Optimal job insertion in the no-wait job shop

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.