# Hybrid Multi-Strategy Improved Butterfly Optimization Algorithm
## 3 Improvement of the BOA  
### Subsection Optimization of Initial Population Positions  
The traditional initialization method of BOA involves randomly generating the positions of butterflies, which is uncertain and does not guarantee that the generated butterfly population will have thoroughness and diversity. This error can further be reflected in the global search efficiency and solution accuracy of the Butterfly Optimization Algorithm. Introducing a chaotic function for parameter initialization can reduce the probability of getting stuck in local optima and enhance the intensity of global search and optimization. The Sine-Plane Map (SPM) is a classic chaotic map known for its ergodicity and randomness, and its performance is superior to that of the Logistic chaotic map and the Henon chaotic map [12]. Therefore, this paper uses SPM chaotic mapping with good traversal rows and uniformity to initialize the new population. The calculation formula is as follows:  
\[x(i+1)=\left\{ \begin{array}{c}mod\left(\frac{x(i)}{\eta}+\mu\text{sin}(\pi x (i))+r,1\right),\;0\leq x(i)<\eta\\ mod\left(\frac{x(i)/\eta}{0.5-\eta}+\mu\text{sin}(\pi x(i))+r,1\right),\;\eta \leq x(i)<0.5\\ mod\left(\frac{1-x(i)}{\eta}+\mu\text{sin}(\pi(1-x(i)))+r,1\right),\;0.5\leq x (i)<1-\eta\\ mod\left(\frac{1-x(i)}{\eta}+\mu\text{sin}(\pi(1-x(i)))+r,1\right),\;1-\eta \leq x(i)<1\end{array} \right. \tag{5}\]  
In the formula, x(i) represents the chaotic variable of the i-th individual in the population, \(\eta\in\)(0, 0.5), \(\mu\in\)(0, 1), and r is a random number between 0 and 1. When using the SPM chaotic mapping, a set of initial chaotic sequences \(x_{N\times dim}=\left\{y_{ij},\;i=1,\;2\ldots N,\;j=1,\;2\ldots dim\right\}\) is randomly generated within the search interval (lb, ub). By mapping the chaotic vector to the algorithm's solution space, the chaotic butterfly population \(Q1=\left\{X_{i}^{SPM},i=1,\;2\ldots N\right\}\) is obtained, where dim represents the dimension and N is the population size. The new butterfly individual is generated by the following formula:  
\[X_{i,j}^{SPM}=X_{lb}^{j}+\left(X_{ub}^{j}-X_{lb}^{j}\right)\times x_{ij} \tag{6}\]  
\[X_{i}^{SPM}=\left\{X_{i,j}^{SPM},\;j=1,\;2\ldots dim\right\} \tag{7}\]  
\(X_{ub}^{j}\) and \(X_{lb}^{j}\) represent the lower and upper bounds of the j-th dimension for the i-th butterfly individual in the new population, respectively. Reverse learning is a concept in machine learning that was first proposed in 2005 [13] and has since been widely used to enhance the performance of classical algorithms. Reverse learning primarily involves generating an inverse solution and comparing it with the current solution, selecting the better one. [PERSON] [14] applied reverse learning to the Particle Swarm Optimization algorithm, which not only increased the diversity of particles but also prevented them from falling into local extrema. It has been proven that this implementation method can improve the quality of solutions and reduce time complexity. However, the inverse solution generated by reverse learning has a fixed distance from the current solution and lacks randomness [15]. Therefore, to optimize the quality of the initial population of the algorithm to the greatest extent, this paper introduces a random reverse learning strategy after generating initial solutions with chaotic mapping. According to the current solution, an inverse solution is produced. The formula for generating the inverse solution is as follows:  
\[x_{i}^{obl}=rand\times(lb+ub)-x_{i} \tag{8}\]In the formula, \(x_{i}^{obl}\) represents the reverse solution generated based on the reverse learning strategy, \(x_{i}\) represents the feasible solution generated by mapping the chaotic sequence produced by the Sine-Plane Map (SPM) into the algorithm's solution space, and \(lb\) and \(ub\) represent the lower and upper bounds of x, respectively.