Stochastic scheduling with asymmetric earliness and tardiness penalties under random machine breakdowns

Abstract

We study a stochastic scheduling problem of processing a set of jobs on a single machine. Each job has a random processing time Pi and a random due date Di, which are independently and exponentially distributed. The machine is subject to stochastic breakdowns in either preempt-resume or preempt-repeat patterns, with the uptimes following an exponential distribution and the downtimes (repair times) following a general distribution. The problem is to determine an optimal sequence for the machine to process all jobs so as to minimize the expected total cost comprising asymmetric earliness and tardiness penalties, in the form of E[Σαi max {0, Di - Ci} + βi {0, Ci - Di}]. We find sufficient conditions for the optimal sequences to be V-shaped with respect to {E(Pi)/αi} and {E(Pi)/βi} , respectively, which cover previous results in the literature as special cases. We also find conditions under which optimal sequences can be derived analytically. An algorithm is provided that can compute the best V-shaped sequence.