The impact of gamma usage processes on preventive maintenance policies under two-dimensional warranty

This study considers a manufacturer performing preventive maintenance (PM) on a product according to a one-or two-dimensional (2-D) policy. The one-dimensional PM policy is based on either time or usage, while in the two-dimensional case, PM is scheduled based on both scales. The product carries a 2-D warranty that offers protection for a certain amount of time and usage. Its cumulative usage is continuously monitored by the manufacturer and is assumed to follow a gamma process. In this context, we first propose a doubly stochastic Poisson process model for product failures where the stochastic intensity is influenced by the gamma usage process in an additive manner. We then explicitly derive the expected total costs of the two one-dimensional PM policies using the concepts of first hitting times and gamma bridges. For the 2-D PM policy, we express the associated cost in terms of the value function of a dynamic programming model. In the numerical experiments, we show how the variability of the usage process affects the costs of the three PM policies and find that the optimal 2-D policy degenerates into a one-dimensional policy.