A tabu search algorithm for parallel machine total tardiness problem

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Computers & Operations Research 31 (2004) 397 – 414

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A tabu search algorithm for parallel machine total tardiness problem & Umit Bilge∗ , Furkan K-ra.c, M&ujde Kurtulan, Pelin Pekg&un ˙ Department of Industrial Engineering, Bogazici University, Bebek, 80815 Istanbul, Turkey Received 1 April 2001; received in revised form 1 July 2002

Abstract In this study, we consider the problem of scheduling a set of independent jobs with sequence dependent setups on a set of uniform parallel machines such that total tardiness is minimized. Jobs have non-identical due dates and arrival times. A tabu search (TS) approach is employed to attack this complex problem. In order to obtain a robust search mechanism, several key components of TS such as candidate list strategies, tabu classi
∗

Corresponding author. Tel.: +90-212-263-1500; fax: +90-212-265-1800. & Bilge). E-mail address: [email protected] (U.

0305-0548/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(02)00198-3

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1. Introduction The classical parallel machine total tardiness problem (PMTP) can be stated as follows [1–3]: A set of independent jobs is to be processed on a number of continuously available identical parallel machines. Each machine can process only one job at a time, and each job can be processed on one machine. Each job is ready at the beginning of the scheduling horizon and has a distinct processing time and a distinct due date. The objective is to determine a schedule such that total tardiness is minimized, where tardiness of a job is the amount of time its completion time exceeds its due date. The problem is NP-hard even for a single machine (Du and Leung [4]) and exact methods in which the dimensionality problem is acute are mostly limited to special cases like common due dates 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
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[19] present a TS to minimize makespan in a Oow shop with parallel machines, and employ a neighbourhood based on blocks of operations on a critical path. A similar block approach is used by Liaw [20] for makespan minimization in an open shop. Park and Kim [21] compare simulated annealing and TS for a parallel machine scheduling problem where jobs have equal due dates and equal ready times for minimizing holding costs. When jobs are allowed to have distinct arrival times as well as due dates, diCerent processing rates on machines and sequence dependent setup times, the literature becomes really sparse. There are only two studies reported on this more general problem to our knowledge and both of them deal with minimizing the total earliness-tardiness costs: S.erifoPglu and Ulusoy [22] present a GA with a new crossover operator, while Balakrishnan et al. [23] report a compact mathematical model to solve small sized (up to 10 jobs) problems. The TS algorithm proposed here is tested using the problem set given in Serifoglu and Ulusoy [22] and the results are compared to their results for the case where the weight of the earliness penalty 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 several alternative approaches leading towards a robust TS algorithm tailored to solve the problem at hand, and evaluates the performance of this algorithm through numerical experimentation. The paper concludes with discussion of results and further studies in Section 4. 2. Description of the tabu search approach This section outlines the totally deterministic TS algorithm tailored to the GPMTP by discussing several of the key concepts such as solution encoding, initial solutions, tabu classi
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Machine 1

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n1

Machine 2

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nj

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2

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nm

n2

: : : Machine j : : : Machine m

Fig. 1. Solution representation.

2.3. Neighbourhood generation Insert moves and pairwise exchanges (swaps) are two of the frequently used move types in permutation problems. An insert move identi
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the search process. Since jobs have distinct ready times, diCerent processing times on diCerent types of machines and sequence dependent setup times, calculation of total tardiness for a given move is a tedious task. Although this is implemented in an eScient way by
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2.5. Tabu classi
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403

Fig. 2. Sample screen for ‘WinMeta’.

out a search of a given length from each of these solutions. An elite solution is de
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Table 1 Problem design parameters Number of jobs: n Number of machines: m Maximum setup duration: Amax

20, 40, 60 2, 4 4,8

Pentium 4- 1:6 GHz CPU, Host Bus 133 MHz with 512 MB RAM. The problem set and the results obtained are presented in the next sections. 3.1. Example problems Although WinMeta can be used to generate a new set of problems, in this study the benchmark problem set due to S.erifoPglu and Ulusoy [22] is used. Their test problems were generated using the design in Table 1 with 20 instances for each combination. The details regarding the generation of problems are as follows: The machines considered belong to one of two diCerent types, Types I and II, which have the same characteristics except that they have diCerent processing rates. Type II machines are older technology machines and the processing time of a job on Type II machine is 10 – 20% larger than on a Type I machine. Likewise, setup times on a Type II machine are 20 – 40% larger than 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 the Type II machine, piII , multipliers are chosen from the interval [1:10; 1:20] randomly and applied to the corresponding processing time on machine Type I. Setup times on a Type I machine, sIji , are obtained from the uniform distribution U [1; Amax ], where two levels of maximum setup time, Amax , are utilized. Multipliers are randomly chosen from the interval [1:20; 1:40] and used to obtain the corresponding 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 (pX II + sXII )(n=m − 1), where the
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Table 2 Comparison of the candidate list strategies Problem type

