An optimal predictive inspection and maintenance policy for a multi-state system: A belief-based SMDP approach

Inspection and maintenance (IM) are crucial for guaranteeing the functionality of engineered systems. In existing studies, an equidistant decision framework is commonly adopted, assuming periodic inspections and immediate maintenance actions (if needed). This assumption limits the search for globally optimal IM decisions. Moreover, the scenario of self-announcing failures and non-negligible IM durations that lead to non-equidistant decision intervals has not been investigated. In this study, we consider the aforementioned factors and propose a novel predictive IM policy that enables decision-makers to conduct non-periodic inspections and perform postponed maintenance actions after an inspection, thereby maximizing the system’s long-run profit rate. First, a belief-based Semi-Markov decision process (SMDP) is formulated to characterize a sequential IM decision-making problem based on the belief about the system state, which is then transformed into an equivalent belief-based MDP. Next, we derive the structural properties of the optimal solution to the transformed MDP, including the existence of the control limits. We further demonstrate that these results remain valid for the hidden failure scenario. Then, we demonstrate that when the minimal decision interval in the proposed sequential IM policy is sufficiently short, the policy is equivalent to a predictive IM policy. For computational efficiency, we develop an improved value iteration algorithm that iteratively reduces the minimum decision interval in the belief-based SMDP until convergence. A case study of an industrial water-filter system demonstrates both the performance superiority of the proposed predictive IM policy and the computational efficiency of the proposed algorithm.