Optimal common due-date with completion time tolerance

The common due-date problem involves minimizing the absolute deviation around a common due-date. An interesting variation modifies the problem so that no penalty costs are incurred for jobs completing within the tolerance interval of the common due-date. For tolerance intervals that are less than half the execution time of the shortest job, solution techniques have been developed. However, the techniques do not extend to tolerance intervals of any arbitrarily chosen size. In this paper, for a variety of penalty functions we develop a polynomial time solution algorithm for any size tolerance interval. We also demonstrate how optimizing techniques developed in this paper can be applied to a common due-date problem with a partial set of fixed jobs and a predetermined common due-date.