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Unified optimality conditions for set-valued optimizations
Single-machine bi-criterion scheduling with release times and exponentially time-dependent learning effects
1. | School of Economics and Management, Shenyang Aerospace University, Shenyang 110136, China |
2. | School of Science, Shenyang Aerospace University, Shenyang 110136, China |
This paper deals with a single machine bi-criterion scheduling problem with exponentially time-dependent learning effects and non-zero job release times. The goal is to minimize the weighted sum of the makespan and the total completion time. First, a branch-and-bound algorithm incorporating with some dominance properties and three lower bounds is developed for the optimal solutions. Then heuristic and particle swarm optimization algorithms are presented for near-optimal solutions. Finally, computational experiments are conducted to evaluate the performances of the proposed algorithms. Computational results indicate that the algorithms perform well in either solving the problem or efficiently generating near-optimal solutions.
References:
[1] |
F. Ahmadizar and L. Hosseini,
Bi-criteria single machine scheduling with a time-dependent learning effect and release times, Applied Mathematical Modelling, 36 (2012), 6203-6214.
doi: 10.1016/j.apm.2012.02.002. |
[2] |
D. Biskup,
A state-of-the-art review on scheduling with learning effects, European Journal of Operational Research, 188 (2008), 315-329.
doi: 10.1016/j.ejor.2007.05.040. |
[3] |
T.C.E. Cheng, W.-H. Kuo and D.-L. Yang,
Scheduling with a position-weighted learning effect based on sum-of-logarithm-processing-times and job position, Information Sciences, 221 (2013), 490-500.
doi: 10.1016/j.ins.2012.09.001. |
[4] |
T.C.E. Cheng, W.-H. Kuo and D.-L. Yang,
Scheduling with a position-weighted learning effect, Optimization Letters, 8 (2014), 293-306.
doi: 10.1007/s11590-012-0574-5. |
[5] |
M. Cheng, P.-R. Tadikamall, J. Shang and B. Zhang,
Single machine scheduling problems with exponentially time-dependent learning effects, Journal of Manufacturing Systems, 34 (2015), 60-65.
doi: 10.1016/j.jmsy.2014.11.001. |
[6] |
C. Chu,
Efficient heuristics to minimize total flow time with release dates, Operations Research Letters, 12 (1992), 321-330.
doi: 10.1016/0167-6377(92)90092-H. |
[7] |
M.-I. Dessouky and J.-S. Deogun,
Sequencing jobs with unequal ready times to minimize mean flow time, SIAM Journal on Computing, 10 (1981), 192-202.
doi: 10.1137/0210014. |
[8] |
B. Haddar, M. Khemakhem, S. Hanafi and C. Wilbaut,
A hybrid quantum particle swarm optimization for the multidimensional knapsack problem, Engineering Applications of Artificial Intelligence, 55 (2016), 1-13.
doi: 10.1016/j.engappai.2016.05.006. |
[9] |
N. Hosseini and R. Tavakkoli-Moghaddam,
Two meta-heuristics for solving a new two-machine flowshop scheduling problem with the learning effect and dynamic arrivals, The International Journal of Advanced Manufacturing Technology, 65(5) (2013), 771-786.
doi: 10.1007/s00170-012-4216-y. |
[10] |
A. Janiak, M.-Y. Kovalyov and M. Lichtenstein,
Strong NP-hardness of scheduling problems with learning or aging effect, Annals of Operations Research, 206 (2013), 577-583.
doi: 10.1007/s10479-013-1364-x. |
[11] |
W.-H. Kuo,
Single-machine group scheduling with time-dependent learning effect and position-based setup time learning effect, Annals of Operations Research, 196 (2012), 349-359.
doi: 10.1007/s10479-012-1111-8. |
[12] |
W.-C. Lee, C.-C. Wu and P.-H. Hsu,
A single-machine learning effect scheduling problem with release times, Omega, 38(1-2) (2010), 3-11.
doi: 10.1016/j.omega.2009.01.001. |
[13] |
Y.-Y. Lu, F. Teng, and Z.-X. Feng, Scheduling jobs with truncated exponential sum-of-logarithm-processing-times based and position-based learning effects,
Asia-Pacific Journal of Operational Research, 32(4) (2015), 1550026.
