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A superlinearly convergent hybrid algorithm for solving nonlinear programming
Scheduling jobs with controllable processing time, truncated job-dependent learning and deterioration effects
1. | School of Science, Shenyang Aerospace University, Shenyang 110136, China |
2. | Business School, Hunan University, Changsha 410082, Hunan, China |
3. | Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China |
In this paper, we consider single machine scheduling problems with controllable processing time (resource allocation), truncated job-dependent learning and deterioration effects. The goal is to find the optimal sequence of jobs and the optimal resource allocation separately for minimizing a cost function containing makespan (total completion time, total absolute differences in completion times) and/or total resource cost. For two different processing time functions, i.e., a linear and a convex function of the amount of a common continuously divisible resource allocated to the job, we solve them in polynomial time respectively.
References:
[1] |
A. Bachman, A. G. Janiak, I. B. Alidaee and N. K. Womer, Scheduling deteriorating jobs dependent on resources for the makespan minimization, In Operations Research Proceedings 2000: Selected Papers of the Symposium on Operations Research (OR 2000), Dresden: Springer, (2001), 29-34. |
[2] |
J. Bai, Z.-R. Li and X. Huang,
Single-machine group scheduling with general deterioration and learning effects, Applied Mathematical Modelling, 36 (2012), 1267-1274.
doi: 10.1016/j.apm.2011.07.068. |
[3] |
J. Bai, M.-Z. Wang and J.-B. Wang,
Single machine scheduling with a general exponential learning effect, Applied Mathematical Modelling, 36 (2012), 829-835.
doi: 10.1016/j.apm.2011.07.002. |
[4] |
D. Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research, 115 (1999), 173-178. Google Scholar |
[5] |
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. |
[6] |
T. C. E. Cheng, S.-R. Cheng, W.-H. Wu, P.-H. Hsu and C.-C. Wu, A two-agent single-machine scheduling problem with truncated sum-of-processing-times-based learning considerations, Computers & Industrial Engineering, 60 (2011), 534-541. Google Scholar |
[7] |
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. |
[8] |
S. Gawiejnowicz, Time-Dependent Scheduling, Springer-Verlag Berlin Heidelberg, 2008. |
[9] |
R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan,
Optimization and approximation in deterministic sequencing and scheduling: A survey, Annals of Discrete Mathematics, 5 (1979), 287-326.
doi: 10.1016/S0167-5060(08)70356-X. |
[10] |
P. Guo, W. Cheng and Y. Wang,
A general variable neighborhood search for single-machine total tardiness scheduling problem with step-deteriorating jobs, Journal of Industrial and Management Optimization, 10 (2014), 1071-1090.
doi: 10.3934/jimo.2014.10.1071. |
[11] | G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1976. Google Scholar |
[12] |
H. Hoogeveen,
Multicriteria scheduling, European Journal of Operational Research, 167 (2005), 592-623.
doi: 10.1016/j.ejor.2004.07.011. |
[13] |
I. Kacem and E. Levner,
An improved approximation scheme for scheduling a maintenance and proportional deteriorating jobs, Journal of Industrial and Management Optimization, 12 (2016), 811-817.
doi: 10.3934/jimo.2016.12.811. |
[14] |
J. J. Kanet, Minimizing variation of flow time in single machine systems, Management Science, 27 (1981), 1453-1459. Google Scholar |
[15] |
G. Li, M.-L. Luo, W.-J. Zhang and X.-Y. Wang, Single-machine due-window assignment scheduling based on common flow allowance, learning effect and resource allocation, International Journal of Production Research, 54 (2015), 1228-1241. Google Scholar |
[16] |
G. Mosheiov and J. B. Sidney,
Scheduling with general job-dependent learning curves, European Journal of Operational Research, 147 (2003), 665-670.
doi: 10.1016/S0377-2217(02)00358-2. |
[17] |
Y.-P. Niu, J. Wang and N. Yin,
Scheduling problems with effects of deterioration and truncated job-dependent learning, Journal of Applied Mathematics and Computing, 47 (2015), 315-325.
