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Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center
School of Mathematical and Physical Sciences, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia |
A railway network is an indispensable part of the public transportation system in many major cities around the world. In order to provide a safe and reliable service, a fleet of passenger trains must undergo regular maintenance. These maintenance operations are lengthy procedures, which are planned for one year or a longer period. The planning specifies the dates of trains' arrival at the maintenance center and should take into account the uncertain duration of maintenance operations, the periods of validity of the previous maintenance, the desired number of trains in service, and the capacity of the maintenance center. The paper presents a nonlinear programming formulation of the considered problem and several optimization procedures which were compared by computational experiments using real world data. The results of these experiments indicate that the presented approach is capable to be used in real world planning process.
References:
[1] |
S. Ahmed, Two-stage stochastic integer programming: A brief introduction, in Wiley Encyclopedia of Operations Research and Management Science (eds. J.J. Cochran, L.A. Cox, P. Keskinocak, J.P. Kharoufeh and J.C. Smith), John Wiley & Sons, (2011).
doi: 10.1002/9780470400531.eorms0092. |
[2] |
G. Bayraksan and D. P. Morton, Assessing solution quality in stochastic programs via sampling, in Informs 2009 Tutorials in Operations Research, (2009), 102-122.
doi: 10.1287/educ.1090.0065. |
[3] |
P. Billingsley, Probability and Measure, 3$^rd$ edition, John Wiley & Sons, New York, 1995. |
[4] |
W. Biscarri, S. D. Zhao and R. J. Brunner,
A simple and fast method for computing the Poisson binomial distribution function, Computational Statistics & Data Analysis, 122 (2018), 92-100.
doi: 10.1016/j.csda.2018.01.007. |
[5] |
K. Doganay and M. Bohlin, Maintenance plan optimization for a train fleet, in Computers in Railways XII, (eds. B. Ning and C.A. Brebbia), WIT Press, (2010), 349-358.
doi: 10.2495/CR100331. |
[6] |
B. Fortz, M. Labbé, F. Louveaux and M. Poss,
Stochastic binary problems with simple penalties for capacity constraints violations, Mathematical Programming, 138 (2013), 199-221.
doi: 10.1007/s10107-012-0520-4. |
[7] |
G. L. Giacco, D. Carillo, A. D'Ariano, D. Pacciarelli and A. G. Marín,
Short-term rail rolling stock rostering and maintenance scheduling, Transportation Research Procedia, 3 (2014), 651-659.
doi: 10.1016/j.trpro.2014.10.044. |
[8] |
T. Homem-de Mello and G. Bayraksan,
Monte Carlo sampling-based methods for stochastic optimization, Surveys in Operations Research and Management Science, 19 (2014), 56-85.
doi: 10.1016/j.sorms.2014.05.001. |
[9] |
Y. C. Lai, D. C. Fang and K. L. Huang,
Optimizing rolling stock assignment and maintenance plan for passenger railway operations, Computers & Industrial Engineering, 85 (2015), 284-295.
doi: 10.1016/j.cie.2015.03.016. |
[10] |
B. Lin, J. Wu, R. Lin, J. Wang, H. Wang and X. Zhang,
Optimization of high-level preventive maintenance scheduling for high-speed trains, Reliability Engineering & System Safety, 183 (2019), 261-275.
doi: 10.1016/j.ress.2018.11.028. |
[11] |
H. R. Lourenço, O. C. Martin, and T. Stützle, Iterated local search: framework and applications, in Handbook of Metaheuristics (eds. M. Gendreau and J. Potvin), 2$^nd$ edition, Springer, Boston, (2010), 363-397. |
[12] |
D. G. Malcolm, J. H. Rooseboom, C. E. Clark and W. Fazar,
Application of a technique for research and development program evaluation, Operations Research, 7 (1959), 646-669.
doi: 10.1287/opre.7.5.646. |
[13] |
J. G. Pérez, M. Martín, C. García and M. Granero,
Project management under uncertainty beyond beta: The generalized bicubic distribution, Operations Research Perspectives, 3 (2016), 67-76.
doi: 10.1016/j.orp.2016.09.001. |
[14] |
S. Mitchell, M. O'Sullivan and I. Dunning, PuLP: a Linear Programming Toolkit for Python, 2011. Available from: http://www.optimization-online.org/DB\_FILE/2011/09/3178.pdf. |
[15] |
C. Sriskandarajah, A. K. S. Jardine and C. K. Chan,
Maintenance scheduling of rolling stock using a genetic algorithm, Journal of the Operational Research Society, 49 (1998), 1130-1145.
