## 6 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{6}{c}{**Numerical Performance of**} \\ \hline  & \multicolumn{2}{c}{**MA**} & \multicolumn{2}{c}{**KHA**} & \multicolumn{2}{c}{**MAKHA**} \\ \hline
**Objective function** & \(f_{calc}\) & \(\sigma\) & \(f_{calc}\) & \(\sigma\) & \(f_{calc}\) & \(\sigma\) \\ \hline
**Ackley** & 4.8E\(-\)8 & 0 & 0.00129 & 0 & **0** & 0 \\
**Beale** & 0.084 & 0.125 & 0.0508 & 0.193 & **0** & 0 \\
**[PERSON]** & \(-\)105.326 & 1.45 & \(-\)103.52 & 7.4 & \(-\)**106.7645** & 0 \\
**[PERSON]** & 0.179 & 0.172 & 1.28E\(-\)12 & 0 & **0** & 0 \\
**Bukin 6** & 3.487 & 1.77 & 0.074 & 0.029 & **0.0267** & 0.0157 \\
**Carrom table** & \(-\)23.9138 & 0.436 & \(-\)**24.1568** & 0 & \(-\)**24.1568** & 0 \\
**Cross-leg table** & \(-\)0.002 & 4E\(-\)3 & \(-\)0.00035 & 0 & \(-\)**0.9985** & 0 \\
**Generalized egg holder** & \(-\)949.58 & 0 & \(-\)862.1 & 0 & \(-\)**959.641** & 0 \\
**Goldstein-Price** & 3.051 & 0.055 & **3** & 0 & **3** & 0 \\
**Himmelblau** & 0.179 & 0.187 & 7.4E\(-\)13 & 0 & **3.7E\(-\)31** & 0 \\
**Levy 13** & 0.0616 & 0.08 & 2.1E\(-\)7 & 0 & **1.35E\(-\)31** & 0 \\
**Schaffer** & **0** & 0 & 1.7E\(-\)6 & 0 & **0** & 0 \\
**Zettl** & \(-\)0.0037 & 1E\(-\)3 & \(-\)**0.00379** & 0 & \(-\)**0.00379** & 0 \\
**Helical valley** & 136 & 250 & 9.8E\(-\)5 & 3E\(-\)3 & **0** & 0 \\
**Powell** & 18.46 & 49 & 1.9E\(-\)5 & 0 & **0** & 0 \\
**[PERSON]** & 113.6 & 256 & 0.698 & 1.6 & **0** & 0 \\
**Extended Cube** & 3.568 & 0.8 & 1.658 & 5.28 & **0** & 0 \\
**Shekel 5** & \(-\)10.139 & 0.06 & \(-\)5.384 & 3.1 & \(-\)**10.1532** & 0 \\
**Sphere** & 1.4E\(-\)14 & 0 & 1E\(-\)10 & 0 & **0** & 0 \\
**Hartman 6** & \(-\)2.7499 & 0.2 & \(-\)3.2587 & 0.06 & \(-\)**3.2627** & 0.06 \\
**Griewank** & 0.0165 & 0.032 & 0.0769 & 0.095 & **8.4E\(-\)8** & 0 \\
**[PERSON]** & **1.3E-9** & 0 & 89.38 & 48.38 & 6.47E\(-\)6 & 0 \\
**[PERSON]** & 47.45 & 7.5 & 52.593 & 18.97 & **1E\(-\)5** & 0 \\
**Sine envelope sine wave** & **3.3E\(-\)11** & 0 & 16.107 & 5.94 & 3.46E\(-\)6 & 0 \\
**[PERSON]** & \(-\)1916.84 & 28.1 & \(-\)1645.42 & 36.9 & \(-\)**1958.31** & 0 \\
**Trigonometric** & 1.35E\(-\)5 & 0 & 285.43 & 545.5 & **0** & 0 \\
**[PERSON]** & **1.46E\(-\)10** & 0 & 1E\(-\)3 & 5.5E\(-\)3 & 8.4E\(-\)6 & 0 \\ \hline \end{tabular}
* Bold numbers represent the best global minimum value obtained for each function.  
\end{table} 
Table 4: Values of the mean minima (\(f_{calc}\)) and standard deviations (\(\sigma\)) obtained by the MAKHA, MA and KHA algorithms for the benchmark problems used in this study. *The results of new proposed hybrid were promising that could be attributed to the combination of some of the mechanisms and processes of MA and KHA together to produce a reliable algorithm with appreciated performance. The procedures of both algorithms include exploration/diversification and exploitation/intensification features as follows:  
(a) Exploration or diversification feature: The watch-jump process (MA), physical random diffusion (KHA), the somersault process (MA), and genetic operators (KHA). (b) Exploitation or intensification feature: The climb process (MA), the watch-jump process (MA), the induced motion (KHA), and the foraging activity (KHA). Both algorithms (MA and KHA) make a good compromise and balance between exploration/diversification and exploitation/intensification features. However, MAKHA uses the best setup of efficient operators, which are capable of producing the previous promising results.