# Reliability Analysis of Vehicle Braking System Based on Hyperellipsoidal Dynamic Bayesian Network
## 2 Basic Theory of Reliability Analysis
### Fuzzy Theory
The specific calculation process is as follows:  
If \(n\) experts evaluate the probability parameter \(P^{l}y^{b_{j}}\) in the T-S gate rule l for a non-bottom event, and synthesize the expert evaluation information, the trapezoidal fuzzy number \(F_{k,l,y^{b_{j}}}=\left(F_{k,l,y^{b_{j}}}^{1},F_{k,l,y^{b_{j}}}^{2},F_{k,l,y ^{b_{j}}}^{3},F_{k,l,y^{b_{j}}}^{4}\right)\) of the evaluation information is, where \(k=1,2,\ldots,\) n.  
First the arithmetic mean \(F_{a}=\left(F_{a}^{1},F_{a}^{2},F_{a}^{3},F_{a}^{4},\right)\) of the n trapezoidal fuzzy numbers is calculated, see Equation (12) below:  
\[F_{a}^{1}=\frac{1}{n}\sum_{k=1}^{n}F_{k,l,y^{b_{j}}}^{1},F_{a}^{2}=\frac{1}{n} \sum_{k=1}^{n}F_{k,l,y^{b_{j}}}^{2},F_{a}^{3}=\frac{1}{n}\sum_{k=1}^{n}F_{k, l,y^{b_{j}}}^{3},F_{a}^{4}=\frac{1}{n}\sum_{k=1}^{n}F_{k,l,y^{b_{j}}}^{4} \tag{12}\]  
Then, use Equation (13) to calculate the distance \(d\left(F_{k,l,y^{b_{j}}},F_{a}\right)\) between the expert evaluation information \(F_{k,l,y^{b_{j}}}\) and the average evaluation information \(F_{a}\):  
\[d(F_{k,l,y^{b_{j}}},F_{a})=\frac{1}{4}(\left|F_{k,l,y^{b_{j}}}^{1}-F_{a}^{1} \right|+\left|F_{k,l,y^{b_{j}}}^{2}-F_{a}^{2}\right|+\left|F_{k,l,y^{b_{j}}} ^{3}-F_{a}^{3}\right|+\left|F_{k,l,y^{b_{j}}}^{4}-F_{a}^{4}\right|) \tag{13}\] \begin{table}
\begin{tabular}{c c c} \hline
**Serial Number** & **Language of Evaluation** & **Fuzzy Number Form** \\ \hline
1 & Very Low & \(f_{1}=(0,0,0.1,0.2)\) \\
2 & Low & \(f_{2}=(0.1,0.2,0.2,0.3)\) \\
3 & Fairly Low & \(f_{3}=(0.2,0.3,0.4,0.5)\) \\
4 & Medium & \(f_{4}=(0.4,0.5,0.5,0.6)\) \\
5 & Fairly High & \(f_{5}=(0.5,0.6,0.7,0.8)\) \\
6 & High & \(f_{6}=(0.7,0.8,0.8,0.9)\) \\
7 & Very High & \(f_{7}=(0.8,0.9,1,1)\) \\ \hline \end{tabular}
\end{table} 
Table 1: Fuzzy number form. The similarity \(s\left(F_{k,l,y^{b_{j}}},F_{a}\right)\) between the expert evaluation information \(F_{k,l,y^{b_{j}}}\) and the average evaluation information \(F_{a}\) is calculated by Equation (14):  
\[s(F_{k,l,y^{b_{j}}},F_{a})=1-\frac{d(F_{k,l,y^{b_{j}}},F_{a})}{\sum\limits_{k=1 }^{n}d(F_{k,l,y^{b_{j}}},F_{a})} \tag{14}\]  
The evaluation weight value of each expert can be calculated by dividing the evaluation similarity of each expert by the sum of the similarities of all experts. This can be achieved by using the following Equation (15):  
\[w(F_{k,l,y^{b_{j}}},F_{a})=\frac{s(F_{k,l,y^{b_{j}}},F_{a})}{\sum\limits_{k=1 }^{n}s(F_{k,l,y^{b_{j}}},F_{a})} \tag{15}\]  
The synthesized expert evaluation information, based on the evaluation weights assigned by each expert, is obtained from the following Equation (16). \[W_{l,y^{b_{j}}}=\sum\limits_{k=1}^{n}w_{k,l,y^{b_{j}}}F_{k,l,y^{b_{j}}}=\sum \limits_{k=1}^{n}\frac{s(F_{k,l,y^{b_{j}}},F_{a})}{\sum\limits_{k=1}^{n}s(F_{ k,l,y^{b_{j}}},F_{a})} \tag{16}\]  
In this context, \(W_{k,l,y^{b_{j}}}\) represents the calculated evaluation weights of each expert. Defuzzification  
In accordance with the meticulous derivation of the fuzzy extension principle, the output of the aforementioned comprehensive expert evaluation process continues to manifest as a fuzzy probability, encapsulated within the form of a fuzzy number. Such fuzzy probabilities cannot be directly applied as probabilistic parameters in subsequent precise calculations or decision-making processes due to their inherent uncertainty. Accordingly, in order to effectively transform this ambiguity into an operationally precise value, the technical means of defuzzification is required, i.e., the conversion of fuzzy numbers into clear and unambiguous values. Among the many defuzzification methods, the area-mean method, the weighted area method, the maximum affiliation method, and the gravity method, each with its own characteristics, are widely used in different fields of fuzzy evaluation processing [14]. This paper presents a detailed analysis of the most commonly used and relatively robust area-mean method for defuzzification of fuzzy numbers.