Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems
Abstract
:1. Introduction and Targeted Contribution
2. Literature Review
3. Problem Description
4. Mathematical Model
4.1. Parameters
4.2. Results Obtained on Mixed Blocking Constrained Jobshop Problems
5. Evaluation Function for Meta-Heuristics
5.1. Bierwirth Vector
5.2. Evaluation Function
5.3. Meta-Heuristics Proposed
6. Benchmarks and Computing Results
7. Conclusions and Perspectives
Author Contributions
Funding
Conflicts of Interest
References
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Jobs | Machines | Blocking Constraints | |||
---|---|---|---|---|---|
A1 | A2 | A3 | A4 | ||
5 | 5 | 0.04 | 0.03 | 0.02 | 0.04 |
10 | 5 | 67.77 | 33.78 | 47.44 | 40.37 |
10 | 10 | 1174.30 | 618.32 | 605.60 | 84.9 |
15 | 5 | >1 h | >1 h | >1 h | >1 h |
j | m | Meta-Heuristic | Parameters | A1 | A2 | A3 | A4 |
---|---|---|---|---|---|---|---|
5 | 5 | PSO | ctbvi = 10; nb_indiv = 10 | 22.7 | 9.6 | 10.4 | 21.5 |
PSO | ctbvi = 100; nb_indiv = 100 | 14.8 | 5.8 | 5.2 | 17.0 | ||
GA | ctbvi = 100; nb_indiv = 100 | 12.6 | 2.4 | 2.3 | 12.3 | ||
10 | 5 | PSO | ctbvi = 10; nb_indiv = 10 | 63.3 | 59.2 | 52.4 | 66.2 |
PSO | ctbvi = 100; nb_indiv = 100 | 49.0 | 47.6 | 40.3 | 51.6 | ||
GA | ctbvi = 100; nb_indiv = 100 | 40.8 | 33.3 | 30.8 | 22.7 |
j | m | GA (%) | GA (s) | PSO (%) | PSO (s) |
---|---|---|---|---|---|
10 | 5 | 5.91 | 0.6 | 9.47 | 2.0 |
10 | 10 | 8.24 | 1.6 | 12.88 | 6.6 |
15 | 5 | 1.28 | 1.2 | 3.17 | 4.2 |
15 | 10 | 16.59 | 3.4 | 19.90 | 16.8 |
15 | 15 | 14.56 | 8.4 | 18.71 | 39.2 |
20 | 5 | 0.31 | 2.4 | 2.54 | 6.4 |
20 | 10 | 16.06 | 8.0 | 18.91 | 26.6 |
30 | 10 | 5.60 | 13.8 | 9.26 | 77.2 |
Average: | 8.57 | 11.86 |
j | m | Blocking Matrix | Execution Time | j | m | Blocking Matrix | Execution Time | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A1 | A2 | A3 | A4 | A1 | A2 | A3 | A4 | A1 | A2 | A3 | A4 | A1 | A2 | A3 | A4 | ||||
10 | 5 | 1512 | 1167 | 1217 | 1127 | 3 | 7 | 3 | 4 | 30 | 10 | 7792 | 5101 | 5295 | 4421 | 53 | 67 | 63 | 88 |
10 | 5 | 1503 | 1140 | 1172 | 1086 | 5 | 3 | 4 | 3 | 30 | 10 | 8204 | 5742 | 5289 | 4836 | 83 | 58 | 79 | 121 |
10 | 5 | 1278 | 1050 | 1086 | 1004 | 2 | 7 | 2 | 2 | 30 | 10 | 7814 | 5126 | 4949 | 4819 | 43 | 94 | 161 | 53 |
10 | 5 | 1212 | 1025 | 973 | 903 | 4 | 3 | 7 | 4 | 30 | 10 | 7710 | 5545 | 4842 | 4630 | 49 | 43 | 75 | 150 |
10 | 5 | 1189 | 945 | 864 | 872 | 3 | 3 | 5 | 3 | 30 | 10 | 7657 | 4700 | 5083 | 4945 | 48 | 165 | 143 | 82 |
10 | 10 | 2524 | 1679 | 1983 | 1827 | 8 | 11 | 9 | 6 | 30 | 15 | 8101 | 6097 | 5505 | 5029 | 102 | 43 | 368 | 306 |
10 | 10 | 2296 | 1722 | 1611 | 1437 | 6 | 8 | 7 | 9 | 30 | 15 | 7988 | 5291 | 5563 | 4991 | 68 | 239 | 256 | 384 |
10 | 10 | 2434 | 1569 | 1834 | 1692 | 19 | 9 | 14 | 6 | 30 | 15 | 7552 | 5767 | 5668 | 4884 | 198 | 133 | 89 | 179 |
10 | 10 | 2605 | 1819 | 1762 | 1812 | 15 | 8 | 18 | 6 | 30 | 15 | 8163 | 5336 | 5681 | 4962 | 87 | 196 | 120 | 193 |
10 | 10 | 2704 | 1689 | 1750 | 1676 | 6 | 11 | 6 | 17 | 30 | 15 | 8186 | 5486 | 5893 | 4993 | 80 | 241 | 130 | 307 |
15 | 5 | 2173 | 1718 | 1644 | 1546 | 8 | 5 | 4 | 4 | 30 | 20 | 8143 | 6261 | 5853 | 5776 | 302 | 186 | 420 | 243 |
15 | 5 | 1938 | 1527 | 1364 | 1456 | 12 | 5 | 12 | 10 | 30 | 20 | 8220 | 6060 | 6058 | 5435 | 139 | 334 | 285 | 399 |
