## 6 Experimental Scheme Design and Optimization Results Discussion
### Gray Relational Grade  
The GRA is a method proposed by [PERSON] [54] in 1982 that uses GRG to represent the grade of approximation between design scheme and ideal scheme, which provides an effective mathematical method for dealing with problems of inadequate information, indigent information, and uncertain information. Due to the order of magnitude difference, the experimental results in Table 5 need to be preprocessed before gray correlation analysis. According to the characteristics of the corresponding data, the experimental results are scaled to between 0 and 1 using the different normalization methods. If the objective has the feature of 'bigger is better', the normalization method is expressed as  
\[x_{i}^{*}(k)=\frac{x_{i}(k)-\min_{k}x_{i}(k)}{\max_{k}x_{i}(k)-\min_{k}x_{i}(k)} \tag{11}\]  
If the objective has the feature of'smaller is better', the normalization method is expressed as  
\[x_{i}^{*}(k)=\frac{\max_{k}x_{i}(k)-x_{i}(k)}{\max_{k}x_{i}(k)-\min_{k}x_{i}(k)} \tag{12}\]  
where \(x_{i}^{*}(k)\) is the normalized value of the \(i\)th response in the \(k\)th objective function; \(x_{i}(k)\) is the initial value of the objective function; \(\max_{k}x_{i}(k)\) and \(\min_{k}x_{i}(k)\) are the maximum and minimum values of the \(k\)th objective function, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & **Level 1** & **Level 2** & **Level 3** & **Level 4** \\ \hline \(H\) & 950 & 1000 & 1050 & 1100 \\ \(\beta\) & 20 & 25 & 30 & 35 \\ \(\alpha\) & 80 & 85 & 90 & 95 \\ \(R\) & 800 & 850 & 950 & 1000 \\ \(r\) & 500 & 600 & 650 & 700 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Influencing factors and levels. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Number** & \(H\) & \(\beta\) & \(\alpha\) & \(R\) & \(r\) & \(f_{1}\)(\(\alpha\))/N & \(f_{2}\)(\(\alpha\))/(\({}^{\circ}\)) & \(f_{3}\)(\(\alpha\))/(\({}^{\circ}\)) \\ \hline
1 & 1 & 1 & 1 & 1 & 1 & 19.46 & 2.441 & 3.19 \\
2 & 1 & 2 & 2 & 2 & 2 & 19.32 & 1.95 & 2.557 \\
3 & 1 & 3 & 3 & 3 & 3 & 20.01 & 1.845 & 2.291 \\
4 & 1 & 4 & 4 & 4 & 4 & 20.25 & 1.706 & 2.084 \\
5 & 2 & 1 & 2 & 3 & 4 & 18.51 & 1.694 & 2.131 \\
6 & 2 & 2 & 1 & 4 & 3 & 18.69 & 1.926 & 2.41 \\
7 & 2 & 3 & 4 & 1 & 2 & 17.42 & 1.879 & 2.546 \\
8 & 2 & 4 & 3 & 2 & 1 & 18.34 & 2.595 & 3.312 \\
9 & 3 & 1 & 3 & 4 & 2 & 18.21 & 2.047 & 2.45 \\
10 & 3 & 2 & 4 & 3 & 1 & 18.36 & 2.541 & 2.975 \\
11 & 3 & 3 & 1 & 2 & 4 & 15.08 & 1.609 & 2.403 \\
12 & 3 & 4 & 2 & 1 & 3 & 14.81 & 1.713 & 2.633 \\
13 & 4 & 1 & 4 & 2 & 3 & 15.64 & 1.774 & 2.322 \\
14 & 4 & 2 & 3 & 1 & 4 & 14.58 & 1.655 & 2.313 \\
15 & 4 & 3 & 2 & 4 & 1 & 17.22 & 2.945 & 3.518 \\
16 & 4 & 4 & 1 & 3 & 2 & 15.46 & 2.248 & 3.021 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Experimental layout and experimental results. The above normalization method is also called 'gray generation', and the corresponding gray relational coefficient (GRC) is calculated as  
\[\gamma(x_{0}^{*}(k),x_{i}^{*}(k))=\frac{\Delta_{\min}+\zeta\Delta_{\max}}{\Delta _{0i}(k)+\zeta\Delta_{\max}} \tag{13}\]  
where \(x_{0}^{*}(k)\) is the ideal experimental scheme defined by ideal; \(x_{i}^{*}(k)\) is the designed experimental scheme; \(\Delta_{0i}(k)\) is the absolute difference between \(x_{0}^{*}(k)\) and \(x_{i}^{*}(k)\); \(\Delta_{\max}\) and \(\Delta_{\min}\) are the maximum and minimum values of \(\Delta_{0i}(k)\), respectively; \(\zeta\) is the discrimination coefficient, \(\zeta\)\(\in\) [0, 1]. In this article, \(x_{0}^{*}(k)\) = 1 is selected as the reference result, and the GRC of the experimental results is shown in Table 6.  
The GRG is calculated by weighted summation of the GRC of each objective function, and the calculation formula of GRG is  
\[\Gamma(x_{0}^{*},x_{i}^{*})=\sum_{k=1}^{n}\omega_{k}\gamma(x_{0}^{*}(k),\ x_{i}^{*}(k)) \tag{14}\]  
where \(n\) is the number of objective functions; \(\omega_{k}\) is the weight coefficient of the \(k\)th objective function, \(\sum_{k=1}^{n}\omega_{k}=1\).