Uncertain Random Block Replacement Policy with No Replacement at Failure

The classic block replacement problem assumes the lifetime of component as a random variable, and the downtime cost caused by failed component is a constant. However, when a new type of unit is used in a block replacement system, its lifetime is indeterminate and its distribution cannot be obtained by probability theory due to the lack of historical operational data. And if the component is not replaced immediately when it fails, the downtime cost per unit time is not fixed but always effected by some stochastic factors such as market. Thus, uncertainty and randomness coexist in a block replacement system of new components. In this paper, the lifetime of the component is regarded as an uncertain variable and the downtime cost per unit time is considered as a random variable, and an uncertain random programming model of block replacement with no replacement at failure is developed for a system composed of new type of components, and the formulated model aims to find an optimal replacement time to minimize the expected cost rate in one replacement cycle. An implicit solution of the optimal replacement time is obtained, and the condition ensuring the existence and uniqueness of the optimal replacement time is given. Finally, a numerical example is presented to prove feasibility of the proposed model.