Single machine scheduling to maximize the number of on-time jobs with generalized due-dates

In scheduling problems with generalized due dates (gdd), the due dates are specified according to their position in the sequence, and the j-th due date is assigned to the job in the j-th position. We study a single-machine problem with generalized due dates, where the objective is maximizing the number of jobs completed exactly on time. We prove that the problem is NP-hard in the strong sense. To our knowledge, this is the only example of a scheduling problem where the job-specific version has a polynomial-time solution, and the gdd version is strongly NP-hard. A branch-and-bound (B&B) algorithm, an integer programming (IP)-based procedure, and an efficient heuristic are introduced and tested. Both the B&B algorithm and the IP-based solution procedure can solve most medium-sized problems in a reasonable computational effort. The heuristic serves as the initial step of the B&B algorithm and in itself obtains the optimum in most cases. We also study two special cases: max-on-time for a given job sequence and max-on-time with unit-execution-time jobs. For both cases, polynomial-time dynamic programming algorithms are introduced, and large-sized problems are easily solved.