doi: 10.1007/s12190-014-0777-2. |
[18] |
J. Qian and G. Steiner,
Fast algorithms for scheduling with learning effects and time-dependent processing times on a single machine, European Journal of Operational Research, 225 (2013), 547-551.
doi: 10.1016/j.ejor.2012.09.013. |
[19] |
D. Shabtay and G. Steiner,
A survey of scheduling with controllable processing times, Discrete Applied Mathematics, 155 (2007), 1643-1666.
doi: 10.1016/j.dam.2007.02.003. |
[20] |
J.-B. Wang and M.-Z. Wang,
Minimizing makespan in three-machine flow shops with deteriorating jobs, Computers & Operations Research, 30 (2013), 1350022, 14 pp.
doi: 10.1142/S021759591350022X. |
[21] |
X.-R. Wang and J.-J. Wang,
Single-machine scheduling with convex resource dependent processing times and deteriorating jobs, Applied Mathematical Modelling, 37 (2013), 2388-2393.
doi: 10.1016/j.apm.2012.05.025. |
[22] |
J.-B. Wang, M.-Z. Wang and P. Ji,
Scheduling jobs with processing times dependent on position, starting time and allotted resource, Asia-Pacific Journal of Operational Research, 29 (2012), 1250030 (15 pages).
doi: 10.1142/S0217595912500303. |
[23] |
X.-R. Wang, J.-B. Wang, J. Jin and P. Ji,
Single machine scheduling with truncated job-dependent learning effect, Optimization Letters, 8 (2014), 669-677.
doi: 10.1007/s11590-012-0579-0. |
[24] |
D. Wang, M.-Z. Wang and J.-B. Wang, Single-machine scheduling with learning effect and resource-dependent processing times, Computers & Industrial Engineering, 59 (2010), 458-462. Google Scholar |
[25] |
J.-B. Wang, X.-Y. Wang, L.-H. Sun and L.-Y. Sun,
Scheduling jobs with truncated exponential learning functions, Optimization Letters, 7 (2013), 1857-1873.
doi: 10.1007/s11590-011-0433-9. |
[26] |
X.-Y. Wang, Z. Zhou, X. Zhang, P. Ji and J.-B. Wang,
Several flow shop scheduling problems with truncated position-based learning effect, Computers & Operations Research, 40 (2013), 2906-2929.
doi: 10.1016/j.cor.2013.07.001. |
[27] |
C.-M. Wei, J.-B. Wang and P. Ji,
Single-machine scheduling with time-and-resource-dependent processing times, Applied Mathematical Modelling, 36 (2012), 792-798.
doi: 10.1016/j.apm.2011.07.005. |
[28] |
C.-C. Wu, Y. Yin and S.-R. Cheng, Some single-machine scheduling problems with a truncation learning effect, Computers & Industrial Engineering, 60 (2011), 790-795. Google Scholar |
[29] |
C.-C. Wu, Y. Yin and S.-R. Cheng, Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions, Journal of the Operation Research Society, 64 (2013), 147-156. Google Scholar |
[30] |
C.-C. Wu, Y. Yin, W.-H. Wu and S.-R. Cheng, Some polynomial solvable single-machine scheduling problems with a truncation sum-of-processing-times based learning effect, European Journal of Industrial Engineering, 6 (2012), 441-453. Google Scholar |
[31] |
W.-H. Wu, Y. Yin, W.-H. Wu, C.-C. Wu and P.-H. Hsu,
A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents, Journal of Industrial and Management Optimization, 10 (2014), 591-611.
doi: 10.3934/jimo.2014.10.591. |
[32] |
D. Xu, K. Sun and H. Li,
Parallel machine scheduling with almost periodic maintenance and non-preemptive jobs to minimize makespan, Computers & Operations Research, 35 (2008), 1344-1349.
