## 5 Multiobjective Optimization of Main Beam Structure Based on RSM
### Multiobjective Optimization of Main Beam Structure
#### 5.2.3 Construction of Response Surface Model for Multiobjective Optimization of Main Beam  
According to the above optimization variables and model optimization requirements, the mathematical model of design variables, objective functions, and constraints was constructed, and the response surface model was built and solved. The mathematical representation of the design variables was as follows  
\[X=[x_{1},x_{2},x_{3},x_{4}]^{\top} \tag{23}\]  
where \(x_{i}\) represents the no. \(i\) design variable, \(i=1,\ 2,\ 3,\ 4\), corresponding to the parameter \(P_{1}-P_{4}\) of the main beam. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Max. Deformation**} & \multirow{2}{*}{**Max. Equivalent**} & \multirow{2}{*}{**Minimum Fatigue**} & \multirow{2}{*}{**First Order Natural**} & \multirow{2}{*}{**Quality (kg)**} \\  & & **(mm)** & & & & \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 10: True response value of sample point objective function. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**No. **} & **P1 (mm)** & **P2 (mm)** & **P3 (mm)** & **P4 (mm)** \\ \hline
1 & 25.0 & 25.0 & 30.0 & 30.0 \\
2 & 22.5 & 25.0 & 30.0 & 30.0 \\
3 & 27.5 & 25.0 & 30.0 & 30.0 \\
4 & 25.0 & 22.5 & 30.0 & 30.0 \\
5 & 25.0 & 27.5 & 30.0 & 30.0 \\
6 & 25.0 & 25.0 & 27.0 & 30.0 \\
7 & 25.0 & 25.0 & 23.0 & 30.0 \\
8 & 25.0 & 25.0 & 30.0 & 27.0 \\... &... &... &... &... \\
14 & 23,239 & 23,239 & 32.113 & 27.887 \\
15 & 26.760 & 23.239 & 32.112 & 27.887 \\
16 & 23.239 & 26.760 & 32.112 & 27.887 \\
17 & 26.760 & 26.760 & 32.112 & 27.887 \\
18 & 23.239 & 23.239 & 27.887 & 32.112 \\
19 & 26.760 & 23.239 & 27.887 & 32.112 \\
20 & 23.239 & 26.760 & 27.887 & 32.112 \\
21 & 26.760 & 26.760 & 27.887 & 32.112 \\
22 & 23.239 & 23.239 & 32.112 & 32.112 \\
23 & 26.760 & 23.239 & 32.112 & 32.112 \\
24 & 23.239 & 26.760 & 32.112 & 32.112 \\
25 & 26.760 & 26.760 The value range of design variables was expressed as:  
\[u_{i}\leq x_{i}\leq v_{i},\ i=1,\ 2,\ 3,\ 4 \tag{24}\]  
where \(u_{i}\) and \(v_{i}\) represent the lower and upper limit values of design variables, respectively, and their specific values are shown in Table 7.  
The optimization aimed to minimize the mass, equivalent stress, deformation, first-order natural frequency, and minimum fatigue life of the main beam. Its objective function was defined as  
\[ \begin{cases}f_{1}(x_{i})=\min S(x_{i})(i=1,2,3,4)\\ f_{2}(x_{i})=\min M(x_{i})(i=1,2,3,4)\\ f_{3}(x_{i})=\min D(x_{i})(i=1,2,3,4)\\ f_{4}(x_{i})=\max F(x_{i})(i=1,2,3,4)\\ f_{5}(x_{i})=\max L(x_{i})(i=1,2,3,4)\end{cases}  \tag{25}\]  
where \(S(x_{i})\) is the maximum equivalent stress of the main beam, \(M(x_{i})\) is the mass of the main beam, \(D(x_{i})\) is the maximum deformation of the main beam, \(F(x_{i})\) is the first-order natural frequency of the main beam, and \(L(x_{i})\) is the minimum service life of the main beam. The maximum equivalent stress of the main beam was less than the allowable stress of the material, so the constraint condition was expressed as  
\[\sigma(x_{i})\leq[\sigma],i=1,2,3,4 \tag{26}\]  
where \(\sigma(x_{i})\) is the maximum equivalent stress of main beam and \([\sigma]\) is the allowable stress of material. According to the above definition of the design variables, constraints and objective functions, the multiobjective optimization mathematical model of the main beam can be expressed as:  
\[ \begin{cases}&x=[x_{1},x_{2},x_{3},x_{4}]^{\top}\\ &f_{1}(x_{i})=\min S(x_{i})\\ &f_{2}(x_{i})=\min M(x_{i})\\ &f_{3}(x_{i})=\min D(x_{i})\\ &f_{4}(x_{i})=\max F(x_{i})\\ &f_{5}(x_{i})=\max L(x_{i})\\ &u_{i}\leq x_{i}\leq v_{i}(i=1,2,3,4)\\ &\sigma(x_{i})\leq[\sigma]\end{cases}  \tag{27}\]  
The response surface model represents the functional relationship between the sample point and its target function response value.