Abstract
In this paper, we study scheduling with step-learning, i.e., a setting where the processing times of the jobs started after their job-dependent learning-dates are reduced. The goal is to minimize makespan on a single machine. We focus first on the case that idle times between consecutive jobs are not allowed. We prove that the problem is NP-hard, implying that no polynomial-time solution exists and, consequently, propose a pseudo-polynomial time dynamic programming algorithm. An extensive numerical study is provided to examine the running time of the algorithm with different learning-dates and job processing time ranges. The special case of a common learning-date for all the jobs is also studied, and a (more efficient) pseudo-polynomial dynamic programming is introduced and tested numerically. In the last part of the paper, the more complicated setting in which idle times are allowed is studied. An appropriate dynamic programming is introduced and tested as well.
Similar content being viewed by others
References
Azzouz, A., Ennigrou, M., & Ben Said, L. (2018). Scheduling problems under learning effects: Classification and cartography. International Journal of Production Research, 56(4), 1642–1661.
Azzouz, A., Pan, P. A., Hsu, P. H., Lin, W. C., Liu, S., Said, L. B., & Wu, C. C. (2020). A two-stage three-machine assembly scheduling problem with a truncation position-based learning effect. Soft Computing, 24, 10515–10533.
Cheng, M., Xiao, S., Luo, R., & Lian, Z. (2018). Single-machine scheduling problems with a batch-dependent aging effect and variable maintenance activities. International Journal of Production Research, 56(23), 7051–7063.
Ding, J., Shen, L., Lü, Z., & Peng, B. (2019). Parallel machine scheduling with completion-time-based criteria and sequence-dependent deterioration. Computers & Operations Research, 103, 35–45.
Fu, Y., Zhou, M., Guo, X., & Qi, L. (2019). Artificial-molecule-based chemical reaction optimization for flow shop scheduling problem with deteriorating and learning effects. IEEE Access, 7, 53429–53440.
Gao, F., Liu, M., Wang, J. J., & Lu, Y. Y. (2018). No-wait two-machine permutation flow shop scheduling problem with learning effect, common due date and controllable job processing times. International Journal of Production Research, 56(6), 2361–2369.
Gawiejnowicz, S. (2008). Time-dependent scheduling. Springer Science & Business Media.
Gawiejnowicz, S. (2020a). Models and algorithms of time-dependent scheduling (p. 538). Springer.
Gawiejnowicz, S. (2020b). A review of four decades of time-dependent scheduling: Main results, new topics, and open problems. Journal of Scheduling, 23(1), 3–47.
Gawiejnowicz, S., & Kurc, W. (2020). New results for an open time-dependent scheduling problem. Journal of Scheduling, 23(6), 733–744.
Geng, X. N., Wang, J. B., & Bai, D. (2019). Common due date assignment scheduling for a no-wait flowshop with convex resource allocation and learning effect. Engineering Optimization, 51(8), 1301–1323.
Graham, R. L., Lawler, E. L., & Lenstra, J. K. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.
Liu, W., Yao, Y., & Jiang, C. (2019). Single-machine resource allocation scheduling with due-date assignment, deterioration effect and position-dependent weights. Engineering Optimization, 52(4), 701–714.
Miao, C., & Zhang, Y. (2019). Scheduling with step-deteriorating jobs to minimize the makespan. Journal of Industrial & Management Optimization, 15(4), 1955–1964.
Mor, B., Mosheiov, G., & Shapira, D. (2020). Flowshop scheduling with learning effect and job rejection. Journal of Scheduling, 23(6), 631–641.
Mosheiov, G. (1995). Scheduling jobs with step-deterioration; minimizing makespan on a single-and multi-machine. Computers & Industrial Engineering, 28(4), 869–879.
Mousavi, S. M., Mahdavi, I., & Rezaeian, J. (2018). An efficient bi-objective algorithm to solve re-entrant hybrid flow shop scheduling with learning effect and setup times. Operational Research International Journal, 18, 123–158.
Pei, J., Wang, X., Fan, W., Pardalos, P. M., & Liu, X. (2019). Scheduling step-deteriorating jobs on bounded parallel-batching machines to maximise the total net revenue. Journal of the Operational Research Society, 70(10), 1830–1847.
Renna, P. (2019). Flexible job-shop scheduling with learning and forgetting effect by multi-agent system. International Journal of Industrial Engineering Computations, 10(4), 521–534.
Rostami, M., Nikravesh, S., & Shahin, M. (2018). Minimizing total weighted completion and batch delivery times with machine deterioration and learning effect: A case study from wax production. Operational Research International Journal. https://doi.org/10.1007/s12351-018-0373-6
Soper, A. J., & Strusevich, V. A. (2020). Refined conditions for V-shaped optimal sequencing on a single machine to minimize total completion time under combined effects. Journal of Scheduling, 23(6), 665–680.
Strusevich, V. A., & Rustogi, K. (2017). Scheduling with time-changing effects and rate-modifying activities. Springer International Publishing.
Sun, X., & Geng, X. N. (2019). Single-machine scheduling with deteriorating effects and machine maintenance. International Journal of Production Research, 57(10), 3186–3199.
Sun, X., Geng, X. N., Wang, J. B., & Liu, F. (2019). Convex resource allocation scheduling in the no-wait flowshop with common flow allowance and learning effect. International Journal of Production Research, 57(6), 1873–1891.
Wang, H., Huang, M., & Wang, J. (2019a). An effective metaheuristic algorithm for flowshop scheduling with deteriorating jobs. Journal of Intelligent Manufacturing, 30(7), 2733–2742.
Wang, J. B., Liu, F., & Wang, J. J. (2019b). Research on m-machine flow shop scheduling with truncated learning effects. International Transactions in Operational Research, 26(3), 1135–1151.
Woo, Y. B., & Kim, B. S. (2018). Matheuristic approaches for parallel machine scheduling problem with time-dependent deterioration and multiple rate-modifying activities. Computers & Operations Research, 95, 97–112.
Wu, C. C., Azzouz, A., Chung, I. H., Lin, W. C., & Ben Said, L. (2019). A two-stage three-machine assembly scheduling problem with deterioration effect. International Journal of Production Research, 57(21), 6634–6647.
Wu, C. C., Wang, D. J., Cheng, S. R., Chung, I. H., & Lin, W. C. (2018). A two-stage three-machine assembly scheduling problem with a position-based learning effect. International Journal of Production Research, 56(9), 3064–3079.
Wu, C. C., Zhang, X., Azzouz, A., Shen, W. L., Cheng, S. R., Hsu, P. H., & Lin, W. C. (2020). Metaheuristics for two-stage flow-shop assembly problem with a truncation learning function. Engineering Optimization. https://doi.org/10.1080/0305215X.2020.1757089
Yan, P., Wang, J. B., & Zhao, L. Q. (2019). Single-machine bi-criterion scheduling with release times and exponentially time-dependent learning effects. Journal of Industrial & Management Optimization, 15(3), 1117–1131.
Acknowledgements
The second author was supported by the Israel Science Foundation (Grant No. 884/22). The third author was supported by the Israel Science Foundation (Grant No. 2505/19) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation – Project Number 452470135).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Atsmony, M., Mor, B. & Mosheiov, G. Single machine scheduling with step-learning. J Sched 27, 227–237 (2024). https://doi.org/10.1007/s10951-022-00763-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10951-022-00763-5