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Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem
Tabu search guided by reinforcement learning for the max-mean dispersion problem
1. | School of Management, Northwestern Polytechnical University, 710072, Xi'an, China |
2. | Department of Computer science and Technology, Xidian University, 710071, Xi'an, China |
We present an effective hybrid metaheuristic of integrating reinforcement learning with a tabu-search (RLTS) algorithm for solving the max–mean dispersion problem. The innovative element is to design using a knowledge strategy from the $ Q $-learning mechanism to locate promising regions when the tabu search is stuck in a local optimum. Computational experiments on extensive benchmarks show that the RLTS performs much better than state-of-the-art algorithms in the literature. From a total of 100 benchmark instances, in 60 of them, which ranged from 500 to 1, 000, our proposed algorithm matched the currently best lower bounds for all instances. For the remaining 40 instances, the algorithm matched or outperformed. Furthermore, additional support was applied to present the effectiveness of the combined RL technique. The analysis sheds light on the effectiveness of the proposed RLTS algorithm.
References:
[1] |
R. Aringhieri, R. Cordone and A. Grosso,
Construction and improvement algorithms for dispersion problems, European J. Oper. Res., 242 (2015), 21-33.
doi: 10.1016/j.ejor.2014.09.058. |
[2] |
R. Aringhieri and R. Cordone,
Comparing local search metaheuristics for the maximum diversity problem, J. Oper. Res. Soc., 62 (2011), 266-280.
doi: 10.1057/jors.2010.104. |
[3] |
J. Boyan and A. W. Moore,
Learning evaluation functions to improve optimization by local search, J. Machine Learning Research, 1 (2000), 77-112.
doi: 10.1162/15324430152733124. |
[4] |
J. Brimberg, N. Mladenović, R. Todosijević and D. Urošević,
Less is more: Solving the max-mean diversity problem with variable neighborhood search, Information Sciences, 382 (2017), 179-200.
doi: 10.1016/j.ins.2016.12.021. |
[5] |
E. K. Burke, G. Kendall and E. Soubeiga,
A tabu-search hyperheuristic for timetabling and rostering, J. Heuristics, 9 (2003), 451-470.
doi: 10.1023/B:HEUR.0000012446.94732.b6. |
[6] |
R. Carrasco, A. Pham, M. Gallego, F. Gortázar, R. Martí and A. Duarte,
Tabu search for
the Max–Mean Dispersion Problem, Knowledge-Based Systems, 85 (2015), 256-264.
doi: 10.1016/j.knosys.2015.05.011. |
[7] |
F. C. De Lima Júnior, A. D. D. Neto and J. D. De Melo, Hybrid metaheuristics using reinforcement learning applied to salesman traveling problem, in Traveling Salesman Problem, Theory and Applications, IntechOpen, 2010.
doi: 10.5772/13343. |
[8] |
F. Della Croce, M. Garraffa and F. Salassa, A hybrid heuristic approach based on a quadratic knapsack formulation for the max-mean dispersion problem, in Combinatorial Optimization, Lecture Notes in Comput. Sci., 8596, Springer, Cham, 2014,186–194.
doi: 10.1007/978-3-319-09174-7_16. |
[9] |
F. Della Croce, M. Garraffa and F. Salassa,
A hybrid three-phase approach for the max-mean dispersion problem, Comput. Oper. Res., 71 (2016), 16-22.
doi: 10.1016/j.cor.2016.01.003. |
[10] |
F. Della Croce, A. Grosso and M. Locatelli,
A heuristic approach for the max-min diversity problem based on max-clique, Comput. Oper. Res., 36 (2009), 2429-2433.
doi: 10.1016/j.cor.2008.09.007. |
[11] |
P. Galinier, Z. Boujbel and M. Coutinho Fernandes,
An efficient memetic algorithm for the graph partitioning problem, Ann. Oper. Res., 191 (2011), 1-22.
doi: 10.1007/s10479-011-0983-3. |
[12] |
M. Garraffa, F. Della Croce and F. Salassa,
An exact semidefinite programming approach for the max-mean dispersion problem, J. Comb. Optim., 34 (2017), 71-93.
doi: 10.1007/s10878-016-0065-1. |
[13] |
A. Gosavi,
Reinforcement learning: A tutorial survey and recent advances, INFORMS J. Comput., 21 (2009), 178-192.
doi: 10.1287/ijoc.1080.0305. |
[14] |
X. Lai, D. Yue, J.-K. Hao and F. Glover,
Solution-based tabu search for the maximum min-sum dispersion problem, Inform. Sci., 441 (2018), 79-94.
doi: 10.1016/j.ins.2018.02.006. |
[15] |
X. Lai and J. K. Hao,
A tabu search based memetic algorithm for the max-mean dispersion problem, Comput. Oper. Res., 72 (2016), 118-127.
doi: 10.1016/j.cor.2016.02.016. |
[16] |
P. Larranaga, A review on estimation of distribution algorithms, in Estimation of Distribution Algorithmn, Genetic Algorithms and Evolutionary Computation, 2, Springer, Boston, 2002, 57–100.
doi: 10.1007/978-1-4615-1539-5_3. |
[17] |
Z. Lu, F. Glover and J.-K. Hao, Neighborhood combination for unconstrained binary quadratic programming, MIC 2009: The VIII Metaheuristics International Conference, Hamburg, Germany, 2009. |
[18] |
R. Martí and F. Sandoya,
GRASP and path relinking for the equitable dispersion problem, Comput. Oper. Res., 40 (2013), 3091-3099.
doi: 10.1016/j.cor.2012.04.005. |
[19] |
V. V. Miagkikh and W. F. Punch, Global search in combinatorial optimization using reinforcement learning algorithms, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, Washington, DC, 1999.
doi: 10.1109/CEC.1999.781925. |
[20] |
D. Nijimbere, S. Zhao, H. Liu, B. Peng and A. Zhang, A hybrid metaheuristic of integrating estimation of distribution algorithm with tabu search for the max-mean dispersion problem, Math. Probl. Eng., 2019 (2019), 16pp.
doi: 10.1155/2019/7104702. |
[21] |
D. C. Porumbel, J.-K. Hao and F. Glover,
A simple and effective algorithm for the MaxMin diversity problem, Ann. Oper. Res., 186 (2011), 275-293.
doi: 10.1007/s10479-011-0898-z. |
[22] |
O. A. Prokopyev, N. Kong and and D. L. Martinez-Torres,
The equitable dispersion problem, European J. Oper. Res., 197 (2009), 59-67.
doi: 10.1016/j.ejor.2008.06.005. |
[23] |
A. P. Punnen, S. Taghipour, D. Karapetyan and B. Bhattacharyya,
The quadratic balanced optimization problem, Discrete Optim., 12 (2014), 47-60.
doi: 10.1016/j.disopt.2014.01.001. |
[24] |
I. Sghir, J. K. Hao, I. B. Jaafar and K. Ghédira,
A multi-agent based optimization method applied to the quadratic assignment problem, Expert Systems Appl., 42 (2015), 9252-9262.
doi: 10.1016/j.eswa.2015.07.070. |
[25] |
J. A. Torkestani and M. R. Meybodi,
A cellular learning automata-based algorithm for solving the vertex coloring problem, Expert Systems Appl., 38 (2011), 9237-9247.
doi: 10.1016/j.eswa.2011.01.098. |
[26] |
Y. Wang, Q. Wu and F. Glover,
Effective metaheuristic algorithms for the minimum differential dispersion problem, European J. Oper. Res., 258 (2017), 829-843.
doi: 10.1016/j.ejor.2016.10.035. |
[27] |
Y. Wang, J.-K. Hao, F. Glover and Z. Lü,
A tabu search based memetic algorithm for the maximum diversity problem, Engineering Appl. Artificial Intell., 27 (2014), 103-114.
doi: 10.1016/j.engappai.2013.09.005. |
[28] |
Y. Xu, D. Stern and H. Samulowitz, Learning adaptation to solve constraint satisfaction problems. Available from: https://www.microsoft.com/en-us/research/wp-content/uploads/2009/01/lion2009.pdf. |
[29] |
T. Yu and W.-G. Zhen,
A multi-step $ Q(\lambda)$ learning approach to power system stabilizer, IFAC Proceedings Volumes, 43 (2010), 220-224.
doi: 10.3182/20100826-3-tr-4015.00042. |
[30] |
Y. Zhou, J.-K. Hao and and B. Duval,
Reinforcement learning based local search for grouping problems: A case study on graph coloring, Expert Systems Appl., 64 (2016), 412-422.
doi: 10.1016/j.eswa.2016.07.047. |
show all references
References:
[1] |
R. Aringhieri, R. Cordone and A. Grosso,
Construction and improvement algorithms for dispersion problems, European J. Oper. Res., 242 (2015), 21-33.
doi: 10.1016/j.ejor.2014.09.058. |
[2] |
R. Aringhieri and R. Cordone,
Comparing local search metaheuristics for the maximum diversity problem, J. Oper. Res. Soc., 62 (2011), 266-280.
doi: 10.1057/jors.2010.104. |
[3] |
J. Boyan and A. W. Moore,
Learning evaluation functions to improve optimization by local search, J. Machine Learning Research, 1 (2000), 77-112.
doi: 10.1162/15324430152733124. |
[4] |
J. Brimberg, N. Mladenović, R. Todosijević and D. Urošević,
Less is more: Solving the max-mean diversity problem with variable neighborhood search, Information Sciences, 382 (2017), 179-200.
