## 5 Experimental Results and Analysis
### Experimental Results and Discussion on Algorithms Comparison  
To evaluate the performance of our proposed MOFECO-MS algorithm, we compare it with eight other intelligent optimization algorithms based on different mechanisms. These include the classic Non-Dominated Sorting Genetic Algorithm II (NSGAII) [26], the Multi-objective Particle Swarm Optimization (MOPSO) [33], the Multi-Objective Whale Optimization Algorithm (MOWOA) [34], the Multi-objective Grey Wolf \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline SM1 & \(2.99\times 10^{11}\) & \(1.62\times 10^{11}\) & \(2.36\times 10^{11}\) & \(2.26\times 10^{11}\) & \(4.99\times 10^{10}\) \\ SM2 & \(2.99\times 10^{11}\) & \(1.49\times 10^{11}\) & \(2.31\times 10^{11}\) & \(2.29\times 10^{11}\) & \(5.00\times 10^{10}\) \\ SM3 & \(3.11\times 10^{11}\) & \(2.07\times 10^{11}\) & \(2.56\times 10^{11}\) & \(2.61\times 10^{11}\) & \(3.09\times 10^{10}\) \\ SM4 & \(3.25\times 10^{11}\) & \(2.79\times 10^{11}\) & \(2.93\times 10^{11}\) & \(2.98\times 10^{11}\) & \(1.25\times 10^{10}\) \\ \hline \end{tabular}
\end{table} 
Table 13: Statistical results of HV indicator obtained by four different search methods. Figure 7: The pareto front corresponding to the median HV values obtained by MOFECO-MS algorithm with four search methods. Optimization (MOGWO) [35], the Multi-objective Aquila Optimizer (MOAO) [36], and the Multi-Objective Honey Badger algorithm (MOHBA) [37], all proposed in recent years. Additionally, we include the basic Multi-Objective Five-element Cycle Optimization algorithm (MOFECO) [24] and a Local Search Many-Objective Five-element Cycle optimization algorithm (LSMOFECO) [25]. MOWOA mimics humpback whale hunting behavior, while MOGWO simulates gray wolf group predation behavior. MOAO and MOHBA, proposed in the last two years, are inspired by eagles' hunting and honey badgers' foraging behavior, respectively. The parameters for MOWOA, MOGWO, MOAO, and MOHBA remain consistent with their original literature. NSGAII employs the same crossover and mutation operators as MOFECO-MS, with crossover and mutation probabilities set to 0.9 and 0.1, respectively. MOFECO and LSMOFECO utilize parameters identical to those in MOFECO-MS. All eight algorithms share a population size of \(N=100\). The termination criterion for each algorithm is when the number of objective function evaluations reaches 100,000 (\(c_{\max}=N\times 1000\)). We conduct 15 independent runs for each algorithm on the BITTP instances in Table 5. The experimental results are presented in Tables 14-40, Figures 8-16. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Best** & **Worst** & **Median** & **Average** & **StdDev** \\ \hline NSGAII & \(5.94\times 10^{9}\) & \(2.51\times 10^{9}\) & \(3.83\times 10^{9}\) & \(3.93\times 10^{9}\) & \(1.02\times 10^{9}\) \\ MOPSO & \(7.53\times 10^{8}\) & \(2.88\times 10^{8}\) & \(5.92\times 10^{8}\) & \(5.69\times 10^{8}\) & \(1.38\times 10^{8}\) \\ MOAO & \(9.97\times 10^{8}\) & \(3.06\times 10^{8}\) & \(5.12\times 10^{8}\) & \(5.34\times 10^{8}\) & \(1.68\times 10^{8}\) \\ MOHBA & \(7.41\times 10^{8}\) & \(1.70\times 10^{8}\) & \(4.48\times 10^{8}\) & \(4.42\times 10^{8}\) & \(1.37\times 10^{8}\) \\ MOWOA & \(7.92\times 10^{8}\) & \(2.31\times 10^{8}\) & \(4.35\times 10^{8}\) & \(4.79\times 10^{8}\) & \(1.59\times 10^{8}\) \\ MOGWO & \(6.99\times 10^{8}\) & \(1.59\times 10^{8}\) & \(4.39\times 10^{8}\) & \(4.28\times 10^{8}\) & \(1.37\times 10^{8}\) \\ MOFECO & \(5.04\times 10^{9}\) & \(2.00\times 10^{9}\) & \(3.57\times 10^{9}\) & \(3.48\times 10^{9}\) & \(7.97\times 10^{8}\) \\ LSMOFECO & \(9.46\times 10^{9}\) & \(4.56\times 10^{9}\) & \(9.00\times 10^{9}\) & \(8.47\times 10^{9}\) & \(1.30\times 10^{9}\) \\ MOFECO-MS & \(9.57\times 10^{9}\) & \(7.71\times 10^{9}\) & \(8.64\times 10^{9}\) & \(8.76\times 10^{9}\) & \(4.70\times 10^{8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Statistical results of HV indicator of the eight comparison algorithms for _eil_51_n250.