## 5 Experimental Results and Analysis
### Parameter Setting of Morfeco-Ms
#### 5.1.1 Determining the Parameters \(N\) and \(c_{max}\)  
To facilitate a more equitable comparison with other optimization algorithms, we have set the termination criterion for our iterative algorithm, MOFECO-MS, not by the number of iterations but by the number of objective function evaluations. Since different optimization algorithms employ varying update strategies at each iteration, using the number of objective function evaluations as the termination criterion ensures a fairer comparison.  
Therefore, under this termination condition, we set the number of iterations to a relatively large value to accommodate the required number of objective function evaluations. The number of objective function evaluations is determined based on the population size. The population size settings were referenced from [22], and we configured different \(N\) and corresponding \(c_{max}\) values, as shown in Table 6.  
Table 6 presents the results of experiments conducted with different combinations of \(N\) and \(c_{max}\). The performance metrics considered are the average hypervolume \(HV_{Ave}\) and the computation time \(t\).  
The combination of \(N=100\), \(c_{max}=1000\times N\) was selected in Table 6 due to its superior performance across the metrics. Specifically:  
* Hypervolume (\(HV_{Ave}\)): The average hypervolume value of \(2.98\times 10^{11}\) indicates a high-quality solution set, demonstrating better convergence and diversity compared to other configurations.
* Computation time (\(t\)): With a computation time of 155.75, this configuration offers a reasonable balance between solution quality and computational efficiency. While there are configurations with slightly higher hypervolume values, they require significantly more computation time, making \(N=100\), \(c_{max}=1000\times N\) the most efficient choice overall.  
In addition, Figure 5 shows Pareto fronts under different \(N\) while keeping \(c_{max}=500\times N\). The Pareto fronts are generated by running the algorithm 15 times, with the HV metric taken as the median value. Among the tested values of \(N(50,100,200)\), \(N=100\) is chosen because it likely provides a good balance between convergence speed and solution quality, as indicated by the Pareto front.  
Figure 6 shows Pareto fronts under different \(c_{max}\) with \(N=100\). The Pareto fronts are generated by varying \(c_{max}\). By observing the Pareto fronts, \(c_{max}=1000\times N\) is selected because it demonstrates a better trade-off between the objective functions (in this case, profit and time). The chosen \(c_{max}\) value ensures that the algorithm can effectively explore the solution space and find optimal solutions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(N\) & \(c_{max}\) & \(HV_{Ave}\) & \(t\) \\ \hline
50 & \(500\times N\) & \(1.23\times 10^{11}\) & 35.18 \\
50 & \(1000\times N\) & \(1.46\times 10^{11}\) & 76.46 \\
50 & \(2000\times N\) & \(1.53\times 10^{11}\) & 187.01 \\
100 & \(500\times N\) & \(1.94\times 10^{11}\) & 71.02 \\
100 & \(1000\times N\) & \(2.98\times 10^{11}\) & 155.75 \\
100 & \(2000\times N\) & \(3.05\times 10^{11}\) & 365.49 \\
200 & \(500\times N\) & \(1.81\times 10^{11}\) & 175.31 \\
200 & \(1000\times N\) & \(2.88\times 10^{11}\) & 357.59 \\
200 & \(2000\times N\) & \(5.65\times 10^{11}\) & 895.15 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Evaluation of \(N\) and \(c_{max}\) on performance metrics.  
Therefore, by selecting this combination, we ensure that our algorithm achieves high-quality solutions within a manageable computational budget, thereby providing a robust and efficient optimization framework.