## 2 Methodology
### Fuzzy Comprehensive Evaluation
#### 2.3.3 Fuzzy Failure Probability  
To convert the expert scoring matrix into a fuzzy number, the linguistic expression needs to be quantified, and then the fuzzy number is calculated by fuzzy theory. The key to quantifying linguistic expression is to determine the fuzzy membership functions. According to the expert scoring matrix, this paper uses the trapezoid-triangular membership function recommended by the engineering to describe the expert language [36]. Figure 5 shows the fuzzy membership functions and the fuzzy range corresponding to fuzzy languages rating, in which the abscissa indicates the fuzzy range corresponding to the fuzzy languages rating of the basic events, and the ordinate indicates the fuzzy membership function corresponds to the fuzzy languages rating. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Grade** & **Score** \\ \hline \multirow{4}{*}{Job title} & Professor, professor-level senior engineer & 8 \\  & Associate Professor, Senior Engineer & 6 \\  & Lecturer, Engineer & 4 \\  & Worker & 1 \\ \hline \multirow{4}{*}{Service time (Year)} & \(>20\) & 9 \\  & 15–20 & 7 \\  & 10–15 & 5 \\  & 5–10 & 3 \\ \hline \multirow{4}{*}{Education level} & PhD & 8 \\  & Master & 5 \\ \cline{1-1}  & Bachelor & 3 \\ \cline{1-1}  & others & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experts score table. Figure 4: Expert scoring risk matrix. Very Low (VL), Low (L), Fairly Low (FL), Medium (M), Fairly High (FH), High (H) and Very High (VH). The membership function corresponding to every fuzzy language is:  
\[f_{VL}(x)=\left\{ \begin{array}{ll}1&(0<x\leq 0.1)\\ \frac{0.2-x}{0.1}&(0.1<x\leq 0.2)\\ 0&\textit{other}\end{array} \right., \tag{2}\]  
\[f_{L}(x)=\left\{ \begin{array}{ll}\frac{x-0.1}{0.1}&(0.1<x\leq 0.2)\\ \frac{0.3-x}{0.1}&(0.2<x\leq 0.3)\\ 0&\textit{other}\end{array} \right., \tag{3}\]  
\[f_{FL}(x)=\left\{ \begin{array}{ll}\frac{x-0.2}{0.1}&(0.2<x\leq 0.3)\\ 1&(0.3<x\leq 0.4)\\ \frac{0.5-x}{0.1}&(0.4<x\leq 0.5)\\ 0&\textit{other}\end{array} \right., \tag{4}\]  
\[f_{M}(x)=\left\{ \begin{array}{ll}\frac{x-0.4}{0.1}&(0.4<x\leq 0.5)\\ \frac{0.6-x}{0.1}&(0.5<x\leq 0.6)\\ 0&\textit{other}\end{array} \right., \tag{5}\]  
\[f_{FH}(x)=\left\{ \begin{array}{ll}\frac{x-0.5}{0.1}&(0.5<x\leq 0.6)\\ 1&(0.6<x\leq 0.7)\\ \frac{0.8-x}{0.1}&(0.7<x\leq 0.8)\\ 0&\textit{other}\end{array} \right., \tag{6}\]  
\[f_{H}(x)=\left\{ \begin{array}{ll}\frac{x-0.7}{0.1}&(0.7<x\leq 0.8)\\ \frac{0.9-x}{0.1}&(0.8<x\leq 0.9)\\ 0&\textit{other}\end{array} \right., \tag{7}\]  
\[f_{VH}(x)=\left\{ \begin{array}{ll}\frac{x-0.8}{0.1}&(0.8<x\leq 0.9)\\ 1&(0.9<x\leq 1)\\ 0&\textit{other}\end{array} \right. \tag{8}\]  
Let the \(\lambda\)-cut sets of the membership functions represented by the fuzzy language be [37]:  
\(VL_{\lambda}=[v_{1},v_{2}]\), \(L_{\lambda}=[l_{1},l_{2}]\), \(FL_{\lambda}=[f_{1},f_{2}]\), \(M_{\lambda}=[m_{1},m_{2}]\), \(FH_{\lambda}=[\,h_{1},h_{2}\,]\), \(H_{\lambda}=[\,g_{1},g_{2}\,]\), \(VH_{\lambda}=[\,b_{1},b_{2}. ]\) Among them, \(v_{1}\), \(v_{2}\),..., \(b_{1}\) and \(b_{2}\) are the upper and lower limits of the \(\lambda\)-cut sets of the above formula, respectively. Let \(\lambda=\left(x-0.2\right)/0.1\), then \(v=0.2-0.1\lambda\);  
Figure 5: Diagram of fuzzy membership functions. Very Low (VL), Low (L), Fairly Low (FL), Medium (M), Fairly High (FH), High (H) and Very High (VH). Let \(\lambda=\left(x-0.1\right)/0.1\), then \(l_{1}=0.1\lambda+0.1\), \(l_{2}=0.3-0.1\lambda\). Similarly:  
\[f_{1}=0.1\lambda+0.2,\ f_{2}=0.5-0.1\lambda;\]  
\[m_{1}=0.1\lambda+0.4,\ m_{2}=0.6-0.1\lambda;\]  
\[h_{1}=0.1\lambda+0.5,\ h_{2}=0.8-0.1\lambda;\]  
\[g_{1}=0.1\lambda+0.7,\ g_{2}=0.9-0.1\lambda;\]  
\[b=0.1\lambda+0.8.\]  
Then the fuzzy set theory is applied to solve the total fuzzy set \(z\) for the basic events:  
\[