Minimizing Makespan on an Unbounded Parallel Batch Processor with Rejection and Release Times

In this paper, we consider the following scheduling model: jobs can either be scheduled on an unbounded parallel batch processor or be rejected. Jobs arrive at different times. The scheduled jobs will be processed together as a batch the processing time of which is the greatest processing time of its members. For each rejected job, there is a corresponding penalty cost. The objective is to minimize the sum of the makespan of the scheduled jobs and the total penalties of the rejected ones. We show that the problem is polynomial time solvable when the number of release times is fixed, and present a PTAS for the general case.