## 2 Methodology
### Chaos Game Optimization
For the seed located at the \(i\)th candidate point \(X_{i}\), a die with three red and three green faces is used. If the green face appears upon rolling the die, the seed moves towards the global best solution \(GB\); if the red face appears, it moves towards the \(i\)th mean group \(MG_{i}\). This process is mathematically represented by generating two random integers between 0 and 1, with the possibility of generating two equal integers taken into account, allowing the seed to move along the line connecting \(GB\) and \(MG_{i}\). This aspect is illustrated in Figure 2a, with its mathematical representation provided in Equation (3). \[ \begin{split} Seed_{i}^{1}=X_{i}+\alpha_{i}\times(\beta_{i} \times GB-\gamma_{i}\times MG_{i}),i=1,2,\ldots,n\end{split}  \tag{3}\]  
where \(Seed_{i}^{1}\) represents the seed positioned at the \(i\)th solution candidate point; \(\alpha_{i}\) denotes the movement limitation factor; \(\beta_{i}\) and \(\gamma_{i}\) are vectors containing randomly generated numbers within the range [0, 1]. For the seed positioned at the global best solution point \(GB\), a die with three red and three blue faces is used. Upon rolling the die, if a blue face appears, the seed moves towards the \(i\)th candidate \(X_{i}\); if a red face appears, it moves towards the \(i\)th mean group \(MG_{i}\). Considering the possibility of generating two equal integers, the seed can also move along the line connecting \(X_{i}\) and \(MG_{i}\). This concept is illustrated in Figure 2b, while the mathematical representation is shown in Equation (4). \[ \begin{split} Seed_{i}^{2}=GB+\alpha_{i}\times(\beta_{i}\times X _{i}-\gamma_{i}\times MG_{i}),i=1,2,\ldots,n\end{split}  \tag{4}\]  
where \(Seed_{i}^{2}\) represents the seed positioned at the global best point \(GB\). To adjust the seed located at the mean group point \(MG_{i}\), a die with three blue and three green faces is employed. Upon rolling the die, if the blue face appears, the seed moves towards the \(i\)th candidate \(X_{i}\); if the green face appears, it moves towards the global best \(GB\). Accounting for the potential occurrence of two equal integers, the seed may also traverse the line connecting \(X_{i}\) and \(GB\). This concept is illustrated in Figure 2c, with its mathematical representation provided in Equation (5). \[ \begin{split} Seed_{i}^{3}=MG_{i}+\alpha_{i}\times(\beta_{i} \times X_{i}-\gamma_{i}\times GB),i=1,2,\ldots,n\end{split}  \tag{5}\]  
To improve the mutation phase of the CGO, a fourth seed positioned at the \(i\)th candidate point \(X_{i}\) is introduced for position updating. This seed is allowed to move freely and randomly within the search space. This concept is illustrated in Figure 2d, with the mathematical representation shown in Equation (6). \[ \begin{split} Seed_{i}^{4}=X_{i}\Big{(}x_{i}^{k}=x_{i}^{k}+ \delta\Big{)},k=1,2,\ldots,d\end{split}  \tag{6}\]  
where \(\delta\) is a vector with random number in the range of 0 and 1. The movement limitation factor \(\alpha_{i}\) is intricately integrated into the position updating process to regulate the exploration and exploitation rates of the CGO. This factor is randomly determined by selecting one of the following scenarios:  
\[\alpha_{i}=\left\{ \begin{array}{l}Rand\\ 2\times Rand\\ (\rho\times Rand)+1\\ (\varepsilon\times Rand)+\varepsilon\end{array} \right. \tag{7}\]  
where \(\rho\) and \(\varepsilon\) are two random integers in the range of [0, 1].