# Hardware Reliability Analysis of a Coal Mine Gas Monitoring System Based on Fuzzy-FTA
## 2 The Principle of the Fuzzy-FTA Analysis
### Application of the Fuzzy-FTA Method
#### 2.3.3 Interval Analysis of the Occurrence Probability of the Top Event  
According to the principle of fuzzy mathematics, the occurrence probability value of each basic event of the gas monitoring and prediction system is processed into a triangular fuzzy number [21], and the reference function is:  
\[\left\{\begin{array}{cc}L(\frac{m-x}{a})=\max\left(0,\frac{m-x}{a}\right)&x \leq m,\alpha>0\\ R\left(\frac{x-m}{\beta}\right)=\max\left(0,\frac{x-m}{\beta}\right)&x>m, \alpha>0\end{array}\right. \tag{12}\]  
The membership function \(A\) corresponding to the triangular fuzzy number is:  
\[\mu_{A(x)}=\left\{\begin{array}{cc}0&x<m-\alpha\\ 1-\frac{m-x}{a}&m-\alpha\leq x\leq m\\ 1-\frac{x-m}{\beta}&m<x\leq m+\beta\\ 0&x>m+\beta\end{array}\right. \tag{13}\]\(A=(\alpha,m,\beta)\), where \(m\) is the mean, \(\alpha,\beta\) respectively are the upper and lower limits of the confidence interval. The distribution of \(\alpha\), \(\beta\) is larger, \(A\) is more blurred. When \(A\) is equal to 0, \(A\) is the non-fuzzy number. The \(\lambda\) cut set of \(A\) can be expressed as:  
\(A_{\sim_{\lambda}}=\left[(m-\alpha)+\alpha\lambda,(m+\beta)-\beta\lambda\right]\)  
According to the principle of fuzzy mathematics, the probability value of each basic event in a gas monitoring and forecasting system is treated as a triangular fuzzy number [22].  
Let \(s_{1}\) and \(s_{2}\) be the input events of the AND/OR gate, and s the output event, then, \(\underline{E}_{s_{1}}\) and \(\underline{E}_{s_{2}}\) represent the fuzzy failure probabilities of \(s_{1}\) and \(s_{2}\), respectively, as shown in Figure 2.  
Therefore, the fuzzy failure probability \(\underline{E}_{s}\) of the output event s of the AND gate/OR gate in the fault tree can be obtained by the following formula:  
(1)  
AND gate structure  
\(F_{\sim_{\alpha_{2}}}=\left[(m-\alpha_{1})+\alpha_{1}\lambda,(m+\beta_{1})- \beta_{1}\lambda\right]\), \(F_{\sim_{\alpha_{2}}}=\left[(m-\alpha_{2})+\alpha_{2}\lambda,(m+\beta_{2})- \beta_{2}\lambda\right]\)  
\(F^{AND}_{\sim}=\underset{\tau_{\alpha_{1}}\tau_{\beta_{1}}\sim_{\alpha_{2}}}{ \mathbb{F}}=\left[(m-\alpha_{1})+\alpha_{1}\lambda,(m+\beta_{1})-\beta_{1} \lambda\right]\)\([(m-\alpha_{2})+\alpha_{2}\lambda,(m+\beta_{2})-\beta_{2}\lambda]\)  
Formula (14) introduced the AND gate relationship between the output \(\underline{E}_{s}\) with the inputs \(\underline{E}_{s_{1}}\) and \(\underline{E}_{s_{2}}\).  
(2)  
OR gate structure  
\(F^{OR}_{\sim}=1-\binom{1-F_{s1}}{\sim}\left(1-F_{s2}\right)=[1,1]-\frac{2}{ \prod}\{[1,1]-[(m-\alpha_{i})+\alpha_{i}\lambda,(m+\beta_{i})-\beta_{i} \lambda]\}\)  
\(=\left[1-\prod\limits_{1}^{2}[(m-\alpha_{1})+\alpha_{1}\lambda],1-\frac{2}{ \prod}[(m+\beta_{1})-\beta_{1}\lambda]\right]\)  
Formula (15) introduced the OR gate relationship between the output \(\underline{E}_{s}\) with the inputs \(\underline{E}_{s_{1}}\) and \(\underline{E}_{s_{2}}\).  
In this way, according to the structural function of the fault tree, the \(\lambda\) cut interval of the occurrence probability of the top event can be calculated. By assigning different values to \(\lambda\), the probability interval of the top event occurrence under different fuzzy degrees can be obtained [23].  
Figure 2: AND gate and OR gate structure chart.