Single machine scheduling with common due date and controllable processing times

In this paper, we consider a single machine scheduling problem with a common due date. Job processing times are controllable to the extent that they can be reduced, up to a certain limit, at a cost proportional to the reduction. The common due date, along with the associated job schedule that minimizes a certain cost function, are to be determined. This function is made up of costs associated with the common due date, processing time reduction as well as job absolute value in lateness. We show that the problem can be formulated as an assignment problem and thus can be solved with well-known algorithms. For the case that where all the jobs have a common difference between normal and crash processing time and an equal unit compression penalty, we present an O(n log n) algorithm to obtain the optimal solution.