## 2 Materials and Methods
### Experimental Design
\begin{cases}S_{a}=1.8221\times n^{0.0830}\times f_{z}^{0.3404}\times a_{p}^{0.6230} \\ D_{s}=2.1839\times n^{0.0059}\times f_{z}^{0.0125}\times a_{p}^{0.0181}\\ D_{e}=0.0247\times n^{0.3490}\times f_{z}^{0.3076}\times a_{p}^{0.2630}\end{cases}  \tag{30}\]  
The main effects of the process parameters on the surface roughness, fractal dimension, and dimension error are shown in Figures 14  
The dimensional error shows an overall increasing trend with the enlargement of the process parameters. This is due to the fact that with the increase in the feed per tooth and cutting depth, the cutting forces exhibit an increasing trend. This leads to greater deformation of thin-walled micro-structures in the vertical feed direction, ultimately resulting in an increase in dimensional error. Conversely, as the spindle speed increases, the heat generated during cutting also increases, making the workpiece material prone to thermal deformation. Consequently, this leads to an augmentation in dimensional error. The reasons for small deformation errors are as follows: The micro-milling force is relatively small, and previous studies have shown that the average cutting force for the micro-milling of titanium alloys is within 5 N [31]; the positioning accuracy of the KERN EVO machining center is 1 \(\upmu\)m, with a repeatability accuracy of \(\pm\) 0.5 \(\upmu\)m, exhibiting excellent machining precision; and Ti-6 Al-4V possesses high strength, and under the influence of relatively small cutting forces, noticeable deformation errors are not likely to occur. The main effects of the process parameters on the SCE are shown in Figure 17. It can be observed that the SCE significantly decreases with the increase in feed per tooth and cutting depth because their impacts on the material removal rate do not significantly affect power consumption. However, the multiplication of the spindle speed leads to a rapid increase in power consumption. The relationship between the SCE and MRR is illustrated in Figure 18, with a Pearson correlation coefficient of \(-\)0.793, which exceeds 0.7, indicating a significant negative correlation. This implies that enhancing the machining efficiency can reduce specific energy consumption. In order to overcome the problem of intelligent optimization algorithms easily getting trapped in local optima, this paper investigated and compared the optimal combination of process parameters using three evolutionary algorithms. Since NSGA-II and MOPSO perform poorly when dealing with four or more optimization objectives, the optimization objectives were first compressed. To retain and interpret the original variable information as  
Figure 17: Main effects of process parameters on SCE. Figure 18: Correlation between MRR and SCE. much as possible, principal component analysis (PCA) was applied to the three indicators representing the machining quality. First, the standardization of \(S_{a}\), \(D_{s}\), and \(D_{e}\) was performed according to Equation (31). \[ \begin{split} X_{mn}^{\prime}=\left(X_{mn}-\overline{X}_{m} \right)/\sigma_{m}\\ \overline{X}_{m}=\sum\limits_{n=1}^{p}X_{mn}/p\\ \sigma_{m}=\sqrt{\sum\limits_{n=1}^{p}\left(X_{mn}-\overline{X}_ {m}\right)^{2}/p}\end{split}  \tag{31}\]  
where \(X_{mn}^{\prime}\) represents the \(n\)-th sampled value of the \(m\)-th objective function after normalization; \(X_{mn}\) represents the \(n\)-th sampled value of the \(m\)-th objective function; \(p\) represents the total number of samples; and \(\overline{X}_{m}\) and \(\sigma_{m}\) represent the mean and standard deviation, respectively. The surface roughness, fractal dimension, and dimensional error correlation coefficients are shown in Table 9. It can be observed that the surface quality indices, \(S_{a}\) and \(D_{s}\), exhibit a strong negative correlation, while the correlation between surface quality and dimensional error is relatively weak.