# Reliability Analysis of an Air Traffic Network: From Network Structure to Transport Function
## 3 Reliability Measurement of the Air Traffic Network
### Measurement of the CORN, TTRN, and CARN
#### 3.3.1 Description of Uncertainty Transformation Model
.3.2 Reliability Evaluation of the Air Traffic Network Based on the Uncertainty Transformation Model  
The architecture of the reliability evaluation framework is illustrated in Figure 4.  
The specific description of the evaluation algorithm is as follows:  
Establish the evaluation object's metric set and comment set. Establish the metric set \(U=\{\textit{CORN},\textit{TTRN},\textit{CARN}\}\) and comment set \(V=\{\textit{very unreliable},\textit{ unreliable},\textit{less reliable},\textit{ relatively reliable},\textit{very reliable}\}\). The evaluation rule is shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{Ex}\) & \(\mathbf{En}\) & \(\mathbf{He}\) \\ \hline + & \(Ex_{1}+Ex_{2}\) & \(\sqrt{Ln_{1}^{2}+En_{2}^{2}}\) & \(\sqrt{He_{1}^{2}+He_{2}^{2}}\) \\ - & \(Ex_{1}-Ex_{2}\) & \(\sqrt{En_{1}^{2}+En_{2}^{2}}\) & \(\sqrt{He_{1}^{2}+He_{2}^{2}}\) \\ \(\times\) & \(Ex_{1}\times Ex_{2}\) & \(\sqrt{\left(\frac{En_{1}}{En_{1}}\right)^{2}+\left(\frac{En_{1}}{En_{1}} \right)^{2}}\) & \(\sqrt{\left(\frac{He_{1}}{Ex_{1}}\right)^{2}+\left(\frac{He_{2}}{Ex_{2}} \right)^{2}}\) \\ \(\div\) & \(Ex_{1}\div Ex_{2}\) & \(\left|\frac{Ex_{1}}{Ex_{1}}\right|\times\sqrt{\left(\frac{En_{1}}{Ex_{1}} \right)^{2}+\left(\frac{En_{2}}{Ex_{2}}\right)^{2}}\) & \(\left|\frac{Ex_{1}}{Ex_{2}}\right|\times\sqrt{\left(\frac{En_{1}}{Ex_{1}} \right)^{2}+\left(\frac{En_{2}}{Ex_{2}}\right)^{2}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Computing rules of the uncertainty transformation model. Figure 4: The architecture of reliability evaluation based on the uncertainty transformation model. 2. Compute the reliability indexes. Calculate the CORN, TTRN, and CARN of the air traffic network based on the methods in Section 3.2. 3. Compute the weight uncertainty transformation models \(W=\{W_{1},W_{2},W_{3}\}\). \(W_{1}\) represents the weight uncertainty transformation model of the CORN, \(W_{2}\) represents the weight uncertainty transformation model of the TTRN, and \(W_{3}\) represents the weight uncertainty transformation model of the CARN. Use the Delphi method to score the weights of the CORN, TTRN, and CARN. The score given by experts is a numerical range because of the fuzziness of reliability; Table 3 shows an example. Then, the uncertainty transformation models for each index are calculated based on the converse fuzzy set generator. There are two fuzzy sets for each index, \(C_{\text{max}}(E_{\text{max}},E_{\text{max}},H_{\text{max}})\) and \(C_{\text{min}}(E_{\text{min}},E_{\text{min}},H_{\text{min}})\). The weight uncertainty transformation model \(W_{i}(E_{xi},E_{ni},H_{ei})\) can be calculated using the following formulas [34]:  
\[E_{xi}=\frac{C_{\text{min},i}+C_{\text{max},i}}{2} \tag{13}\]  
\[E_{ni}=\frac{C_{\text{min},i}-C_{\text{max},i}}{6} \tag{14}\]  
\[H_{ei}=k_{i} \tag{15}\]  
where \(k_{i}\) is a constant and can be adjusted according to the fuzziness of the comments. d. Compute the actual uncertainty transformation models. Calculate the uncertainty transformation models \(R=\{R_{1},R_{2}R_{3}\}\) using the converse fuzzy set generator based on the values of the CORN, TTRN, and CARN calculated in step b. \(R_{1}\) represents the actual uncertainty transformation model of the CORN, \(R_{2}\) represents the actual uncertainty transformation model of the TTRN, and \(R_{3}\) represents the actual uncertainty transformation model of the CARN. e. Calculate the comprehensive evaluation uncertainty transformation model \(RC\) based on the following formula:  
\[RC=\sum_{i=1}^{n}R_{i}\times W_{i} \tag{16}\]  
f. Compare the similarity between \(RC\) and the reliability rule, determine the most similar fuzzy set, and determine the evaluation results.