# 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  
Since the grid mechanism and selection leader component maintain the diversity of the archive during optimization [13], three parameters were selected (\(Grida\): grid inflation parameter, \(Grid\beta\): leader selection pressure parameter, and \(Gridn\): number of grids per each dimension) to perform a sensitivity analysis before evaluating the performance of the IMOGWO. Figure 5 illustrates the variation of three performance metrics--IGD, Spacing, and HV--with respect to different values of these parameters. In the left subplot, changes in \(Grida\) (from 0.05 to 0.15) show that both IGD and Spacing achieve optimal values around \(Grida\) = 0.1, indicating that this setting offers a good balance between solution spread and convergence accuracy. HV remains relatively stable, suggesting the algorithm's robustness to small variations in this parameter. In the center subplot, increasing the leader selection pressure \(Grid\beta\) from 2 to 6 results in a sharp decline in HV and an increase in IGD and Spacing. This suggests that excessive pressure on the best solutions can reduce diversity and lead to premature convergence. Overall, the results demonstrate that the IMOGWO maintains stable and high-quality performance within a reasonable range of parameter values, with optimal settings around \(Grida=0.1\), \(Grid\beta=4\), and \(Gridn=10\). In the right subplot, the number of grid divisions \(Gridn\) has a more pronounced effect. Increasing \(Gridn\) from 4 to 12 significantly improves IGD and Spacing, implying enhanced selection granularity and improved leader discrimination. However, overly fine grid resolution (e.g., \(Gridn=16\)) may lead to slight performance degradation, likely due to overfitting the selection process. In order to evaluate the performance of the IMOGWO, three algorithms (the non-dominated sorting genetic algorithm II (NSGA II), multi-objective grey wolf optimizer (MOGWO), and multi-objective multi-verse optimizer (MOMVO)) are selected for comparison. The maximum iterations are set by \((D*10,000)/N\), where \(D\) and \(N\) represent the dimensions of the problem and the population size, respectively [2]. The value of \(N\) is currently set to 100. Figure 6 shows the true and calculated Pareto-optimal fronts of four multi-objective algorithms for ZDT1, which is a convex problem. Generally, the calculated Pareto-optimal fronts of the IMOGWO and MOMVO are much closer to the true Pareto-optimal fronts compared to those of MOGWO and NSGA II. MOGWO can only obtain a few solutions on the true Pareto-optimal front, which indicates there is still room for MOGWO to improve the convergence accuracy. MOMVO can only get several discrete optimal solutions, which means it fails to search the whole solution space. Solutions obtained by NSGA II are far away from the true Pareto-optimal front. Therefore, the IMOGWO can obtain the true Pareto front of ZDT1, which shows the best performance of the four algorithms. Figure 7 shows the true and calculated Pareto-optimal fronts of four multi-objective algorithms for ZDT2, which is a concave problem. The performance of the IMOGWO and MOMVO is also better than that of MOGWO and NSGA II. MOGWO and NSGA II cannot even capture the shape of the true Pareto-optimal front. The results of MOGWO cluster in a small region of the first objective, which indicates the archive updating fails to search the whole solution space. In addition, NSGA II obtains a convex Pareto-optimal front. For MOMVO, it can only obtain part of the true Pareto front, which in turn demonstrates that using associative learning is able to enhance the diversity of solution sets. Although the calculated Pareto-optimal front of the IMOGWO has a margin of error in certain regions, it still outperforms the other algorithms. Figure 5: The effects of \(Grida\), \(Grid\beta\), and \(Gridn\) for the IMOGWO and MOGWO. Figure 8 shows the true and calculated Pareto-optimal fronts of four multi-objective algorithms for ZDT3, which has a number of disconnected Pareto-optimal fronts.