The NP-Hardness of Minimizing the Total Late Work on an Unbounded Batch Machine

We consider the problem of minimizing the total late work ( n ∑ j=1 V j) on an unbounded batch processing machine, where V j = min{Tj, p j} and Tj = max{C j−d j,0}. The batch processing machine can process up to B (B≥ n) jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time, respectively. For a batch of jobs, the processing time of the batch is equal to the largest processing time among the jobs in this batch. In this paper, we prove that the problem 1|B≥ n| n ∑ j=1 V j is NP-hard.