Optimal Replacement Policy for Multi-State System Under Imperfect Maintenance

A multi-state system (MSS) has more than two discrete states corresponding to different performance rates. Usually, MSS is viewed as in a failure state once its performance rate falls below user demand, and maintenance is carried out immediately. Generally, the repaired system cannot be regarded as good as new, and oftentimes the system restoration is stochastic. We introduce an optimal replacement policy for MSSs, called policy. Under this policy, a MSS is replaced whenever its failure number reaches. We assess the dynamic element state probabilities of each aging multi-state element (MSE) using a stochastic process model which is identified as a non-homogeneous continuous time Markov model (NHCTMM), and we evaluate the state distribution of the entire MSS via the combination of the stochastic process, and the universal generating function (UGF). To quantify the quality of imperfect maintenance, a quasi-renewal process is used to describe the stochastic behavior of each individual MSE after repair. Moreover, we derive an explicit expression of the long-run expected profit per unit time, and determine the optimal failure number to replace the entire system. The proposed models are demonstrated via an illustrative case, followed by some comparative studies.