Single machine scheduling with controllable processing times and an unavailability period to minimize the makespan

We study a single machine scheduling problem, where job processing times are controllable, and there is a fixed machine unavailability interval. We assume that the job processing time is a convex decreasing function of the amount of resource allocated to its processing operation. We further assume that there is a budget restriction on the total resource allocation cost. Our aim is to find a job schedule that minimizes the makespan. We prove that the problem is NP-hard and develop both a constant factor approximation algorithm and a fully polynomial time approximation scheme (FPTAS) for solving it. The FPTAS is obtained despite the fact that we could not design a pseudo-polynomial time algorithm for finding the optimal solution.