## 4 Engineering Example Calculation
### TOPSIS Method for Construction Quality Evaluation  
The data in Table 4 were normalized to establish the initial evaluation matrix \(A\). Due to the document width limitation, \(A^{T}\) is used here to represent the \(A\) matrix more intuitively. \[A=\left(A^{T}\right)^{T}= \begin{bmatrix}1&0.9&0.78&0.6&0.4&0\\ 1&0.9&0.78&0.6&0.4&0\\ 1&0.9&0.78&0.6&0.4&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&0.9&0.78&0.6&0.4&0\\ 1&0.9&0.78&0.6&0.4&0\\ \end{bmatrix} ^{T}\]  
According to Equation (20), the weighted decisionalization matrix, \(B\), was obtained as follows:  
\[B=\left(B^{T}\right)^{T}= \begin{bmatrix}0.0921&0.0829&0.0718&0.0552&0.0368&0 \\ 0.0419&0.0377&0.0327&0.0251&0.0168&0\\ 0.0501&0.0451&0.0391&0.0301&0.0201&0\\ 0.0279&0.0251&0.0217&0.0167&0.0111&0\\ 0.0400&0.0360&0.0312&0.0240&0.0160&0\\ 0.0500&0.0450&0.0390&0.0300&0.0200&0\\ 0.0414&0.0373&0.0323&0.0249&0.0166&0\\ 0.0498&0.0448&0.0389&0.0299&0.0199&0\\ 0.0485&0.0437&0.0378&0.0291&0.0194&0\\ 0.0571&0.0514&0.0445&0.0343&0.0228&0\\ 0.0358&0.0322&0.0279&0.0215&0.0143&0\\ 0.0884&0.0796&0.0690&0.0530&0.0354&0\\ 0.0862&0.0776&0.0672&0.0517&0.0345&0\\ 0.0663&0.0597&0.0517&0.0398&0.0265&0\\ 0.0496&0.0447&0.0387&0.0298&0.0198&0\\ 0.0386&0.0348&0.0301&0.0232&0.0154&0\\ 0.0397&0.0357&0.0310&0.0238&0.0159&0\\ 0.0609&0.0548&0.0475&0.0366&0.0244&0\\ 0.0356&0.0320&0.0278&0.0214&0.0142&0\\ \end{bmatrix} \]  
From Equations (21) and (22), the positive and negative ideal solutions of the weighted decision matrix were obtained as \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator Meaning** & \multicolumn{5}{c}{**Calculated Values**} \\ \hline  & \(-0.0921\)\(-0.0829\) & \(-0.0718\) & \(-0.0552\) & \(-0.0368\) & \(0\) \\  & \(-0.0419\)\(-0.0377\) & \(-0.0327\) & \(-0.0251\) & \(-0.0168\) & \(0\) \\  & \(-0.0501\)\(-0.0451\) & \(-0.0391\) & \(-0.0301\) & \(-0.0201\) & \(0\) \\  & \(-0.0279\)\(-0.0251\) & \(-0.0217\) & \(-0.0167\) & \(-0.0111\) & \(0\) \\  & \(-0.0400\)\(-0.0360\) & \(-0.0312\) & \(-0.0240\) & \(-0.0160\) & \(0\) \\  & \(-0.0500\)\(-0.0450\) & \(-0.0390\) & \(-0.0300\) & \(-0.0200\) & \(0\) \\  & \(-0.0414\)\(-0.0373\) & \(-0.0323\) & \(-0.0249\) & \(-0.0166\) & \(0\) \\  & \(-0.0498\)\(-0.0448\) & \(-0.0389\) & \(-0.0299\) & \(-0.0199\) & \(0\) \\  & \(-0.0485\)\(-0.0437\) & \(-0.0378\) & \(-0.0291\) & \(-0.0194\) & \(0\) \\ \(\left(\min H_{ij}-H_{ij}\right)\) & \(-0.0571\)\(-0.0514\) & \(-0.0445\) & \(-0.0343\) & \(-0.0228\) & \(0\) \\  & \(-0.0358\)\(-0.0322\) & \(-0.0279\) & \(-0.0215\) & \(-0.0143\) & \(0\) \\  & \(-0.0884\)\(-0.0796\) & \(-0.0690\) & \(-0.0530\) & \(-0.0354\) & \(0\) \\  & \(-0.0862\)\(-0.0766\) & \(-0.0672\) & \(-0.0517\) & \(-0.0345\) & \(0\) \\  & \(-0.0663\)\(-0.0597\) & \(-0.0517\) & \(-0.0398\) & \(-0.0265\) & \(0\) \\  & \(-0.0496\)\(-0.0447\) & \(-0.0387\) & \(-0.0298\) & \(-0.0198\) & \(0\) \\  & \(-0.0386\)\(-0.0348\) & \(-0.0301\) & \(-0.0232\) & \(-0.0154\) & \(0\) \\  & \(-0.0397\)\(-0.0357\) & \(-0.0310\) & \(-0.0238\) & \(-0.0159\) & \(0\) \\  & \(-0.0609\)\(-0.0548\) & \(-0.0475\) & \(-0.0366\) & \(-0.0244\) & \(0\) \\  & \(-0.0356\)\(-0.0320\) & \(-0.0278\) & \(-0.0214\) & \(-0.0142\) & \(0\) \\  & \(0.0564\) & \(0.0457\) & \(0.0343\) & \(0.0203\) & \(0.0090\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Construction quality evaluation results of the special bridge across the Yellow River (dimensionless).