Uncertainty Importance Measure of Individual Components in Multi-State Systems

Traditional reliability importance measures have been successfully extended from binary-state models to multi-state models. The calculation of these measures typically relies on the true reliabilities of components. In reality, however, the true values of component reliabilities are usually unknown, and they are generally approximated by their estimates generated from testing or field failure data. The accuracy of the estimates is limited by the available data. Research on uncertainty importance measures (UIMs) has emerged on this account to rank components based on their potentials to reduce the uncertainty about the estimated system reliability. The UIMs of components for binary-state models are well studied, but there is a lack of studies dedicated to multi-state models. In this paper, the reliability estimator and the corresponding uncertainty (characterized by the variance estimator) are derived for multi-state systems with structures such as serial, parallel, bridge, and their more complex combinations. The derivation process utilizes multinomial reliability testing and the universal generating function method. With the help of the derived estimators, we extend uncertainty importance research to multi-state models through a newly defined variance-based measure. Examples are provided to demonstrate the proposed ideas.