52 Chapter 2. Automatic Inference of Fault Tree Models via Multi-Objective Evolutionary Algorithms so that the number of unique observations of BE matches the space complexity O(2w), where w is the number of unique BE for a given FT. 2.7.2 Case studies To establish a reliable ground truth, we utilised existing FTs from the literature across di!erent applications. Our selection criteria included the number of elements in the FT, the number of MCSs, and their orders. Table 2.5 lists the selected case studies, providing details on the number of BE, the total number of AND, OR, and VoTgates, the number of MCSs, their orders, and the space complexity, measured as O(2w). Since we only work with complete datasets, the space complexity in Table 2.5 also reflects the size of the failure dataset. We distinguish between the number of unique BE (w) and the total number of BE (W) because some FTs contain shared BEs. 2.7.3 Key findings of the FT-MOEA algorithm To illustrate our findings and main contributions, we use the Mono-propellant propulsion system (MPPS) case study from Table 2.5. We first generate the failure dataset as described in Section 2.7.1, containing N=250,000 data points. This dataset is then used as input for the FT-MOEA algorithm, with the following initial parameters: ps =400, ng =100, and uc =20. We compare the evolutionary process across generations for two m.o.f.s, d and sdc, to contrast the approach by Linard, Bucur, and Stoelinga, 2019 (FT-EA, minimising only ωd) with our multi-objective optimisation process (FT-MOEA, minimising ωs, ωd, and ωc). Figure 2.5(a) shows the results across generations when minimising solely ωd. A rapid decrease in ωd is observed in the initial generations, but from the 10th generation onwards, there is a sharp increase in FTs size (ωs) (up to ωs =100) without further reduction in ωd, while the ground truth FT has ωs =23. Conversely, using FT-MOEA (Figure 2.5(b)) results in a more balanced decrease in all directions. Moreover, FT-MOEA reaches the global optimum (i.e., ωd =ωc =0.0) by the 20th generation with ωs =14 (red circle with an arrow in 2.5(b)), an equivalent compressed version of the ground truth FT.
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