A tabu search algorithm for parallel machine total tardiness problem

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<ul><li><p>Available online at www.sciencedirect.com</p><p>Computers &amp; Operations Research 31 (2004) 397414www.elsevier.com/locate/dsw</p><p>A tabu search algorithm for parallel machine total tardinessproblem</p><p>&amp;Umit Bilge, Furkan K-ra.c, M&amp;ujde Kurtulan, Pelin Pekg&amp;unDepartment of Industrial Engineering, Bogazici University, Bebek, 80815 Istanbul, Turkey</p><p>Received 1 April 2001; received in revised form 1 July 2002</p><p>Abstract</p><p>In this study, we consider the problem of scheduling a set of independent jobs with sequence dependentsetups on a set of uniform parallel machines such that total tardiness is minimized. Jobs have non-identicaldue dates and arrival times. A tabu search (TS) approach is employed to attack this complex problem. In orderto obtain a robust search mechanism, several key components of TS such as candidate list strategies, tabuclassi</p></li><li><p>398 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>1. Introduction</p><p>The classical parallel machine total tardiness problem (PMTP) can be stated as follows [13]: Aset of independent jobs is to be processed on a number of continuously available identical parallelmachines. Each machine can process only one job at a time, and each job can be processed on onemachine. Each job is ready at the beginning of the scheduling horizon and has a distinct processingtime and a distinct due date. The objective is to determine a schedule such that total tardiness isminimized, where tardiness of a job is the amount of time its completion time exceeds its duedate. The problem is NP-hard even for a single machine (Du and Leung [4]) and exact methodsin which the dimensionality problem is acute are mostly limited to special cases like common duedates and equal processing times (i.e. Root [5], Lawler [6], Elmaghraby and Park [7], Dessouky [8]).A large class of heuristics is based on list scheduling where the jobs are </p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 399</p><p>[19] present a TS to minimize makespan in a Oow shop with parallel machines, and employ aneighbourhood based on blocks of operations on a critical path. A similar block approach is usedby Liaw [20] for makespan minimization in an open shop. Park and Kim [21] compare simulatedannealing and TS for a parallel machine scheduling problem where jobs have equal due dates andequal ready times for minimizing holding costs.When jobs are allowed to have distinct arrival times as well as due dates, diCerent processing</p><p>rates on machines and sequence dependent setup times, the literature becomes really sparse. Thereare only two studies reported on this more general problem to our knowledge and both of them dealwith minimizing the total earliness-tardiness costs: S.erifoPglu and Ulusoy [22] present a GA witha new crossover operator, while Balakrishnan et al. [23] report a compact mathematical model tosolve small sized (up to 10 jobs) problems.The TS algorithm proposed here is tested using the problem set given in Serifoglu and Ulusoy</p><p>[22] and the results are compared to their results for the case where the weight of the earlinesspenalty is zero (In this case their problem also reduces to total tardiness problem).The next section describes the key aspects of the TS approach used. Section 3 compares sev-</p><p>eral alternative approaches leading towards a robust TS algorithm tailored to solve the problem athand, and evaluates the performance of this algorithm through numerical experimentation. The paperconcludes with discussion of results and further studies in Section 4.</p><p>2. Description of the tabu search approach</p><p>This section outlines the totally deterministic TS algorithm tailored to the GPMTP by discussingseveral of the key concepts such as solution encoding, initial solutions, tabu classi</p></li><li><p>400 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>Machine 1 1 2 3 .. .. .. .. .. .. n1</p><p>Machine 2 1 2 3 .. .. .. .. .. .. .. n2</p><p>:</p><p>:</p><p>:</p><p>Machine j 1 2 3 .. .. .. .. nj:</p><p>:</p><p>:</p><p>Machine m 1 2 3 .. .. .. nm</p><p>Fig. 1. Solution representation.</p><p>2.3. Neighbourhood generation</p><p>Insert moves and pairwise exchanges (swaps) are two of the frequently used move types inpermutation problems. An insert move identi</p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 401</p><p>the search process. Since jobs have distinct ready times, diCerent processing times on diCerent typesof machines and sequence dependent setup times, calculation of total tardiness for a given move is atedious task. Although this is implemented in an eScient way by </p></li><li><p>402 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>2.5. Tabu classi</p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 403</p><p>Fig. 2. Sample screen for WinMeta.