An Optimal Age Replacement Policy for Multi-State Systems

We consider an age replacement policy (ARP) for a system which consists of several multi-state elements. These elements can be in different states with performance levels ranging from perfectly functioning (the highest state) to total failure (the zero state). The multi-state system (MSS) is considered to be in a failure or unacceptable state if its performance level, determined by the multiple elements and the configuration, falls below the user demand level, and is considered as in a working or acceptable state if its performance level is greater than or equal to the user demand level. Under an ARP, a multi-state system is replaced at a failure, or at age, whichever comes first. The deterioration of the multi-state element is assumed to follow the non-homogenous continuous-time Markov chain (NHCTMC). We use the recursion to solve the Chapman-Kolmogorov's (C-K) forward equation to obtain the time-dependent state probabilities of each element of the system. Then we compute the state probabilities of the entire system by using the Lz-transform method. Finally, we derive the expected cost and profit functions, and determine the cost minimization or profit maximization ARPs. The multi-state model under the ARP is a generalization of the classic two-state maintenance model, and can be applied to analyze more complex aging systems. Numerical examples are presented to demonstrate our results.