# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 3 Results and Discussion
### Evaluation of Benchmark Functions
#### 3.1.2 Discussion of the Results
For the three  
Figure 9: The true and calculated Pareto fronts of Vienet2. objective optimization problem (Vinnet2-3), the performance of these four algorithms can be ranked as follows: IMOGWO > MOMVO > MOGWO > NSGA II. Furthermore, Table 5 presents the statistical results of the spread metric for the four algorithms--IMOGWO, MOGWO, NSGA II, and MOMVO--across five benchmark problems: ZDT1-ZDT3 and Viennet2-Viennet3. The spread metric measures the uniformity of solution distribution along the Pareto front, where lower values indicate better distribution quality. The IMOGWO demonstrates competitive and stable performance, particularly on ZDT1 (mean = 0.876), ZDT2 (mean = 1.103), and Viennet2 (mean = 1.072), where it achieves both a relatively low mean and standard deviation. Notably, on Viennet3, the IMOGWO outperforms the other methods in terms of both mean spread (1.203) and stability (std = 0.139), suggesting its ability to maintain good diversity on more complex landscapes. Compared to its predecessor MOGWO, the IMOGWO exhibits lower standard deviations across all problems, indicating improved robustness in maintaining spread. While MOMVO shows strong performance in certain cases (e.g., Viennet3), its larger standard deviations (e.g., std = 0.274) point to less consistent diversity control. These results confirm that the IMOGWO achieves a good trade-off between convergence and diversity. Figure 13 presents a comparison of computing time between the proposed IMOGWO and the original MOGWO across five multi-objective benchmark problems: ZDT1, ZDT2, ZDT3, Viennet2, and Viennet3. For the relatively simple problems (ZDT1-ZDT3), both algorithms exhibit low and comparable computational costs, with the IMOGWO requiring only slightly more time due to additional steps such as Bloch coordinate initialization and archive perturbation. On the more complex test problems (Viennet2 and Viennet3), the computing time increases significantly for both methods due to the increased dimensionality and function complexity. Importantly, although the IMOGWO introduces several enhancements to improve convergence and diversity, its computational time remains close to that of MOGWO, with a moderate decrease observed. This confirms that the proposed modifications maintain computational efficiency while offering performance improvements, making the algorithm suitable for practical multi-objective optimization tasks. Figure 10: The true and calculated Pareto fronts of Viennet3. Figure 11: Boxplot of the statistical results for GD, IGD, Spacing, and HV—two objectives. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline. & & IMOGWO & MOGWO & NSGA II & MOMVO & & & IMOGWO & MOGWO & NSGA II & MOMVO \\ \hline \multirow{4}{*}{ZDT1} & max & 1.144 & 1.119 & 0.986 & 1.023 & & max & 1.144 & 1.438 & 0.994 & 1.294 \\  & min & 0.638 & 0.788 & 0.968 & 0.738 & & min & 0.989 & 0.860 & 0.992 & 0.997 \\  & mean & 0.876 & 0.971 & 0.975 & 0.872 & & mean & 1.072 & 1.089 & 0.993 & 1.123 \\  & std & 0.158 & 0.111 & 0.006 & 0.089 & & std & 0.057 & 0.171 & 0.001 & 0.124 \\ \hline \multirow{4}{*}{ZDT2} & max & 1.439 & 1.216 & 0.987 & 1.067 & & & & & \\  & min & 0.773 & 0.734 & 0.968 & 0.830 & & & & & \\  & mean & 1.103 & 0.996 & 0.976 & 0.933 & & & & & \\  & std & 0.253 & 0.120 & 0.007 & 0.076 & & & & & \\ \hline \multirow{4}{*}{ZDT3} & max & 1.044 & 1.218 & 1.000 & 1.948 & & max & 1.360 & 0.987 & 1.190 & 1.724 \\  & min & 0.749 & 0.911 & 0.975 & 0.819 & & min & 0.940 & 0.556 & 0.850 & 1.002 \\ \cline{1-1}  & mean & 0.906 & 1.072 & 0.985 & 1.503 & & mean & 1.203 & 0.834 & 0.980 & 1.323 \\ \cline{1-1}  & std & 0.105 & 0.089 & 0.007 & 0.431 & & std & 0.139 & 0.136 & 0.116 & 0.274 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The statistical results of spread for the different optimization algorithms. Figure 12: Boxplot of the statistical results for GD, IGD, and Spacing—three objectives. Figure 13: A comparison of computing time between the IMOGWO and MOGWO.