54 Chapter 2. Automatic Inference of Fault Tree Models via Multi-Objective Evolutionary Algorithms (a) (b) 0 5 10152025 s 0 0.25 0.5 0.75 1 d Gen. 1 Gen. 2 Gen. 3 Gen. 10 Gen. 20 0 20 40 60 80 100 s 0 0.25 0.5 0.75 1 d Figure 2.5: Evolution of metrics over generations: (a) using the m.o.f. d, and (b) using the m.o.f. sdc, for the MPPS case study (ps =400, ng =100, uc =20). The red circle with an arrow at the bottom of (b) marks the global optimum. In Figure 2.6, we compare m.o.f. d and sdc across generations, focusing on the metrics of the best FT per generation (i.e., the one on the first Pareto front with the smallest errors ωd and ωc). Figure 2.6(a) shows ωd across generations for both objective functions, indicating that the m.o.f. d more quickly minimises ωd than m.o.f. sdc. However, m.o.f. sdc achieves the global optimum by the 20th generation, whereas m.o.f. d does not. 0 10 20 30 40 Generation 0 0.1 0.2 0.3 d sdc (a) d d sdc 0 10 20 30 40 Generation 0 0.2 0.4 0.6 (b) c d sdc s 0 20 40 60 80 100 0 10 20 30 40 Generation (c) d sdc 0 10 20 30 40 Generation 0 50 100 150 200 Cum. Conv. (min) (d) Figure 2.6: Metrics over generations for the best FTs using the m.o.f.s d and sdc. Convergence of (a) εd, (b) εc, (c) εs, (d) cumulative convergence time in minutes. Using the MPPS case study (ps =400, ng =100, uc =20). Figure 2.6(b) compares ωc, showing similar trends. Figure 2.6(c) depicts ωs variation over generations, illustrating that our m.o.f. maintains smaller FT structures. Although the ground truth FT size is 23, FT-MOEA finds one with ωs = 14, an equivalent and compressed version of the original (see Appendix B.3, Figure B.2 for details). Figure 2.6(d) shows the cumulative convergence time (t) for both m.o.f.s. Our algorithm finds the optimal solution in about 20 minutes, whereas minimising only ωd takes about 4 hours without reaching the global optimum. Details on the
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