Parallel Machine Scheduling with Rejection under Special Conditions

We consider a version of parallel machine scheduling with rejection. There are m machines and n jobs, and each job is either accepted and then processed by one of the machines, or rejected and then a rejected fee is paid. The goal is to find a way of partitioning the set of jobs into two subsets, the subset of accepted and the subset of rejected jobs, the subset of accepted and the subset of rejected jobs, and to schedule all the accepted jobs to m machines such that the sum of the makespan and the total rejection fees of the rejected jobs is minimized. In this paper, we focus on a special case of this NP-hard problem where the processing time of each job is a power of 2. We show that if one more condition is satisfied, namely, if the ratio of rejection fee and processing time of each job can be indexed in the same way as the order of processing time, then the problem is polynomial time solvable. We therefore present an efficient algorithm for this case. Our result is also applicable to the case in which the processing times are the powers of q, where q is a positive integer.