Condition Fault Tree: An Extension of Traditional Fault Tree to Handle Uncertainty

Fault Tree Analysis (FTA) is a powerful and well-established tool, widely-used to evaluate system reliability. The logical connections between faults and causes in Fault Trees (FT) are assumed to be deterministic and are represented graphically via logical gates (such as AND gate, OR gate, NOT gate, etc.). However, sometimes the causalities can be uncertain. Considering that some of the causal relationships in FTs may be uncertain or non-deterministic, we propose a new model to represent the uncertainties, so called as Condition Fault Tree (CFT). We extend the traditional FTA by introducing a new parameter U, which illustrates the random mechanism of how parent event can cause child event. The probability of U (which is denoted by u = Pr{U}), is used to measure the uncertainty between parent event and child event. By introducing rules of parameter U in CFT, we explore its properties and corollaries. We also introduce a methodology to simplify CFTs based on Contraction, Elimination and Extraction rules. With the simplification rules, the structure of CFT can be simplified and the size of CFT can be significantly reduced. Since CFT is an extension of traditional FT, a qualitative analysis method and a quantitative method are introduced. For qualitative analysis, one can simplify a given CFT into the simplest form with the aforementioned rules, properties, and corollaries. With the simplest form of CFT, one can then get the Minimum Cut Sets with uncertainties, as an extension of Minimum Cut Sets. For quantitative analysis, exact calculation methods based on Inclusion-Exclusion and Disjoint-Sum-of-Product are proposed. Some examples are used to illustrate how CFT works.