## 4 Experimental Results
### The Benchmark Results  
In order to validate the performance in solving complex multi-objective optimization problems with the proposed modified HMOBPSO algorithm, a set of DTLZ and ZDT benchmark problems is employed in this paper. The ZDT benchmark problems were first introduced by [PERSON], [PERSON], and [PERSON] and have since become a standard test suite for multi-objective optimization algorithms [37]. The DTLZ problems are another popular set of benchmark problems for multi-objective optimization, which were introduced by [PERSON], [PERSON], [PERSON], and [PERSON] [38]. The solutions to these problems are known, making it possible to compare the performance of different algorithms. Therefore, these benchmark problems are widely used in the literature to evaluate the effectiveness of multi-objective optimization algorithms. The performance of the algorithm is evaluated based on its ability to find a set of Pareto-optimal solutions that provide a trade-off between the conflicting objectives. The proposed modified HMOBPSO algorithm is compared with other state-of-the-art multi-objective optimization algorithms, such as NSGA-II [16], MOPSO [33], CMOPSO [34], and MOBOA [35], on the ZDT and DTLZ benchmark problems to demonstrate its effectiveness. To compare the optimization performance among the tested algorithms, the Inverted Generational Distance (IGD) is employed as the commonly used metric for evaluating the quality of a set of solutions obtained using a multi-objective optimization algorithm [39]. It measures the distance between the obtained solutions and the true Pareto-optimal front. The lower the IGD value, the better the quality of the solutions obtained. In addition to IDG, other performance metrics such as the hypervolume and spread are also used in the literature [39; 40]. The IGD value is calculated using the following expression:  
\[IGD=\frac{\sum_{v_{i}\in P_{A}}d(v_{i},P_{A})}{|P^{*}|}, \tag{45}\]  
where \(d(\cdot)\) represents the smallest possible Euclidean distance between \(v_{i}\) and the non-dominated solution set's points \(P_{A}\), which are provided by the tested algorithm, \(P^{*}\) is a set of uniformly distributed points along the true Pareto-optimal front, and \(|\cdot|\) denotes the operator that returns the number of the elements in the set. Ten thousand reference points were chosen at random from the resulting Pareto fronts to construct IGD metrics. A Comparative Analysis of Statistical Performance  
The Wilcoxon Signed-Rank and Friedman tests are commonly used statistical tests to compare the performance of different algorithms based on their IGD values [41]. The Wilcoxon Signed-Rank test is a non-parametric test used to compare two related samples, while the Friedman test is a non-parametric test used to compare more than two related samples. Since both of these tests are non-parametric tests, they do not require assumptions regarding the distribution of the data. These tests help to determine if there is a statistically significant difference between the algorithms' performances or if they perform similarly. Overall, the use of performance metrics and statistical tests is essential in determining the effectiveness of multi-objective optimization algorithms and selecting the most suitable algorithm for a given problem. In order to carry out statistical analysis in this paper, the experiments were conducted independently for a total of 30 iterations, and the standard deviation of the aforementioned performance metrics was computed. The statistical tests were performed with a significance level of \(\alpha=0.05\). The Wilcoxon signed-rank test was applied in order to compare the performance of the two algorithms and reveal if the first algorithm taken for comparison outperforms the second method statistically.