## 5 Numerical Experiments

Table 2: Decision variables, global optimum of benchmark functions and number of iterations used for testing the performance of MA, KHA, and MAKHA. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Objective Function** & **NV** & \begin{tabular}{c} **Search** \\ **Domain** \\ \end{tabular} & \begin{tabular}{c} **Global** \\ **Minimum** \\ \end{tabular} & \
\begin{tabular}{c} **Iterations** \\ **MA** \\ \end{tabular} &
\begin{tabular}{c} **MAKHA** \\ \end{tabular} \\ \hline Ackley & 2 & [\(-\)35, 35] & 0 & 25 & 3000 & 1000 \\ Beale & 2 & [\(-\)4.5, 4.5] & 0 & 25 & 3000 & 1000 \\ [PERSON] & 2 & [\(-\)2\(\pi\), 2\(\pi\)] & \(-\)106.765 & 25 & 3000 & 1000 \\ Booth & 2 & [\(-\)10, 10] & 0 & 25 & 3000 & 1000 \\ Bukin 6 & 2 & [\(-\)15, 3] & 0 & 25 & 3000 & 1000 \\ Carrom table & 2 & [\(-\)10, 10] & \(-\)24.15681 & 25 & 3000 & 1000 \\ Cross-leg table & 2 & [\(-\)10, 10] & \(-\)1 & 25 & 3000 & 1000 \\ Generalized egg holder & 2 & [\(-\)512, 512] & \(-\)959.64 & 124 & 15,000 & 5000 \\ Goldstein-Price & 2 & [\(-\)2, 2] & 3 & 25 & 3000 & 1000 \\ Himmelblau & 2 & [\(-\)5, 5] & 0 & 25 & 3000 & 1000 \\ Levy 13 & 2 & [\(-\)10, 10] & 0 & 25 & 3000 & 1000 \\ Schaffer & 2 & [\(-\)100, 100] & 0 & 199 & 15,000 & 8000 \\ Zettl & 2 & [\(-\)5, 5] & \(-\)0.003791 & 25 & 3000 & 1000 \\ Helical valley & 3 & [\(-\)1000, 1000] & 0 & 25 & 3000 & 1000 \\ \hline \hline \end{tabular}  
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Powell & 4 & [\(-\)1000, 1000] & 0 & 50 & 6000 & 2000 \\ [PERSON] & 4 & [\(-\)1000, 1000] & 0 & 25 & 3000 & 1000 \\ Extended Cube & 5 & [\(-\)100, 100] & 0 & 25 & 3000 & 1000 \\ [PERSON] 5 & 4 & [0, 10] & \(-\)10.1532 & 25 & 3000 & 1000 \\ Sphere & 5 & [\(-\)100, 100] & 0 & 75 & 9000 & 1000 \\ Hartman 6 & 6 & [0,1] & \(-\)3.32237 & 25 & 3000 & 1000 \\ [PERSON] & 50 & [\(-\)600, 600] & 0 & 124 & 15000 & 5000 \\ [PERSON] & 50 & [\(-\)5.12, 5.12] & 0 & 124 & 15000 & 5000 \\ [PERSON] & 50 & [\(-\)50, 50] & 0 & 124 & 15000 & 5000 \\ Sine envelope sine wave & 50 & [\(-\)100, 100] & 0 & 124 & 15000 & 5000 \\ [PERSON] & 50 & [\(-\)5,5] & \(-\)1958.2995 & 124 & 15000 & 5000 \\ [PERSON] & 50 & [\(-\)1000, 1000] & 0 & 124 & 15000 & 5000 \\ [PERSON] & 50 & [\(-\)5,10] & 0 & 25 & 3000 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Condition** & **Parameter** & **Selected value** \\ \hline  & & \(b\) & 1 \\  & R \(\geq\) 100 & \(b\) & 10 \\  & & \(c\) & \(-\)1 \\
**MA** & & \(d\) & 1 \\  & R \(\geq\) 500 & \(c\) & \(-\)10 \\  & R \(\geq\) 500 & \(d\) & 30 \\  & & \(N_{C}\) & 30 \\ \hline  & & \(D_{max}\) & [0.002, 0.01] \\  & & \(C_{t}\) & 0.5 \\
**KHA** & & \(V_{f}\) & 0.02 \\  & & \(N_{max}\) & 0.01 \\  & & \(w_{f}\) and \(w_{N}\) & [0.1, 0.8] \\ \hline  & & \(b\) & 1 \\  & R \(<\) 2 & \(b\) & 0.5*R \\  & R \(\geq\) 100 & \(b\) & 10 \\  & & \(c\) & \(-\)0.1 \\
**MAKHA I** & & \(d\) & 0.1 \\  & & \(D_{max}\) & 0 \\  & & \(C_{t}\) & 0.5 \\  & & \(V_{f}\) & 0.2 \\  & & \(w_{f}\) & 0.1 \\ \hline  & & \(b\) & 0.3*R \\  & & \(c\) & \(-\)R \\  & & \(d\) & R \\
**MAKHA II** & & \(D_{max}\) & [0.002, 0.01] \\
**(NV = 50)** & & \(C_{t}\) & 0.5 \\  & & \(V_{f}\) & 0.02 \\  & & \(w_{f}\) & [0.1, 0.8] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Selected values of the parameters used in the implementation of MA, KHA and MAKHA. The twenty-seven problems constitute a comprehensive testing of the reliability and effectiveness of the hybrid algorithm. Thirteen functions have two variables only, yet some of them are very difficult to optimize. Surface plots of eight of the two-variable functions are shown in Figure 4. Each problem was solved 30 times by each of the three algorithms. The best value at each iteration was recorded and the means of the best values were calculated amongst the 30 run as a function of the iteration number. The plots of the mean best values _versus_ the number of function for the three algorithms provide a clear comparison of both the reliability and the efficiency of the algorithms.