## 2 Methodology  
### Artificial Electric Field Algorithm (AEFA)  
The artificial electric field algorithm (AEFA) imitates how charged particles flow in an electrostatic field. The repulsive force between charged particles is ignored by AEFA, and only the attractive force between particles is considered. This enables surrounding low-charge particles to be drawn by high-charge particles in the electric field. Each charged particle in the search area represents a potential solution. The charge associated with a particle determines the quality of the solution. The solution is closer to the ideal solution as the charge increases. Figure 1 illustrates the motion of the charged particles where the circles represent the several particles as they are located across the electric field and the circle sizes represent the quantity values of the charges that the particles carry. Particle \(C_{1}\) experiences a force named Force in Figure 1 and it accelerates in the direction of the other three charged particles as a result of being attracted towards them. As can be seen,\(C_{4}\) has the highest charge and it strongly attract \(C_{1}\). As a result, the direction of the force acting on \(C_{1}\) is more closely drawn towards the direction of \(C_{4}\). Particles in the search area with low charges would therefore gravitate towards those with high charges throughout the AEFA iteration phase, enabling the algorithm to reach to the best outcome [22; 38]. Equations (1) and (2) explains the theoretical formulation of the AEFA [39]. \[Vel_{i}^{t+1}=\ \mathrm{r}\ \times Vel_{i}^{t}+accl_{i}^{t} \tag{1}\]  
\[P_{i}^{t+1}=P_{i}^{t}+Vel_{i}^{t} \tag{2}\]  
The \(i\)th charged particle's velocity is given as \(Vel_{i}^{t+1}\) and the location of that particle is represented with \(P_{i}^{t+1}\) using Equations (1) and (2). The velocity and current position can be updated at iteration \(t+1\); \(r\) denotes random number in the range \([0,1]\). The acceleration of particle at the \(t\) iteration is denoted by \(accl_{i}^{t}\). The formula of acceleration when a particle is exposed to an electric fields' force is defined in Equation (3) as follows by [PERSON]'s second law,  
\[accl_{i}^{t}=\frac{C_{i}^{t}\times E{F_{i}^{t}}}{m_{i}^{t}} \tag{3}\]  
where \(m_{i}^{t}\) denote mass of a particle and \(C_{i}^{t}\) represent the charge of a particle. The electric field intensity of the \(i\)th particle at iteration \(t\) is expressed as,  
\[E{F_{i}^{t}}=\frac{Force_{i}^{t}}{C_{i}^{t}} \tag{4}\]  
The charged particle's resulting force in the search space is given as,  
\[Force_{i}^{t}=\sum_{j=1,j\
eq i}^{N}\mathrm{r}\times Force_{ij}^{t} \tag{5}\]  
According to the Coulomb's force, the resulting force exacted on particle \(i\) by \(j\) is computed as follows. \[Force_{ij}^{t}=K^{t}\times\frac{C_{i}^{t}\times C_{j}^{t}\times\left(P_{ \mathrm{best}}^{t}-P_{i}^{t}\right)}{R_{ij}^{t}+\varepsilon} \tag{6}\]  
\[R_{ij}=\left\|P_{i}(t),P_{j}(t)\right\|_{2} \tag{7}\]  
Figure 1: Illustration of the forces exerted on charges. \[K^{t}=K_{0}\times\exp\left(-\alpha\times\frac{\text{iteration}}{\text{Max}_{ \text{iteration}}}\right) \tag{8}\]  
In Equation (7), \(R_{ij}\) denote the distance between two agents \(i\) and \(j\), \(P_{\text{best}}^{t}\) is the particle with ideal solution at the iteration \(i\), and \(\varepsilon\) denote a very tiny positive value. The Coulomb constant \(K_{0}\) is initially set at 500 and drops by a factor of 30 which is denoted as \(\alpha\) in Equation (8) at every iteration. This progressive decline enables the algorithm to start with a global search and switch to a local search later on. Each particles charge \(C_{i}^{t}\) at a given iteration as given as follows,  
\[C_{i}^{t}=\frac{cq_{i}^{t}}{\sum_{i=1}^{N}cq_{i}^{t}} \tag{9}\]  
\[P_{i}^{t+1}=P_{i}^{t}+Vel_{i}^{t} \tag{2}\]0  
In Equation (9), the total number of particles which is represented by \(N\), the normalization of \(cq_{i}^{t}\) is denoted by \(C_{i}^{t}\), where \(cq_{i}^{t}\) is the charge of the \(i\)th particle.