Scheduling of parallel identical machines to maximize the weighted number of just-in-time jobs

We study the problem of nonpreemptively scheduling n jobs on m identical machines in parallel to maximize the weighted number of jobs that are completed exactly at their due dates. We show that this problem is solvable in polynomial time even if positive set-up times are allowed. Moreover, we show that if due date tolerances are permitted, then already the single-machine case is NP-hard even if all set-up times are zero and all weights are the same.