## 3 The Proposed Evaluation Method
### Weights Determination
#### 3.4.1 Weight Determination of the Sub-Criterion Layer (TOPSIS)  
Step 1: Standardize the original data. For \(m\) evaluation schemes with \(n\) indicators each, the original data matrix \(\mathbf{A}\) is given in Equation (30). \[\mathbf{A}= \begin{bmatrix}x_{11}&x_{12}&\dots&x_{1n}\\ x_{21}&x_{22}&\dots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\dots&x_{mn}\end{bmatrix}  \tag{30}\]  
where \(x_{ij}\) is the original data. After standardizing matrix \(\mathbf{A}\) (Section 3.3), the resulting standardized matrix \(\mathbf{A}\prime\) is given in Equation (31). \[\mathbf{A}^{\prime}= \begin{bmatrix}x_{11}^{\prime}&x_{12}^{\prime}&\dots&x_{1 n}^{\prime}\\ x_{21}^{\prime}&x_{22}^{\prime\prime}&\dots&x_{2n}^{\prime\prime}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}^{\prime}&x_{m2}^{\prime}&\dots&x_{mn}^{\prime}\end{bmatrix}  \tag{31}\]  
where \(x_{ij}\prime\) is the standardized data. Step 2: Dimensionless processing of standardized data. After applying Equation (30) to remove dimensions from standardized matrix \(\mathbf{A}^{\prime}\), the resulting dimensionless matrix \(\mathbf{A}^{\prime\prime}\) is given in Equation (33). \[x_{ij}^{\prime\prime}=\frac{x_{ij}\prime}{\sqrt{\sum\limits_{i=1}^{m}x_{ij}^{ \prime 2}}} \tag{32}\]  
\[\mathbf{A}^{\prime\prime}= \begin{bmatrix}x_{11}^{\prime\prime}&x_{12}^{\prime \prime}&\dots&x_{1n}^{\prime\prime}\\ x_{21}^{\prime\prime}&x_{22}^{\prime\prime}&\dots&x_{2n}^{\prime\prime}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}^{\prime\prime}&x_{m2}^{\prime\prime}&\dots&x_{mn}^{\prime\prime}\end{bmatrix}  \tag{33}\]  
where \(x_{ij}^{\prime\prime}\) is the dimensionless data. Step 3: Calculate the weighting coefficient matrix. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \hline \multicolumn{6}{c}{(a)} \\ \hline \multicolumn{6}{c}{Evaluation indicators} & \multicolumn{6}{c}{Yes (existence or good)} & \multicolumn{6}{c}{No (inexistence or bad)} \\ \hline \multicolumn{6}{c}{Significance} & \multicolumn{6}{c}{1} \\ \hline \multicolumn{6}{c}{(b)} \\ \hline \(n\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline Five levels & Worst & – & Bad & – & General & – & Good & – & Best \\ \hline Seven levels & Worst & Worse & Bad & – & General & – & Good & Better & Best \\ \hline Nine levels & Worst & Worse & Bad & Worse than general & General & Better than general & Good & Better & Best \\ \hline \hline \end{tabular}
\end{table} 
Table 1: (a) Numerical qualitative evaluation indicators. (b) Numerical qualitative evaluation indicators. Evaluation index weights are calculated using Equations (28)-(30). These weights form diagonal matrix \(\mathbf{W}\) (Equation (37)), which is then used with Equation (27) to compute weighting coefficient matrix \(\mathbf{Y}\) (Equation (38)). \[\overline{x_{j}}=\frac{1}{m}\underset{i=1}{\overset{m}{\underset{i=1}{\overset{ m}{\sum}}}}\,x_{ij}^{\prime\prime} \tag{34}\]  
\[S_{j}=\sqrt{\frac{1}{m-1}\underset{i=1}{\overset{m}{\sum}}\left(x_{ij}^{ \prime\prime}-\overline{x_{j}}\right)^{2}} \tag{35}\]  
\[w_{j}=\frac{S_{j}}{\left|\overline{x_{j}}\right|},j=1,2,\ldots,n \tag{36}\]  
\[\mathbf{W}= \begin{bmatrix}w_{1}&0&\ldots&0\\ 0&w_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&w_{n}\end{bmatrix}  \tag{37}\]  
\[\mathbf{Y}=\mathbf{A}^{\prime\prime}\bullet\mathbf{W}\] \[= \begin{bmatrix}x_{11}^{\prime\prime}&x_{12}^{\prime\prime}& \ldots&x_{1n}^{\prime\prime}\\ x_{21}^{\prime\prime}&x_{22}^{\prime\prime}&\ldots&x_{2n}^{\prime\prime}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}^{\prime\prime}&x_{m2}^{\prime\prime}&\ldots&x_{mn}^{\prime\prime} \end{bmatrix}\begin{bmatrix}w_{1}&0&\ldots&0\\ 0&w_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&w_{n}\end{bmatrix}  \tag{38}\]  
where \(i=1,2,\ldots,m\); \(j=1,2,\ldots,n\). Step 4: Calculate the positive and negative ideal solutions.