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Resource allocation flowshop scheduling with learning effect and slack due window assignment
Department of Basic, Shenyang Sport University, Shenyang 110102, China |
We study flowshop scheduling problems with respect to slack due window assignments, which are operations in which jobs are assigned an individual due window. We combine learning effect and controllable processing times, in which the flowshop has a two-machine no-wait setup. The goal is to determine job sequence, slack due window based on common flow allowance, due window size, and resource allocation. We provide a bicriteria analysis for the scheduling and resource consumption costs. We show that the two costs can be solved in polynomial time utilizing three different combinations.
References:
[1] |
A. Allahverdi,
A survey of scheduling problems with no-wait in process, European J. Oper. Res., 255 (2016), 665-686.
doi: 10.1016/j.ejor.2016.05.036. |
[2] |
A. Azzouz, M. Ennigrou and L. B. Said,
Scheduling problems under learning effects: Classification and cartography, Internat. J. Prod. Res., 56 (2018), 1642-1661.
doi: 10.1080/00207543.2017.1355576. |
[3] |
D. Biskup,
A state-of-the-art review on scheduling with learning effects, European J. Oper. Res., 188 (2008), 315-329.
doi: 10.1016/j.ejor.2007.05.040. |
[4] |
F. Gao, M. Liu, J.-J. Wang and Y.-Y. Lu,
No-wait two-machine permutation flow shop scheduling problem with learning effect, common due date and controllable job processing times, Internat. J. Prod. Res., 56 (2018), 2361-2369.
doi: 10.1080/00207543.2017.1371353. |
[5] |
X.-N. Geng, J.-B. Wang and D. Bai,
Common due date assignment scheduling for a no-wait flowshop with convex resource allocation and learning effect, Eng. Optim., 51 (2019), 1301-1323.
doi: 10.1080/0305215X.2018.1521397. |
[6] |
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, Ann. Discrete Math., 5 (1979), 287-326.
doi: 10.1016/S0167-5060(08)70356-X. |
[7] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.
![]() |
[8] |
A. Janiak, W. A. Janiak, T. Krysiak and T. Kwiatkowski,
A survey on scheduling problems with due windows, European J. Oper. Res., 242 (2015), 347-357.
doi: 10.1016/j.ejor.2014.09.043. |
[9] |
S. Khalilpourazari and M. Mohammadi, A new exact algorithm for solving single machine scheduling problems with learning effects and deteriorating jobs, preprint, arXiv: 1809.03795. Google Scholar |
[10] |
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, Internat. J. Prod. Res., 53 (2015), 1228-1241.
doi: 10.1080/00207543.2014.954057. |
[11] |
X.-X. Liang, M. Liu, Y.-B. Feng, J.-B. Wang and L.-S. Wen, Solution algorithms for single-machine resource allocation scheduling with deteriorating jobs and group technology, Engineering Optimization, (2019).
doi: 10.1080/0305215X.2019.1638920. |
[12] |
Y. Liu and Z. Feng,
Two-machine no-wait flowshop scheduling with learning effect and convex resource-dependent processing times, Comput. Industrial Engineering, 75 (2014), 170-175.
doi: 10.1016/j.cie.2014.06.017. |
[13] |
Z. Liu, Z. Wang and Y.-Y. Lu, A bicriteria approach for single machine scheduling with resource allocation, learning effect and a deteriorating maintenance activity, Asia-Pac. J. Oper. Res., 34 (2017), 16pp.
doi: 10.1142/S0217595917500117. |
[14] |
Y.-Y. Lu, G. Li, Y.-B. Wu and P. Ji,
Optimal due-date assignment problem with learning effect and resource-dependent processing times, Optim. Lett., 8 (2014), 113-127.
doi: 10.1007/s11590-012-0467-7. |
[15] |
Y.-Y. Lu, J.-B. Wang, P. Ji and H. He,
A note on resource allocation scheduling with group technology and learning effects on a single machine, Eng. Optim., 49 (2017), 1621-1632.
doi: 10.1080/0305215X.2016.1265305. |
[16] |
M. Mohammadi and S. Khalilpourazari, Minimizing makespan in a single machine scheduling problem with deteriorating jobs and learning effects, Proceedings of the 6th International Conference on Software and Computer Applications, 2017,310-315.
