Single-Machine Scheduling with Step-Deteriorating Jobs and Rejection

In this paper, we consider the single-machine scheduling with step-deteriorating jobs and rejection. Each job is either rejected by paying a rejection penalty, or accepted and processed on the single machine, and the actual processing time of each accepted job is a step function of its starting time and the common deteriorating date. The objective is to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs. For the case of common deteriorating penalty, we first show that the problem is NP-hard in the ordinary sense. Then we present two pseudo-polynomial algorithms and a 2-approximation algorithm. Furthermore, we propose a fully polynomial time approximation scheme. For the case of common normal processing time, we present two pseudo-polynomial time algorithms, a 2-approximation algorithm and a fully polynomial time approximation scheme.