Minimizing the number of late jobs on a single machine under due date uncertainty
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
We study the problem of minimizing the number of late jobs on a single machine where job processing times are known precisely and due dates are uncertain. The uncertainty is captured through a set of scenarios. In this environment, an appropriate criterion to select a schedule is to find one with the best worst-case performance, which minimizes the maximum number of late jobs over all scenarios. For a variable number of scenarios and two distinct due dates over all scenarios, the problem is proved NP-hard in the strong sense and non-approximable in pseudo-polynomial time with approximation ratio less than 2. It is polynomially solvable if the number s of scenarios and the number v of distinct due dates over all scenarios are given constants. An O(nlog n) time s-approximation algorithm is suggested for the general case, where n is the number of jobs, and a polynomial 3-approximation algorithm is suggested for the case of unit-time jobs and a constant number of scenarios. Furthermore, an O(n s+v−2/(v−1) v−2) time dynamic programming algorithm is presented for the case of unit-time jobs. The problem with unit-time jobs and the number of late jobs not exceeding a given constant value is solvable in polynomial time by an enumeration algorithm. The obtained results are related to a min-max assignment problem, an exact assignment problem and a multi-agent scheduling problem.
- Agnetis, A., Mirchandani, P. B., Pacciarelli, D., & Pacifici, A. (2004). Scheduling problems with two competing agents. Operations Research, 52, 229–242. CrossRef
- Aissi, H., Bazgan, C., & Vanderpooten, D. (2005). Complexity of the min-max and min-max regret assignment problem. Operations Research Letters, 33, 634–640. CrossRef
- Aloulou, M. A., & Della-Croce, F. (2008). Complexity of one machine scheduling problems under scenario-based uncertainty. Operations Research Letters, 36, 338–342. CrossRef
- Cheng, T. C. E., Ng, C. T., & Yuan, J. J. (2006). Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs. Theoretical Computer Science, 362, 273–281. CrossRef
- Deineko, V., & Woeginger, G. J. (2006). On the robust assignment problem under a fixed number of cost scenarios. Operations Research Letters, 34, 175–179. CrossRef
- Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: Freeman.
- Karzanov, A. V. (1987). Maximum matching of given weight in complete and complete bipartite graphs. Cybernetics, 23, 8–13 Translation from Kibernetika, 1, 7–11 (1987) (in Russian). CrossRef
- Kouvelis, P., & Yu, G. (1997). Robust discrete optimization and its applications. Dordrecht: Kluwer Academic.
- Moore, J. M. (1968). A n job, one machine scheduling algorithm for minimizing the number of late jobs. Management Science, 15, 102–109. CrossRef
- Mulmuley, K., Vazirani, U. V., & Vazirani, V. V. (1987). Matching is as easy as matrix inversion. Combinatorica, 7, 105–103. CrossRef
- Papadimitriou, C. H., & Yannakakis, M. (1982). The complexity of restricted spanning tree problems. Journal of the ACM, 29, 285–309. CrossRef
- Minimizing the number of late jobs on a single machine under due date uncertainty
Journal of Scheduling
Volume 14, Issue 4 , pp 351-360
- Cover Date
- Print ISSN
- Online ISSN
- Springer US
- Additional Links
- Single machine
- Min-max approach
- Author Affiliations
- 1. LAMSADE, Paris Dauphine University and CNRS, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France
- 2. United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova 6, 220012, Minsk, Belarus