50 Chapter 2. Automatic Inference of Fault Tree Models via Multi-Objective Evolutionary Algorithms (iv) BE-disconnect: Selects a basic event in the set BE of a given FT and disconnects it. (v) BE-connect: Takes a disconnected basic event from a given FT and randomly places it under a gate in the set G. (vi) BE-swap: Moves a basic event in the set BE of a given FT to a di!erent parent gate in the set G. (vii) Crossover: Randomly selects two FTs in the o!spring population, then chooses an element in the set Vof each FT to exchange. 2.6.5 Step 4 - Multi-objective function Metrics calculation Our multi-objective function considers three metrics: fault tree size (ωs), error based on failure data (ωd), and error based on MCSs (ωc). - Fault Tree Size (ωs): Number of elements Vin an FT (Eq. 2.1): ωs =|V| =|BE| +|G|, (2.1) here, ωs ⇒2 since every FT has at least one BE and one G(i.e., the TE). - Error Based on Failure Data (ωd): Calculated using a vector Pwith Nvalues, where Nis the number of data points. Pcontains the TE values for a given set BE. The corresponding ground truth top event (TE↓) is provided in the failure dataset (Section 2.5). ωd is computed as: ωd =1→) Ni=1 xi N * xi =1, if Pi =TE↓i . xi =0, Otherwise. (2.2) ωd ranges from 0 to 1, where 0 indicates perfect mapping of BE to the corresponding TE in the failure dataset. - Error Based on MCSs (ωc): Computed using the RV-coe!cient (Robert and Escoufier, 1976), which generalises the squared Pearson correlation coe!cient to measure similarity between the MCS matrix from the failure dataset (MD) and the MCS matrix of a given FT (M F ). M F is derived from the disjunctive normal form(DNF), which requires transforming an FT into its DNF to identify MCSs and construct M F . This transformation is computationally intensive for large FT sizes. MD and M F are matrices of sizes p↘w and q ↘w respectively, where w is the number of unique BEs, and p and q are the numbers of MCSs in the failure dataset and FT, respectively. The computation of ωc is given by: ωc =1→ tr(MDMT F M F MT D) +tr(MDMT D) 2 tr(M F MT F )2 (2.3)
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