## 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
Its general formula is shown as follows  
\[y=\beta_{0}+\sum_{i=1}^{n}\beta_{i}x_{i}+\sum_{i=2}^{n}\sum_{j=1}^{i-1}\beta_ {i}x_{i}x_{j}+\sum_{i=1}^{n}\beta_{i}x_{i}^{2}+\varepsilon \tag{28}\]  
where \(y\) is the objective function, \(\beta\) is the regression equation coefficient, \(x_{i}\) is the design variable, \(n\) is the number of design variables, and \(\varepsilon\) is the error term. In the multiobjective optimization of the main beam, the response outputs of the design variables \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) were, respectively, the maximum deformation \(D(x)\), the maximum equivalent stress \(S(x)\), the minimum fatigue life \(L(x)\), the first-order natural frequency \(F(x)\), and the mass \(M(x)\) of the main beam. Therefore, the test had four factors and five responses, and the response surface model was  
\[ \begin{split} y(x)=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta _{3}x_{3}+\beta_{4}x_{4}+\beta_{11}x_{1}^{2}+\beta_{11}x_{1}^{2}+\beta_{22}x_{ 2}^{2}+\beta_{33}x_{3}^{2}+\beta_{44}x_{4}^{2}+\beta_{12}x_{1}x_{2}+\beta_{13} x_{1}x_{3}\\ +\beta_{14}x_{1}x_{4}+\beta_{23}x_{2}x_{3}+\beta_{24}x_{2}x_{4}+\beta_{34}x_ {3}x_{4}+\varepsilon\end{split}  \tag{29}\]The above formula can be expressed in matrix as follows  
\[y(x)=\alpha^{T}\beta+\varepsilon \tag{30}\]  
where the definition of \(\alpha\) and \(\beta\) are as follows  
\[\alpha=[1,x_{1},x_{2},x_{3},\ldots,x_{3}x_{4}] \tag{31}\]  
\[\beta=[\beta_{0},\beta_{1},\beta_{2},\beta_{3},\ldots,\beta_{34}]^{T} \tag{32}\]  
When there is \(k\) groups of tests, Formula (30) is expressed as follows  
\[Y=X\cdot B+\varepsilon \tag{33}\]  
In Formula (33), matrix \(B\) is an unknown matrix representing the relationship between an independent variable and a dependent variable. The unknown matrix B [46] was calculated by the least square method, and its calculation formula was as follows  
\[B=\left(X^{T}X\right)^{-1}X^{T}Y \tag{34}\]  
The response surface models of the maximum deformation \(D(x)\), the maximum equivalent stress \(S(x)\), the minimum fatigue life \(L(x)\), the first-order natural frequency \(F(x)\), and the mass \(M(x)\) of the main beam were obtained by calculating the data of sample points and their response values in Tables 9 and 10 using the above method, as shown in Formulas (35), (36), (37), (38), and (39), respectively. \[ \begin{array}{rl}D(x)=22.88-&8.14\times 10^{2}x_{1}-9.91\times 10^{2}x_{2}-0.505 x_{3}-0.33x_{4}+4.03\times 10^{4}x_{2}-2.34\times 10^{-17}x_{1}x_{3}\\ &-2.79\times 10^{-17}x_{1}x_{4}-3.36\times 10^{4}x_{2}x_{3}-3.36\times 10^{4}x_{ 2}x_{4}-3.36\times 10^{4}x_{3}x_{4}+1.34\times 10^{3}x_{1}^{2}\\ &+2.14\times 10^{3}x_{2}^{2}+5.37\times 10^{3}x_{3}^{2}+3.15\times 10^{3}x_{4}^{2} \end{array}  \tag{35}\]  
\[ \begin{array}{rl}S(x)=6.06\times&10^{3}-42.01x_{1}-47.26x_{2}-89.52x_{3}-2. 14\times 10^{2}x_{4}-0.12x_{1}x_{2}+0.11x_{1}x_{3}-0.11x_{1}x_{4}\\ &+0.11x_{2}x_{3}-5.91\times 10^{-2}x_{2}x_{4}+0.68x_{3}x_{4}+0.89x_{1}^{2}+0.97x_{2}^{ 2}+0.99x_{3}^{2}+31.82x_{4}^{2}\end{array}  \tag{36}\]  
\[ \begin{array}{rl}L(x)=-4.57\times 10^{6}+2.59\times 10^{4}x_{1}+3.45\times 10^{4}x _{2}+6.02\times 10^{4}x_{3}+1.88\times 10^{5}x_{4}+1.62\times 10^{2}x_{1}x_{2} \\ &-1.31\times 10^{2}x_{1}x_{3}+1.71\times 10^{2}x_{1}x_{4}-1.38\times 10^{2}x_{2}x_{3} +71.96x_{2}x_{4}-31.35x_{3}x_{4}\\ &-6.16\times 10^{2}x_{1}^{2}-7.29\times 10^{2}x_{2}^{2}-8.19\times 10^{2}x_{3}^{2}-3.11\times 10^{3}x_{4}^{2}\end{array}  \tag{37}\]  
\[ \begin{array}{rl}F(x)=7.62-1.79\times&10^{-2}x_{1}+2.21\times 10^{-2}x_{2}+ 0.67x_{3}+0.54x_{4}+1.81\times 10^{-3}x_{1}x_{2}+1.68\times 10^{-4}x_{1}x_{3} \\ &+1.68\times 10^{-4}x_{1}x_{4}--1.68\times 10^{-4}x_{2}x_{3}+5.01\times 10^{-4}x_{ 2}x_{4}-3.51\times 10^{-3}x_{3}x_{4}\\ &-2.58\times 10^{-3}x_{1}^{2}-3.38\times 10^{-3}x_{2}^{2}-7.34\times 10^{-3}x_{3}^{2} -6.79\times 10^{-3}x_{4}^{2}\end{array}  \tag{38}\]  
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