American Institute of Mathematical Sciences

Advanced
September  2021, 17(5): 2817-2835. doi: 10.3934/jimo.2020096

Resource allocation flowshop scheduling with learning effect and slack due window assignment

Shuang Zhao ,

Department of Basic, Shenyang Sport University, Shenyang 110102, China

*Corresponding author

Received  October 2019 Revised  February 2020 Published  September 2021 Early access  May 2020

Fund Project: This work was supported by the Liaoning Province Universities and Colleges Basic Scientific Research Project of Youth Project, Education Department of Liaoning (China) (Grant no. LQN2017ST04)

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.

Keywords: Scheduling, flowshop, learning effect, resource allocation, slack due window assignment.
Mathematics Subject Classification: Primary: 90B35; Secondary: 90C26.
Citation: Shuang Zhao. Resource allocation flowshop scheduling with learning effect and slack due window assignment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2817-2835. doi: 10.3934/jimo.2020096
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.

[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. 

[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.

[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. 

[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.

Figure 1.  A two-machine no-waiting flowshop scheduling
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Table 1.  List of notations
notations definitions
$ n $ the total number of jobs
$ J_{j} $ $ (j = 1,2,\ldots,n) $ index of job
$ \overline{p}_{j} $ normal processing time of job $ J_{j} $
$ {p}_{j} $ actual processing time of job $ J_{j} $
$ a_{j} $ learning index of job $ J_{j} $
$ b_{j} $ compression rate of job $ J_{j} $
$ u_{j} $ amount of resource that can be allocated to job $ J_{j} $
$ v_{j} $ per time unit cost associated with resource allocated
for job $ J_{j} $
$ \bar{d}_j $ due date for job $ J_j $
$ a $ learning index
$ \eta $ positive constant
$ M_{i} $ $ (i = 1,2) $ index of machine
$ O_{j,i} $ operation of job $ J_j $ processed on machine $ M_i $
$ \overline{p}_{j,i} $ normal processing time of operation $ O_{j,i} $
$ {p}_{j,i} $ actual processing time of operation $ O_{j,i} $
$ a_{j,i} $ learning index of operation $ O_{j,i} $
$ u_{j,i} $ amount of resource that can be allocated to
operation $ O_{j,i} $
$ v_{j,i} $ per time unit cost associated with resource allocated
for operation $ O_{j,i} $
$ C_{j,i} $ completion time of operation $ O_{j,i} $
$ [d_j, d'_j] $ due window of job $ J_j $
$ d_j $ starting time of due window for job $ J_j $
$ d'_j $ finishing time of due window for job $ J_j $
$ D' $ due window size
$ C_j = C_{j,2} $ completion time of job $ J_{j} $
$ W_j $ waiting time of job $ J_{j} $
$ E_j $ earliness of job $ J_{j} $
$ T_j $ tardiness of job $ J_{j} $
$ C_{\max} $ makespan
$ \sum_{j = 1}^n C_j $ total completion time
$ \sum_{j = 1}^n W_j $ waiting completion time
$ \sum_{j = 1}^n\sum_{h = j}^n|C_j-C_h| $ total absolute differences in completion times
$ \sum_{j = 1}^n\sum_{h = j}^n|W_j-W_h| $ total absolute differences in waiting times
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notations definitions
$ n $ the total number of jobs
$ J_{j} $ $ (j = 1,2,\ldots,n) $ index of job
$ \overline{p}_{j} $ normal processing time of job $ J_{j} $
$ {p}_{j} $ actual processing time of job $ J_{j} $
$ a_{j} $ learning index of job $ J_{j} $
$ b_{j} $ compression rate of job $ J_{j} $
$ u_{j} $ amount of resource that can be allocated to job $ J_{j} $
$ v_{j} $ per time unit cost associated with resource allocated
for job $ J_{j} $
$ \bar{d}_j $ due date for job $ J_j $
$ a $ learning index
$ \eta $ positive constant
$ M_{i} $ $ (i = 1,2) $ index of machine
$ O_{j,i} $ operation of job $ J_j $ processed on machine $ M_i $
$ \overline{p}_{j,i} $ normal processing time of operation $ O_{j,i} $
$ {p}_{j,i} $ actual processing time of operation $ O_{j,i} $
$ a_{j,i} $ learning index of operation $ O_{j,i} $
$ u_{j,i} $ amount of resource that can be allocated to
operation $ O_{j,i} $
$ v_{j,i} $ per time unit cost associated with resource allocated
for operation $ O_{j,i} $
$ C_{j,i} $ completion time of operation $ O_{j,i} $
