## 4 Experimental Design and Results
### Case 2: The Performance of HEMOGWO on SCOS Problems  
This section focuses on evaluating the performance of the proposed HEMOGWO algorithm in solving SCOS of varying scales. A real-world case from a printing equipment manufacturer in Shaanxi Province, China, serves as the application background. To advance its digital transformation, the enterprise aims to integrate five categories of core industrial software: ERP, MES, WMS, DCS, and PLM. Upon receiving the integration request, the platform decomposes it into five sub-tasks and performs service matching and composition. Assuming each sub-task has 20 candidate services, this scenario can be abstracted as a SCOS instance of scale \"5-20\". Prior to computing the aggregated QoS metric, the weights for the four attributes--execution time, cost, reputation, and service quality--are uniformly set to 0.25. Parameter settings for all comparison algorithms are configured according to Table 4.  
As the number of sub-tasks \(n\) and candidate services \(m\) increases, the solution space grows exponentially, imposing significant challenges on the algorithm's global search capability and Pareto dominance handling. To further assess the generalizability and scalability of HEMOGWO, nine SCOS problem settings of different sizes are constructed: \(n\in\{\)5, 10, 15\(\}\) and \(m\in\{\)20, 50, 100\(\}\), forming combinations such as 5-20, 5-50, 5-100, 10-20, 10-50, 10-100, 15-20, 15-50, and 15-100. Given that no PFtrue exists for real-world industrial composition optimization problems, this study adopts a widely used heuristic approximation method from existing research. Specifically, all algorithms are independently executed 30 times on each test instance, and the resulting solution sets are combined. A global non-dominated sorting is then applied to extract the set of globally optimal solutions, which is treated as the approximate PFtrue. This approach maximizes coverage of the potential solution space and is extensively applied in multi-objective optimization research [8; 9], effectively supporting fair performance evaluation across algorithms. To further validate the quality of the constructed PFtrue, the distribution of solutions in the objective space is manually examined. Additionally, the consistency of GD and IGD metrics across algorithms confirms that the approximate PFtrue is both representative and reliable for comparative analysis. The GD and IGD metrics are again used to evaluate convergence and distribution performance. To mitigate the influence of stochastic variations, each algorithm is indepen \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Metric** & **Algorithm** & **Mean Rank** & **Final Priority** & **N** & **df** & **Iman-Davenport F** & \(p\)**-Value** \\ \hline \multirow{4}{*}{GD} & HEMOGWO & 1.41 & 1 & & & & \\  & SMOGWO & 2.00 & 2 & & & & \\  & MOGWO & 2.76 & 3 & & & & \\  & MOEAD\_DE & 4.71 & 5 & & & 6.46 \(\times\) 10\({}^{1}\) & 1.18 \(\times\) 10\({}^{-26}\) \\  & MOPSO & 5.59 & 6 & & & & \\  & NSGA-III & 4.53 & 4 & & & & \\  & HEMOGWO & 1.24 & 1 & 17 & (5, 80) & & \\  & SMOGWO & 1.94 & 2 & & & & \\  & MOGWO & 2.82 & 3 & & & & \\  & MOEAD\_DE & 4.82 & 5 & & & 1.42 \(\times\) 10\({}^{2}\) & 2.79 \(\times\) 10\({}^{-38}\) \\  & MOPSO & 5.76 & 6 & & & & \\  & NSGA-III & 4.41 & 4 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: [PERSON] test results for GD and GD metrics on the benchmark functions (significant level is 0.05). dently executed 30 times. The mean and standard deviation of GD and IGD values are reported in Tables 8 and 9, respectively. For clarity, the best-performing values in each metric are highlighted in bold. According to the statistical results, HEMOGWO consistently outperforms the competing algorithms across various problem scales. The specific analysis is as follows. Tables 8 and 9 present the GD and IGD results of all algorithms, respectively, for evaluating their overall distribution performance.