## 4 Multi-Objective Optimization Design
### Design Results of MOOD Problem  
Building upon the RSM model established in Section 2.3.1, the MOOD takes the minimum displacement response and the minimum acceleration response as its objectives and formulates the modal problem of the small-radius curved tunnel under the action of seismic excitation as:  
\[\left\{ \begin{array}{c}Max\ D=\sqrt{d_{S}\times d_{A}}\\ s.t.\ 10\ \mathrm{m}\leq depth\leq 20\ \mathrm{m}\\ 45\leq angle\leq 60\\ 300\ \mathrm{m}\leq R\ \leq 350\ \mathrm{m}\end{array} \right. \tag{2}\]  
\[d_{s}=\left(\frac{s(depth,angle,R)-s^{L}}{s^{U}-s^{L}}\right)^{W_{1}} \tag{3}\]  
\[d_{a}=\left(1-\frac{a(depth,angle,R)-a^{L}}{a^{U}-a^{L}}\right)^{W_{2}} \tag{4}\]  
where, \(s^{U},s^{L}\) and \(a^{U},a^{L}\) denote the upper and lower bounds for the displacement response and acceleration response, respectively. \(W_{1}\) and \(W_{2}\) are the weighting coefficients around the displacement response and acceleration response, respectively. The multi-objective optimization problem is solved by applying an optimization algorithm to vary the buried depth and curve angle to calculate the optimal configuration of the small-radius curved tunnel through the Design-Expert software. For both objectives, the same importance parameter (\(r\)) is set. After many calculations, it is more reasonable that a displacement objective weight of 5 and an acceleration objective weight of 4 were specified within the weight setting range, to achieve an emphasis on the displacement objective. The Design-Expert software can search for one or more points in the factor domain depending onthe objective optimization characteristics to maximize the value of the objective function. Figure 9 shows the relationship between the expectation function and the design variables of buried depth and curve angle. It can be seen that the value of the objective function increases with increasing buried depth: when the buried depth reaches 10-17 m, the expected value increases slowly, and when the buried depth is 17-20 m, the expected value increases sharply. The expected value of the expectation function increases as the angle decreases, but the overall rate of increase is slow. Table 9 gives the geometrical parameters values of point A of the small-radius curved tunnel under seismic excitation and with minimum displacement and acceleration conditions. In Table 9, the geometrical parameters that meet the requirements of the multi-objective optimization are the maximum buried depth, the minimum curve angle and the maximum curve radius within the domain. The expected value of the results is 0.859, which is generally in line with expectations. The relationship between the B-point expectation function and the design variables of buried depth and curve angle of curved tunnel is shown in Figure 10. It can be seen from Figure 10 that the expected value of the expectation function increases with increasing buried depth: when the angle is taken as 45-52\({}^{\circ}\), the expected value increases with the increase in buried depth, and the function value of the expectation function decreases slightly when the buried depth is 19 m-20 m. When the angle is taken as 52-60\({}^{\circ}\), the expectation increases monotonically with increasing buried depth. Once both the buried depth and the angle are taken to their maximum values, the expected value of the expectation function reaches its maximum value, which is 0.725. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Depth (m)** & **Angle (deg)** & **\(R\) (m)** & **\(s\) (m)** & **\(a\) (m/s\({}^{2}\))** & **Expected Value** \\ \hline
20.000 & 45.002 & 350.000 & 0.114 & 2.363 & 0.859 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Design-Expert optimization design results for point A.  
Figure 9: Expectation functions and design variables in relation to surface for point A. Table 10 gives the values of the geometrical parameters for the small-radius curved tunnel under seismic excitation and with conditions of minimum displacement and acceleration.