## 2 Materials and Methods
### Outline of Computational Approach
#### 2.5.1 Particle Swarm Optimization (PSO)  
PSO is a widely recognized optimization algorithm inspired by the collective behavior of natural systems, such as the synchronized movement of bird flocks and fish schools. First proposed by [PERSON] and [PERSON] [41], PSO has become a popular and effective method for solving complex optimization problems. As a stochastic, population-based optimization technique, it is particularly well-suited for addressing continuous optimization challenges. The algorithm works by simulating a group of particles, each representing a potential solution within the problem space. These particles move through the solution space, interacting with one another to identify the best possible solution. Each particle updates its position and velocity based on two primary factors--its own best-known position (known as the cognitive component) and the best-known position of the entire swarm (referred to as the social component). The cognitive component guides the particle toward its personal best solution, while the social component encourages it to move toward the swarm's global best solution. By combining these two influences, particles effectively balance individual exploration and collective knowledge, enabling them to converge toward optimal solutions [41]. At its core, PSO relies on an iterative process where particles continuously refine their positions and velocities based on their own experiences and the swarm's collective performance. This process repeats until a predefined stopping condition, such as reaching  
Figure 2: Workflow of the optimization process (Source: author’s own elaboration). a maximum number of iterations or achieving a satisfactory solution, is met. Thanks to its simplicity, efficiency, and adaptability, PSO has been successfully applied in various fields, including engineering, machine learning, and finance. However, the algorithm's performance is highly sensitive to parameter settings, such as inertia weight and acceleration coefficients, which may need to be adjusted depending on the specific problem being solved [41]. The mathematical expressions of the PSO algorithm, including Equations (4)-(8), were adapted from [41]. The function to be optimized is denoted as \(g(y)\), where \(y\) represents the candidate solution vector, \(K\) signifies the dimensionality of the vector, and \(\tau\) stands for the iteration index. Each entity \(Q_{i}(\tau)\) corresponds to a particular position within the search space. This relationship is expressed in Equation (4) [41]. \[Q_{i}(\tau)=(y_{i,1}(\tau),y_{i,2}(\tau),\ldots,y_{i,K}(\tau)) \tag{4}\]  
The velocity \(S_{i}(\tau)\) and position \(Q_{i}(\tau)\) of each entity are modified following the expressions given in Equations (5) and (6). \[S_{i,k}(\tau+1)=\alpha\cdot S_{i,k}(\tau)+\beta_{1}\cdot\gamma_{1,k}\cdot\left[ Q_{i,k}^{best}(\tau)-Y_{i,k}(\tau)\right]+\beta_{2}\cdot\gamma_{2,k}\cdot\left[ Q_{k}^{global}(\tau)-Y_{i,k}(\tau)\right] \tag{5}\]  
\[Y_{i,k}(\tau+1)=Y_{i,k}(\tau)+S_{i,k}(\tau+1) \tag{6}\]  
In these expressions, \(\alpha\) represents the inertia weight factor, \(\beta_{1}\) and \(\beta_{2}\) are the individual and collective learning coefficients, \(\gamma_{1}\) and \(\gamma_{2}\) are randomly generated values within the range \([0,1]\), \(Q_{i}^{best}\) denotes the best-known position of entity \(i\), and \(Q_{k}^{global}\) represents the \(k\)th component of the optimal position within the group [41]. To monitor the individual performance of each entity and determine the optimal solution within the group, the individual best position \(Q_{i}^{best}\) and the global best position \(Q_{k}^{global}\) are iteratively updated according to Equations (7) and (8) [41].