Average % improvement over EDD-based initial solution for 10 problem instances Candidate list strategy

60 jobs—2 machines 60 jobs—4 machines

Low

High

Closeness

53.43 77.19

50.21 67.90

51.32 63.97

The
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Table 3 Final results for problems with 40 jobs and 2 machines Cycle string Strategies

M None

M Low

Non-improving iterations

5000

5000

Problem

Result

% Dev

Result

40241 40242 40243 40244 40245 40246 40247 40248 40249 402410 402411 402412 402413 402414 402415 402416 402417 402418 402419 402420

14 079 3946 3335 10 758 19 703 26 767 18 565 37 513 1142 1270 1726 8288 8382 5860 21 977 43 502 15 816 6391 27 258 2934

0.00 0.00 0.00 6.16 0.04 1.48 0.00 0.00 7.62 18.27 0.00 1.07 0.00 0.00 1.88 0.00 0.00 8.21 0.00 1.60

14 079 4013 3335 10 095 19 748 26 372 18 565 37 658 1055 1038 1835 8331 8382 5869 22 378 43 502 15 816 5866 27 258 2934

Average % deviation Average CPU

LMMSMM Low+ dynamic 5000

M Low+ diversi
M Low+ intensi
% Dev

Result

% Dev

Result

% Dev

Result

% Dev

0.00 1.67 0.00 0.00 0.27 0.00 0.00 0.39 0.00 0.00 5.94 1.58 0.00 0.15 3.64 0.00 0.00 0.00 0.00 1.60

14 079 3946 3335 10 095 19 722 26 372 19 324 37 789 1055 1038 1869 8465 8382 5869 22 134 43 502 15 976 6430 28 192 2974

0.00 0.00 0.00 0.00 0.14 0.00 3.93 0.73 0.00 0.00 7.65 3.14 0.00 0.15 2.58 0.00 1.00 8.77 3.31 2.93

14 079 3946 3335 10 095 19 748 26 372 18 565 37 658 1055 1038 1726 8331 8382 5869 22 134 43 502 15 816 5866 27 258 2934

0.00 0.00 0.00 0.00 0.27 0.00 0.00 0.39 0.00 0.00 0.00 1.58 0.00 0.15 2.58 0.00 0.00 0.00 0.00 1.60

14 079 3946 3335 10 095 19 703 26 372 18 565 37 658 1055 1038 1726 8199 8382 5860 22 190 43 502 15 816 5866 27 258 2934

0.00 0.00 0.00 0.00 0.04 0.00 0.00 0.39 0.00 0.00 0.00 0.00 0.00 0.00 2.83 0.00 0.00 0.00 0.00 1.60

2.32

0.76

1.72

0.33

0.24

18.50

14.20

11.50

19.05

22.50

Hence, in all the tables below presenting the experimental results, the “% dev” column indicates the value ((result − best-known)=result × 100), where “best-known” is the solution given in Table 7 and “result” is the solution given by TS under the corresponding strategy. Those problems with zero initial solution are skipped and this situation is indicated by a “—” in the percent deviation columns. The CPU times are reported in seconds. The second column in Tables 3–6, referred as “None”, represents the case when there is no candidate list strategy, while the third column represents the case when the candidate list strategy is “Low”. The results indicate that the “Low” candidate list strategy is very powerful, not only improving the performance but also dramatically decreasing the CPU time. As problem size is increased from 40 jobs-2 machines (Table 3) to 60 jobs-4 machines (Table 6) the complexity increases in both sequencing and allocation aspects of the problem. Average percentage of deviation from the best solution known comply with this fact.

