Markov Maintenance Models

ON OPTIMAL POLICIES FOR MULTI·REPAIR·TYPE MARKOV MAINTENANCE MODELS Yukio Hatoyama Tokyo Institute of Technology (Received April 6, 1978) This paper treats an extension of an optimal machine maintenance model with Markovian deterioration introduced by Derman. The system consists of an operating machine whose deterioration is Markovian, a finite number of identical spare machines, and several types of repair facilities where machines to be repaired are sent depending on the types of repair work required. At eacl1 period of time, a decision is made on an operating machine whether it is repaired or not, knowing its degree of deterioration, the type of repair work required if the repair decision is chosen, and the number of machines in each type of repair facility. Here, the repair time distributions, material costs, and labor cost all depend on the type of repair work required on the machine. Sufficient conditions which result in the optimality of control limit policies of some kind are obtained. In this study a discrete time finite state Markov maintenance model with several types of repair shops is presented. Because of their wide applicability in the practical world, a number of authors have studied optimization problems for machine maintenance models when changes of states are Markovian. Their main concern has been on the structure of an optimal policy, and a simple repair rule called a control limit policy has been introduced. In 1963 Derman [1] introduced the basic maintenance model of this type. Assuming a simple cost structure and the IFR property on the transition probabilities, Derman showed the optimality of a control limit policy. Kolesar [7] extended the basic model by introducing state dependent operating costs, and Kalymon [4] further generalized the cost structure by allowing replacement costs to be stochastic. Klein [6] expanded Derman's model to include a costly inspection, or an imperfect information, which has been developed by Taylor [10] and Rosenfield [8] with emphasis on trying to find simple types of optimal policies. The aforementioned models have the properties that the amount of time 106 © 1979 The Operations Research Society of Japan Multi-RepairType Maintenance Models 107 needed for t:he repair of a machine is one unit of time, and an unlimited supply of new spares is available. In that sense, they are replacement models rather than repair models. In 1973 Kao [5] introduced a semi-Markovian approaeh to the basic model. In his model, the repair time of a machine takes some random time according to its semi-Markovian nature, while the supply of new spares is kept unlimited. In this report we develop a general repair model in the sense that the repair time is a random variable, and that the supply of spare machines is limited. 1. Description of the Model Consider a basic mu1ti-repair-type Markov machine maintenance model. The flow of machines in the system is schematica11y shown in Fig. 1. There are an operating machine, 8 (8 2: 1) identical spare machines and K (K 2: 1) kinds of repair shops in the system. The system is observed periodically and at the beginning of each period an operating machine is classified as being in one of I + 1 (I 2: 1) states, with each state showing the degree of deterioration. 0 represents a state of a machine in its best condition, while I denotes its failure. When a machine is operating, two choices are available at the beginning of each period: to let it keep operating, or to repair it. If the operating spare machine units _I type I 1 1 I J type 2 I 1 . J