|
[16] |
Sydney Trains, Sydney Trains Annual Report 2017-18, 2018. Available from: https://www.transport.nsw.gov.au/news-and-events/reports-and-publications/sydney-trains-annual-reports. |
[17] |
M. Waskom and the seaborn development team, mwaskom/seaborn, 2020. Available from: https://doi.org/10.5281/zenodo.592845. |
show all references
References:
[1] |
S. Ahmed, Two-stage stochastic integer programming: A brief introduction, in Wiley Encyclopedia of Operations Research and Management Science (eds. J.J. Cochran, L.A. Cox, P. Keskinocak, J.P. Kharoufeh and J.C. Smith), John Wiley & Sons, (2011).
doi: 10.1002/9780470400531.eorms0092. |
[2] |
G. Bayraksan and D. P. Morton, Assessing solution quality in stochastic programs via sampling, in Informs 2009 Tutorials in Operations Research, (2009), 102-122.
doi: 10.1287/educ.1090.0065. |
[3] |
P. Billingsley, Probability and Measure, 3$^rd$ edition, John Wiley & Sons, New York, 1995. |
[4] |
W. Biscarri, S. D. Zhao and R. J. Brunner,
A simple and fast method for computing the Poisson binomial distribution function, Computational Statistics & Data Analysis, 122 (2018), 92-100.
doi: 10.1016/j.csda.2018.01.007. |
[5] |
K. Doganay and M. Bohlin, Maintenance plan optimization for a train fleet, in Computers in Railways XII, (eds. B. Ning and C.A. Brebbia), WIT Press, (2010), 349-358.
doi: 10.2495/CR100331. |
[6] |
B. Fortz, M. Labbé, F. Louveaux and M. Poss,
Stochastic binary problems with simple penalties for capacity constraints violations, Mathematical Programming, 138 (2013), 199-221.
doi: 10.1007/s10107-012-0520-4. |
[7] |
G. L. Giacco, D. Carillo, A. D'Ariano, D. Pacciarelli and A. G. Marín,
Short-term rail rolling stock rostering and maintenance scheduling, Transportation Research Procedia, 3 (2014), 651-659.
doi: 10.1016/j.trpro.2014.10.044. |
[8] |
T. Homem-de Mello and G. Bayraksan,
Monte Carlo sampling-based methods for stochastic optimization, Surveys in Operations Research and Management Science, 19 (2014), 56-85.
doi: 10.1016/j.sorms.2014.05.001. |
[9] |
Y. C. Lai, D. C. Fang and K. L. Huang,
Optimizing rolling stock assignment and maintenance plan for passenger railway operations, Computers & Industrial Engineering, 85 (2015), 284-295.
doi: 10.1016/j.cie.2015.03.016. |
[10] |
B. Lin, J. Wu, R. Lin, J. Wang, H. Wang and X. Zhang,
Optimization of high-level preventive maintenance scheduling for high-speed trains, Reliability Engineering & System Safety, 183 (2019), 261-275.
doi: 10.1016/j.ress.2018.11.028. |
[11] |
H. R. Lourenço, O. C. Martin, and T. Stützle, Iterated local search: framework and applications, in Handbook of Metaheuristics (eds. M. Gendreau and J. Potvin), 2$^nd$ edition, Springer, Boston, (2010), 363-397. |
[12] |
D. G. Malcolm, J. H. Rooseboom, C. E. Clark and W. Fazar,
Application of a technique for research and development program evaluation, Operations Research, 7 (1959), 646-669.
doi: 10.1287/opre.7.5.646. |
[13] |
J. G. Pérez, M. Martín, C. García and M. Granero,
Project management under uncertainty beyond beta: The generalized bicubic distribution, Operations Research Perspectives, 3 (2016), 67-76.
doi: 10.1016/j.orp.2016.09.001. |
[14] |
S. Mitchell, M. O'Sullivan and I. Dunning, PuLP: a Linear Programming Toolkit for Python, 2011. Available from: http://www.optimization-online.org/DB\_FILE/2011/09/3178.pdf. |
[15] |
C. Sriskandarajah, A. K. S. Jardine and C. K. Chan,
Maintenance scheduling of rolling stock using a genetic algorithm, Journal of the Operational Research Society, 49 (1998), 1130-1145.