doi: 10.1016/j.ins.2016.12.021. |
[5] |
E. K. Burke, G. Kendall and E. Soubeiga,
A tabu-search hyperheuristic for timetabling and rostering, J. Heuristics, 9 (2003), 451-470.
doi: 10.1023/B:HEUR.0000012446.94732.b6. |
[6] |
R. Carrasco, A. Pham, M. Gallego, F. Gortázar, R. Martí and A. Duarte,
Tabu search for
the Max–Mean Dispersion Problem, Knowledge-Based Systems, 85 (2015), 256-264.
doi: 10.1016/j.knosys.2015.05.011. |
[7] |
F. C. De Lima Júnior, A. D. D. Neto and J. D. De Melo, Hybrid metaheuristics using reinforcement learning applied to salesman traveling problem, in Traveling Salesman Problem, Theory and Applications, IntechOpen, 2010.
doi: 10.5772/13343. |
[8] |
F. Della Croce, M. Garraffa and F. Salassa, A hybrid heuristic approach based on a quadratic knapsack formulation for the max-mean dispersion problem, in Combinatorial Optimization, Lecture Notes in Comput. Sci., 8596, Springer, Cham, 2014,186–194.
doi: 10.1007/978-3-319-09174-7_16. |
[9] |
F. Della Croce, M. Garraffa and F. Salassa,
A hybrid three-phase approach for the max-mean dispersion problem, Comput. Oper. Res., 71 (2016), 16-22.
doi: 10.1016/j.cor.2016.01.003. |
[10] |
F. Della Croce, A. Grosso and M. Locatelli,
A heuristic approach for the max-min diversity problem based on max-clique, Comput. Oper. Res., 36 (2009), 2429-2433.
doi: 10.1016/j.cor.2008.09.007. |
[11] |
P. Galinier, Z. Boujbel and M. Coutinho Fernandes,
An efficient memetic algorithm for the graph partitioning problem, Ann. Oper. Res., 191 (2011), 1-22.
doi: 10.1007/s10479-011-0983-3. |
[12] |
M. Garraffa, F. Della Croce and F. Salassa,
An exact semidefinite programming approach for the max-mean dispersion problem, J. Comb. Optim., 34 (2017), 71-93.
doi: 10.1007/s10878-016-0065-1. |
[13] |
A. Gosavi,
Reinforcement learning: A tutorial survey and recent advances, INFORMS J. Comput., 21 (2009), 178-192.
doi: 10.1287/ijoc.1080.0305. |
[14] |
X. Lai, D. Yue, J.-K. Hao and F. Glover,
Solution-based tabu search for the maximum min-sum dispersion problem, Inform. Sci., 441 (2018), 79-94.
doi: 10.1016/j.ins.2018.02.006. |
[15] |
X. Lai and J. K. Hao,
A tabu search based memetic algorithm for the max-mean dispersion problem, Comput. Oper. Res., 72 (2016), 118-127.
doi: 10.1016/j.cor.2016.02.016. |
[16] |
P. Larranaga, A review on estimation of distribution algorithms, in Estimation of Distribution Algorithmn, Genetic Algorithms and Evolutionary Computation, 2, Springer, Boston, 2002, 57–100.
doi: 10.1007/978-1-4615-1539-5_3. |
[17] |
Z. Lu, F. Glover and J.-K. Hao, Neighborhood combination for unconstrained binary quadratic programming, MIC 2009: The VIII Metaheuristics International Conference, Hamburg, Germany, 2009. |
[18] |
R. Martí and F. Sandoya,
GRASP and path relinking for the equitable dispersion problem, Comput. Oper. Res., 40 (2013), 3091-3099.
doi: 10.1016/j.cor.2012.04.005. |
[19] |
V. V. Miagkikh and W. F. Punch, Global search in combinatorial optimization using reinforcement learning algorithms, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, Washington, DC, 1999.
doi: 10.1109/CEC.1999.781925. |
[20] |
D. Nijimbere, S. Zhao, H. Liu, B. Peng and A. Zhang, A hybrid metaheuristic of integrating estimation of distribution algorithm with tabu search for the max-mean dispersion problem, Math. Probl. Eng., 2019 (2019), 16pp.
doi: 10.1155/2019/7104702. |
[21] |
D. C. Porumbel, J.-K. Hao and F. Glover,
A simple and effective algorithm for the MaxMin diversity problem, Ann. Oper. Res., 186 (2011), 275-293.
doi: 10.1007/s10479-011-0898-z. |
[22] |
O. A. Prokopyev, N. Kong and and D. L. Martinez-Torres,
The equitable dispersion problem, European J. Oper. Res., 197 (2009), 59-67.
doi: 10.1016/j.ejor.2008.06.005. |
[23] |
A. P. Punnen, S. Taghipour, D. Karapetyan and B. Bhattacharyya,
The quadratic balanced optimization problem, Discrete Optim., 12 (2014), 47-60.
doi: 10.1016/j.disopt.2014.01.001. |
[24] |
I. Sghir, J. K. Hao, I. B. Jaafar and K. Ghédira,
A multi-agent based optimization method applied to the quadratic assignment problem, Expert Systems Appl., 42 (2015), 9252-9262.
doi: 10.1016/j.eswa.2015.07.070. |
[25] |
J. A. Torkestani and M. R. Meybodi,
A cellular learning automata-based algorithm for solving the vertex coloring problem, Expert Systems Appl., 38 (2011), 9237-9247.
doi: 10.1016/j.eswa.2011.01.098. |
[26] |
Y. Wang, Q. Wu and F. Glover,
Effective metaheuristic algorithms for the minimum differential dispersion problem, European J. Oper. Res., 258 (2017), 829-843.
doi: 10.1016/j.ejor.2016.10.035. |
[27] |
Y. Wang, J.-K. Hao, F. Glover and Z. Lü,
A tabu search based memetic algorithm for the maximum diversity problem, Engineering Appl. Artificial Intell., 27 (2014), 103-114.
doi: 10.1016/j.engappai.2013.09.005. |
[28] |
Y. Xu, D. Stern and H. Samulowitz, Learning adaptation to solve constraint satisfaction problems. Available from: https://www.microsoft.com/en-us/research/wp-content/uploads/2009/01/lion2009.pdf. |
[29] |
T. Yu and W.-G. Zhen,
A multi-step $ Q(\lambda)$ learning approach to power system stabilizer, IFAC Proceedings Volumes, 43 (2010), 220-224.
doi: 10.3182/20100826-3-tr-4015.00042. |
[30] |
Y. Zhou, J.-K. Hao and and B. Duval,
Reinforcement learning based local search for grouping problems: A case study on graph coloring, Expert Systems Appl., 64 (2016), 412-422.
