## 5 Discussion
### Solution Comparison  
This section focuses on the solutions obtained from different methods by comparing the three objective function values of the final non-dominated solutions. As previously  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & **Category Preference** & **Objective Importance** & **Time Budget** & **Departure** & **Final Arrival** \\  & **Weights** & **Weights** & **Time Budget** & **Location** & **Location** \\ \hline
1 & [0.3, 0.2, 0.5] & [0.4, 0.3, 0.3] & 9:00–22:00 & [121.60, 38.91] & [121.60, 38.91] \\
2 & [0.25, 0.5, 0.25] & [0.5, 0.3, 0.2] & 8:00–18:30 & [121.62, 38.90] & [121.33, 38.83] \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\
19 & [0.4, 0.4, 0.2] & [0.3, 0.3, 0.4] & 9:30–19:00 & [121.64, 38.93] & [121.64, 38.93] \\
20 & [0.25, 0.25, 0.5] & [0.333, 0.333, 0.333] & 9:00–22:00 & [121.59, 38.91] & [121.59, 38.91] \\ \hline \hline \end{tabular}  
* The complete version is in Table 2 in the Appendix B.  
\end{table}
Table 4: Personalized information of the 20 designed tourist cases.  
Figure 11: The comparison of average, maximum, and minimum HV values for 20 different tourists (INSGA-II, NSGA-II, MOPSO, and MOACO).  
stated, for each tourist and method, the final non-dominated solutions are obtained according to the dominance relation of all non-dominated solutions from 20 independent experiments. Figure 12 depicts the final non-dominated solutions of five methods in the three-dimensional space of objectives. Obviously, compared with the baseline methods, our INSGA-II obtains more feasible non-dominated solutions, and the solutions from INSGA-II are closer to the minimum value of each objective, demonstrating that the INSGA-II algorithm can obtain more optimal solutions and a better Pareto front for the proposed multi-objective problem.  
To quantitatively evaluate the final non-dominant solutions from different methods for all tourists, the distance to reference point (DRP) is calculated in Equation (25), denoted as the Euclidean distance between a final non-dominated solution (representing a feasible route) and the reference point (\(O^{\prime}\)). The three coordinate axes values of \(O^{\prime}\) are the minimum of the three objectives in the tourist's final non-dominated solutions. Lower \(DPR\) indicates that the solution is more aligned to minimize the three objectives.  
\[DRP_{r}=\sum_{obj\in\{C,V,D\}}\left[Nor\Big{(}f_{obj,r}\Big{)}-Nor\Big{(}f_{ obj,ref}\Big{)}\right]^{2} \tag{25}\]  
In Figure 13, for each tourist, with the \(DPR\) of all final non-dominated solutions from each algorithm calculated, the upper boundary, lower boundary, and middle point of the vertical bar indicates the maximum, minimum, and average \(DPR\) of these final non-dominated solutions, respectively. In Table 5, further information is extracted from Figure 13. Compared with baseline methods, the INSGA-II significantly obtains the lowest average \(DPR\) (70%) and the minimum \(DPR\) (95%) for most tourists. Therefore, the INSGA-II outperforms baseline methods in terms of optimality of solutions and can obtain the optimal solution in most cases.  
Figure 12: The final non-dominated solutions from different methods (an example of tourist #12).