Scheduling jobs with agreeable processing times and due dates on a single batch processing machine

In this paper we study the problems of scheduling jobs with agreeable processing times and due dates on a single batch processing machine to minimize total tardiness, and weighted number of tardy jobs. We prove that the problem of minimizing total tardiness is NP-hard even if the machine capacity is two jobs and we develop a pseudo-polynomial-time algorithm for an NP-hard special case of this problem. We also develop a pseudo-polynomial-time algorithm for the NP-hard problem of minimizing weighted number of tardy jobs, which suggests that this problem cannot be strongly NP-hard unless P=NP.