doi: 10.1016/j.eswa.2016.07.047. |




Instance( | GRASP-PR[18] | HH[8] | HHP[9] | TP-TS[6] | MAMMDP[15] | EDA[16] | RLTS | |||
MDPI1(500) | 78.61 | 81.25 | 81.28 | 81.28 | 81.2770 | 81.2770 | 81.2770 | 81.2770 | 81.2770 | 81.2770 |
MDPI2(500) | 76.87 | 77.45 | 77.79 | 77.60 | 78.6102 | 78.6102 | 78.6102 | 78.6102 | 78.6102 | 78.6102 |
MDPI3(500) | 75.69 | 75.31 | 76.30 | 75.65 | 76.3008 | 76.3008 | 76.3008 | 76.3008 | 76.3008 | 76.3008 |
MDPI4(500) | 81.81 | 82.28 | 82.33 | 81.47 | 82.3321 | 82.3321 | 82.3321 | 82.3321 | 82.3321 | 82.3321 |
MDPI5(500) | 78.57 | 80.01 | 80.08 | 79.92 | 80.3540 | 80.3540 | 80.3540 | 80.3540 | 80.3540 | 80.3540 |
MDPI6(500) | 79.64 | 81.12 | 81.25 | 79.93 | 81.2486 | 81.2486 | 81.2486 | 81.2486 | 81.2486 | 81.2486 |
MDPI7(500) | 75.50 | 78.09 | 78.16 | 77.71 | 78.1645 | 78.1645 | 78.1645 | 78.1645 | 78.1645 | 78.1645 |
MDPI8(500) | 76.98 | 79.01 | 79.06 | 78.70 | 79.1399 | 79.1399 | 79.1399 | 79.1399 | 79.1399 | 79.1399 |
MDPI9(500) | 75.72 | 76.98 | 77.36 | 77.15 | 77.4210 | 77.4210 | 77.4210 | 77.4210 | 77.4210 | 77.4210 |
MDPI10(500) | 80.38 | 81.24 | 81.25 | 81.02 | 81.3099 | 81.3099 | 81.3099 | 81.3099 | 81.3099 | 81.3099 |
MDPII1(500) | 108.15 | 109.16 | 109.38 | 109.33 | 109.6101 | 109.6101 | 109.6101 | 109.6101 | 109.6101 | 109.6101 |
MDPII2(500) | 103.29 | 105.06 | 105.33 | 104.81 | 105.7175 | 105.7175 | 105.7175 | 105.7175 | 105.7175 | 105.7175 |
MDPII3(500) | 106.30 | 107.64 | 107.79 | 107.18 | 107.8217 | 107.8217 | 107.8217 | 107.8217 | 107.8217 | 107.8217 |
MDPII4(500) | 104.62 | 105.37 | 106.10 | 105.69 | 106.1001 | 106.1001 | 106.1001 | 106.1001 | 106.1001 | 106.1001 |
MDPII5(500) | 103.61 | 106.37 | 106.55 | 106.59 | 106.8572 | 106.8572 | 106.8572 | 106.8572 | 106.8572 | 106.8572 |
MDPII6(500) | 104.81 | 105.52 | 105.77 | 106.17 | 106.2980 | 106.2980 | 106.2980 | 106.2980 | 106.2980 | 106.2980 |
MDPII7(500) | 104.50 | 106.61 | 107.06 | 106.92 | 107.1494 | 107.1494 | 107.1494 | 107.1494 | 107.1494 | 107.1494 |
MDPII8(500) | 100.02 | 103.41 | 103.78 | 103.49 | 103.7792 | 103.7792 | 103.7792 | 103.7792 | 103.7792 | 103.7792 |
MDPII9(500) | 104.93 | 106.20 | 106.24 | 105.97 | 106.6198 | 106.6198 | 106.6198 | 106.6198 | 106.6198 | 106.6198 |
MDPII10(500) | 103.50 | 103.79 | 104.15 | 103.56 | 104.6515 | 104.6515 | 104.6515 | 104.6515 | 104.6515 | 104.6515 |
MDPI1(750) | – | – | – | 95.86 | 96.6507 | 96.6507 | 96.6507 | 96.6507 | 96.6507 | 96.6507 |
MDPI2(750) | – | – | – | 97.42 | 97.5649 | 97.5649 | 97.5649 | 97.5649 | 97.5649 | 97.5649 |
MDPI3(750) | – | – | – | 96.97 | 97.7989 | 97.7989 | 97.7989 | 97.7989 | 97.7989 | 97.7989 |
MDPI4(750) | – | – | – | 95.21 | 96.0414 | 96.0414 | 96.0414 | 96.0414 | 96.0414 | 96.0414 |
MDPI5(750) | – | – | – | 96.65 | 96.7619 | 96.7619 | 96.7619 | 96.7619 | 96.7619 | 96.7619 |
MDPI6(750) | – | – | – | 99.25 | 99.8613 | 99.8613 | 99.8613 | 99.8613 | 99.8613 | 99.8613 |
MDPI7(750) | – | – | – | 96.26 | 96.5454 | 96.5454 | 96.5454 | 96.5454 | 96.5454 | 96.5454 |
MDPI8(750) | – | – | – | 96.46 | 96.7270 | 96.7270 | 96.7270 | 96.7270 | 96.7270 | 96.7270 |
MDPI9(750) | – | – | – | 96.78 | 98.0584 | 98.0584 | 98.0584 | 98.0584 | 98.0584 | 98.0584 |
MDPI10(750) | – | – | – | 99.85 | 100.0642 | 100.0642 | 100.0642 | 100.0642 | 100.0642 | 100.0642 |
MDPII1(750) | – | – | – | 127.69 | 128.8637 | 128.8637 | 128.8637 | 128.8637 | 128.8637 | 128.8637 |
MDPII2(750) | – | – | – | 130.79 | 130.9544 | 130.9544 | 130.9544 | 130.9544 | 130.9544 | 130.9544 |
MDPII3(750) | – | – | – | 129.40 | 129.7825 | 129.7825 | 129.7825 | 129.7825 | 129.7825 | 129.7825 |
MDPII4(750) | – | – | – | 125.68 | 126.5823 | 126.5823 | 126.5823 | 126.5823 | 126.5823 | 126.5823 |
MDPII5(750) | – | – | – | 128.13 | 129.1229 | 129.1229 | 129.1229 | 129.1229 | 129.1229 | 129.1229 |
MDPII6(750) | – | – | – | 128.55 | 129.0252 | 129.0252 | 129.0252 | 129.0252 | 129.0252 | 129.0252 |
MDPII7(750) | – | – | – | 124.91 | 125.6467 | 125.6467 | 125.6467 | 125.6467 | 125.6467 | 125.6467 |
MDPII8(750) | – | – | – | 130.66 | 130.9405 | 130.9405 | 130.9405 | 130.9405 | 130.9405 | 130.9405 |
MDPII9(750) | – | – | – | 128.89 | 128.8899 | 128.8899 | 128.8899 | 128.8899 | 128.8899 | 128.8899 |
MDPII10(750) | – | – | – | 132.99 | 133.2653 | 133.2653 | 133.2653 | 133.2653 | 133.2653 | 133.2653 |
MDPI1(1, 000) | – | – | – | 118.76 | 119.1741 | 119.1741 | 119.1741 | 119.1741 | 119.1741 | 119.1741 |
MDPI2(1, 000) | – | – | – | 113.22 | 113.5248 | 113.5248 | 113.5248 | 113.5248 | 113.5248 | 113.5248 |
MDPI3(1, 000) | – | – | – | 114.51 | 115.1386 | 115.1386 | 115.1386 | 115.1386 | 115.1386 | 115.1386 |
MDPI4(1, 000) | – | – | – | 110.53 | 111.1504 | 111.1504 | 111.1504 | 111.1504 | 111.1504 | 111.1504 |
MDPI5(1, 000) | – | – | – | 111.24 | 112.7232 | 112.7232 | 112.7232 | 112.7232 | 112.7232 | 112.7232 |
MDPI6(1, 000) | – | – | – | 112.08 | 113.1987 | 113.1987 | 113.1987 | 113.1987 | 113.1987 | 113.1987 |
MDPI7(1, 000) | – | – | – | 110.94 | 111.5555 | 111.5555 | 111.5555 | 111.5555 | 111.5555 | 111.5555 |
MDPI8(1, 000) | – | – | – | 110.29 | 111.2632 | 111.2632 | 111.2632 | 111.2632 | 111.2632 | 111.2632 |
MDPI9(1, 000) | – | – | – | 115.78 | 115.9588 | 115.9588 | 115.9588 | 115.9588 | 115.9588 | 115.9588 |
MDPI10(1, 000) | – | – | – | 114.29 | 114.7316 | 114.7316 | 114.7316 | 114.7316 | 114.7316 | 114.7316 |
MDPII1(1, 000) | – | – | – | 145.46 | 147.9362 | 147.9362 | 147.9362 | 147.9362 | 147.9362 | 147.9362 |
MDPII2(1, 000) | – | – | – | 150.49 | 151.3800 | 151.3800 | 151.3800 | 151.3800 | 151.3800 | 151.3800 |
MDPII3(1, 000) | – | – | – | 149.36 | 150.7882 | 150.7882 | 150.7882 | 150.7882 | 150.7882 | 150.7882 |
MDPII4(1, 000) | – | – | – | 147.91 | 149.1780 | 149.1780 | 149.1780 | 149.1780 | 149.1780 | 149.1780 |
MDPII5(1, 000) | – | – | – | 150.23 | 151.5208 | 151.5208 | 151.5208 | 151.5208 | 151.5208 | 151.5208 |
MDPII6(1, 000) | – | – | – | 147.29 | 148.3434 | 148.3434 | 148.3434 | 148.3434 | 148.3434 | 148.3434 |
MDPII7(1, 000) | – | – | – | 148.41 | 148.7424 | 148.7424 | 148.7424 | 148.7424 | 148.7424 | 148.7424 |
