A MATRIX ANALYTIC MODEL FOR MACHINE MAINTENANCE
David A Green,
Department of Applied Mathematics,
University of Adelaide,
5005, Australia.
Andrew V. Metcalfe,
Department of Applied Mathematics,
University of Adelaide,
5005, Australia.
David C. Swailes,
Department of Engineering Mathematics,
Newcastle University,
United Kingdom
Abstract
In this paper, we consider a production line consisting of machines
working in series, at the same speed with independent exponential
times before failure and times for repair. It can be shown that this
production line has an exponential time before failure with rate equal
to the sum of the individual machine failure rates. The repair time
for the line is distributed as a mixture of exponentials
(hyper-exponential). We compare an analysis using non-linear cost
functions, based on the hyper-exponential distribution of repair times
for the line with an approximation that assumes an exponential repair
time. The approximate exponential repair time has a mean equal to a
weighted average of the individual repair times, with weights
proportional to the failure rates. Non-linear cost functions allow for
differences between costs of: overtime, extra shifts, and failure to
meet deadlines; or for deterioration of a product if it is left
standing on a production line. We use a realistic example to
demonstrate that the approximate analysis can underestimate costs by
over 10%. We also present models for two production lines with one or
two repair crews. In the instance of a single repair crew, we
distinguish between a zero priority and a priority for line one case,
and show, in contrast to a single line, that the stationary
distributions are slightly different from approximations which assume
the line repair times are exponentially distributed