## 2 Methodology
### Severity Impact Analysis by Interval-Valued Pythagorean Fuzzy Analytic Hierarchy Process  
AHP ascertains the weights of criteria and the priorities of alternatives via pairwise comparisons [48; 49]. AHP integrates both rational and intuitive values. By decomposing complex decisions into a manageable hierarchy, AHP allows decision-makers to consider both quantitative data and qualitative insights. This hierarchical structure is particularly valuable when dealing with numerous structural factors distributed across multiple systems and subsystems, where coupling effects exist between factors. Nonetheless, the pair wise comparison process can introduce imprecision as it relies on expert judgments. In real-world scenarios, experts may struggle to score crisp values due to constraints in useful information or cognitive capacity [50]. To address this imprecision, this work uses IVPF-AHP to rank the identified failure-causing factors based on their fault severity impact. Integrating AHP and FST allows for more robust reflections of DMs' judgments. The procedures of the IVPF-AHP method are shown below:  
1. Establishment of the hierarchical structure. A typical structure of AHP hierarchy has the objective at the top position, followed by factors and alternatives at the bottom position. If it is necessary to divide too many factors into multistep, a sub-criterion can be made. The structure of AHP analysis for our work to study transportation safety is consistent with the FT structure. A BE is treated as a factor. 2. Construction of the Interval-Valued Pythagorean pairwise comparison matrix \(A\). The DMs are invited to finish the questionnaire-form survey to set the relative importance weight of factors. The pairwise comparison matrix \(A=\left(a_{ij}\right)_{m\times m}\) is constructed using the linguistic terms given in Table 1.  
3. The calculation of the difference matrix \(D=\left(d_{ij}\right)_{m\times m}\) is as follows:  
\[d_{ijL}={\mu_{ijL}}^{2}-{v_{ijL}}^{2} \tag{18}\]  
\[d_{ijL}={\mu_{ijL}}^{2}-{v_{ijL}}^{2}. \tag{19}\]  
4. The calculation of the multiplicative matrix \(S=\left(s_{ij}\right)_{m\times m}\) is as follows:  
\[s_{ijL}=\sqrt{1000^{d_{ijL}}} \tag{20}\]  
\[s_{ijL}=\sqrt{1000^{d_{ijL}}}. \tag{21}\]  
5. The calculation of the indeterminacy matrix \(H=\left(h_{ij}\right)_{m\times m}\) matrix is as follows:  
\[h_{ij}=1-\left({\mu_{ijL}}^{2}-{\mu_{ijL}}^{2}\right)-\left({v_{ijL}}^{2}-{v_{ ijL}}^{2}\right). \tag{22}\]  
6. The calculation of the unnormalized weights \(T=\left(t_{ij}\right)_{m\times m}\) is as follows:  
\[t_{ij}=h_{ij}\left(s_{ijL}+s_{ijL}\right)/2. \tag{23}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Linguistic Terms**} & \multicolumn{4}{c}{**Interval-Valued Pythagorean Numbers**} \\ \cline{2-5}  & \(\mu_{L}\) & \(\mu_{U}\) & \(\
u_{L}\) & \(\
u_{U}\) \\ \hline Certainly Low Importance (CLI) & 0.0 & 0.0 & 0.9 & 1.0 \\ Very Low Importance (VLI) & 0.1 & 0.2 & 0.8 & 0.9 \\ Low Importance (LI) & 0.2 & 0.35 & 0.65 & 0.8 \\ Below Average Importance (BLI) & 0.35 & 0.45 & 0.55 & 0.65 \\ Equal Importance (EI) & 0.1965 & 0.1965 & 0.1965 & 0.1965 \\ Average Importance (AI) & 0.45 & 0.55 & 0.45 & 0.55 \\ Above Average Importance (AHI) & 0.55 & 0.65 & 0.35 & 0.45 \\ High Importance (HI) & 0.65 & 0.8 & 0.2 & 0.35 \\ Very High Importance (VHI) & 0.8 & 0.9 & 0.1 & 0.2 \\ Certainly High Importance (CHI) & 0.9 & 1.0 & 0.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Linguistic scale for the evaluations [51]. The determination of the criterion weight \(w_{i}\) for each factor is as follows:  
\[w_{i}=\sum_{i=1}^{m}t_{ij}/\sum_{i=1}^{m}\sum_{j=1}^{m}t_{ij}\,. \tag{24}\]  
Factor Ranking Using Results from Fault Tree Analysis and Interval-Valued Pythagorean Fuzzy Analytic Hierarchy Process  
Applying FTA and IVPF-AHP generates two types of relative importance weights and weight rankings. An adjustment weight is introduced to ensure a balanced evaluation that considers both fault severity impact and occurrence probability impact.