## 2 Materials and Methods
### Outline of Computational Approach
#### 2.5.7 The Colony-Based Search Algorithm (CSA)  
The CSA is a novel evolutionary computation method developed by [PERSON] and [PERSON] [52] in 2024 to solve numerical optimization problems. This algorithm aims to provide a more effective search process by utilizing different mutation and crossover strategies. In evolutionary algorithms, the ability to maintain diversity within the population is a decisive factor in overall performance. CSA addresses this issue by employing a specialized mechanism that can preserve the numerical diversity of the population over extended periods. The algorithm creates a Clan Matrix containing pattern vectors randomly selected from the Colony Matrix in each iteration, thereby ensuring the preservation of numerical differences among solutions [52]. One of the most notable features of CSA is its use of a multi-component mutation strategy. This method generates new solutions through the random combination of three distinct components with different characteristics. Additionally, the algorithm employs an efficient approach for scaling the guidance vector, optimizing the direction of numerical evolution. Unlike common methods, such as standard differential evolution (DE) and particle swarm optimization (PSO), CSA updates the guidance vectors based on a bijective evolutionary search direction. As a result, each pattern vector evolves toward another pattern vector, enabling the algorithm to explore a broader search space [52]. The mathematical expressions of the CSA, including Equations (54)-(63), were adapted from [52]. The main population of the CSA is referred to as the Colony Matrix (\(Q\)), which comprises a randomly initialized solution vector of size \(N\times T\), corresponding to \(T\) times the size of the Clan Matrix (\(Q\)). The initial Colony Matrix (\(Q^{0}\)) and the objective function values of the colony patterns (\(fitQ^{0}\)) are generated using the procedure described in Equation (54). Here, \(K\) denotes the dimension of the problem, \(G\) represents the objective function, while \(\Lambda_{low,k}\), \(\Lambda_{up,k}\) define the lower and upper boundaries of the search space, respectively [52]. \[\left\{ \begin{array}{l}Q^{0}_{i=1:T\times N,k=1:K}\sim U(\Lambda_{low,k}, \Lambda_{up,k})\\ fitQ^{0}=G(Q^{0})\end{array} \right. \tag{54}\]  
The generation process of the swarm clan, \(Q\) and the objective function values of the patterns \(fitQ^{0}\) in are expressed Equation (55). Here, \(index=randperm(T\cdot N,N)\) represents a random permutation function that selects \(N\) elements from the initial colony. \[[Q,fitQ]=[Q^{0}(index,:),fitQ^{0}(index)] \tag{55}\]  
The scale factor (\(scale\)) used in the CSA to regulate the amplitude of the direction vector is computed using Equation (56). Here \(\Phi\sim LevyDist(\alpha,\beta,N,1)\), follows a Levy distribution, and \(\oslash\) denotes the scaling vector and Hadamard division operator, respectively. The \(rand()\) function in Equation (57) generates a new real number from a uniform distribution \(U(0,1)\) at each call. Therefore, the two calls in the expression \(rand()<rand()\) are independent of each other [52]. \[scale= \begin{cases}(rand(N,\zeta)-0.50)\oslash(rand(N,\zeta)-0.50),&\text{if }\zeta =1\\ sign(rand(N,1)-0.5)^{\circ}\Phi,&\text{if }\zeta=K\end{cases}  \tag{56}\]\[\zeta= \begin{cases}1,&\text{if }rand()<rand()\\ K,&\text{otherwise}\end{cases}  \tag{57}\]  
Levy flights introduce the necessary exploration capability to escape local optima and discover diverse, high-quality solutions within complex search spaces. By enabling occasional long jumps, Levy flights improve global search efficiency, increasing robustness and preventing the algorithm from becoming trapped. The Levy distribution is a probability distribution characterized by heavy tails, which means it has a higher likelihood of generating extreme values compared to a normal distribution [52]. The probability density function (PDF) of the Levy distribution is defined in Equation (58).