Approximate calculations of age-based random replacement times

It is always difficult to discuss the analytical formulas, in which optimum solutions of replacement policies satisfy, due to the mathematical complexity. This paper proposes several approximate calculations for age-based random replacement policies, such that the unit is replaced at time T, at random time Y-j (j = 1, 2, ...) over time T, and at random time Y-N for Y-j (j = 1, 2, ..., N) We use the inequality properties of the formulas and the bounds of the extended failure rate functions to find the approximate calculations of replacement times in analytical ways. It is shown from numerical examples that these approximate calculations would become useful references for the optimum age-based random replacement times.