MDPII8(1, 000) | – | – | – | 145.87 | 147.8268 | 147.8268 | 147.8268 | 147.8268 | 147.8268 | 147.8268 |
MDPII9(1, 000) | – | – | – | 145.67 | 147.0839 | 147.0839 | 147.0839 | 147.0839 | 147.0839 | 147.0839 |
MDPII10(1, 000) | – | – | – | 148.40 | 150.0461 | 150.0461 | 150.0461 | 150.0461 | 150.0461 | 150.0461 |
Instance( | GRASP-PR[18] | HH[8] | HHP[9] | TP-TS[6] | MAMMDP[15] | EDA[16] | RLTS | |||
MDPI1(500) | 78.61 | 81.25 | 81.28 | 81.28 | 81.2770 | 81.2770 | 81.2770 | 81.2770 | 81.2770 | 81.2770 |
MDPI2(500) | 76.87 | 77.45 | 77.79 | 77.60 | 78.6102 | 78.6102 | 78.6102 | 78.6102 | 78.6102 | 78.6102 |
MDPI3(500) | 75.69 | 75.31 | 76.30 | 75.65 | 76.3008 | 76.3008 | 76.3008 | 76.3008 | 76.3008 | 76.3008 |
MDPI4(500) | 81.81 | 82.28 | 82.33 | 81.47 | 82.3321 | 82.3321 | 82.3321 | 82.3321 | 82.3321 | 82.3321 |
MDPI5(500) | 78.57 | 80.01 | 80.08 | 79.92 | 80.3540 | 80.3540 | 80.3540 | 80.3540 | 80.3540 | 80.3540 |
MDPI6(500) | 79.64 | 81.12 | 81.25 | 79.93 | 81.2486 | 81.2486 | 81.2486 | 81.2486 | 81.2486 | 81.2486 |
MDPI7(500) | 75.50 | 78.09 | 78.16 | 77.71 | 78.1645 | 78.1645 | 78.1645 | 78.1645 | 78.1645 | 78.1645 |
MDPI8(500) | 76.98 | 79.01 | 79.06 | 78.70 | 79.1399 | 79.1399 | 79.1399 | 79.1399 | 79.1399 | 79.1399 |
MDPI9(500) | 75.72 | 76.98 | 77.36 | 77.15 | 77.4210 | 77.4210 | 77.4210 | 77.4210 | 77.4210 | 77.4210 |
MDPI10(500) | 80.38 | 81.24 | 81.25 | 81.02 | 81.3099 | 81.3099 | 81.3099 | 81.3099 | 81.3099 | 81.3099 |
MDPII1(500) | 108.15 | 109.16 | 109.38 | 109.33 | 109.6101 | 109.6101 | 109.6101 | 109.6101 | 109.6101 | 109.6101 |
MDPII2(500) | 103.29 | 105.06 | 105.33 | 104.81 | 105.7175 | 105.7175 | 105.7175 | 105.7175 | 105.7175 | 105.7175 |
MDPII3(500) | 106.30 | 107.64 | 107.79 | 107.18 | 107.8217 | 107.8217 | 107.8217 | 107.8217 | 107.8217 | 107.8217 |
MDPII4(500) | 104.62 | 105.37 | 106.10 | 105.69 | 106.1001 | 106.1001 | 106.1001 | 106.1001 | 106.1001 | 106.1001 |
MDPII5(500) | 103.61 | 106.37 | 106.55 | 106.59 | 106.8572 | 106.8572 | 106.8572 | 106.8572 | 106.8572 | 106.8572 |
MDPII6(500) | 104.81 | 105.52 | 105.77 | 106.17 | 106.2980 | 106.2980 | 106.2980 | 106.2980 | 106.2980 | 106.2980 |
MDPII7(500) | 104.50 | 106.61 | 107.06 | 106.92 | 107.1494 | 107.1494 | 107.1494 | 107.1494 | 107.1494 | 107.1494 |
MDPII8(500) | 100.02 | 103.41 | 103.78 | 103.49 | 103.7792 | 103.7792 | 103.7792 | 103.7792 | 103.7792 | 103.7792 |
MDPII9(500) | 104.93 | 106.20 | 106.24 | 105.97 | 106.6198 | 106.6198 | 106.6198 | 106.6198 | 106.6198 | 106.6198 |
MDPII10(500) | 103.50 | 103.79 | 104.15 | 103.56 | 104.6515 | 104.6515 | 104.6515 | 104.6515 | 104.6515 | 104.6515 |
MDPI1(750) | – | – | – | 95.86 | 96.6507 | 96.6507 | 96.6507 | 96.6507 | 96.6507 | 96.6507 |
MDPI2(750) | – | – | – | 97.42 | 97.5649 | 97.5649 | 97.5649 | 97.5649 | 97.5649 | 97.5649 |
MDPI3(750) | – | – | – | 96.97 | 97.7989 | 97.7989 | 97.7989 | 97.7989 | 97.7989 | 97.7989 |
MDPI4(750) | – | – | – | 95.21 | 96.0414 | 96.0414 | 96.0414 | 96.0414 | 96.0414 | 96.0414 |
MDPI5(750) | – | – | – | 96.65 | 96.7619 | 96.7619 | 96.7619 | 96.7619 | 96.7619 | 96.7619 |
MDPI6(750) | – | – | – | 99.25 | 99.8613 | 99.8613 | 99.8613 | 99.8613 | 99.8613 | 99.8613 |
MDPI7(750) | – | – | – | 96.26 | 96.5454 | 96.5454 | 96.5454 | 96.5454 | 96.5454 | 96.5454 |
MDPI8(750) | – | – | – | 96.46 | 96.7270 | 96.7270 | 96.7270 | 96.7270 | 96.7270 | 96.7270 |
MDPI9(750) | – | – | – | 96.78 | 98.0584 | 98.0584 | 98.0584 | 98.0584 | 98.0584 | 98.0584 |
MDPI10(750) | – | – | – | 99.85 | 100.0642 | 100.0642 | 100.0642 | 100.0642 | 100.0642 | 100.0642 |
MDPII1(750) | – | – | – | 127.69 | 128.8637 | 128.8637 | 128.8637 | 128.8637 | 128.8637 | 128.8637 |
MDPII2(750) | – | – | – | 130.79 | 130.9544 | 130.9544 | 130.9544 | 130.9544 | 130.9544 | 130.9544 |
MDPII3(750) | – | – | – | 129.40 | 129.7825 | 129.7825 | 129.7825 | 129.7825 | 129.7825 | 129.7825 |
MDPII4(750) | – | – | – | 125.68 | 126.5823 | 126.5823 | 126.5823 | 126.5823 | 126.5823 | 126.5823 |
MDPII5(750) | – | – | – | 128.13 | 129.1229 | 129.1229 | 129.1229 | 129.1229 | 129.1229 | 129.1229 |
MDPII6(750) | – | – | – | 128.55 | 129.0252 | 129.0252 | 129.0252 | 129.0252 | 129.0252 | 129.0252 |
MDPII7(750) | – | – | – | 124.91 | 125.6467 | 125.6467 | 125.6467 | 125.6467 | 125.6467 | 125.6467 |
MDPII8(750) | – | – | – | 130.66 | 130.9405 | 130.9405 | 130.9405 | 130.9405 | 130.9405 | 130.9405 |
MDPII9(750) | – | – | – | 128.89 | 128.8899 | 128.8899 | 128.8899 | 128.8899 | 128.8899 | 128.8899 |
MDPII10(750) | – | – | – | 132.99 | 133.2653 | 133.2653 | 133.2653 | 133.2653 | 133.2653 | 133.2653 |
MDPI1(1, 000) | – | – | – | 118.76 | 119.1741 | 119.1741 | 119.1741 | 119.1741 | 119.1741 | 119.1741 |
MDPI2(1, 000) | – | – | – | 113.22 | 113.5248 | 113.5248 | 113.5248 | 113.5248 | 113.5248 | 113.5248 |
MDPI3(1, 000) | – | – | – | 114.51 | 115.1386 | 115.1386 | 115.1386 | 115.1386 | 115.1386 | 115.1386 |
MDPI4(1, 000) | – | – | – | 110.53 | 111.1504 | 111.1504 | 111.1504 | 111.1504 | 111.1504 | 111.1504 |
MDPI5(1, 000) | – | – | – | 111.24 | 112.7232 | 112.7232 | 112.7232 | 112.7232 | 112.7232 | 112.7232 |
MDPI6(1, 000) | – | – | – | 112.08 | 113.1987 | 113.1987 | 113.1987 | 113.1987 | 113.1987 | 113.1987 |
MDPI7(1, 000) | – | – | – | 110.94 | 111.5555 | 111.5555 | 111.5555 | 111.5555 | 111.5555 | 111.5555 |
MDPI8(1, 000) | – | – | – | 110.29 | 111.2632 | 111.2632 | 111.2632 | 111.2632 | 111.2632 | 111.2632 |
MDPI9(1, 000) | – | – | – | 115.78 | 115.9588 | 115.9588 | 115.9588 | 115.9588 | 115.9588 | 115.9588 |
MDPI10(1, 000) | – | – | – | 114.29 | 114.7316 | 114.7316 | 114.7316 | 114.7316 | 114.7316 | 114.7316 |
MDPII1(1, 000) | – | – | – | 145.46 | 147.9362 | 147.9362 | 147.9362 | 147.9362 | 147.9362 | 147.9362 |
MDPII2(1, 000) | – | – | – | 150.49 | 151.3800 | 151.3800 | 151.3800 | 151.3800 | 151.3800 | 151.3800 |
MDPII3(1, 000) | – | – | – | 149.36 | 150.7882 | 150.7882 | 150.7882 | 150.7882 | 150.7882 | 150.7882 |
MDPII4(1, 000) | – | – | – | 147.91 | 149.1780 | 149.1780 | 149.1780 | 149.1780 | 149.1780 | 149.1780 |
