# Optimal job insertion in the no-wait job shop

## 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

- Adams J, Balas E, Zawack D (1988) The shifting bottleneck procedure for job shop scheduling. Manag Sci 34(3):391–401 MathSciNetMATHCrossRefGoogle Scholar
- 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
- Cook WJ, Cunningham WH, Pulleyblank WR, Schrijver A (1997) Combinatorial optimization. Wiley-Interscience, New York CrossRefGoogle Scholar
- 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
- Gröflin H, Klinkert A (2009) A new neighborhood and tabu search for the blocking job shop. Discrete Appl Math 157:3643–3655 MathSciNetMATHCrossRefGoogle Scholar
- Kis T (2001) Insertion techniques for job shop scheduling. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne Google Scholar
- Kis T, Hertz A (2003) A lower bound for the job insertion problem. Discrete Appl Math 128:395–419 MathSciNetMATHCrossRefGoogle Scholar
- Lawrence S (1984) Supplement to resource constrained project scheduling: an experimental investigation of heuristic scheduling techniques. GSIA, Carnegie Mellon University, Pittsburgh Google Scholar
- Schrijver A (2003) Combinatorial optimization, polyhedra and efficiency. Springer, Berlin MATHGoogle Scholar
- Schuster C (2006) No-wait job shop scheduling: tabu search and complexity of subproblems. Math Methods Oper Res 63(3):473–491 MathSciNetMATHCrossRefGoogle Scholar
- 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
- van den Broek J (2009) MIP-based approaches for complex planning problems. Ph.D. thesis, Technische Universiteit Eindhoven Google Scholar
- 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
- 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
- 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