# A Novel Binary QUasi-Affine TRansformation Evolutionary (QUATRE) Algorithm
## 3 Proposed Binary QUATRE Algorithm
### New Scaling Factor Based on Exploration and Exploitation  
The convergence process of meta-heuristic algorithm can be regarded as two stages: exploration and exploitation. In the early stage of the algorithm, the exploration ability is conducive to searching more space and jumping out of the local optimal solution. In the later stage of the algorithm, the approximate optimal solution has been found, and the development ability can help to find the optimal solution. Similarly, for the binary version, the position needs to be able to switch between 0 and 1 quickly in the beginning and slowly in the end. In continuous QUATRE, [PERSON] proposed a linear decrease scale factor shown as Equation (10) [48], where \(F_{max}\) and \(F_{min}\) are the predetermined maximum and minimum values of scale factor F. Usually, we set \(F\in[0,2]\), then \(F_{max}=2\) and \(F_{min}=0\). \(G\) is the current generation number, and \(MAXG\) is the maximum generation number. \[F=F_{max}-(F_{max}-F_{min})\times\frac{G}{MAXG} \tag{10}\]  
However, in the binary version, things are different, as proved by [PERSON] [54] and [PERSON] [65], as the transfer function also determines exploration and exploitation. The slope of the transfer function can be calculated to evaluate the velocity trend, for the slope indicates the switching speed of position. Figure 5 shows the curve of original \(S_{1}(x)\), \(V_{1}(x)\), \(U_{1}(x)\), and \(Z_{1}(x)\) functions described in Tables 2-5. We can find that the slope is small when the value of \(|x|\) is large, while the slope is large when the value of \(|x|\) is small. Since the transfer function and scale factor \(F\) combined determine the probability of conversion in the proposed binary QUATRE, a linearly increasing scale factor is presented in this manuscript as shown in Equation (11). \[F=(F_{max}-F_{min})\times\frac{G}{MAXG} \tag{11}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Transfer Function** \\ \hline \(V_{1i}(x)\) & \(T_{1i}(x)=|erf(\frac{\sqrt{\pi}}{2}(x-0.5))|/erf(\frac{\sqrt{\pi}}{2}(2.5))\) \\ \(V_{2i}(x)\) & \(T_{2i}(x)=|\tanh(x-0.5)|/\tanh(2.5)\) \\ \(V_{3i}(x)\) & \(0.5)/\sqrt{1+(x-0.5)^{2}}(2.5)/\sqrt{1+(2.5)^{2}}\) \\ \(V_{4i}(x)\) & \(|\frac{2}{\pi}\arctan(\frac{2}{\pi}(x-0.5))|/\frac{2}{\pi}\arctan(\frac{2}{\pi} (2.5))\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The expressions of the improved V-shaped transfer functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Transfer Function** \\ \hline \(U_{1i}(x)\) & \(T_{1i}(x)=\min\left(|(x-0.5)^{1.5}|,1\right)\) \\ \(U_{2i}(x)\) & \(T_{2i}(x)=\min\left(|(x-0.5)^{2}|,1\right)\) \\ \(U_{3i}(x)\) & \(T_{3i}(x)=\min\left(|(x-0.5)^{3}|,1\right)\) \\ \(U_{4i}(x)\) & \(T_{4i}(x)=\min\left(|(x-0.5)^{4}|,1\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The expressions of the improved U-shaped transfer functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Transfer Function** \\ \hline \(Z_{1i}(x)\) & \(T_{1i}(x)=\sqrt{1-2^{x-0.5}}/\sqrt{1-2^{-2.5}}\) \\ \(Z_{2i}(x)\) & \(T_{2i}(x)=\sqrt{1-5^{x-0.5}}/\sqrt{1-5^{-2.5}}\) \\ \(Z_{3i}(x)\) & \(T_{3i}(x)=\sqrt{1-8^{x-0.5}}/\sqrt{1-8^{-2.5  
matrices are binary, we only denote the continuous matrix, vector or number, for example, \(X^{cont}\) means \(X\) is a continuous matrix. ```
1:The dimension \(D\), population size \(ps\), max iterations \(MAXG\), fitness function \(f(X)\), mutation schema to calculate \(B\), and original transfer function \(T(x)\). Initialization:
2:Initialize searching space \(V\), \(G=1\), position matrix \(X_{G}\)\(X_{pbest,G}=X_{G}\) and calculation \(X_{gbest,G}\). Iteration:
3:while\(G<MAXG|istopCriterion\)do
4: Generate matrix \(M\) by Equation (5), calculate \(\overline{M}\)
5:\(F=(F_{max}-F_{min})\times\frac{G}{MAXC}\)
6: Calculation mutation Matrix B according to mutation schema. 7:\(X_{G+1}^{cont}=M\otimes X_{G}+\overline{M}\otimes B\)
8: Convert every \(x_{ik,G+1}^{cont}\) to probability value \(T(x_{ik,G+1}^{cont})\) by original transfer function.