## 2 Methodology
### Optimization Process
2:Evaluation: Compute objective functions: OBJ1, OBJ2, and OBJ3. 3:Ranking: Sort non dominated fronts and identify elite leaders from first front. 4:for each non-elite individual do
5: Select three random elite solutions. 6: Generate a donor vector via bitwise logic (XOR/AND) from selected individuals. 7: Perform crossover and mutation to generate a trial solution. 8: Re-evaluate trial on all objectives. 9:if trial solution dominates current then
10: Replace original with trial. 11:endif
12:endfor
13:Return: Updated binary population representing optimized thinning configurations. ```  
**Algorithm 1** MO-BDE for Antenna Array Thinning  
Equation (6) defines the \(OBJ_{3}\) function, a penalty term designed to regulate the number of active elements in the antenna array. In this formulation, \(N_{T}\) denotes the number of active elements in the current configuration, while \(N\) represents the desired number of elements for array thinning. The term \((N_{T}-N)^{2}\) imposes a penalty on deviations from the target sparsity level, thereby guiding the optimization process toward solutions with a controlled number of active elements. The inclusion of a constant term (1) ensures that the function remains strictly positive, maintaining its significance within the multi-objective optimization framework. The use of \(OBJ_{3}\) enables solutions with various thinning ratiosto be evaluated through non-dominated sorting, facilitating the exploration of diverse trade-offs between array sparsity and radiation performance. MO-BDE is employed for its effectiveness in binary-state optimization and its strong exploratory capability within the solution space. In contrast, MO-PSO is well-suited for continuous parameter tuning such as amplitude weights, with demonstrated efficiency and convergence in large-scale configurations. This hybrid strategy exploits the strengths of both algorithms while maintaining low computational overhead. ```
1:Initialization: Create swarm of particles with integer amplitude weights [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] for active elements. 2:Evaluation: Compute objective functions: OBJ1 and OBJ2. 3:Ranking: Sort non dominated fronts and identify elite leaders. 4: Assign initial personal best (\(pBest\)) for each particle and select global best (\(gBest\)) from elite set. 5:for each non-elite particle do
6: Update velocity using inertia weight \(w\), cognitive component \(c_{1}\cdot(pBest-position)\), and social component \(c_{2}\cdot(gBest-position)\). 7: Apply stochastic mutation to velocity for exploration. 8: Update particle position and re-evaluate fitness. 9: Update \(pBest\) if improved. 10:endfor
11:Adaptation: Adjust parameters (\(w\), \(c_{1}\), \(c_{2}\)) based on convergence rate. 12:Return: Population with optimized excitation weights for active elements. ```  
**Algorithm 2** MO-PSO for Amplitude Excitation Tuning  
Figure 4 outlines the complete framework implementation. The multi-objective strategy is demonstrated in two design cases, reinforcing the structural transformation properties. This approach enables optimized array configurations that balance performance trade-offs across frequency bands, ensuring both spectral efficiency and stable radiation patterns. While the Eisenstein fractal geometry inherently supports dense packing with uniform inter-element distances, a lower bound on spacing is typically im  
Figure 4: Framework optimization flowchart for proposed designs. coupling effects, which can distort the radiation pattern performance. In the present implementation, amplitude excitation weights are rounded to integer values between 1 and 10 to minimize quantization errors and simplify calibration. Moreover, the optimization framework employs amplitude tapering alone, excluding phase shift control to avoid the complexity of the feeding network on such large scale arrays.