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Table 4 Final results for problems with 40 jobs and 4 machines Cycle string Strategies

M None

M Low

Non-improving iterations

5000

5000

Problem

Result

% Dev

Result

40441 40442 40443 40444 40445 40446 40447 40448 40449 404410 404411 404412 404413 404414 404415 404416 404417 404418 404419 404420

0 0 0 0 0 0 1155 166 129 0 0 0 2807 3456 1388 0 0 0 0 0

0 0 0 — 0 0 20.87 71.08 100.00 0 0 — 0.00 21.76 0.00 0 — 0 — 0

0 0 0 0 0 0 922 68 0 0 0 0 2851 2704 1388 0 0 0 0 0

Average % deviation Average CPU

LMMSMM Low+ dynamic 5000

M Low+ diversi
M Low+ intensi
% Dev

Result

% Dev

Result

% Dev

Result

% Dev

0 0 0 — 0 0 0.87 29.41 0 0 0 — 1.54 0.00 0.00 0 — 0 — 0

0 0 0 0 0 0 1216 79 0 0 0 0 2919 2704 1886 0 0 0 0 0

0 0 0 — 0 0 24.84 39.24 0 0 0 — 3.84 0.00 26.41 0 — 0 — 0

0 0 0 0 0 0 922 68 0 0 0 0 2851 2704 1388 0 0 0 0 0

0 0 0 — — 0 0.87 29.41 0 0 0 — 1.54 0.00 0.00 0 — 0 — 0

0 0 0 0 0 0 914 66 0 0 0 0 2851 2704 1388 0 0 0 0 0

0 0 0 — — 0 0.00 27.27 0 0 0 — 1.54 0.00 0.00 0 — 0 — 0

13.36

1.99

5.89

1.99

1.80

7.5

4.25

2.65

4.8

5.2

Hence, the TS algorithm with the “Low” candidate list strategy is a successful solution method for GPMTP. The studies from this point onward will aim to
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Table 5 Final results for problems with 60 jobs and 2 machines Cycle string Strategies

M None

M Low

Non-improving iterations

5000

5000

Problem

Result

% Dev

Result

60241 60242 60243 60244 60245 60246 60247 60248 60249 602410 602411 602412 602413 602414 602415 602416 602417 602418 602419 602420

14 205 6990 18 094 74 054 35 690 53 173 27 152 8493 17 576 21 577 11 453 14 216 12 806 6951 20 502 24 281 13 554 40 459 351 24 544

0.00 6.61 4.41 2.23 2.94 5.04 1.81 5.31 4.47 2.94 2.17 0.96 0.00 1.11 2.37 1.64 9.83 3.73 53.28 4.20

14 366 6704 18 352 73 113 37 265 50 975 26 804 8270 17 803 22 172 11 694 14 080 13 237 7069 20 017 24 047 13 877 40 632 256 24 813

Average % deviation Average CPU

LMMSMM Low+ dynamic 5000

M Low+ diversi
M Low+ intensi
% Dev

Result

% Dev

Result

% Dev

Result

% Dev

1.12 2.63 5.75 0.97 7.04 0.95 0.54 2.76 5.69 5.54 4.19 0.00 3.26 2.76 0.00 0.68 11.93 4.14 35.94 5.24

14 677 6990 17 749 73 389 35 543 52 825 26 776 8998 17 254 21 434 11 860 14 991 13 303 6941 20 068 23 883 12 222 40 237 300 26 500

3.22 6.61 2.55 1.34 2.54 4.42 0.43 10.62 2.69 2.29 5.53 6.08 3.74 0.97 0.25 0.00 0.00 3.20 45.33 11.27

14 360 6570 17 593 73 113 35 488 50 975 26 804 8270 17 336 22 172 11 694 14 080 12 978 7069 20 017 24 047 13 419 40 632 256 24 813

1.08 0.64 1.69 0.97 2.39 0.95 0.54 2.76 3.15 5.54 4.19 0.00 1.33 2.76 0.00 0.68 8.92 4.14 35.94 5.24

14 350 6662 18 352 73 113 36 406 50 975 26 804 8087 17 695 21 518 11 334 14 080 13 185 7048 20 017 24 047 13 874 39 989 256 23 612

1.01 2.01 5.75 0.97 4.85 0.95 0.54 0.56 5.11 2.67 1.15 0.00 2.87 2.47 0.00 0.68 11.91 2.60 35.94 0.42

5.75

5.06

5.65

4.14

4.12

58.45

51.90

39.50

64.85

77.15

for [2 × medium tenure] iterations. In all experiments the candidate list strategy is “Low” and the stopping criterion is 5000 non-improving iterations. It is concluded that the LMMSMM string with kS = 0:35; kM = 0:5 and kL = 0:8 is the best among these structures. The solutions for the selected dynamic tenure structure are presented in the fourth columns of Tables 3–6. Incorporating the dynamic tenure structure into the base TS algorithm improves the solution quality only for 60 jobs-4 machines case. It does, however, diminish the CPU time signi
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Table 6 Final results for problems with 60 jobs and 4 machines Cycle string Strategies

