## 2 Multi-Objective Optimization Modeling in Material Cutting Stage of CNC Lathe
### Model Building
In order to verify the reliability of the established model, five groups of test data were selected for verification, and the errors of the turning specific energy \(SEC\) and surface roughness \(Ra\) obtained via formula fitting and testing were compared. The results of the verification of the turning process are shown in Table 3. Among the five sets of test data, the average error of the \(SEC\) is \(0.17\,\mathrm{J/mm^{3}}\), with an average error percentage of 5.89%, and the average error of Ra is 0.22 \(\upmu\)m, with an average error percentage of 4.86%. These results show that the developed model has high accuracy, effectively predicting both turning specific energy \(SEC\) and surface roughness \(Ra\). According to the mathematical expressions of cutting specific energy and surface roughness, as well as the production conditions that must be met by turning parameters,  
Figure 5: Comparison of fitted results. \begin{table}
\begin{tabular}{c} \(\mathbf{Ra\ Actual}\) \\ **Value** \\ **(j/mm\({}^{3}\))** \\ \end{tabular} \\ \hline
1 & 40 & 0.3 & 0.7 & 475 & 3.12 & 3.39 & 4.34 & 4.61 \\
2 & 40 & 0.4 & 0.5 & 377 & 2.99 & 2.83 & 6.29 & 6.63 \\
3 & 50 & 0.4 & 0.5 & 461 & 2.65 & 2.77 & 4.35 & 4.14 \\
4 & 50 & 0.2 & 0.7 & 336 & 3.07 & 2.88 & 1.85 & 1.77 \\
5 & 60 & 0.4 & 0.7 & 678 & 2.31 & 2.42 & 3.12 & 3.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Verification results for turning test. the multi-objective optimization model of a CNC lathe in the material-cutting stage can be determined as follows:  
\[ \begin{array}{l}\minSEC=16.75v_{c}^{-0.552}f^{-0.264}a_{p}^{-0.106}\\ \minRa=7977v_{c}^{-1.658}f^{1.190}a_{p}^{-0.09}\\ \left\{\begin{array}{l}v_{\min}\leq v_{c}\leq v_{\max}\\ f_{\min}\leq f\leq f_{\max}\\ a_{p\min}\leq a_{p}\leq a_{p\max}\\ P_{c}=C_{\text{r}}v_{c}^{\text{r}+1}f^{\text{q}}a_{p}^{\text{z}}K_{\text{r}} \leq 7.5\text{kW}\end{array}\right.\end{array}  \tag{15}\]  
The weighted summation method is frequently employed to convert multi-objective optimization problems into their single-objective counterparts. Due to the different magnitudes of cutting specific energy and surface roughness, it is necessary to normalize the two objective functions. Different weight coefficients are set for each optimization objective according to the different efficiency goals pursued. The larger the coefficient, the more emphasis is placed on the optimization of specific cutting energy and, conversely, the lower the machined surface roughness. The weighted optimization objective function is shown in Equation (16):  
\[\min F=\min(cSEC^{*}+(1-c)Ra^{*}) \tag{16}\]  
where c and \(1-\text{c}\) are the weighting coefficients for cutting specific energy and machined surface roughness, respectively; \(c=0.1\), \(0.2\), \(\ldots\), \(0.9\); and \(SEC^{*}\) and \(Ra^{*}\) are the normalized cutting specific energy and machined surface roughness functions, respectively. The normalization process is defined as follows:  
\[f_{i}^{*}=\frac{f_{i}-f_{i,\min}}{f_{i,\max}-f_{i,\min}} \tag{17}\]  
where \(f_{i,\max}\) and \(f_{i,\min}\) are the maximum and minimum values of each objective function for single-objective optimization, respectively.