MDPII5(1, 000) | – | – | – | 150.23 | 151.5208 | 151.5208 | 151.5208 | 151.5208 | 151.5208 | 151.5208 |
MDPII6(1, 000) | – | – | – | 147.29 | 148.3434 | 148.3434 | 148.3434 | 148.3434 | 148.3434 | 148.3434 |
MDPII7(1, 000) | – | – | – | 148.41 | 148.7424 | 148.7424 | 148.7424 | 148.7424 | 148.7424 | 148.7424 |
MDPII8(1, 000) | – | – | – | 145.87 | 147.8268 | 147.8268 | 147.8268 | 147.8268 | 147.8268 | 147.8268 |
MDPII9(1, 000) | – | – | – | 145.67 | 147.0839 | 147.0839 | 147.0839 | 147.0839 | 147.0839 | 147.0839 |
MDPII10(1, 000) | – | – | – | 148.40 | 150.0461 | 150.0461 | 150.0461 | 150.0461 | 150.0461 | 150.0461 |
Instance |
TP-TS | MAMMDP | EDA | RLTS | ||||||||
$ f_{best} $ | $ f_{avg} $ | $ time $ | $ f_{best} $ | $ f_{avg} $ | time | $f_{best} $ | $ f_{avg} $ | $ time $ | ||||
MDPI1(3, 000) | 188.0953 | 189.049 | 189.049 | 88.36 | 189.049 | 189.049 | 69.35 | 189.049 | 189.049 | 47.45 | ||
MDPI2(3, 000) | 186.4730 | 187.3873 | 187.3873 | 60.71 | 187.3873 | 187.3873 | 53.64 | 187.3873 | 187.3873 | 66.65 | ||
MDPI3(3, 000) | 184.3414 | 185.6668 | 185.6551 | 352.85 | 185.6668 | 185.6453 | 399.44 | 185.6668 | 185.6622 | 426.7 | ||
MDPI4(3, 000) | 185.5882 | 186.1637 | 186.1536 | 300.37 | 186.1637 | 186.1637 | 240.45 | 186.1637 | 186.1637 | 137.8 | ||
MDPI5(3, 000) | 186.2349 | 187.5455 | 187.5455 | 61.29 | 187.5455 | 187.5455 | 94.16 | 187.5455 | 187.5455 | 54.23 | ||
MDPI6(3, 000) | 189.0935 | 189.4313 | 189.4313 | 51.99 | 189.4313 | 189.4313 | 48.21 | 189.4313 | 189.4313 | 23.15 | ||
MDPI7(3, 000) | 187.4512 | 188.2426 | 188.2426 | 86.57 | 188.2426 | 188.2426 | 120.95 | 188.2426 | 188.2426 | 65.45 | ||
MDPI8(3, 000) | 185.7358 | 186.7968 | 186.7968 | 48.04 | 186.7968 | 186.7968 | 38.7 | 186.7968 | 186.7968 | 31.2 | ||
MDPI9(3, 000) | 187.1076 | 188.2313 | 188.2313 | 151.78 | 188.2313 | 188.2313 | 82.66 | 188.2313 | 188.2313 | 47.25 | ||
MDPI10(3, 000) | 184.6866 | 185.6825 | 185.6238 | 228.72 | 185.6825 | 185.6719 | 510.41 | 185.6825 | 185.6822 | 467.1 | ||
MDPII1(3, 000) | 252.1818 | 252.3204 | 252.3204 | 59.7 | 252.3204 | 252.3204 | 42.42 | 252.3204 | 252.3204 | 51.2 | ||
MDPII2(3, 000) | 248.6972 | 250.0621 | 250.0621 | 220.1 | 250.0621 | 250.0617 | 248.1 | 250.0621 | 250.0576 | 514.4 | ||
MDPII3(3, 000) | 250.5303 | 251.9063 | 251.9063 | 146.32 | 251.9063 | 251.9063 | 139.36 | 251.9063 | 251.9063 | 78.1 | ||
MDPII4(3, 000) | 253.0963 | 253.941 | 253.9406 | 370.76 | 253.941 | 253.9406 | 352.17 | 253.941 | 253.941 | 184.05 | ||
MDPII5(3, 000) | 252.5621 | 253.2604 | 253.2604 | 374 | 253.2604 | 253.2603 | 308.8 | 253.2604 | 253.2608 | 344.15 | ||
MDPII6(3, 000) | 249.7160 | 250.6778 | 250.6778 | 55.35 | 250.6778 | 250.6778 | 55.9 | 250.6778 | 250.6778 | 37.25 | ||
MDPII7(3, 000) | 249.5939 | 251.1344 | 251.1344 | 74.72 | 251.1344 | 251.1344 | 88.61 | 251.1344 | 251.1344 | 59.85 | ||
MDPII8(3, 000) | 252.0565 | 252.9996 | 252.9996 | 79.82 | 252.9996 | 252.9996 | 123.56 | 252.9996 | 252.9996 | 61.72 | ||
MDPII9(3, 000) | 251.3625 | 252.4258 | 252.4258 | 90.27 | 252.4258 | 252.4258 | 58.5 | 252.4258 | 252.4258 | 94.4 | ||
MDPII10(3, 000) | 251.1169 | 252.3966 | 252.3966 | 13.18 | 252.3966 | 252.3966 | 8.73 | 252.3966 | 252.3966 | 12.02 | ||
MDPI1(5, 000) | 236.3332 | 4395.321 | 4395.24 | 2914.9 | 4395.321 | 4395.288 | 3084.12 | 4395.321 | 4395.312 | 2804.12 | ||
MDPI2(5, 000) | 239.0143 | 219.7661 | 219.762 | 145.745 | 219.7661 | 219.7644 | 154.206 | 219.7661 | 219.7656 | 140.206 | ||
MDPI3(5, 000) | 238.4742 | 240.1625 | 240.1029 | 312.13 | 240.1625 | 240.0434 | 905.58 | 240.1625 | 240.0519 | 1061.45 | ||
MDPI4(5, 000) | 237.3972 | 241.8274 | 241.793 | 1244.36 | 241.8274 | 241.768 | 958.6 | 241.8274 | 241.7649 | 1046.76 | ||
MDPI5(5, 000) | 240.0439 | 240.8908 | 240.8882 | 810.48 | 240.8908 | 240.8494 | 992.57 | 240.8908 | 240.8518 | 1040.85 | ||
MDPI6(5, 000) | 238.0015 | 240.9972 | 240.9768 | 653.64 | 240.9972 | 240.9281 | 1240.33 | 240.9925 | 240.9129 | 910.6 | ||
MDPI7(5, 000) | 239.7444 | 242.4801 | 242.4759 | 735.16 | 242.4801 | 242.4419 | 1104.64 | 242.4801 | 242.4593 | 895.02 | ||
MDPI8(5, 000) | 237.9150 | 240.3229 | 240.3063 | 976.02 | 240.376 | 240.2726 | 992.36 | 240.376 | 240.2698 | 1072.8 | ||
MDPI9(5, 000) | 235.9103 | 242.8149 | 242.775 | 259.5 | 242.8201 | 242.7624 | 965.79 | 242.8201 | 242.7778 | 1036.81 | ||
MDPI10(5, 000) | 241.8043 | 241.195 | 241.1618 | 1148.6 | 241.195 | 241.1291 | 909.39 | 241.195 | 241.134 | 1036.05 | ||
MDPII1(5, 000) | 316.7478 | 239.7606 | 239.6676 | 1219.71 | 239.7606 | 239.5363 | 1001.4 | 239.7606 | 239.6131 | 1441.87 | ||
MDPII2(5, 000) | 323.6829 | 243.4737 | 243.373 | 457.28 | 243.4737 | 243.3442 | 981.99 | 243.4737 | 243.3673 | 758.61 | ||
MDPII3(5, 000) | 321.9291 | 322.2359 | 322.1813 | 1519.05 | 322.2359 | 322.1594 | 1114.21 | 322.2359 | 322.1691 | 1089.2 | ||
MDPII4(5, 000) | 317.6767 | 327.3019 | 327.0063 | 1103.13 | 327.3019 | 327.1024 | 731.65 | 327.3019 | 327.2153 | 858.95 | ||
MDPII5(5, 000) | 317.7479 | 324.8135 | 324.8016 | 955.81 | 324.8135 | 324.777 | 1043.24 | 324.8135 | 324.7917 | 1039.47 | ||
MDPII6(5, 000) | 319.3890 | 322.2376 | 322.1973 | 664.1 | 322.2277 | 322.1171 | 1059.47 | 322.2376 | 322.1448 | 1106.05 | ||
MDPII7(5, 000) | 319.9806 | 322.4912 | 322.3807 | 1014.9 | 322.5012 | 322.3717 | 971.06 | 322.5012 | 322.3647 | 1151.2 | ||
MDPII8(5, 000) | 318.8545 | 322.9505 | 322.7039 | 352.88 | 322.9505 | 322.6878 | 1405.72 | 322.9505 | 322.7466 | 937.45 | ||
MDPII9(5, 000) | 320.4376 | 322.8504 | 322.7931 | 714.31 | 322.8504 | 322.7615 | 1164.19 | 322.8504 | 322.8136 | 1052.93 | ||
MDPII10(5, 000) | 320.8530 | 323.1121 | 323.0533 | 879.48 | 323.1121 | 322.8815 | 1168.86 | 323.1121 | 322.8943 | 1015.75 |
Instance |
TP-TS | MAMMDP | EDA | RLTS | ||||||||
$ f_{best} $ | $ f_{avg} $ | $ time $ | $ f_{best} $ | $ f_{avg} $ | time | $f_{best} $ | $ f_{avg} $ | $ time $ | ||||
MDPI1(3, 000) | 188.0953 | 189.049 | 189.049 | 88.36 | 189.049 | 189.049 | 69.35 | 189.049 | 189.049 | 47.45 | ||
