## 3 Mathematical Representation of the Optimization Technique
### Jellyfish Optimization Algorithm:
\[a\ =\ -1+t\bigg{(}\frac{-1}{MaxIt}\bigg{)} \tag{35}\]  
\[v\ =\ (a-1)\times rand+1 \tag{36}\]  
The constant, \(a\), exhibits a linear reduction from \(-1\) to \(-2\) during the iterative process. This well-constructed temporal change ensures both exploration and exploitation in the MFO algorithm. **C. Hybrid Jellyfish Search-Moth Flame Optimizer (JS-MFO)**  
In the artificial JS optimizer, the movement toward an ocean current represents the exploration phase, while the movement within a jellyfish swarm depicts the exploitation phase. A time control mechanism alternates between these two processes [28]. For passive motion (type A) within the jellyfish swarm, each jellyfish updates its position (new solution) relative to its current location based on (24), exploiting the current solution without focusing on the best solutions found nearby. However, exploiting the best solutions in the vicinity of the current solution can help to discover more promising solutions. For active motion (type B), referring to the initial position of jellyfish (i) (current solution (i)), a jellyfish (j) is randomly selected to determine the step size and direction of movement. This updates the position of jellyfish (i) (solution (i)) using (28). This random selection of jellyfish (j) neglects the focus on the best positions (i.e., best solutions corresponding to the highest food availability), causing the algorithm to explore newly discovered solutions randomly without exploiting the best solutions or the best global solution found so far, potentially limiting its ability to find promising solutions. These two shortcomings of passive (A) and active (B) motions reduce the efficiency of the JS algorithm in the exploitation phase. On the other hand, and based on the literature review, the MFO algorithm signals a useful exploitation capability by carrying out a good search in the proximity of the best solutions. This performance is achieved by using a spiral path around each exploited solution. However, the MFO algorithm presents a drawback; it is unable to share the best global solution obtained so far between agents during the search stage. These characteristics lead to a powerful exploitation mechanism and weak exploration process of the MFO algorithm. Achieving a new efficient optimization approach is the reason behind our hybridization of the JS and MFO algorithms, which benefits from both advantages of the two algorithms, as shown in Figure 1. In this work, the exploitation capability of the basic JS algorithm is improved by introducing the spiral movement of MFO around the best solutions during the search process. In the early iterations of JS and just after the swarm is formed, the search space is not sufficiently explored, so in this case we try to exploit the good solutions found so far (in order to obtain promising solutions) in the first half-course of iterations (t varies from 0 to MaxIt/2); the elite solutions (which represent 30% of the sorted population of solutions, from the best to the worst) are selected to be exploited, and hence they may be imposed on the spiral movement using (37) and (38). After the first half-course of iterations, the search space is relatively well explored, so in the second half-course of iterations (t changes from MaxIt/2 to MaxIt) we concentrate on the exploitation of the best global solution found so far, which must be imposed on the spiral movement using (39) and (40). Figure 1: Flowchart of the JS-MFO algorithm. In such a manner, the elite solutions are exploited to achieve promising solutions in the early iterations, and inversely the best global solution is more exploited in the later iterations to improve the quality of the best solution. \[D_{i} = |X_{i}(t)-X_{\text{{\em elite}}}| \tag{37}\] \[X_{i}(t+1) = D_{i}.e^{\text{{\em plt}}}.cos(2\pi t)+X_{\text{{\em elite}}}\] (38) \[D_{i} = |X_{i}(t)-X^{*}|\] (39) \[X_{i}(t+1) = D_{i}.e^{\text{{\em plt}}}.cos(2\pi t)+X^{*} \tag{40}\]