## 3 Proposed Hybrid JADE-DO Algorithm
### Exploration and Exploitation Behavior  
The hybrid JADEDO algorithm is designed to balance exploration and exploitation effectively throughout the optimization process. Exploration refers to the algorithm's ability to investigate diverse regions of the search space, while exploitation focuses on refining solutions in promising areas.  
In the _mutation and crossover_ phase of JADE, the generation of mutant vectors \(V(i,:)\) introduces diversity by combining information from the current population and top-performing individuals. Specifically, Equation (24) utilizes the scaling factor \(F(i)\) and differences between individuals to explore new regions:  
\[V(i,:)=P(i,:)+F(i)\cdot(X_{\text{pbest}}-P(i,:))+F(i)\cdot(P1-P2) \tag{24}\]  
Figure 1: Flowchart of the hybrid JADEDO algorithm.  
where \(V(i,:)\) is the mutant vector for individual \(i\); \(P(i,:)\) is the current individual; \(F(i)\) is the scaling factor; \(X_{\text{best}}\) is a randomly selected individual from the top \(p0\%\) best individuals; and \(P1\) and \(P2\) are randomly selected individuals from the population and the archive \(A\), respectively.  
The _adaptation_ mechanism in JADE adjusts the mean crossover rate \(uCR\) and mean scaling factor \(uF\) based on successful parameters from previous generations, enhancing exploitation by fine-tuning the parameters to focus on promising regions. This is shown in Equations (25) and (26):  
\[uCR=(1-c)\cdot uCR+c\cdot\text{mean}(\text{Scr}) \tag{25}\]  
where \(uCR\) is the updated mean crossover rate; \(c\) is the learning rate; and \(\text{mean}(\text{Scr})\) is the mean of successful crossover rates Scr.  
\[uF=(1-c)\cdot uF+c\cdot\left(\frac{\Sigma(\text{Sf}^{2})}{\Sigma\text{Sf}}\right) \tag{26}\]  
where \(uF\) is the updated mean scaling factor; and Sf is the set of successful scaling factors.  
The integration of the dandelion optimizer introduces additional exploration and exploitation behaviors. In the _rising stage_ (Equation (27)), the algorithm simulates the dispersal of dandelion seeds to explore new areas:  
\[P(i,:)=P(i,:)+\alpha\cdot v_{x}\cdot v_{y}\cdot\text{LogN}(\lambda)\cdot( \text{Best}-P(i,:)) \tag{27}\]  
where \(P(i,:)\) is the \(i\)-th individual in the population; \(\alpha\) is a scaling factor; \(v_{x}\) and \(v_{y}\) are velocity components in the \(x\) and \(y\) directions; \(\text{LogN}(\lambda)\) is a log-normal random Xwith parameter \(\lambda\); and Best is the best individual found so far.  
In the _decline stage_ (Equation (28)), the algorithm exploits the information gathered by converging toward the mean position \(P_{\text{mean}}\):  
\[P(i,j)=P(i,j)-\beta(i,j)\cdot\alpha\cdot(P_{\text{mean}}(j)-\beta(i,j)\cdot \alpha\cdot P(i,j)) \tag{28}\]  
where \(P(i,j)\) is the \(j\)-th component of individual \(i\); \(\beta(i,j)\) is a random factor between 0 and 1; \(\alpha\) is the scaling factor; and \(P_{\text{mean}}(j)\) is the \(j\)-th component of the mean position of the population.  
Finally, the _landing stage_ (Equation (29)) employs a Levy flight mechanism to refine solutions, enhancing exploitation:  
\[P(i,j)=\text{Elite}(j)+\text{Step\_length}(i,j)\cdot\alpha\cdot\left(\text{ Elite}(j)-P(i,j)\cdot\frac{2l}{\text{Max\_iter}}\right) \tag{29}\]  
where \(\text{Elite}(j)\) is the \(j\)-th component of the elite (best) individual; \(\text{Step\_length}(i,j)\) is a step length determined by the Levy distribution; \(\alpha\) is the scaling factor; \(l\) is the current iteration number; and \(\text{Max\_iter}\) is the maximum number of iterations.