## 2 Methodology
### Uncertainty Reduction by Interval-Valued Pythagorean Fuzzy Set
The relevant concepts of IVPFS are briefly introduced as follows:  
**Definition 1**. : \(P\) _is defined as PFS as follows:_  
\[P=\big{\{}\left(x,P(\mu_{P}(x),\
u_{P}(x))\right);x\in X\big{\}} \tag{1}\]  
_where \(X\) function is a constant set and \(\mu_{P}(x):X\mapsto[0,1]\) and \(\
u_{P}(x):X\mapsto[0,1]\) represent the membership and non-membership degree of \(x\in X\) belonging to \(P\). \(\mu_{P}(x)\) and \(\
u_{P}(x)\) should satisfy the relationship as follows:_  
\[0\leq\mu_{P}(x)^{2}+\
u_{P}(x)^{2}\leq 1.x\in X. \tag{2}\]  
_The indeterminacy degree of PFS relative to \(P\) can be calculated as follows:_  
\[\pi_{P}(x)=\sqrt{1-\mu_{P}(x)^{2}-\
u_{P}(x)^{2}}. \tag{3}\]  
**Definition 2**. : \(\widetilde{P}\) _is defined as an IVPFS in \(X\) as follows:_  
\[\widetilde{P}=\bigg{\{}\bigg{\langle}x,\widetilde{P}\bigg{(}\bigg{[}\mu_{ \widetilde{P}_{L}}(x),\mu_{\widetilde{P}_{U}}(x)\bigg{]},\bigg{[}\
u_{ \widetilde{P}_{L}}(x),\
u_{\widetilde{P}_{U}}(x)\bigg{]}\bigg{)}\bigg{\rangle} ;x\in X\bigg{\}} \tag{4}\]  
_where \(\bigg{\langle}\bigg{[}\mu_{\widetilde{P}_{L}}(x),\mu_{\widetilde{P}_{U}}(x) \bigg{]},\bigg{[}\
u_{\widetilde{P}_{L}}(x),\
u_{\widetilde{P}_{U}}(x)\bigg{]} \bigg{\rangle}\) is called an Interval-Valued Pythagorean Fuzzy number, \(0\leq\mu_{\widetilde{P}_{L}}(x)\leq\mu_{\widetilde{P}_{U}}(x)\leq\
u_{ \widetilde{P}_{L}}(x)\leq\
u_{\widetilde{P}_{U}}(x)\). \(\mu_{\widetilde{P}_{L}}(x)\) and \(\
u_{\widetilde{P}_{L}}(x)\) should satisfy the relationship as_  
\[0\leq\mu_{\widetilde{P}_{L}}(x)^{2}+\
u_{\widetilde{P}_{L}}(x)^{2}\leq 1.x \in X. \tag{5}\]  
_The indeterminacy value of PFS relative to \(\widetilde{P}\) can be calculated as follows:_  
\[\pi_{\widetilde{P}}(x)=\bigg{[}\pi_{\widetilde{P}_{L}}(x),\pi_{\widetilde{P}_ {U}}(x)\bigg{]}=\bigg{[}\sqrt{1-\mu_{\widetilde{P}_{U}}(x)^{2}-\
u_{\widetilde {P}_{U}}(x)^{2}},\sqrt{1-\mu_{\widetilde{P}_{L}}(x)^{2}-\
u_{\widetilde{P}_{L} }(x)^{2}}\bigg{]}. \tag{6}\]  
relationships between the TE and BEs based on \"AND\" and \"OR\" logic. In quantitative assessment, BEs and their logical relationships are identified to find the logical expression of TE. The occurrence probability of TE can be quantitatively estimated based on the occurrence probability of each risk factor. This work uses FTA to analyze the causal relationships of the identified factors and ranks these factors based on their likelihood of occurrence. The probability analysis in FTA offers distinct advantages as it calculates the occurrence probability of basic events based on real maintenance data, thereby accurately reflecting the actual frequency of different failure factors in operational conditions. The structured approach of FTA in establishing causal relationships and quantifying probabilities based on empirical data provides a level of analytical rigor and practical relevance that is another advantage. To calculate the contribution of each factor to the likelihood of public transportation defects, FTA enables the establishment of the logical relationships between the public transportation safety problem (TE), the critical systems (Immediate Events), and the failure-causing factors (BEs). In this work, the failure probability of BEs can be extracted from the collected maintenance records. The probability of a TE is estimated using Equations (14)-(16), which are derived based on the principles of Boolean algebra. \[P_{OR}=1-\prod_{i=1}^{n}(1-P_{i}) \tag{14}\]  
\[P_{AND}=\prod_{i=1}^{n}P_{i} \tag{15}\]  
\[P_{TE}=\prod_{j\in M}\left(1-\prod_{BE_{i}\in Q_{j}}^{n}(1-P_{i})\right) \tag{16}\]  
where \(P_{i}\) is the probability of \(BE_{i}\), and \(Q_{j}\) is a group of \(BE_{i}\). To assess the importance of each BE, their contributions to the likelihood of the TE occurring are determined. This information holds significant value for decision-makers (DMs) as it allows them to identify the vulnerable points within a system. By identifying these vulnerable points, DMs can effectively pinpoint the easily fault-prone factors that require more attention.