## 3 Methodology
### Model Construction: Two-Stage Multi-Objective Model for TTDP-CD  
#### 3.4.1 Model Objectives  
As stated in the preceding section, this study aims to design personalized tour routes with a high level of experience value for tourists. The spatial structure of the route is optimized based on the multi-objective orienteering problem with time windows (MOOPTW) model, balancing the total crowding, value, and distance within a given time budget (\(T_{max}\)). The perceived crowding at each attraction depends on the crowding at the visiting time and the length of stay. With the tourist's visiting time determined, the total perceived crowding during the tourist's stay in \(A_{i}\), denoted as \(TPC_{i}\), can be defined as:  
\[TPC_{i}=\left[\sum_{w_{k}\in W}\left(t_{i,w_{k}}\cdot\overline{F}_{i,w_{k}} \right)\right]/t_{i}^{s} \tag{7}\]  
where \(t_{i,w_{k}}\) denotes the stay time in time windows \(w_{k}\); \(\overline{F}_{i,w_{k}}\) is the crowd flow indicator; \(t_{i}^{s}\) is the total stay time in \(A_{i}\). To meet the tourist's personalized preferences, the stay time in attraction \(A_{i}\), denoted as \(t_{i}^{s}\), is adjusted from the historical average stay time \(t_{i}\) based on the tourist's preference for the category of \(A_{i}\). The tourist's stay time at \(A_{i}\), denoted as \(t_{i}^{s}\), is calculated as follows:  
\[\Delta t=\lambda\bigg{(}\alpha_{i}-\frac{\alpha_{n}+\alpha_{c}+\alpha_{e}}{N_{ t}}\bigg{)} \tag{8}\]  
\[t_{i}^{s}=t_{i}+\Delta t \tag{9}\]where \(\alpha_{i}\) denotes the tourist's preference for the category of \(A_{i}\), which is an element of tourist preference weight; \(\lambda\) denote the adjustment scaling factor; \(N_{t}\) denotes the number of categories, and \(N_{t}=3\). Generally, the value of an attraction is affected by objective factors (e.g., the attraction's grade \(G_{i}\)) and subjective factors (e.g., the tourist's preference \(\alpha_{i}\)). The attraction's grade reflects the quality of the landscape, environment, and service. There are five grades, including 5A, 4A, 3A, 2A, and no grade, corresponding to \(G_{i}=5,4,3,2,1\), respectively, according to the official standard of rating for the quality of tourist attractions in China. As for the preference, numerous studies have demonstrated that a tourist's preference for a POI can significantly affect the value obtained at that particular POI [59,60]. In addition to grade and preference, visitors' historical travel patterns are also substantially informative for determining the value of attractions. The attraction's status value (\(S_{i}\)) out of the attractions in the city can be reflected with its degree of centrality (\(D_{i,in}\) and \(D_{i,out}\)) as crowd interaction indicator and denoted as Equation (10). A higher status value indicates greater importance of the attraction node, such as a core transit attraction [40,46]. Moreover, the average stay time (\(t_{i}\)) in the attraction \(A_{i}\) can reflect the value of \(A_{i}\). The more extended the stay at the attraction, the more appealing and worthwhile the attraction is. Therefore, the comprehensive value of attractions can be expressed as Equation (11), where \(\beta\) denotes a constant scaling factor and \(\varepsilon\) represents the natural logarithm. \[S_{i}=D_{i,in}+D_{i,out} \tag{10}\]  
\[V_{i}=\beta\cdot G_{i}\cdot S_{i}\cdot t_{i}\cdot e^{\alpha_{i}} \tag{11}\]  
TTDP-CD's objectives include minimizing the perceived crowding at the attractions along the itinerary (Equation (12)), maximizing the total value of the attractions in the itinerary (Equation (13)), and minimizing the total distance travelled, with the departure and final arrival locations determined by tourists (Equation (14)). To facilitate the calculation, the maximization objective function of value is converted into a minimization one by evaluating the opposite number.