|
[16] |
Sydney Trains, Sydney Trains Annual Report 2017-18, 2018. Available from: https://www.transport.nsw.gov.au/news-and-events/reports-and-publications/sydney-trains-annual-reports. |
[17] |
M. Waskom and the seaborn development team, mwaskom/seaborn, 2020. Available from: https://doi.org/10.5281/zenodo.592845. |

Train type |
|||||
non-PH | PH | ||||
1 | 25 | 4 | 3 | 1 | |
2 | 5 | 5 | 2 | 1 | |
3 | 5 | 5 | 1 | 1 |
Train type |
|||||
non-PH | PH | ||||
1 | 25 | 4 | 3 | 1 | |
2 | 5 | 5 | 2 | 1 | |
3 | 5 | 5 | 1 | 1 |
Train type | Minimum | Most likely | Maximum | Distribution |
1 | 20 | 25 | 40 | beta-PERT |
2 | 27 | 30 | 46 | beta-PERT |
3 | 29 | 30 | 52 | beta-PERT |
Train type | Minimum | Most likely | Maximum | Distribution |
1 | 20 | 25 | 40 | beta-PERT |
2 | 27 | 30 | 46 | beta-PERT |
3 | 29 | 30 | 52 | beta-PERT |
Case | Case | |||||
1 | 1 | 1000 | 6 | 1 | 100 | |
2 | 1 | 300 | 7 | 1 | 50 | |
3 | 1 | 200 | 8 | 1 | 10 | |
4 | 1 | 180 | 9 | 1 | 1 | |
5 | 1 | 150 |
Case | Case | |||||
1 | 1 | 1000 | 6 | 1 | 100 | |
2 | 1 | 300 | 7 | 1 | 50 | |
3 | 1 | 200 | 8 | 1 | 10 | |
4 | 1 | 180 | 9 | 1 | 1 | |
5 | 1 | 150 |
Case | LB | MIPM | MILP | |||||
Time | Obj. | %Rel | Time | Obj. | %Rel | |||
1 | 156, 744 | 2, 078 | 206, 060 | 31.46 | 7 | 201, 203 | 28.36 | |
2 | 62, 612 | 235 | 77, 764 | 24.20 | 7 | 80, 353 | 28.33 | |
3 | 48, 629 | 68 | 59, 178 | 21.69 | 8 | 59, 103 | 21.54 | |
4 | 45, 768 | 100 | 55, 368 | 20.98 | 8 | 56, 322 | 23.06 | |
5 | 40, 190 | 54 | 48, 447 | 20.54 | 11 | 49, 369 | 22.84 | |
6 | 30, 789 | 46 | 36, 245 | 17.72 | 8 | 36, 572 | 18.78 | |
7 | 20, 940 | 41 | 23, 624 | 12.82 | 8 | 23, 337 | 11.45 | |
8 | 10, 594 | 39 | 10, 753 | 1.50 | 8 | 10, 818 | 2.11 | |
9 | 6, 706 | 38 | 6, 725 | 0.28 | 8 | 6, 748 | 0.63 | |
Average | 46, 997 | 300 | 58, 240 | 16.80 | 8 | 58, 203 | 17.46 |
Case | LB | MIPM | MILP | |||||
Time | Obj. | %Rel | Time | Obj. | %Rel | |||
1 | 156, 744 | 2, 078 | 206, 060 | 31.46 | 7 | 201, 203 | 28.36 | |
2 | 62, 612 | 235 | 77, 764 | 24.20 | 7 | 80, 353 | 28.33 | |
3 | 48, 629 | 68 | 59, 178 | 21.69 | 8 | 59, 103 | 21.54 | |
4 | 45, 768 | 100 | 55, 368 | 20.98 | 8 | 56, 322 | 23.06 | |
5 | 40, 190 | 54 | 48, 447 | 20.54 | 11 | 49, 369 | 22.84 | |
6 | 30, 789 | 46 | 36, 245 | 17.72 | 8 | 36, 572 | 18.78 | |
7 | 20, 940 | 41 | 23, 624 | 12.82 | 8 | 23, 337 | 11.45 | |
8 | 10, 594 | 39 | 10, 753 | 1.50 | 8 | 10, 818 | 2.11 | |
