Optimal maintenance policy for systems subject to random shocks: a semi-Markov decision process approach

This paper studies an average-cost maintenance optimization problem for a single-component system subject to continuous degradation and random shocks. The degradation process is modeled by a Gamma process, while random shocks arrive according to a Poisson process. A system failure occurs when either the degradation level reaches or exceeds its failure threshold, or the accumulated number of shocks reaches the shock failure threshold. Since failures are non-self-announcing, both the failure state and the system condition can only be revealed through inspections. To reduce the long-run operating cost, we propose a state-dependent non-periodic inspection and replacement policy, where the next inspection interval and the replacement action are jointly optimized according to the current degradation level and accumulated shock state. A numerical algorithm based on state-space discretization and bisection search is developed to compute the optimal policy. Numerical results show that the proposed policy consistently outperforms the optimized periodic inspection and replacement benchmark, with average-cost-rate reductions ranging from 8.60% to 29.33% across the tested parameter settings. Sensitivity analysis further demonstrates that the proposed policy consistently outperforms the benchmark policy under different cost, shock, and degradation parameter settings. The results confirm that dynamically adjusting inspection intervals according to the system state can effectively balance inspection cost, downtime cost, and replacement cost.