</p><p>out a search of a given length from each of these solutions. An elite solution is de</p></li><li><p>404 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>Table 1Problem design parameters</p><p>Number of jobs: n 20, 40, 60Number of machines: m 2, 4Maximum setup duration: Amax 4,8</p><p>Pentium 4- 1:6 GHz CPU, Host Bus 133 MHz with 512 MB RAM. The problem set and the resultsobtained are presented in the next sections.</p><p>3.1. Example problems</p><p>Although WinMeta can be used to generate a new set of problems, in this study the benchmarkproblem set due to S.erifoPglu and Ulusoy [22] is used. Their test problems were generated using thedesign in Table 1 with 20 instances for each combination. The details regarding the generation ofproblems are as follows: The machines considered belong to one of two diCerent types, Types Iand II, which have the same characteristics except that they have diCerent processing rates. Type IImachines are older technology machines and the processing time of a job on Type II machine is 1020% larger than on a Type I machine. Likewise, setup times on a Type II machine are 2040% largerthan the corresponding setup times on a Type I machine. The processing times on a Type I machine,pIi , are obtained from the uniform distribution U [4,20], and to generate the processing times for theType II machine, pIIi , multipliers are chosen from the interval [1:10; 1:20] randomly and applied tothe corresponding processing time on machine Type I. Setup times on a Type I machine, sIji, are ob-tained from the uniform distribution U [1; Amax], where two levels of maximum setup time, Amax, areutilized. Multipliers are randomly chosen from the interval [1:20; 1:40] and used to obtain the corre-sponding setup time on machine Type II. Ready times, ri, have the uniform distribution U [1; Rmax],where Rmax is the maximum ready time. Rmax is computed as ( XpII + XsII)(n=m 1), where the </p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 405</p><p>Table 2Comparison of the candidate list strategies</p><p>Problem type Average % improvement over EDD-based initial solution for 10 problem instances</p><p>Candidate list strategy</p><p>Low High Closeness</p><p>60 jobs2 machines 53.43 50.21 51.3260 jobs4 machines 77.19 67.90 63.97</p><p>The </p></li><li><p>406 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>Table 3Final results for problems with 40 jobs and 2 machines</p><p>Cycle string M M LMMSMM M MStrategies None Low Low+ Low+ Low+</p><p>dynamic diversi</p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 407</p><p>Table 4Final results for problems with 40 jobs and 4 machines</p><p>Cycle string M M LMMSMM M MStrategies None Low Low+ Low+ Low+</p><p>dynamic diversi</p></li><li><p>408 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>Table 5Final results for problems with 60 jobs and 2 machines</p><p>Cycle string M M LMMSMM M MStrategies None Low Low+ Low+ Low+</p><p>dynamic diversi</p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 409</p><p>Table 6Final results for problems with 60 jobs and 4 machines</p><p>Cycle string M M LMMSMM M MStrategies None Low Low+ Low+ Low+</p><p>dynamic diversi</p></li><li><p>410 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>Table 7The best solutions known for the benchmark problems</p><p>Problem Best-known Problem Best-known Problem Best-known Problem Best-known</p><p>40241 14 079 40441 0 60241 14 205 60441 040242 3946 40442 0 60242 6528 60442 273740243 3335 40443 0 60243 17 296 60443 15540244 10 095 40444 0 60244 72 406 60444 040245 19 695 40445 0 60245 34 640 60445 259140246 26 372 40446 0 60246 50 492 60446 33940247 18 565 40447 914 60247 26 660 60447 474440248 37 513 40448 48 60248 8042 60448 040249 1055 40449 0 60249 16 790 60449 0402410 1038 404410 0 602410 20 943 604410 4626402411 1726 404411 0 602411 11 204 604411 4423402412 8199 404412 0 602412 14 080 604412 0402413 8382 404413 2807 602413 12 806 604413 0402414 5860 404414 2704 602414 6874 604414 0402415 21 563 404415 1388 602415 20 017 604415 0402416 43 502 404416 0 602416 23 883 604416 58402417 15 816 404417 0 602417 12 222 604417 0402418 5866 404418 0 602418 38 948 604418 0402419 27 258 404419 0 602419 164 604419 0402420 2887 404420 0 602420 23 514 604420 0</p><p>When those 60-job problems that give non-zero solutions under base TS are examined, it is observedthat nine out of 29 are improved in this way. Therefore, this is an eScient way of using the timesaved by applying the Low candidate list strategy especially for larger sized problems.