## 4 Experiment and Results
### Optimization Results
#### 4.4.1 Initial Route Solutions  
The algorithm is executed using the parameters listed in Table 3. Each run yields a set of non-dominated solutions. The initial stage was executed 20 times independently to avoid random errors and one-time occasionality. We extracted one of the 20 independent experiments and plotted the non-dominated solutions after the convergence in the three-dimensional space composed of three objectives, shown in Figure 8. These non-dominated solutions can be fitted into a three-dimensional surface and distributed evenly on the three targets, demonstrating that our method can obtain a well-distributed Pareto front.  
The final non-dominated solutions are obtained by reevaluating the 20 sets of non-dominated solutions based on the dominance relationship. Each of the final non-dominant solutions can meet the needs of tourists with extensive objective values and has its emphasis on the three objectives. In this paper, a satisfaction evaluation function is developed to select a reference solution from the final non-dominated solutions that best satisfies the importance of different objectives for the tourist. With the tourist's objective-importance weights for three objectives determined, the satisfaction of route \(r\) is calculated as follows:  
\[S_{r}=100\times\left\{1-\left[w_{C}\frac{f_{C,r}-f_{C,min}}{f_{C,max}-f_{C,min }}+w_{V}\frac{f_{V,r}-f_{V,min}}{f_{V,max}-f_{V,min}}+w_{D}\frac{f_{D,r}-f_{D, min}}{f_{D,max}-f_{D,min}}\right]\right\} \tag{24}\]  
where \(f_{C,r}\), \(f_{V,r}\), \(f_{D,r}\) denote the objective function values of route \(r\). A larger value of \(S_{r}\) indicates that the route is better considering the three objectives simultaneously according to the tourist's needs.  
Figure 8: Non-dominated solutions of initial route solutions of one independent experiment. The three axes correspond to the three minimized objective function values. The satisfaction evaluation \(S_{r}\) of each non-dominated solution (represented as a dot) can refer to the values on the color scale according to the dot’s color.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{P}\) & \(\mathbf{P_{c}}\) & \(\mathbf{P_{m}}\) & \(\mathbf{P_{A}}\) & \(\mathbf{G}\) \\ \hline Values & 50 & 0.7 & 0.2 & 500 & 2000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the INSGA-II algorithm.  
Figure 9 depicts routes with three, four, and five spots of the selected reference solutions, with the red areas representing the scope of the to-visit attractions and the line charts representing the crowding flow indicator (\(\overline{F}_{i}\)) of the attractions involved in the routes. In the three routes, there is one 5A attraction (#92 Tiger Beach Ocean Park) and two 4A attractions (#12 Hengshan Temple and #67 Dalian Forest Zoo), which indicates that attractions with higher grades are more likely to be selected due to their high value. The remaining five attractions are not highly rated but are well located (close to highly rated attractions), have a high status in the attraction network, or fall within the tourist's preferred category (cultural sightseeing category). For example, #27 Yanwo Mountain is 1.5 km away from the 5A-rated attraction #92 Tiger Beach Ocean Park and 3.4 km away from the 4A-rated attraction #67 Dalian Forest Zoo; and #107 Zhengjue Lecture Temple and #142 Jade Buddha Temple are cultural sightseeing attractions. Moreover, in the line charts of Figure 9, by correlating the visiting schedule and peak crowding period of each attraction, it can be concluded that the initial routes avoid the historical crowding peak in most cases.