**Lili Zuo 1, Zhenxia Sun 1, Lingfa Lu 1,\* and Liqi Zhang <sup>2</sup>**


Received: 8 July 2019; Accepted: 24 July 2019; Published: 26 July 2019

**Abstract:** In this paper, we study two scheduling problems on a single machine with rejection and an operator non-availability interval. In the operator non-availability interval, no job can be started or be completed. However, a *crossover* job is allowed such that it can be started before this interval and completed after this interval. Furthermore, we also assume that job rejection is allowed. That is, each job is either accepted and processed in-house, or is rejected by paying a rejection cost. Our task is to minimize the sum of the makespan (or the total weighted completion time) of accepted jobs and the total rejection cost of rejected jobs. For two scheduling problems with different objective functions, by borrowing the previous algorithms in the literature, we propose a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme (FPTAS), respectively.

**Keywords:** scheduling with rejection; machine non-availability; operator non-availability; dynamic programming; FPTAS
