## 2 Related Evaluation Methods
### TOPSIS Method  
The TOPSIS method is a widely used multi-criteria decision-making (MCDM) method [32; 33; 34; 35]. It evaluates alternatives based on their relative closeness to the ideal solution (best possible performance across all criteria) and the negative-ideal solution (worst possible performance). The procedural workflow of this evaluation method is illustrated in Figure 1.  
The basic steps of its evaluation are as follows.  
Step 1: Construct the initial evaluation matrix \(\mathbf{C}\).  
The assumed evaluation scenario comprises \(m\) schemes, each with \(n\) evaluation indicators. Equation (1) provides the initial evaluation matrix \(\mathbf{C}\).  
\[\mathbf{C}=\left[c_{ij}\right]_{m\times n}=\begin{bmatrix}c_{11}&c_{12}&\dots& c_{1n}\\ \vdots&\vdots&\ddots&\vdots\\ c_{m1}&c_{m2}&\dots&c_{mn}\end{bmatrix} \tag{1}\]  
where \(c_{ij}\) is the \(j\)th indicator of the \(i\)th plan.  
Step 2: Construct the dimensionless initial evaluation matrix \(\mathbf{Z}=\left[z_{ij}\right]_{m\times n}\).  
Figure 1: The procedural workflow of TOPSIS evaluation method.  
Step 3: Construct the normalized matrix \(\mathbf{R}\).  
\[\mathbf{R}=\begin{bmatrix}r_{i1}&\dots&r_{in}\\ \vdots&\ddots&\vdots\\ r_{m1}&\dots&r_{mm}\end{bmatrix} \tag{2}\]  
where the expression of \(r_{ij}\) is shown in Equation (3).  
\[r_{ij}=\frac{z_{ij}}{\sqrt{\sum\limits_{i=1}^{n}z_{ij}^{2}}} \tag{3}\]  
Step 4: Construct the weighted identity matrix \(\mathbf{X}\).  
By multiplying the normalized matrix \(\mathbf{R}\) obtained in Step 3 with the calculated weight vector \(w=(w_{1},w_{2},\dots,w_{m})\) of each evaluation index, the weighted evaluation matrix \(\mathbf{X}\) is expressed as follows:  
\[x_{ij}=r_{ij}\bullet w_{m} \tag{4}\]  
Step 5: Calculate the Euclidean distance.  
For the evaluation problem with \(m\) feasible solutions \(x_{i}\), the expression is  
\[x_{i}=(x_{i1},x_{i2},\dots,x_{in}) \tag{5}\]  
where \(i=1,2,\dots,m\).  
Equation (6) is the positive ideal solution (\(x_{i}^{+}\)).  
\[x_{i}^{+}=\max\
olimits_{j}\bigl{(}x_{ij}\bigr{)} \tag{6}\]  
where \(i=1,2,\dots,m;j=1,2,\dots,n\).  
The Euclidean distance between any feasible solution \(x_{i}\) and \(x_{i}^{+}\) is \(S_{i}^{+}\), as defined in Equation (7).  
\[S_{i}^{+}=\sqrt{\sum\limits_{j=1}^{n}\Bigl{(}x_{ij}-x_{j}^{+}\Bigr{)}^{2}} \tag{7}\]  
where \(i=1,2,\dots,m\).  
Equation (8) is the negative ideal solution (\(x_{i}^{-}\)).  
\[x_{i}^{-}=\min\
olimits_{j}\bigl{(}x_{ij}\bigr{)} \tag{8}\]  
where \(i=1,2,\dots,m;j=1,2,\dots,n\).  
The Euclidean distance between any feasible solution \(x_{i}\) and \(x_{i}^{-}\) is \(S_{i}^{-}\), as defined in Equation (9).  
\[S_{i}^{-}=\sqrt{\sum\limits_{j=1}^{n}\Bigl{(}x_{ij}-x_{j}^{-}\Bigr{)}^{2}} \tag{9}\]  
where \(i=1,2,\dots,m\).  
Step 6: Calculate the relative closeness \(C_{i}\).  
\[C_{i}=\frac{S_{i}^{-}}{S_{i}^{-}+S_{i}^{+}} \tag{10}\]  
In multi-criteria decision analysis, a solution classified as the positive ideal solution achieves a closeness coefficient (\(C_{i}\)) of 1. The closer an alternative's \(C_{i}\) value is to 1, the more it aligns with the ideal benchmark. Conversely, a solution labeled as the negative ideal solution yields a \(C_{i}\) of 0. The closer an alternative's \(C_{i}\) value is to 0, the more it deviates from the ideal benchmark. According to the above steps, it is possible to sort and finally find a satisfactory solution.