## 2 Materials and Methods  
### 2.1. Moth-Flame Optimization Algorithm  
[PERSON] introduced this nature-inspired algorithm in the year 2015. It relies on the navigation strategy utilized by moths, referred to as transverse orientation [21]. Moths employ this method for nocturnal navigation, maintaining a constant angle to the moon; this allows them to travel in a direct trajectory over extended distances [22; 23]. However, when the moths find artificial lights, they become trapped in a spiral path; this occurs when the lights are exceptionally close compared to the moon [24; 25]. The moths then try to keep a similar angle to the source of light, which leads to a deadly spiral fly path [22; 26]. The moth acts as a search agent or particle, while the artificial light (the flame) represents a local solution. In this way, the search agent (the moth) spirals around the local solution (the flame), getting closer to it as time passes [27; 28]. The MFO algorithm employs a population denoted as \(X\), which simulates the foraging actions of \(N\) moths under the influence of simulated light. A particle \(x_{i}\) represents each moth, so the population is expressed as \(X=\{x_{1},x_{2},...,x_{N}\}\). When the MFO algorithm is initiated, the set of particles \(X^{0}\) is created to form the initial population. This set of particles is defined as \(X^{0}=\{x_{1}^{0},x_{2}^{0},...,x_{N}^{0}\}\), and each initial particle \(X_{i}^{0}=\{x_{i,1}^{0},x_{i,2}^{0},...,x_{i,d}^{0}\}\) is randomly implemented within a low limit (\(lb\)) and a superior limit (\(ub\)) in the dimensional space research. Equations (5) and (6) show the initialization of the particles. \[x_{ij}^{0}=lb_{j}+rand_{ij}(ub_{j}-lb_{j}) \tag{5}\]  
\[i=1,2,...,N;\quad j=1,2,...,d \tag{6}\]where \(x_{ij}^{0}\) is the initialization of particle \(i\) in its \(j\) dimension, the limits are shorted with \(lb_{j}\) and \(ub_{j}\), which are the inferior and the superior limits for the \(j\) dimension, respectively. \(Rand_{ij}\) is a random value corresponding to the particle \(i\) in its dimension \(j\), whose value is in the interval [0,1]. At the beginning of the algorithm, the algorithm parameters are defined, such as the number of particles \(N\), the number of dimensions \(d\), the search station limits \(lb\) and \(ub\), and the maximum number of iterations \(k_{max}\). Each particle moves in the search space by using transverse orientation; this is modeled in a logarithmic spiral \(s_{i}\), which is defined as follows:  
\[s_{i}=D_{i}e^{br}cos(2\pi r)+F_{j} \tag{7}\]  
where \(b\) is a constant representing the shape of the logarithmic spiral; in this algorithm, it has a value of 1 [29; 30]. \(D_{i}\) represents the absolute value of the distance between the particle \(x_{i}\) and the local solution \(F_{j}\), which is expressed as follows [31; 32]:  
\[D_{i}=\mid F_{j}-x_{i}\mid \tag{8}\]  
The parameter \(r\) is a random number between \(-2\) and \(1\). It determines how close the position of the particle will be from a local solution; this value is calculated for every dimension as follows:  
\[r=(a-1)*rand+1 \tag{9}\]  
where \(a\) is a convergence constant used to emphasize exploitation as iterations progress. The defined convergence constant holds a value of  
\[a=-1+k\bigg{(}\frac{N-1}{k_{max}}\bigg{)} \tag{10}\]  
In addition to harnessing the convergence constant to facilitate a seamless transition between the exploration and exploitation stages, an additional strategic dimension comes into play within the MFO algorithm; this involves deliberately managing the number of localized solutions, often denoted as \"flames\", throughout each iteration. This population of localized solutions is systematically whittled down through a gradual and calculated reduction until only a solitary solution endures. This strategic control, executed in tandem with the convergence above constant, enhances the algorithm's convergence trajectory and bolsters its overall efficacy.