doi: 10.1145/3056662.3056715. |
[17] |
G. Mosheiov and D. Oron, Job-dependent due-window assignment based on a common flow allowance, Foundations Comput. Decision Sciences, 35 (2010), 185-195. Google Scholar |
[18] |
D. Shabtay and G. Steiner,
A survey of scheduling with controllable processing times, Discrete Appl. Math., 155 (2007), 1643-1666.
doi: 10.1016/j.dam.2007.02.003. |
[19] |
H.-B. Shi and J.-B. Wang,
Research on common due window assignment flow shop scheduling with learning effect and resource allocation, Eng. Optim., 52 (2020), 669-686.
doi: 10.1080/0305215X.2019.1604698. |
[20] |
X. Sun, X. Geng, J.-B. Wang and F. Liu,
Convex resource allocation scheduling in the no-wait flowshop with common flow allowance and learning effect, Internat. J. Prod. Res., 57 (2019), 1873-1891.
doi: 10.1080/00207543.2018.1510559. |
[21] |
Y. Tian, M. Xu, C. Jiang, J.-B. Wang and X.-Y. Wang,
No-wait resource allocation flowshop scheduling with learning effect under limited cost availability, Comput. J., 62 (2019), 90-96.
doi: 10.1093/comjnl/bxy034. |
[22] |
J.-B. Wang, X.-N. Geng, L. Liu, J.-J. Wang and Y.-Y. Lu,
Single machine CON/SLK due date assignment scheduling with controllable processing time and job-dependent learning effects, Comput. J., 61 (2018), 1329-1337.
doi: 10.1093/comjnl/bxx121. |
[23] |
J.-B. Wang, M. Gao, J.-J. Wang, L. Liu and H. He, Scheduling with a position-weighted learning effect and job release dates, Engineering Optimization, (2019).
doi: 10.1080/0305215X.2019.1664498. |
[24] |
J.-B. Wang and L. Li,
Machine scheduling with deteriorating jobs and modifying maintenance activities, Comput. J., 61 (2018), 47-53.
doi: 10.1093/comjnl/bxx032. |
[25] |
J.-B. Wang, F. Liu and J.-J. Wang,
Research on $m$-machine flow shop scheduling with truncated learning effects, Int. Trans. Oper. Res., 26 (2019), 1135-1151.
doi: 10.1111/itor.12323. |
[26] |
J.-B. Wang, and M.-Z. Wang, Single-machine due-window assignment and scheduling with learning effect and resource-dependent processing times, Asia-Pac. J. Oper. Res., 31 (2014), 15pp.
doi: 10.1142/S0217595914500365. |
[27] |
J.-B. Wang and J.-J. Wang,
Research on scheduling with job-dependent learning effect and convex resource-dependent processing times, Internat. J. Prod. Res., 53 (2015), 5826-5836.
doi: 10.1080/00207543.2015.1010746. |
[28] |
D. Wang, M.-Z. Wang and J.-B. Wang,
Single-machine scheduling with learning effect and resource-dependent processing times, Comput. Industrial Engineering, 59 (2010), 458-462.
doi: 10.1016/j.cie.2010.06.002. |
[29] |
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, J. Oper. Res. Soc., 65 (2014), 1-13.
doi: 10.1057/jors.2012.161. |
[30] |
Y. Yin, D. 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, J. Oper. Res. Soc., 67 (2016), 1169-1183.
doi: 10.1057/jors.2016.14. |
show all references
References:
[1] |
A. Allahverdi,
A survey of scheduling problems with no-wait in process, European J. Oper. Res., 255 (2016), 665-686.
doi: 10.1016/j.ejor.2016.05.036. |
[2] |
A. Azzouz, M. Ennigrou and L. B. Said,
Scheduling problems under learning effects: Classification and cartography, Internat. J. Prod. Res., 56 (2018), 1642-1661.
doi: 10.1080/00207543.2017.1355576. |
[3] |
D. Biskup,
A state-of-the-art review on scheduling with learning effects, European J. Oper. Res., 188 (2008), 315-329.
doi: 10.1016/j.ejor.2007.05.040. |
[4] |
F. Gao, M. Liu, J.-J. Wang and Y.-Y. Lu,
No-wait two-machine permutation flow shop scheduling problem with learning effect, common due date and controllable job processing times, Internat. J. Prod. Res., 56 (2018), 2361-2369.
doi: 10.1080/00207543.2017.1371353. |
[5] |
X.-N. Geng, J.-B. Wang and D. Bai,
Common due date assignment scheduling for a no-wait flowshop with convex resource allocation and learning effect, Eng. Optim., 51 (2019), 1301-1323.
doi: 10.1080/0305215X.2018.1521397. |
[6] |
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, Ann. Discrete Math., 5 (1979), 287-326.
doi: 10.1016/S0167-5060(08)70356-X. |
[7] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.
