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 O(n2 ·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.