## 4 Experimental Study and Performance Benchmarking
### Number of Times the Algorithm Converged to the Optimal Solution  
To ensure that the algorithms could find multiple solutions consistently, the number of simulations in which the algorithm converged to the best-known solution (\(N_{C}\)) was also recorded. Since none of the algorithms converged to majority of the best-known solution for the Reeves problems, the best optimal solution found was used instead. For example, the best-known solution for Rec-01 was 1247, while the best optimal solution found by the algorithms was 1249, therefore, 1249 was used to calculate \(N_{C}\). The results for \(N_{C}\) are given in Table 4. A complete analysis for HEIGA and IEGA could not be done for this metric as these results were not reported by the authors (this is indicated by 'NA' in Table 4). However, some inferences could be made based on the results reported. For example, we could assume that both HEIGA and IEGA always converged to the best-known solution for Car-01 and Car-02, as the mean values were equal to the best-known solution and the standard deviations were zero. Since CM had the worst optimal solution of the three algorithms for the Reeves problems, its \(N_{C}\)'s were 0. Similarly, sharing fitness was unable to find the best optimal solution for all but problem 1, and therefore, its \(N_{C}\) for those problems was 0. The proposed algorithm had the best performance of all algorithms (where results were available). It had a \(N_{C}\) of 10 for problems 7 and 9, 1 for problems 5, 13, and 17, 2 for problems 1 and 15, 4 for problem 11, and 8 for problem 3. The proposed algorithm had an \(N_{C}\) of 10 for all Carlier problems, indicating that it had a 100% converge rate to the best-known solution for those problems. Sharing fitness also had an \(N_{C}\) of 10 for problems, 1, 2, and 8, and had an \(N_{C}\) of 4, 9, 3, 3, and 8 for the remaining problems. CM had the worst performance as it was unable to converge to the best-known solution for problems 2, 3, and 4. It had an \(N_{C}\) of 2, 6, and 9 for problems 1, 5, and 6, respectively, and had an \(N_{C}\) of 10 for problems 7 and 8. Lastly, for the Heller problem, the proposed algorithm had an \(N_{C}\) of 10 while the other two algorithms had an \(N_{C}\) of 0. These results further solidified the argument that the proposed algorithm had the best performance of the three algorithms, as it was able to consistently converge to the best optimal solution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{2}{c}{**Proposed**} & \multirow{2}{*}{**HEIGA**} & \multirow{2}{*}{**IEGA**} & \multicolumn{2}{c}{**Sharing**} & \multirow{2}{*}{**CM**} \\  & & **Algorithm** & & & **Fitness** & \\ \hline Rec-01 & **2** & NA & NA & 1 & 0 \\ Rec-03 & **8** & NA & NA & 0 & 0 \\ Rec-05 & **1** & NA & NA & 0 & 0 \\ Rec-07 & **10** & NA & NA & 0 & 0 \\ Rec-09 & **10** & NA & NA & 0 & 0 \\ Rec-11 & **4** & NA & NA & 0 & 0 \\ Rec-13 & **1** & NA & NA & 0 & 0 \\ Rec-15 & **2** & NA & NA & 0 & 0 \\ Rec-17 & **1** & NA & NA & 0 & 0 \\ \hline Car-01 & **10** & **10** & **10** & **10** & 2 \\ Car-02 & **10** & **10** & NA & **10** & 0 \\ Car-03 & **10** & NA & NA & 4 & 0 \\ Car-04 & **10** & 10 & NA & 9 & 0 \\ Car-05 & **10** & NA & NA & 3 & 6 \\ Car-06 & **10** & NA & NA & 3 & 9 \\ Car-07 & **10** & 0 & NA & 8 & **10** \\ Car-08 & **10** & **10** & **10** & **10** & **10** \\ \hline Hel-02 & **10** & NA & NA & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of simulations in which the algorithms were able to converge to the best optimal solution (\(N_{C}\)). where \(F_{i}\) is the feature matrix of individual \(i\) and \(F_{j}\) is the feature matrix of individual \(j\). For the hybrid algorithm, the value of \(\sigma_{share}\) was changed to 10. The performance of the hybrid algorithm was then compared to the performances of sharing fitness. The proposed algorithm and the results for the four indicators are given in Tables 6-9.