A maintenance activity scheduling with time-and-position dependent deteriorating effects

We deal with a single-machine scheduling problem with an optional maintenance activity (denoted by $ ma $), where the actual processing time of a job is a function of its starting time and position. The optional $ ma $ means that the machine will perform a $ ma $, after $ ma $ is completed, the machine will return to the initial state. The objective is to determine an optimal job sequence and the location of the maintenance activity such that makespan is to be minimized. Based on some properties of an optimal sequence, we introduce a polynomial time algorithm to solve the problem, and the time complexity is $ O({n}^4) $, where $ {n} $ is the number of jobs.