![]() |
[8] |
A. Janiak, W. A. Janiak, T. Krysiak and T. Kwiatkowski,
A survey on scheduling problems with due windows, European J. Oper. Res., 242 (2015), 347-357.
doi: 10.1016/j.ejor.2014.09.043. |
[9] |
S. Khalilpourazari and M. Mohammadi, A new exact algorithm for solving single machine scheduling problems with learning effects and deteriorating jobs, preprint, arXiv: 1809.03795. Google Scholar |
[10] |
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, Internat. J. Prod. Res., 53 (2015), 1228-1241.
doi: 10.1080/00207543.2014.954057. |
[11] |
X.-X. Liang, M. Liu, Y.-B. Feng, J.-B. Wang and L.-S. Wen, Solution algorithms for single-machine resource allocation scheduling with deteriorating jobs and group technology, Engineering Optimization, (2019).
doi: 10.1080/0305215X.2019.1638920. |
[12] |
Y. Liu and Z. Feng,
Two-machine no-wait flowshop scheduling with learning effect and convex resource-dependent processing times, Comput. Industrial Engineering, 75 (2014), 170-175.
doi: 10.1016/j.cie.2014.06.017. |
[13] |
Z. Liu, Z. Wang and Y.-Y. Lu, A bicriteria approach for single machine scheduling with resource allocation, learning effect and a deteriorating maintenance activity, Asia-Pac. J. Oper. Res., 34 (2017), 16pp.
doi: 10.1142/S0217595917500117. |
[14] |
Y.-Y. Lu, G. Li, Y.-B. Wu and P. Ji,
Optimal due-date assignment problem with learning effect and resource-dependent processing times, Optim. Lett., 8 (2014), 113-127.
doi: 10.1007/s11590-012-0467-7. |
[15] |
Y.-Y. Lu, J.-B. Wang, P. Ji and H. He,
A note on resource allocation scheduling with group technology and learning effects on a single machine, Eng. Optim., 49 (2017), 1621-1632.
doi: 10.1080/0305215X.2016.1265305. |
[16] |
M. Mohammadi and S. Khalilpourazari, Minimizing makespan in a single machine scheduling problem with deteriorating jobs and learning effects, Proceedings of the 6th International Conference on Software and Computer Applications, 2017,310-315.
doi: 10.1145/3056662.3056715. |
[17] |
G. Mosheiov and D. Oron, Job-dependent due-window assignment based on a common flow allowance, Foundations Comput. Decision Sciences, 35 (2010), 185-195. Google Scholar |
[18] |
D. Shabtay and G. Steiner,
A survey of scheduling with controllable processing times, Discrete Appl. Math., 155 (2007), 1643-1666.
doi: 10.1016/j.dam.2007.02.003. |
[19] |
H.-B. Shi and J.-B. Wang,
Research on common due window assignment flow shop scheduling with learning effect and resource allocation, Eng. Optim., 52 (2020), 669-686.
doi: 10.1080/0305215X.2019.1604698. |
[20] |
X. Sun, X. Geng, J.-B. Wang and F. Liu,
Convex resource allocation scheduling in the no-wait flowshop with common flow allowance and learning effect, Internat. J. Prod. Res., 57 (2019), 1873-1891.
doi: 10.1080/00207543.2018.1510559. |
[21] |
Y. Tian, M. Xu, C. Jiang, J.-B. Wang and X.-Y. Wang,
No-wait resource allocation flowshop scheduling with learning effect under limited cost availability, Comput. J., 62 (2019), 90-96.
doi: 10.1093/comjnl/bxy034. |
[22] |
J.-B. Wang, X.-N. Geng, L. Liu, J.-J. Wang and Y.-Y. Lu,
Single machine CON/SLK due date assignment scheduling with controllable processing time and job-dependent learning effects, Comput. J., 61 (2018), 1329-1337.
doi: 10.1093/comjnl/bxx121. |
[23] |
J.-B. Wang, M. Gao, J.-J. Wang, L. Liu and H. He, Scheduling with a position-weighted learning effect and job release dates, Engineering Optimization, (2019).
doi: 10.1080/0305215X.2019.1664498. |
[24] |
J.-B. Wang and L. Li,
Machine scheduling with deteriorating jobs and modifying maintenance activities, Comput. J., 61 (2018), 47-53.
