## 4 Experimental Study and Performance Benchmarking
### Number of Times the Algorithm Converged to the Optimal Solution
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{2}{c}{**Best Optimal Solution Found**} & \multicolumn{2}{c}{**Best Optimal Solution Found**} \\  & **Using the New Hybrid Algorithm** & **Using Previous Three Algorithms** \\ \hline Rec-01 & 1280 & **1249** \\ Rec-03 & 1124 & **1111** \\ Rec-05 & 1249 & **1245** \\ Rec-07 & 1584 & **1566** \\ Rec-09 & 1590 & **1537** \\ Rec-11 & 1457 & **1431** \\ Rec-13 & 2003 & **1954** \\ Rec-15 & 2012 & **1962** \\ Rec-17 & 2008 & **1919** \\ \hline Car-01 & **7038** & **7038** \\ Car-02 & **7166** & **7166** \\ Car-03 & **7312** & **7312** \\ Car-04 & **8803** & **8803** \\ Car-05 & **7720** & **7720** \\ Car-06 & **8505** & **8505** \\ Car-07 & **6590** & **6590** \\ Car-08 & **8366** & **8366** \\ \hline Hel-02 & 139 & **137** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the best optimal solution found using the new hybrid algorithm and the previous three algorithms. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{2}{c}{**Best Mean Value and Standard**} & **Best Mean Value and Standard** \\  & **Deviation of the Optimal** & **Deviation of the Optimal** \\  & **Solution Found Using the New Hybrid Algorithm** & **Devious Three Algorithms** \\ \hline Rec-01 & 1310.60 \(\pm\) 17.66 & **1249.00 \(\pm\) 0.00** \\ Rec-03 & 1139.80 \(\pm\) 13.10 & **1111.00 \(\pm\) 0.00** \\ Rec-05 & 1269.60 \(\pm\) 11.28 & **1245.00 \(\pm\) 0.00** \\ Rec-07 & 1623.90 \(\pm\) 29.44 & **1579.00 \(\pm\) 8.07** \\ Rec-09 & 1617.70 \(\pm\) 18.60 & **1567.70 \(\pm\) 16.54** \\ Rec-11 & 1457.20 \(\pm\) 17.76 & **1440.80 \(\pm\) 11.78** \\ Rec-13 & 2042.20 \(\pm\) 21.90 & **1956.90 \(\pm\) 2.28** \\ Rec-15 & 2047.60 \(\pm\) 30.31 & **1974.30 \(\pm\) 6.31** \\ Rec-17 & 2028.00 \(\pm\) 13.72 & **1944.30 \(\pm\) 11.62** \\ \hline Car-01 & 7038.00 \(\pm\) 0.00 & **7038.00 \(\pm\) 0.00** \\ Car-02 & 7187.00 \(\pm\) 66.41 & **7166.00 \(\pm\) 0.00** \\ Car-03 & 7382.10 \(\pm\) 27.89 & **7312.00 \(\pm\) 0.00** \\ Car-04 & 8803.00 \(\pm\) 0.00 & **8803.00 \(\pm\) 0.00** \\ Car-05 & 7745.70 \(\pm\) 18.37 & **7720.00 \(\pm\) 0.00** \\ Car-06 & 8561.20 \(\pm\) 71.27 & **8505.00 \(\pm\) 0.00** \\ Car-07 & 6605.90 \(\pm\) 25.60 & **6590.00 \(\pm\) 0.00** \\ Car-08 & 8366.00 \(\pm\) 0.00 & **8366.00 \(\pm\) 0.00** \\ \hline Hel-02 & 143.10 \(\pm\) 1.66 & 137.00 \(\pm\) 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of the mean value and standard deviation of the optimal solutions found using the new hybrid algorithm and from the previous three algorithms. As can be seen from the results above, the new hybrid algorithm was able to achieve the same best optimal makespan for the Carlier problems but not for the Reeves and Heller problems. In terms of the mean value and standard deviation of the optimal solution found, the new hybrid algorithm had a similar performance to the proposed algorithm for the Carlier problems 1, 4, and 8, while it had a higher mean value and standard deviation for rest of the benchmark problems. When comparing \(N_{c}\), the new hybrid algorithm was able to converge to the best optimal solutions in all 10 simulations for Carlier problems 1, 4, and 8, but was unable to achieve similar results for the remaining problems. In terms of \(N_{O}\), the new hybrid algorithm was unable to obtain more optimal solutions for any benchmark problem. These results indicate that the new hybrid algorithm did not have a superior performance to the proposed algorithm.