Scheduling jobs with step-deterioration; Minimizing makespan on a single- and multi-machine

Certain maintenance procedures fail to perform prior to a pre-specified deadline and need extra time for successful accomplishment. Processing times under this specific deterioriation are represented by a step function. We study the problem of makespan minimization of step-wise deteriorating jobs. We introduce compact integer programming formulations. We then focus on heuristic methods, since even the case of single machine single-step deterioration is NP-complete. These methods are extended to settings of multi-machine and multi-step deterioration. Our numerical tests indicate that the proposed algorithms are highly accurate and efficient in the context of a numerical study discussed in this paper.