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[[sum ]αi
max{0,Di −
Ci} + βi
max{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.