## 7 Butterfly Optimization Algorithm and Improvements
### Chaos Enhanced BOA Algorithm  
Literature shows that the traditional BOA method has high local search ability, but it often prematurely converges to local optima, which degrades optimization efficiency, particularly when dealing with the complicated and multi-modal optimization [37]. Therefore there is a need to modify original BOA and improve optimization performance. This is why some academics have recommended incorporating chaos into BOA in order to boost overall optimization efficiency and avoid the issues in performance [36]. In nonlinear dynamic systems, we can observe a bounded dynamic behavior called chaos, which is described with chaos maps, which generate a limited series of random integers in response to a given starting condition. By analyzing Equation (43), we see that the butterfly's ability to detect the scents emitted off by other butterflies is one of the most crucial aspects in BOA, since it directs their flight throughout the search phase. Thus, when solving complex optimization problems, parameter \(c\) should not be kept constant during the optimization process. Searching with a small \(c\) may cause premature convergence, while a large constant value of \(c\) early in the search phase increases the risk of missing the region of global optimum, which affects optimization performance. Therefore, we propose an adaptive mechanism to change value of sensory fragrance \(c\) during the optimization process which is based on the use of chaos maps, which can be written in the following form  
\[c^{(G)}=e^{-\frac{c}{c_{\max}}}\cdot x_{c}^{(G)} \tag{48}\]  
where \(x_{c}^{(G)}\) represents the chaotic motion produced by chosen chaos map in the \(G\)th generation. The Equation (48) describes a parameter which has the chaotic motion but is upper bounded by the factor \(e^{-\frac{c}{c_{\max}}}\). Numerous distinct chaos maps may be found in literature. The five most popular chaotic maps have been utilised in the current study to change the value of parameter \(c\) and to identify which one is most appropriate for the task at hand. Piecewise map: The Piecewise map is described as  
\[x_{c}^{(G+1)}=\left\{ \begin{array}{ll}\frac{x_{c}^{(G)}}{P},&0 \leq x_{c}^{(G)}<P\\ \frac{x_{c}^{(G)}-P}{0.5-P},&P\leq x_{c}^{(G)}\leq 0.5\\ \frac{1-P-x_{c}^{(G)}}{0.5-P},&0.5\leq x_{c}^{(G)}\leq 1-P\\ \frac{1-x_{c}^{(G)}}{P},&1-P\leq x_{c}^{(G)}\leq 1\end{array} \right. \tag{49}\] \[G=1,2,\ldots,G_{max}. \tag{50}\]  
Iterative map: The expression for the Iterative map can be written as  
\[x_{c}^{(G+1)}=\sin(\frac{P\pi}{x_{c}^{(G)}}),\quad G=1,2,\ldots,G_{max}. \tag{51}\]  
where \(P\) is the parameters choosen as \(P\in(0,1)\). Logistic map: The Logistic map is written as  
\[x_{c}^{(G+1)}=Px_{c}^{(G)}\Big{(}1-x_{c}^{(G)}\Big{)},\quad G=1,2,\ldots,G_{max}. \tag{52}\]  
Sine map: The expression for Sine map can be written as  
\[x_{c}^{(G+1)}=\frac{a}{4}\sin\Big{(}\pi x_{c}^{(G)}\Big{)},\quad G=1,2,\ldots,G _{max}. \tag{53}\]  
where the chaotic motion is achieved for \(a\in(0,4)\). Tent map: The Tent map is written according to the equation  
\[x_{c}^{(G+1)}=\left\{ \begin{array}{cc}\frac{x_{c}^{(G)}}{0.7},&x_{c}^{(G)}< 0.7\\ \frac{10}{3}\Big{(}1-x_{c}^{(G)}\Big{)},&x_{c}^{(G)}\geq 0.7\end{array} \right.G=1,2, \ldots,G_{max}. \tag{54}\]  
In this respect, Figure 3 illustrates the changes of the employed chaos maps given in Equations (49)-(54), with the increase in generations, and the plot chaos based sensory fragrance \(c^{(G)}\) from Equation (48) with the employment of each chaos map expression for setting the value of \(x_{c}^{(G)}\). From the Figure 3, we find that the considered chaos maps generate a number sequence that is both ergodic and irregular, but each considered map has unique properties that are visible on the graph. Moreover, we observe that the exponential factor \(e^{-G/G_{\max}}\) serves as the top boundary for the chaos-based sensory scent \(c^{(G)}\).