# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020096

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

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

*Corresponding author

Received  October 2019 Revised  February 2020 Published  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.

Citation: Shuang Zhao. Resource allocation flowshop scheduling with learning effect and slack due window assignment. Journal of Industrial & Management Optimization, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
A two-machine no-waiting flowshop scheduling
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
 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
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
 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
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
 $J_j$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $\overline{p}_{j,1}$ 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
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 8.3066 7.3485 6.1644 4.6904 ${{\vartheta '}_r}$ 7.4833 8 8.3066 7.3485 6.1644 4.6904 2.4495
 $r$ 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 8.3066 7.3485 6.1644 4.6904 ${{\vartheta '}_r}$ 7.4833 8 8.3066 7.3485 6.1644 4.6904 2.4495
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 60.404 59.0879 $2$ 320.499 285.009 266.227 253.712 244.441 237.138 231.148 $3$ 275.066 226.625 202.438 186.898 175.69 167.045 160.077 $4$ 251.123 217.096 199.533 188.01 179.571 172.982 167.617 $5$ 564.196 463.98 413.929 381.771 358.582 340.699 326.287 $6$ 694.177 631.646 597.724 574.772 557.58 543.917 532.628 $7$ 396.498 333.047 300.801 279.855 264.629 252.813 243.239
 ${j\backslash r}$ 1 2 3 4 5 6 7 $1$ 79.0185 71.0153 66.8296 64.0572 62.0114 60.404 59.0879 $2$ 320.499 285.009 266.227 253.712 244.441 237.138 231.148 $3$ 275.066 226.625 202.438 186.898 175.69 167.045 160.077 $4$ 251.123 217.096 199.533 188.01 179.571 172.982 167.617 $5$ 564.196 463.98 413.929 381.771 358.582 340.699 326.287 $6$ 694.177 631.646 597.724 574.772 557.58 543.917 532.628 $7$ 396.498 333.047 300.801 279.855 264.629 252.813 243.239
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
 $J_j$ $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $\overline{p}_{j,1}$ 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
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 8.3066 7.3485 6.1644 4.6904 ${{\vartheta '}_r}$ 7.4833 8 8.3066 7.3485 6.1644 4.6904 2.4495 ${\Psi'} _r$ 14.5544 13.479 13.09 11.8643 9.7939 7.5856 4.8381
 $r$ 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 8.3066 7.3485 6.1644 4.6904 ${{\vartheta '}_r}$ 7.4833 8 8.3066 7.3485 6.1644 4.6904 2.4495 ${\Psi'} _r$ 14.5544 13.479 13.09 11.8643 9.7939 7.5856 4.8381
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
 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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