On a Calculation Method and Stochastic Bounds for the Birnbaum Importance Measure of a Component

In modern society, complex and large-scale systems perform their own tasks to sustain various infrastructures. In this context, it is crucial to maintain and improve the reliability of these systems to ensure safe and sustainable social lives. To improve the reliability of these systems, we prefer to improve components with higher importance in the system. Various importance measures have been proposed [1–7]. The Birnbaum importance measure [1] is fundamental in the reliability engineering and is usually defined as a partial differentiation of the reliability function under the assumption of stochastic independence among the components. In this paper, we examine in detail an algorithm [8] for deriving the Birnbaum importance measure from the minimal path and cut state vectors, where the importance measure is defined as the probability of the set of all the critical state vectors. The definition shows that the Birnbaum importance measure can be defined without the independence assumption, a non-empty condition is required to use the inclusion and exclusion method in the algorithm and stochastic bounds for the measure are also given when a joint performance probability of the components is associated.