doi: 10.1142/S0217595915500268. |
[14] |
F. Marini and B. Walcza,
Particle swarm optimization (PSO). A tutorial, Chemometrics and Intelligent Laboratory Systems, 149 (2015), 153-165.
doi: 10.1016/j.chemolab.2015.08.020. |
[15] |
H. Melo and J. Watada,
Gaussian-PSO with fuzzy reasoning based on structural learning for training a neural network, Neurocomputing, 172 (2016), 405-412.
doi: 10.1016/j.neucom.2015.03.104. |
[16] |
Y.-P. Niu, L. Wan and J.-B. Wang, A note on scheduling jobs with extended sum-of-processing-times-based and position-based learning effect,
Asia-Pacific Journal of Operational Research, 32 (2015), 1550001 (12 pages).
doi: 10.1142/S0217595915500013. |
[17] |
I. Pan and S. Das,
Fractional order fuzzy control of hybrid power system with renewable generation using chaotic PSO, SA Transactions, 62 (2016), 19-29.
doi: 10.1016/j.isatra.2015.03.003. |
[18] |
R. Rudek,
The single processor total weighted completion time scheduling problem with the sum-of-processing-time based learning model, Information Sciences, 199 (2012), 216-229.
doi: 10.1016/j.ins.2012.02.043. |
[19] |
Y.-R. Shiau, M.-S. Tsai, W.-C. Lee and T.C.E. Cheng,
Two-agent two-machine flowshop scheduling with learning effects to minimize the total completion time, Computers & Industrial Engineering, 87 (2015), 580-589.
doi: 10.1016/j.cie.2015.05.032. |
[20] |
M.-R. Singh and S.-S. Mahapatra,
A quantum behaved particle swarm optimization for flexible job shop scheduling, Computers & Industrial Engineering, 93 (2016), 36-44.
doi: 10.1016/j.cie.2015.12.004. |
[21] |
L.-H. Sun, K. Cui, J.-H. Chen, J. Wang and X.-C. He,
Some results of the worst-case analysis for flow shop scheduling with a learning effect, Annals of Operations Research, 211 (2013), 481-490.
doi: 10.1007/s10479-013-1368-6. |
[22] |
J.-B. Wang and M.-Z. Wang,
Worst-case behavior of simple sequencing rules in flow shop scheduling with general position-dependent learning effects, Annals of Operations Research, 191 (2011), 155-169.
doi: 10.1007/s10479-011-0923-2. |
[23] |
J.-B. Wang and J.-J. Wang,
Single-machine scheduling with precedence constraints and position-dependent processing times, Applied Mathematical Modelling, 37 (2013), 649-658.
doi: 10.1016/j.apm.2012.02.055. |
[24] |
J.-J. Wang and B.-H. Zhang,
Permutation flowshop problems with bi-criterion makespan and total completion time objective and position-weighted learning effects, Computers & Operations Research, 58 (2015), 24-31.
doi: 10.1016/j.cor.2014.12.006. |
[25] |
C.-C. Wu, P.-H. Hsu, J.-C. Chen and N.-S. Wang,
Genetic algorithm for minimizing the total weighted completion time scheduling problem with learning and release times, Computers & Operations Research, 38 (2011), 1025-1034.
doi: 10.1016/j.cor.2010.11.001. |
[26] |
C.-C. Wu and C.-L. Liu,
Minimizing the makespan on a single machine with learning and unequal release times, Computers & Industrial Engineering, 59 (2010), 419-424.
doi: 10.1016/j.cie.2010.05.014. |
show all references
References:
[1] |
F. Ahmadizar and L. Hosseini,
Bi-criteria single machine scheduling with a time-dependent learning effect and release times, Applied Mathematical Modelling, 36 (2012), 6203-6214.
doi: 10.1016/j.apm.2012.02.002. |
[2] |
D. Biskup,
A state-of-the-art review on scheduling with learning effects, European Journal of Operational Research, 188 (2008), 315-329.
doi: 10.1016/j.ejor.2007.05.040. |
[3] |
T.C.E. Cheng, W.-H. Kuo and D.-L. Yang,
Scheduling with a position-weighted learning effect based on sum-of-logarithm-processing-times and job position, Information Sciences, 221 (2013), 490-500.
doi: 10.1016/j.ins.2012.09.001. |
[4] |
T.C.E. Cheng, W.-H. Kuo and D.-L. Yang,
Scheduling with a position-weighted learning effect, Optimization Letters, 8 (2014), 293-306.
