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Journal of Scheduling

, Volume 19, Issue 5, pp 609–616 | Cite as

High-multiplicity scheduling on one machine with forbidden start and completion times

  • Michaël Gabay
  • Christophe Rapine
  • Nadia Brauner
Article

Abstract

We are interested in a single machine scheduling problem where jobs can neither start nor end on some specified instants, and the aim is to minimize the makespan. This problem models the situation where an additional resource, subject to unavailability constraints, is required to start and to finish a job. We consider in this paper the high-multiplicity version of the problem, when the input is given using a compact encoding. We present a polynomial time algorithm for large diversity instances (when the number of different processing times is greater than the number of forbidden instants). We also show that this problem is fixed-parameter tractable when the number of forbidden instants is fixed, regardless of jobs characteristics.

Keywords

Scheduling High-multiplicity Availability constraints Parameterized complexity 

Notes

Acknowledgments

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR—11-LABX-0025).

References

  1. Billaut, J. C., & Sourd, F. (2009). Single machine scheduling with forbidden start times. 4OR, 7(1), 37–50.CrossRefGoogle Scholar
  2. Brauner, N., Crama, Y., Grigoriev, A., & Van De Klundert, J. (2005). A framework for the complexity of high-multiplicity scheduling problems. Journal of Combinatorial Optimization, 9(3), 313–323.CrossRefGoogle Scholar
  3. Brauner, N., Finke, G., Lehoux-Lebacque, V., Rapine, C., Kellerer, H., Potts, C., et al. (2009). Operator non-availability periods. 4OR, 7(3), 239–253.Google Scholar
  4. Chen, Y., Zhang, A., & Tan, Z. (2013). Complexity and approximation of single machine scheduling with an operator non-availability period to minimize total completion time. Information Sciences, 251, 150–163.CrossRefGoogle Scholar
  5. Clifford, J. J., & Posner, M. E. (2001). Parallel machine scheduling with high multiplicity. Mathematical Programming, 89(3), 359–383.CrossRefGoogle Scholar
  6. Eisenbrand, F. (2003). Fast integer programming in fixed dimension. In G. Battista & U. Zwick (Eds.), Algorithms—ESA 2003 (Vol. 2832, pp. 196–207)., Lecture notes in computer science Berlin: Springer.CrossRefGoogle Scholar
  7. Filippi, C., & Agnetis, A. (2005). An asymptotically exact algorithm for the high-multiplicity bin packing problem. Mathematical Programming, 104(1), 21–37.CrossRefGoogle Scholar
  8. Filippi, C., & Romanin-Jacur, G. (2009). Exact and approximate algorithms for high-multiplicity parallel machine scheduling. Journal of Scheduling, 12(5), 529–541.CrossRefGoogle Scholar
  9. Hochbaum, D. S., & Shamir, R. (1991). Strongly polynomial algorithms for the high multiplicity scheduling problem. Operations Research, 39(4), 648–653.CrossRefGoogle Scholar
  10. Lenstra, H. W. (1983). Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4), 538–548.CrossRefGoogle Scholar
  11. Rapine, C., & Brauner, N. (2013). A polynomial time algorithm for makespan minimization on one machine with forbidden start and completion times. Discrete Optimization, 10(4), 241–250.CrossRefGoogle Scholar
  12. Rapine, C., Brauner, N., Finke, G., & Lebacque, V. (2012). Single machine scheduling with small operator-non-availability periods. Journal of Scheduling, 15(2), 127–139.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Michaël Gabay
    • 1
    • 2
  • Christophe Rapine
    • 3
  • Nadia Brauner
    • 1
    • 2
  1. 1.Université Grenoble Alpes, G-SCOPGrenobleFrance
  2. 2.CNRS, G-SCOPGrenobleFrance
  3. 3.Université de Lorraine, Laboratoire LGIPM, Ile du SaulcyMetzFrance

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