Decomposition Methods for the Parallel Machine Scheduling Problem with Setups

We study the unrelated parallel machine scheduling problem with sequence and machine dependent setup times and the objective of makespan minimization. Two exact decomposition-based methods are proposed based on logic-based Benders decomposition and branch-and-check. These approaches are hybrid models that make use of a mixed integer programming master problem and a specialized solver for travelling salesman subproblems. The master problem is used to assign jobs to machines, while the subproblems find optimal schedules on each machine given the master problem assignments. Computational results show that the decomposition models are able to find optimal solutions up to four orders-of-magnitude faster than the existing state of the art as well as solve problems six times larger than an existing mixed integer programming model. We further investigate the solution quality versus run-time trade-off for large problem instances for which the optimal solutions cannot be found and proved in a reasonable time. We demonstrate that the branch-and-check hybrid algorithm is able to produce better schedules in less time than the state-of-the-art metaheuristic, while also providing an optimality gap.