## 2 System Model and Problem Formulation
### Standard ChOA  
The ChOA is a nature-inspired metaheuristic algorithm modeled on the social and hunting behaviors of chimpanzees [42]. This algorithm captures the complex strategies chimps use in the wild to hunt prey, balancing exploration and exploitation. By mimicking how chimps work together in roles such as attackers, barriers, chasers, and drivers, the ChOA provides a robust framework for solving optimization problems in various fields. The algorithm introduces chaotic variables to introduce randomness, ensuring that the solution space is thoroughly explored. In the ChOA, each of the four chimp roles plays a crucial part in the hunting process, mimicking how real chimpanzees work together to catch prey. Attackers are the chimps that directly approach the prey, getting as close as possible to capture it, while barriers create obstacles or block the prey's escape routes, limiting its movement. Chasers pursue the prey from different angles, ensuring that it remains within the hunting zone and reducing its chances of escape. Finally, drivers strategically move the prey toward the attackers by directing its path or frightening it into specific areas. Together, these roles ensure a coordinated effort to trap and capture the prey, balancing exploration and exploitation in the optimization process by working in a cooperative and strategic manner. Equations (13)-(17) present the mathematical model for determining the distance between chimps and their prey [42]. \[d=\Big{|}c\cdot X_{prey}(t)-m\cdot X_{chimp}(t)\Big{|}, \tag{13}\]  
\[X_{chimp}\ (t+1)=X_{prey}\ (t)-a\cdot d, \tag{14}\]  
\[a=2\cdot f\cdot r_{1}-f, \tag{15}\]  
\[c=2\cdot r_{2}, \tag{16}\]  
\[m=Chaotic\_value \tag{17}\]  
where \(X_{prey}(t)\) is the prey's position vector; \(X_{chimp}(t)\) denotes the chimp's position vector; \(r_{1}\) and \(r_{2}\) are the random vectors \(\in[0,1]\); \(a,c\), and \(m\) are the coefficient vectors; \(m\) indicates a chaotic vector; and \(f\) is the dynamic vector \(\in[0,2.5]\). Figure 3 illustrates the two main phases of the hunting process. In Figure 3a, the hunting process in the exploration phase is depicted. Here, chimps assume diverse roles: drivers move prey towards attackers, barriers trap it, and chasers close in. The roles are dynamically updated to encourage broader exploration. As the algorithm moves into exploitation, chimps converge towards the optimal solution. Figure 3b shows the transition into this phase, where the positions of the attackers, barriers, drivers, and chasers become more synchronized to corner the prey. Equations (18)-(20) update the positions of the chimps based on their distance from the prey, allowing them to exploit the most promising areas in the solution space [42]. \[\left\{ \begin{aligned} & d_{Attacker}=|c_{1}.X_{Attacker}-m_{1}.X|\\ & d_{Barrier}=|c_{2}.X_{Barrier}-m_{2}.X|\\ & d_{Chaser}=|c_{3}.X_{Chaser}-m_{3}.X|\\ & d_{Dririer}=|c_{4}.X_{Dririer}-m_{4}.X|\\ \end{aligned} \right., \tag{18}\]\[ \begin{cases}X_{1}=X_{Attacker}-a_{1}(d_{Attacker})\\ X_{2}=X_{Barrier}-a_{2}(d_{Barrier})\\ X_{3}=X_{Chaser}-a_{3}(d_{Chaser})\\ X_{4}=X_{Driver}-a_{4}(d_{Driver})\end{cases} , \tag{19}\]  
\[X\left(t+1\right)=\frac{X_{1}+X_{2}+X_{3}+X_{4}}{4} \tag{20}\]  
where \(X_{Attacker}\) presents the best search agent, \(X_{Barrier}\) is the second-best search agent, \(X_{Chaser}\) denotes the third-best search agent, \(X_{Driver}\) is the fourth-best search agent, and \(X(t+1)\) is the updated position of each chimp. Figure 4 visually summarizes the movement and coordination of chimps in the ChOA, where multiple factors like the calculated distance and roles dictate how the chimps close in on their prey. Each type of chimp adjusts its position based on the others, ultimately converging towards the best solution after the exploration and exploitation phases. Finally, Equation (21) introduces a probabilistic approach, where the chimps' next position is influenced by both chaotic variables and prey movements, thus ensuring randomness and diversity in exploration when necessary.