## 4 A Modified Chaotic Based PSO with Uniform Initialization
### Personal Learning Strategy  
Alongside social learning (\(R_{1}\)) and cognitive learning (\(R_{2}\)), velocity updating is also affected by personal and global learning strategies. In practice, the particles learn from their personal best (_Pbest_) and global best (_Gbest_) positions. The advantage of this learning strategy means PSO has certain benefits such as fast convergence, high reliability and strong global exploration ability. However, behind the fast convergence remains one drawback, which is premature convergence. To overcome this drawback, we utilized the stochastic personal best (_SPbest_) as proposed by [47]. SPbest allows the particles to learn from other excellent individuals in the population and makes the movement of particles more heterogeneous. By enhancing the heterogeneity of the population, we are able to overcome the premature convergence in the PSO algorithm. SPbest is, however, contradictive with standard Pbest, for which particles learn only from their own best personal best. The overall three combination strategies proposed in this study are described in Algorithm 3.  
Equations (5) and (6) express the _SPbest_ technique. _SPbest_ works as follows: first, from the population, two mutually different personal best (_Pbest_) particles in each iteration are randomly selected. Second, based on their fitness value, these selected _Pbest_ are compared to determine the candidate personal best solution (_CPbest_). Second, the better _CPbest_ is compared with the current personal best (_Pbest_) (see Equation (6)). As formulated in Equation (6), if _CPbest_ is less than _Pbest_, then _Spbest_ is equal to _Cpbest_, and _Spbest_ is equal to _Pbest_, otherwise.  
\[\begin{array}{c}CPbest(t)=argmin\{fit(\textit{Pbest}_{a}(t)),\;fit(\textit{ Pbest}_{b}(t))\},\\ a\
eq b\in\{1,\,2,\ldots,N\}\end{array} \tag{5}\]  
where _CPbest(t)_ is the candidate personal best solution, \(fit(\textit{Pbest}_{a})\) is fitness value of \(\textit{Pbest}_{a}\), \(fit(\textit{Pbest}_{b})\) is fitness value of \(\textit{Pbest}_{b}\), \(N\) is number of particles.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Map** & **Function** & **Range** \\ \hline Gauss & \(x_{k+1}=\left\{\dfrac{0,\quad x_{k}=0}{x_{k}}\right.\) & (0,1) \\ \(\dfrac{1}{x_{k}}mod\,1,\quad x_{k}\
eq 0,\;x_{0}=0.7\) & (0,1) \\ \hline Singer & \(x_{k+1}=1.07(7.86x_{k}-23.3x_{k}^{2}+28.75x_{k}^{3}-13.3x_{k}^{4})\) & (0,1) \\ \hline Logistic & \(x_{k+1}=4x_{k}(1-x_{k}),\;x_{0}\
ot\in(0.0,\,0.25,\,0.5,\,0.75,\,1.0)\) & (0,1) \\ \hline Circle & \(x_{k+1}=x_{k}+b-\left(\frac{a}{2\pi}\right)\sin(2\pi x_{k})mod(1)\) & (0,1) \\  & where \(a=0.5,\;b=0.2,\) and the chaotic generate sequence is in \((0,1)\) \\ \hline Sine & \(x_{k+1}=\frac{a}{4}\sin(\pi x_{k}),\;0<a\leq 4\) & (0,1) \\ \hline Sinusoidal & \(x_{k+1}=ax_{k}^{2}\sin(\pi x_{k})\) & \\  & where \(a=2.3,\) and \(x_{0}=0.7.\) It has the following simplified form: & (0,1) \\  & \(x_{k+1}=\sin(\pi x_{k})\) & (\(-1,1\)) \\ \hline Chebyshev & \(x_{k+1}=\cos(k\,cos^{-1}(x_{k}))\) & (\(-1,1\)) \\ \hline Bernoulli shift & \(x_{k+1}=\left\{\begin{array}{ll}\frac{x_{k}}{1-x_{k}},0<x_{k}\leq 1-\lambda \\ \frac{x_{k}-(1-\lambda)}{\lambda},1-\lambda<x_{k}<1\end{array}\right.\) & (0,1) \\  & where \(\lambda=1/2\) as treated in [20]. & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Eight well-known chaotic maps.  
Figure 1: Chaotic value distributions during 100 iterations.  
\[SPbest_{i}(t)=\left\{\begin{array}{l}CPbest(t)fit(CPbest)<fit(Pbest_{i})\\ \mathit{Pbest_{i}(t)otherwise}\end{array}\right. \tag{6}\]  
\[\mathit{Pbest_{i}(t)=argmin\{fit(X_{i}(1)),\ fit(X_{i}(2)),\ldots,fit(X_{i}(t))\}} \tag{7}\]  
Based on this recognition, the standard velocity updating in Equation (1) is replaced by Equation (8) where \(\xi_{1}\) and \(\xi_{2}\) are the chaotic map values.  
\[V_{i}(t+1)=\omega(t)V_{i}(t)+C_{1}\xi_{1}*(SPbest_{i}(t)-X_{i}(t))+C_{2}\xi_{2}* (\mathit{Gbest_{i}(t)-X_{i}(t))} \tag{8}\]  
\[X_{i}(t+1)=X_{i}(t)+V_{i}(t+1) \tag{9}\]