July  2019, 15(3): 1117-1131. doi: 10.3934/jimo.2018088

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

* Corresponding author

Received  October 2016 Revised  March 2018 Published  July 2018

Fund Project: This research is partly supported by National Natural Science Foundation (Grant No. U1433124), the Humanities and Social Science Fund of the Ministry of Education (Grant No. 17YJA630139), Doctor Foundation of Liaoning Province (Grant No. 20170520175) and Social Science Planning Fund of Liaoning Province (Grant No. L16DGL007). Ji-Bo Wang was partially supported by the National Natural Science Foundation of China (Grant No. 71471120), the Support Program for Innovative Talents in Liaoning University (grant no. LR2016017) and the Liaoning BaiQianWan Talents Program.

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.

Citation: Ping Yan, Ji-Bo Wang, Li-Qiang Zhao. Single-machine bi-criterion scheduling with release times and exponentially time-dependent learning effects. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1117-1131. doi: 10.3934/jimo.2018088
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.  Google Scholar

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M. ChengP.-R. TadikamallJ. 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.  Google Scholar

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A. JaniakM.-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.  Google Scholar

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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.  Google Scholar

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W.-C. LeeC.-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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

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L.-H. SunK. CuiJ.-H. ChenJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

C.-C. WuP.-H. HsuJ.-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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T.C.E. ChengW.-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.  Google Scholar

[4]

T.C.E. ChengW.-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.  Google Scholar

[5]

M. ChengP.-R. TadikamallJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. HaddarM. KhemakhemS. 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.  Google Scholar

[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.  Google Scholar

[10]

A. JaniakM.-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.  Google Scholar

[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.  Google Scholar

[12]

W.-C. LeeC.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y.-R. ShiauM.-S. TsaiW.-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.  Google Scholar

[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.  Google Scholar

[21]

L.-H. SunK. CuiJ.-H. ChenJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

C.-C. WuP.-H. HsuJ.-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.  Google Scholar

[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.  Google Scholar

Table 1.  Conversion process from a job permutation to a position vector of a particle
Dimension 1 2 3 4 5
Job permutation 4 1 3 5 2
${s_{HA, j}}$ 2 5 3 1 4
${x_{HA, j}}$ 2.09 -3.88 0.33 3.15 -2.11
Dimension 1 2 3 4 5
Job permutation 4 1 3 5 2
${s_{HA, j}}$ 2 5 3 1 4
${x_{HA, j}}$ 2.09 -3.88 0.33 3.15 -2.11
Table 2.  The performance of the branch-and-bound algorithm
$n$ $\alpha$ 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
$n$ $\alpha$ 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
Table 3.  The performance of the heuristic and PSO algorithms with $n$ = 20, 25, 30
$n$ $\alpha$ Error percentage cpu time
$H{A_1}$ $H{A_2}$ $H{A_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$
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
$n$ $\alpha$ Error percentage cpu time
$H{A_1}$ $H{A_2}$ $H{A_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$
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
Table 4.  The performance of the heuristic and PSO algorithms for large-size jobs
$n$ $\alpha$ Relative deviance percentage cpu time
$H{A_1}$ $H{A_2}$ $H{A_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$
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
$n$ $\alpha$ Relative deviance percentage cpu time
$H{A_1}$ $H{A_2}$ $H{A_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$ $PS{O_1}$ $PS{O_2}$ $PS{O_3}$
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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