9 | 6, 706 | 38 | 6, 725 | 0.28 | 8 | 6, 748 | 0.63 | |
Average | 46, 997 | 300 | 58, 240 | 16.80 | 8 | 58, 203 | 17.46 |
Case | MIPM-LS1 | MILP-LS1 | |||||
LS Time | Obj. | %Rel | LS Time | Obj. | %Rel | ||
1 | 46.32 | 189, 551 | 20.93 | 19.59 | 192, 637 | 22.90 | |
2 | 44.29 | 72, 584 | 15.93 | 32.72 | 75, 368 | 20.37 | |
3 | 36.11 | 56, 483 | 16.15 | 46.72 | 55, 102 | 13.31 | |
4 | 28.95 | 53, 125 | 16.07 | 60.71 | 51, 999 | 13.61 | |
5 | 67.76 | 42, 884 | 6.70 | 46.84 | 44, 547 | 10.84 | |
6 | 22.06 | 32, 971 | 7.09 | 13.01 | 35, 356 | 14.83 | |
7 | 7.44 | 22, 948 | 9.59 | 26.11 | 22, 905 | 9.38 | |
8 | 10.56 | 10, 752 | 1.49 | 15.11 | 10, 793 | 1.88 | |
9 | 6.89 | 6, 724 | 0.27 | 6.85 | 6, 732 | 0.39 | |
Average | 30.04 | 54, 225 | 10.47 | 29.74 | 55, 049 | 11.95 |
Case | MIPM-LS1 | MILP-LS1 | |||||
LS Time | Obj. | %Rel | LS Time | Obj. | %Rel | ||
1 | 46.32 | 189, 551 | 20.93 | 19.59 | 192, 637 | 22.90 | |
2 | 44.29 | 72, 584 | 15.93 | 32.72 | 75, 368 | 20.37 | |
3 | 36.11 | 56, 483 | 16.15 | 46.72 | 55, 102 | 13.31 | |
4 | 28.95 | 53, 125 | 16.07 | 60.71 | 51, 999 | 13.61 | |
5 | 67.76 | 42, 884 | 6.70 | 46.84 | 44, 547 | 10.84 | |
6 | 22.06 | 32, 971 | 7.09 | 13.01 | 35, 356 | 14.83 | |
7 | 7.44 | 22, 948 | 9.59 | 26.11 | 22, 905 | 9.38 | |
8 | 10.56 | 10, 752 | 1.49 | 15.11 | 10, 793 | 1.88 | |
9 | 6.89 | 6, 724 | 0.27 | 6.85 | 6, 732 | 0.39 | |
Average | 30.04 | 54, 225 | 10.47 | 29.74 | 55, 049 | 11.95 |
Case | MIPM-SLS | MILP-SLS | |||||
LS Time | Obj. | %Rel | LS Time | Obj. | %Rel | ||
1 | 879 | 189, 551 | 20.93 | 2, 366 | 190, 584 | 21.59 | |
2 | 871 | 72, 584 | 15.93 | 3, 593 | 72, 868 | 16.38 | |
3 | 884 | 56, 483 | 16.15 | 931 | 54, 766 | 12.62 | |
4 | 802 | 53, 125 | 16.07 | 3, 245 | 51, 208 | 11.89 | |
5 | 862 | 42, 884 | 6.70 | 3, 589 | 44, 173 | 9.91 | |
6 | 838 | 32, 971 | 7.09 | 3, 592 | 34, 911 | 13.39 | |
7 | 3, 196 | 22, 631 | 8.08 | 899 | 22, 905 | 9.38 | |
8 | 699 | 10, 752 | 1.49 | 687 | 10, 793 | 1.88 | |
9 | 708 | 6, 724 | 0.27 | 749 | 6, 732 | 0.39 | |
Average | 1, 082 | 54, 189 | 10.30 | 2, 183 | 54, 327 | 10.82 |
Case | MIPM-SLS | MILP-SLS | |||||
LS Time | Obj. | %Rel | LS Time | Obj. | %Rel | ||
1 | 879 | 189, 551 | 20.93 | 2, 366 | 190, 584 | 21.59 | |
2 | 871 | 72, 584 | 15.93 | 3, 593 | 72, 868 | 16.38 | |
3 | 884 | 56, 483 | 16.15 | 931 | 54, 766 | 12.62 | |
4 | 802 | 53, 125 | 16.07 | 3, 245 | 51, 208 | 11.89 | |
