## 5 Experimental Results
### Parameters Determination of the Interval Ratio of PTBO  
In our PTBO algorithm, there are two parameters, the alpha and the beta, that need to be allocated to determine the critical intervals about the three phases. In a natural system, we can observe that the elements in the middle meta-stable phase account for the majority, and the elements in the unstable and stable phases occupy only a small proportion. This phenomenon is consistent with the two-eight law (or the 1/5 th rule) [2]. For the case of simplicity, we give a value of 0.8 to beta, which is the proportion of the elements in the meta-stable phase. So, the elements in the unstable and stable phases account for 0.2 in total. The specific interval ratio settings of the three phases are shown in Table 5.  
In order to determine which interval ratio is the most suitable for the PTBO algorithm, we conducted some compared experiments with 50 independent runs according to Table 5. The compared results of different interval ratios are listed in Table 6. In Table 6, the mean values are listed in the first line, and the standard deviations are displayed in the second line. We can intuitively observe that the ratio of prop4 has the best accuracy results compared with the other three ratios. Hence, in the subsequent experiments, we choose the value of beta to be 0.8, and the proportion of the unstable and stable phases is a random ratio of 0.2 in total.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Function ID** & **Functions Name** \\ \hline \multirow{6}{*}{Uni-modal Functions} & F01 & Sphere Function \\  & F02 & Rotated High Conditioned Elliptic Function \\  & F03 & Rotated Bent Cigar Function \\  & F04 & Rotated Discus Function \\  & F05 & Different Powers Function \\ \hline \multirow{6}{*}{Multimodal Functions} & F06 & Rotated [PERSON]’s Function \\  & F07 & Rotated Schaffers F7 Function \\  & F08 & Rotated Ackley’s Function \\  & F09 & Rotated Weierstrass Function \\  & F10 & Rotated Griewank’s Function \\  & F11 & Rastrigin’s Function \\  & F12 & Rotated Rastrigin’s Function \\  & F13 & Non-Continuous Rotated Rastrigin’s Function \\  & F14 & Schwefel’s Function \\  & F15 & Rotated Schwefel’s Function \\  & F16 & Rotated Katsuura Function \\  & F17 & Lunacek Bi\_Rastrigin Function \\  & F18 & Rotated Lunacek Bi\_Rastrigin Function \\  & F19 & Expanded Griewank’s plus Rosenbrock’s Function \\  & F20 & Expanded Scaffer’s F6 Function \\ \hline \multirow{6}{*}{Composition Functions} & F21 & Composition Function 1 (\(n=5\), Rotated) \\  & F22 & Composition Function 2 (\(n=3\), Unrotated) \\ \cline{1-1}  & F23 & Composition Function 3 (\(n=3\), Rotated) \\ \cline{1-1}  & F24 & Composition Function 4 (\(n=3\), Rotated) \\ \cline{1-1}  & F25 & Composition Function 5 (\(n=3\), Rotated) \\ \cline{1-1}  & F26 & Composition Function 6 (\(n=5\), Rotated) \\ \cline{1-1}  & F27 & Composition Function 7 (\(n=5\), Rotated) \\ \cline{1-1}  & F28 & Composition Function 8 (\(n=5\), Rotated) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Benchmark functions tested by PTBO (CEC2013).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ratio Method** & **No.** & **Stable Phase** & **Meta-Stable Phase** & **Unstable Phase** \\ \hline \multirow{3}{*}{Fixed ratio} & Prop1 & 0.05 & 0.8 & 0.15 \\  & Prop2 & 0.1 & 0.8 & 0.1 \\  & Prop3 & 0.15 & 0.8 & 0.05 \\ \hline Random ratio & Prop4 & 0.2 * rand & 0.8 & 0.2 * (1 \(-\) rand) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Different interval ratio settings of the three phases.