doi: 10.1016/j.cor.2006.08.015. |
[33] |
D. Xu, L. Wan, A. Liu and D.-L. Yang, Single machine total completion time scheduling problem with workload-dependent maintenance duration, Omega-The International Journal of Management Science, 52 (2015), 101-106. Google Scholar |
[34] |
D.-L. Yang, T. C. E. Cheng and S.-J. Yang, Parallel-machine scheduling with controllable processing times and rate-modifying activities to minimise total cost involving total completion time and job compressions, International Journal of Production Research, 52 (2014), 1133-1141. Google Scholar |
[35] |
D.-L. Yang and W.-H. Kuo, Some scheduling problems with deteriorating jobs and learning effects, Computers & Industrial Engineering, 58 (2010), 25-28. Google Scholar |
[36] |
Y. Yin, S. -R. Cheng, J. Y. Chiang, J. C. H. Chen, X. Mao and C. -C. Wu, Scheduling problems with due date assignment, Discrete Dynamics in Nature and Society, 2015 (2015), Article ID 683269 (2 pages). Google Scholar |
[37] |
Y. Yin, T. C. E. Cheng, L. Wan, C.-C. Wu and J. Liu, Two-agent singlemachine scheduling with deteriorating jobs, Computers & Industrial Engineering, 81 (2015), 177-185. Google Scholar |
[38] |
Y. Yin, T. C. E. Cheng and C. -C. Wu, Scheduling with time-dependent processing times, Mathematical Problems in Engineering, 2015 (2015), Article ID 367585 (2 pages). Google Scholar |
[39] |
Y. Yin, T. C. E. Cheng, C.-C. Wu and S.-R. Cheng, Single-machine due window assignment and scheduling with a common flow allowance and controllable job processing time, Journal of the Operation Research Society, 65 (2014), 1-13. Google Scholar |
[40] |
N. Yin, L. Kang and X.-Y. Wang,
Single-machine group scheduling with processing times dependent on position, starting time and allotted resource, Applied Mathematical Modelling, 38 (2014), 4602-4613.
doi: 10.1016/j.apm.2014.03.014. |
[41] |
Y. Yin, D. -J. Wang, T. C. E. Cheng and C. -C. Wu, Bi-criterion single-machine scheduling and due window assignment with common flow allowances and resource-dependent processing times Journal of the Operation Research Society, (2016).
doi: 10.1057/jors.2016.14. |
[42] |
C. Zhao, C.-J. Hsu, W.-H. Wu, S.-R. Cheng and C.-C. Wu, Note on a unified approach to the single-machine scheduling problem with a deterioration effect and convex resource allocation, Journal of Manufacturing Systems, 38 (2016), 134-140. Google Scholar |
show all references
References:
[1] |
A. Bachman, A. G. Janiak, I. B. Alidaee and N. K. Womer, Scheduling deteriorating jobs dependent on resources for the makespan minimization, In Operations Research Proceedings 2000: Selected Papers of the Symposium on Operations Research (OR 2000), Dresden: Springer, (2001), 29-34. |
[2] |
J. Bai, Z.-R. Li and X. Huang,
Single-machine group scheduling with general deterioration and learning effects, Applied Mathematical Modelling, 36 (2012), 1267-1274.
doi: 10.1016/j.apm.2011.07.068. |
[3] |
J. Bai, M.-Z. Wang and J.-B. Wang,
Single machine scheduling with a general exponential learning effect, Applied Mathematical Modelling, 36 (2012), 829-835.
doi: 10.1016/j.apm.2011.07.002. |
[4] |
D. Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research, 115 (1999), 173-178. Google Scholar |
[5] |
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. |
[6] |
T. C. E. Cheng, S.-R. Cheng, W.-H. Wu, P.-H. Hsu and C.-C. Wu, A two-agent single-machine scheduling problem with truncated sum-of-processing-times-based learning considerations, Computers & Industrial Engineering, 60 (2011), 534-541. Google Scholar |
[7] |
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. |
[8] |
S. Gawiejnowicz, Time-Dependent Scheduling, Springer-Verlag Berlin Heidelberg, 2008. |
[9] |
R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan,
Optimization and approximation in deterministic sequencing and scheduling: A survey, Annals of Discrete Mathematics, 5 (1979), 287-326.
doi: 10.1016/S0167-5060(08)70356-X. |
[10] |
P. Guo, W. Cheng and Y. Wang,
A general variable neighborhood search for single-machine total tardiness scheduling problem with step-deteriorating jobs, Journal of Industrial and Management Optimization, 10 (2014), 1071-1090.
