Scheduling jobs on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties

Abstract

This article addresses the problem of scheduling n jobs with a common due date on a machine subject to stochastic breakdowns to minimize absolute early-tardy penalties. We investigate the problem under the conditions that the uptimes follow an exponential distribution, and the objective measure in detail is to minimize the expected sum of the absolute deviations of completion times from the common due date. We proceed to study in two versions (the downtime follows an exponential distribution or is a constant entailed for the repeat model job), one of which is the so-called preempt-resume version, the other of which is the preempt-repeat version. Three terms of work have been done. (i) Formulations and Preliminaries. A few of necessary definitions, relations and basic facts are established. In particular, the conclusion that the expectation of the absolute deviation of the completion time about a job with deterministic processing time t from a due date is a semi-V-shape function in t has been proved. (ii) Properties of Optimal Solutions. A few characteristics of optimal solutions are established. Most importantly, the conclusion that optimal solutions possess semi-V-shape property has been proved. (iii) Algorithm. Some computing problems on searching for optimal solutions are discussed.