## 4 Conclusions  
The conducted study has shown, that the usage of metamodels may greatly increase the performance of the optimization algorithms. The authors were able to tune the optimization parameters to increase the efficiency of the proposed algorithms by respectively 12% and 5% for single- and multi-island versions of the hybrid algorithm (comparing the resultant optimal average objective function to the one obtained as an average for all of the analyzed runs). Moreover, the proposed approach enables reducing the total number of parameter combinations to study, as the experiments are divided into single- and multi-population implementation of the genetic algorithm separately (however, sequentially). To conduct this study, one does not necessarily require an excessive sampling plan for the metamodel training phase. The authors have shown, that even for a significant number of variables, the performance of the metamodels trained on smaller data sets is mildly decreased (R\({}^{2}\) drop of 10% for the response \(F_{LIMIT}\)) or almost the same (R\({}^{2}\) reduced by 2% for response Mass) compared to the four times bigger sampling plan \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Average Best** & **Single Population** & **Multi-Island** & **Worst Case** \\
**Individual** & **GA + EVOL** & **GA + EVOL** & **GA + EVOL** \\ \hline After GA & 0.358 & 0.318 & 0.385 \\ After GA + EVOL & 0.318 & 0.301 & 0.380 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the average best individual objective function values obtained from different algorithm parameters using FEM-based optimization. Figure 20: The comparison of the final design mass and shape: 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. used in the considered work. Nonetheless, the performance of the metamodel is strongly dependent on the underlying nature of the problem. The authors presented the nonlinear buckling example, which included both geometrical and material nonlinearities, making the constraint function extremely sensitive to the design variables. Compared to other studies described in the literature, this one handles both the evolutionary operators as well as penalty-related ones (as the constrained-optimization problem is considered). This can lead to surprising conclusions, such as the fact that a proper penalty function may have a more significant impact on the optimization efficiency, than the number of individuals or generations of the genetic algorithm. However, this observation may be problem-related, as the authors considered a fixed amount of objective function calls, which may not guarantee the attainment of the global optimum. Yet, in real-world engineering optimization problems, one cannot find a single algorithm that will assure this global optimum each time. Nonetheless, the presented method of parameters tuning is universal in its nature and may be adopted in almost all types of optimization algorithms. In stochastic processes, such as in genetic algorithms and evolution strategies, it is beneficial to increase the number of algorithm repetitive runs, which is relatively costless when working on quick-to-run metamodels rather than the actual FEM-based or other time-consuming simulations. In this case, the noise associated with the natural spread of evolutionary methods is mitigated, allowing for drawing valid conclusions regarding the algorithm optimization process. Finally, tests performed on the actual FEM-based optimization with the proposed hybrid approach show, that the conclusions withdrawn from the metamodel-based study are valid and they indeed helped in increasing the efficiency of the optimization process.