## 4 The Improved NSGA-II
### Comparative Verification  
This paper selects four test functions of multi-objective algorithms, including ZDT1, ZDT2, ZDT3, and ZDT6, to assess the effectiveness of the improved NSGA-II [31], which are shown in Table 4. These functions cover convex, concave, and discontinuous features, which can effectively evaluate the comprehensive performance of most multi-objective evolutionary algorithms. The performance of the proposed algorithm is compared with other multi-objective optimization algorithms, such as the multi-objective particle swarm optimization algorithm (MOPSO), the multi-objective evolutionary algorithm based on decomposition (MOEA/D), the strength pareto evolutionary algorithm 2(SPEA2), KNSGA-II [25], and AONSGA-II [26].  
The standard NSGA-II is set population size \(N=100\), iteration times \(gen=300\), crossover probability \(p_{c}=0.8\), and mutation probability \(p_{m}=0.1\). The improved NSGA-II is set with population size \(N=100\), iteration times \(gen=300\), learning rate of \(\alpha=0.06\), discount rate \(\gamma=0.85\), and greediness rate \(\varepsilon=0.6\).  
To avoid making the article too lengthy, only the Pareto optimal frontiers of the traditional NSGA-II and the improved NSGA-II are shown in Figure 4 for each test function. Other optimization algorithms were only compared in terms of the mean inverted generational distance (IGD) and hypervolume (HV) values, as shown in Tables 5 and 6.  
Figure 3: Flow chart of the improved NSGA-II.  
\begin{table}
\begin{tabular}{c c c} \hline
**Test Function** & **Representation** & **Variable Range** \\ \hline ZDT1 & \(f_{1}(x)=x_{1}\) & \(n=30\) \\  & \(f_{2}(x)=h(1-\sqrt{f_{1}/h})\) & \(0\leq x_{i}\leq 1,(i=1,2,\cdots,n)\) \\  & \(h(x)=1+9\sum\limits_{i=2}^{n}x_{i}/(n-1)\) & \\ ZDT2 & \(f_{1}(x)=x_{1}\) & \(n=30\) \\  & \(f_{2}(x)=h(1-(f_{1}/h)^{2})\) & \(0\leq x_{i}\leq 1,(i=1,2,\cdots,n)\) \\  & \(h(x)=1+9\sum\limits_{i=2}^{n}x_{i}/(n-1)\) & \\ ZDT3 & \(f_{1}(x)=x_{1}\) & \(n=30\) \\  & \(f_{2}(x)=h(1-\sqrt{f_{1}/h}-(f_{1}/h)\sin(10\pi f_{1}))\) & \(0\leq x_{i}\leq 1,(i=1,2,\cdots,n)\) \\  & \(h(x)=1+9\sum\limits_{i=2}^{n}x_{i}/(n-1)\) & \\ ZDT6 & \(f_{1}(x)=1-\exp(-4\pi_{1})\sin^{6}(6\pi x_{1})\) & \(n=10\) \\  & \(f_{2}(x)=h(1-(f_{1}/h)^{2})\) & \(0\leq x_{i}\leq 1,(i=1,2,\cdots,n)\) \\  & \(h(x)=1+9(\sum\limits_{i=2}^{n}x_{i}/(n-1))\) & \\ \hline \end{tabular}
\end{table}
Table 4: Representations of four test functions.  
Figure 4: Comparison chart of the Pareto optimal frontier of the test function: (**a**) ZDT1 Pareto optimal frontier comparison chart; (**b**) ZDT2 Pareto optimal frontier comparison chart; (**c**) ZDT3 Pareto optimal frontier comparison chart; (**d**) ZDT6 Pareto optimal frontier comparison chart.  
The IGD and HV mean values obtained from running each function ten times are shown in Tables 5 and 6. The bold portion represents the algorithms with the best results in terms of IGD and HV. To ensure fairness in the comparison, the same population size and number of iterations were used for the other multi-objective algorithms, and the remaining parameters were set to commonly used values.  
According to the above results, although the MOEA/D algorithm has a lower average IGD value on the ZDT6 function, it does not demonstrate greater advantages on other test functions. KNSGA-II has higher average HV values, but its IGD results are not very good. Through comparison, it can be found that the proposed improved NSGA-II has better comprehensive performance and better search capability than other optimization algorithms, providing a foundation for the subsequent trajectory planning and enabling the search for better trajectories.