doi: 10.1093/comjnl/bxx032. |
[25] |
J.-B. Wang, F. Liu and J.-J. Wang,
Research on $m$-machine flow shop scheduling with truncated learning effects, Int. Trans. Oper. Res., 26 (2019), 1135-1151.
doi: 10.1111/itor.12323. |
[26] |
J.-B. Wang, and M.-Z. Wang, Single-machine due-window assignment and scheduling with learning effect and resource-dependent processing times, Asia-Pac. J. Oper. Res., 31 (2014), 15pp.
doi: 10.1142/S0217595914500365. |
[27] |
J.-B. Wang and J.-J. Wang,
Research on scheduling with job-dependent learning effect and convex resource-dependent processing times, Internat. J. Prod. Res., 53 (2015), 5826-5836.
doi: 10.1080/00207543.2015.1010746. |
[28] |
D. Wang, M.-Z. Wang and J.-B. Wang,
Single-machine scheduling with learning effect and resource-dependent processing times, Comput. Industrial Engineering, 59 (2010), 458-462.
doi: 10.1016/j.cie.2010.06.002. |
[29] |
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, J. Oper. Res. Soc., 65 (2014), 1-13.
doi: 10.1057/jors.2012.161. |
[30] |
Y. Yin, D. 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, J. Oper. Res. Soc., 67 (2016), 1169-1183.
doi: 10.1057/jors.2016.14. |

notations | definitions |
|
the total number of jobs |
index of job | |
normal processing time of job |
|
actual processing time of job |
|
learning index of job |
|
compression rate of job |
|
amount of resource that can be allocated to job |
|
per time unit cost associated with resource allocated | |
for job |
|
due date for job |
|
learning index | |
positive constant | |
index of machine | |
operation of job |
|
normal processing time of operation |
|
actual processing time of operation |
|
learning index of operation |
|
amount of resource that can be allocated to | |
operation |
|
per time unit cost associated with resource allocated | |
for operation |
|
completion time of operation |
|
due window of job |
|
starting time of due window for job |
|
finishing time of due window for job |
|
due window size | |
completion time of job |
|
waiting time of job |
|
earliness of job |
|
tardiness of job |
|
makespan | |
total completion time | |
waiting completion time | |
total absolute differences in completion times | |
total absolute differences in waiting times |
notations | definitions |
|
the total number of jobs |
index of job | |
normal processing time of job |
|
actual processing time of job |
|
learning index of job |
|
compression rate of job |
|
amount of resource that can be allocated to job |
|
per time unit cost associated with resource allocated | |
for job |
|
due date for job |
|
learning index | |
positive constant | |
index of machine | |
operation of job |
|
normal processing time of operation |
|
actual processing time of operation |
|
learning index of operation |
|
amount of resource that can be allocated to | |
operation |
|
per time unit cost associated with resource allocated | |
for operation |
|
completion time of operation |
|
due window of job |
|
starting time of due window for job |
|
finishing time of due window for job |
|
due window size | |
completion time of job |
|
waiting time of job |
|
earliness of job |
|
tardiness of job |
|
makespan | |
total completion time | |
waiting completion time | |
total absolute differences in completion times | |
total absolute differences in waiting times |
problem | complexity | reference |
$F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}\right|A+ \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Liu and Feng [12] |
$F2|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|A$ | $O(n^3)$ | Tian et al. [21] |
$F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, A\leq Y\right|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Tian et al. [21] |
$F2|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}} \right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|A$ | $O(n \log n)$ | Tian et al. [21] |
