Minimizing number of tardy jobs on a batch processing machine with incompatible job families

In this paper we consider the problem of minimizing number of tardy jobs on a single batch processing machine. The batch processing machine is capable of processing up to B jobs simultaneously as a batch. We are given a set of n jobs which can be partitioned into m incompatible families such that the processing times of all jobs belonging to the same family are equal and jobs of different families cannot be processed together. We show that this problem is NP-hard and present a dynamic programming algorithm which has polynomial time complexity when the number of job families and the batch machine capacity are fixed. We also show that when the jobs of a family have a common due date the problem can be solved by a pseudo-polynomial time procedure.