5 | 862 | 42, 884 | 6.70 | 3, 589 | 44, 173 | 9.91 | |
6 | 838 | 32, 971 | 7.09 | 3, 592 | 34, 911 | 13.39 | |
7 | 3, 196 | 22, 631 | 8.08 | 899 | 22, 905 | 9.38 | |
8 | 699 | 10, 752 | 1.49 | 687 | 10, 793 | 1.88 | |
9 | 708 | 6, 724 | 0.27 | 749 | 6, 732 | 0.39 | |
Average | 1, 082 | 54, 189 | 10.30 | 2, 183 | 54, 327 | 10.82 |
Case | In. Obj. | Time | Hybrid ILS with LS1 | Hybrid ILS with |
|||
Obj. | %Rel | Obj. | %Rel | ||||
1 | 206, 060 | 2, 685 | 189, 487 | 20.89 | 180, 530 | 15.17 | |
2 | 77, 764 | 864 | 71, 715 | 14.54 | 67, 092 | 7.16 | |
3 | 59, 178 | 581 | 55, 624 | 14.38 | 51, 344 | 5.58 | |
4 | 55, 368 | 932 | 50, 780 | 10.95 | 47, 803 | 4.45 | |
5 | 48, 447 | 805 | 42, 416 | 5.54 | 41, 666 | 3.67 | |
6 | 36, 245 | 472 | 32, 828 | 6.62 | 32, 371 | 5.14 | |
7 | 23, 624 | 817 | 22, 494 | 7.42 | 22, 948 | 9.59 | |
8 | 10, 753 | 483 | 10, 752 | 1.49 | 10, 752 | 1.49 | |
9 | 6, 725 | 355 | 6, 724 | 0.27 | 6, 724 | 0.27 | |
Average | 58, 240 | 888 | 53, 647 | 9.12 | 51, 248 | 5.84 |
Case | In. Obj. | Time | Hybrid ILS with LS1 | Hybrid ILS with |
|||
Obj. | %Rel | Obj. | %Rel | ||||
1 | 206, 060 | 2, 685 | 189, 487 | 20.89 | 180, 530 | 15.17 | |
2 | 77, 764 | 864 | 71, 715 | 14.54 | 67, 092 | 7.16 | |
3 | 59, 178 | 581 | 55, 624 | 14.38 | 51, 344 | 5.58 | |
4 | 55, 368 | 932 | 50, 780 | 10.95 | 47, 803 | 4.45 | |
5 | 48, 447 | 805 | 42, 416 | 5.54 | 41, 666 | 3.67 | |
6 | 36, 245 | 472 | 32, 828 | 6.62 | 32, 371 | 5.14 | |
7 | 23, 624 | 817 | 22, 494 | 7.42 | 22, 948 | 9.59 | |
8 | 10, 753 | 483 | 10, 752 | 1.49 | 10, 752 | 1.49 | |
9 | 6, 725 | 355 | 6, 724 | 0.27 | 6, 724 | 0.27 | |
Average | 58, 240 | 888 | 53, 647 | 9.12 | 51, 248 | 5.84 |
Case | In. Obj. | Time | hybrid ILS with SLS | hybrid ILS with |
|||
Obj. | %Rel | Obj. | %Rel | ||||
1 | 206, 060 | 3, 600 | 187, 992 | 19.94 | 179, 847 | 14.74 | |
2 | 77, 764 | 2, 482 | 70, 845 | 13.15 | 66, 967 | 6.96 | |
3 | 59, 178 | 1, 640 | 55, 100 | 13.31 | 51, 324 | 5.54 | |
4 | 55, 368 | 2, 281 | 51, 459 | 12.43 | 47, 785 | 4.41 | |
5 | 48, 447 | 2, 496 | 42, 193 | 4.98 | 41, 432 | 3.09 | |
6 | 36, 245 | 1, 634 | 32, 968 | 7.08 | 32, 371 | 5.14 | |
7 | 23, 624 | 1, 413 | 22, 613 | 7.99 | 22, 948 | 9.59 | |
8 | 10, 753 | 1, 191 | 10, 752 | 1.49 | 10, 752 | 1.49 | |
9 | 6, 725 | 1, 078 | 6, 724 | 0.27 | 6, 724 | 0.27 | |
Average | 58, 240 | 1, 979 | 53, 405 | 8.96 | 51, 128 | 5.69 |
Case | In. Obj. | Time | hybrid ILS with SLS | hybrid ILS with |
