## 2 Multi-Objective Optimization Modeling in Material Cutting Stage of CNC Lathe  
### Modeling Process  
The complete machining process of a CNC lathe can be divided into a standby stage, a spindle-idling stage, a tool-carrier-moving stage, and a cutting material stage. Depending on the source of energy consumption, the energy consumption of each stage is related to the working power of the parts involved in the work. The power of each stage is expressed as the standby power \(P_{standby}\), the idle power of the main shaft \(P_{idle}\), the moving power of the tool holder \(P_{tool}\), and the cutting material power \(P_{cutting}\). The power compositions of each stage are shown in Figure 1.  
It can be seen that the standby and no-load phases primarily involve the operation of auxiliary systems and the rotation of the spindle. After stable operation, \(P_{standby}\) and \(P_{idle}\) are basically fixed values, and the optimization space is small [16]. Although the tool resting stage occurs throughout the entire turning process, the power of the feed shaft motor driving the tool rest movement is low, and \(P_{tool}\) is proportional to the feed rate, so optimization is of little significance. \(P_{cutting}\) changes dramatically during the cutting material phase, wherein the power is directly influenced by the turning force \(F_{c}\). In case of cutting, \(F_{c}\) is related to the cutting speed \(v_{c}\) (m/min), feed rate \(f\) (mm/r), and depth of cut \(a_{p}\) (mm). Therefore, the power \(P_{c}\) when turning materials can be expressed as follows:  
\[P_{c}=P_{cutting}-P_{tool}=F_{c}v_{c}=C_{F}v_{c}^{x}f^{y}a_{p}^{z}K_{F} \tag{1}\]  
Here, \(C_{F}\), \(x\), \(y\), \(z\), and \(K_{F}\) are coefficients related to machine tools and materials [17].  
The specific energy of cutting (\(SEC\)), an index used to measure processing energy consumption [17], is expressed by Formula (2):  
\[SEC=\frac{E}{V} \tag{2}\]  
Figure 1: Power compositions of each stage of machine tool processing.  
where \(E\) is the total energy consumption of the cutting material, and \(V\) is the volume of material removed.  
As can be seen from the above equation, if the cutting specific energy in a machining operation is lower, less energy will be consumed when cutting the same volume of material because the volume of processed material is not exactly the same in actual processing. Therefore, power cannot be used as the energy consumption index. Therefore, the energy consumption per unit volume of material, as indicated by the specific energy of cutting (\(SEC\)), was adopted as the energy consumption index. Under a certain processing amount, the lower the cutting energy, the less electrical energy consumed by the machine tool.  
By dividing the numerator and the denominator of Equation (3) by the processing time \(t\) at the same time, we obtain  
\[SEC=\frac{E/t}{V/t}=\frac{P_{c}}{MRR} \tag{3}\]  
where \(MRR\) (mm\({}^{3}\)/s) is the material removal rate, indicating the volume of material removed per second, and is expressed by Equation (4) as follows:  
\[MRR=\frac{1000v_{c}fa_{p}}{60} \tag{4}\]  
By combining Equations (1), (3), and (4), an expression in the form of the cutting specific energy index can be obtained, as shown in Equation (5):  
\[SEC=Kv_{c}^{m}f^{n}a_{p}^{r} \tag{5}\]  
Here, \(K=\frac{60C_{F}K_{F}}{1000}\), \(m=x-1\), \(n=y-1\), and \(r=z-1\).  
In addition to energy consumption, machining quality is also a goal that producers focus on, and surface roughness is commonly used in engineering to measure machining quality [17]. The complex relationship between surface roughness and cutting parameters can be approximated in exponential form, as shown in Equation (6):  
\[Ra=k_{0}v_{c}^{k_{1}}f^{k_{2}}a_{p}^{k_{3}} \tag{6}\]