Optimal maintenance of systems with Markovian mission and deterioration

We consider the maintenance of a mission-based system that is designed to perform missions consisting of a random sequence of phases or stages with random durations. A …nite state Markov process describes the mission process. The age or deterioration process of the system is described by another …nite state Markov process whose generator depends on the phases of the mission. We discuss optimal replacement and optimal repair problems and characterize the optimal policies under some monotonicity assumptions. We also provide numerical illustrations to demonstrate the structure of the optimal policies. Key words: Optimal maintenance, mission-based system, Markovian deterioration 1. Introduction Maintenance actions are vital for companies to increase reliability and availability of the production system and to decrease production costs. A signi…cant portion of the total work force of a company is employed in maintenance departments and a great amount of money is spent for maintenance by companies every year. Therefore, optimal maintenance problems which address the obvious trade-o¤ between maintenance costs and productivity are very important for both researchers and company managers. In this study, we analyze optimal replacement and repair problems for systems which are designed to perform missions consisting of di¤erent stages or phases. Such systems are called mission-based systems or phased-mission systems in the literature. The sequence and the duration of the phases can be deterministic or random, and all failure properties of the components as well as the con…guration of the system change dramatically from phase to phase. These kind of systems were …rst introduced as phased-mission systems by Esary and Ziehms (1975) and a vast literature has accumulated since then. There are various models that involve systems with repairable and non-repairable components with deterministic or random phase durations and sequences. We refer the reader to Veatch (1986), Kim and Park (1994), Mura and Bondavalli (2001) and references cited in these papers for di¤erent models. We assume that the mission process is a Markov process. In other words, the system under consideration performs a mission whose successive phases follow a Markov chain and all phase durations are exponentially distributed. We also assume that the system is subject to Markovian Corresponding Author. Tel: +90(212)338-1723; Fax: +90(212)338-1548 Email addresses: [email protected] (Bora Çekyay), [email protected] (Süleyman Özekici) Preprint submitted to European Journal of Operational Research May 26, 2010 aging or deterioration. In other words, the successive deterioration levels of the system follow a Markov chain and holding times in each deterioration level are exponentially distributed during any phase. The most important point is that the generator of the deterioration process of the system (the transition probability matrix and rates of the holding times) depends on the mission process. This implies that the deterioration process is a Markov process modulated by another Markov process (i.e., the mission process). We characterize optimal replacement and repair policies which minimize the expected total discounted cost by considering phase dependent maintenance costs and state occupancy costs. Optimal maintenance of systems subject to Markovian deterioration has been studied extensively in OR literature. One of the earliest and basic models where the deterioration process is described by a Markov chain is analyzed by Derman (1970). It is shown that the optimal policies minimizing both expected total discounted cost and expected average cost are controllimit type under some monotonicity assumptions on the costs and the deterioration process. In particular, there exists a critical deterioration level above which the optimal decision is “replace” and below which the optimal decision is “do nothing”. The same model with sate occupancy costs incurred each time that the system is inspected is analyzed by Kolesar (1966) and the optimality of a control-limit policy is proved. A similar model with state dependent replacement costs is analyzed in Kawai et al. (2002). A generalization of the model in Kolesar (1966) is analyzed in Wood (1988) by considering the case where the replacement action may fail with some probability and the occupancy costs are not paid during replacement. The analysis is done by applying uniformization techniques by which a continuous-time Markov decision process can be converted into an equivalent discrete-time Markov decision process. The optimal policy is control-limit type provided that model parameters satisfy some monotonicity conditions. Özekici and Günlük (1992) provide some su¢ cient conditions which make the lifetime of a system with Markovian deterioration increasing failure rate on average (IFRA), and also show that these conditions imply the optimality of a control-limit policy if the replacement cost does not depend on the deterioration level of the system. They also analyze optimal repair problems considering several di¤erent cost structures. In a related work, Özekici (1995) considers the maintenance problem of a device operating in a random environment and provides characterizations of optimal maintenance policies when the device ages intrinsically. In most of the literature, it is generally assumed that the system is working under a …xed environment or phase. Therefore, this study extends this line of research by examining systems with Markovian deterioration modulated by an external mission process. For more information on optimal maintenance problems, the interested readers are referred to the surveys by Jardine and Buzacott (1985), Cho and Parlar (1991), Wang (2002), and Nicolai and Dekker (2007). In Section 2, we describe the stochastic structure of the mission and deterioration processes in detail. We analyze the optimal replacement problem in Section 3. In Section 4, the optimal repair problem is discussed and some special repair cost structures are further analyzed in Section 5. A number of illustrations are also provided in the Appendix. 2. Mission and Deterioration Processes Let Xt be the phase of the mission which is performed at time t. We assume that the mission process X = fXt; t 0g is a Markov process with a …nite state space E, transition probability matrix Q; and transition rate vector . We also suppose that the deterioration level or age of the system takes values in some …nite set F = f0; 1; ;Mg where 0 stands for a