doi: 10.3934/jimo.2014.10.1071. |
[11] | G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1976. Google Scholar |
[12] |
H. Hoogeveen,
Multicriteria scheduling, European Journal of Operational Research, 167 (2005), 592-623.
doi: 10.1016/j.ejor.2004.07.011. |
[13] |
I. Kacem and E. Levner,
An improved approximation scheme for scheduling a maintenance and proportional deteriorating jobs, Journal of Industrial and Management Optimization, 12 (2016), 811-817.
doi: 10.3934/jimo.2016.12.811. |
[14] |
J. J. Kanet, Minimizing variation of flow time in single machine systems, Management Science, 27 (1981), 1453-1459. Google Scholar |
[15] |
G. Li, M.-L. Luo, W.-J. Zhang and X.-Y. Wang, Single-machine due-window assignment scheduling based on common flow allowance, learning effect and resource allocation, International Journal of Production Research, 54 (2015), 1228-1241. Google Scholar |
[16] |
G. Mosheiov and J. B. Sidney,
Scheduling with general job-dependent learning curves, European Journal of Operational Research, 147 (2003), 665-670.
doi: 10.1016/S0377-2217(02)00358-2. |
[17] |
Y.-P. Niu, J. Wang and N. Yin,
Scheduling problems with effects of deterioration and truncated job-dependent learning, Journal of Applied Mathematics and Computing, 47 (2015), 315-325.
doi: 10.1007/s12190-014-0777-2. |
[18] |
J. Qian and G. Steiner,
Fast algorithms for scheduling with learning effects and time-dependent processing times on a single machine, European Journal of Operational Research, 225 (2013), 547-551.
doi: 10.1016/j.ejor.2012.09.013. |
[19] |
D. Shabtay and G. Steiner,
A survey of scheduling with controllable processing times, Discrete Applied Mathematics, 155 (2007), 1643-1666.
doi: 10.1016/j.dam.2007.02.003. |
[20] |
J.-B. Wang and M.-Z. Wang,
Minimizing makespan in three-machine flow shops with deteriorating jobs, Computers & Operations Research, 30 (2013), 1350022, 14 pp.
doi: 10.1142/S021759591350022X. |
[21] |
X.-R. Wang and J.-J. Wang,
Single-machine scheduling with convex resource dependent processing times and deteriorating jobs, Applied Mathematical Modelling, 37 (2013), 2388-2393.
doi: 10.1016/j.apm.2012.05.025. |
[22] |
J.-B. Wang, M.-Z. Wang and P. Ji,
Scheduling jobs with processing times dependent on position, starting time and allotted resource, Asia-Pacific Journal of Operational Research, 29 (2012), 1250030 (15 pages).
doi: 10.1142/S0217595912500303. |
[23] |
X.-R. Wang, J.-B. Wang, J. Jin and P. Ji,
Single machine scheduling with truncated job-dependent learning effect, Optimization Letters, 8 (2014), 669-677.
doi: 10.1007/s11590-012-0579-0. |
[24] |
D. Wang, M.-Z. Wang and J.-B. Wang, Single-machine scheduling with learning effect and resource-dependent processing times, Computers & Industrial Engineering, 59 (2010), 458-462. Google Scholar |
[25] |
J.-B. Wang, X.-Y. Wang, L.-H. Sun and L.-Y. Sun,
Scheduling jobs with truncated exponential learning functions, Optimization Letters, 7 (2013), 1857-1873.
doi: 10.1007/s11590-011-0433-9. |
[26] |
X.-Y. Wang, Z. Zhou, X. Zhang, P. Ji and J.-B. Wang,
Several flow shop scheduling problems with truncated position-based learning effect, Computers & Operations Research, 40 (2013), 2906-2929.
doi: 10.1016/j.cor.2013.07.001. |
[27] |
C.-M. Wei, J.-B. Wang and P. Ji,
Single-machine scheduling with time-and-resource-dependent processing times, Applied Mathematical Modelling, 36 (2012), 792-798.
