## 3 Numerical Results
### Design of Experiment--Hybrid Algorithm Parameters Optimization
#### 3.2.2 Multi-Island Genetic Algorithm and Evolution Strategy Results
Figure 16: The convergence of the proposed hybrid optimization algorithm (10 runs)—multi-island case. Figure 15: The optimized parameters for the hybrid algorithm (multi-island case). 2.3. Results Comparison of the Hybrid Algorithm vs. Genetic Algorithm with the Same Number of the Total Function Calls  
The proposed hybrid approach consists of two evolutionary algorithms which are called one after another. To justify this connection, which is rather uncommon, the authors decided to compare the proposed method with a single genetic algorithm, but with the increased number of allowed function calls--from 1600 to 2000 (i.e., equal to the total function calls used in the hybrid approach), to achieve the direct comparison between hybrid and purely genetic approach. The study was conducted for both--the single population and the multi-island version of the genetic algorithm, and similar to the previous examples, both settings were checked using 10 repetitions of the optimization process. As the limit of function calls is increased to match the one from the hybrid approach, two cases are examined in the study--one, in which the total number of generations is increased by 20%, and the second, in which the total number of individuals per population is increased by 20%. The overview of the resultant average best individuals (among 10\(\times\) repetitions) is shown in Table 4.  
The above results show that the hybrid approach outperformed the application of a single genetic algorithm, even though  
Implementation of the Tuned Hybrid Optimization Algorithm Parameters to the FEM-Based Constrained Mass Minimization Problem  
The last part of the study was the verification of the newly obtained optimization scenario in the FEM-based mass minimization problem with limit load constraint. Both of the tuned hybrid algorithms were examined, as the actual simulation-based coarse response may differ from the smooth metamodel-based study. Additionally, for comparison purposes, the authors ran the optimization with parameters corresponding to the lowest efficiency of the hybrid approach, to determine whether this trend will be correctly captured by the FEM-based optimization. The convergence of all of the analyzed cases is shown below. To minimize the effect of randomness, each FEM-based study was repeated three times. As all algorithms have a different number of individuals and generations, the only common feature is the number of function calls (individuals calculated); therefore, it was chosen as the X-scale in Figure 19.  
Figure 12:0 The convergence of the FEM-based optimization: H_SP_GA—hybrid approach with a single population genetic algorithm, H_MI_GA—hybrid approach with a multi-island genetic algorithm, and H_WORST—hybrid approach with properties corresponding to the least-performing algorithm in accordance with the metamodel-based study. Figure 12:1 The convergence of the multi-island genetic algorithm with 2000 function calls. The resultant designs of the bracket being optimized differ in terms of the final shape and mass. The average values of the best individuals after each stage of the hybrid optimization are listed in Figure 12:2  
The resultant shape of the bracket cannot be averaged; therefore, the best designs from the H_SP_GA and H_MI_GA are shown together (Figure 20) with one of the resultant H_WORST designs. It can be seen, that for the two first cases, the bracket design is similar (thickness, general contour), with some differences observed mostly in the lower section. On the other hand, the design from the H_WORST algorithm (which is the least performing) is overly heavy, with a very thick lower part connected with the upper region via an hourglass-shaped contour, which from a strength perspective should be avoided.