$F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, A\leq Y\right|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n \log n)$ | Tian et al. [21] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta} \right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Gao et al. [4] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Gao et al. [4] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)$ | $O(n^3)$ | Geng et al. [5] |
$F2|NW, CON, $ $p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Geng et al. [5] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)$ | $O(n \log n)$ | Geng et al. [5] |
$F2|NW, CON, $ $p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Geng et al. [5] |
$F2\left|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, $ $ \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|\sum_{j = 1}^n(\alpha E_j+\beta T_j+ q)$ | $O(n^3)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, $ $ \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Sun et al. [20] |
$F2\left|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n \log n)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)$ | $O(n \log n)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Sun et al. [20] |
$F2\left|NW, CON-DW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a_{j, i}}}{u_{j, i}} \right)^{\eta}\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\delta d +\gamma D)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Shi and Wang [19] |
$F2\left|NW, CON-DW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\delta d +\gamma D)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Shi and Wang [19] |
P1, P2, P3 | $O(n^3)$ | Theorem 1 |
P1($Z$), P2($Z$), P3($Z$) | $O(n\log n)$ | Theorem 2 |
problem | complexity | reference |
$F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}\right|A+ \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Liu and Feng [12] |
$F2|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|A$ | $O(n^3)$ | Tian et al. [21] |
$F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, A\leq Y\right|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Tian et al. [21] |
$F2|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}} \right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|A$ | $O(n \log n)$ | Tian et al. [21] |
$F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, A\leq Y\right|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n \log n)$ | Tian et al. [21] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta} \right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Gao et al. [4] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Gao et al. [4] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)$ | $O(n^3)$ | Geng et al. [5] |
$F2|NW, CON, $ $p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Geng et al. [5] |
$F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)$ | $O(n \log n)$ | Geng et al. [5] |
$F2|NW, CON, $ $p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Geng et al. [5] |
$F2\left|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, $ $ \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|\sum_{j = 1}^n(\alpha E_j+\beta T_j+ q)$ | $O(n^3)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, $ $ \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Sun et al. [20] |
$F2\left|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n \log n)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)$ | $O(n \log n)$ | Sun et al. [20] |
$F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Sun et al. [20] |
$F2\left|NW, CON-DW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a_{j, i}}}{u_{j, i}} \right)^{\eta}\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\delta d +\gamma D)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n^3)$ | Shi and Wang [19] |
$F2\left|NW, CON-DW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\delta d +\gamma D)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ | $O(n\log n)$ | Shi and Wang [19] |
P1, P2, P3 | $O(n^3)$ | Theorem 1 |
P1($Z$), P2($Z$), P3($Z$) | $O(n\log n)$ | Theorem 2 |
|
|||||||
|
4 | 6 | 8 | 9 | 10 | 5 | 18 |
1 | 3 | 5 | 6 | 4 | 17 | 3 | |
2 | 5 | 3 | 1 | 6 | 7 | 4 | |
3 | 4 | 2 | 4 | 5 | 6 | 8 | |
-0.1 | -0.2 | -0.25 | -0.12 | -0.3 | -0.15 | -0.24 | |
