## 3 Pelican Optimization Algorithm  
We found a population-based optimization algorithm [22] inspired by pelicans, the POA. The algorithm simulates evolutionary processes in an ecosystem by treating pelicans as individuals in a population. Each individual represents a potential solution and provides optimization suggestions, which are derived from setting the problem variable to the location of each individual in the search space. In the process of population initialization, in order to ensure the diversity of the population and the global search ability, each member is randomly initialized within the specified upper and lower bounds of the problem, as shown in Equation (1). \[x_{(i,j)}=l_{j}+rand\cdot(u_{j}-l_{j}),i=1,2,\ldots,N,j=1,2,\ldots,m, \tag{1}\]  
In the given equation, the variable \(x_{i,j}\) represents the value of the \(j\)th variable in the \(i\)-th candidate solution. \(N\) is the total number of members. The number of problem variables is denoted by \(m\), indicating the quantity of features or parameters to be optimized. It is important to note that the term \"rand\" in this context represents a random number generator that produces random numbers between 0 and 1, introducing randomness during the algorithm's execution. The variables \(l_{j}\) and \(u_{j}\) represent the lower and upper bounds of the \(j\)th problem variable, respectively, and these bounds are significant for controlling the range of the solution space. To update potential solutions, the algorithm (POA) mimics the tactics and behaviors that pelicans use when attacking and hunting prey. This hunting technique consists of two parts:  
(i) Approaching prey while in the exploring phase. In the first stage, the method by which pelicans approach when they spot prey is simulated and mathematically reproduced in Formula (2). \[x_{i,j}^{P_{1}}= \begin{cases}x_{i,j+rand\cdot(p_{j}-l,x_{i,j})},&\text{if }F_{p}<F_{i}\\ x_{i,j+rand\cdot(x_{i,j}-p_{j})},&\text{else}\end{cases}  \tag{2}\]  
In the context of Equation (3), we can observe the importance of the variable \(x_{i}^{P_{1}}\), which represents an updated state of the pelican in dimension \(j\)th due to the result of stage 1, which can be the \(i\)th pelican. To introduce further diversity and exploration, the value of \(I\) is introduced as a random number that ranges between one and two. Furthermore, the variable \(p_{j}\) is employed to denote the position of the prey in the \(j\)th dimension, while \(F_{p}\) represents the objective function value of the prey. By incorporating Equation (3), we are able to effectively simulate and model this process. \[X_{i}= \begin{cases}X_{i}^{P_{1}},&\text{if }F_{i}^{P_{1}}<F_{i}\\ X_{i},&\text{else}\end{cases}  \tag{3}\]  
In the given context, \(F_{i}^{P_{1}}\) refers to the objective function value obtained during phase 1, while \(X_{i}^{P_{1}}\) represents the updated status of the \(i\)th pelican after phase 1. (ii) Winging on the water surface (exploitation phase). In the later stages, when hunting begins, the pelican serves as a storage function for fish after collection through a unique pouch on its neck. This action allows the pelicans to efficiently capture the fish and store them for consumption. The equation models the movement and interaction of the pelicans with the fish during this phase, taking into account factors such as the position and velocity of the pelicans, the behavior of the fish, and the surrounding environment. The mathematical representation provides a means to simulate and analyze the hunting behavior of pelicans in a quantitative manner. \[x_{i,j}^{P_{2}}=x_{i,j}+R\cdot\left(1-\frac{t}{T}\right)\cdot\left(2\cdot rand -1\right)\cdot\ x_{i,j} \tag{4}\]  
In Equation (5), the variable \(X_{i,j}^{P_{2}}\) denotes the updated state of the \(i\)th pelican in the \(j\)th dimension, which depends on phase 2. The constant \(R\) is set to 0.2, and \(R\cdot\left(1-t/T\right)\) represents the radius of the immediate neighborhood around \(x_{i,j}\).