Optimal replacement in a proportional hazards model with cumulative and dependent risks

This paper deals with a new maintenance problem with two competing dependent risks: minor failures and major failures. The cumulative number of minor failures is incorporated into the proportional hazards model as a covariant process of the hazard function of major failures. We introduce the “Time Discrete Markovian Approximation” (TDMA) technique to solve the “curse of dimensionality” in Markov Decision Processes (MDP) and simplify the high-dimensional integration when computing the average cost by renewal theory. We develop a new optimal control limit policy with a “mixed hazards function” as the threshold and reveal its agreement with the solution from MDP. Additionally, a corresponding iterative algorithm is developed to produce a sequence converging to the optimal solution faster than the policy iteration. •Optimal replacement for a system subject to multiple dependent and cumulative risks.•Semi-MDP framework with a novel approximation addressing computation challenges.•The optimal policy exhibits a structure of control limit with “mixed thresholds”.