A novel integer programing formulation for scheduling with family setup times on a single machine to minimize maximum lateness

Abstract This paper focuses on the problem of scheduling n jobs with family setup times on a single machine with the objective of minimizing the maximum lateness. To solve this problem we develop a novel ordered-batch integer programing formulation. The formulation utilizes two properties of optimal solutions. Firstly, we observe that there is a restricted set of batches that we need to consider to find the optimal solution. Secondly, we know the order in which batches should be processed if they occur in an optimal solution. Using branch and bound, our integer program finds optimal schedules for significantly larger problem instances than are reported in the literature. In contrast to existing algorithms, our formulation is strongest for problem instances with many families and large setup times. For example, we are able to find optimal solutions to problems with 1080 jobs and 270 families. We attribute this improvement to our linear programing relaxation being tighter than existing bounds for these cases. To explain this performance, we analyze the theoretical tightness of this formulation. We show that if the number of jobs in each family is bounded then the gap between a heuristic rounding and the lower bound produced by the linear programing increases at most sub-linearly with the number of jobs. The optimality gaps of prior approximation algorithms grow linearly with the number of jobs. Our work improves on these prior results.