doi: 10.1007/s11590-012-0574-5. |
[5] |
M. Cheng, P.-R. Tadikamall, J. Shang and B. Zhang,
Single machine scheduling problems with exponentially time-dependent learning effects, Journal of Manufacturing Systems, 34 (2015), 60-65.
doi: 10.1016/j.jmsy.2014.11.001. |
[6] |
C. Chu,
Efficient heuristics to minimize total flow time with release dates, Operations Research Letters, 12 (1992), 321-330.
doi: 10.1016/0167-6377(92)90092-H. |
[7] |
M.-I. Dessouky and J.-S. Deogun,
Sequencing jobs with unequal ready times to minimize mean flow time, SIAM Journal on Computing, 10 (1981), 192-202.
doi: 10.1137/0210014. |
[8] |
B. Haddar, M. Khemakhem, S. Hanafi and C. Wilbaut,
A hybrid quantum particle swarm optimization for the multidimensional knapsack problem, Engineering Applications of Artificial Intelligence, 55 (2016), 1-13.
doi: 10.1016/j.engappai.2016.05.006. |
[9] |
N. Hosseini and R. Tavakkoli-Moghaddam,
Two meta-heuristics for solving a new two-machine flowshop scheduling problem with the learning effect and dynamic arrivals, The International Journal of Advanced Manufacturing Technology, 65(5) (2013), 771-786.
doi: 10.1007/s00170-012-4216-y. |
[10] |
A. Janiak, M.-Y. Kovalyov and M. Lichtenstein,
Strong NP-hardness of scheduling problems with learning or aging effect, Annals of Operations Research, 206 (2013), 577-583.
doi: 10.1007/s10479-013-1364-x. |
[11] |
W.-H. Kuo,
Single-machine group scheduling with time-dependent learning effect and position-based setup time learning effect, Annals of Operations Research, 196 (2012), 349-359.
doi: 10.1007/s10479-012-1111-8. |
[12] |
W.-C. Lee, C.-C. Wu and P.-H. Hsu,
A single-machine learning effect scheduling problem with release times, Omega, 38(1-2) (2010), 3-11.
doi: 10.1016/j.omega.2009.01.001. |
[13] |
Y.-Y. Lu, F. Teng, and Z.-X. Feng, Scheduling jobs with truncated exponential sum-of-logarithm-processing-times based and position-based learning effects,
Asia-Pacific Journal of Operational Research, 32(4) (2015), 1550026.
doi: 10.1142/S0217595915500268. |
[14] |
F. Marini and B. Walcza,
Particle swarm optimization (PSO). A tutorial, Chemometrics and Intelligent Laboratory Systems, 149 (2015), 153-165.
doi: 10.1016/j.chemolab.2015.08.020. |
[15] |
H. Melo and J. Watada,
Gaussian-PSO with fuzzy reasoning based on structural learning for training a neural network, Neurocomputing, 172 (2016), 405-412.
doi: 10.1016/j.neucom.2015.03.104. |
[16] |
Y.-P. Niu, L. Wan and J.-B. Wang, A note on scheduling jobs with extended sum-of-processing-times-based and position-based learning effect,
Asia-Pacific Journal of Operational Research, 32 (2015), 1550001 (12 pages).
doi: 10.1142/S0217595915500013. |
[17] |
I. Pan and S. Das,
Fractional order fuzzy control of hybrid power system with renewable generation using chaotic PSO, SA Transactions, 62 (2016), 19-29.
doi: 10.1016/j.isatra.2015.03.003. |
[18] |
R. Rudek,
The single processor total weighted completion time scheduling problem with the sum-of-processing-time based learning model, Information Sciences, 199 (2012), 216-229.
doi: 10.1016/j.ins.2012.02.043. |
[19] |
Y.-R. Shiau, M.-S. Tsai, W.-C. Lee and T.C.E. Cheng,
Two-agent two-machine flowshop scheduling with learning effects to minimize the total completion time, Computers & Industrial Engineering, 87 (2015), 580-589.
doi: 10.1016/j.cie.2015.05.032. |
[20] |
M.-R. Singh and S.-S. Mahapatra,
A quantum behaved particle swarm optimization for flexible job shop scheduling, Computers & Industrial Engineering, 93 (2016), 36-44.