-0.3 | -0.1 | -0.35 | -0.25 | -0.22 | -0.13 | -0.32 |
|
|||||||
|
4 | 6 | 8 | 9 | 10 | 5 | 18 |
1 | 3 | 5 | 6 | 4 | 17 | 3 | |
2 | 5 | 3 | 1 | 6 | 7 | 4 | |
3 | 4 | 2 | 4 | 5 | 6 | 8 | |
-0.1 | -0.2 | -0.25 | -0.12 | -0.3 | -0.15 | -0.24 | |
-0.3 | -0.1 | -0.35 | -0.25 | -0.22 | -0.13 | -0.32 |
|
1 | 2 | 3 | 4 | 5 | 6 | 7 |
|
50 | 58 | 63 | 48 | 32 | 16 | 0 |
-2 | 1 | 21 | 22 | 22 | 22 | 6 | |
7.0711 | 7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | |
7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | 2.4495 |
|
1 | 2 | 3 | 4 | 5 | 6 | 7 |
|
50 | 58 | 63 | 48 | 32 | 16 | 0 |
-2 | 1 | 21 | 22 | 22 | 22 | 6 | |
7.0711 | 7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | |
7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | 2.4495 |
${j\backslash r}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
$1$ | 79.0185 | 71.0153 | 66.8296 | 64.0572 | 62.0114 | 60.4040 | 59.0879 |
$2$ | 320.4994 | 285.0093 | 266.2270 | 253.7116 | 244.4413 | 237.1383 | 231.1477 |
$3$ | 275.0662 | 226.6246 | 202.4384 | 186.8978 | 175.6899 | 167.0453 | 160.0771 |
$4$ | 251.1229 | 217.0961 | 199.5329 | 188.0100 | 179.5710 | 172.9821 | 167.6170 |
$5$ | 564.1964 | 463.9798 | 413.9291 | 381.7710 | 358.5816 | 340.6988 | 326.2867 |
$6$ | 694.1769 | 631.6463 | 597.7243 | 574.7719 | 557.5800 | 543.9173 | 532.6284 |
$7$ | 396.4977 | 333.0474 | 300.8013 | 279.8553 | 264.6294 | 252.8128 | 243.2393 |
${j\backslash r}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
$1$ | 79.0185 | 71.0153 | 66.8296 | 64.0572 | 62.0114 | 60.4040 | 59.0879 |
$2$ | 320.4994 | 285.0093 | 266.2270 | 253.7116 | 244.4413 | 237.1383 | 231.1477 |
$3$ | 275.0662 | 226.6246 | 202.4384 | 186.8978 | 175.6899 | 167.0453 | 160.0771 |
$4$ | 251.1229 | 217.0961 | 199.5329 | 188.0100 | 179.5710 | 172.9821 | 167.6170 |
$5$ | 564.1964 | 463.9798 | 413.9291 | 381.7710 | 358.5816 | 340.6988 | 326.2867 |
$6$ | 694.1769 | 631.6463 | 597.7243 | 574.7719 | 557.5800 | 543.9173 | 532.6284 |
$7$ | 396.4977 | 333.0474 | 300.8013 | 279.8553 | 264.6294 | 252.8128 | 243.2393 |
|
|||||||
|
4 | 6 | 5 | 7 | 8 | 7 | 6 |
2 | 8 | 10 | 14 | 10 | 7 | 5 | |
2 | 4 | 4 | 4 | 5 | 5 | 5 | |
4 | 3 | 2 | 2 | 4 | 5 | 6 | |
8 | 24 | 20 | 28 | 40 | 35 | 30 |
|
|||||||
|
4 | 6 | 5 | 7 | 8 | 7 | 6 |
2 | 8 | 10 | 14 | 10 | 7 | 5 | |
2 | 4 | 4 | 4 | 5 | 5 | 5 | |
4 | 3 | 2 | 2 | 4 | 5 | 6 | |
8 | 24 | 20 | 28 | 40 | 35 | 30 |
|
1 | 2 | 3 | 4 | 5 | 6 | 7 |
|
50 | 58 | 63 | 48 | 32 | 16 | 0 |
-2 | 1 | 21 | 22 | 22 | 22 | 6 | |
7.0711 | 7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | |
7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | 2.4495 | |
14.5544 | 13.4790 | 13.0900 | 11.8643 | 9.7939 | 7.5856 | 4.8381 |
|
1 | 2 | 3 | 4 | 5 | 6 | 7 |
|
50 | 58 | 63 | 48 | 32 | 16 | 0 |
-2 | 1 | 21 | 22 | 22 | 22 | 6 | |
7.0711 | 7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | |
7.4833 | 8.0000 | 8.3066 | 7.3485 | 6.1644 | 4.6904 | 2.4495 | |
14.5544 | 13.4790 | 13.0900 | 11.8643 | 9.7939 | 7.5856 | 4.8381 |
jobs ( |
Min | Mean | Max |
40 | 916 | 992.10 | 1,060 |
60 | 1,368 | 1,491.60 | 1,991 |
80 | 2,352 | 2,671.20 | 2,956 |
100 | 3,972 | 4,232.70 | 4,618 |
120 | 5,182 | 5,979.85 | 6,223 |
140 | 7,775 | 7,868.50 | 8,171 |
160 | 8,229 | 9,142.60 | 9,915 |
180 | 10,892 | 11,514.70 | 12,325 |
200 | 12,884 | 13,715.60 | 14,191 |
220 | 14,134 | 15,125.50 | 16,119 |
240 | 16,185 | 17,815.60 | 18,181 |
260 | 18,347 | 19,752.40 | 20,282 |
280 | 20,981 | 21,156.70 | 22,941 |
300 | 23,823 | 24,715.90 | 25,671 |
jobs ( |
Min | Mean | Max |
40 | 916 | 992.10 | 1,060 |
60 | 1,368 | 1,491.60 | 1,991 |
80 | 2,352 | 2,671.20 | 2,956 |
100 | 3,972 | 4,232.70 | 4,618 |
120 | 5,182 | 5,979.85 | 6,223 |
140 | 7,775 | 7,868.50 | 8,171 |
160 | 8,229 | 9,142.60 | 9,915 |
180 | 10,892 | 11,514.70 | 12,325 |
200 | 12,884 | 13,715.60 | 14,191 |
220 | 14,134 | 15,125.50 | 16,119 |
240 | 16,185 | 17,815.60 | 18,181 |
260 | 18,347 | 19,752.40 | 20,282 |
280 | 20,981 | 21,156.70 | 22,941 |
300 | 23,823 | 24,715.90 | 25,671 |
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