Two-machine open-shop scheduling with rejection to minimize the makespan

In this paper, we consider the two-machine open-shop scheduling problem with rejection. The objective is to minimize the sum of the makespan of accepted jobs and the total rejection cost of rejected jobs. We show that the problem is NP-hard even in three special cases and then provide a pseudo-polynomial time algorithm for this problem, which means that the problem is actually NP-hard in the ordinary sense. This answers an open problem posed by Shabtay et al. in (J Schedul 16:3–28, 2013). Finally, a 2-approximation algorithm and an FPTAS are designed for the problem.