Minimizing weighted earliness and tardiness penalties about a common due date on single machine with exponential processing times

A problem of scheduling n jobs with exponential processing times on a single machine is discussed, the objective is to find an optimal schedule to minimize the expectation of total weighted absolute deviations of completion times about a deterministic common due date. This problem is a typical scheduling model in just-in-time manufacturing system where both earliness penalties and tardiness penalties are considered. The deterministic equivalent of the objective function is derived, and /spl Lambda/-shaped property, with respect to the products of jobs' weights and their processing time rates, of the optimal schedules of this problem is established. The /spl Lambda/-shaped property of the optimal schedule can reduce the candidates of optimal schedule from n! to 2/sup n/.