doi: 10.1016/j.cie.2015.12.004. |
[21] |
L.-H. Sun, K. Cui, J.-H. Chen, J. Wang and X.-C. He,
Some results of the worst-case analysis for flow shop scheduling with a learning effect, Annals of Operations Research, 211 (2013), 481-490.
doi: 10.1007/s10479-013-1368-6. |
[22] |
J.-B. Wang and M.-Z. Wang,
Worst-case behavior of simple sequencing rules in flow shop scheduling with general position-dependent learning effects, Annals of Operations Research, 191 (2011), 155-169.
doi: 10.1007/s10479-011-0923-2. |
[23] |
J.-B. Wang and J.-J. Wang,
Single-machine scheduling with precedence constraints and position-dependent processing times, Applied Mathematical Modelling, 37 (2013), 649-658.
doi: 10.1016/j.apm.2012.02.055. |
[24] |
J.-J. Wang and B.-H. Zhang,
Permutation flowshop problems with bi-criterion makespan and total completion time objective and position-weighted learning effects, Computers & Operations Research, 58 (2015), 24-31.
doi: 10.1016/j.cor.2014.12.006. |
[25] |
C.-C. Wu, P.-H. Hsu, J.-C. Chen and N.-S. Wang,
Genetic algorithm for minimizing the total weighted completion time scheduling problem with learning and release times, Computers & Operations Research, 38 (2011), 1025-1034.
doi: 10.1016/j.cor.2010.11.001. |
[26] |
C.-C. Wu and C.-L. Liu,
Minimizing the makespan on a single machine with learning and unequal release times, Computers & Industrial Engineering, 59 (2010), 419-424.
doi: 10.1016/j.cie.2010.05.014. |
Dimension | 1 | 2 | 3 | 4 | 5 |
Job permutation | 4 | 1 | 3 | 5 | 2 |
2 | 5 | 3 | 1 | 4 | |
2.09 | -3.88 | 0.33 | 3.15 | -2.11 |
Dimension | 1 | 2 | 3 | 4 | 5 |
Job permutation | 4 | 1 | 3 | 5 | 2 |
2 | 5 | 3 | 1 | 4 | |
2.09 | -3.88 | 0.33 | 3.15 | -2.11 |
Node | cpu time | ||||
Mean | Max | Mean | Max | ||
20 | 0.6 | 156866 | 373724 | 3.462 | 8.001 |
0.7 | 143956 | 399559 | 3.403 | 8.112 | |
0.8 | 119154 | 438721 | 2.963 | 8.898 | |
0.9 | 231118 | 708148 | 5.446 | 14.991 | |
25 | 0.6 | 16365534 | 120686189 | 476.151 | 3476.91 |
0.7 | 6657363 | 24736334 | 226.272 | 682.129 | |
0.8 | 2057701 | 7619907 | 69.852 | 228.938 | |
0.9 | 1654445 | 5476902 | 61.673 | 197.13 | |
30 | 0.6 | 57205995.4 | 271309428 | 2034.489 | 9230.5 |
0.7 | 163983353 | 793164929 | 3551.163 | 16301.3 | |
0.8 | 166549081.2 | 792800370 | 3535.774 | 14842.375 | |
0.9 | 183125835.4 | 781099881 | 4101.858 | 13852.2 |
Node | cpu time | ||||
Mean | Max | Mean | Max | ||
20 | 0.6 | 156866 | 373724 | 3.462 | 8.001 |
0.7 | 143956 | 399559 | 3.403 | 8.112 | |
0.8 | 119154 | 438721 | 2.963 | 8.898 | |
