## 2 Materials and Methods
### Experimental Design
[PERSON]'s sphericity test and the Kaiser-Meyer-Olkin (KMO) sampling adequacy test were conducted to assess the suitability of PCA, as shown in Table 10. The results indicate that [PERSON]'s sphericity test was significant (\(p=0.000<0.05\)), and the KMO value was 0.681, which falls between 0.6 and 0.7. Therefore, PCA was appropriate. Table 11 presents the common factor variance matrix, revealing that the \(S_{a}\) and \(D_{s}\) indicators for surface quality exhibit sufficient information extraction, and the information extraction content of \(D_{e}\) is relatively small. Higher eigenvalues indicate greater potential energy and are more crucial for the analysis. In Figure 19, only the first eigenvalue is greater than one, which indicates only one factor is needed to extract from the original features. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Sampling Adequacy KMO Measure**} & **0.681** \\ \hline [PERSON]’s test of sphericity & Approximate chi-square & 22.351 \\ \hline df & 3 \\ \hline Sig. & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: KMO and [PERSON]’s tests. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(S_{a}\) & \(D_{s}\) & \(D_{e}\) \\ \hline \(S_{a}\) & \(1\) & \(-0.832\) & 0.579 \\ \hline \(D_{s}\) & \(-0.832\) & \(1\) & \(-0.628\) \\ \hline \(D_{e}\) & \(0.579\) & \(-0.628\) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Correlations between the optimization objectives. The explained total variance is shown in Table 12. The variance contribution of the first component is 78.851%, allowing for the extraction of original features with minimal information loss. The factor loading matrix is shown in Table 13, which illustrates the contributions of the principal components to each variable. According to the principal component matrix, the linear combinations of machining quality variables can be obtained as Equation (32). This principal component can be interpreted as the machining quality. \[P_{1}=0.386(Sa)-0.394(Ds)+0.345(De) \tag{32}\]  
Therefore, the objective function optimized with NSGA-II and MOPSO can be expressed as Equation (33), and the objective function optimized with NSGA-III can be expressed as Equation (34). \[\text{min}f_{11}(X)= \begin{cases}f_{1}(X)=P_{1}\\ f_{2}(X)=MRR^{-1}\\ f_{3}(X)=SEC\end{cases}  \tag{33}\]  
\[\text{min}f_{21}(X)= \begin{cases}f_{1}(X)=Sa\\ f_{2}(X)=Ds^{-1}\\ f_{3}(X)=De\\ f_{4}(X)=MRR^{-1}\\ f_{5}(X)=SEC\end{cases}  \tag{34}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & \multicolumn{2}{c}{**Initial Eigenvalue**} & \multicolumn{2}{c}{**Extract Loading Sum of Squares**} \\ \hline  & **Eigenvalue** & **Initial Variance \%** & **Cumulative Variance \%** & **Eigenvalue** & **Initial Variance \%** & **Cumulative Variance \%** \\ \hline
1 & 2.366 & 78.851 & 78.851 & 2.366 & 78.851 & 78.851 \\ \hline
2 & 0.470 & 15.654 & 94.505 & & \\ \hline
3 & 0.165 & 5.495 & 100.000 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Explained total variance. Figure 19: Scree plot of PCA results. The decision variables can be represented as X = [\(n,f_{z},a_{p}\)] T. The ranges of the process parameters are as follows: 50 \(\mu\)m \(\leq a_{p}\leq\) 150 \(\mu\)m, 15,000 rpm \(\leq n\leq\) 45,000 rpm, and 0.5 mm/t \(\leq\)\(f_{z}\leq\) 3.5 mm/t. The initial population size and number of iterations for each optimization algorithm were set as 500 and 1500, respectively. A membership function was used to select results from the Pareto solution set, as shown in Equation (35). \[p_{i}=\frac{f_{\text{max}}-f_{i}}{f_{\text{max}}-f_{\text{imin}}} \tag{35}\]  
where \(f_{\text{imax}}\) and \(f_{\text{imin}}\) represent the maximum and minimum values of the \(i\)-th objective function, respectively, and \(f_{i}\) denotes the value of the \(i\)-th objective function. The optimal solution is selected by maximizing the average \(p\)-value, as shown in Equation (36).