## 2 Methodology
### Severity Impact Analysis by Interval-Valued Pythagorean Fuzzy Analytic Hierarchy Process
This adjustment weight serves to modify the relative importance of these two aspects, generating a revised ranking of factors. Based on the new ranking, the more important factors at the top of the ranking are selected for further analysis. The modified ranking is calculated as follows:  
\[MR_{i}=w\cdot R_{i}^{1}+(1-w)\cdot R_{i}^{2} \tag{25}\]  
where \(MR_{i}\) represents the combination ranking value of factor \(i\), \(R_{i}^{1}\) denotes the ranking of factor \(i\) based on FTA, and \(R_{i}^{2}\) denotes the ranking of factor \(i\) based on IVPF-AHP. \(w\) is defined as the adjustment weight. Subsequently, the failure-causing factors can be ranked again based on the calculated combination ranking values. Cause-Effect Relationship Analysis by Interval-Valued Pythagorean Fuzzy Decision-Making Trial and Evaluation Laboratory  
DEMATEL aims to analyze and resolve complex, interrelated problem groups [52; 53]. Compared to traditional MCDM methods, DEMATEL provides two notable advantages. Unlike conventional techniques that assume independence among elements, DEMATEL identifies interdependencies within a system by using structural modeling and causal diagrams. Furthermore, while traditional methods may struggle with decision quality and complexity when confronted with numerous criteria, DEMATEL is particularly well suited to handle such situations, ensuring both decision quality and manageability. However, this people-oriented process is often unclear and uncertain. To mitigate the ambiguity of human assessments, this work adopts the IVPF-DEMATEL method to analyze the cause-effect relationships among the selected failure-causing factors. The following steps can represent the procedures involved in IVPF-DEMATEL:  
1. Build of the Interval-Valued Pythagorean initial direct relation matrix \(A\). Experts compare the factors to generate an Interval-Valued Pythagorean initial direct relation matrix \(A=\big{(}a_{ij}\big{)}_{m\times m}\) where \(a_{ij}\) indicates the factor \(i\) on factor \(j\) impact. The transformation of Interval-Valued Pythagorean Fuzzy semantics is shown in Table 2.  
2. Defuzzification of the Interval-Valued Pythagorean initial direct relation matrix \(A\) for a crisp direct relation matrix \(B=\big{(}d_{ij}\big{)}_{m\times m}\) is carried out according to Definition 6. \begin{table}
\begin{tabular}{c c} \hline
**Linguistic Terms** & **Interval-Valued Pythagorean Fuzzy Number** \\ \hline Very high influence & ([0.80, 0.95], [0.00, 0.10]) \\ High influence & ([0.60, 0.80], [0.10, 0.30]) \\ Low influence & ([0.40, 0.60], [0.30, 0.50]) \\ Very low influence & ([0.20, 0.40], [0.50, 0.70]) \\ No influence & ([0.00, 0.20], [0.70, 0.90]) \\ \hline \end{tabular}
\end{table} 
Table 2: Linguistic term and the corresponding Interval-Valued Pythagorean Fuzzy numbers [54]. 3. The normalization of the crisp direct relation matrix \(N=\left(n_{ij}\right)_{m\times m}\) is as follows: \[N=S\times B\] (26) where \(SS=min\bigg{\{}1/\min\limits_{1\leq i\leq m}\sum_{j=1}^{m}b_{ij},1/\max\limits_ {1\leq j\leq m}\sum_{i=1}^{m}b_{ij}\bigg{\}},i,j=1,2,\cdots,m\). 4. Computation of the total relation matrix \(T\) is as follows: \[T=N(I-N)^{-1}\] (27) where \(I\) is an \(n\times n\) identity matrix. 5. The influence degree, influenced degree, centrality degree, and cause degree are calculated. According to the total relation matrix \(T\), the influence degree \(D\) and influenced degree \(R\), which are the sum of rows and the sum of columns, are computed by applying Equations (28) and (29). The centrality degree is calculated as \((D+R)\). The cause degree is calculated as \((D-R)\). \[R=\left\lceil\sum_{i=1}^{n}t_{ij}\right\rceil_{1\times n} \tag{28}\] \[D=\left\lceil\sum_{j=1}^{n}t_{ij}\right\rceil_{n\times 1}. \tag{29}\]  
\((D+R)\), named \"centrality degree\", represents the degree of relative importance of each factor. The factor with a higher \((D+R)\) value is more important that with a lower value.