## 4 Experimental Results
### The Benchmark Results  
In this section, the optimization performance of the proposed MHPSODE algorithm has been evaluated and compared with well-known multi-objective evolutionary algorithms, such as NSGA-II [42], MO_Ring_PSO_SCD [22], and DEMO [43] on CEC2009 benchmark problems. The considered multi-objective optimization test instances released in CEC2009 benchmark consist of 10 unconstrained optimization problems and 3 problems that deal with constrained optimization [45]. In the following simulations the first 10 functions are taken into consideration. Furthermore, the DTLZ-I [46] test problem has also employed for analysis. To analyze and compare the optimization performance of multiple algorithms, in this paper the Inverted Generational Distance (IGD) is employed as the performance metrics [45; 47]. The IGD metric can be calculated based on the expression  
\[IGD=\frac{\sum_{v_{i}\in P_{A}}d(v_{i},P_{A})}{|P^{*}|}, \tag{50}\]  
Figure 8: Pareto frontier for center distance and gearbox efficiency for different ISO grade oils. where \(d(v_{i},P_{A})\) denotes the minimum Euclidean distance between \(v_{i}\) and the points in the non-dominated solution set \(P_{A}\) obtained by the tested algorithm, and \(P^{*}\) is the set of uniformly distributed points in the solution space. To calculate IGD metrics, the 10,000 reference points have been sampled from the obtained Pareto fronts. It should be noted that smaller values of IGD metric points to better performing algorithm. To obtain results for the statistical analysis all the experiments are independently run for 30 times and the standard deviation of the above-mentioned performance metrics has been calculated. To perform the statistical analysis, and determine the statistically significant differences between the obtained results, in this paper Wilcoxon Signed-Rank Test and Friedman test have been performed with the significance level \(\alpha=0.05\). First, to determine the significant differences between the performance of two algorithms, the Wilcoxon signed-rank test has been applied. Here, the sum of ranks for the problem, where first algorithm outperforms the second is denoted with \(R^{+}\), while the sum of ranks, where second algorithm outperforms the first one is symbolized with \(R^{-}\). In this regard, for the null hypothesis of the Wilcoxon signed-rank test it is assumed that: \"there is no difference between the mean results of the two samples\" [48]. Conversely, the alternative hypothesis is that: \"there is a difference in the mean results of the two samples\" [48]. The \(p\) value has been employed and compared to the significance level in the statistical analysis to test the hypothesis. Therefore, the null hypothesis can be rejected when \(p\leq\alpha\). According to the obtained results of the statistical analysis one of the following three signs \((+,-,\approx)\) have been given to the comparison of the algorithms. In this regard, the sign \((+)\) denotes that the first algorithm significantly outperforms the second algorithm, \((-)\) sign means that the first algorithm performed significantly worse than the first one, and \((\approx)\) denotes that the two algorithms taken for consideration have comparable performance. Secondly, the Friedman test [48] has been performed to determine the significant differences in performance between all considered algorithms, which is achieved by comparing the ranks of all algorithms. In this way, the algorithm with a minimum rank value is denoted as the best performing algorithm, while the one with the highest rank is considered to be the worst performing algorithm. For the null hypothesis of the Friedman test it is assumed that: \"there is no difference among the performance of all algorithms\", while the alternative hypothesis is \"there is a difference among the performance of all algorithms\" [48].