## 5 Results and Discussions
### Numerical Analysis on Benchmark Functions
#### 5.3.4 Population Sizes Change  
Additionally, in this study, population size changes were observed while solving benchmark problems from SET-2 and SET-3 with 10 and 30 variables. Figure 10 shows the change of the COBRA-f, COBRA-fas and COBRA-SHA component population sizes during the optimization process on three functions from SET-2 with 10 variables, namely [PERSON]'s function (the first column), [PERSON]'s function (the second column) and [PERSON]'s function (the third column), with the best-found fuzzy-controller parameters. The figures on the first row demonstrate the original fuzzy-controlled COBRA-f tuning procedure behavior, the figures on the second row show the COBRA-fas modification, and finally the figures on the third row show the proposed COBRA-SHA approach. The behavior of these three tuning methods is quite different. The standard COBRA-f tends to give all resources to one component (which can be seen for [PERSON]'s and [PERSON]'s functions). However, for [PERSON]'s function, which is a complex optimization problem with many local minima, there was competition between the PSO and BA approaches for resources while the FFA component still had the biggest population size. The COBRA-fas modification demonstrated similar behavior for [PERSON]'s function (CSA had the largest amount of resources while FSS and BA competed for \"second place\"). However, for [PERSON]'s function all the components had population sizes with the number of individuals within the range [22; 23] during the optimization process. It should be noted that the same solutions were found by the COBRA-f and COBRA-fas approaches, but while COBRA-f was able to find a solution with 300 individuals throughout all populations, the COBRA-fas modification used only 133. Finally, the proposed algorithm COBRA-SHA increased all population sizes simultaneously (but differently): each population contained at least 20 individuals. Nevertheless, the largest amounts of resources were usually given to two components: for example, in the case of [PERSON]'s function the winners were the FFA and BA algorithms, but for [PERSON]'s function, they were the WPS and FSS algorithms. It should be noted that while solving [PERSON]'s problem by COBRA-SHA all components had 49-51 individuals in their populations. Figure 9: Results of the [PERSON] statistical test for SET-3. Next, Figure 11 shows the change of the COBRA-f, COBRA-fas and COBRA-SHA component population sizes during the optimization process on three functions from SET-2 with 30 variables, namely the Sphere function (the first column), the Quartic function (the second column) and [PERSON]'s function (the third column) with the best-found fuzzy-controller parameters. The algorithms demonstrate the same behavior as in the previous case (benchmark problems from SET-2 but with 10 variables). However, let us consider the optimization process while solving the Quartic problem with the COBRA-fas approach. First, the BA algorithm appeared to be the best choice for the fuzzy controller. Thus, it increased the BA's population to 40 individuals, when other populations had minimal sizes. After that, the population sizes did not change, and only after more than \(250,000\) calculations was the FFA approach able to improve the optimization process, with its population size starting to increase gradually. Therefore, in the end FFA had the largest amount of resources. Finally, Figure 12 shows the change in the COBRA-f, COBRA-fas and COBRA-SHA component population sizes during the optimization process on three functions from SET-3, namely Rotated Discus Function (the first column), Shifted and Rotated HGBat Function (the second column) and Shifted and Rotated Expanded Scaffer's F6 Function (the third column) with the best-found fuzzy-controller parameters. The first problem is unimodal, and the others are multi-modal. Figure 10: Population size changes for SET-2 with 10 variables. The standard COBRA-f tuning method usually makes multiple oscillations, but the winning component is changed over time.