A multi-stage imperfect maintenance strategy for multi-state systems with variable user demands

The main purpose of this paper is to schedule an imperfect maintenance policy for a single unit multi-state system (MSS) exhibiting more than two output states in the duration of its multiple operation stages. Firstly, a nonhomogeneous continuous time Markov chain (NHCTMC) is introduced to describe the state evolution process when the number of external shocks is modeled by a nonhomogeneous Poisson process (NHPP), and state residence probabilities of the Markov process are derived via a proposed recursive algorithm (RA) in solving the Chapman-Kolmogorov (CK) differential equations. Based on those modeling approaches, reliability measures such as the survival function and mean time to first failure (MTTFF) are evaluated where the randomly varied user demands reflecting real industrial applications are taken into consideration. We subsequently put forward a multi-stage imperfect maintenance policy by dividing system output states into three subsets concerning different degrees of degradation, incorporating a condition-based maintenance (CBM) action and a corrective maintenance (CM) activity in each operation stage. It is assumed that the CBM is a minimal repair and the effectiveness of CM is imperfect. The system will not be completely replaced by a brand new one until the number of failures reaches N in light of the investigated maintenance policy, where the expected profit in per renewal cycle is maximized to seek the optimal N analytically. An illustrative example of a bulk power electric system is presented to verify the proposed idea numerically, in which three common types of demand distributions are introduced.