|||
Obj. | %Rel | Obj. | %Rel | ||||
1 | 206, 060 | 3, 600 | 187, 992 | 19.94 | 179, 847 | 14.74 | |
2 | 77, 764 | 2, 482 | 70, 845 | 13.15 | 66, 967 | 6.96 | |
3 | 59, 178 | 1, 640 | 55, 100 | 13.31 | 51, 324 | 5.54 | |
4 | 55, 368 | 2, 281 | 51, 459 | 12.43 | 47, 785 | 4.41 | |
5 | 48, 447 | 2, 496 | 42, 193 | 4.98 | 41, 432 | 3.09 | |
6 | 36, 245 | 1, 634 | 32, 968 | 7.08 | 32, 371 | 5.14 | |
7 | 23, 624 | 1, 413 | 22, 613 | 7.99 | 22, 948 | 9.59 | |
8 | 10, 753 | 1, 191 | 10, 752 | 1.49 | 10, 752 | 1.49 | |
9 | 6, 725 | 1, 078 | 6, 724 | 0.27 | 6, 724 | 0.27 | |
Average | 58, 240 | 1, 979 | 53, 405 | 8.96 | 51, 128 | 5.69 |
Case | multi-start ILS with LS1 | multi-start ILS with |
|||||||
Time | In. Obj. | Obj. | %Rel | Time | In. Obj. | Obj. | %Rel | ||
1 | 2, 904 | 731, 900 | 200, 646 | 28.01 | 3, 600 | 701, 192 | 221, 792 | 41.50 | |
2 | 3, 206 | 246, 939 | 76, 713 | 22.52 | 3, 600 | 228, 459 | 82, 564 | 31.87 | |
3 | 3, 525 | 205, 225 | 56, 717 | 16.63 | 3, 600 | 161, 630 | 62, 662 | 28.86 | |
4 | 3, 101 | 157, 225 | 52, 039 | 13.70 | 3, 600 | 141, 340 | 57, 548 | 25.74 | |
5 | 2, 946 | 155, 164 | 46, 623 | 16.01 | 3, 600 | 115, 812 | 48, 889 | 21.65 | |
6 | 2, 996 | 110, 872 | 35, 309 | 14.68 | 3, 600 | 97, 424 | 39, 585 | 28.57 | |
7 | 2, 736 | 80, 850 | 23, 722 | 13.29 | 3, 600 | 60, 620 | 29, 355 | 40.19 | |
8 | 2, 548 | 50, 200 | 11, 109 | 4.86 | 3, 600 | 29, 562 | 14, 635 | 38.15 | |
9 | 2, 386 | 40, 485 | 6, 917 | 3.14 | 3, 600 | 30, 494 | 11, 311 | 68.67 | |
Average | 2, 928 | 197, 651 | 56, 644 | 14.76 | 3, 600 | 174, 059 | 63, 149 | 36.13 |
Case | multi-start ILS with LS1 | multi-start ILS with |
|||||||
Time | In. Obj. | Obj. | %Rel | Time | In. Obj. | Obj. | %Rel | ||
1 | 2, 904 | 731, 900 | 200, 646 | 28.01 | 3, 600 | 701, 192 | 221, 792 | 41.50 | |
2 | 3, 206 | 246, 939 | 76, 713 | 22.52 | 3, 600 | 228, 459 | 82, 564 | 31.87 | |
3 | 3, 525 | 205, 225 | 56, 717 | 16.63 | 3, 600 | 161, 630 | 62, 662 | 28.86 | |
4 | 3, 101 | 157, 225 | 52, 039 | 13.70 | 3, 600 | 141, 340 | 57, 548 | 25.74 | |
5 | 2, 946 | 155, 164 | 46, 623 | 16.01 | 3, 600 | 115, 812 | 48, 889 | 21.65 | |
6 | 2, 996 | 110, 872 | 35, 309 | 14.68 | 3, 600 | 97, 424 | 39, 585 | 28.57 | |
7 | 2, 736 | 80, 850 | 23, 722 | 13.29 | 3, 600 | 60, 620 | 29, 355 | 40.19 | |
8 | 2, 548 | 50, 200 | 11, 109 | 4.86 | 3, 600 | 29, 562 | 14, 635 | 38.15 | |
