# High-Fidelity Surrogate Based Multi-Objective Optimization Algorithm
## 6 Results and Discussion  
Multiple benchmark multi-objective optimization problems and a real-world engineering design challenge were used to test the proposed technique. Promising outcomes were gathered using the suggested algorithm. There were numerous computational optimization runs (approximately five runs). As noted earlier, Table 2 presents samples of the outcomes from five separate runs. The actual, real-world example (geometry of the airfoil in a wind turbine) required between 986 and 1172 iterations, with 242 serving as the median. The median number of function evaluations is 1213, with a range of 16,582 to 19,934. RBF surrogate model is the winner in the majority of test runs. The median number of convergent Pareto points, which varies from 729 to 793, is 760. A comparison of the complete airfoil geometry was carried out, as seen in Figure 3. Solid lines depict the geometry of the initial (Datum) unoptimized airfoil. The unoptimized airfoil geometry is shown by the continuous line, while the airfoil shape generated by the AMSP optimizer is shown by the dashed line. The most obvious inference from this graph is that the geometry of the airfoil in the second half of the chord, from maximum thickness to TE, has been significantly affected by the optimization process. An enhanced airfoil shape is produced when three surrogates are combined into an ensemble. This indicates better pressure distribution and greatly increases the lifting force and lowers the drag force of the turbomachinery blades. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & F1 & & & & & F2 & & & \\ \hline Run & Ite. & Eval & \#Par.P & Surr. & RMSE & Run & Ite. & Eval & Par.P & Suur. & RMSE \\ \hline
1 & 37 & 75 & 48 & RBF & 0.9901 & 1 & 24 & 246 & 137 & RBF & 0.9721 \\
2 & 32 & 61 & 45 & RSF & 0.9899 & 2 & 22 & 281 & 188 & KRG & 0.9704 \\
3 & 23 & 56 & 35 & KRG & 0.9879 & 3 & 23 & 275 & 167 & RBF & 0.9569 \\
4 & 30 & 66 & 43 & RBF & 0.9744 & 4 & 21 & 458 & 309 & RSF & 0.9565 \\
5 & 33 & 71 & 47 & RSF & 0.9904 & 5 & 20 & 410 & 357 & RBF & 0.9777 \\ \hline \multicolumn{10}{c}{T1} & \multicolumn{10}{c}{T2} \\ \hline Run & Ite. & Eval & \#Par.P & Surr. & RMSE & Run & Ite. & Eval & Par.P & Suur. & RMSE \\ \hline
1 & 19 & 446 & 363 & RBF & 0.9812 & 1 & 27 & 58 & 39 & RSF & 0.9744 \\
2 & 19 & 443 & 358 & RSF & 0.9963 & 2 & 29 & 59 & 37 & RBF & 0.9730 \\
3 & 16 & 368 & 306 & KRG & 0.9811 & 3 & 34 & 73 & 44 & KRG & 0.9773 \\
4 & 17 & 379 & 313 & RSF & 0.9792 & 4 & 28 & 65 & 44 & RSF & 0.9302 \\
5 & 15 & 332 & 264 & RBF & 0.9971 & 5 & 38 & 87 & 46 & RBF & 0.9565 \\ \hline \multicolumn{10}{c}{T3} & \multicolumn{10}{c}{P1} \\ \hline Run & Ite. & Eval & \#Par.P & Surr. & RMSE & Run & Ite. & Eval & Par. P & Suur. & RMSE \\ \hline
1 & 250 & 1216 & 760 & RSF & 0.9325 & 1 & 1172 & 18,857 & 3464 & RSF & 0.9772 \\
2 & 234 & 1213 & 765 & RBF & 0.9355 & 2 & 1218 & 19,934 & 3615 & RSF & 0.9821 \\
3 & 273 & 1294 & 793 & RBF & 0.9781 & 3 & 986 & 16,582 & 3146 & RSF & 0.9763 \\
4 & 232 & 1170 & 752 & RSF & 0.9932 & 4 & 1019 & 19,121 & 3623 & RBF & 0.9775 \\
5 & 219 & 1172 & 729 & RBF & 0.9946 & 5 & 997 & 18,587 & 3455 & RBF & 0.9801 \\ \hline \hline \end{tabular} \# Par. P = Number of Pareto points.  
\end{table} 
Table 2: Test results of the introduced algorithm on multi-objective test functions. Figures 4-7 present the performance space and the evaluated points for test functions F1, F2, T1, T2, and T3 respectively. Figure 4a displays the sampling points coordinates in the feasible design space, Figure 4b shows sampling design at which the sampling design were evaluated using the objective functions. Figure 4c displays the Pareto design coordinates and Figure 4d displays the Pareto frontier after all Pareto coordinates were converged iteration after iteration to form the Pareto frontier. The same can be seen in Figures 5-8 for test functions F1, F2, T1, T2, and T3.