Optimal Control‐Limit Structure of Nonperiodic Inspection and Delayed Replacement Policies for Stochastic Degradation Systems

For many engineering systems, continuous condition monitoring is often infeasible due to technical and cost constraints, and system condition information is only available at inspection times. Delayed replacement, implemented after a certain waiting time, can help reduce inspection costs. Motivated by this, we study an optimal inspection and maintenance problem aimed at minimizing the long‐run average cost rate, in which system degradation can only be revealed through inspections. At each inspection epoch, the decision maker chooses whether to schedule a nonperiodic inspection, implement a delayed replacement, or perform an immediate replacement to mitigate the risk of failure. By formulating the problem as an average‐cost semi‐Markov decision process, we establish the existence and uniqueness of the optimal average cost rate and characterize an optimal stationary policy. Furthermore, the optimal policy is shown to exhibit a control‐limit structure, under which the optimal waiting times across different actions follow an ordered pattern. Numerical experiments illustrate the theoretical results and demonstrate that the proposed policy consistently outperforms commonly used benchmark strategies. A comprehensive sensitivity analysis further reveals a clear separation of roles: degradation dynamics primarily influence the length of optimal waiting times, whereas cost parameters govern the partition of inspection and replacement action regions.