MDPI2(3, 000) | 186.4730 | 187.3873 | 187.3873 | 60.71 | 187.3873 | 187.3873 | 53.64 | 187.3873 | 187.3873 | 66.65 | ||
MDPI3(3, 000) | 184.3414 | 185.6668 | 185.6551 | 352.85 | 185.6668 | 185.6453 | 399.44 | 185.6668 | 185.6622 | 426.7 | ||
MDPI4(3, 000) | 185.5882 | 186.1637 | 186.1536 | 300.37 | 186.1637 | 186.1637 | 240.45 | 186.1637 | 186.1637 | 137.8 | ||
MDPI5(3, 000) | 186.2349 | 187.5455 | 187.5455 | 61.29 | 187.5455 | 187.5455 | 94.16 | 187.5455 | 187.5455 | 54.23 | ||
MDPI6(3, 000) | 189.0935 | 189.4313 | 189.4313 | 51.99 | 189.4313 | 189.4313 | 48.21 | 189.4313 | 189.4313 | 23.15 | ||
MDPI7(3, 000) | 187.4512 | 188.2426 | 188.2426 | 86.57 | 188.2426 | 188.2426 | 120.95 | 188.2426 | 188.2426 | 65.45 | ||
MDPI8(3, 000) | 185.7358 | 186.7968 | 186.7968 | 48.04 | 186.7968 | 186.7968 | 38.7 | 186.7968 | 186.7968 | 31.2 | ||
MDPI9(3, 000) | 187.1076 | 188.2313 | 188.2313 | 151.78 | 188.2313 | 188.2313 | 82.66 | 188.2313 | 188.2313 | 47.25 | ||
MDPI10(3, 000) | 184.6866 | 185.6825 | 185.6238 | 228.72 | 185.6825 | 185.6719 | 510.41 | 185.6825 | 185.6822 | 467.1 | ||
MDPII1(3, 000) | 252.1818 | 252.3204 | 252.3204 | 59.7 | 252.3204 | 252.3204 | 42.42 | 252.3204 | 252.3204 | 51.2 | ||
MDPII2(3, 000) | 248.6972 | 250.0621 | 250.0621 | 220.1 | 250.0621 | 250.0617 | 248.1 | 250.0621 | 250.0576 | 514.4 | ||
MDPII3(3, 000) | 250.5303 | 251.9063 | 251.9063 | 146.32 | 251.9063 | 251.9063 | 139.36 | 251.9063 | 251.9063 | 78.1 | ||
MDPII4(3, 000) | 253.0963 | 253.941 | 253.9406 | 370.76 | 253.941 | 253.9406 | 352.17 | 253.941 | 253.941 | 184.05 | ||
MDPII5(3, 000) | 252.5621 | 253.2604 | 253.2604 | 374 | 253.2604 | 253.2603 | 308.8 | 253.2604 | 253.2608 | 344.15 | ||
MDPII6(3, 000) | 249.7160 | 250.6778 | 250.6778 | 55.35 | 250.6778 | 250.6778 | 55.9 | 250.6778 | 250.6778 | 37.25 | ||
MDPII7(3, 000) | 249.5939 | 251.1344 | 251.1344 | 74.72 | 251.1344 | 251.1344 | 88.61 | 251.1344 | 251.1344 | 59.85 | ||
MDPII8(3, 000) | 252.0565 | 252.9996 | 252.9996 | 79.82 | 252.9996 | 252.9996 | 123.56 | 252.9996 | 252.9996 | 61.72 | ||
MDPII9(3, 000) | 251.3625 | 252.4258 | 252.4258 | 90.27 | 252.4258 | 252.4258 | 58.5 | 252.4258 | 252.4258 | 94.4 | ||
MDPII10(3, 000) | 251.1169 | 252.3966 | 252.3966 | 13.18 | 252.3966 | 252.3966 | 8.73 | 252.3966 | 252.3966 | 12.02 | ||
MDPI1(5, 000) | 236.3332 | 4395.321 | 4395.24 | 2914.9 | 4395.321 | 4395.288 | 3084.12 | 4395.321 | 4395.312 | 2804.12 | ||
MDPI2(5, 000) | 239.0143 | 219.7661 | 219.762 | 145.745 | 219.7661 | 219.7644 | 154.206 | 219.7661 | 219.7656 | 140.206 | ||
MDPI3(5, 000) | 238.4742 | 240.1625 | 240.1029 | 312.13 | 240.1625 | 240.0434 | 905.58 | 240.1625 | 240.0519 | 1061.45 | ||
MDPI4(5, 000) | 237.3972 | 241.8274 | 241.793 | 1244.36 | 241.8274 | 241.768 | 958.6 | 241.8274 | 241.7649 | 1046.76 | ||
MDPI5(5, 000) | 240.0439 | 240.8908 | 240.8882 | 810.48 | 240.8908 | 240.8494 | 992.57 | 240.8908 | 240.8518 | 1040.85 | ||
MDPI6(5, 000) | 238.0015 | 240.9972 | 240.9768 | 653.64 | 240.9972 | 240.9281 | 1240.33 | 240.9925 | 240.9129 | 910.6 | ||
MDPI7(5, 000) | 239.7444 | 242.4801 | 242.4759 | 735.16 | 242.4801 | 242.4419 | 1104.64 | 242.4801 | 242.4593 | 895.02 | ||
MDPI8(5, 000) | 237.9150 | 240.3229 | 240.3063 | 976.02 | 240.376 | 240.2726 | 992.36 | 240.376 | 240.2698 | 1072.8 | ||
MDPI9(5, 000) | 235.9103 | 242.8149 | 242.775 | 259.5 | 242.8201 | 242.7624 | 965.79 | 242.8201 | 242.7778 | 1036.81 | ||
MDPI10(5, 000) | 241.8043 | 241.195 | 241.1618 | 1148.6 | 241.195 | 241.1291 | 909.39 | 241.195 | 241.134 | 1036.05 | ||
MDPII1(5, 000) | 316.7478 | 239.7606 | 239.6676 | 1219.71 | 239.7606 | 239.5363 | 1001.4 | 239.7606 | 239.6131 | 1441.87 | ||
MDPII2(5, 000) | 323.6829 | 243.4737 | 243.373 | 457.28 | 243.4737 | 243.3442 | 981.99 | 243.4737 | 243.3673 | 758.61 | ||
MDPII3(5, 000) | 321.9291 | 322.2359 | 322.1813 | 1519.05 | 322.2359 | 322.1594 | 1114.21 | 322.2359 | 322.1691 | 1089.2 | ||
MDPII4(5, 000) | 317.6767 | 327.3019 | 327.0063 | 1103.13 | 327.3019 | 327.1024 | 731.65 | 327.3019 | 327.2153 | 858.95 | ||
MDPII5(5, 000) | 317.7479 | 324.8135 | 324.8016 | 955.81 | 324.8135 | 324.777 | 1043.24 | 324.8135 | 324.7917 | 1039.47 | ||
MDPII6(5, 000) | 319.3890 | 322.2376 | 322.1973 | 664.1 | 322.2277 | 322.1171 | 1059.47 | 322.2376 | 322.1448 | 1106.05 | ||
MDPII7(5, 000) | 319.9806 | 322.4912 | 322.3807 | 1014.9 | 322.5012 | 322.3717 | 971.06 | 322.5012 | 322.3647 | 1151.2 | ||
MDPII8(5, 000) | 318.8545 | 322.9505 | 322.7039 | 352.88 | 322.9505 | 322.6878 | 1405.72 | 322.9505 | 322.7466 | 937.45 | ||
MDPII9(5, 000) | 320.4376 | 322.8504 | 322.7931 | 714.31 | 322.8504 | 322.7615 | 1164.19 | 322.8504 | 322.8136 | 1052.93 | ||
MDPII10(5, 000) | 320.8530 | 323.1121 | 323.0533 | 879.48 | 323.1121 | 322.8815 | 1168.86 | 323.1121 | 322.8943 | 1015.75 |
Names | Range | Debugging Intervals | Final values |
Greedy factor | [0.5, 1) | 0.05 | 0.7 |
Learning factor | (0, 1) | 0.1 | 0.5 |
Discount factor | [0, 1) | 0.1 | 0.5 |
Names | Range | Debugging Intervals | Final values |
Greedy factor | [0.5, 1) | 0.05 | 0.7 |
Learning factor | (0, 1) | 0.1 | 0.5 |
Discount factor | [0, 1) | 0.1 | 0.5 |
Instances ($ n $) / $ \epsilon $ | 0.5 | 0.55 | 0.6 | 0.65 | 0.7 | 0.75 | 0.8 | 0.85 | 0.9 | 0.95 |
MDPI1(5, 000) | -0.122851 | -0.081087 | -0.130172 | -0.128802 | -0.056058 | -0.129919 | -0.041805 | -0.155847 | -0.044694 | -0.105110 |
MDPI2(5, 000) | -0.041683 | -0.044909 | -0.042706 | -0.031403 | -0.032596 | -0.058119 | -0.012692 | -0.060083 | -0.056580 | -0.054879 |
MDPI3(5, 000) | -0.067597 | -0.050908 | -0.053767 | -0.106229 | -0.068687 | -0.076957 | -0.105697 | -0.085361 | -0.115044 | -0.051265 |
MDPI4(5, 000) | -0.133373 | -0.042282 | -0.115774 | -0.091527 | -0.064438 | -0.134837 | -0.063109 | -0.100193 | -0.061781 | -0.071073 |
MDPI5(5, 000) | -0.042153 | -0.058244 | -0.073869 | -0.027768 | -0.041856 | -0.064060 | -0.072030 | -0.051006 | -0.032583 | -0.059985 |
MDPI6(5, 000) | -0.040458 | -0.030610 | -0.054468 | -0.079938 | -0.042002 | -0.093500 | -0.018872 | -0.071949 | -0.053420 | -0.030553 |
