## 4 Tested Cases
### Engineering Application  
In this section, an underwater vehicle base design problem is utilized to verify the effectiveness of the proposed method. The base is a braced structure for vibration devices in the hull of an \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Functions** & **Initial Sample Size** & **EI** & **WEI** & **LCB** & **PLCB** & **EW-LCB** \\ \hline \multirow{4}{*}{SA} & \(n=5d\) & 27.55/4 & 28.98/5 & 27.02/3 & 24.12/2 & **23.34/1** \\  & \(n=10d\) & 32.12/3 & 36.15/5 & 34.88/4 & 31.23/2 & **27.92/1** \\  & \(n=15d\) & 41.67/4 & 43.98/5 & 41.12/3 & 40.20/2 & **38.03/1** \\ \cline{1-1}  & \(n=5d\) & 393.6/3 & 524.5/5 & 513.4/4 & **142.5/1** & 173.2/2 \\ \cline{1-1}  & \(n=10d\) & 300.4/3 & 534.6/4 & 540.4/5 & **167.1/1** & 199.2/2 \\ \cline{1-1}  & \(n=15d\) & 403.4/3 & 525.4/4 & 536.5/5 & **166.6/1** & 206.1/2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of different initial sample points for functions SA and GF. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Metrics** & **EI** & **WEI** & **LCB** & **PLCB** & **EW-LCB** \\ \hline \multirow{2}{*}{Average rank} & FEmean & 3.70 & 4.00 & 3.22 & 3.10 & 1.10 \\  & FEstd & 2.50 & 3.40 & 2.90 & 3.70 & 2.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average ranking results for all AKBDO approaches considering all numerical cases. \begin{table}
\begin{tabular}{c c c} \hline \hline
**i** & **Hypothesis** & **\(p\)-Values** \\ \hline
1 & EI vs. EW-LCB & 0.0028 \\
2 & WEI vs. EW-LCB & 0.0001 \\
3 & LCB vs. EW-LCB & 0.0016 \\
4 & PLCB vs. EW-LCB & 0.0056 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The \(p\)-values obtained in the numerical examples by the difference significance test. underwater vehicle. The main capability of the base provides a platform for the installation of some imported vibration equipment and avoids the vibration transmitting to the hull directly. Meanwhile, the mechanical impedance of the base has a determination effect in reducing the level of noise. Specifically, the mechanical impedance is expected at a high level under all computational frequencies. The structural profile of the base adjoined to the hull of the underwater vehicle is depicted in Figure 6. The fixed structural and material parameters of the cylindrical shell and the base are listed in Table 7.  
In this work, the objective is to maximize the minimum difference of the impedance between the scheme in design and the required impedance value under the same frequency. Simultaneously, the weight of the optimized scheme should be less than that of the allowable one. Therefore, the optimization problem can be described as,  
\[ \begin{split} find\ \mathbf{x}=[x_{1},\ldots,x_{6}]\\ \max f(\mathbf{x})=\min\{I(\mathbf{x},\omega_{i})-I(\mathbf{x}_{0},\omega_{i} );i=1,2,\ldots,k\}\\ s.t.\ \ g(\mathbf{x})=W(\mathbf{x})-W(\mathbf{x}_{0})<0\end{split}  \tag{38}\]  
where \(\mathbf{x}\) represents the vector of the design variables, which is a six-dimensional vector.\(\omega_{i}\) is the \(i^{\text{th}}\) computational frequency. In detail, the design variables of this problem are the thickness of the panels of the base. Figure 7 plots the schematic diagram and Table 8 lists the meanings and value space of the design variables, respectively. \(I(\mathbf{x},\omega_{i})\) and \(W(\mathbf{x})\) represent the impedance value under a specific computational frequency and the weight of the base, respectively.\(I(\mathbf{x},\omega_{i})\) and \(W(\mathbf{x}_{0})\) are the required impedance value at the \(i^{\text{th}}\) computational frequency and the allowable base weight, respectively.