# Hardware Reliability Analysis of a Coal Mine Gas Monitoring System Based on Fuzzy-FTA
## 3 Reliability Analysis Process of Gas Monitoring System Based on Fuzzy-FTA
### Quantitative Analysis of Fault Tree
#### 3.4.2 Calculation of the Fuzzy Probability of the Top Event  
The \(\lambda\) cut set is shown in Table 1 according to the triangular fuzzy probability of each basic event. After obtaining the cut-off set of each basic event, the probability of the top event is calculated using Equation [26]. The probability of the top event is calculated as follows:  
\[g=\left\{[1,1]-\prod_{i=1}^{16}([1,1]-q_{i})\right\}\times\left\{[1,1]-([1,1] -q_{17})([1,1]-q_{18})([1,1]-q_{19})\right\} \tag{19}\]  
The fuzzy probability of the top event is obtained by evaluating the structural function of the fault tree. Moreover, the high-order items ignored in the calculation process are obtained. The \(\lambda\) cut set form of the occurrence probability of the top event is calculated as follows:  
\[T=[0.000095+0.000020\lambda,0.000135-0.000022\lambda] \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Basic Event** & **Event Name** & **Suggested Value** & **Order of Magnitude** & **Suggested Order of Magnitude** & **Linguistic Variable Value** & **Linguistic Variable Value** & **Remark** \\ \hline X17 & Lack of responsibility of the staff & \(10^{-4}\) & & & & \\ X18 & System lacks spare parts & \(10^{-2}\) & & & & \\ X19 & Insufficient maintenance & & & & & \\  & technology & & & & & \\ \hline \multicolumn{8}{c}{**Remark**: (1) “Guistic variable value” column: 1: Impossible, 3: Seldom, 5: Occasional, 7: Probable, 9: Frequently. (2) “Inguistic variable value” column: 2: Surely, 4: Nearly, 6: This may be a little better, 8: It is hard to determine. (3) “Order of magnitude” column, the quantity value comes from the relevant data. If you think it is inappropriate, please fill in the value you think is suitable in the back column.} \\ \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Basic Event** & **Event Name** & **Graduate** & **Probability of Occurrence** \\ \hline X1 & Industrial computer host & \(q_{1}\) & \([4.60+0.52\lambda,6.52-0.52\lambda]\times 10^{-5}\) \\ X2 & Display failure & \(q_{2}\) & \([4.96+0.44\lambda,6.44-0.44\lambda]\times 10^{-5}\) \\ X3 & Monitor substation failure & \(q_{3}\) & \([2.36+0.24\lambda,2.84-0.24\lambda]\times 10^{-5}\) \\ X4 & Gas sensor failure & \(q_{4}\) & \([4.08+0.52\lambda,5.12-0.52\lambda]\times 10^{-3}\) \\ X5 & Temperature sensor fault & \(q_{5}\) & \([2.92+0.28\lambda,3.48-0.28\lambda]\times 10^{-5}\) \\ X6 & Wind speed sensor fault & \(q_{6}\) & \([5.20+0.60\lambda,6.40-0.60\lambda]\times 10^{-5}\) \\ X7 & Power sensor failure & \(q_{7}\) & \([3.88+0.32\lambda,4.52-0.32\lambda]\times 10^{-5}\) \\ X8 & Switch sensor failure & \(q_{8}\) & \([4.86+0.54\lambda,5.94-0.54\lambda]\times 10^{-5}\) \\ X9 & Breaker failure & \(q_{9}\) & \([4.56+0.64\lambda,5.84-0.64\lambda]\times 10^{-5}\) \\ X10 & PLC controller failure & \(q_{10}\) & \([4.36+0.24\lambda,4.84-0.24\lambda]\times 10^{-6}\) \\ X11 & Mechanically break the line & \(q_{11}\) & \([1.88+0.32\lambda,2.52-0.32\lambda]\times 10^{-5}\) \\ X12 & Self-aging short circuit & \(q_{12}\) & \([4.44+0.56\lambda,5.56-0.56\lambda]\times 10^{-5}\) \\ X13 & Poor line contact & \(q_{13}\) & \([3.80+0.40\lambda,4.60-0.40\lambda]\times 10^{-5}\) \\ X14 & Abnormal power failure & \(q_{14}\) & \([3.10+0.30\lambda,3.70-0.30\lambda]\times 10^{-3}\) \\ X15 & High communication error rate & \(q_{15}\) & \([3.00+0.40\lambda,3.80-0.40\lambda]\times 10^{-8}\) \\ X16 & Short circuit to ground & \(q_{16}\) & \([3.78+0.42\lambda,4.62-0.42\lambda]\times 10^{-4}\) \\ X17 & Lack of responsibility of the staff & \(q_{17}\) & \([4.48+0.52\lambda,5.52-0.52\lambda]\times 10^{-4}\) \\ X18 & System lacks spare parts & \(q_{18}\) & \([2.70+0.30\lambda,3.30-0.30\lambda]\times 10^{-2}\) \\ X19 & Insufficient maintenance & \(q_{19}\) & \([3.68+0.32\lambda,4.32-0.32\lambda]\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Probability of occurrence of the fault tree event table.