## 5 Numerical Experiments  
Twenty-seven classical benchmark functions were used to evaluate the performance of MAKHA as compared to the original MA and KHA. Tables 1 and 2 show the benchmark functions used along with their names, number of variables, variable limits and the value of the global minimum. Table 3 lists the parameters used in the three algorithms in which their values were set to give the best attainable results for each algorithm. A new parameter for MAKHA was defined as the midpoint between the lower boundary and the upper boundary of the decision variables \(X\). It is calculated from this formula:  
\[R=0.5\sum_{L=1}^{N^{\prime}}\left(UB_{L}-LB_{L}\right) \tag{23}\]  
in which, according to the value R, certain values were assigned to MAKHA's parameters, as seen in Table 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Objective Function** \\ \hline [PERSON] [34] & \(f_{1}=20\left(1-e^{-\alpha\cdot\beta\left(\beta\left(x_{1}-x_{2}\right)\right)} \right)-e^{-\beta\cdot\left(\alpha\left(\alpha\left(\alpha\left(\alpha\left(\alpha \right)\right)\right)\right)}+e^{\beta}\right.\) \\ [PERSON] [35] & \(f_{2}=\left(1.5-x_{1}+x_{2}x_{2}\right)^{2}+\left(2.25-x_{1}+x_{2}x_{2}\right) ^{2}+\left(2.625-x_{1}+x_{2}x_{2}\right)^{2}\) \\ [PERSON] [36] & \(f_{3}=\sin(x_{1})e^{\left(1-\cos x_{1}\right)^{2}}+\cos(x_{2})e^{\left(1-\sin x _{1}\right)^{2}}+\left(x_{1}-x_{2}\right)^{2}\) \\ Booth [35] & \(f_{4}=\left(x_{1}+2x_{2}-7\right)^{2}+\left(2x_{1}+x_{2}-5\right)^{2}\) \\ Bukin 6 [35] & \(f_{5}=100\sqrt{\left|x_{2}-0.01x_{1}^{2}\right|}+0.01\left|x_{1}+10\right|\) \\ Carrom table [37] & \(f_{6}=-\left[\cos x_{1}\cos x_{2}e^{\left|-\left|-\beta\left(x_{1}^{2}+x_{2}^ {2}\right|/4\right)\right|^{2}}/30\right.\) \\ Cross-leg table [37] & \(f_{7}=-\left[\left|\sin\left(x_{1}\right)\sin\left(x_{2}\right)e^{\left|00- \beta\left(x_{1}^{2}+x_{2}^{2}\right|/4\right|}+1\right]^{-0.1}\right.\) \\ Generalized egg holder [35] & \(f_{8}=\sum\limits_{i=1}^{m-1}\left[-\left(x_{i+1}+47\right)\sin\left(\sqrt{ \left|x_{i+1}+x_{i}\right./2+47\right|}\right)+\right]\) \\ \(f_{9}=\left(1+\left(x_{1}+x_{2}+1\right)^{2}\left(19-14x_{1}+3x_{1}^{2}-14x_{ 2}+6x_{1}x_{2}+3x_{2}^{2}\right)\right)\) \\ Goldstein–Price [38] & \(\left(30+\left(2x_{1}-3x_{2}\right)^{2}\left(18-32x_{1}+12x_{1}^{2}+48x_{2}-36 x_{1}x_{2}+27x_{2}^{2}\right)\right)\) \\ [PERSON] [39] & \(f_{10}=\left(x_{1}^{2}+x_{2}-11\right)^{2}+\left(x_{2}^{2}+x_{1}-7\right)^{2}\) \\ [PERSON] 13 [40] & \(f_{11}=\sin^{2}\left(3\pi x_{1}\right)+\left(x_{1}-1\right)^{2}\left[1+\sin^{ 2}\left(3\pi x_{2}\right)\right]+\left(x_{2}-1\right)^{2}\left[1+\sin^{2}\left( 2\pi x_{2}\right)\right]\ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Sphere [47] & \(f_{19}=\sum\limits_{i=1}^{m}x_{i}^{\ 2}\) \\ [PERSON] 6 * [48] & \(f_{20}=-\sum\limits_{i=1}^{4}\alpha_{i}\exp(-\sum\limits_{j=1}^{6}A_{ij}(x_{j}- O_{ij})^{2})\) \\ [PERSON] [49] & \(f_{21}=\frac{1}{4000}\Bigg{[}\sum\limits_{i=1}^{m}\big{(}x_{i}-100\big{)}^{2} \Bigg{]}-\Bigg{[}\prod\limits_{i=1}^{m}\cos\Bigg{(}\frac{x_{i}-100}{\sqrt{i}} \Bigg{)}\Bigg{]}+1\) \\ [PERSON] [50] & \(f_{22}=\sum\limits_{i=1}^{m}\big{(}x_{i}^{\ 2}-10\cos\big{(}2\pi x_{i}\big{)}+10\big{)}\) \\ [PERSON] [51] & \(f_{23}=\sum\limits_{i=1}^{m-1}100\big{(}x_{i+1}-x_{i}^{\ 2}\big{)}^{2}+\big{(}x_{i}-1\big{)}^{2}\) \\ Sine envelope sine wave [37] & \(f_{24}=\sum\limits_{i=1}^{m-1}\Bigg{\{}0.5+\frac{sin^{2}\Big{[}\sqrt{x_{i+1}^ {\ 2}+x_{i}^{\ 2}}\Big{]}-0.5}{\Big{[}0.001\big{(}x_{i+1}^{\ 2}+x_{i}^{\ 2}\big{)}+1\Big{]}^{2}}\Bigg{\}}\) \\ [PERSON]–Tang [52] & \(f_{25}=0.5\sum\limits_{i=1}^{m}(x_{i}^{\ 4}-16x_{i}^{\ 2}+5x_{i})\) \\ Trigonometric [53] & \(f_{26}=\sum\limits_{i=1}^{m}\Bigg{[}m+i\ \big{(}1-\cos x_{i}\big{)}-\sin x_{i}- \sum\limits_{j=1}^{m}\cos x_{j}\Bigg{]}^{2}\) \\ [PERSON] [54] & \(f_{27}=\sum\limits_{i=1}^{m}x_{i}^{\ 2}+\Bigg{(}\sum\limits_{i=1}^{m}0.5\,i\ x_{i} \Bigg{)}^{2}+\Bigg{(}\sum\limits_{i=1}^{m}0.5\,i\ x_{i}\Bigg{)}^{4}\) \\ \hline \hline \end{tabular}  
* Shekel and Hartman parameters were obtained from [36].  
\end{table}