## 2 Materials and Methods
### Design Process
#### 2.2.3 Multi-Level Comprehensive Evaluation
The scalar product of each row is calculated according to Equations (9) and (10), and then its geometric mean is determined. \[a_{J}=\sqrt[k]{M_{J}}(J=1,2,3,\ldots,n) \tag{9}\]  
\[W_{J}=\frac{a_{J}}{\sum_{J=1}^{k}a_{J}} \tag{10}\]  
(3) Calculate the maximum characteristic value of each evaluation index [19]. \[\lambda\ \text{max}{=}\ \frac{1}{n}\text{$\sum_{J=1}^{n}\left(\sum_{J=1}^{n}a_{J }a_{J}/a_{J}\right)$} \tag{11}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Scale** & **Implication** \\ \hline
1 & by comparison, the two elements are equally important \\
3 & by comparison, one element is slightly more important than the other \\
5 & by comparison, one element is obviously more important than the other \\
7 & by comparison, one element is strongly more important than the other \\
9 & by comparison, one element is extremely more important than the other \\
2, 4, 6, 8 & the middle value between each of the above two adjacent scales \\ \hline \hline \end{tabular}
\end{table} 
Table 9: One to nine scaling method. Here, a\({}_{\text{jn}}\) is the nth component of the vector a\({}_{\text{j}}\) and n is the number of steps. (4) Consistency test. Calculating the consistency ratio:  
\[\text{CI}=\frac{\lambda_{max}-r}{n-1} \tag{12}\]  
\[\text{CR}=\frac{\text{CI}}{\text{RI}} \tag{13}\]  
Here, RI is the average random consistency index; CR is the consistency ratio. CR \(\leq\) 0.1 indicates that the consistency test is passed, and vice versa, it is failed. Step 7: Entropy weighting method to calculate objective weights. (1) Convert the scoring matrix into a normalized matrix  
The assessment indicators in the matrix are normalized according to Equation (14):  
\[P_{jk}=\frac{b_{jk}-minb_{jk}}{\underset{j}{\text{max}}xb_{jk}-\underset{j}{ \text{min}}b_{jk}} \tag{14}\]  
Here, \(P_{jk}\) is the normalized decision matrix, is the original matrix value, \(J=1\), 2, \(\ldots\), m, \(k=1\), 2, \(\ldots\), m, \(k=1\), 2, \(\ldots\), n.  
(2) Determine the entropy of each evaluation index. Calculate the entropy value of the kth indicator according to Equation (15):  
\[Y_{k}=k\underset{j=1}{\overset{m}{\sum}}P_{jk}\text{ln}\,P_{jk},k=1/\text{ln }\,m \tag{15}\]  
Here, \(Y_{k}\) is the information entropy of each indicator, and the smaller \(Y_{k}\) is, the higher dispersion of the data under \(k\) indicators and the greater the amount of information provided. (3) Calculate entropy weights. After defining the entropy of the \(k\) indicators, the entropy weights can be obtained as follows:  
\[\omega_{k}=\frac{1-Y_{k}}{n-\underset{k=1}{\overset{m}{\sum}}Y_{k}} \tag{16}\]  
Step 8: Gray weighted composite weight calculation. Calculate the average of the correlation coefficients between each indicator and the reference series at a certain value according to Equation (17):  
\[r^{\prime}{}_{0J}=\frac{\underset{k=1}{\overset{m}{\sum}}\xi_{jk}}{m} \tag{17}\]  
Here, \(\xi_{jk}\) is the correlation between the comparison series and the reference series; \(m\) is the number of evaluation indicators. The gray weighted correlation is calculated and ranked according to Equation (18), and the top-ranked scheme is the preferred scheme. The indicators with a high weight and low score in the design scheme need to be optimized to a high degree [19]. \[r_{0J}=\frac{\underset{k=1}{\overset{m}{\sum}}\omega_{k}\xi_{jk}}{m} \tag{18}\]  
Here, \(\omega_{k}\) is the weight value of the kth assessment index; \(\xi_{jk}\) is the correlation between the kth assessment index of the Jth product and the reference series.