## 5 Experimental Results and Cases Analysis
### Evaluation and Comparison  
This section endeavors to provide a comparative analysis of the proposed IMOAHDO algorithm alongside other widely utilized multi-objective optimization algorithms, specifically the fast Non-Dominated Sorting Genetic Algorithm (NSGA-II), the Multi-Objective Human Learning Optimization Algorithm (MOHLO) and the Speed-Constrained Multi-Objective Particle Swarm Optimization Algorithm (SMPSO). To compare the performance of the aforementioned multi-objective algorithms, Hypervolume (HV), Inverted Generational Distance (IGD), and Generational Distance (GD) are employed as evaluation metrics. The calculation method of HV is shown in Section 5.1. The detailed introductions of IGD and GD are as follows. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Group** & \(pr\) & **Mean HV** \\ \hline  & \(pr\) & 0.8 \\ IMOAHDO & \(pi\) & 1.0 \\  & \(k_{pr}\) & 0.6 \\  & \(k_{pi}\) & 0.8 \\ \hline NSGA-II & \(pc\) & 0.8 \\  & \(pr\) & 0.2 \\ \hline MOHLO & \(pr\) & 0.3 \\  & \(pi\) & 0.7 \\ \hline  & \(w\) & 0.7 \\  & \(c1\) & 0.2 \\  & \(c2\) & 0.2 \\  & \(sc\) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A summary of the algorithm parameters. Figure 9: The algorithm HV indicator under different parameters. Figure 10 shows the block plots and line charts of the average HV metric for 30 independent runs of four multi-objective optimization algorithms under 100 test cases. For clarity, the 100 cases are sorted according to the metric results of the IMOAHLO algorithm. It is observed that the IMOAHLO algorithm outperformed other algorithms, achieving better results than NSGA-II and MOHLO, and significantly better than SMPSO, which indicates that IMOAHLO has better performance in multi-objective optimization problems. Figures 11 and 12 show the IGD and GD metrics in the same way, respectively, clearly illustrating the differences among the four algorithms. The smaller the IGD and GD values, the closer the solution set generated by the algorithm is to the true Pareto front, indicating the better the performance of the algorithm. In terms of IGD and GD metrics, IMOAHLO surpasses the other three algorithms significantly. This success is attributed to the algorithm's diverse search strategy and its adaptive mechanism, which continuously adjusts individual search strategies throughout the entire iteration process. Table 3 summarizes the performance comparison of four multi-objective optimization algorithms--IMOAHLO, NSGA-II, MOHLO, and SMPSO--using three commonly accepted metrics. Each entry in the table reports the mean value \(\pm\) standard deviation over 30 independent runs. HV measures the volume of objective space dominated by the obtained Pareto front relative to a reference point, with larger values indicating a better convergence and diversity. In Table 3, IMOAHLO achieves an average HV of 0.5636 \(\pm\) 0.2892, which is higher than NSGA-II (0.5040 \(\pm\) 0.2676), MOHLO (0.4913 \(\pm\) 0.3023), and SMPSO (0.3870 \(\pm\) 0.2193). This result demonstrates that IMOAHLO consistently finds Pareto-front solutions that cover a larger portion of the desirable objective space with a greater spread. IGD quantifies the average distance from points on the true Pareto front to the approximated front; lower IGD values indicate that the algorithm's solutions closely approximate the true Pareto front. IMOAHLO's IGD is 0.1271 \(\pm\) 0.1948, which is substantially lower than MOHLO (0.3477 \(\pm\) 0.3263), SMPSO (1.4313 \(\pm\) 1.7409), and NSGA-II (1.8104 \(\pm\) 2.2911). The much smaller mean and tighter standard deviation for IMOAHLO imply a more accurate  
Figure 12: Comparative results of the four algorithms on the GD metric.