Two-phase optimal and heuristic algorithms for flow shop scheduling with reworks under overlapped queue time limits

This study addresses flow shop scheduling in which each job is reworked when one of its overlapped queue time limits is violated. The problem is to determine the start times of jobs at each stage and rework setups/operations if occur. After decomposing the problem into the job process route selection and the resulting flow shop or reentrant flow shop scheduling sub-problems, a two-phase optimal branch and bound (TP-B&B) algorithm is proposed for the three-stage makespan problem. The algorithm consists of generating the job process routes using a two-level tree, and finding the resulting optimal schedules using a B&B algorithm while reducing the search space by a dominance property and the best upper bound obtained by solving the resulting scheduling problems. Moreover, for the multi-stage makespan problem, a basic two-phase variable neighborhood search (TP-VNS) algorithm is proposed that solves the two sub-problems iteratively, where each sub-problem is solved using a shaking and local search improvement method with different neighborhood structures. Then, it is extended to a general TP-VNS algorithm with a variable neighborhood descent method. Computational results show that the TP-B&B algorithm gave 21.5% more optimal solutions than the Gurobi while requiring much less computation times for the small sized test instances. Also, the TP-VNS algorithms outperform the existing one significantly. Specifically, the basic (general) TP-VNS algorithm gave 0.7% (1.2%) improvement in the overall optimality gap for small sized test instances, and 1.2% (2.9%) improvement in the overall relative performance ratio for medium-to-large sized test instances.