# High-Fidelity Surrogate Based Multi-Objective Optimization Algorithm
## 6 Results and Discussion
It is quite noticeable that the proposed algorithm successfully converged to optimal Pareto set and successfully identified the Pareto frontier for all tested benchmark multi-objective functions and most importantly for the practical problem, which is the wind turbine blade shape/profile optimization. The algorithm was capable of identifying the Pareto frontier for a real-life practical problem, seen in Figure 8, with three objective functions and 40 design variables. By identifying the optimal design variables \(x_{1},\ y_{1},\ldots,x_{10},\ y_{10}\), the optimal airfoil profile was generated that minimizes the loss values \(L_{1}\), \(L_{2}\), and \(L_{3}\). Figure 4: Computational simulations Pareto frontier determination of test function F1 (**a**) coordinates of sampling points; (**b**) sampling points in feasible design space; (**c**) coordinates of Pareto points; (**d**) the seeking Pareto frontier. Figure 3: Airfoil shape design before and after optimization process. Figure 5: Computational simulations Pareto frontier determination of test function F2 (**a**) coordinates of sampling points; (**b**) sampling points in feasible design space; (**c**) coordinates of Pareto points; (**d**) the seeking Pareto frontier. Figure 6: Computational simulations Pareto frontier determination of test function T1 (**a**) coordinates of sampling points; (**b**) sampling points in feasible design space; (**c**) coordinates of Pareto points; (**d**) the seeking Pareto frontier. Table 3 reports the total number of iterations, the total number of objective function evaluations, and the total number of Pareto points in the ideal Pareto frontier. The efficient Pareto set was successfully identified with a reasonable number of Pareto set points and a low computing cost using RBF, which is devoted to the sample points, and Kriging, which interpolates the sample points and gives good accuracy. Table 3 displays the variation range, median, and Pareto set points for the iterations, function evaluations, and number of iterations. Figure 8: Computational simulations Pareto frontier determination of test function T3 (**a**) coordinates od sampling points; (**b**) sampling points in feasible design space; (**c**) coordinates of Pareto points; (**d**) the seeking Pareto frontier. Figure 7: Computational simulations Pareto frontier determination of test function T2 (**a**) coordinates od sampling points; (**b**) sampling points in feasible design space; (**c**) coordinates of Pareto points; (**d**) the seeking Pareto frontier. For each test problem, more than five runs have been completed because LHD creates sample points in the region of interest at random. The best outcome from those several runs is represented by these results. For five distinct runs, the quantity of iterations, total quantity of evaluated points, and quantity of converged frontier points were noted. It worth mentioning that the task can be completed in a relatively small number of iterations. All the evaluated points are plotted together with the feasible performance space, as shown in Figures 4-8 to better observe the accuracy of converged frontier points. The converged frontier points are extremely close to the genuine Pareto frontier, as can be seen by the graphs. Benchmark test functions were used to compare the proposed algorithm's performance to that of other MOO algorithms. Table 4 shows that AMSP outperforms the other MOO techniques, such as Pareto Set Pursuing (PSP) and Multi-objective Optimization Genetic Algorithm (MOGA). In terms of the number of fitness evaluations, AMSP outperformed the other algorithms in terms of efforts and resources needed. In addition, the cost of computation was lowered reflected in reduction of number of function evaluations and computation time. Furthermore, when AMSP was utilized, more Pareto set points were obtained. As a result, a more accurate and smoother Pareto border was identified.