A random maintenance last model with preventive maintenance for the product under a random warranty

Although renewing pro-rate replacement warranty (RPRW) can help producers obtain some compensation from users, there seldom exists a two-dimensional random RPRW with a refund (2D-RRPRW with R) where a refund can guarantee the fairness of users. In addition, although random periodic replacement last (RPRL) can extend the service span after the expiry of the warranty, RPRL considering preventive maintenance (PM) has been seldom modeled to further lengthen the service span after the expiry of the warranty. In view of these, a 2D-RRPRW with R is devised to guarantee the fairness of users by integrating the limited job cycles and a refund into RPRW. Under the case where 2D-RRPRW with R war-rants products with job cycles, a RPRL with PM is modeled to further lengthen the service span after the expiry of the warranty and reduce the failure frequency. It shows that to shorten the warranty period can makes the warranty cost of 2D-RRPRW with R to be less than the warranty cost of classic RPRW; and the performance of RPRL with PM outperforms the performance of classic RPRL.