## 3 Results
### Results and Analysis of Multi-Objective Optimization  
In material processing, raising the feed rate can significantly improve milling efficiency; however, it also leads to an increase in the milling force applied to the tool, which in turn accelerates wear. As a result, there is a certain level of conflict among material removal rate, milling force, and tool wear. To effectively tackle this challenge, it is essential to find a balance between these three elements using multi-objective optimization methods. The prediction model of the wear width of the rake face and the milling force are as follows:  
\[\left\{ \begin{array}{l}b=6.68\times 10^{-8}n^{1.043}v_{\rm f}^{0.505}a_{\rm p }^{0.152}\\ F=33.681-2.980\times 10^{-3}n+2.738\times 10^{-2}v_{\rm f}-0.873a_{\rm p}-4.217 \times 10^{-7}nv_{\rm f}+5.736\times 10^{-4}na_{\rm p}\\ +1.917\times 10^{-4}v_{\rm f}a_{\rm p}+5.481\times 10^{-8}n^{2}-4.5\times 10^{-6}v_{ \rm f}^{2}-0.307a_{\rm p}^{2}\end{array} \right. \tag{10}\]  
Material removal rate, \(Q\), can be represented as [27]:  
\[Q=\frac{1}{1000}v_{\rm f}\cdot a_{\rm p}\cdot a_{e} \tag{11}\]  
The following outlines the objective function intended for optimization:  
Figure 8: The variation relationship between the milling force, \(F\), and the spindle speed, \(n\), the feed rate, \(v_{\rm f}\), as well as the axial milling depth, \(a_{\rm p}\). (**a**) The \(F\)-\(v_{\rm f}\) variation curve; (**b**) the \(F\)-\(a_{\rm p}\) variation curve; (**c**) the \(F\)-\(n\) variation curve. \[\left\{ \begin{array}{l}\min(1)=6.68\times 10^{-8}\mathrm{x}_{1}^{1.043}\cdot \mathrm{x}_{2}^{0.505}\cdot\mathrm{x}_{3}^{0.152}\\ \min(2)=33.681-2.980\times 10^{-3}\mathrm{x}_{1}+2.738\times 10^{-2}\mathrm{x}_{2}-0.873 \mathrm{x}_{3}-4.217\times 10^{-7}\mathrm{x}_{1}\mathrm{x}_{2}+5.736\times 10^{-4} \mathrm{x}_{1}\mathrm{x}_{3}\\ \qquad+1.917\times 10^{-4}\mathrm{x}_{2}\mathrm{x}_{3}+5.481\times 10^{-8} \mathrm{x}_{1}^{2}-4.5\times 10^{-6}\mathrm{x}_{2}^{2}-0.307\mathrm{x}_{3}^{2} \\ \min(3)=-\frac{1}{1000}\cdot\mathrm{x}_{2}\cdot\mathrm{x}_{3}\end{array} \right. \tag{12}\]  
The constraints are as follows:  
\[\left\{ \begin{array}{l}12000\leq\mathrm{x}_{1}\leq 18000\\ 1000\leq\mathrm{x}_{2}\leq 2000\\ 3\leq\mathrm{x}_{3}\leq 5\end{array} \right. \tag{13}\]  
where y(1), y(2), and y(3) denote the wear width, \(b\), of the rake face, the milling force, \(F\), and the material removal rate, \(-Q\), respectively. The reason why y(3) equals the material removal rate, \(Q\), multiplied by \(-1\) is that, during milling, our goal is to maximize the material removal rate. However, the genetic algorithm tool, gamultiobj, is designed for solving multi-objective minimization problems. By multiplying the material removal rate by \(-1\), we can convert the maximization problem into a minimization problem, which can be effectively optimized using the genetic algorithm. Similarly, x\({}_{1}\), x\({}_{2}\), and x\({}_{3}\) represent spindle speed, \(n\), feed rate, \(v_{t}\), and axial milling depth, \(a_{\mathrm{P}}\). The radial milling depth, \(a_{\mathrm{e}}\), is set to 1 mm. This is because an excessively large \(a_{\mathrm{e}}\) can compromise the surface quality of the workpiece, while an overly small \(a_{\mathrm{e}}\) may reduce machining efficiency. Consequently, following the factory's recommended value, \(a_{\mathrm{e}}\) is ultimately set to 1 mm. The multi-objective genetic algorithm NSGA-II was adopted in MATLAB to optimize the objective functions. The Pareto fractions were set at 0.3, the population size was 200, the number of generations was 300, the number of stall generations was 200, and the function tolerance was established at \(1\times 10^{-10}\). When the algorithm was executed, the variations in the Pareto front related to the wear width, \(b\), of the rake face, the milling force, \(F\), and the material removal rate, \(Q\), were acquired as shown in Figure 9. The figure reveals that there exists a striped scattering distribution among the milling force, the wear width, and the material removal rate.