# 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} \hline \hline
**Function** & **QUATRE1-S2** & **QUATRE1-V2** & **QUATRE1-U2** & **QUATRE1-Z2** & **QUATRE2-S2i** & **QUATRE2-V2i** & **QUATRE2-U2i** & **QUATRE2-Z2i** \\ \hline f1 & = & + & + & + & + & + & + & + \\ f2 & – & + & + & + & + & + & + & + \\ f3 & – & + & + & + & + & + & + & + \\ f4 & = & = & – & – & – & – & – & – \\ f6 & = & + & + & + & + & + & + \\ f7 & – & + & + & + & + & + & + \\ \hline f8 & – & – & – & + & + & + & + & + \\ f9 & – & + & + & + & + & + & + & + \\ f10 & = & + & + & + & + & + & + & + \\ f11 & = & + & + & + & + & + & + & + \\ f12 & = & + & + & + & + & + & + & + \\ f23 & = & = & = & = & = & = & = \\ \hline \hline \end{tabular}
\end{table} 
Table 15: The \(t\)-test results of the compared algorithms (QUATRE1-S2, QUATRE1-V2, QUATRE1-U2, QUATRE1-Z2, QUATRE2-S2i, QUATRE2-V2i, QUATRE2-U2i, and QUATRE2-Z2i) on ABGWO-V3a. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Function** & **BDE** & **ABGWO-V3a** & **RQUATE1-S2** & **BQUATE1-V2** & **BQUATE1-U2** & **BQUATE2-S4i** & **BQUATE2-V4i** & **BQUATE2-V4i** & **BQUATE2-V4i** \\  & & & & & & & & & & \\ \hline \(\Omega\) & 0.62744 & 16.2599 & 3.4680 & 3.4155 & 5.3599 & 3.6765 & 6.7833 & 6.3142 & 5.5937 & 6.6418 \\ \(\Omega\) & 0.70993 & 16.7380 & 2.5646 & 3.2855 & 5.2737 & 3.7985 & 6.6544 & 6.3128 & 6.0057 & 6.5947 \\ \(\Omega\) & 2.1763 & 17.2274 & 3.1963 & 3.8897 & 6.1890 & 4.607 & 7.6322 & 7.0619 & 6.1531 & 7.3415 \\ \(\Omega\) & 0.6887 & 16.7796 & 2.6008 & 3.3246 & 3.5231 & 3.9412 & 6.0011 & 6.2222 & 5.6701 & 6.5871 \\ \(\Omega\) & 0.8110 & 17.6210 & 2.7793 & 3.3000 & 5.4401 & 4.2954 & 6.9998 & 6.3239 & 7.7365 & 6.5544 \\ \(\Omega\) & 0.7087 & 15.4679 & 2.5756 & 3.2588 & 5.3466 & 4.4473 & 7.2255 & 6.2489 & 5.8564 & 6.6404 \\ \(\Omega\) & 0.7183 & 15.7941 & 2.5968 & 3.2864 & 5.5784 & 4.2904 & 6.7943 & 6.2623 & 5.6009 & 6.6486 \\ \hline \(\Omega\) & 0.7162 & 14.9921 & 2.5967 & 3.4029 & 5.3499 & 4.3990 & 7.5326 & 6.2789 & 5.7329 & 6.4835 \\ \(\Omega\) & 0.7261 & 17.7570 & 2.9605 & 3.2833 & 5.3474 & 4.4991 & 82.8188 & 6.3021 & 6.6065 & 6.6877 \\ \(\Omega\) & 0.7281 & 15.5744 & 2.6108 & 3.2774 & 5.4029 & 3.7106 & 7.8801 & 6.3021 & 6.4057 & 6.6676 \\ \(\Omega\) & 0.7276 & 14.5106 & 2.6747 & 3.2942 & 5.3558 & 3.2791 & 8.3999 & 6.5746 & 6.3031 & 6.6952 \\ \(\Omega\) & 1.2497 & 14.7383 & 3.0666 & 3.5307 & 5.6124 & 4.0569 & 7.9435 & 6.4522 & 6.6766 & 7.2990 \\ \(\Omega\) & 1.2632 & 14.7070 & 3.1742 & 3.6262 & 5.9242 & 4.0865 & 8.1302 & 6.4171 & 3.6563 & 6.8991 \\ \(\Omega\) & 0.6415 & 1.8261 & 1.4972 & 1.0698 & 1.4449 & 1.2067 & 1.8882 & 1.5145 & 1.3213 & 1.5724 \\ \hline \(\Omega\) & 0.6694 & 2.0747 & 0.5584 & 0.6493 & 1.1020 & 0.7056 & 1.3389 & 1.0139 & 1.0575 & 1.0099 \\ \(\Omega\) & 0.5553 & 1.0338 & 0.4320 & 0.3566 & 0.6607 & 0.4244 & 0.5298 & 0.5916 & 0.6631 & 0.6532 \\ \(\Omega\) & 0.5438 & 1.0066 & 0.3377 & 0.3458 & 0.5490 & 0.4188 & 0.5847 & 0.5966 & 0.6930 & 0.6113 \\ \(\Omega\) & 0.5592 & 1.0166 & 1.5641 & 0.3535 & 0.5985 & 0.4312 & 0.5705 & 0.3900 & 0.5966 & 0.6280 \\ \(\Omega\) & 0.7773 & 1.6071 & 1.7176 & 0.3457 & 1.4079 & 0.6622 & 0.4831 & 0.5914 & 0.6882 & 0.5999 \\ \(\Omega\) & 0.7973 & 3.0584 & 2.4736 & 0.6864 & 1.4021 & 1.9917 & 1.4648 & 1.5124 & 1.5258 & 1.2942 \\ \(\Omega\) & 0.8798 & 2.1235 & 1.6826 & 0.6974 & 1.0730 & 0.8097 & 1.1294 & 1.1596 & 1.1616 & 1.2014 \\ \(\Omega\) & 0.8969 & 2.1611 & 1.3334 & 0.7354 & 1.2336 & 0.8218 & 1.1837 & 1.2024 & 1.2066 & 1.2449 \\ \(\Omega\) & 1.0245 & 2.2511 & 1.1589 & 0.8008 & 1.1845 & 0.8869 & 1.4182 & 1.2899 & 1.2973 & 1.3347 \\ \hline \hline \end{tabular}
\end{table} 
Table 17. The time consumption of the algorithms(BDE, ABGWO-V3a, BQUATE1-S2, BQUATE1-V2, BQUATE1-U2, BQUATE1-22, BQUATE2-S4i, BQUATE2-V4i, BQUATE2-U4i, and BQUATE2-Z4i).