journal of Operational Research, vol. 165, pp. 479-494, 2005.

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