Abstract
This paper considers machine scheduling that integrates machine deterioration caused by jobs and, consequently, maintenance activities. The maintenance state of the machine is represented by a maintenance level which drops by a certain, possibly job-dependent amount while jobs are processed. A maintenance level of less than zero is associated with the machine’s breakdown and is therefore forbidden. Hence, maintenance activities that raise the maintenance level may become necessary and have to be scheduled additionally. We consider the objective to minimize the makespan throughout the paper. For the single machine case, we provide a linear-time approximation algorithm with worst-case a bound of 5/4, and comment on how an FPTAS from previous literature can be employed to apply to our problem. Due to problem similarity, these results also apply to the minimum subset sum problem, and the 5/4 linear-time approximation algorithm is an improvement over the 5/4 quadratic-time approximation algorithm of Güntzer and Jungnickel. For the general problem with multiple machines, we provide approximability results, two fast heuristics, an approximation algorithm with an instance-dependent approximation factor and a computational study evaluating the heuristics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bock, S., Briskorn, D., & Horbach, A. (2012). Scheduling flexible maintenance activities subject to job-dependent machine deterioration. Journal of Scheduling, 15(5), 565–578.
Brucker, P. (2004). Scheduling Algorithms. Berlin: Springer.
Chen, B. (1991). Tighter bound for MULTIFIT scheduling on uniform processors. Discrete Applied Mathematics, 31, 227–260.
Chen, J.-S. (2008). Scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan. European Journal of Operational Research, 190, 90–102.
Coffman, E. G., Garey, M. R., & Johnson, D. S. (1978). An application of bin-packing to multiprocessor scheduling. SIAM Journal on Computing, 7, 1–17.
Friesen, D., & Langston, M. (1983). Bounds for multifit scheduling on uniform processors. SIAM Journal on Computing, 12, 60–69.
Garey, M. R., & Johnson, D. S. (1979). Computers and Intractability - A guide to the theory of NP-completeness. New York: W.H Freemand and Company.
Gens, G., & Levner, E. (1981). Fast approximation algorithm for job sequencing with deadlines. Discrete Applied Mathematics, 3, 313–318.
Graham, R. L., Lawler, E. L., Lenstra, J. K., & Kan, A. G. H. R. (1979). Optimisation and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 236–287.
Grigoriu, L., & Friesen, D. K. (2010). Scheduling on same-speed processors with at most one downtime on each machine. Discrete Optimization, 7, 212–221.
Güntzer, M., & Jungnickel, D. (2000). Approximate minimization algorithms for the 0/1 knapsack and subset-sum problem. Operations Research Letters, 26, 55–66.
Hochbaum, D. S., & Shmoys, D. B. (1987). Using dual approximation algorithms for scheduling problems theoretical and practical results. Journal of the ACM, 34(1), 144–162.
Janiak, A., & Kovalyov, M. (1996). Single machine scheduling subject to deadlines and resource dependent processing times. European Journal of Operational Research, 94, 284–291.
Kellerer, H., Mansini, R., & Speranza, M. (2000). Two linear approximation algorithms for the subset-sum problem. European Journal of Operational Research, 120, 289–296.
Kellerer, H., Pferschy, U., & Pisinger, D. (2004). Knapsack Problems. Berlin: Springer.
Kovalyov, M. (1996). A rounding technique to construct approximation algorithms for knapsack and partition type problems. Applied Mathematics and Computer Science, 6, 101–113.
Lee, C.-Y. (2004). Machine scheduling with availability constraints. In: J. Y.-T. Leung (Eds.), Handbook of scheduling: Algorithms, models and performance analysis (pp. 22-1–22-13). Boca Raton, FL: Chapman and Hall/CRC.
Lodree, E. J, Jr., & Geiger, C. D. (2010). A note on the optimal sequence position for a rate-modifying activity under simple linear deterioration. European Journal of Operational Research, 201, 644–648.
Mor, B., & Mosheiov, G. (2012). Scheduling a maintenance activity and due-window assignment based on common flow allowance. International Journal of Production Economics, 135, 222–230.
Mosheiov, G., & Sarig, A. (2009). Scheduling a maintenance activity and due-window assignment on a single machine. Computers & Operations Research, 36, 2541–2545.
Pinedo, M. (2012). Scheduling: Theory, Algorithms, and Systems. Berlin: Springer.
Rustogi, K., & Strusevich, V. A. (2012a). Single machine scheduling with general positional deterioration and rate-modifying maintenance. Omega, 40, 791–804.
Rustogi, K., & Strusevich, V. A. (2012b). Simple matching vs linear assignment in scheduling models with positional effects: A critical review. European Journal of Operational Research, 222, 393–407.
Rustogi, K., & Strusevich, V. A. (2014). Combining time and position dependent effects on a single machine subject to rate-modifying activities. Omega, 42, 166–178.
Shabtay, D., & Steiner, G. (2007). A survey of scheduling with controllable processing times. Discrete Applied Mathematics, 155, 1643–1666.
Sun, K., & Li, H. (2010). Scheduling problems with multiple maintenance activities and non-preemptive jobs on two identical parallel machines. International Journal of Production Economics, 124, 151–158.
Yang, D.-L., & Yang, S.-J. (2013). Unrelated parallel-machine scheduling problems with multiple rate-modifying activities. Information Sciences, 235, 280–286.
Yang, D.-L., Cheng, T. C. E., & Yang, S.-J. (2014). Parallel-machine scheduling with controllable processing times and rate-modifying activities to minimise total cost involving total completion time and job compressions. International Journal of Production Research, 52, 1133–1141.
Yang, S.-J. (2010). Single-machine scheduling problems with both start-time dependent learning and position dependent aging effects under deteriorating maintenance consideration. Applied Mathematics and Computation, 217, 3321–3329.
Yang, S.-J., & Yang, D.-L. (2010a). Minimizing the makespan on single-machine scheduling with aging effect and variable maintenance activities. Omega, 38, 528–533.
Yang, S.-J., & Yang, D.-L. (2010b). Minimizing the total completion time in single-machine scheduling with aging/deteriorating effects and deteriorating maintenance activities. Computers and Mathematics with Applications, 60, 2161–2169.
Yue, M. (1990). On the exact upper bound of the multifit processor scheduling algorithm. Annals of Operations Research, 24, 233–259.
Zhao, C.-L., & Tang, H.-Y. (2010). Single machine scheduling with general job-dependent aging effect and maintenance activities to minimize makespan. Applied Mathematical Modelling, 34, 837–841.
Acknowledgements
The authors wish to thank the anonymous reviewers and the associate editor for their constructive comments and suggestions, according to which this paper was significantly improved.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grigoriu, L., Briskorn, D. Scheduling jobs and maintenance activities subject to job-dependent machine deteriorations. J Sched 20, 183–197 (2017). https://doi.org/10.1007/s10951-016-0502-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10951-016-0502-0