## 3 Methodology
### Model Construction: Two-Stage Multi-Objective Model for TTDP-CD
#### 3.4.3 Two-Stage Route Strategy  
The unanticipated and uncontrollable crowd surge within the attraction would substantially impact the experience and safety. Therefore, this paper proposes a two-stage route strategy that dynamically adjusts the routes as tourists travel along the initial tour route designed by the constructed multi-objective model. At the end of the visit to an attraction, if the next attraction is overcrowded, the model will generate a candidate set consisting of alternative attractions to the overcrowded attraction and re-plan the remaining route for the tourist according to the three proposed objectives. Figure 3 depicts the detailed process of the two-stage route strategy.  
Figure 3: Process of the two-stage route strategy.  
In Part I, the tour commences in accordance with the initial route, where the grey attractions represent completed trips. At the decision point, the dynamic adjustment stage will be activated if the crowding at the next attraction B exceeds the crowd threshold; otherwise, the tourist will complete the remaining itinerary as planned and continue to judge at each subsequent decision point. With \(MAX(flow_{i})\) and scale factor \(\varphi\) determined, the crowd threshold \(FThred_{i}\) is calculated as follows:  
\[FThred_{i}=\varphi\cdot MAX(flow_{i}) \tag{21}\]  
When the dynamic adjustment stage is activated, it is necessary to find a candidate set to replace attraction B so that the adjusted route retains similar characteristics to the initial one. The rules for determining the candidate set are as follows: (a) the same category as B; (b) the crowding is less than \(FThred_{i}\) currently; and (c) the crowd structure is similar to that of B, indicating similar audience groups. The final candidate set consists of the top-15 attractions regarding the crowd structure similarity calculated in Section 3.3.  
In Part II, all possible route solutions for the dynamic adjustment stage are presented, where a complete route is a route from the root node to the leaf nodes. These routes include unvisited attractions, such as C and D, and a candidate set of B (e.g., B\({}^{\prime}\), B\({}^{\prime\prime}\)). Our goal is to search for the optimal solution from these potential solutions.  
In Part III, the objectives of the adjusted stage model are identical to those of the initial route stage in Equations (12)-(14), while the mathematical models of the two stages vary in constraints. The change in the distance is inevitable after the adjustment, but an excessive increase in distance indicates a failed adjustment. Therefore, the constraint of the adjusted distance is determined by Equation (22), where \(D_{ini}\) and \(D_{adj}\) are the total route distances before and after the adjustment. Meanwhile, \(\lambda_{d}\) is the distance scaling factor related to the location distribution of attractions and the spatial scale of the city, which can refer to the distances of past tourist trips from mobile tracking data.  
\[D_{adj}<\lambda_{d}D_{ini} \tag{22}\]  
Finally, the optimal adjusted route is determined using the adjusted-stage model and the multi-objective optimization algorithm. The to-visit attractions include attractions C and D from the initial route and an alternate attraction, B. Furthermore, the next segment of transfer will also become the decision point for the next dynamic adjustment.