# Optimal job insertion in the no-wait job shop

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## 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.

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