0.9 | 231118 | 708148 | 5.446 | 14.991 | |
25 | 0.6 | 16365534 | 120686189 | 476.151 | 3476.91 |
0.7 | 6657363 | 24736334 | 226.272 | 682.129 | |
0.8 | 2057701 | 7619907 | 69.852 | 228.938 | |
0.9 | 1654445 | 5476902 | 61.673 | 197.13 | |
30 | 0.6 | 57205995.4 | 271309428 | 2034.489 | 9230.5 |
0.7 | 163983353 | 793164929 | 3551.163 | 16301.3 | |
0.8 | 166549081.2 | 792800370 | 3535.774 | 14842.375 | |
0.9 | 183125835.4 | 781099881 | 4101.858 | 13852.2 |
Error percentage | cpu time | ||||||||||||||||||
Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | ||
20 | 0.6 | 44.876 | 81.861 | 0.019 | 0.192 | 3.404 | 5.749 | 0 | 0 | 0 | 0 | 0 | 0 | 0.218 | 0.234 | 0.212 | 0.234 | 0.217 | 0.219 |
0.7 | 61.42 | 100.792 | 0.016 | 0.163 | 4.101 | 9.411 | 0 | 0 | 0 | 0 | 0 | 0 | 0.218 | 0.234 | 0.215 | 0.234 | 0.217 | 0.234 | |
0.8 | 59.038 | 104.588 | 0.066 | 0.618 | 3.054 | 5.542 | 0 | 0 | 0 | 0 | 0 | 0 | 0.219 | 0.234 | 0.214 | 0.234 | 0.217 | 0.219 | |
0.9 | 56.364 | 87.247 | 0.03 | 0.242 | 3.316 | 7.171 | 0 | 0 | 0 | 0 | 0 | 0 | 0.226 | 0.234 | 0.225 | 0.249 | 0.223 | 0.234 | |
25 | 0.6 | 58.287 | 78.022 | 0 | 0 | 2.622 | 4.439 | 0 | 0 | 0 | 0 | 0 | 0 | 0.3 | 0.312 | 0.295 | 0.312 | 0.296 | 0.312 |
0.7 | 50.031 | 64.443 | 0 | 0 | 3.474 | 5.486 | 0 | 0 | 0 | 0 | 0 | 0 | 0.304 | 0.327 | 0.298 | 0.312 | 0.301 | 0.327 | |
0.8 | 53.712 | 108.917 | 0.01 | 0.08 | 3.184 | 4.685 | 0 | 0 | 0 | 0 | 0 | 0 | 0.301 | 0.312 | 0.296 | 0.312 | 0.296 | 0.312 | |
0.9 | 55.843 | 69.099 | 0.051 | 0.364 | 3.007 | 5.37 | 0 | 0 | 0 | 0 | 0 | 0 | 0.3056 | 0.312 | 0.303 | 0.312 | 0.306 | 0.328 | |
30 | 0.6 | 61.811 | 92.105 | 0 | 0 | 2.77 | 3.85 | 0 | 0 | 0 | 0 | 0 | 0 | 0.43 | 0.483 | 0.418 | 0.468 | 0.424 | 0.468 |
0.7 | 67.222 | 85.626 | 0 | 0 | 2.827 | 3.469 | 0 | 0 | 0 | 0 | 0 | 0 | 0.399 | 0.406 | 0.39 | 0.405 | 0.393 | 0.406 | |
0.8 | 69.096 | 88.61 | 0 | 0 | 2.552 | 5.058 | 0 | 0 | 0 | 0 | 0 | 0 | 0.393 | 0.406 | 0.387 | 0.405 | 0.377 | 0.405 | |
0.9 | 64.549 | 85.241 | 0.01 | 0.026 | 2.787 | 3.487 | 0 | 0 | 0 | 0 | 0 | 0 | 0.412 | 0.421 | 0.399 | 0.406 | 0.399 | 0.406 |
Error percentage | cpu time | ||||||||||||||||||
Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | ||
20 | 0.6 | 44.876 | 81.861 | 0.019 | 0.192 | 3.404 | 5.749 | 0 | 0 | 0 | 0 | 0 | 0 | 0.218 | 0.234 | 0.212 | 0.234 | 0.217 | 0.219 |
0.7 | 61.42 | 100.792 | 0.016 | 0.163 | 4.101 | 9.411 | 0 | 0 | 0 | 0 | 0 | 0 | 0.218 | 0.234 | 0.215 | 0.234 | 0.217 | 0.234 | |
