## 4 Validation of Surface Roughness Model and Process Parameter Optimization
### Parameters Optimization Based on Taguchi-Grey Correlation Method  
The analysis of signal-noise ratio SNR is a meritocratic process. By selecting the optimal level combination of all parameters in the system, the designed product has a strong resistance to disturbances and robustness, and the fluctuation of the product quality characteristic is gentle. In this section, the surface roughness and the milling force are selected as the corresponding objectives. It is expected to improve the cutting performance and obtain high machining accuracy by designing for small milling forces and surface \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & \(\mathbf{V_{c}}\) **(m/min)** & \(\mathbf{a_{e}}\) **(mm)** & \(\mathbf{a_{p}}\) **(mm)** & \(\mathbf{f_{z}}\) **\(\mathbf{\mu}\)m/z** \\ \hline
1 & 14.14 & 0.5 & 0.5 & 10 \\
2 & 14.14 & 1 & 1 & 15 \\
3 & 14.14 & 1.5 & 1.5 & 20 \\
4 & 14.14 & 2 & 2 & 25 \\
5 & 23.56 & 0.5 & 1 & 20 \\
6 & 23.56 & 1 & 0.5 & 25 \\
7 & 23.56 & 1.5 & 2 & 10 \\
8 & 23.56 & 2 & 1.5 & 15 \\ \hline
9 & 32.99 & 0.5 & 1.5 & 25 \\
10 & 32.99 & 1 & 2 & 20 \\
11 & 32.99 & 1.5 & 0.5 & 15 \\
12 & 32.99 & 2 & 1 & 10 \\
13 & 42.41 & 0.5 & 2 & 15 \\
14 & 42.41 & 1 & 1.5 & 10 \\
15 & 42.41 & 1.5 & 1 & 25 \\
16 & 42.41 & 2 & 0.5 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The orthogonal experiments. Figure 27: The comparison between the theoretical and experimental values. roughness. According to the desired characteristics of the corresponding objectives, the signal-noise ratio SNR can be expressed as [50; 51]:  
\[\text{SNR}=-10\log\left(\frac{1}{n}\sum_{i=1}^{n}{y_{i}}^{2}\right) \tag{35}\]  
where \(n\) denotes the number of experimental repetitions, and \(y_{i}\) denotes the objective value of the \(i\)-th experiment. According to the experimental results of the corresponding objectives, the SNR for the surface roughness and milling force is shown in Table 5.  
The variance analysis based on the Taguchi method is suitable for single-objective optimization. Multiobjective optimization can be performed by transforming multiple response objectives into a single gray correlation degree value using gray correlation theory, which includes three steps of normalization, calculation of gray correlation coefficient, and calculation of gray correlation degree. The SNR results are taken as the original data and quantified as follows [52]:  
\[x_{i}(k)=\frac{\underset{i}{maxy_{i}(k)-y_{i}(k)}}{\underset{i}{maxy_{i}(k)- \underset{i}{miny_{i}(k)}}} \tag{36}\]  
where, \(x_{i}(k)\) denotes the standardized response objective, and \(y_{i}(k)\) denotes the objective value in the \(i\)-th group of experiments. The gray correlation coefficients of the response objectives are calculated as follows [52]:  
\[\xi_{i}(k)=\frac{\underset{i}{min}\left|x_{i}^{0}-x_{i}(k)\right|+\xi \underset{i}{max}\left|x_{i}^{0}-x_{i}(k)\right|}{\left|x_{i}^{0}-x_{i}(k) \right|+\xi\underset{i}{max}\left|x_{i}^{0}-x_{i}(k)\right|} \tag{37}\]  
where \(\xi_{i}(k)\) denotes the number of gray correlation coefficients, and \(x_{i}^{0}\) denotes the ideal normalized value in the \(i\)th group of experiments, which is equal to \(1\) in the calculation. \(\xi\) is the discrimination coefficient, which is equal to \(0.5\). The gray correlation is calculated as follows [52]:  
\[\gamma_{i}=\frac{1}{n}\sum_{i=1}^{n}{\xi_{i}(k)} \tag{38}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multicolumn{2}{c}{**Surface Roughness \(S_{a}\)**} & \multicolumn{2}{c}{**Milling Force F**} \\ \cline{2-5}  & **Results/\(\mu\)m** & **SNR** & **Results/N** & **SNR** \\ \hline
1 & 1.196 & \(-1.555\) & \(5.531\) & \(-14.856\) \\
2 & 1.635 & \(-4.270\) & \(18.901\) & \(-25.530\) \\
3 & 2.072 & \(-6.328\) & \(43.251\) & \(-32.720\) \\
4 & 2.196 & \(-6.833\) & \(74.268\) & \(-37.416\) \\where \(n\) denotes the total number of experiments to be arranged, and \(\gamma_{i}\) denotes the gray correlation.