## 2 Multi-Objective Optimization Modeling in Material Cutting Stage of CNC Lathe
### Model Building  
Equations (5) and (6) are nonlinear equations. Equations (5) and (6) can be transformed into linear forms by computing the logarithms of both sides and considering the new unknowns equal to the logarithm of the corresponding originals. Take Equation (7) as an example:  
\[\ln SEC=\ln C+(x-1)\ln v_{c}+(y-1)\ln f+(z-1)\ln a_{p} \tag{7}\]  
Let us set \(y=\ln SEC\), \(\beta_{0}=\ln C\), \(\beta_{1}=x-1\), \(\beta_{2}=y-1\), \(\beta_{3}=z-1\), \(x_{1}=\ln v_{c}\), \(x_{2}=\ln f\), and \(x_{3}=\ln a_{p}\); then, Equation (8) can be converted into a linear equation:  
\[y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3} \tag{8}\]  
\(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\) are the regression coefficients. The regression coefficients are the solution to a linear system with nine equations and four unknowns, whose matrix form is as follows, as shown in Equation (9):  
\[\mathbf{Y}=\mathbf{X}\mathbf{\beta} \tag{9}\]  
\[\text{where }\mathbf{Y}= \begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{9}\end{bmatrix} ,\mathbf{X}= \begin{bmatrix}1&x_{1,1}&x_{1,2}&x_{1,3}\\ 1&x_{2,1}&x_{2,2}&x_{2,3}\\ \vdots&\vdots&\vdots&\vdots\\ 1&x_{9,1}&x_{9,2}&x_{9,3}\end{bmatrix} ,\text{and }\mathbf{\beta}= \begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \beta_{2}\\ \beta_{3}\end{bmatrix} .\]  
The estimating parameters can be set using the least squares method \(\mathbf{\beta}\). Let \(b_{0}\), \(b_{1}\), and \(b_{2}\) be the least squares estimates of the parameters \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\), respectively, and \(\hat{y}\) be the regression value of the test result \(y_{i}\); then, the regression Equation (10) is as follows:  
\[\hat{y}_{i}=b_{0}+b_{1}x_{i,1}+b_{2}x_{i,2}+b_{3}x_{i,3} \tag{10}\]  
According to the principle of the least squares method, the sum of squares of a random error \(\varepsilon_{i}\) should be minimized, and \(\varepsilon_{i}\) is the difference between the test result \(y_{i}\) and the regression value \(\hat{y}\). Then, the sum of squares of the random errors of the nine groups of tests is expressed as \(\mathbf{Q}\):  
\[\mathbf{Q}=\sum_{i=1}^{9}\varepsilon_{i}=\sum_{i=1}^{9}\left(y_{i}-\hat{y}_{i} \right)^{2}=\sum_{i=1}^{9}\left(y_{i}-b_{0}-b_{1}x_{i,1}-b_{2}x_{i,2}-b_{3}x_{ i,3}\right)^{2} \tag{11}\]  
Set \(\mathbf{b}=\left[b_{0}\ b_{1}\ b_{2}\ b_{3}\right]^{T}\), according to the extreme value principle; calculate the first derivative of \(\mathbf{Q}\) versus \(\mathbf{b}\); and set the derivative equal to zero. After simplification, we can finally obtain the following:  
\[\mathbf{b}= \begin{bmatrix}b_{0}\\ b_{1}\\ b_{2}\\ b_{3}\end{bmatrix} =(\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}^{T}\mathbf{Y}= \begin{bmatrix}2.819\\ -0.552\\ -0.264\\ -0.106\end{bmatrix}  \tag{12}\]  
Since \(b_{0}=\ln C=2.819\), \(C=16.75\). Thus, the final turning specific energy model of the cutting parameters is shown in Equation (13):  
\[SEC=16.75v_{c}^{-0.552}f^{-0.264}a_{p}^{-0.106} \tag{13}\]  
Similarly, the function expression of surface roughness \(Ra\) is shown in Equation (14):  
\[Ra=0.965v_{c}^{0.565}f^{0.4}a_{p}^{-0.035} \tag{14}\]Comparisons between the fitting value of the turning specific energy and surface roughness model and the test value are shown in Figures 5a and 5b, respectively. The correlation coefficient \(R\) is equal to 0.98, and \(R^{2}\) is equal to 0.97, between the fitting value of the turning specific energy and the actual value; regarding model significance, \(P=0.01<0.05\). For the surface roughness model, \(R=0.983\) and \(R^{2}=0.967\), while the model significance corresponds to \(P=0.02<0.05\). This means that the cutting specific energy calculated using Equations (13) and (14) has a strong linear relationship with the test value, indicating that the established model meets the fitting requirements.