0.8 | 59.038 | 104.588 | 0.066 | 0.618 | 3.054 | 5.542 | 0 | 0 | 0 | 0 | 0 | 0 | 0.219 | 0.234 | 0.214 | 0.234 | 0.217 | 0.219 | |
0.9 | 56.364 | 87.247 | 0.03 | 0.242 | 3.316 | 7.171 | 0 | 0 | 0 | 0 | 0 | 0 | 0.226 | 0.234 | 0.225 | 0.249 | 0.223 | 0.234 | |
25 | 0.6 | 58.287 | 78.022 | 0 | 0 | 2.622 | 4.439 | 0 | 0 | 0 | 0 | 0 | 0 | 0.3 | 0.312 | 0.295 | 0.312 | 0.296 | 0.312 |
0.7 | 50.031 | 64.443 | 0 | 0 | 3.474 | 5.486 | 0 | 0 | 0 | 0 | 0 | 0 | 0.304 | 0.327 | 0.298 | 0.312 | 0.301 | 0.327 | |
0.8 | 53.712 | 108.917 | 0.01 | 0.08 | 3.184 | 4.685 | 0 | 0 | 0 | 0 | 0 | 0 | 0.301 | 0.312 | 0.296 | 0.312 | 0.296 | 0.312 | |
0.9 | 55.843 | 69.099 | 0.051 | 0.364 | 3.007 | 5.37 | 0 | 0 | 0 | 0 | 0 | 0 | 0.3056 | 0.312 | 0.303 | 0.312 | 0.306 | 0.328 | |
30 | 0.6 | 61.811 | 92.105 | 0 | 0 | 2.77 | 3.85 | 0 | 0 | 0 | 0 | 0 | 0 | 0.43 | 0.483 | 0.418 | 0.468 | 0.424 | 0.468 |
0.7 | 67.222 | 85.626 | 0 | 0 | 2.827 | 3.469 | 0 | 0 | 0 | 0 | 0 | 0 | 0.399 | 0.406 | 0.39 | 0.405 | 0.393 | 0.406 | |
0.8 | 69.096 | 88.61 | 0 | 0 | 2.552 | 5.058 | 0 | 0 | 0 | 0 | 0 | 0 | 0.393 | 0.406 | 0.387 | 0.405 | 0.377 | 0.405 | |
0.9 | 64.549 | 85.241 | 0.01 | 0.026 | 2.787 | 3.487 | 0 | 0 | 0 | 0 | 0 | 0 | 0.412 | 0.421 | 0.399 | 0.406 | 0.399 | 0.406 |
Relative deviance percentage | cpu time | ||||||||||||||||||
Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | ||
100 | 0.6 | 80.067 | 102.419 | 0 | 0 | 0.884 | 1.017 | 2.767 | 6.564 | 0 | 0 | 0.024 | 0.092 | 10.863 | 11.092 | 10.842 | 11.169 | 10.869 | 11.138 |
0.7 | 88.275 | 116.801 | 0.002 | 0.009 | 0.84 | 1.251 | 2.361 | 5.691 | 0 | 0 | 0.035 | 0.081 | 10.603 | 10.814 | 10.561 | 10.856 | 10.575 | 10.86 | |
0.8 | 84.937 | 108.155 | 0 | 0 | 0.912 | 1.124 | 3.762 | 7.213 | 0 | 0 | 0.025 | 0.04 | 10.066 | 10.564 | 9.964 | 10.381 | 10.034 | 10.422 | |
0.9 | 83.932 | 96.221 | 0 | 0 | 0.781 | 1.18 | 4.207 | 10.818 | 0 | 0 | 0.03 | 0.062 | 9.039 | 9.599 | 9.347 | 9.989 | 9.191 | 9.615 | |
150 | 0.6 | 87.157 | 100.618 | 0 | 0 | 0.599 | 0.759 | 6.623 | 9.635 | 0 | 0 | 0.082 | 0.126 | 22.663 | 23.141 | 22.515 | 23.06 | 22.424 | 22.87 |
0.7 | 88.804 | 100.072 | 0 | 0 | 0.602 | 0.677 | 6.866 | 10.731 | 0 | 0 | 0.091 | 0.113 | 21.9 | 22.214 | 21.759 | 22.277 | 21.755 | 22.386 | |
0.8 | 86.592 | 98.456 | 0 | 0 | 0.583 | 0.745 | 6.783 | 9.836 | 0 | 0 | 0.089 | 0.161 | 21.435 | 21.862 | 21.38 | 21.98 | 21.633 | 22.23 | |
0.9 | 84.431 | 106.413 | 0.003 | 0.005 | 0.593 | 0.687 | 6.664 | 9.043 | 0 | 0 | 0.08 | 0.147 | 20.213 | 20.688 | 20.404 | 21.097 | 20.466 | 21.138 | |
