2 2.8. Discussion and conclusions 57 Fault tree complexity Figure 2.8(d) depicts convergence time. Generally, for all m.o.f.s, case studies ordered by convergence time from longest to shortest are ddFT, MPPS, COVID-19, CSD, PT, and SMS. This suggests a relationship between the complexity of the underlying FT model and the time required for the algorithm to find it. We hypothesise that this complexity is influenced by the number of MCSs and their orders (see O-MCSs in Table 2.5). For example, the ddFT case study, with six MCSs and orders between 3 and 6, typically took the longest to converge. In contrast, the SMS case study, with 13 MCSs all of order 1, converged almost immediately. Thus, a higher number of MCSs and their orders may increase the time needed to reach the global optimum. Further research is needed to quantify this complexity. Influence of superfluous Basic Events 0 0 1 2 3 4 5 6 25 50 75 100 s -6 0 1 2 3 4 5 6 -4 -2 0 2 4 6 BEs d sdc d sdc (a) (b) Figure 2.9: Influence of number of superfluous Basic Events (ϑ) on (a) FT size (εs), (b) additional or missing number of BEs (± BEs). Using the m.o.f.s sdcandd, and the MPPS case study (ps =400, ng =100, uc =20). Real-world datasets may include varying numbers of BEs, not all contributing to system failure. Superfluous variables, which do not a!ect the TE regardless of their state, are termed superfluous BEs. We evaluate number of superfluous BEs (ε) ranging from 0 to 6 using the MPPS case study with ps =400, ng =100, uc =20, and m.o.f.s sdc and d. Figure 2.9(a) illustrates ωs across di!erent ε values. When using m.o.f. sdc, ωs is smaller than the ground truth (dashed horizontal red line). In contrast, with m.o.f. d, ωs increases with higher ε values. Figure 2.9(b) shows the additional or missing number of BEs (±BEs) for di!erent ε values. The MPPS case study has 7 unique BEs; we subtract this from the unique BEs of each inferred FT. The m.o.f. sdc consistently yields an FT with 7BEs (±BEs =0), indicating that superfluous BEs are e!ectively removed during optimisation. Conversely, m.o.f. d shows variability and performs less e!ectively as ε increases. 2.8 Discussion and conclusions We demonstrate that e"cient and interpretable Fault Tree (FT) structures can be inferred using Multi-Objective Evolutionary Algorithms. However, several challenges must be addressed before real-world applications can be considered. Although our algorithm outperforms alternative approaches, its scalability presents
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