206 Appendix B. Appendixes: FT-MOEA Delete Front 3 (d) d s Pass to the next generation Max. population size (ps) Error Front Sorted fault trees F1 F2 F1 F2 (e) F3 F3 F4 Front 1 Front 3 Front 2 Front 4 (a) (b) (c) Front 1 d s d s Figure B.1: Conceptual visualisation of the Non-Dominated Sorting Genetic Algorithms (NSGA-II) and Crowding-Distance in the context of automatic inference of FTs. In (a) error based on the failure dataset (εd) versus fault tree size (εs), in (b) Pareto fronts, (c) details of the first front, (d) influence of Crowding-Distance, and (e) criteria for acceptance and rejection of FTs between generations. Figure B.1(d) shows the e!ect of the Crowding-Distance. Suppose that only four of the five FTs of the third front can pass to the next generation. Therefore, it is necessary to “break” the front. To do so, we compute the Crowding-Distance metric (di) (Section B.2.2). Those solutions in the front with a large di value have priority to pass to the next generation. Conversely, those with a small di value have a lower priority because it means that they are similar to other solutions in the front. Therefore, in Figure B.1(d) the FTs marked with the arrow “Delete” must have similar features in ωs and ωd compared to its neighbours, thus becoming the candidate to be deleted from the third front. Figure B.1(e) represents the process to select the FTs that pass to the next generation, where first the FTs within each front are ordered from the minimum to the maximum error (or errors when considering the minimisation of both 3c and 3d, here we sum them up and sort them up), then only the first ps FTs pass to the next generation. Here we can observe that one FTs from the third front and the only FTs from the fourth front did not manage to pass to the next generation.
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