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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 warrants 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.

The practical application of time-variant reliability analysis is limited by its computationally expensive models which describe the structural system behavior. This paper presents a new adaptive PC-Kriging (APCK) approach to accurately and efficiently assess the time-variant reliabilities. Time interval is firstly discretized with a series of time instants and then the stochastic process is reconstructed by standard normal random variables and deterministic function of time. PC-Kriging (PCK) models are built at each time instant to predict the instantaneous responses of performance function. To improve the accuracy and efficiency, a new update strategy based on the integration of U- and H- learning functions is developed to refine the PCK models of instantaneous responses. One or two best samples are identified by the proposed learning criterion for updating the PCK models. Finally, Monte Carlo simulation (MCS) is used to estimate the time-variant reliability based on the updated PCK models. Four examples are used to validate the accuracy and efficiency of the proposed method.