## 2 Methodology
### Fuzzy Comprehensive Evaluation
#### 2.3.3 Fuzzy Failure Probability
\begin{array}{c}f_{EXP1@EXP2@EXP3@EXP4@EXP5@EXP6@EXP6@EXP7@EXP8@EXP9@EXP10 5@EXP11@EXP12}\\ =\max\left|W_{1}f_{EXP1}(x)\wedge W_{2}f_{EXP2}(x)\wedge W_{3}f_{EXP3}(x) \wedge W_{4}f_{EXP4}(x)\wedge W_{5}f_{EXP4}(x)\wedge W_{6}f_{EXP6}(x)\\ \wedge W_{7}f_{EXP7}(x)\wedge W_{8}f_{EXP8}(x)\wedge W_{9}f_{EXP9}(x)\wedge W _{10}f_{EXP10}(x)\wedge W_{11}f_{EXP11}(x)\wedge W_{12}f_{EXP12}(x)\right| \\ =\left|\sum\limits_{i=1}^{12}W_{i}(a_{EXPi}\lambda+b_{EXPi}),\sum\limits_{i= 1}^{12}W_{i}(c_{EXPi}-d_{EXPi}\lambda)\right|\\ =\left|a\lambda+b,c-d\lambda\right|\end{array}  \tag{9}\]  
Where \"\(@\)\" is the bounded operator, \(f_{M@N}(z)=\max(f_{\rm M}(z)\wedge f_{\rm N}(z))\), \"\(\wedge\)\" means a small value is taken, _EXPi_ is the score of the \(i\)-th expert on the basic event \(X\), Wi is the weight of the \(i\)-th expert(\(0<Wi<1\)), _fEXPi_ is the membership function corresponding to the \(i\)-th expert's score, _aEXPi_ and _bEXPi_ are the coefficients corresponding to the upper limit of the membership function \(\lambda\)-cut sets corresponding to the \(i\)-th expert's score, and they are positive real numbers, _cEXPi_ and _dEXPi_ are the coefficients corresponding to the lower limit. Then, these were solved to obtain the membership function of the average fuzzy number W, where a, b, c and d are positive real numbers. Let \(W=\left|(z_{1},z_{2})\right|=\left|(a\lambda+b),\left(c-d\lambda\right)\right|\), then \(\lambda=\left(z_{1}-b\right)/a\) and \(\lambda=\left(c-z_{2}\right)/d\). Therefore, the membership function of the average fuzzy number W is:  
\[f_{W}(z)=\left\{ \begin{array}{ll}\frac{z_{1}-b}{a}&\left(b<z\leq a+\ b \right)\\ 1&\left(a+b<z\leq c-d\right)\\ \frac{c-z_{2}}{d}&\left(c-d<z_{2}\leq c\right)\\ 0&\mbox{\emph{other}}\end{array} \right.. \tag{10}\]  
The membership function of the average fuzzy number \(W\) is used to solve the probability of every basic event. Firstly, the fuzzy possibility score (_FPS_) is needed [37]. After obtaining the membership function of the average fuzzy number W, the fuzzy maximum set and fuzzy minimum set are defined respectively. The fuzzy number is converted into a left and right _FPS_ (\(w\)) by the left and right fuzzy ranking method. First, define the fuzzy maximum set and fuzzy minimum set:  
\[f_{\max}(x)=\left\{ \begin{array}{ll}x&\left(0<x<1\right)\\ 1&\left(\mbox{\emph{other}}\right)\end{array} \right., \tag{11}\]  
\[f_{\min}(x)=\left\{ \begin{array}{ll}1-x&\left(0<x<1\right)\\ 0&\left(\mbox{\emph{other}}\right)\end{array} \right.. \tag{12}\]  
The left and right fuzzy possibility of the fuzzy numbers are:  
\[FPS_{R}(w)=\sup_{x}[f_{w}(x)\wedge f_{\max}(x)], \tag{13}\]\[FPS_{L}(w)=\sup_{x}[f_{w}(x)\wedge f_{\min}(x)]. \tag{14}\]  
where \"sup\" means the minimum upper bound of a set, \(FPS_{R}(w)\)is the right fuzzy possibility, \(FPS_{L}(w)\)is the left fuzzy possibility. The fuzzy possibility \(FPS(w)\)is:  
\[FPS(w)=\frac{[FPS_{R}(w)+1-FPS_{L}(w)]}{2}. \tag{15}\]  
Finally, the probability of every basic event can be obtained by converting the \(FPS\), that is, the fuzzy failure probability (\(FFR\)) [37]. A function developed by [PERSON] [38] is used for converting \(FPS\) to \(FFR\):  
\[FFR=\left\{ \begin{array}{cc}\frac{1}{10^{k}}&(FPS\
eq 0)\\ 0&(FPS=0)\end{array} \right.. \tag{16}\]  
where:  
\[k=[\frac{1-FPS}{FPS}]^{\frac{1}{3}}\times\log(\frac{1}{E_{mM}}). \tag{17}\]  
In the formula, \(E_{mM}\) is the most likely failure rate, which is a reference value obtained from the statistical data of the top event, or an empirical value provided by experts. \(FFR\) is the fuzzy probability of basic events obtained by expert evaluation and fuzzy theory.