## 4 Experimental Results
### The Benchmark Results
The present notation designates the summation of ranks for the scenario in which the first algorithm exhibits superior performance compared to the second as \(R^{+}\), whereas the summation of ranks for the scenario in which the second algorithm outperforms the first is represented by \(R^{-}\). In this regard, the null hypothesis of the Wilcoxon signed-rank test states that \"it is assumed that there is no significant difference between the performance of the two algorithms\" [41], and the alternative hypothesis states that there is a significant difference between the performance of the two tested algorithms. If the tested \(p\)-value is less than the significance \(\alpha\), then the null hypothesis is rejected. On the other hand, if the \(p\)-value is greater than \(\alpha\), the null hypothesis is accepted, and it can be concluded that there is no significant difference between the performance of the two algorithms. The Friedman test, on the other hand, is used to compare the performance of more than two related samples and determine if there is a significant difference between them. Here, the algorithm that possesses the minimum rank value is identified as the most effective algorithm, whereas the algorithm with the highest rank is deemed to be the least effective algorithm. Therefore, the null hypothesis of the Friedman test is that there is \"no significant difference between the performances of the tested algorithms\" [41], and the alternative hypothesis is that \"there is a significant difference between the performance of tested algorithms\". The Friedman test uses the test statistic \(Q\), which is calculated as the sum of squared deviations of the ranks of each algorithm from their mean rank, divided by a constant factor. The \(p\)-value obtained from the test is compared to the significance level \(\alpha\) to determine if the null hypothesis should be rejected or not. Table 6 presents the outcomes of numerical simulation conducted across 30 independent runs for each of the ZDT test problems under consideration. The results are presented in the form of mean and standard deviation values of IGD metrics. From the results presented in Table 6, it is observed that the HMOBPSO algorithm showed the best performance on a number of considered problems, including DTLZ1, DTLZ3, DTLZ5, DTLZ6, and DTLZ7, while on other it had a performance similar to that of the best-performing algorithm. To achieve better insight into the obtained data presented in Table 6, the appropriate chart is created with the logarithm of the mean of IGD metric on the y-axis, which is depicted in Figure 10.  
The [PERSON] signed-rank test was used to undertake a statistical pair-wise comparison of the optimization performance of the proposed HMOBPSO and other examined algorithms. Table 7 displays the statistical results obtained from the application of [PERSON]'s signed-rank test on ZDT benchmark problems within the present context. The statistical analysis provided results that were used to assign one of three signs \((+,-,\approx)\) to the comparison of the algorithms. Regarding this matter, the symbol \((+)\) signifies that the initial algorithm exhibits a significantly superior performance compared to the second algorithm, while the symbol \((-)\) indicates that the first algorithm's performance is significantly inferior to that of the second algorithm. The symbol \((\approx)\) denotes that the two algorithms being evaluated demonstrate comparable performance.