Simultaneous optimisation of test and maintenance intervals of given components using non-homogenous semi-Markov models
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Simultaneous optimisation of test and maintenance intervals of given components using non-homogenous semi-Markov models G Becker1, J Nagel1, L Camarinopoulos2 and G Zioutas3* 1
RISA Sicherheitsanalysen GmbH, Berlin; 2University of Piraeus, Greece; and 3Aristotelian University of Thessaloniki, Greece
A model was developed for simultaneous optimisation of test and maintenance intervals of components, where failures are not self-annunciated but observable only by tests or true demands. Costs caused by maintenance, tests and repairs constitute a complicated objective function for optimisation. An average non-availability, which must not be exceeded for safety reasons, is given as the main condition. In order to compute the objective function, a suitable non-homogeneous semi-Markov process was taken as the mathematical model and the descriptive integral equations were solved numerically. This was possible by using frequency densities hrj
t rather than transition probabilities pij
t as a base for calculation. Hence, for a process with N states, a formulation as an initial value problem with N dependent integral equations can be developed as opposed to N 2 equations, which normally are required. Applicability has been shown with a practical example (emergency diesel generators). Keywords: maintenance; Markov processes; reliability; stochastic processes
Introduction The ®nite-state Markov decision process is a simple and tractable model for decision making under uncertainty. It has been applied for machine maintenance following various solution procedures for several different types of Markov decision processes, many of them based on dynamic programming. The problem objective is to select the optimal strategy that optimises cost or pro®t, with respect to all strategies. This objective function depends on several performance measures, for example long-run proportion of time machines are broken, long-run average number of working machines or long-run average number of failures. All these models are based on the Markov property, which assumes independent exponential operating and repair times. There are various performance measures that can be derived from Markov process models depending on the complexity of the maintained system. If the Markov process satis®es some conditions, several methods have been provided for evaluating some performance measures and the long-run average cost. Generally, evaluation of performance measures for Markov process models of complex systems may be dif®cult. Therefore, specialised algorithms have been proposed (Grassmar,1 Fox2). *Correspondence: G Zioutas, Div. of Comp. Methods & Comp. Progr. General Department Faculty of Technology, University of Thessaloniki, 54006 Thessaloniki, Greece. E-mail: [email protected]
If the stochastic process of the presented maintained system exhibits only some lack of memory it is not completely Markovian. It is a semi-Markovian discretestate process a
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