## 2 Materials and Methods
### Description of the Multi-Objective Optimization Problem  
In the material removal process, improving a specific technical index by adjusting the process parameters may lead to the deterioration of performance in other areas. Therefore, the objective of this study was to balance the relationship between sustainability, machining quality, and efficiency in micro-milling processes, which is illustrated in Figure 5.  
In this study, the optimization objectives included surface roughness (\(S_{a}\)), surface fractal dimension (\(D_{s}\)), and size error (\(D_{c}\)) as measures of machining quality; the material removal rate (MRR) considered for evaluating machining efficiency; and the specific cutting  
Figure 4: The power loss of drive motor. Figure 5: The multi-objective optimization problem description. energy consumption (SCE) employed for energy efficiency assessment. Their definitions are shown in Equations (16)-(20). \[S_{a}=\iint\limits_{S}|\Phi(x,y)-\xi(x,y)|dxdy/S \tag{16}\]  
where \(\Phi(x,y)\) denotes the height information of the surface, and \(\xi(x,y)\) corresponds to the reference plane of the surface. \[D_{s}=\frac{\ln N(\varepsilon)}{\ln(\varepsilon^{-1})} \tag{17}\]  
where \(\varepsilon\) represents the side length of the box, and \(N(\varepsilon)\) represents the number of boxes. The value of the surface fractal dimension represents the richness of details on the surface, reflecting the complexity and finesse of the surface contour in space, as well as its ability to fill space. In comparison with parameters of surface roughness, the surface fractal dimension is more sensitive to the absence of surface microstructures. Therefore, selecting the surface fractal dimension to characterize surface quality holds a certain significance [26; 27]. \[D_{e}=|D_{a}-D_{i}| \tag{18}\]  
where \(D_{e}\) is the actual dimension, \(D_{a}\) is the actual dimension, and \(D_{i}\) is the design dimension. The dimensional errors (\(D_{e}\)) were measured using pixel analysis of the SEM-measured picture. In comparison with image scales, three positions on the thin-walled microstructure were selected for averaging; thus, the actual dimensions were determined, followed by subtracting the design dimensions to obtain the dimensional errors. \[MRR=a_{p}\cdot a_{e}\cdot f \tag{19}\]  
where \(a_{p}\), \(a_{e}\), and \(f\) are the cutting depth, cutting width, and feed speed, respectively. \[SCE=\frac{E_{\text{total}}}{V_{\text{material}}}=\frac{P\cdot T}{MRR\cdot T}= \frac{p}{MRR} \tag{20}\]  
where \(SCE\), \(E_{\text{total}}\), \(V_{\text{material}}\), \(T\), and \(P\) are the specific cutting energy, cutting energy, volume of removal material, machining time, and machining power, respectively. The three process parameters to be optimized are the spindle speed \(n\), feed per tooth \(f_{z}\), and cutting depth \(a_{p}\). Considering the main technical requirements of thin-walled microstructures, the optimal combination of process parameters can be obtained with the constraints shown in Equations (21) and (22). \[\left\{ \begin{array}{l}n_{\text{min}}\leq n\leq n_{\text{max}}\\ f_{z\text{min}}\leq f_{z}\leq f_{z\text{max}}\\ a_{\text{p}\text{min}}\leq a_{\text{p}}\leq a_{\text{p}\text{max}}\end{array} \right. \tag{21}\]  
\[\left\{ \begin{array}{l}\min Sa\big{(}n,f_{z},a_{p}\big{)}\\ \max Ds\big{(}n,f_{z},a_{p}\big{)}\\ \min De\big{(}n,f_{z},a_{p}\big{)}\\ \max MRR\big{(}n,f_{z},a_{p}\big{)}\\ \min SEC\big{(}n,f_{z},a_{p}\big{)}\end{array} \right. \tag{22}\]  
To eliminate the negative impacts caused by the improper selection of process parameters, it is essential to adopt appropriate optimization methods, which generally fall into two categories: traditional methods and intelligent methods. Traditional methods typically rely on experimental design and gradient-based search, the limitation of which lies in the fact that the convergence can be greatly influenced by the nonlinearity of the objective function and initial value settings and the global optimality is hard to ensure.