9 | 2, 386 | 40, 485 | 6, 917 | 3.14 | 3, 600 | 30, 494 | 11, 311 | 68.67 | |
Average | 2, 928 | 197, 651 | 56, 644 | 14.76 | 3, 600 | 174, 059 | 63, 149 | 36.13 |
Case | multi-start ILS with SLS | multi-start ILS with |
|||||||
Time | In. Obj. | Obj. | %Rel | Time | In. Obj. | Obj. | %Rel | ||
1 | 3, 600 | 704, 669 | 207, 738 | 32.53 | 3, 600 | 792, 732 | 229, 339 | 46.31 | |
2 | 3, 600 | 234, 564 | 76, 228 | 21.75 | 3, 600 | 242, 8870 | 80, 410 | 28.43 | |
3 | 3, 600 | 176, 729 | 55, 596 | 14.33 | 3, 600 | 149, 560 | 61, 766 | 27.01 | |
4 | 3, 600 | 165, 457 | 52, 054 | 13.73 | 3, 600 | 139, 882 | 56, 795 | 24.09 | |
5 | 3, 600 | 141, 360 | 45, 025 | 12.03 | 3, 600 | 112, 386 | 50, 796 | 26.39 | |
6 | 3, 600 | 112, 397 | 35, 946 | 16.75 | 3, 600 | 109, 447 | 43, 065 | 39.87 | |
7 | 3, 600 | 78, 051 | 23, 870 | 13.99 | 3, 600 | 59, 009 | 29, 110 | 39.02 | |
8 | 3, 600 | 57, 798 | 11, 192 | 5.64 | 3, 600 | 30, 803 | 15, 081 | 42.35 | |
9 | 3, 600 | 35, 437 | 6, 903 | 2.93 | 3, 600 | 28, 582 | 9, 512 | 41.85 | |
Average | 3, 600 | 189, 607 | 57, 173 | 14.85 | 3, 600 | 185, 030 | 63, 986 | 35.04 |
Case | multi-start ILS with SLS | multi-start ILS with |
|||||||
Time | In. Obj. | Obj. | %Rel | Time | In. Obj. | Obj. | %Rel | ||
1 | 3, 600 | 704, 669 | 207, 738 | 32.53 | 3, 600 | 792, 732 | 229, 339 | 46.31 | |
2 | 3, 600 | 234, 564 | 76, 228 | 21.75 | 3, 600 | 242, 8870 | 80, 410 | 28.43 | |
3 | 3, 600 | 176, 729 | 55, 596 | 14.33 | 3, 600 | 149, 560 | 61, 766 | 27.01 | |
4 | 3, 600 | 165, 457 | 52, 054 | 13.73 | 3, 600 | 139, 882 | 56, 795 | 24.09 | |
5 | 3, 600 | 141, 360 | 45, 025 | 12.03 | 3, 600 | 112, 386 | 50, 796 | 26.39 | |
6 | 3, 600 | 112, 397 | 35, 946 | 16.75 | 3, 600 | 109, 447 | 43, 065 | 39.87 | |
7 | 3, 600 | 78, 051 | 23, 870 | 13.99 | 3, 600 | 59, 009 | 29, 110 | 39.02 | |
8 | 3, 600 | 57, 798 | 11, 192 | 5.64 | 3, 600 | 30, 803 | 15, 081 | 42.35 | |
9 | 3, 600 | 35, 437 | 6, 903 | 2.93 | 3, 600 | 28, 582 | 9, 512 | 41.85 | |
Average | 3, 600 | 189, 607 | 57, 173 | 14.85 | 3, 600 | 185, 030 | 63, 986 | 35.04 |
Tables | EQUAL | Winner | ||
7 | 1 | 6 | 2 | MIPM + ILS + |
8 | 1 | 6 | 2 | MIPM + ILS + |
9 | 9 | 0 | 0 | multi-start ILS + |
10 | 9 | 0 | 0 | multi-start ILS + |
Tables | EQUAL | Winner | ||
7 | 1 | 6 | 2 | MIPM + ILS + |
8 | 1 | 6 | 2 | MIPM + ILS + |
9 | 9 | 0 | 0 | multi-start ILS + |
10 | 9 | 0 | 0 | multi-start ILS + |
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