Optimal Replacement Rules when Changes of State are Semi-Markovian

Abstract

This paper investigates the use of a discrete-time semi-Markov process to model a system that deteriorates in usage. Replacement rules that are (1) state-dependent, (2) state-age-dependent, and (3) age-dependent are proposed. The system operating costs and replacement costs are functions of the underlying states. The optimization criterion is the expected average cost per unit time. Under the first two replacement rules, the paper generates semi-Markov decision processes so that optimal policies can be obtained by the policy-iteration method. Sufficient conditions for the existence of an optimal control-limit state-dependent replacement rule are derived. For the age-dependent policy, the objective function is obtained so that the minimization can be carried out over the integers. An illustrative example is given at the end.

Optimality of Control Limit Maintenance Policies Under Nonstationary Deterioration

“…The left-hand side of inequality (a) is decreasing by Condition 3.4, while the righthand side is increasing by Condition 3.3 and Lemma l. Therefore, inequality (4) holds for any (*,t) wrth m > i and s 2 t, thus proving that the optimal policy is a CLP.…”

“…In order to take the aging process into account in a more general way, Kao [4] developed a semi-Markovian process. He discussed "stage-age" replacement policies, in which the optimal replacement time depends on both the state and the sojourn time in it.…”

Maintenance optimization for a Markovian deteriorating system with population heterogeneity

“…Bobos and Protonotarios (1978) propose a model formulation that allows for multiple maintenance activities-e.g., replacement and multiple types of repair-and Kurt and Kharoufeh (2010) introduce a limit on the number of repairs. Semi-Markovian deterioration (Kao, 1973) and age-dependent deterioration (Benyamini and Yechiali, 1999) have been considered as generalizations of the deterioration behavior. A comprehensive review of these variations and extensions is given in Çekyay and Özekici (2011).…”

Real-time health prognosis and dynamic preventive maintenance policy for equipment under aging Markovian deterioration

“…Under the deterioration models, some papers are concerned with optimal PM policies (e.g. Kao 1973); some focus on optimal inspection schedules (e.g. Milioni and Pliska 1988) and others consider simultaneous optimization of both inspection and PM schedules (e.g.…”

“…That is, the probability to make transition to a less healthy state does not increase with the age. In the above-mentioned literature, the semi-Markovian model is often used to capture the aging effect by incorporating a random sojourn time in each state (see Kao 1973). Different from the semi-Markovian model, we introduce an aging factor that discounts the probabilities of transitions to healthier states while increasing the probabilities of transitions to less healthy states.…”