# A Novel Binary QUasi-Affine TRansformation Evolutionary (QUATRE) Algorithm
## 4 Experimental Results and Analysis
### Benchmark Function
Fist of all, the S-shaped an V-shaped families of transfer functions are compared, that is the first approach of BQUATRE algorithm with \(S_{2}\) transfer function (record as BQUATRE1-S2), the first approach of BQUATRE algorithm with \(V_{2}\) transfer function (record as BQUATRE1-V2), the second approach of BQUATRE algorithm with the improved transfer function \(S_{2i}\) (record as BQUATRE2-S2i) and the second approach of BQUATRE algorithm with the improved transfer function \(V_{2i}\) (record as BQUATRE2-V2i). Since QUATRE is the improved DE, the proposed binary QUATRE is compared with Binary DE(record as BDE). The binary PSO is the most primitive binary algorithm, so we also used it in our experiment (record as BPSO). The third compared algorithm is the Advanced Binary Grey Wolf Optimizer algorithm with the V3a transfer function (record as ABGWO-V3a), which is proposed in Reference [54] by [PERSON] ABGWO-V3a can obtain better performance than many other binary meta-heuristic algorithms. Table 13 shows the experimental results of BDE, ABGWO-V3a, BQUATRE1-S2, BQUATRE1-V2, BQUATRE2-S2i, BQUATRE2-V2i, and BPSO. In detail, all algorithms in the experiment run 30 times, 500 iterations and 30 individuals on each benchmark function. Among them, AVG and STD are the mean and standard deviation of the results of 30 times, respectively. The results in red and in blue are the best result of the seven algorithms. The blue results means that all algorithms obtained the optimal solution. The last line refers to the number of times the algorithm gets the optimal result. From Table 13, we can see that the seven algorithms get the same solutions on 6 benchmark functions, which are all multimodal functions in low dimensions. BQUATRE1-V2, BQUATRE2-S2i and BQUATRE2-V2i obtained 14, 15 and 14 red results, respectively. However, BDE, ABGWO-V3a, and BPSO only obtained 6, 3 and 1 red results, respectively. Therefore, BQUATRE1-V2, BQUATRE2-S2i and BQUATRE2-V2i are superior to BDE, ABGWO-V3a, and BPSO. At the same time, we can see that BQUATRE1-S2 does not perform well. However, BQUATE2-S2i obtained the best results compared to the other five algorithms. This result shows Equation (14) is effective for BQUATE2. Figures 6-8 are the visualization results of Table 13. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Function** & **Search Space** & **D** & \(f_{min}\) \\ \hline Fifth of Dejong & \(f_{14}(x)=(\frac{1}{500}\sum_{j=1}^{25}\frac{1}{j+\sum_{i+1}(x_{i}-a_{ij})})^{ -1}\) & [–65,65] & 2 & 1 \\ Kowalik & \(f_{15}(x)=\sum_{i=1}^{11}[a_{i}-\frac{x_{i}(0j+x_{i}x_{i})}{b_{i}+b_{i}x_{i}}]^{ 2}\) & [–5,5] & 4 & 0.00030 \\ Six-hump camel back & \(f_{16}(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{4}{3}x_{1}^{4}+x_{1}x_{2}-4x_{2}^{2}+4x_ {2}^{4}\) & [–5,5] & 2 & \(-1.0316\) \\ Branins & \((x_{2}-\frac{5.1}{4\pi^{2}}+\frac{5}{4\pi}x_{1}-6)^{2}+10(1-\frac{1}{8\pi})\cos x _{1}+10\) & [–5,5] & 2 & 0.398 \\  & \(f_{18}(x)=[1+(x_{1}+x_{2}+1)^{2}(19-14x_{1}+3x_{1}^{2}-14x_{2}\) & & \\ Goldstein-Price & \(+6x_{1}x_{2}+3x_{2}^{2})\big{]}\times[30+(2x_{1}-3x_{2})^{2}\times(18-32x_{1}\) & [–2,2] & 2 & 3 \\  & \(+12x_{1}^{2}+48x_{2}-36x_{1}x_{2}+27x_{2}^{2})\) & & & \\ [PERSON] 1 & \(f_{19}(x)=-\sum_{i=1}^{4}c_{i}\exp(-\sum_{j=1}^{4}a_{ij}(x_{j}-p_{ij})^{2})\) & [1,3] & 3 & \(-3.86\) \\ [PERSON] 2 & \(f_{20}(x)=-\sum_{i=1}^{4}c_{i}\exp(-\sum_{j=1}^{4}a_{ij}(x_{j}-p_{ij})^{2})\) & [0,1] & 6 & \(-3.32\) \\ [PERSON] 1 & \(f_{21}(x)=-\sum_{j=1}^{5}[(X-a_{i})(X-a_{i})^{T}+c_{i}]^{-1}\) & [0,10] & 4 & \(-10.1532\) \\ [PERSON] 2 & \(f_{22}(x)=-\sum_{i=1}^{5}[(X-a_{i})(X-a_{i})^{T}+c_{i}]^{-1}\) & [0,10] & 4 & \(-10.4028\) \\ [PERSON] 3 & \(f_{23}(x)=-\sum_{i=1}^{10}[(X-a_{i})(X-a_{i})^{T}+c_{i}]^{-1}\) & [0,10] & 4 & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Multimodal benchmark functions in low dimension.