Part 1 Probability of Failure Event Occurrence

Fuzzy calculation in fault trees - Given the critical event or undesired condition (top event), a fault tree can be developed using backward logic to create a network of intermediate events linked by logic operators (usually AND and OR operators) down to the basic events. The fault tree itself is the logic structure relating the top event to the primary events. These primary/basic events may be related to human error (operators, design or maintenance), hardware or software failures, environmental conditions or operational conditions. 

The probability of an event defined as a fuzzy set is developed as below (Bowles and Paltiez (1995)). Let 5 be a sample space and P a probability measure defined on 5. Then, 

Zadeh has observed that P( ) can be viewed as the expected value of the characteristic function that defines the set E (Zadeh (1987)). By analogy, he defines the probability of the fuzzy set A as the expected value of the membership function for A  

Intuitively, Equations (6.8) and (6.9) define the probability of a fuzzy event as the summation over all elements, of the probability the event occurs weighted by the degree to which the element is a member of the event. Alternatively, it can be viewed as the probability of the possibility of the fuzzy event. 

The following properties of the probability of ordinary events also hold for the probabilities of fuzzy events (Terano et al. (1992)). 

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