Optimal preventive maintenance policies for products with multiple failure modes after geometric warranty expiry

In this paper, the optimal maintenance strategy for the warranty product subject to multiple failure modes and repair time limit is studied from the customer's perspective. When the product fails, a minimal repair based on the failure type is conducted. If the product can not be repaired within a given time limit, it will be replaced with a new one by the manufacturer during the warranty period. Different from the traditional renewing warranty policy, a geometric warranty policy is adopted in this work, under which once the warranty product is replaced; the length of warranty period is geometrically renewed. After the warranty expires, the product undergoes preventive maintenance (PM) periodically and the PM effect is modeled by reducing the virtual age of different failure modes by different degree. The objective is to determine the optimal interval time for PM and the optimal number of PMs by minimizing the expected life cycle cost rate of product. The existence and uniqueness of the optimal PM policy are proved theoretically, and the optimal bivariate policy can be obtained by using a recursive seeking algorithm. Numerical examples illustrate the effectiveness of geometric warranty policy and the proposed PM model.