## 4 Experimental Study and Performance Benchmarking
### The Best Solution Found  
As mentioned above, 10 independent simulations were ran for each test problem using the three algorithms. The first method used to compare the performance of these algorithms included observing the best solution obtained during 10 simulations. The results obtained by these algorithms were also compared to the best-known solution, as well as the best-known solutions obtained using different algorithms in the available literature. These results are shown in Table 2.  
According to Table 2, the proposed algorithm, sharing fitness algorithm, and HIEGA were able to find the best-known solutions for all Carlier benchmark problems. IEGA was able to converge to the best-known solution for all Carlier problems besides Car-03, while CM was able to find the best-known solution for Car-01, 05, 06, 07, and 08. For the Reeves problems, the HIEGA had the best performance, as it was able to converge to the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Generations** & **Population Size** & **Algorithm Specific Parameters** \\ \hline Proposed Algorithm & 1000 & 200 & Number of clusters = 3 \\ Sharing Fitness & 1000 & 200 & \(\sigma_{share}=0\) & \(\beta=1\) \\ CM & 1000 & 200 & \(\sigma_{share}=50\) & Maximum niche size = 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters used for the three algorithms.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & **Best-Known Solution** & **Proposed Algorithm** & **HIEGA** & **IEGA** & **Sharing Fitness** & **CM** \\ \hline Rec-01 & 1247 & 1249 & **1247** & 1312 & 1249 & 1379 \\ Rec-03 & 1109 & 1111 & **1109** & 1111 & 1114 & 1187 \\ Rec-05 & 1242 & 1245 & 1245 & 1257 & 1245 & 1307 \\ Rec-07 & 1566 & **1566** & **1566** & 1622 & 1589 & 1733 \\ Rec-09 & 1537 & **1537** & **1537** & 1616 & 1600 & 1706 \\ Rec-11 & 1431 & **1431** & **1431** & 1565 & 1475 & 1614 \\ Rec-13 & 1930 & 1954 & **1930** & 2020 & 1980 & 2145 \\ Rec-15 & 1950 & 1962 & 1959 & 2043 & 2007 & 2156 \\ Rec-17 & 1902 & 1919 & 1935 & 2030 & 1989 & 2178 \\ \hline Car-01 & 7038 & **7038** & **7038** & **7038** & **7038** & **7038** \\ Car-02 & 7166 & **7166** & **7166** & **7166** & **7166** & **7166** & 7565 \\ Car-03 & 7312 & **7312** & **7312** & **7366** & **7312** & 7399 \\ Car-04 & 8003 & **8003** & **8003** & **8003** & **8003** & 8410 \\ Car-05 & 7720 & **7720** & **7720** & **7720** & **7720** & **7720** \\ Car-06 & 8505 & **8505** & **8505** & **8505** & **8505** & **8505** \\ Car-07 & 6590 & **6590** & **6590** & **6590** & **6590** & **6590** \\ Car-08 & 8366 & **8366** & **8366** & **8366** & **8366** & **8366** \\ \hline Hel-02 & 136 & 137 & **136** & 140 & 139 & 154 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Best optimal solution obtained by the different algorithms.  
best-known solution for 6 out of the 9 problems. The proposed algorithm was able to obtain the best-known solution for Rec-07, Rec-09, and Rec-11, and was able to obtain a better solution than HEIGA for Rec-17. It was also able to converge within 1% of the best-known solution for the remainder of the problems. Sharing fitness, IEGA, and CM were unable to converge to any of the best-known solutions for the Reeves problems and were within 4% and 15% of the best-known solutions. Similarly, only HIEGA converged to the best-known solution of the Heller problem. However, the proposed algorithm was within 0.7% of the best solution, while sharing fitness, IEGA, and CM were unable to do so.