## 3 Bridge Construction Quality Evaluation Method and Evaluation Process
### TOPSIS Method  
The technique for order preference by similarity to an ideal solution (TOPSIS) method, also known as the multi-attribute decision method for approximating ideal points, is a comprehensive evaluation technique that analyzes and judges based on distance. It evaluates results by calculating the distance between each evaluation object and the optimal solution and the worst solution, then ranking their relative proximity to the idealized target. The optimal solution of the evaluation result must be the solution closest to the optimal solution and the solution farthest from the worst solution [35]. The TOPSIS method has no clear restrictions and requirements in terms of the number of evaluation indicators, sample content, and data, and it can analyze the situation of each indicator comprehensively, reflecting the overall situation and having universal applicability. Therefore, this paper adopted this method to analyze the construction quality of the mega bridge across the Yellow River. The specific steps are as follows:(1) Construct the decision attribute matrix. Assume that a system to be evaluated has \(n\) objects to be evaluated \(A=\{A_{1},A_{2},\ldots,A_{n}\}\); each object to be evaluated has \(m\) evaluation indicators \(B=\{B_{1},B_{2},\ldots,B_{m}\}\); its evaluation value constitutes the decision matrix \(E=\big{(}x_{ij}\big{)}_{m\times n}\).  
The normalized matrix \(F=\big{(}X_{ij}\big{)}_{m\times n}\) is obtained after normalizing the original data matrix using the polarization method;  
(2) Calculate the weighted normalized decision matrix. The values of the weighted normalized decision matrix are as follows:  
\[H_{ij}=X_{ij}\times w_{j} \tag{20}\]  
where \(X_{ij}\) is the value of each evaluation index score after standardization and \(w_{j}\) is the weight value of each evaluation index calculated by the combined weighting method;  
(3) Determine the positive and negative ideal solutions. The positive ideal solution, \(H^{+}\), consists of the maximum value in each column in \(H\), i.e.,  
\[H^{+}=\{\text{max}H_{i1},\text{max}H_{i2},\ldots\text{max}H_{in}\} \tag{21}\]  
The negative ideal solution, \(H^{-}\), consists of the smallest value in each column in \(H\), i.e.,  
\[H^{-}=\{\text{min}H_{i1},\text{min}H_{i2},\ldots\text{min}H_{in}\} \tag{22}\]  
(4) Calculate the distance to the positive ideal point \(S_{i}^{+}\) and the distance to the negative ideal point \(S_{i}^{-}\) for each solution:  
\[S_{i}^{+}=\sqrt{\sum_{j=1}^{m}\bigl{(}\text{max}H_{ij}-H_{ij}\bigr{)}^{2}},1 \leq i\leq n \tag{23}\]  
\[S_{i}^{-}=\sqrt{\sum_{j=1}^{m}\bigl{(}\text{min}H_{ij}-H_{ij}\bigr{)}^{2}},1 \leq i\leq n \tag{24}\]  
(5) Calculate the relative proximity of the alternative solution to the positive ideal solution. The relative closeness of the alternative to the positive ideal solution, \(G_{i}\), is calculated as follows:  
\[G_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}} \tag{25}\]  
In the above equation, \(G_{i}\to 1\) means that the evaluation object is closer to the positive ideal solution, and \(G_{i}\to 0\) means that the evaluation object is closer to the negative ideal solution.