## 3 Numerical Results
### Design of Experiment--Hybrid Algorithm Parameters Optimization
#### 3.2.2 Multi-Island Genetic Algorithm and Evolution Strategy Results  
Once the genetic operators and penalty function parameters were fixed, the second study, this time on the multi-island version of the genetic algorithm, was conducted. Again, the analysis of the fitted full-quadratic response surface shows the dependencies between the factors and resultant objective function values (Figure 13). A strongly nonlinear effect can be seen in all factors besides the first parameter--the number of autonomous populations (islands). On the other hand, this parameter has the strongest effect on the objective function value, despite being almost linear in its nature. Combining the results from the first and second study, one may conclude, that for this particular example, the total number of function calls is severely limited (in accordance with the given calculation time bounds); therefore, introducing multiple autonomous populations strongly limits the number of generations (or the number of individuals per each population). In effect, this diminishes the chance of converging to the optimum values, due to insufficient evolution of the population(s). Specifically, the desired evolution could be obtained only once the total number of generations or individuals (or both) is increased. Figure 11: The optimized parameters for the hybrid algorithm (single-island case). Figure 12: The convergence of the proposed hybrid optimization algorithm (10 runs)—single-population case. Nevertheless, this study delivered multiple interesting observations regarding the mutual interactions between the studied factors (see Figure 14 for details). First, the most significant interaction (statistically) is the one between the number of autonomous populations (islands) and the migration rate. For a relatively small rate of migration (5%), increasing the number of islands helped in the attainment of a lower resultant objective function value. For higher values, i.e., 12.5% and 20%, more islands were an obstacle to the efficient minimization of the objective function. The second most important interaction is the one between the number of generations and the interval of migration between the islands. It not only shows the strongly nonlinear  
Figure 14: The interactions between considered factors for the MIGA + EVOL case. Figure 13: The main effect plot for the hybrid optimization using MIGA + EVOL approach (mean value of the fitted RSM shown in the Y axis). relationship between the factors, but also the opposite trend for extreme values of the parameters. For frequent migration (interval = 4), the population with a higher number of individuals and a lower number of generations is preferred. On the other hand, when the migration interval is increased to 20, a significantly higher number of generations is needed to minimize the objective function. Once again, based on the fitted quadratic response surface, the authors adjusted the considered parameters to produce the lowest average objective function utilizing only a limited number of simulations. The results of this optimization are presented in Figure 15. The most efficient optimization is anticipated when high randomization is introduced to the population, i.e., there are only two islands, each with forty randomly generated individuals, and the migration occurs for 20% of the individuals per every four generations (with forty generations in total). This produces a very diverse population, as gene exchange is frequent and broad. Similar to the single-population version of the evolution algorithm study, the convergence of 10 runs is shown in Figure 16. This time, the average objective function is 0.285, with min and max values of 0.266 and 0.305, which in total yields a lower spread, compared to the previous study. As the average objective functions differ only by 0.001, both algorithms may be considered equally efficient within the considered scenario, with the multi-island approach having a more narrow span of the resultant best-fitted individuals.