Burn-in Time and Estimation of Change-Point with Weibull-Exponential Mixture Distribution

Abstract

Many industrial products have three phases in their product lives: infant-mortality, normal, and wear-out phases. In the infant-mortality phase, the failure rate is high, but decreasing; in the normal phase, the failure rate remains constant; and in the wear-out phase, the failure rate is increasing. A burn-in procedure may be used to reduce early failures before shipping a product to consumers. A cost model is formulated to find the optimal burn-in time, which minimizes the expected sum of manufacturing cost, burn-in cost, and warranty cost incurred by failed items found during the warranty period. A mixture of Weibull hyperexponential distribution with shape parameter less than one and exponential distribution is used to describe the infant-mortality and the normal phases of the product life. The product under consideration can be either repairable or non-repairable. When the change-point of the product life distribution is unknown, it is estimated by using the maximum-likelihood estimation method. The effects of sample size on estimation error and the performance of the model are studied, and a sensitivity analysis is performed to study the effects of several parameters of the W-E distribution and costs on the optimal burn-in time.