## 3 Materials and Methods
### Problem Description and Formulation
For fair comparison, each low-level heuristic was used in isolation and terminated after 1600 \(\times\) 100 evaluation functions. For each dataset, 30 independent trials were run for each algorithm with different random seeds and a population size of 32 and 50 generations in each run. For all three low-level heuristics, the crossover and mutation probabilities were set to 0.9 and 1/24, respectively, and the corresponding distribution indices were set to 10 and 20, respectively. Distance sharing \(\sigma\) for the UD metric and MOGA was set to 0.01 in the normalized space. This setting was used for SSC and UD as a feedback indicator in the ranking scheme of HHMO_CF_GDA and as a performance measure for comparison. All the algorithms were implemented with the same common sub-functions using Microsoft Visual Studio C++ platform and Ubuntu on Windows 10 with Intel Core i7 computer system. Furthermore, all the experimental parameters were the same as those commonly used in the scientific literature for discrete problems [37]. Table 6 presents the parameter settings used to solve the module cluster problem. ## 5 Results and Discussion

We conducted various experiments to achieve an optimal or near-optimal solution for the multi-objective module cluster problem. We repeated each experiment 30 times for each dataset and calculated the average, maximum, minimum, and standard deviation of the results. For all the performance metrics, a higher value indicates better performance. For each dataset, HHMO_CF_GDA exhibited a better performance compared to the individual low-level algorithms with respect to RNI, SSC, and UD, as presented in Table 7. The results revealed that HHMO_CF_GDA yielded a better solution than the other algorithms. The performance results are also displayed as box plots in Figures 3-5 to provide a clear visualization of the distributions of the simulation data of the 30 independent runs. The results revealed that HHMO_CF_GDA achieved the maximum possible cohesion and minimum coupling, considering MQ, to keep the result as balanced as possible compared to the standalone algorithms. A higher SSC value indicates a better quality of \begin{table}
\begin{tabular}{c c} \hline \hline
**HHMO\_CF Parameters** & **Settings** \\ \hline Population size & 32 \\ \hline Total number of generations & 50 \\ \hline Total number of iterations of HHMO\_CF\_GDA & 100 \\ \hline Stopping criteria & Max iterations \\ \hline Length of individual chromosome & Based on total number of components \\ \hline Number of independent runs & 30 \\ \hline Random seed for NSGAII2 & 1.0 \\ \hline Number of independent runs for NSGAII2 & Based on ranking scheme \\ \hline Number of independent runs for SPEA2 & Based on ranking scheme \\ \hline Random seed for SPEA2 & 11 \\ \hline Number of independent runs for MOGA & Based on ranking scheme \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameter settings of HHMO_CF_GDA and low-level heuristics. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Total Number of Components** & **Total Number of Edges** \\ \hline mutins & 20 & 57 \\ \hline telnet2 & 28 & 81 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Dataset information. non-dominated space, that is, a smaller distance from the actual Pareto optimal front. The SSC metric calculates the size of the objective function space occupied by the solutions surrounding the Pareto optimal front. The higher the UD value, the better the quality of the non-dominated set, that is, the non-dominated front is spread widely along the Pareto optimal front. The UD metric evaluates the distribution of non-dominated individuals over the Pareto optimal front. RNI values lie between 0 and 1, and the metric evaluates the fraction of non-dominated individuals in the population and indicates the quality of the solution. A RNI value of 0 indicates that none of the individuals in a population are non-dominated, whereas a RNI value of 1 indicates that all the individuals in a population are non-dominated.