Single machine scheduling problem with unexpected failures under flexible maintenance

In real industrial environments, the assumption that preventive maintenance completely eliminates unexpected failures may be too idealistic. Therefore, this work incorporates unexpected failures into the single machine scheduling problem with sequence-dependent setup times under a flexible maintenance scheme, where corrective maintenance is taken when unexpected failures occur to minimize the makespan. To this end, two lemmas and valid inequalities are obtained and totally four models are proposed. Specifically, the unexpected failures of the first two models are represented by individual jobs, and by integrating jobs within each batch, respectively, and the first two models are extended to obtain the latter two models by adding valid inequalities. In addition, we design a metaheuristic algorithm based on multistart strategy for larger instances. Finally, we perform extensive numerical experiments to efficiency validate the model, as well as to verify the performance of the valid inequalities. Managerial insights are also derived that considering unexpected failures under flexible maintenance can achieve a trade-off between preventive maintenance and corrective maintenance, thereby reducing both the makespan and unexpected failures of the machine. In addition, this work provides a new perspective on controlling the degree of fluctuation in job completion time when unexpected failures occur in real production.