44 Chapter 2. Automatic Inference of Fault Tree Models via Multi-Objective Evolutionary Algorithms Cut Sets (MCSs)—minimal combinations of component failures leading to system failure. Smaller MCSs highlight system vulnerabilities. Quantitative analysis calculates dependability metrics, such as systemReliability, Availability, and Mean-Time-to-Failure. These calculations require the FT leaves to have assigned failure probabilities. For formal definitions and terminology related to Fault Trees, see Section I.4.1. Figures 2.2(a) and 2.2(b) illustrate the event and gate symbols used in constructing the FT model. Figure 2.2(c) presents an example of an FT for a Container Seal Design, adapted from NASA, 2002. In this FT, the top event, sealing function fails, occurs either due to a common cause seal failure or independent seal failures. The former requires both contamination tape failure and a basic cause seal failure, while the latter necessitates failures in the metal-to-metal seal, fused plug, and at least two of the three compression seals. 2.3 Multi-Objective Evolutionary Algorithms Evolutionary Algorithms (EAs) are population-based search strategies inspired by natural selection, where the most fit individuals are more likely to reproduce and pass on their traits to subsequent generations (Ojha, Singh, Chakraborty, et al., 2019). When EAs are used to optimise several conflicting objective functions simultaneously in a multi-dimensional space, they are termed Multi-Objective Evolutionary Algorithms (MOEAs) (Deb, 2011). MOEAs yield a set of solutions with trade-o!s, known as Pareto-optimal solutions, from which users can select based on higher-level qualitative considerations (Deb, 2005). To address the challenge of automatically inferring FTs from a failure dataset while optimising di!erent metrics, we chose to employ the Elitist Non-dominated Sorting Genetic Algorithm(NSGA-II) (Section 2.3.1) and the Crowding-Distanceapproach (Section 2.3.2), both of which are widely used in multi-objective optimisation. 2.3.1 Elitist Non-dominated Sorting Genetic Algorithms The Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) (Deb, Pratap, Agarwal, et al., 2002) is designed to find multiple Pareto-optimal solutions. NSGA-II employs the elitist principle, a diversity-preserving mechanism that focuses on non-dominated solutions (Deb, 2005). This principle ensures solution quality by allowing the best individual(s) of the current generation to advance to the next. Non-dominated MOEAs rely on the concept of dominance, comparing two solutions to determine if one dominates the other. Non-dominated sorting is important for identifying elitist e"cient solutions in MOEAs, but it is computationally intensive due to the numerous comparisons required (Long, X. Wu, and C. Wu, 2021). A set of solutions that do not dominate each other forms a non-dominated front.
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