Stochastic scheduling on a single machine with exponentially distributed common due date

The problem of minimizing total weighted deviations of job completion times from a common due date on a single machine is a typical scheduling model in a just-in-time production environment. It is NP-hard. However, an LTP (largest processing time) job schedule is optimal for the case where job weights are proportional to processing times. In this paper, the stochastic counterpart of this problem with proportional weights in which the processing times and the due date are random variables is considered. When the due date is exponentially distributed with an arbitrary positive rate, the optimal solution of the problem is derived. Under some conditions, the result is extended to the situation where the machine is subject to stochastic breakdowns.