Verification of the Multi-Stage Optimization

The analysis of the convergence behavior of the multi-stage optimization environment is performed in two steps. In the first step, the simulation results of the successively applied hybrid ES-PS method are compared with the standard single-stage ES-PS method. In the second step, the use of SA as a preliminary stage to the ES-PS method is compared with the standard single-stage ES-PS method. This approach allows for evaluating separately the advantages of the second hybrid optimization stage and those of the pre-connected SA stage. For this comparison, the described exemplary design optimization is simulated 60 times with each of the different methods and the medians, means, dispersions, and variances of the resulting fitness functions are analyzed.

In Figure 5a, the distribution of the fitness values of the single-stage ES-PS method are plotted in comparison with the two-stage ES-PS method. In the single-stage method, all optimization parameters from Table 4 are varied simultaneously, and the number of generations is *g* = 50 and the population size is *p* = 200. In the two-stage method, the number of optimization parameters in each stage is less than in the one-stage method. Due to this resulting reduction of the solution space in the both stages, the number of generations and populations in the ES method, and thus the computation effort, can be reduced compared to the single-stage method. For the two-stage method, it is *g* = 20 and *p* = 50 in both stages. The reduction of the solution space also causes a reduction of the dispersion and variance, as can be seen in Figure 5a, with a smaller mean value and median. This shows that using the two-stage ES-PS method results in more stable convergence behavior while increasing the speed of convergence.

In Figure 5b, the results of a second single-stage ES-PS method are plotted in comparison to the combination of SA and the single-stage ES-PS method. The single-stage ES-PS method is performed with the same settings as in the previous comparison. This allows an indication of the robustness of the convergence behavior of this method. As shown in Figure 5a,b, the median, the mean value, the dispersion, and the variance are in a very similar range in both cases, suggesting a robust convergence behavior of the method. Combining this method with the SA results in a significant reduction of the median and mean value as shown in Figure 5b. The dispersion and variance also decrease when SA is used. This shows that the previously conducted global search using SA also improves the convergence behavior of the optimization compared with the standard single-stage ES-PS method. The computational effort increases with the use of SA. However, this can be counteracted by ANN as explained in the following section. Since the multi-stage optimization environment runs sequentially, an improvement in the convergence behavior by the SA and the two-stage ES-PS method compared to the single-stage one also means that the convergence behavior of the entire optimization environment is improved.
