Optimal age‐replacement time with minimal repair based on cumulative repair‐cost limit for a system subject to shocks

An operating system is subject to random shocks that arrive according to a non‐homogeneous Poisson process and cause the system failed. System failures experience to be divided into two categories: a type‐I failure (minor), rectified by a minimal repair; or a type‐II failure (catastrophic) that calls for a replacement. An age‐replacement model is studied by considering both a cumulative repair‐cost limit and a system’s entire repair‐cost history. Under such a policy, the system is replaced at age T, or at the k‐th type‐I failure at which the accumulated repair cost exceeds the pre‐determined limit, or at any type‐II failure, whichever occurs first. The object of this article is to study analytically the minimum‐cost replacement policy for showing its existence, uniqueness, and the structural properties. The proposed model provides a general framework for analyzing the maintenance policies, and presents several numerical examples for illustration purposes.