doi: 10.1016/j.apm.2011.07.005. |
[28] |
C.-C. Wu, Y. Yin and S.-R. Cheng, Some single-machine scheduling problems with a truncation learning effect, Computers & Industrial Engineering, 60 (2011), 790-795. Google Scholar |
[29] |
C.-C. Wu, Y. Yin and S.-R. Cheng, Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions, Journal of the Operation Research Society, 64 (2013), 147-156. Google Scholar |
[30] |
C.-C. Wu, Y. Yin, W.-H. Wu and S.-R. Cheng, Some polynomial solvable single-machine scheduling problems with a truncation sum-of-processing-times based learning effect, European Journal of Industrial Engineering, 6 (2012), 441-453. Google Scholar |
[31] |
W.-H. Wu, Y. Yin, W.-H. Wu, C.-C. Wu and P.-H. Hsu,
A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents, Journal of Industrial and Management Optimization, 10 (2014), 591-611.
doi: 10.3934/jimo.2014.10.591. |
[32] |
D. Xu, K. Sun and H. Li,
Parallel machine scheduling with almost periodic maintenance and non-preemptive jobs to minimize makespan, Computers & Operations Research, 35 (2008), 1344-1349.
doi: 10.1016/j.cor.2006.08.015. |
[33] |
D. Xu, L. Wan, A. Liu and D.-L. Yang, Single machine total completion time scheduling problem with workload-dependent maintenance duration, Omega-The International Journal of Management Science, 52 (2015), 101-106. Google Scholar |
[34] |
D.-L. Yang, T. C. E. Cheng and S.-J. Yang, Parallel-machine scheduling with controllable processing times and rate-modifying activities to minimise total cost involving total completion time and job compressions, International Journal of Production Research, 52 (2014), 1133-1141. Google Scholar |
[35] |
D.-L. Yang and W.-H. Kuo, Some scheduling problems with deteriorating jobs and learning effects, Computers & Industrial Engineering, 58 (2010), 25-28. Google Scholar |
[36] |
Y. Yin, S. -R. Cheng, J. Y. Chiang, J. C. H. Chen, X. Mao and C. -C. Wu, Scheduling problems with due date assignment, Discrete Dynamics in Nature and Society, 2015 (2015), Article ID 683269 (2 pages). Google Scholar |
[37] |
Y. Yin, T. C. E. Cheng, L. Wan, C.-C. Wu and J. Liu, Two-agent singlemachine scheduling with deteriorating jobs, Computers & Industrial Engineering, 81 (2015), 177-185. Google Scholar |
[38] |
Y. Yin, T. C. E. Cheng and C. -C. Wu, Scheduling with time-dependent processing times, Mathematical Problems in Engineering, 2015 (2015), Article ID 367585 (2 pages). Google Scholar |
[39] |
Y. Yin, T. C. E. Cheng, C.-C. Wu and S.-R. Cheng, Single-machine due window assignment and scheduling with a common flow allowance and controllable job processing time, Journal of the Operation Research Society, 65 (2014), 1-13. Google Scholar |
[40] |
N. Yin, L. Kang and X.-Y. Wang,
Single-machine group scheduling with processing times dependent on position, starting time and allotted resource, Applied Mathematical Modelling, 38 (2014), 4602-4613.
doi: 10.1016/j.apm.2014.03.014. |
[41] |
Y. Yin, D. -J. Wang, T. C. E. Cheng and C. -C. Wu, Bi-criterion single-machine scheduling and due window assignment with common flow allowances and resource-dependent processing times Journal of the Operation Research Society, (2016).
