## 3 Materials and Methods
### Evaluation Method (GRA-TOPSIS)  
The purpose of the GRA technique is to measure the relationship between elements on the basis of similarity [60]. It has significant advantages in solving complex decision-making problems characterized by vague, incomplete, and inaccurate information [61]. TOPSIS is a management decision-making method for comparing and ranking multidimensional vectors, in which Euclidean distance calculation is the core of the model. This method can systematically analyze the gap between the object and the ideal state, truly reflect the problem, and provide clear information for decision makers [62]. However, TOPSIS cannot accurately reflect the changing trend of data series, and GRA ignores the correlation situation between the scheme and the ideal solution [63]. By defining the gray association coefficient of GRA, the limitations of TOPSIS can be improved, and the purpose of correcting the relative proximity can be achieved [35]. The procedure of the GRA-TOPSIS method is as follows.  
1. Determine the positive ideal solution \(Z^{+}\) and negative ideal solution \(Z^{-}\) of the weighted normalized matrix. \[Z^{+}=\left({{z_{1}}^{+},{z_{2}}^{+},\ldots,{{z_{n}}^{+}}}\right)=\omega\] (20) \[Z^{-}=\left({{z_{1}}^{-},{z_{2}}^{-},\ldots,{{z_{n}}^{-}}}\right)=0\] (21) where \({{z_{j}}^{+}}=max{{z_{ij}}}=\omega_{j},\;{{z_{j}}^{-}}=minz_{ij}=0,\;j\in N\).
2. Calculate the Euclid distance to the positive ideal solution \(di^{+}\) and negative ideal solution \(di^{-}\) for each scheme. \[di^{+}=\sqrt{\sum_{j=1}^{n}(zij-{{z_{j}}^{+}})^{2}}\text{ and }di^{-}=\sqrt{\sum_{j=1}^{n}(zij-{{z_{j}}^{-}})^{2}}\] (22)Calculate the gray correlation coefficient matrix between each scheme and the positive \(R^{+}\) and negative ideal solution \(R^{-}\).  
\[R^{+}=\left(r_{ij}^{+}\right)_{m\times n}\text{ and }R^{-}=\left(r_{ij}^{-}\right)_{m \times n} \tag{23}\]  
\[r_{ij}^{+}=\frac{\underset{i}{minmin}\left|z_{j}^{+}-z_{ij}\right|+\underset{ i}{maxmax}\left|z_{j}^{+}-z_{ij}\right|}{\left|z_{j}^{+}-z_{ij}\right|+ \underset{i}{maxmax}\left|z_{j}^{+}-z_{ij}\right|} \tag{24}\]  
\[r_{ij}^{-}=\frac{\underset{i}{minmin}\left|z_{j}^{-}-z_{ij}\right|+\underset{ i}{maxmax}\left|z_{j}^{-}-z_{ij}\right|}{\left|z_{j}^{-}-z_{ij}\right|+ \underset{i}{maxmax}\left|z_{j}^{-}-z_{ij}\right|} \tag{25}\]  
where \(\varepsilon\) is the resolution coefficient (\(\varepsilon\in[0,\,1]\)). We take the value of 0.5 [64].  
Calculate the gray correlation degree of each scheme with the positive ideal solution \(r^{+}\) and the negative ideal solution \(r^{-}\).  
\[r^{+}=\frac{1}{n}\sum_{j=1}^{n}r_{ij}^{+}\text{ and }r^{-}=\frac{1}{n}\sum_{j=1}^ {n}r_{ij}^{-} \tag{26}\]  
5. Dimensionless processing is performed on Euclid distance and correlation degree.  
\[D_{i}{}^{+}=\frac{d_{i}{}^{+}}{maxd_{i}{}^{+}}\text{ and }D_{i}{}^{-}=\frac{d_{i}{}^{-} }{maxd_{i}{}^{-}} \tag{27}\]  
\[R_{i}{}^{+}=\frac{r_{i}{}^{+}}{maxr_{i}{}^{+}}\text{ and }R_{i}{}^{-}=\frac{r_{i}{}^{-} }{maxr_{i}{}^{-}} \tag{28}\]  
6. Combine dimensionless distance and dimensionless correlation. The larger the values of \(D_{i}{}^{-}\) and \(R_{i}{}^{+}\), the closer the scheme is to the positive ideal solution, and so the combined formula can be determined as Equation (30).  
\[P_{i}{}^{+}=\alpha D^{-}+\beta R^{+}\text{ and }P_{i}{}^{-}=\alpha D^{+}+\beta R ^{-} \tag{29}\]  
\[Qi^{+}=\frac{P_{i}{}^{+}}{P_{i}{}^{+}+P_{i}{}^{-}} \tag{30}\]  
where \(\alpha=\beta=\frac{1}{2}\). Sorting the cases according to the size of the relative closeness, the closer the degree is to one, the better the scheme is.