STANDARD-DEVIATION AS AN ALTERNATIVE TO FUZZINESS IN FAULT-TREE MODELS

Construction of fault-tree models almost always involves some engineering judgment or estimation when it comes to assignment of probabilities for the primary events in the fault tree. The top event probability in a fault tree is, of course, a function of the probabilities of occurrence of the primary events. A clearly desirable goal is to characterize fluctuations in the top-event probability that might be caused by variations or uncertainty in the probabilities of the primary events. Numerous authors have borrowed ideas from the field of fuzzy-set theory to introduce fuzzy computations into fault-tree algorithms. Primary events have fuzzy probabilities, and the goal is then to compute top-event probability as a fuzzy number. A more useful way to introduce uncertainty in such models is through treatment of primary event probabilities as random variables whose standard deviation provides a measure of the uncertainty in the model. This paper shows that useful probabilistic information can be obtained by bounding the standard deviation of the top-event probability. This approach provides probabilistic information not inherent in fuzzy methods. A corresponding caveat, of course, is that this approach requires specific knowledge of the standard deviation of the probabilities of occurrence of the primary events, and this knowledge is not assumed in the fuzzy approaches which instead assume known possibility functions for the primary events. Even if precise probability distributions for all primary events are known, computational difficulties thwart any effort to determine the exact probability distribution of the top event. Feasible extensions of existing algorithms can be used to bound the standard deviation, with little increase in computational complexity.