doi: 10.1057/jors.2016.14. |
[42] |
C. Zhao, C.-J. Hsu, W.-H. Wu, S.-R. Cheng and C.-C. Wu, Note on a unified approach to the single-machine scheduling problem with a deterioration effect and convex resource allocation, Journal of Manufacturing Systems, 38 (2016), 134-140. Google Scholar |
10 | 8 | 11 | 18 | 9 | 16 | |
2 | 1 | 3 | 2 | 3 | 4 | |
3 | 2 | 3 | 1 | 2 | 2 | |
10 | 8 | 12 | 11 | 14 | 9 | |
-0.25 | -0.15 | -0.2 | -0.1 | -0.3 | -0.25 |
10 | 8 | 11 | 18 | 9 | 16 | |
2 | 1 | 3 | 2 | 3 | 4 | |
3 | 2 | 3 | 1 | 2 | 2 | |
10 | 8 | 12 | 11 | 14 | 9 | |
-0.25 | -0.15 | -0.2 | -0.1 | -0.3 | -0.25 |
57.2076 | 43.3110 | 32.7497 | 22.2915 | 14.3500 | 7.0000 | |
54.4152 | 39.8396 | 29.2421 | 20.4850 | 12.8824 | 6.1146 | |
49.6038 | 39.1831 | 35.2680 | 26.2806 | 16.3438 | 7.7000 | |
119.8304 | 92.7490 | 69.5101 | 49.3994 | 31.4144 | 15.0473 | |
48.4057 | 35.2400 | 27.8993 | 19.8607 | 12.9150 | 6.3000 | |
72.4152 | 48.1385 | 35.9187 | 28.4465 | 22.9600 | 11.2000 |
57.2076 | 43.3110 | 32.7497 | 22.2915 | 14.3500 | 7.0000 | |
54.4152 | 39.8396 | 29.2421 | 20.4850 | 12.8824 | 6.1146 | |
49.6038 | 39.1831 | 35.2680 | 26.2806 | 16.3438 | 7.7000 | |
119.8304 | 92.7490 | 69.5101 | 49.3994 | 31.4144 | 15.0473 | |
48.4057 | 35.2400 | 27.8993 | 19.8607 | 12.9150 | 6.3000 | |
72.4152 | 48.1385 | 35.9187 | 28.4465 | 22.9600 | 11.2000 |
10 | 8 | 11 | 18 | 9 | 1 | |
10 | 8 | 12 | 11 | 14 | 9 | |
-0.25 | -0.15 | -0.2 | -0.1 | -0.3 | -0.25 |
10 | 8 | 11 | 18 | 9 | 1 | |
10 | 8 | 12 | 11 | 14 | 9 | |
-0.25 | -0.15 | -0.2 | -0.1 | -0.3 | -0.25 |
77.1456 | 64.1292 | 55.1751 | 47.3852 | 40.7772 | 32.0996 | |
57.2925 | 49.8783 | 44.0898 | 38.5982 | 32.7021 | 25.2778 | |
92.8312 | 78.9720 | 68.8700 | 59.7165 | 50.2195 | 38.6263 | |
121.6433 | 108.3767 | 97.1029 | 85.8274 | 73.2596 | 56.9729 | |
89.9964 | 73.1030 | 62.0516 | 54.9075 | 47.5698 | 37.4467 | |
98.3754 | 81.7770 | 70.3588 | 60.4252 | 51.9987 | 40.9331 | |
The bold numbers are the optimal solution |
77.1456 | 64.1292 | 55.1751 | 47.3852 | 40.7772 | 32.0996 | |
57.2925 | 49.8783 | 44.0898 | 38.5982 | 32.7021 | 25.2778 | |
92.8312 | 78.9720 | 68.8700 | 59.7165 | 50.2195 | 38.6263 | |
121.6433 | 108.3767 | 97.1029 | 85.8274 | 73.2596 | 56.9729 | |
89.9964 | 73.1030 | 62.0516 | 54.9075 | 47.5698 | 37.4467 | |
98.3754 | 81.7770 | 70.3588 | 60.4252 | 51.9987 | 40.9331 | |
The bold numbers are the optimal solution |
Theorem 3.3 | ||
Theorem 4.4 | ||
Theorem 4.6 | ||
Theorem 4.9 | ||
Theorem 4.10 | ||
Theorem 4.13 | ||
Theorem 4.14 | ||
Theorem 4.15 | ||
Theorem 4.16 |
Theorem 3.3 | ||
Theorem 4.4 | ||
Theorem 4.6 | ||
Theorem 4.9 | ||
Theorem 4.10 | ||
Theorem 4.13 | ||
Theorem 4.14 | ||
Theorem 4.15 | ||
Theorem 4.16 |
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