Two Discrete-time Age-based Replacement Problems with/without Discounting

This paper considers two classical age-based replacement models within a discrete-time framework: a standard age replacement model and an opportunistic age replacement model. Specifically, our analysis incorporates the concept of replacement priority in situations where failure replacement and preventive replacement occur at a given age or opportunity. We explore two priority cases within each replacement model. First, we formulate optimal preventive replacement policies aimed at minimizing the associated expected cost rate in the age replacement model and the opportunistic age replacement model by the familiar renewal reward argument. Next, we extend the findings presented earlier to scenarios involving discounting. We develop formulations for the expected total discounted costs over an infinite time horizon and obtain optimal preventive replacement policies minimizing these total expected costs. Additionally, we explore unified models incorporating probabilistic priority. To provide practical insights, we present numerical illustrations using real failure data from pole air switches, comparing the performance of these optimal preventive policies.