An $(\underline{s},\overline{s},S)$ optimal maintenance policy for systems subject to shocks and progressive deterioration

TL;DR

This paper develops an optimal maintenance policy for deteriorating systems affected by shocks and progressive wear, using impulse control and viscosity solutions to the HJB equation, with numerical examples and future research directions.

Contribution

It introduces a novel $( ext{underline}s, ext{overline}s,S)$ policy framework for systems with shock and progressive deterioration, characterizes the value function, and explores related singular control problems.

Findings

Optimal $( ext{underline}s, ext{overline}s,S)$ maintenance policy proven to be optimal.

Value function characterized as the unique viscosity solution of the HJB equation.

Numerical examples illustrate the policy's effectiveness and potential applications.

Abstract

We define a model of a system that deteriorate as a result of (i) shocks, modeled as a compound Poisson process and (ii) deterministic, state dependent progressive rate, with variable and fixed maintenance cost. We define maintenance strategies based on an impulse control model where time and size of interventions are executed according the the system state, which is obtained from permanent monitoring. We characterize the value function as the unique viscosity solution of the HJB equation and prove that a $(\underline{s},\overline{s},S)$ policy is optimal. We also provide numerical examples. Finally, a singular control problem is proposed when there is no fixed cost, which study and relation with the former problem is open for future discussion.