## 2 Evaluation Method
### Objective Empowerment of Cloud Entropy Weight Method  
The basic idea of the entropy weighting method is to determine the indicator weights based on the magnitude of the variability of the risk indicators, i.e., the more frequently a value appears, the greater its contribution to the qualitative concept [25]. In the actual evaluation process, expert opinions fluctuate around a certain value, i.e., a random number with a stable tendency can be used to replace the value, which is highly similar to the central idea of the cloud model and entropy weight method. Since the risk evaluation index system of the Yellow River bridge studied in this paper has many factors and is relatively independent, it is inconvenient to cross-compare them, and the traditional weight calculation method only considers the expectation _Ex_ in the cloud model and the utilization of the parameter _En; He_ is not sufficient, while the cloud entropy _En_ reflects the consensus of experts on the factor, which is significant for the calculation of index weights [26]. Therefore, in this paper, the cloud model (combined with the entropy weight method for the evaluation of the variability of indicators) is used to transform the subjective judgment of experts into parameters representing uncertainty through the cloud model and is then combined with the cloud entropy weight method's weight improvement model to calculate the objective weight value of each indicator, which reflects the importance of each indicator as scientifically as possible [27].  
There are \(m\) experts and \(n\) risk evaluation indicators and _xij_ is the score of the _i_-th expert on evaluation index \(j\); then, according to the calculation formula of the reverse cloud generator, the cloud model's parameter value of the _j_-th evaluation index is obtained as in Formulas (15)-(17).  
\[Ex_{j}=\overline{x_{j}}=\frac{1}{m}\sum\limits_{i=1}^{m}x_{ij} \tag{15}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Order** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline R.I. & 0 & 0 & 0.52 & 0.89 & 1.12 & 1.26 & 1.36 & 1.41 & 1.46 & 1.49 \\ \hline \end{tabular}
\end{table}
Table 2: Average random consistency index _R.I._\[En_{j}=\sqrt{\frac{\pi}{2}}\times\frac{1}{m}{\sum\limits_{i=1}^{m}{\left|{x_{ij}- Ex_{j}}\right|}} \tag{16}\]  
\[He_{j}=\sqrt{\frac{1}{m-1}{\sum\limits_{i=1}^{m}{\left({x_{ij}-Ex_{j}}\right)^{2 }-En_{j}^{2}}}} \tag{17}\]  
In the formula, \(i=1,2,\cdots,m:j=1,2,\cdots,n\), and \(Ex_{j}\), \(En_{j}\), and \(He_{j}\) are the expectation, entropy, and super-entropy of the \(j\)-th evaluation index, respectively.  
Then, use the traditional weight calculation method to determine the weight of the \(j\)-th evaluation index; see Formula (18).  
\[\theta_{j}=\frac{Ex_{j}}{\sum\limits_{j=1}^{n}{Ex_{j}}} \tag{18}\]  
When \(m\) experts have the same expectation \(Ex_{j}\) for the importance of an indicator, the weights obtained by using the weight calculation method of Formula (18) are consistent. In fact, the changes in the scores among experts are not the same, that is, there is a difference between entropy \(En_{j}\) and super-entropy \(He_{j}\), so this method cannot guarantee the comprehensiveness and accuracy of weight calculation. Therefore, this paper proposes an improved cloud model entropy weight method to replace the above traditional weight calculation method; see Formula (19).  
\[\beta_{j}=\left\{{\begin{array}{*{20}{c}}\frac{Ex_{j}}{\ln(1+En_{j})+1}\frac {1}{\sum\limits_{j=1}^{n}{\frac{Ex_{j}}{\ln(1+En_{j})+1}}}&\left({En_{j}\
eq 0} \right)\\ \frac{Ex_{j}}{\sum\limits_{j=1}^{n}{Ex_{j}}}&\left({En_{j}=0}\right)\\ \end{array}}\right. \tag{19}\]  
In the formula, \(\beta_{j}\) is the objective weight value of the \(j\)-th index determined by the improved cloud model entropy weight method.