## 5 Results and Discussions
### Numerical Analysis on Benchmark Functions
#### 5.3.3 Numerical Results for SET-3  
The next step was to test and compare the stated biology-inspired algorithms and their modifications by using benchmark functions from SET-3. The 16 functions with \(D=30\) used in SET-3 were taken from the CEC 2014 competition [51]. All these functions are minimization problems with a shifted and rotated global optimum, which is randomly distributed in \([-80,80]\). The search range for all problems was \([-100,100]\). The statistical results in terms of mean, standard deviation and best solution of different algorithms for functions from CEC 2014 are listed in Table 10. The best results are shown in bold. From Table 7, it can be observed that the COBRA-SHA approach in most cases performs better than the other algorithms included in the comparison in terms of the mean value. To be more specific, this happened for the first three unimodal functions \(f_{1}\), \(f_{2}\) and \(f_{3}\). Moreover, for function \(f_{2}\) COBRA-SHA outperformed the other algorithms by all criteria. However, for \(f_{1}\) and \(f_{3}\) the best results (out of 51 program runs) were found by the COBRA-fas approach. Figure 6: Results of the [PERSON] statistical test for SET-2 (\(D=10\)). Figure 7: Results of the [PERSON] statistical test for SET-2 (\(D=30\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(f\) & \multicolumn{2}{c}{ISO} & \multicolumn{2}{c}{WFS} & \multicolumn{2}{c}{FSS} & \multicolumn{2}{c}{CA} & \multicolumn{2}{c}{FFA} & \multicolumn{2}{c}{ESA} & \multicolumn{2}{c}{FFA} & \multicolumn{2}{c}{ESA-a} & \multicolumn{2}{c}{ESA-a} & \multicolumn{2}{c}{COBRA-4} & \multicolumn{2}{c}{COBRA-As} & \multicolumn{2}{c}{COBRA-SHIA} \\ \hline  & mean & 1.38976 & 0.691141 & 1.29891 & 0.526562 & 3.64401 & 0.54154 & 2.3909  
Regarding the multi-modal functions \(f_{4}\), \(f_{8}\), \(f_{9}\), \(f_{13}\) and \(f_{16}\), COBRA-SHA was able to outperform all the biology-inspired algorithms included in the comparison in terms of mean and best values. The COBRA-SHA modification performed better than the other algorithms for the rest of the multi-modal functions (namely \(f_{6}\), \(f_{7}\), \(f_{10}\), \(f_{11}\), \(f_{12}\), \(f_{14}\) and \(f_{15}\)) except the fifth benchmark problem, but it was able to find the best value for \(f_{5}\). The fuzzy-controlled COBRA-f found the best solution for function \(f_{14}\) and gave the best mean value for function \(f_{5}\). As with the COBRA-SHA modification, COBRA-fas gave the best values for functions \(f_{7}\) and \(f_{10}\). For functions \(f_{6}\) and \(f_{11}\), the best values were found by the WPS algorithm, while for function \(f_{15}\) it was found by the FFA algorithm. Finally, the modification CSA-a was able to achieve the best value for function \(f_{12}\). The results of the comparison between COBRA-SHA and the other mentioned algorithms according to the Mann-Whitney statistical test with significance level \(p=0.01\) are presented in Table 8 (the same notations are used). The results of the Mann-Whitney statistical test are presented in Figure 8. Then all the stated algorithms were compared with the proposed modification COBRA-SHA according to the Friedman statistical test. The obtained results are demonstrated in Figure 9 (the same notations as in Figures 6 and 7 are used). Thus, it was established that the results obtained by the proposed approach are statistically better according to the [PERSON] and Mann-Whitney tests than the results obtained by the mentioned biology-inspired algorithms (PSO, WPS, FSS, FFA, CSA and BA) and their modifications with the external archive (FFA-a, CSA-a, BA-a). Moreover, COBRA-SHA statically outperformed the fuzzy-controlled COBRA-f. Furthermore, the experimental results for the benchmark problems from SET-3 showed that the COBRA-SHA approach is more useful for solving complex multi-modal optimization problems than the similar modification COBRA-fas. Therefore, the workability and usefulness of the proposed COBRA-SHA algorithm were demonstrated.