## 2 Methodology
### Zebra Optimization Algorithm  
Metaheuristic optimization algorithms, by simulating various phenomena and behaviors in nature, provide powerful tools for solving complex optimization problems and play a significant role in scientific research and engineering applications [13]. The zebra optimization algorithm (ZOA) used in this study is a novel swarm intelligence optimization algorithm that features robust optimization capabilities and rapid convergence [14]. Zebras are primarily found in the African savanna and are renowned for their distinctive black and white stripes. They are highly social animals, and their social behavior exhibits a high degree of organization and cooperation. ZOA mimics the collaborative behaviors of zebra groups when searching for food and responding to predators.  
In the ZOA, each zebra represents a potential solution, and the grassland where the zebras reside represents the solution space of the problem. The optimization steps of ZOA are as follows:  
(1) Initialization: An initial population is randomly generated within the solution space, and the position of each zebra is set, as shown in Equation (2).  
\[x_{i,j}=lb_{j}+r\cdot\left(ub_{j}-lb_{j}\right) \tag{2}\]where \(x_{i,j}\) denotes the position of the \(i\)-th zebra in the \(j\)-th dimension; \(ub_{j}\) and \(lb_{j}\) are the upper and lower bounds of optimization, respectively; and \(r\) is a random number within the range [0, 1].  
(2) First Phase (Foraging Behavior): This is the global search phase of ZOA. Zebras update their positions based on their foraging behavior. In ZOA, the leading zebra is considered the best-performing member of the population, guiding other members toward better positions within the search space. The mathematical representation of the foraging behavior is given by Equations (3) and (4).  
\[x_{i,j}^{new,P1}=x_{i,j}+r\cdot\left(PZ_{j}-I\cdot x_{i,j}\right) \tag{3}\]  
\[X_{i}=\begin{cases}X_{i}^{new,P1},&F_{i}^{new,P1}<F_{i}\\ X_{i},&else\end{cases} \tag{4}\]  
where \(X_{i}^{new,P1}\) is the new position of the \(i\)-th zebra during the first phase, \(x_{i,j}^{new,P1}\) is its \(j\)-th dimension value, \(PZ_{j}\) represents the position of the leading zebra in the \(j\)-th dimension, \(I\) is the population variation control parameter, and \(F_{i}\) is the fitness value of the \(i\)-th zebra.  
(3) Second Phase (Defense Strategies Against Predators): This is the local search phase of ZOA. Zebras exhibit different reactions when facing various predators. When encountering large predators (e.g., lions), zebras choose to flee; when encountering smaller predators (e.g., hyenas), zebras gather to confuse or intimidate the predator. In ZOA, these two behaviors occur with equal probability. The mathematical description of the position update is provided in Equations (5) and (6).  
\[x_{i,j}^{new,P2}=\begin{cases}S_{1}:x_{i,j}+R\cdot\left(2r-1\right)\cdot\left( 1-\frac{t}{T}\right)\cdot x_{i,j},&P_{s}\leq 0.5\\ S_{2}:x_{i,j}+r\cdot\left(AZ_{j}-I\cdot x_{i,j}\right),&else\end{cases} \tag{5}\]  
\[X_{i}=\begin{cases}X_{i}^{new,P2},&F_{i}^{new,P2}<F_{i}\\ X_{i},&else\end{cases} \tag{6}\]  
where \(X_{i}^{new,P2}\) is the new position of the \(i\)-th zebra during the second phase, \(x_{i,j}^{new,P2}\) is its \(j\)-th dimension value, \(R\) is a constant (value of 0.01), \(t\) is the current iteration number, \(T\) is the maximum number of iterations, \(P_{s}\) is the switching probability between the two strategies, and \(AZ_{j}\) is the position of the attacked zebra in the \(j\)-th dimension.