# Health State Assessment of Road Tunnel Based on Improved Extension Cloud Model
## 3 Model Establishment
### Construction of Improved Extension Cloud Model  
Currently, the fuzziness and randomness at the boundary of grades are not fully considered by the traditional tunnel evaluation method, which makes use of interval division for classifying the tunnel health level. Therefore, the extension cloud model is introduced to solve the above problem. The cloud model mainly reflects the fuzziness and randomness in natural language concepts. The mapping between qualitative and quantitative aspects is formed by utilizing cloud generators, thereby enabling the mutual transformation between qualitative expression and quantitative calculation. The cloud generators are divided into two categories: forward cloud generator (FCG) and backward cloud generator (BCG). The forward cloud generator generates cloud droplets by inputting three digital features of the cloud to realize the transformation from qualitative to quantitative. On the contrary, the backward cloud generator regards the data as cloud droplets, and outputs three digital features of the cloud to realize the transformation from quantitative to qualitative. It can effectively express the fuzzy uncertainty and random uncertainty of knowledge concepts [25]. The principle of the cloud generator is shown in Figure 3.  
Replacing the value \(R=(N,C,V)\) in the \((Ex,En,He)\) physical element with the from the normal cloud model to obtain the extension cloud model, as shown in Equation (4). \[R=(N,C,V)= \begin{bmatrix}N&C_{1}&(Ex_{1},En_{1},He_{1})\\ &C_{2}&(Ex_{2},En_{2},He_{2})\\ &\vdots&\vdots\\ &C_{n}&(Ex_{n},En_{n},He_{n})\end{bmatrix}  \tag{4}\]  
where \(N\) represents the road tunnel sample to be evaluated. \(C\) represents the health indicators of the tunnel structure to be evaluated. \((Ex_{n},En_{n},He_{n})\) represents the cloud description of the corresponding level of the indicator. \(Ex\) is the expected value, which is the value that embodies qualitative concepts, reflecting the health level of the indicator. \(En\) is the entropy which measures the fuzziness and randomness in qualitative concepts, reflecting the range of health levels, \(He\) is the hyper-entropy, reflecting the uncertainty of correlation. According to Equation (4), the key to solving the extension cloud model lies in calculating the digital characteristic value of the cloud, denoted by \((Ex,En,He)\). By definition of expectation, the concept of this level is reasonably represented by the center value of the constrained interval. Therefore, the calculation formula is \(Ex=(c_{\min}+c_{\max})/2\), where  
Figure 3: Cloud generator principle. \(c_{\text{min}}\) and \(c_{\text{max}}\) are the level boundaries, the hyper-entropy \(He\) calculation formula is generally \(He\), which can also be adjusted based on experience and actual situations [16; 33]. The cloud entropy \(En\), which reflects the fuzziness of the indicator state level, is the most critical cloud characteristic parameter. The value of the cloud entropy affects the size of the health level range. Moreover, it also affects the accuracy of the results. Currently, there are mainly two cloud entropy calculation methods [37] (Equations (5) and (6)):  
Cloud entropy calculation method based on the \"\(3 En\)\" rule:  
\[En^{\prime}=\frac{c_{\text{max}}-c_{\text{min}}}{6} \tag{5}\]  
Cloud entropy calculation method based on the \"50% correlation\" rule:  
\[En^{\prime\prime}=\frac{c_{\text{max}}-c_{\text{min}}}{2.355} \tag{6}\]  
The cloud entropy of the standard-level cloud model is calculated from different perspectives by the above two methods. The \"\(3 En\)\" rule embodies the clarity of level division, while the \"50% correlation\" rule focuses more on the fuzziness of level division. The level correlation may conflict with the conclusion of the health status evaluation depending on the different cloud entropy calculation results.