## 4 Analysis of Experimental Results
### Cognitive Radio Application
#### 4.2.1 IGEO Problem Solving Model
After 30 experiments, the average of all outcomes is calculated. This investigation used the genetic algorithm (GA) [50], the particle swarm algorithm (PSO) [4], the butterfly optimization algorithm (BOA) [13], and the cat swarm optimization (CSO) [51] as comparison methods. Figure 8 shows the system solution's average maximum system benefit for N = 10 and M = 10. The figure indicates that GA exhibits the best system performance, followed by the IGEO algorithm, while the GEO algorithm exhibits the worst performance. This observation highlights the effectiveness of the improved strategy when utilized in the spectrum allocation model. Figure 9 presents the fairness of each algorithm in 30 distinct channel environments. The results demonstrate that the IGEO algorithm yields the highest user fairness, whereas the GA algorithm results in the worst user fairness. Overall, the IGEO algorithm achieves the best maximization of both system benefits and user fairness. These results suggest that IGEO can more effectively be used to resolve the spectrum allocation problem, compared to the other algorithms. Figure 8: Maximize the system benefit of the algorithms. Figure 9: Fairness under different parameter environments. To investigate the impact of different numbers of users on the average system benefit, in this study, the number of available channels is maintained at a constant M = 10 in the environment, while increasing the number of SUs N from 10 to 30, with an increment of 2. The relationship between the number of users and the average benefit is analyzed, as shown in Figure 10. The results indicate that the average benefit of the cognitive radio system gradually decreases with increasing numbers of SUs. However, the IGEO algorithm outperforms the GA, PSO, BOA, CSA, and GEO algorithms in terms of the average system benefit value obtained. On the other hand, this study looks into how different channel counts affect the overall value of the system. The average advantage of the system for different numbers of channels is obtained while increasing M from 10 to 30, with an increment of 2, as shown in Figure 11. Our results show that when the number of possible spectra in the region rises, the average benefit of the channel gradually increases. Notably, with the exception of BOA, the average benefit of the IGEO algorithm is higher than that of other methods. This underlines once more how well the improved algorithm allocates spectrum. Figure 11: The association between average benefit and frequency spectrum count. Figure 10: The association between average benefit and user count. yield an optimal value of 0.001644. However, the key difference lies in the very small standard deviation of IGEO, indicating its high stability in solving the problem. #### 4.3.3 Car Side Impact Design Problem

The car side impact design problem is a static constrained optimization problem with a single objective and multiple variables. Its main objective is to minimize the total weight of the vehicle. The mathematical model is presented in Appendix B, and the meanings of the specific parameters can be found in [57]. The performances of different algorithms when designing car side collisions are compared in Table 16. Among the compared algorithms, which included GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS, IGEO achieved the best result, with the minimum value of 21.8829. #### 4.3.4 Cantilever Beam Design Problem

The cantilever beam optimization problem involves five variables and a vertical displacement constraint, where each variable has a constant thickness, and the objective is to minimize the weight of the beam. A schematic diagram for this problem is presented in Appendix A. The mathematical model for this problem is given in Appendix B. After applying various optimization algorithms such as GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS, IGEO was used to optimize the cantilever beam design problem. The results obtained using these algorithms are presented in Table 17, where it can be observed that IGEO achieved the best result, with a value of 1.179635.