## 3 Results and Analysis
### Simulation Experiment and Result Analysis
#### 3.3.2 Performance Comparison in Different Dimensions
**} & \multirow{2}{*}{**Algorithm**} & \multicolumn{3}{c}{**D = 10**} & \multicolumn{3}{c}{**D = 100**} & \multicolumn{3}{c}{**D = 200**} \\ \cline{3-10}  & & **Mean** & **Standard** & **Time** & **Mean** & **Standard** & **Time** & **Mean** & **Standard** & **Time** \\ \hline \multirow{4}{*}{\(f_{1}\)} & MDBAS & \(4.5\times 10^{-32}\) & \(1.91\times 10^{-32}\) & \(1.008\) & \(9.16\times 10^{2}\) & \(3.98\times 10^{2}\) & 3.481 & \(4.91\times 10^{4}\) & \(7.16\times 10^{3}\) & 7.045 \\  & AFS & \(0.627\) & \(0.154\) & \(94.87\) & \(3.08\times 10^{2}\) & 51.4 & 149.8 & \(4.43\times 10^{3}\) & \(5.14\times 10^{2}\) & 213.4 \\  & AFS-MDBAS & \(8.37\times 10^{-28}\) & \(2.28\times 10^{-27}\) & 13.25 & \(7.51\) & 3.64 & 68.92 & \(2.0\times 10^{3}\) & \(2.99\times 10^{2}\) & 82.17 \\  & AFS-MMSBAS & \(\mathbf{4.5\times 10^{-106}}\) & \(\mathbf{4.99\times 10^{-106}}\) & 17.27 & **0.00814** & **0.00454** & 72.97 & **19.1** & **4.67** & 81.85 \\ \hline \multirow{4}{*}{\(f_{2}\)} & MDBAS & \(4.62\times 10^{2}\) & \(9.69\times 10^{2}\) & 1.341 & \(3.92\times 10^{6}\) & \(4.23\times 10^{6}\) & 4.911 & \(5.21\times 10^{8}\) & \(1.58\times 10^{6}\) & 7.65 \\  & AFS & \(\mathbf{67.3}\) & \(\mathbf{18.6}\) & \(105.5\times 10^{2}\) & \(3.57\times 10^{4}\) & \(2.29\times 10^{4}\) & 166.4 & \(1.84\times 10^{4}\) & \(3.99\times 10^{5}\) & 253.7 \\  & AFS-MDBAS & \(1.25\times 10^{2}\) & \(3.12\times 10^{2}\) & 4.46 & \(1.74\times 10^{3}\) & \(1.23\times 10^{3}\) & 83.17 & \(1.58\times 10^{6}\) & \(6.05\times 10^{3}\) & 130.7 \\  & AFS-MMSBAS & \(90.2\times 10^{2}\) & \(8.071\) & \(\mathbf{4.08\times 10^{2}}\) & \(\mathbf{3.99\times 10^{2}}\) & 86.12 & \(\mathbf{5.18\times 10^{3}}\) & \(\mathbf{1.69\times 10^{3}}\) & 133.7 \\ \hline \multirow{4}{*}{\(f_{3}\)} & MDBAS & \(0.0659\) & \(0.454\) & \(1.333\) & \(3.19\times 10^{9}\) & \(2.25\times 10^{10}\) & 4.979 & \(1.94\times 10^{7}\) & \(7.36\times 10^{7}\) & 7.355 \\  & AFS & \(2.01\) & \(0.345\) & \(131.9\times 10^{2}\) & 3.56 & 130.1 & \(2.67\times 10^{2}\) & 5.52 & 180.0 \\  & AFS-MDBAS & \(\mathbf{4.99\times 10^{-14}}\) & \(\mathbf{1.26\times 10^{-13}}\) & 15.71 & 38.1 & 4.93 & 23.63 & \(1.41\times 10^{2}\) & 9.85 & 332.26 \\  & AFS-MMSBAS & \(0.0925\) & \(0.0823\) & 29.39 & 2.74 & **0.92** & 32.33 & **10.3** & **2.19** & 42.11 \\ \hline \multirow{4}{*}{\(f_{4}\)} & MDBAS & \(0.195\) & \(0.118\) & \(1.561\) & 23.3 & 4.8 & 4.958 & \(2.59\times 10^{2}\) & 48.0 & 7.638 \\  & AFS & \(0.388\) & \(0.142\) & \(134.5\times 10^{2}\) & 71.1 & 18.5 & 176.7 & \(3.21\times 10^{2}\) & 81.9 & 317.8 \\  & AFS-MDBAS & \(0.196\) & \(0.0986\) & \(9.239\) & 13.6 & 3.47 & 15.24 & 64.1 & 11.5 & 28.99 \\  & AFS-MMSBAS & \(\mathbf{0.0361}\) & \(\mathbf{0.0244}\) & 13.04 & **1.18** & **0.311** & 20.11 & **3.43** & **0.7**  
At high dimensions (D = 100, 200), the averages of the MDBAS algorithm and the AFS algorithm are far greater than the optimal value, and the standard deviation is also large, indicating that the optimization ability and stability of these two algorithms are poor. For most functions, the averages and standard deviations of the AFS-MDBAS hybrid algorithm are better than those of the separate MDBAS and AFS algorithms, which shows that the hybrid algorithm improves the optimization ability and stability of the MDBAS and AFS algorithms. The AFS-MMSBAS algorithm is an improvement of the AFS-MDBAS algorithm. Although the algorithm is not the best at optimizing the functions f7, f9, f10, and f11, it is not far behind the algorithm with the best performance. In general, the AFS-MMSBAS algorithm proposed in this paper has better optimization ability and stability with respect to high-dimensional problems. Finally, under the same number of iterations, it can be seen that the running time of the algorithm increases with the increase in dimensions. Moreover, whether at low or high dimensions, the hybrid algorithm greatly shortens the running time compared with the AFS algorithm. However, compared with the BAS algorithm, it greatly increases the running time.