## 5 Results and Discussion
### Algorithm Performance Evaluation
#### 5.1.2 Convergence Performance Analysis  
The whale optimization algorithm (WOA) is recognized for its robust global search capability, which is inspired by the hunting behavior of whales. Its ability to explore large search spaces effectively makes it particularly suitable for complex optimization problems. The sparrow search algorithm (SSA) is known for its ability to balance exploration and exploitation, mimicking the social behavior of sparrows to dynamically adjust its search strategy. This flexibility allows SSA to navigate both global and local search spaces. The dung beetle optimization (DBO) algorithm excels in multi-objective optimization, leveraging the dung beetle's behavior to find optimal solutions by focusing on diverse objective functions and handling conflicting goals. To evaluate the performance of the proposed IGWOHHO algorithm, six benchmark test functions were selected and compared with several swarm intelligence algorithms, including WOA, SSA, DBO, GWO, and HHO. All algorithms were implemented under identical computational conditions using MATLAB 2022b, with a population size of 150 and a maximum of 500 iterations. Each algorithm was independently executed 20 times to ensure statistical reliability. Table 5 summarizes the comparative results across all benchmark functions. The optimal value represents the best solution obtained and is used to evaluate solution accuracy. The average value reflects the algorithm's robustness across multiple runs, while the standard deviation indicates the stability of its convergence behavior. The results indicate that, for the unimodal Schwefel 2.22 function, the IGWOHHO algorithm successfully reaches the theoretical optimum of zero and outperforms all other algorithms across all evaluation metrics. In the case of the multimodal Ackley function, IGWOHHO, along with SSA, DBO, and HHO, achieves the theoretical optimal value. However, IGWOHHO demonstrates superior average convergence and a lower standard deviation, indicating better consistency in performance. For the Penalized function, IGWOHHO exhibits a stronger ability to escape local optima while maintaining higher solution stability compared to other algorithms. Regarding the fixed-dimensional multimodal functions, Foxholes and Kowalik, IGWOHHO outperforms the other algorithms in both optimal and average values. This further confirms its overall advantage in tackling complex optimization problems. To further assess the convergence efficiency of IGWOHHO, its performance is compared with several other swarm intelligence algorithms across six benchmark functions. The convergence curves in Figure 11 demonstrate that although both GWO and HHO exhibit effective search capabilities, they show slightly lower stability and adaptability to complex problem landscapes than IGWOHHO. In contrast, algorithms such as SSA, WOA, and DBO are more susceptible to premature convergence, tend to get trapped in local optima, and display larger fluctuations in accuracy, indicating weaker robustness. The dynamic adjustment mechanism of the IGWOHHO algorithm is further analyzed through the exploration and exploitation distribution illustrated in Figure 12. For unimodal functions, local exploitation increases rapidly as the iterations progress, indicating the algorithm's strong refinement capability. In contrast, for multimodal functions, the algorithm maintains a high level of global exploration during the early stages and transitions smoothly to local exploitation in later iterations. This behavior reflects a well-balanced and adaptive search strategy that effectively responds to different landscape characteristics.