Condition-based Maintenance Optimization for Multistate Systems with Semi-Markov Decision Process

The deterioration law of the system will usually be assumed to be Markovian, that is, the future process of the system depends only on its state at the present time and not on its past history. This model has been suggested by Klein [1]. There are at least two good reasons for describing deterioration by a Markov model. Firstly, if each component is approximately subject to an exponential failure law, the complete system can be described approximately by a Markov process. Secondly, the first order approximating description of many physical systems contain no predictive value. Markov process is the stochastic equivalent of this type of process. The theory of Markov process is the most popular way to model equipment deterioration and maintenance under the CBM strategies, it generally classified as discrete time Markov chains and continuous-time Markov chains. Christer [2]developed the delay time model, and Jardine [3] used the proportional hazards model to optimize a mine haul truck wheel motor’ condition monitoring program, and Grall et al. [4] used the models based on gamma process. But there is little attention has been paid to CBM modeling of deteriorating systems with multiple different components which have multistate. In fact, dividing the system deterioration into several discrete states is more practical than describing the deterioration condition by a single scalar continuous variable. Endrenyi et al.[5] classified the equipment deterioration into four stages: initial, minor deterioration, major deterioration, and failure. Later the continuous Markov chain process can be generalized in a natural way as follows. According to Tai et al.[6], failure can be distinguished into random failure and deterioration failure. For most equipment, deterioration failure grows gradually with time, occurres due to deterioration or aging mechanisms. However random failure results from other causes not associated with typical aging. In this study, both random failure and failure due to deterioration are considered. In the case of a multistate system, the concept corresponding to the concept of reliability in a binary system is the state distribution. Many practical systems can perform their intended functions at more than two different levels, ranging from working to completely failed. Sim et al.[7] introduced the exponential device failure in a condition maintenance model. For most aforementioned modeling strategy, the stochastic behavior of individual component without repairing is modeled by homogenous Markov process with the assumption that the transition time between any pair of states is exponentially distributed. However, this is not generalized case and transition time between component states may follow other distributions. For this reason, we formulate the semi-Markov decision process (SMDP) model for the condition based maintenance problem. This model based on the assumption that the component state deteriorating process can be captured by Markovian models, and the sojourn time of the state is not exponentially distributed.