MDPI7(5, 000) | -0.007039 | -0.040785 | -0.022835 | -0.025523 | -0.011432 | -0.022520 | -0.000364 | -0.015116 | 0.004182 | -0.010404 |
MDPI8(5, 000) | -0.048888 | -0.064965 | -0.040440 | -0.050873 | -0.058277 | -0.058174 | -0.040082 | -0.054745 | -0.050569 | -0.078426 |
MDPI9(5, 000) | -0.129854 | -0.078371 | -0.117872 | -0.097018 | -0.076916 | -0.176947 | -0.060925 | -0.175825 | -0.084588 | -0.097384 |
MDPI10(5, 000) | -0.033717 | -0.043167 | -0.010259 | -0.039291 | -0.035665 | -0.062412 | -0.042826 | -0.063820 | -0.018992 | -0.030091 |
MDPII1(5, 000) | -0.005740 | -0.033777 | -0.059031 | -0.047182 | -0.044300 | -0.066763 | -0.060716 | -0.031978 | -0.014357 | -0.039396 |
MDPII2(5, 000) | 0.147624 | 0.167856 | 0.178805 | 0.132264 | 0.164916 | 0.118441 | 0.083772 | 0.211362 | 0.147558 | 0.088298 |
MDPII3(5, 000) | -0.023621 | -0.023856 | -0.045420 | -0.022302 | -0.009195 | -0.030358 | -0.013416 | -0.023812 | -0.027812 | -0.027001 |
MDPII4(5, 000) | -0.166333 | -0.094202 | -0.120204 | -0.096423 | -0.085864 | -0.121264 | -0.120666 | -0.188106 | -0.240823 | -0.147536 |
MDPII5(5, 000) | -0.075976 | -0.096680 | -0.044135 | -0.004540 | -0.033960 | -0.043347 | -0.077273 | -0.032238 | -0.036416 | -0.031091 |
MDPII6(5, 000) | -0.017563 | 0.058637 | 0.005761 | -0.098428 | -0.080234 | 0.015629 | -0.008452 | 0.021738 | 0.033528 | -0.012549 |
MDPII7(5, 000) | -0.015845 | -0.074944 | -0.060637 | -0.081458 | -0.021804 | -0.056874 | -0.053205 | -0.017029 | -0.017963 | -0.067988 |
MDPII8(5, 000) | -0.178216 | -0.143775 | -0.177740 | -0.229257 | -0.176900 | -0.163983 | -0.205498 | -0.172191 | -0.244947 | -0.188872 |
MDPII9(5, 000) | 0.014833 | -0.094614 | -0.111384 | -0.118775 | 0.068232 | -0.047735 | -0.119738 | -0.069007 | -0.124918 | -0.130199 |
MDPII10(5, 000) | 0.017633 | -0.048853 | -0.107008 | -0.162074 | 0.012272 | -0.128291 | 0.050908 | 0.021277 | 0.044312 | -0.082684 |
Debugging results for factors $ \alpha $ and $ \gamma $ were omitted. |
Instances ($ n $) / $ \epsilon $ | 0.5 | 0.55 | 0.6 | 0.65 | 0.7 | 0.75 | 0.8 | 0.85 | 0.9 | 0.95 |
MDPI1(5, 000) | -0.122851 | -0.081087 | -0.130172 | -0.128802 | -0.056058 | -0.129919 | -0.041805 | -0.155847 | -0.044694 | -0.105110 |
MDPI2(5, 000) | -0.041683 | -0.044909 | -0.042706 | -0.031403 | -0.032596 | -0.058119 | -0.012692 | -0.060083 | -0.056580 | -0.054879 |
MDPI3(5, 000) | -0.067597 | -0.050908 | -0.053767 | -0.106229 | -0.068687 | -0.076957 | -0.105697 | -0.085361 | -0.115044 | -0.051265 |
MDPI4(5, 000) | -0.133373 | -0.042282 | -0.115774 | -0.091527 | -0.064438 | -0.134837 | -0.063109 | -0.100193 | -0.061781 | -0.071073 |
MDPI5(5, 000) | -0.042153 | -0.058244 | -0.073869 | -0.027768 | -0.041856 | -0.064060 | -0.072030 | -0.051006 | -0.032583 | -0.059985 |
MDPI6(5, 000) | -0.040458 | -0.030610 | -0.054468 | -0.079938 | -0.042002 | -0.093500 | -0.018872 | -0.071949 | -0.053420 | -0.030553 |
MDPI7(5, 000) | -0.007039 | -0.040785 | -0.022835 | -0.025523 | -0.011432 | -0.022520 | -0.000364 | -0.015116 | 0.004182 | -0.010404 |
MDPI8(5, 000) | -0.048888 | -0.064965 | -0.040440 | -0.050873 | -0.058277 | -0.058174 | -0.040082 | -0.054745 | -0.050569 | -0.078426 |
MDPI9(5, 000) | -0.129854 | -0.078371 | -0.117872 | -0.097018 | -0.076916 | -0.176947 | -0.060925 | -0.175825 | -0.084588 | -0.097384 |
MDPI10(5, 000) | -0.033717 | -0.043167 | -0.010259 | -0.039291 | -0.035665 | -0.062412 | -0.042826 | -0.063820 | -0.018992 | -0.030091 |
MDPII1(5, 000) | -0.005740 | -0.033777 | -0.059031 | -0.047182 | -0.044300 | -0.066763 | -0.060716 | -0.031978 | -0.014357 | -0.039396 |
MDPII2(5, 000) | 0.147624 | 0.167856 | 0.178805 | 0.132264 | 0.164916 | 0.118441 | 0.083772 | 0.211362 | 0.147558 | 0.088298 |
MDPII3(5, 000) | -0.023621 | -0.023856 | -0.045420 | -0.022302 | -0.009195 | -0.030358 | -0.013416 | -0.023812 | -0.027812 | -0.027001 |
MDPII4(5, 000) | -0.166333 | -0.094202 | -0.120204 | -0.096423 | -0.085864 | -0.121264 | -0.120666 | -0.188106 | -0.240823 | -0.147536 |
MDPII5(5, 000) | -0.075976 | -0.096680 | -0.044135 | -0.004540 | -0.033960 | -0.043347 | -0.077273 | -0.032238 | -0.036416 | -0.031091 |
MDPII6(5, 000) | -0.017563 | 0.058637 | 0.005761 | -0.098428 | -0.080234 | 0.015629 | -0.008452 | 0.021738 | 0.033528 | -0.012549 |
MDPII7(5, 000) | -0.015845 | -0.074944 | -0.060637 | -0.081458 | -0.021804 | -0.056874 | -0.053205 | -0.017029 | -0.017963 | -0.067988 |
MDPII8(5, 000) | -0.178216 | -0.143775 | -0.177740 | -0.229257 | -0.176900 | -0.163983 | -0.205498 | -0.172191 | -0.244947 | -0.188872 |
MDPII9(5, 000) | 0.014833 | -0.094614 | -0.111384 | -0.118775 | 0.068232 | -0.047735 | -0.119738 | -0.069007 | -0.124918 | -0.130199 |
MDPII10(5, 000) | 0.017633 | -0.048853 | -0.107008 | -0.162074 | 0.012272 | -0.128291 | 0.050908 | 0.021277 | 0.044312 | -0.082684 |
Debugging results for factors $ \alpha $ and $ \gamma $ were omitted. |
$\epsilon~$ | Mean Rank | $\alpha~$ | Mean Rank | $\gamma~$ | Mean Rank |
$\epsilon~$0.5 | 6.2 | $\alpha~$0.1 | 3.8 | $\gamma~$0 | 4.2 |
$\epsilon~$0.55 | 6.1 | $\alpha~$0.2 | 5.05 | $\gamma~$0.1 | 4.15 |
$\epsilon~$0.6 | 4.9 | $\alpha~$0.3 | 4.6 | $\gamma~$0.2 | 4.7 |
$\epsilon~$0.65 | 4.7 | $\alpha~$0.4 | 4.25 | $\gamma~$0.3 | 5.6 |
$\epsilon~$0.7 | 6.9 | $\alpha~$0.5 | 6.45 | $\gamma~$0.4 | 5.4 |
$\epsilon~$0.75 | 3.65 | $\alpha~$0.6 | 4.95 | $\gamma~$0.5 | 7.15 |
$\epsilon~$0.8 | 6.15 | $\alpha~$0.7 | 4.85 | $\gamma~$0.6 | 5.75 |
$\epsilon~$0.85 | 5.3 | $\alpha~$0.8 | 5.95 | $\gamma~$0.7 | 5.85 |
$\epsilon~$0.9 | 6.1 | $\alpha~$0.9 | 5.1 | $\gamma~$0.8 | 6.3 |
$\epsilon~$0.95 | 5 | $\gamma~$0.9 | 5.9 | ||
$ N $ | 20 | $ N $ | 20 | $ N $ | 20 |
Chi-Square | 18.12 | Chi-Square | 13.88 | Chi-Square | 17.193 |
$ df $ | 9 | $ df $ | 8 | $ df $ | 9 |
Asymp. Sig | 0.034 | Asymp. Sig. | 0.085 | Asymp. Sig. | 0.46 |
$\epsilon~$ | Mean Rank | $\alpha~$ | Mean Rank | $\gamma~$ | Mean Rank |
$\epsilon~$0.5 | 6.2 | $\alpha~$0.1 | 3.8 | $\gamma~$0 | 4.2 |
$\epsilon~$0.55 | 6.1 | $\alpha~$0.2 | 5.05 | $\gamma~$0.1 | 4.15 |
$\epsilon~$0.6 | 4.9 | $\alpha~$0.3 | 4.6 | $\gamma~$0.2 | 4.7 |
