Dynamic programming-based exact and heuristic algorithms for single machine scheduling with sequence-dependent setups

This study presents a novel algorithmic framework and an inventory flow mixed integer programming formulation designed to minimize total tardiness and the number of setups. The approach decomposes the problem into three stages: intra-family scheduling, family sequence optimization, and family-switch timing. We propose a specialized heuristic with $O(n^{5}logn)$ complexity efficiently handles intra-family scheduling and is extended to accommodate subfamily groupings. Dynamic programming is employed for family-switch optimization, with state complexity constrained to $2^n+1$. In the last stage of algorithmic framework, we propose a branch-and-bound method to handle family-switch timing, utilizing lower bounds derived from the results of previous stages. Our overall proposed ”branch-and-bound-regulated dynamic programming (B&B-DP)” algorithm excels in solving large-scale scheduling problems, demonstrating superior performance against four benchmark methods across 150 test cases. This algorithmic framework extends the capabilities of single-machine scheduling with family setup times to handle a large number of jobs. In our experiments, we show that the proposed algorithm reduces total tardiness by 10%–25% compared to other methods. This research not only advances the state of the art in single-machine scheduling but also provides a scalable and effective framework for addressing complex production scheduling challenges.