Advertisement

Journal of Combinatorial Optimization

, Volume 26, Issue 2, pp 345–371 | Cite as

Optimal job insertion in the no-wait job shop

  • Reinhard Bürgy
  • Heinz Gröflin
Article

Abstract

The no-wait job shop (NWJS) considered here is a version of the job shop scheduling problem where, for any two operations of a job, a fixed time lag between their starting times is given. Also, sequence-dependent set-up times between consecutive operations on a machine can be present. The NWJS problem consists in finding a schedule that minimizes the makespan.

We address here the so-called optimal job insertion problem (OJI) in the NWJS. While the OJI is NP-hard in the classical job shop, it was shown by Gröflin & Klinkert to be solvable in polynomial time in the NWJS. We present a highly efficient algorithm with running time \(\mathcal {O}(n^{2}\cdot\max\{n,m\})\) for this problem. The algorithm is based on a compact formulation of the NWJS problem and a characterization of all feasible insertions as the stable sets (of prescribed cardinality) in a derived comparability graph.

As an application of our algorithm, we propose a heuristic for the NWJS problem based on optimal job insertion and present numerical results that compare favorably with current benchmarks.

Keywords

No-wait job shop Optimal job insertion Stable sets Comparability graph 

Notes

Acknowledgements

We thank the anonymous referees for their constructive remarks which led to several improvements in the exposition of the paper.

References

  1. Adams J, Balas E, Zawack D (1988) The shifting bottleneck procedure for job shop scheduling. Manag Sci 34(3):391–401 MathSciNetMATHCrossRefGoogle Scholar
  2. Bozejko W, Makuchowski M (2009) A fast tabu search algorithm for the no-wait job shop problem. Comput Ind Eng 56:1502–1509 CrossRefGoogle Scholar
  3. Cook WJ, Cunningham WH, Pulleyblank WR, Schrijver A (1997) Combinatorial optimization. Wiley-Interscience, New York CrossRefGoogle Scholar
  4. Gröflin H, Klinkert A (2007) Feasible insertions in job shop scheduling, short cycles and stable sets. Cent Eur J Oper Res 177:763–785 MATHCrossRefGoogle Scholar
  5. Gröflin H, Klinkert A (2009) A new neighborhood and tabu search for the blocking job shop. Discrete Appl Math 157:3643–3655 MathSciNetMATHCrossRefGoogle Scholar
  6. Kis T (2001) Insertion techniques for job shop scheduling. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne Google Scholar
  7. Kis T, Hertz A (2003) A lower bound for the job insertion problem. Discrete Appl Math 128:395–419 MathSciNetMATHCrossRefGoogle Scholar
  8. Lawrence S (1984) Supplement to resource constrained project scheduling: an experimental investigation of heuristic scheduling techniques. GSIA, Carnegie Mellon University, Pittsburgh Google Scholar
  9. Schrijver A (2003) Combinatorial optimization, polyhedra and efficiency. Springer, Berlin MATHGoogle Scholar
  10. Schuster C (2006) No-wait job shop scheduling: tabu search and complexity of subproblems. Math Methods Oper Res 63(3):473–491 MathSciNetMATHCrossRefGoogle Scholar
  11. Storer RH, Wu SD, Vaccari R (1992) New search spaces for sequencing problems with application to job shop scheduling. Manag Sci 38(10):1495–1509 MATHCrossRefGoogle Scholar
  12. van den Broek J (2009) MIP-based approaches for complex planning problems. Ph.D. thesis, Technische Universiteit Eindhoven Google Scholar
  13. Werner F, Winkler A (1995) Insertion techniques for the heuristic solution of the job shop problem. Discrete Appl Math 58(2):191–211 MathSciNetMATHCrossRefGoogle Scholar
  14. Yamada T, Nakano R (1992) A genetic algorithm applicable to large-scale job-shop problems. In: Parallel problem solving from nature, vol 2, pp 281–290 Google Scholar
  15. Zhu J, Li X, Wang Q (2009) Complete local search with limited memory algorithm for no-wait job shops to minimize makespan. Eur J Oper Res 198(2):378–386 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of FribourgFribourgSwitzerland

Personalised recommendations