Effective Two-Phase Heuristic and Lower Bounds for Multi-Stage Flexible Flow Shop Scheduling Problem with Unloading Times

This paper addresses the flexible flow shop scheduling problem with unloading operations, which commonly occurs in modern manufacturing processes like sand casting. Although only a few related works have been proposed in the literature, the significance of this problem motivates the need for efficient algorithms and the exploration of new properties. One interesting property established is the symmetry of the problem, where scheduling from the first stage to the last or vice versa yields the same optimal solution. This property enhances solution quality. Considering the problem’s theoretical complexity as strongly NP-Hard, approximate solutions are preferable, especially for medium and large-scale instances. To address this, a new two-phase heuristic is proposed, consisting of a constructive phase and an improvement phase. This heuristic builds upon an existing efficient heuristic for the parallel machine-scheduling problem and extends it to incorporate unloading times efficiently. The selection of the two-phase heuristic is justified by its ability to generate high-quality schedules at each stage. Moreover, new efficient lower bounds based on estimating minimum idle time in each stage are presented, utilizing the polynomial parallel machine-scheduling problem with flow time minimization in the previous stage. These lower bounds contribute to assessing the performance of the two-phase heuristic over the relative gap performance measure. Extensive experiments are conducted on benchmark test problems, demonstrating the effectiveness of the proposed algorithms. The results indicate an average computation time of 9.92 s and a mean relative gap of only 2.80% for several jobs up to 200 and several stages up to 10.