# A Novel Binary QUasi-Affine TRansformation Evolutionary (QUATRE) Algorithm
## 4 Experimental Results and Analysis
### Benchmark Function
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**BDE**} & \multicolumn{2}{c}{**ABGWO-V3a**} & \multicolumn{2}{c}{**BQUATE1-U2**} & \multicolumn{2}{c}{**BQUATE1-Z2**} & \multicolumn{2}{c}{**BQUATE2-U2i**} & \multicolumn{2}{c}{**BQUATE2-Z2i**} & \multicolumn{2}{c}{**BPSO**} \\ \cline{2-13}
**Function** & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} & \multicolumn{2}{c}{**AVG**} & \multicolumn{2}{c}{**STD**} \\ \hline \(f1\) & 3.7333 & 1.8742 & 3.3433 & 1.3072 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.1000 & 3.035  
Figure 11: The optimal values of BDE, ABGWO-V3a, BQUATE1-U2, BQUATE1-Z2, BQUATE2-U2i, BQUATE2-Z2i, and BPSO on f14-f23. Figure 10: The optimal values of BDE, ABGWO-V3a, BQUATE1-U2, BQUATE1-Z2, BQUATE2-U2i, BQUATE2-Z2i, and BPSO on f8-f13. Figure 9: The optimal values of BDE, ABGWO-V3a, BQUATE1-U2, BQUATE1-Z2, BQUATE2-U2i, BQUATE2-Z2i, and BPSO on f1-f7. We can see from Table 15 that the four BQUATRE2 algorithms and BQUATRE1 with Z2 transfer function are not inferior to ABGWO-3 Va on all 23 benchmark functions. BQUATRE1 with V2 and U2 transfer function are superior to ABGWO-3 Va on 11 functions, while inferior on 2 functions. Table 16 shows the results between BDE and the eight BQUATRE. From Table 16, QUATRE1-Z2, QUATRE2-S2i, QUATRE2-U2i and QUATRE2-Z2i are not inferior to BDE on all 23 benchmark functions. QUATRE1-V2, QUATRE1-U2 and QUATRE2-V2i are superior to BDE on 9 functions, while inferior on 1 or 2 functions. Finally, we compare the runtime of the 10 algorithms in this experiment, as shown in Table 17. It can be seen that BDE algorithm has the shortest running time and ABGWO-V3A has the longest running time. As a whole, the four BQUATE2 takes longer to run than BQUATE1, as the transfer functions they used are more complex. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Function** & **QUATRE1-S2** & **QUATRE1-V2** & **QUATRE1-U2** & **QUATRE1-Z2** & **QUATRE2-S2i** & **QUATRE2-V2i** & **QUATRE2-U2i** & **QUATRE2-Z2i** \\ \hline f1 & = & + & + & + & + & + & + & + \\ f2 & = & + & + & + & + & + & + & + \\ f3 & = & + & + & + & + & + & + & + \\ f4 & = & = & = & = & = & = & = & = \\ f5 & – & – & – & = & = & – & = & = \\ f6 & = & + & + & + & + & + & + \\ f7 & = & + & + & + & + & + & + & + \\ \hline f8 & – & – & – & – & – & – & – & – & – & – \\ f9 & = & + & + & + & + & + & + & + \\ f10 & = & + & + & + & + & + & + & + \\ f11 & + & + & + & + & + & + & + & + \\ f12 & = & + & + & + & + & + & + & + \\ f13 & = & – & – & = & = & = & = & = \\ \hline f14 & = & – & – & – & – & – & – & – \\ f15 & = & = & = & = & = & = & = & = \\ f16 & = & = & = & = & = & = & = & = \\ f17 & = & = & = & = & = & = & = & = \\ f19 & = & = & = & = & = & = & = & = \\ f20 & = & = & = & = & = & = & = & = \\ f21 & = & = & = & = & = & = & = & = \\ f22 & = & = & = & = & = & = & = & = \\ f23 & = & = & = & = & = & = & = & = \\ \hline \hline \end{tabular}
\end{table} 
Table 15: The \(t\)-test results of the compared algorithms (QUATRE1-S2, QUATRE1-V2, QUATRE1-U2, QUATRE1-Z, QUATRE1-Z2, QUATRE2-S2i, QUATRE2-V2i, QUATRE2-U2i, and QUATRE2-Z2i) on ABGWO-V3a.