15 | 5 | 2101 | 1649 | 1628 | 1441 | 10 | 8 | 9 | 11 | 30 | 20 | 7783 | 5917 | 6195 | 5653 | 302 | 186 | 420 | 243 |
15 | 5 | 2229 | 1772 | 1686 | 1667 | 7 | 6 | 9 | 11 | 30 | 20 | 8252 | 6063 | 6002 | 5462 | 338 | 382 | 393 | 251 |
15 | 5 | 1888 | 1741 | 1544 | 1576 | 12 | 5 | 5 | 10 | 30 | 20 | 7785 | 6017 | 6422 | 5452 | 292 | 194 | 443 | 275 |
15 | 10 | 3852 | 2548 | 2644 | 2546 | 19 | 17 | 26 | 18 | 50 | 15 | 12,195 | 10,046 | 9615 | 8598 | 359 | 560 | 523 | 469 |
15 | 10 | 3592 | 2431 | 2232 | 2487 | 20 | 14 | 18 | 15 | 50 | 15 | 12,184 | 9812 | 9094 | 8165 | 230 | 796 | 420 | 390 |
15 | 10 | 3783 | 2640 | 2779 | 2321 | 25 | 15 | 17 | 11 | 50 | 15 | 11,672 | 9767 | 8946 | 8451 | 430 | 613 | 351 | 648 |
15 | 10 | 3720 | 2471 | 2430 | 2504 | 36 | 14 | 18 | 27 | 50 | 15 | 11,977 | 9512 | 9712 | 7647 | 658 | 898 | 414 | 976 |
15 | 10 | 3749 | 2572 | 2358 | 2310 | 48 | 25 | 29 | 23 | 50 | 15 | 12,108 | 9827 | 8893 | 8116 | 545 | 251 | 653 | 832 |
15 | 15 | 5455 | 3680 | 3390 | 3508 | 31 | 31 | 36 | 29 | 50 | 20 | 13,105 | 10,386 | 10,090 | 9291 | 556 | 1935 | 829 | 741 |
15 | 15 | 6445 | 3874 | 3708 | 3911 | 32 | 32 | 16 | 44 | 50 | 20 | 12,582 | 10,737 | 9616 | 8860 | 754 | 681 | 796 | 981 |
15 | 15 | 5741 | 3019 | 3357 | 3214 | 61 | 50 | 61 | 42 | 50 | 20 | 12,705 | 9823 | 10,344 | 8852 | 747 | 1671 | 327 | 1335 |
15 | 15 | 5404 | 3525 | 3349 | 3527 | 31 | 22 | 44 | 50 | 50 | 20 | 12,579 | 10,399 | 9981 | 9312 | 902 | 717 | 714 | 981 |
15 | 15 | 5856 | 3485 | 3371 | 3761 | 21 | 42 | 35 | 47 | 50 | 20 | 13,342 | 9943 | 9914 | 8850 | 276 | 1231 | 1349 | 1452 |
20 | 5 | 2798 | 2209 | 2153 | 2134 | 12 | 15 | 11 | 10 | 100 | 20 | 22,632 | 22,363 | 20,972 | 18,108 | 5547 | 2517 | 3130 | 3251 |
20 | 5 | 2545 | 1915 | 1840 | 1785 | 8 | 15 | 21 | 15 | 100 | 20 | 23,598 | 21,479 | 20,796 | 17,913 | 3886 | 7498 | 4863 | 7111 |
20 | 5 | 2725 | 2235 | 2211 | 2115 | 8 | 13 | 15 | 16 | 100 | 20 | 23,797 | 20,866 | 20,584 | 17,482 | 3871 | 5529 | 2207 | 6603 |
20 | 5 | 2735 | 2362 | 1923 | 2086 | 14 | 11 | 14 | 14 | 100 | 20 | 23,925 | 21,273 | 20,513 | 19,538 | 4709 | 6505 | 5209 | 4375 |
20 | 5 | 2787 | 2212 | 2201 | 2303 | 12 | 14 | 9 | 13 | 100 | 20 | 24,947 | 23,354 | 21,578 | 19,088 | 3962 | 4323 | 3342 | 5823 |
20 | 10 | 5405 | 3507 | 3777 | 3462 | 23 | 29 | 17 | 25 | ||||||||||
20 | 10 | 5139 | 3610 | 3453 | 3294 | 39 | 37 | 39 | 24 | ||||||||||
20 | 10 | 5175 | 3451 | 3276 | 3599 | 33 | 40 | 47 | 31 | ||||||||||
20 | 10 | 5185 | 3477 | 3150 | 3176 | 24 | 32 | 29 | 38 | ||||||||||
20 | 10 | 5375 | 3907 | 3324 | 3564 | 22 | 21 | 27 | 19 |
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Sauvey, C.; Trabelsi, W.; Sauer, N. Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems. Mathematics 2020, 8, 121. https://doi.org/10.3390/math8010121
Sauvey C, Trabelsi W, Sauer N. Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems. Mathematics. 2020; 8(1):121. https://doi.org/10.3390/math8010121
Chicago/Turabian StyleSauvey, Christophe, Wajdi Trabelsi, and Nathalie Sauer. 2020. "Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems" Mathematics 8, no. 1: 121. https://doi.org/10.3390/math8010121
APA StyleSauvey, C., Trabelsi, W., & Sauer, N. (2020). Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems. Mathematics, 8(1), 121. https://doi.org/10.3390/math8010121