The bounded single-machine parallel-batching scheduling problem with family jobs and release dates to minimize makespan

We consider the problem of scheduling family jobs with release dates on a bounded batching machine. Our objective is to minimize the maximum completion time of the jobs (i.e., the makespan). We deal with two variants of the problem. One is the identical job size model, in which the machine can process up to b jobs simultaneously as a batch. The other is the non-identical job size model, in which each job has a size no more than 1 and the machine can handle a number of jobs simultaneously as a batch as long as the total size of the jobs does not exceed 1. In this paper we first present a 2-approximation algorithm for the former model and a polynomial time approximation scheme (PTAS) for the case when the number of distinct families is constant; and then derive an approximation algorithm with a worst-case ratio of 5/2 for the latter model.