# 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
The IMOGWO provides the best performance dealing with the ZDT3 benchmark function. Although MOGWO can capture the disconnected feature, the error between the true and calculated Pareto-optimal front is relatively large. So, it has a problem of convergence accuracy. For NSGA II, it shows a Pareto-optimal front with a large error. This might be due to the difficulties of the search space of ZDT3 with a number of discontinuous regions that prevent algorithms from providing accurate results on this test problem. Another reason is that the 30-dimensional version of ZDT3 makes it more challenging for multi-objective algorithms. From the above three benchmark functions, the MOMVO has the same problem that it cannot search the whole solution space. Taking advantage of Manta Ray Foraging and associative learning helps to increase population diversity to avoid premature maturation and falling into local optima, which makes the IMOGWO the best performer in these three benchmark functions. Furthermore, it obviously improves the convergence accuracy compared with MOGWO. In order to further evaluate the performance of the proposed IMOGWO, another two three-objective benchmark functions (Vinnet2 and Vinnet3) are selected, they correspond to a discontinuous and continuous problem, respectively. For Vinnet2, both the IMOGWO and MOGWO work well, as shown in Figure 9. IMOGWO outperforms MOGWO in convergence accuracy, but MOGWO outperforms the IMOGWO in the range of optimal solution search. MOMVO can only find a few optimal solutions, this is the same as the two-objective benchmark functions. Although it has been proven that it has good performance for up to three-objective optimization problems, this algorithm still has room for the improvement of its optima searching ability. As for NSGA II, it does not work for Vinnet2. Figure 8: The true and calculated Pareto fronts of ZDT3. Figure 7: The true and calculated Pareto fronts of ZDT2. As for Vinnet3, the IMOGWO captures the true Pareto-optimal front quite well, as shown in Figure 10. The Pareto-optimal front obtained by MOGWO shifts away from the true Pareto-optimal front, which demonstrates poor convergence accuracy. NSGA II can only obtain part of the true Pareto-optimal front. As regards MOMVO, it lacks a diversity of populations, which restricts the capability of the global search. So far, by comparing the calculated and true Pareto-optimal front, the IMOGWO has shown an improvement in global search and convergence accuracy. Now, in order to obtain the statistical characteristics of four algorithms, each algorithm has to be run 20 times independently. Figure 11 shows the boxplot of the statistical results for the performance metrics (GD, IGD, Spacing, and HV), which are employed to quantitatively analyze convergence and coverage. In Figure 11, GD and IGD show that all Pareto-optimal fronts obtained by the IMOGWO are close to the true Pareto-optimal fronts, which indicates the IMOGWO has the best convergence accuracy. The performance of NSGA II is the worst. The statistical results of Spacing show that Pareto-optimal fronts obtained by MOMVO have better distribution coverage. However, the Spacing range is large. Therefore, it is not very robust compared with the IMOGWO, especially for ZDT3. Finally, HV proves that the IMOGWO has the best comprehensive performance. So, for two-objective optimization problems (ZDT1-3), the performance of these four algorithms can be ranked as follows: IMOGWO > MOMVO > MOGWO > NSGA II. Figure 12 presents the statistical results of three benchmark functions for four algorithms. The values of the GD and IGD indicate the IMOGWO has better convergence accuracy compared with MOGWO. Although NSGA II has a small Spacing value, which implies it has a better distribution coverage of the Pareto-optimal front, the GD and IGD show a larger error between the calculated and true Pareto-optimal front. MOMVO shows a highly competitive convergence; however, the results have a poor distribution coverage. Therefore, the IMOGWO still outperforms the other algorithms overall.