# Health State Assessment of Road Tunnel Based on Improved Extension Cloud Model
## 2 Construction of Indicator System
### Calculation of Indicator Weights  
The weight determination is a crucial stage in the evaluation procedure. The scientific dependability of the evaluation conducted for the health status of a tunnel is directly impacted by the rationality of the weight used. Problems of subjective arbitrariness arise in subjective weighting methods based on expert scores, while problems of objective distortion arise in objective weighting methods based on objective data [33]. The weight determined only by the subjective weighting method is the subjective weight, and the weight determined only by the objective weighting method is the objective weight. Subjective and objective weights are combined by the combination weighting method which is a scientific and reasonable method that can make the evaluation results more reasonable. Therefore, the combination of subjective and objective weighting is adopted to make up for the shortcomings of a single weighting method and improve the accuracy of the evaluation results.  
The analytic hierarchy process (AHP) [34] is used for the calculation of subjective weight, while the entropy weight method is adopted for the calculation of objective weight. The AHP is scored by experts on health evaluation indicators at all levels in the evaluation system. The health evaluation judgment matrix is obtained according to the score results. The weight values of evaluation indexes at all levels are finally obtained after the consistency test and normalization. The entropy weight method determines the weight based on the change degree of the indicator. The greater the degree of change, the higher the weight, and vice versa. Within the conventional entropy method calculation process, there may exist a marked deviation despite a relatively minor variation between entropy values, when the entropy value of an indicator approaches 1. To solve this problem, the traditional entropy method has been improved based on previous research results [35; 36], as shown in Equations (2) and (3):  
\[\alpha_{j}=\frac{\exp(\sum\limits_{k=1}^{m}e_{k}+1-e_{j})-\exp e_{j}}{\sum \limits_{l=1}^{m}\left[\exp(\sum\limits_{k=1}^{m}e_{k}+1-e_{l})-\exp e_{l} \right]} \tag{2}\]  
where \(e_{j}\) is the entropy of the indicator, and \(\alpha_{j}\) is the weight of the indicator.  
\[\mathbf{P}=\eta\mathbf{\omega}+(1-\eta)\mathbf{\alpha} \tag{3}\]  
where \(\mathbf{P}\) is the comprehensive weight vector, \(\mathbf{\omega}\) is the subjective weight. is the objective weight. \(\eta\) is the subjective-objective weight ratio factor, which indicates the degree to  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator** & **I** & **II** & **III** & **IV** & **V** \\ \hline Damage degree of maintenance road & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ Abnormality degree of a drainage system & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ Pavement condition index (PCI) & 90–100 & 80–90 & 70–80 & 60–70 & \textless{}60 \\ Riding quality index (RQI) & 90–100 & 80–90 & 70–80 & 60–70 & \textless{}60 \\ Skid resistance index (SRI) & 90–100 & 80–90 & 70–80 & 60–70 & \textless{}60 \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator** & **I** & **II** & **III** & **IV** & **V** \\ \hline Crack quantity change rate & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ Leakage water quantity change rate & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ Pavement condition index (PCI) change rate & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ Riding quality index (RQI) change rate & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ Skid resistance index (SRI) change rate & 0–0.2 & 0.2–0.4 & 0.4–0.6 & 0.6–0.8 & 0.8–1 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Classification of dynamic indicators.  
which expert judgment weighs toward subjective or objective factors. Generally, subjective and objective factors are equally important, so \(\eta\) is set to 0.5.