Evaluating minimal cut sets and the Fussell-Vesely measure of component importance using the Dynamic and Dependant Tree Theory methodology

Dynamic and Dependant Tree Theory (D 2 T 2 ) is a recent advance in fault tree analysis (FTA) which increases its ability to represent features commonly encountered on modern industrial systems. These advances are achieved by integrating the capabilities of Binary Decision Diagrams, Stochastic Petri Nets and Markov models so that the most appropriate modelling technique is used for each part of the assessment. Currently, the D 2 T 2 framework can predict the system failure probability and failure frequency, along with the Birnbaum, Criticality, Risk Achievement Worth and Risk Reduction Worth measures of component importance. In this paper, the focus is on extending the D 2 T 2 methodology to deliver the Fussell-Vesely (FV) measure of component importance. This measure of importance is defined in terms of the probability of the fault tree’s minimal cut sets. Whilst, for traditional FTA, the minimal cut sets are produced at an intermediate stage in the analysis, they are not calculated in the D 2 T 2 methodology. If they are required, additional processing must be performed. It will be demonstrated that the modularisation employed in D 2 T 2 , with each basic event appearing in only one mutually independent module, makes this a very efficient extension. The FV measures can be calculated directly from the minimal cut sets using a fast, accurate, approximation. For situations where summary information, giving the total number of minimal cut sets and the number of each order, is preferred to a full listing, a method, based on a polynomial formulation is presented.