$ [d_j, d'_j] $ due window of job $ J_j $
$ d_j $ starting time of due window for job $ J_j $
$ d'_j $ finishing time of due window for job $ J_j $
$ D' $ due window size
$ C_j = C_{j,2} $ completion time of job $ J_{j} $
$ W_j $ waiting time of job $ J_{j} $
$ E_j $ earliness of job $ J_{j} $
$ T_j $ tardiness of job $ J_{j} $
$ C_{\max} $ makespan
$ \sum_{j = 1}^n C_j $ total completion time
$ \sum_{j = 1}^n W_j $ waiting completion time
$ \sum_{j = 1}^n\sum_{h = j}^n|C_j-C_h| $ total absolute differences in completion times
$ \sum_{j = 1}^n\sum_{h = j}^n|W_j-W_h| $ total absolute differences in waiting times
Table 2.  Summary of Results, $Z$ is the special condition $v_{j, 1} \overline{p}_{j, 1} = v_{j, 2} \overline{p}_{j, 2} = P'_{j}$
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
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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
Table 3.  Data for Example 1
$ J_j $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
$ \overline{p}_{j,1} $ 4 6 8 9 10 5 18
$ \overline{p}_{j,2} $ 1 3 5 6 4 17 3
$ v_{j,1} $ 2 5 3 1 6 7 4
$ v_{j,2} $ 3 4 2 4 5 6 8
$ a_{j,1} $ -0.1 -0.2 -0.25 -0.12 -0.3 -0.15 -0.24
$ a_{j,2} $ -0.3 -0.1 -0.35 -0.25 -0.22 -0.13 -0.32
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$ J_j $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
$ \overline{p}_{j,1} $ 4 6 8 9 10 5 18
$ \overline{p}_{j,2} $ 1 3 5 6 4 17 3
$ v_{j,1} $ 2 5 3 1 6 7 4
$ v_{j,2} $ 3 4 2 4 5 6 8
$ a_{j,1} $ -0.1 -0.2 -0.25 -0.12 -0.3 -0.15 -0.24
$ a_{j,2} $ -0.3 -0.1 -0.35 -0.25 -0.22 -0.13 -0.32
Table 4.  Weights for Example 1
$ r $ 1 2 3 4 5 6 7
$ {W'_r} $ 50 58 63 48 32 16 0
$ {V'_r} $ -2 1 21 22 22 22 6
$ {{\eta '}_r} $ 7.0711 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904
$ {{\vartheta '}_r} $ 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904 2.4495
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$ r $ 1 2 3 4 5 6 7
$ {W'_r} $ 50 58 63 48 32 16 0
$ {V'_r} $ -2 1 21 22 22 22 6
$ {{\eta '}_r} $ 7.0711 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904
$ {{\vartheta '}_r} $ 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904 2.4495
Table 5.  Values $\lambda_{j, r}$ for Example 1
${j\backslash r}$ 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
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${j\backslash r}$ 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
Table 6.  Data for Example 2
$ J_j $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
$ \overline{p}_{j,1} $ 4 6 5 7 8 7 6
$ \overline{p}_{j,2} $ 2 8 10 14 10 7 5
$ v_{j,1} $ 2 4 4 4 5 5 5
$ v_{j,2} $ 4 3 2 2 4 5 6
$ P_{j} $ 8 24 20 28 40 35 30
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$ J_j $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
$ \overline{p}_{j,1} $ 4 6 5 7 8 7 6
$ \overline{p}_{j,2} $ 2 8 10 14 10 7 5
$ v_{j,1} $ 2 4 4 4 5 5 5
$ v_{j,2} $ 4 3 2 2 4 5 6
$ P_{j} $ 8 24 20 28 40 35 30
Table 7.  Weights for Example 2
$ r $ 1 2 3 4 5 6 7
$ {W'_r} $ 50 58 63 48 32 16 0
$ {V'_r} $ -2 1 21 22 22 22 6
$ {{\eta '}_r} $ 7.0711 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904
$ {{\vartheta '}_r} $ 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904 2.4495
$ {\Psi'} _r $ 14.5544 13.4790 13.0900 11.8643 9.7939 7.5856 4.8381
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$ r $ 1 2 3 4 5 6 7
$ {W'_r} $ 50 58 63 48 32 16 0
$ {V'_r} $ -2 1 21 22 22 22 6
$ {{\eta '}_r} $ 7.0711 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904
$ {{\vartheta '}_r} $ 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904 2.4495
$ {\Psi'} _r $ 14.5544 13.4790 13.0900 11.8643 9.7939 7.5856 4.8381
Table 8.  Computation time of Algorithm 1 in ms
jobs ($ n $) Min Mean Max
40 916 992.10 1,060
60 1,368 1,491.60 1,991
80 2,352 2,671.20 2,956
100 3,972 4,232.70 4,618
120 5,182 5,979.85 6,223
140 7,775 7,868.50 8,171
160 8,229 9,142.60 9,915
180 10,892 11,514.70 12,325
200 12,884 13,715.60 14,191
220 14,134 15,125.50 16,119
240 16,185 17,815.60 18,181
260 18,347 19,752.40 20,282
280 20,981 21,156.70 22,941
300 23,823 24,715.90 25,671
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jobs ($ n $) 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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