200 | 0.6 | 83.977 | 98.545 | 0 | 0 | 0.432 | 0.524 | 8.243 | 15.256 | 0 | 0 | 0.117 | 0.165 | 38.573 | 39.729 | 38.005 | 39.17 | 37.532 | 38.825 |
0.7 | 93.933 | 104.681 | 0 | 0 | 0.435 | 0.53 | 8.561 | 16.389 | 0 | 0 | 0.117 | 0.183 | 37.504 | 38.735 | 37.118 | 37.564 | 37.252 | 37.581 | |
0.8 | 96.236 | 105.467 | 0.002 | 0.011 | 0.447 | 0.577 | 8.572 | 16.24 | 0 | 0 | 0.114 | 0.179 | 37.424 | 38.548 | 37.275 | 38.61 | 37.213 | 38.205 | |
0.9 | 90.333 | 100.959 | 0 | 0 | 0.414 | 0.476 | 8.057 | 16.283 | 0 | 0 | 0.132 | 0.189 | 35.678 | 36.28 | 35.481 | 36.42 | 36.527 | 37.766 |
Relative deviance percentage | cpu time | ||||||||||||||||||
Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | Mean | Max | ||
100 | 0.6 | 80.067 | 102.419 | 0 | 0 | 0.884 | 1.017 | 2.767 | 6.564 | 0 | 0 | 0.024 | 0.092 | 10.863 | 11.092 | 10.842 | 11.169 | 10.869 | 11.138 |
0.7 | 88.275 | 116.801 | 0.002 | 0.009 | 0.84 | 1.251 | 2.361 | 5.691 | 0 | 0 | 0.035 | 0.081 | 10.603 | 10.814 | 10.561 | 10.856 | 10.575 | 10.86 | |
0.8 | 84.937 | 108.155 | 0 | 0 | 0.912 | 1.124 | 3.762 | 7.213 | 0 | 0 | 0.025 | 0.04 | 10.066 | 10.564 | 9.964 | 10.381 | 10.034 | 10.422 | |
0.9 | 83.932 | 96.221 | 0 | 0 | 0.781 | 1.18 | 4.207 | 10.818 | 0 | 0 | 0.03 | 0.062 | 9.039 | 9.599 | 9.347 | 9.989 | 9.191 | 9.615 | |
150 | 0.6 | 87.157 | 100.618 | 0 | 0 | 0.599 | 0.759 | 6.623 | 9.635 | 0 | 0 | 0.082 | 0.126 | 22.663 | 23.141 | 22.515 | 23.06 | 22.424 | 22.87 |
0.7 | 88.804 | 100.072 | 0 | 0 | 0.602 | 0.677 | 6.866 | 10.731 | 0 | 0 | 0.091 | 0.113 | 21.9 | 22.214 | 21.759 | 22.277 | 21.755 | 22.386 | |
0.8 | 86.592 | 98.456 | 0 | 0 | 0.583 | 0.745 | 6.783 | 9.836 | 0 | 0 | 0.089 | 0.161 | 21.435 | 21.862 | 21.38 | 21.98 | 21.633 | 22.23 | |
0.9 | 84.431 | 106.413 | 0.003 | 0.005 | 0.593 | 0.687 | 6.664 | 9.043 | 0 | 0 | 0.08 | 0.147 | 20.213 | 20.688 | 20.404 | 21.097 | 20.466 | 21.138 | |
200 | 0.6 | 83.977 | 98.545 | 0 | 0 | 0.432 | 0.524 | 8.243 | 15.256 | 0 | 0 | 0.117 | 0.165 | 38.573 | 39.729 | 38.005 | 39.17 | 37.532 | 38.825 |
0.7 | 93.933 | 104.681 | 0 | 0 | 0.435 | 0.53 | 8.561 | 16.389 | 0 | 0 | 0.117 | 0.183 | 37.504 | 38.735 | 37.118 | 37.564 | 37.252 | 37.581 | |
0.8 | 96.236 | 105.467 | 0.002 | 0.011 | 0.447 | 0.577 | 8.572 | 16.24 | 0 | 0 | 0.114 | 0.179 | 37.424 | 38.548 | 37.275 | 38.61 | 37.213 | 38.205 | |
0.9 | 90.333 | 100.959 | 0 | 0 | 0.414 | 0.476 | 8.057 | 16.283 | 0 | 0 | 0.132 | 0.189 | 35.678 | 36.28 | 35.481 | 36.42 | 36.527 | 37.766 |
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