## 3 Mathematical Representation of the Optimization Technique
### Jellyfish Optimization Algorithm:
In order to simulate this behavior, a time control mechanism is introduced by regulating the movement, switching between jellyfish moving in a swarm and following the ocean current. This time control mechanism is modeled as a time control function, which randomly fluctuates over the time between 0 and 1. Equation (29) formulates the time control function as below:  
\[C(t)\ =\ \left[\left(1-\frac{t}{MaxIt}\right)\times\left(2\times\mathrm{rand }(0,\ 1)-1\right)\right] \tag{29}\]where _MaxIt_ is the maximal number of iterations and _rand_ is a random number in the interval [0, 1]. (d) Initialization of population  
The initial population of jellyfish is generated randomly. To obtain more diversification of the population, a logistic map is developed, which provides more diverse initial populations than random selection and provides a lower probability of premature convergence [28]. Consequently, the logistic map can be used in the initial population based on the following:  
\[X_{i+1}\ =\ \eta\times X_{i}\times(1-X_{i})\ with\ 0\leq X_{o}\leq 1 \tag{30}\]  
where \(X_{i}\) is the chaotic logistic value of the location of the \(i\)-th jellyfish, \(X_{o}\) is used to generate the initial population of jellyfish, and the parameter \(\eta\) is set to 4. (e) Restriction of boundaries  
During the swarm movement, jellyfish can move outside of the boundaries of the search space; in this case, the location is updated by returning to the opposite bound, referring to Equation (31):  
\[ \begin{array}{ccccc}X^{\prime}_{i,d}&=&(X_{i,d}&-U_{b,d})+L_{b}(\text{d})&if \ X_{i,d}>U_{b,d}\\ X^{\prime}_{i,d}&=&(X_{i,d}&-L_{b,d})+U_{b}(\text{d})&if\ X_{i,d}<L_{b,d}\end{array}  \tag{31}\]  
where \(X_{i,d}\) is the location of the \(i\)-th jellyfish in the \(d\)-th dimension; \(X^{\prime}_{i,d}\) is the updated location after checking the boundary constraints; and \(U_{b,d}\) and \(X_{b,d}\) are the upper and lower bounds of the \(d\)-th dimension in search spaces, respectively. **B. ****Moth Flame Optimization Algorithm**  
As insects, moths share important characteristics with butterflies. The navigation strategy used by moths is a particularly intriguing aspect of their behavior and deserves closer examination. To cover great distances in a straight line, they fly at a fixed angle relative to the moon. This efficient technique is known as transverse orientation [30]. The distance from the light source substantially affects the efficacy of transverse orientation. When the light source is close, the moth spirals around it. This spiral flight path guides the moth to light [31]. The Moth Flame Optimization method was introduced by [PERSON] [30] based on observed behaviors and mathematical models. This method employs moths as potential solutions and their locations are presented as a decision variable vector, allowing each moth to move freely within the problem's solution space. \[MO\ =\ \left[\begin{array}{ccccc}m_{0,1,1}m_{0,1}\ldots\ldots\ldots\ldots m_{0,1,D-1}m_{0,1,D}\\ m_{0,2}m_{0,2}\ldots\ldots\ldots\ldots m_{2,D-1}m_{2,D}\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\  
contrast, the flame represents the most optimal solution that has been discovered so far by the moth. To illustrate the helical flight trajectory of moths in proximity to a certain light source, a logarithmic spiral function is used [31].  
\[\text{MO}_{\text{i}}^{x+1}\ =\ |\text{MO}_{\text{i}}^{x}-F_{l}|_{i}|\cdot e^{bv} \cdot\cos(2\pi v)+Fl_{i} \tag{34}\]  
where \(v\) represents a uniformly distributed random variable ranging from \(-1\) to \(1\). This parameter quantifies the proximity of the moth's subsequent location to the associated flame. To improve the efficiency of exploring the solution space during the first iterations and exploiting the solution space during the final iterations, an adaptive approach is suggested to decrease the parameter t throughout the iterations [31].