M None

M Low

Non-improving iterations

5000

5000

Problem

Result

% Dev

Result

60441 60442 60443 60444 60445 60446 60447 60448 60449 604410 604411 604412 604413 604414 604415 604416 604417 604418 604419 604420

0 3451 935 0 3550 828 5468 0 43 7490 4962 0 0 0 0 264 0 0 0 0

0 20.69 83.42 — 27.01 59.06 13.24 0 100.00 38.24 10.86 0 0 0 0 78.03 0 0 0 0

0 3973 512 0 2961 364 5249 0 43 4993 4717 0 0 0 0 123 0 0 0 0

LMMSMM Low+ dynamic 5000

M Low+ diversi
M Low+ intensi
% Dev

Result

% Dev

Result

% Dev

Result

% Dev

0 31.11 69.73 — 12.50 6.87 9.62 0 100.00 7.35 6.23 0 0 0 0 52.85 0 0 0 0

0 4006 155 0 2737 364 5064 0 0 6039 4937 0 0 0 0 90 0 0 0 0

0 31.68 0.00 — 5.33 6.87 6.32 0 0 23.40 10.41 0 0 0 0 35.56 0 0 0 0

0 3697 512 0 2961 364 4775 0 43 4993 4717 0 0 0 0 123 0 0 0 0

0 25.97 69.73 — 12.50 6.87 0.65 0 100.00 7.35 6.23 0 0 0 0 52.85 0 0 0 0

0 3913 512 0 2737 364 5029 0 43 4975 4553 0 0 0 0 123 0 0 0 0

0 30.05 69.73 — 5.3 6.87 5.67 0 100.00 7.02 2.86 0 0 0 0 52.85 0 0 0 0

Average % deviation

22.66

15.59

6.29

14.85

14.76

Average CPU

24.9

14.25

12.55

22.1

21.9

is expressed in number of non-improving iterations that should be completed before concluding that the search has stagnated. The second parameter de
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Table 7 The best solutions known for the benchmark problems Problem

Best-known

Problem

Best-known

Problem

Best-known

Problem

Best-known

40241 40242 40243 40244 40245 40246 40247 40248 40249 402410 402411 402412 402413 402414 402415 402416 402417 402418 402419 402420

14 079 3946 3335 10 095 19 695 26 372 18 565 37 513 1055 1038 1726 8199 8382 5860 21 563 43 502 15 816 5866 27 258 2887

40441 40442 40443 40444 40445 40446 40447 40448 40449 404410 404411 404412 404413 404414 404415 404416 404417 404418 404419 404420

0 0 0 0 0 0 914 48 0 0 0 0 2807 2704 1388 0 0 0 0 0

60241 60242 60243 60244 60245 60246 60247 60248 60249 602410 602411 602412 602413 602414 602415 602416 602417 602418 602419 602420

14 205 6528 17 296 72 406 34 640 50 492 26 660 8042 16 790 20 943 11 204 14 080 12 806 6874 20 017 23 883 12 222 38 948 164 23 514

60441 60442 60443 60444 60445 60446 60447 60448 60449 604410 604411 604412 604413 604414 604415 604416 604417 604418 604419 604420

0 2737 155 0 2591 339 4744 0 0 4626 4423 0 0 0 0 58 0 0 0 0

When those 60-job problems that give non-zero solutions under base TS are examined, it is observed that nine out of 29 are improved in this way. Therefore, this is an eScient way of using the time saved by applying the “Low” candidate list strategy especially for larger sized problems. 3.6. Intensi
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Table 8 Comparison of GA [22] against TS for 40-job problems Problem

GA best

TS (low)

% Impr. of TS over GA

Problem

GA best

TS (low)

% Impr. of TS over GA

40241 40242 40243 40244 40245 40246 40247 40248 40249 402410 402411 402412 402413 402414 402415 402416 402417 402418 402419 402420

25 482 10 039 6224 17 971 34 632 43 730 35 683 61 017 8951 11 097 4071 15 907 24 500 12 755 32 672 56 979 34 456 17 006 35 856 7122

14 079 4013 3335 10 095 19 748 26 372 18 565 37 658 1055 1038 1835 8331 8382 5869 22 378 43 502 15 816 5866 27 258 2934

