Multipurpose machine scheduling with rejection and identical job processing times

We study a set of scheduling problems on a uniform machine setting. We focus on the case of equal processing time jobs with the additional feature of job rejection. Jobs can either be processed on a predefined set of machines or rejected. Rejected jobs incur a rejection penalty and have no effect on the scheduling criterion under consideration. A solution to our problems consists of partitioning the jobs into two subsets, $$A$$A and $$\overline{A}$$A¯, which are the set of accepted and the set of rejected jobs, respectively. In addition, jobs in set $$A$$A have to be scheduled on the $$m$$m machines. We evaluate the quality of a solution by two criteria. The first, $$F_{1}$$F1, can be any regular scheduling criterion, while the latter, $$F_{2}$$F2, is the total rejection cost. We consider two possible types of regular scheduling criteria; the former is a maximization criterion, while the latter is a summation criterion. For each criterion type we consider four different problem variations. We prove that all four variations are solvable in polynomial time for $$any$$any maximization type of a regular scheduling criterion. When the scheduling criterion is of summation type, we show that only one of the four problem variations is solvable in polynomial time. We provide a pseudo-polynomial time algorithms to solve interesting variants of the $$\mathcal {NP}$$NP-hard problems, as well as a polynomial time algorithm that solves various other special cases.