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April  2017, 13(2): 1025-1039. doi: 10.3934/jimo.2016060

Scheduling jobs with controllable processing time, truncated job-dependent learning and deterioration effects

Ji-Bo Wang 1, , Mengqi Liu 2,, , Na Yin 1, and  Ping Ji 3,

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

* Corresponding author

Received  October 2015 Revised  June 2016 Published  October 2016

Fund Project: The work described in this paper was partially supported by the grant from The Hong Kong Polytechnic University (PolyU projects G-YBFE and 4-BCBJ) and the National Natural Science Foundation of China (Grant Nos. 71471120 and 71471057).

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.

Keywords: Scheduling, resource allocation, deterioration job, learning effect.
Mathematics Subject Classification: Primary: 90B35; Secondary: 90C26.
Citation: Ji-Bo Wang, Mengqi Liu, Na Yin, Ping Ji. Scheduling jobs with controllable processing time, truncated job-dependent learning and deterioration effects. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1025-1039. doi: 10.3934/jimo.2016060
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Gawiejnowicz, Time-Dependent Scheduling, Springer-Verlag Berlin Heidelberg, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Gawiejnowicz, Time-Dependent Scheduling, Springer-Verlag Berlin Heidelberg, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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

Table 1.  Data of Example 1
$J_{j}$$J_{1}$$J_{2}$$J_{3}$$J_{4}$$J_{5}$$J_{6}$
$p_{j}$1081118916
$\beta_{j}$213234
$\bar{u}_{j}$323122
$v_{j}$1081211149
$a_{j}$-0.25-0.15-0.2-0.1-0.3-0.25
Table Options

Download as excel

$J_{j}$$J_{1}$$J_{2}$$J_{3}$$J_{4}$$J_{5}$$J_{6}$
$p_{j}$1081118916
$\beta_{j}$213234
$\bar{u}_{j}$323122
$v_{j}$1081211149
$a_{j}$-0.25-0.15-0.2-0.1-0.3-0.25
Table 2.  Values of $\Lambda_{jr}$
${j\backslash r}$${1}$${2}$${3}$${4}$${5}$${6}$
$1$57.207643.311032.749722.291514.35007.0000
$2$54.415239.839629.242120.485012.88246.1146
$3$49.603839.183135.268026.280616.34387.7000
$4$119.830492.749069.510149.399431.414415.0473
$5$48.405735.240027.899319.860712.91506.3000
$6$72.415248.138535.918728.446522.960011.2000
Table Options

Download as excel

${j\backslash r}$${1}$${2}$${3}$${4}$${5}$${6}$
$1$57.207643.311032.749722.291514.35007.0000
$2$54.415239.839629.242120.485012.88246.1146
$3$49.603839.183135.268026.280616.34387.7000
$4$119.830492.749069.510149.399431.414415.0473
$5$48.405735.240027.899319.860712.91506.3000
$6$72.415248.138535.918728.446522.960011.2000
Table 3.  Data of Example 2
$J_{j}$$J_{1}$$J_{2}$$J_{3}$$J_{4}$$J_{5}$$J_{6}$
$p_{j}$108111891
$v_{j}$1081211149
$a_{j}$-0.25-0.15-0.2-0.1-0.3-0.25
Table Options

Download as excel

$J_{j}$$J_{1}$$J_{2}$$J_{3}$$J_{4}$$J_{5}$$J_{6}$
$p_{j}$108111891
$v_{j}$1081211149
$a_{j}$-0.25-0.15-0.2-0.1-0.3-0.25
Table 4.  Values of $\Theta_{jr}$
${j\backslash r}$${1}$${2}$${3}$${4}$${5}$${6}$
$1$77.145664.129255.175147.385240.777232.0996
$2$57.292549.878344.089838.598232.702125.2778
$3$92.831278.972068.870059.716550.219538.6263
$4$121.6433108.376797.102985.827473.259656.9729
$5$89.996473.103062.051654.907547.569837.4467
$6$98.375481.777070.358860.425251.998740.9331
The bold numbers are the optimal solution
Table Options

Download as excel

${j\backslash r}$${1}$${2}$${3}$${4}$${5}$${6}$
$1$77.145664.129255.175147.385240.777232.0996
$2$57.292549.878344.089838.598232.702125.2778
$3$92.831278.972068.870059.716550.219538.6263
$4$121.6433108.376797.102985.827473.259656.9729
$5$89.996473.103062.051654.907547.569837.4467
$6$98.375481.777070.358860.425251.998740.9331
The bold numbers are the optimal solution
Table 5.  Main results of this paper ($\rho\in\{C_{\max},\sum C_j, TADC\}$)
$1|p_{jr}^A(t,u_j)=p_j\max\left\{r^{a_j},b\right\}+c t-\theta_{j} u_{j}|\delta_1 \rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n^3)$Theorem 3.3
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct|\delta_1\rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n^3)$Theorem 4.4
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct|\delta_1\rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n\log n)$Theorem 4.6
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct,\sum_{j=1}^{n}u_{j}\leq U |\rho$$O(n^3)$Theorem 4.9
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct,\sum_{j=1}^{n}u_{j}\leq U|\rho$$O(n\log n)$Theorem 4.10
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct,\rho\leq R|\sum_{j=1}^{n}u_{j}$$O(n^3)$Theorem 4.13
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct,\rho\leq R| \sum_{j=1}^{n}u_{j}$$O(n\log n)$Theorem 4.14
$1|p_{j}^A= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +c t|(\rho,\sum_{j=1}^{n}u_{j}) $$O(n^3)$Theorem 4.15
$1|p_{j}^A= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +c t|(\rho,\sum_{j=1}^{n}u_{j}) $$O(n\log n)$Theorem 4.16
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$1|p_{jr}^A(t,u_j)=p_j\max\left\{r^{a_j},b\right\}+c t-\theta_{j} u_{j}|\delta_1 \rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n^3)$Theorem 3.3
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct|\delta_1\rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n^3)$Theorem 4.4
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct|\delta_1\rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n\log n)$Theorem 4.6
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct,\sum_{j=1}^{n}u_{j}\leq U |\rho$$O(n^3)$Theorem 4.9
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct,\sum_{j=1}^{n}u_{j}\leq U|\rho$$O(n\log n)$Theorem 4.10
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct,\rho\leq R|\sum_{j=1}^{n}u_{j}$$O(n^3)$Theorem 4.13
$1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct,\rho\leq R| \sum_{j=1}^{n}u_{j}$$O(n\log n)$Theorem 4.14
$1|p_{j}^A= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +c t|(\rho,\sum_{j=1}^{n}u_{j}) $$O(n^3)$Theorem 4.15
$1|p_{j}^A= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +c t|(\rho,\sum_{j=1}^{n}u_{j}) $$O(n\log n)$Theorem 4.16

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