# Unified Genetic Algorithm Approach for Solving Flexible Job-Shop Scheduling Problem
## 4 Numerical Experiments
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**eData 6 \(\times\) 6 (mt06)**} \\ \hline \multirow{2}{*}{Algorithm} & \multicolumn{3}{c}{Existing GA} & \multicolumn{3}{c}{u-COGA} \\ \cline{2-7}  & PS & PT & PU & SS & TT & UU \\ \hline \(\overline{C}_{max}\) & 49.95 & 50.3 & 50.85 & 49.05 & 48.9 & 47.9 \\ SD(\(\overline{C}_{max}\)) & 1.43 & 1.72 & 1.93 & 1.67 & 1.48 & 1.33 \\ Rank & 4 & 5 & 6 & 3 & 2 & 1 \\ \hline \multicolumn{7}{c}{**eData 10 \(\times\) 10 (mt10)**} \\ \hline \multirow{2}{*}{Algorithm} & \multicolumn{3}{c}{Existing GA} & \multicolumn{3}{c}{u-COGA} \\ \cline{2-7}  & PS & PT & PU & SS & TT & UU \\ \hline \(\overline{C}_{max}\) & 1094.7 & 1090.1 & 1093.9 & 1078.8 & 1085 & 1050.85 \\ SDC\({}_{max}\) & 24.67 & 27.93 & 22.59 & 27.67 & 28.19 & 22.64 \\ Rank & 6 & 4 & 5 & 2 & 3 & 1 \\ \hline \multicolumn{7}{c}{**eData 20 \(\times\) 5 (mt20)**} \\ \hline \multirow{2}{*}{Algorithm} & \multicolumn{3}{c}{Existing GA} & \multicolumn{3}{c}{u-COGA} \\ \cline{2-7}  & PS & PT & PU & SS & TT & UU \\ \hline \(\overline{C}_{max}\) & 1134.15 & 1135.15 & 1134.75 & 1074.2 & 1101.9 & 1097.4 \\ SD(\(\overline{C}_{max}\)) & 42.50 & 33.31 & 44.20 & 62.89 & 33.86 & 41.03 \\ Rank & 4 & 6 & 5 & 1 & 3 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Experiment results for eData. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**rData 6 \(\times\) 6 (mt06)**} \\ \hline \multirow{2}{*}{Algorithm} & \multicolumn{3}{c}{Existing GA} & \multicolumn{3}{c}{u-COGA} \\ \cline{2-7}  & PS & PT & PU & SS & TT & UU \\ \hline \(\overline{C}_{max}\) & 49.45 & 48 & 47.9 & 48.25 & 47.85 & 47.25 \\ \hline SD(\(\overline{C}_{max}\)) & 1.79 & 1.56 & 1.51 & 1.74 & 1.53 & 1.74 \\ \hline Rank & 6 & 4 & 3 & 5 & 2 & 1 \\ \hline \multicolumn{7}{c}{**rData 10 \(\times\) 10 (mt10)**} \\ \hline \multirow{2}{*}{Algorithm} & \multicolumn{3}{c}{Existing GA} & \multicolumn{3}{c}{u-COGA} \\ \cline{2-7}  & PS & PT & PU & SS & TT & UU \\ \hline \(\overline{C}_{max}\) & 997.95 & 1008.15 & 1002.05 & 996.35 & 997.8 & 962.2 \\ \hline SD(\(\overline{C}_{max}\)) & 28.14 & 23.04 & 19.33 & 29.50 & 25.35 & 26.02 \\ \hline Rank & 4 & 6 & 5 & 2 & 3 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Experiment results for rData. Tables 6-8 provide the following observations. First, u-COGAs generally produce shorter makespan values than existing GAs, which indicates that the u-COGA is a promising approach for solving the FJSP. Second, for seven out of nine instances in Tables 6-8, two instances (mt06 and mt10) of eData, three instances (mt06, mt10, and mt20) of rData, and two instances (mt06 and mt10) of vData, the shortest makespan values were found by the UU-type u-COGA. In other words, uniform crossover was the best crossover strategy for minimizing the makespan of the FJSP. The performances of SS- and TT-type u-COGAs were not as good as that of the UU-type. Third, excepting one benchmark problem instance, i.e., mt06 of vData in Table 8, the worst makespan value for each instance was obtained by one of the existing GAs, which reveals the limitation of the separate evolution strategy for solving the FJSP. In particular, PS- and PU-type GAs produced the worst makespan values for four instances and three instances, respectively. Finally, the u-COGA and existing GA did not demonstrate significant differences in SD(\(\overline{C_{max}}\)) values. Table 9 shows the average rank of the algorithms listed in Table 5. Again, we can see that the UU-type u-COGA has the smallest average rank, which means that it generally produces a shorter makespan than the other algorithms. In contrast, the average ranks of the individual existing GAs are larger than those of the u-COGAs. Consequently, the average rank of all u-COGAs (2.33) is noticeably smaller than that of all existing GAs (4.63), and we can conclude that the unified evolution strategy of the u-COGA helps to find better solutions for the FJSP.