A new replacement warranty policy indexed by the product’s correlated failure and usage time

Abstract

We study a new warranty policy for non-repairable products which is indexed by two correlated random variables, age and usage and covers all failures in (0,t]. Two different warranty costs for the replacement of the failed product are considered, according to its usage being greater or less than a pre-specified level s>0. A bivariate probability distribution function is applied to incorporate the correlation effect of the two variables. Analytical expressions of the probability density function of the total warranty cost and its expected value, the probability distribution functions of the number of the failed products with usage greater or less than s and their corresponding expected values and costs are derived. Limit results are also obtained. The results obtained are useful measures in establishing the compensation policy and the evaluation of its performance under the proposed warranty. Illustrative numerical examples of the expected cost for Paulson, Pareto and Beta Stacy bivariate distributions are presented and discussed. In particular for Paulson’s bivariate probability distribution, closed form expressions for the expected costs are obtained.