44.75 60.03 46.42 43.83 42.98 39.69 47.97 38.28 88.21 90.65 54.93 47.63 65.79 53.99 31.51 23.65 54.10 65.51 23.98 58.80

40441 40442 40443 40444 40445 40446 40447 40448 40449 404410 404411 404412 404413 404414 404415 404416 404417 404418 404419 404420

2980 4259 2002 2422 131 5549 6348 5745 3304 4270 2142 726 12 067 9821 7812 0 2244 3766 581 6008

0 0 0 0 0 0 922 68 0 0 0 0 2851 2704 1388 0 0 0 0 0

100.00 100.00 100.00 100.00 100.00 100.00 85.48 98.82 100.00 100.00 100.00 100.00 76.37 72.47 82.23 — 100.00 100.00 100.00 100.00

51.13

Avg. % impr. over GA

Avg. % impr. over GA

95.55

diversi
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Table 9 Comparison of GA [22] against TS for 60-job problems Problem

GA best

TS (low)

% Impr. of TS over GA

Problem

GA best

TS (low)

% Impr. of TS over GA

60241 60242 60243 60244 60245 60246 60247 60248 60249 602410 602411 602412 602413 602414 602415 602416 602417 602418 602419 602420

72 860 74 948 93 203 127 175 110 234 148 363 59 213 69 940 98 100 91 911 58 755 54 686 102 444 88 232 90 994 84 974 37 049 81 804 55 911 119 553

14 366 6704 18 352 73 113 37 265 50 975 26 804 8270 17 803 22 172 11 694 14 080 13 237 7069 20 017 24 047 13 877 40 632 256 24 813

80.28 91.06 80.31 42.51 66.19 65.64 54.73 88.18 81.85 75.88 80.10 74.25 87.08 91.99 78.00 71.70 62.54 50.33 99.54 79.25

60441 60442 60443 60444 60445 60446 60447 60448 60449 604410 604411 604412 604413 604414 604415 604416 604417 604418 604419 604420

27 626 23 326 40 861 18 057 13 608 9732 22 731 33 076 25 279 36 781 42 430 17 914 30 541 9370 20 035 14 276 32 919 13 761 13 442 29 440

0 3973 512 0 2961 364 5249 0 43 4993 4717 0 0 0 0 123 0 0 0 0

100.00 82.97 98.75 100.00 78.24 96.26 76.91 100.00 99.83 86.43 88.88 100.00 100.00 100.00 100.00 99.14 100.00 100.00 100.00 100.00

75.07

Avg. % impr. over GA

Avg. % impr. over GA

95.34

demonstrated in Tables 8 and 9, the short-term TS with the “Low” candidate list strategy yields much superior results. The diCerence in the performance can be attributed to the possibility of early convergence of the GA. The GA starts with a random population without seeding in any good solution, and most of the time the results are inferior to the solution given by the EDD list scheduling heuristic. It seems that the authors concentrated on the new crossover operator they proposed rather than aggressively searching for the best results to the problem set they have generated. 4. Conclusions In this paper, a robust TS algorithm for the solution of a very complex parallel machine scheduling problem where jobs have sequence dependent setup times, distinct due dates and ready times is investigated. The major components of TS are tackled through extensive experimentation and as a result, a completely deterministic TS algorithm is developed. The performance of the algorithm is tested using an existing set of problems from literature, and the obtained results are far better than those that were previously reported. Moreover, this paper establishes the benchmark solutions for the problem set used under the total tardiness criterion (Table 7). These best-known values are obtained by collating the best tabu search

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results encountered throughout the study including preliminary analyses under any combination of parameters and strategies. The most critical TS component in this algorithm is its context related candidate list strategy. The so-called “Low” candidate list strategy considers job insertions from the machine with the maximum contribution to total tardiness to each of the other machines. The results reveal that this candidate list strategy is very successful in isolating desirable regions of the neighbourhood, thus not only increases the speed of the search, but also improves the solution quality with its power to overcome topological traps and direct the search to good regions. Generally, the proposed intensi
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[13] Bean JC. Genetic algorithms and random keys for sequencing and optimization. ORSA Journal on Computing 1994;6:154–60. [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 & Sons, 1993. [16] Laguna M, Barnes JW, Glover F. Tabu search methods for a single machine scheduling problem. Journal of Intelligent Manufacturing 1991;2:63–74. [17] James RJW, Buchanan JT. Performance enhancements to tabu search for the early/tardy scheduling problem. EJOR 1998;106:254–65. [18] H&ubscher R, Glover F. Applying tabu search with inOuential diversi