$\epsilon~$0.65 | 4.7 | $\alpha~$0.4 | 4.25 | $\gamma~$0.3 | 5.6 |
$\epsilon~$0.7 | 6.9 | $\alpha~$0.5 | 6.45 | $\gamma~$0.4 | 5.4 |
$\epsilon~$0.75 | 3.65 | $\alpha~$0.6 | 4.95 | $\gamma~$0.5 | 7.15 |
$\epsilon~$0.8 | 6.15 | $\alpha~$0.7 | 4.85 | $\gamma~$0.6 | 5.75 |
$\epsilon~$0.85 | 5.3 | $\alpha~$0.8 | 5.95 | $\gamma~$0.7 | 5.85 |
$\epsilon~$0.9 | 6.1 | $\alpha~$0.9 | 5.1 | $\gamma~$0.8 | 6.3 |
$\epsilon~$0.95 | 5 | $\gamma~$0.9 | 5.9 | ||
$ N $ | 20 | $ N $ | 20 | $ N $ | 20 |
Chi-Square | 18.12 | Chi-Square | 13.88 | Chi-Square | 17.193 |
$ df $ | 9 | $ df $ | 8 | $ df $ | 9 |
Asymp. Sig | 0.034 | Asymp. Sig. | 0.085 | Asymp. Sig. | 0.46 |
Instance |
MTS | RLTS | |||
MDPI1(3, 000) | 189.04897 | 189.04897 | 189.04897 | 189.04897 | |
MDPI2(3, 000) | 187.38729 | 187.38729 | 187.38729 | 187.38729 | |
MDPI3(3, 000) | 185.66681 | 185.65159 | 185.66681 | 185.6594 | |
MDPI4(3, 000) | 186.16373 | 186.16373 | 186.16373 | 186.16373 | |
MDPI5(3, 000) | 187.54552 | 187.54552 | 187.54552 | 187.54552 | |
MDPI6(3, 000) | 189.43126 | 189.43126 | 189.43126 | 189.43126 | |
MDPI7(3, 000) | 188.24258 | 188.24258 | 188.24258 | 188.24258 | |
MDPI8(3, 000) | 186.79681 | 186.79681 | 186.79681 | 186.79681 | |
MDPI9(3, 000) | 188.23126 | 188.23126 | 188.23126 | 188.23126 | |
MDPI10(3, 000) | 185.68251 | 185.67237 | 185.68251 | 185.6747 | |
MDPII1(3, 000) | 252.32043 | 252.32043 | 252.32043 | 252.32043 | |
MDPII2(3, 000) | 250.06214 | 250.05474 | 250.06214 | 250.0569 | |
MDPII3(3, 000) | 251.90627 | 251.90627 | 251.90627 | 251.90627 | |
MDPII4(3, 000) | 253.94101 | 253.93968 | 253.94101 | 253.941 | |
MDPII5(3, 000) | 253.26042 | 253.26016 | 253.26042 | 253.2604 | |
MDPII6(3, 000) | 250.67775 | 250.67775 | 250.67775 | 250.67775 | |
MDPII7(3, 000) | 251.13441 | 251.13441 | 251.13441 | 251.13441 | |
MDPII8(3, 000) | 252.99965 | 252.99965 | 252.99965 | 252.99965 | |
MDPII9(3, 000) | 252.42577 | 252.42577 | 252.42577 | 252.42577 | |
MDPII10(3, 000) | 252.39659 | 252.39659 | 252.39659 | 252.39659 | |
MDPI1(5, 000) | 240.14121 | 240.0212 | 240.1594 | 240.01588 | |
MDPI2(5, 000) | 241.81754 | 241.75355 | 241.8274 | 241.7716 | |
MDPI3(5, 000) | 240.89082 | 240.82517 | 240.89082 | 240.8443 | |
MDPI4(5, 000) | 240.97349 | 240.91546 | 240.9972 | 240.9249 | |
MDPI5(5, 000) | 242.48013 | 242.43047 | 242.48013 | 242.4512 | |
MDPI6(5, 000) | 240.32868 | 240.2663 | 240.32868 | 240.26432 | |
MDPI7(5, 000) | 242.82014 | 242.7599 | 242.82014 | 242.7793 | |
MDPI8(5, 000) | 241.14478 | 241.11345 | 241.195 | 241.1323 | |
MDPI9(5, 000) | 239.76056 | 239.51496 | 239.76056 | 239.5929 | |
MDPI10(5, 000) | 243.38549 | 243.34815 | 243.4737 | 243.3598 | |
MDPII1(5, 000) | 322.22322 | 322.1312 | 322.2359 | 322.1659 | |
MDPII2(5, 000) | 327.30191 | 327.07525 | 327.30191 | 327.2223 | |
MDPII3(5, 000) | 324.81083 | 324.79022 | 324.8135 | 324.7931 | |
MDPII4(5, 000) | 322.21229 | 322.1266 | 322.2376 | 322.08809 | |
MDPII5(5, 000) | 322.42081 | 322.30125 | 322.5012 | 322.3782 | |
MDPII6(5, 000) | 322.95049 | 322.61523 | 322.95049 | 322.7672 | |
MDPII7(5, 000) | 322.85044 | 322.7784 | 322.85044 | 322.7757 | |
MDPII8(5, 000) | 323.03384 | 322.87316 | 323.1121 | 322.8885 | |
MDPII9(5, 000) | 323.52271 | 323.27856 | 323.5438 | 323.4081 | |
MDPII10(5, 000) | 324.51991 | 324.29479 | 324.51991 | 324.5191 |
Instance |
MTS | RLTS | |||
MDPI1(3, 000) | 189.04897 | 189.04897 | 189.04897 | 189.04897 | |
MDPI2(3, 000) | 187.38729 | 187.38729 | 187.38729 | 187.38729 | |
MDPI3(3, 000) | 185.66681 | 185.65159 | 185.66681 | 185.6594 | |
MDPI4(3, 000) | 186.16373 | 186.16373 | 186.16373 | 186.16373 | |
MDPI5(3, 000) | 187.54552 | 187.54552 | 187.54552 | 187.54552 | |
MDPI6(3, 000) | 189.43126 | 189.43126 | 189.43126 | 189.43126 | |
MDPI7(3, 000) | 188.24258 | 188.24258 | 188.24258 | 188.24258 | |
MDPI8(3, 000) | 186.79681 | 186.79681 | 186.79681 | 186.79681 | |
MDPI9(3, 000) | 188.23126 | 188.23126 | 188.23126 | 188.23126 | |
MDPI10(3, 000) | 185.68251 | 185.67237 | 185.68251 | 185.6747 | |
MDPII1(3, 000) | 252.32043 | 252.32043 | 252.32043 | 252.32043 | |
MDPII2(3, 000) | 250.06214 | 250.05474 | 250.06214 | 250.0569 | |
MDPII3(3, 000) | 251.90627 | 251.90627 | 251.90627 | 251.90627 | |
MDPII4(3, 000) | 253.94101 | 253.93968 | 253.94101 | 253.941 | |
MDPII5(3, 000) | 253.26042 | 253.26016 | 253.26042 | 253.2604 | |
MDPII6(3, 000) | 250.67775 | 250.67775 | 250.67775 | 250.67775 | |
MDPII7(3, 000) | 251.13441 | 251.13441 | 251.13441 | 251.13441 | |
MDPII8(3, 000) | 252.99965 | 252.99965 | 252.99965 | 252.99965 | |
MDPII9(3, 000) | 252.42577 | 252.42577 | 252.42577 | 252.42577 | |
MDPII10(3, 000) | 252.39659 | 252.39659 | 252.39659 | 252.39659 | |
MDPI1(5, 000) | 240.14121 | 240.0212 | 240.1594 | 240.01588 | |
MDPI2(5, 000) | 241.81754 | 241.75355 | 241.8274 | 241.7716 | |
MDPI3(5, 000) | 240.89082 | 240.82517 | 240.89082 | 240.8443 | |
MDPI4(5, 000) | 240.97349 | 240.91546 | 240.9972 | 240.9249 | |
MDPI5(5, 000) | 242.48013 | 242.43047 | 242.48013 | 242.4512 | |
MDPI6(5, 000) | 240.32868 | 240.2663 | 240.32868 | 240.26432 | |
MDPI7(5, 000) | 242.82014 | 242.7599 | 242.82014 | 242.7793 | |
MDPI8(5, 000) | 241.14478 | 241.11345 | 241.195 | 241.1323 | |
MDPI9(5, 000) | 239.76056 | 239.51496 | 239.76056 | 239.5929 | |
MDPI10(5, 000) | 243.38549 | 243.34815 | 243.4737 | 243.3598 | |
MDPII1(5, 000) | 322.22322 | 322.1312 | 322.2359 | 322.1659 | |
MDPII2(5, 000) | 327.30191 | 327.07525 | 327.30191 | 327.2223 | |
MDPII3(5, 000) | 324.81083 | 324.79022 | 324.8135 | 324.7931 | |
MDPII4(5, 000) | 322.21229 | 322.1266 | 322.2376 | 322.08809 | |
MDPII5(5, 000) | 322.42081 | 322.30125 | 322.5012 | 322.3782 | |
MDPII6(5, 000) | 322.95049 | 322.61523 | 322.95049 | 322.7672 | |
MDPII7(5, 000) | 322.85044 | 322.7784 | 322.85044 | 322.7757 | |
MDPII8(5, 000) | 323.03384 | 322.87316 | 323.1121 | 322.8885 | |
MDPII9(5, 000) | 323.52271 | 323.27856 | 323.5438 | 323.4081 | |
MDPII10(5, 000) | 324.51991 | 324.29479 | 324.51991 | 324.5191 |
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