Minimizing some cost functions related to both burn-in and field use

Burn-in procedure is used to improve the quality of products. In field operation only those components which survive the burn-in process will be used. Various additive cost functions are considered in this paper. One part of the cost function is the expense incurred until the first component surviving burn-in is obtained. The other part of cost function is either (i) the gain proportional to the mean life in field operation or (ii) the expenditure due to replacement at failure during field operation. We assume that the component before undergoing the burn-in procedure has a bathtub-shaped failure rate function with change points t(1) and t(2). It is shown that the optimal burn-in time b* minimizing the cost function is always before t(1). It is also shown that a large initial failure rate justifies burn-in, i.e., b* > 0.