</p><p>3.6. Intensi</p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 411</p><p>Table 8Comparison of GA [22] against TS for 40-job problems</p><p>Problem GA best TS (low) % Impr. of Problem GA best TS (low) % Impr. ofTS over GA TS over GA</p><p>40241 25 482 14 079 44.75 40441 2980 0 100.0040242 10 039 4013 60.03 40442 4259 0 100.0040243 6224 3335 46.42 40443 2002 0 100.0040244 17 971 10 095 43.83 40444 2422 0 100.0040245 34 632 19 748 42.98 40445 131 0 100.0040246 43 730 26 372 39.69 40446 5549 0 100.0040247 35 683 18 565 47.97 40447 6348 922 85.4840248 61 017 37 658 38.28 40448 5745 68 98.8240249 8951 1055 88.21 40449 3304 0 100.00402410 11 097 1038 90.65 404410 4270 0 100.00402411 4071 1835 54.93 404411 2142 0 100.00402412 15 907 8331 47.63 404412 726 0 100.00402413 24 500 8382 65.79 404413 12 067 2851 76.37402414 12 755 5869 53.99 404414 9821 2704 72.47402415 32 672 22 378 31.51 404415 7812 1388 82.23402416 56 979 43 502 23.65 404416 0 0 402417 34 456 15 816 54.10 404417 2244 0 100.00402418 17 006 5866 65.51 404418 3766 0 100.00402419 35 856 27 258 23.98 404419 581 0 100.00402420 7122 2934 58.80 404420 6008 0 100.00</p><p>Avg. % impr. over GA 51.13 Avg. % impr. over GA 95.55</p><p>diversi</p></li><li><p>412 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>Table 9Comparison of GA [22] against TS for 60-job problems</p><p>Problem GA best TS (low) % Impr. of Problem GA best TS (low) % Impr. ofTS over GA TS over GA</p><p>60241 72 860 14 366 80.28 60441 27 626 0 100.0060242 74 948 6704 91.06 60442 23 326 3973 82.9760243 93 203 18 352 80.31 60443 40 861 512 98.7560244 127 175 73 113 42.51 60444 18 057 0 100.0060245 110 234 37 265 66.19 60445 13 608 2961 78.2460246 148 363 50 975 65.64 60446 9732 364 96.2660247 59 213 26 804 54.73 60447 22 731 5249 76.9160248 69 940 8270 88.18 60448 33 076 0 100.0060249 98 100 17 803 81.85 60449 25 279 43 99.83602410 91 911 22 172 75.88 604410 36 781 4993 86.43602411 58 755 11 694 80.10 604411 42 430 4717 88.88602412 54 686 14 080 74.25 604412 17 914 0 100.00602413 102 444 13 237 87.08 604413 30 541 0 100.00602414 88 232 7069 91.99 604414 9370 0 100.00602415 90 994 20 017 78.00 604415 20 035 0 100.00602416 84 974 24 047 71.70 604416 14 276 123 99.14602417 37 049 13 877 62.54 604417 32 919 0 100.00602418 81 804 40 632 50.33 604418 13 761 0 100.00602419 55 911 256 99.54 604419 13 442 0 100.00602420 119 553 24 813 79.25 604420 29 440 0 100.00</p><p>Avg. % impr. over GA 75.07 Avg. % impr. over GA 95.34</p><p>demonstrated in Tables 8 and 9, the short-term TS with the Low candidate list strategy yieldsmuch superior results. The diCerence in the performance can be attributed to the possibility ofearly convergence of the GA. The GA starts with a random population without seeding in any goodsolution, and most of the time the results are inferior to the solution given by the EDD list schedulingheuristic. It seems that the authors concentrated on the new crossover operator they proposed ratherthan aggressively searching for the best results to the problem set they have generated.</p><p>4. Conclusions</p><p>In this paper, a robust TS algorithm for the solution of a very complex parallel machine schedulingproblem where jobs have sequence dependent setup times, distinct due dates and ready times isinvestigated. The major components of TS are tackled through extensive experimentation and as aresult, a completely deterministic TS algorithm is developed. The performance of the algorithm istested using an existing set of problems from literature, and the obtained results are far better thanthose that were previously reported.Moreover, this paper establishes the benchmark solutions for the problem set used under the total</p><p>tardiness criterion (Table 7). These best-known values are obtained by collating the best tabu search</p></li><li><p>&amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414 413</p><p>results encountered throughout the study including preliminary analyses under any combination ofparameters and strategies.The most critical TS component in this algorithm is its context related candidate list strategy. The</p><p>so-called Low candidate list strategy considers job insertions from the machine with the maximumcontribution to total tardiness to each of the other machines. The results reveal that this candidatelist strategy is very successful in isolating desirable regions of the neighbourhood, thus not onlyincreases the speed of the search, but also improves the solution quality with its power to overcometopological traps and direct the search to good regions.Generally, the proposed intensi</p></li><li><p>414 &amp;U . Bilge et al. / Computers &amp; Operations Research 31 (2004) 397414</p><p>[13] Bean JC. Genetic algorithms and random keys for sequencing and optimization. ORSA Journal on Computing1994;6:15460.</p><p>[14] Glover F, Laguna M. Tabu search. London: Kluwer Academic Publishers, 1997.[15] Reeves CR. Modern heuristic techniques for combinatorial problems. New York: John Wiley &amp; Sons, 1993.[16] Laguna M, Barnes JW, Glover F. Tabu search methods for a single machine scheduling problem. Journal of Intelligent</p><p>Manufacturing 1991;2:6374.[17] James RJW, Buchanan JT. Performance enhancements to tabu search for the early/tardy scheduling problem. EJOR</p><p>1998;106:25465.[18] H&amp;ubscher R, Glover F. Applying tabu search with inOuential diversi</p></li></ul>

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