## 2 Problem Description and Modelling
### Problem Modelling  
According to the above definition, the modelling of personalized route planning problem studied in this paper is as follows.  
**Definition 9 (Personalized scenic tourism route planning problem based on urgency \(\Omega\)).**_The problem is defined as \(\Omega=(G,Q,P(u),\) urgency) for a given road network \(G\), user query condition \(Q\), user preference vector \(P(u)\) and urgency, this paper can plan a personalized scenic tourism route \(R\) meeting the urgency for user \(u\). Specifically, this paper plans the travel route \(R\) with different functions for users according to different values of urgency. That is, when users are anxious to reach a destination, it can plan the route with the shortest travel time; when the user is not anxious to reach the destination, it can plan the route with the largest scenic score for the user; if urgency lies between the two, the route with the best quality ratio will be planned for users. The simultaneous introduction of user preference information in the latter two path planning processes can make the scenic features of path \(R\) conform to user preferences as much as possible. Mathematically, the scenic tourism route planning problem based on urgency is a variant of the AOP, as shown below._  
\[f(R)=\left\{\begin{array}{l}\max_{\begin{subarray}{c}\text{travelTime}(R) \\ =\text{max}\sum_{i=0}^{k}\sum_{j=0}^{k}\frac{1}{travelTime(e_{ij})}\times x_{i, j},\end{subarray}}\qquad urgency\geq 0.7\\ \max_{\begin{subarray}{c}\text{max}qualityRatio(R)\\ =\text{max}\sum_{i=0}^{k}\sum_{j=0}^{k}\frac{scenicScore(e_{ij})}{travelTime(e_ {ij})}\times x_{i,j},\end{subarray}}\quad 0.3< urgency<0.7\\ \max_{\begin{subarray}{c}\text{max}scenicScore(R)\\ =\text{max}\sum_{i=0}^{k}\sum_{j=0}^{k}scenicScore(e_{ij})\times x_{i,j}, \end{subarray}}\quad urgency\leq 0.3\end{array}\right. \tag{3}\]  
subject to  
\[x_{i,j}\in\{0,1\} \tag{4}\]  
\[\sum_{j=1}^{k}x_{0,j}=\sum_{i=0}^{k-1}x_{i,k}=1 \tag{5}\]  
\[\sum_{i=0}^{k}x_{i,0}=\sum_{j=0}^{k-1}x_{k,j}=0 \tag{6}\]  
\[\sum_{i=0}^{k}x_{i,j}=\sum_{j=0}^{k}x_{i,j}\leq 1 \tag{7}\]  
\[\text{dt}(n_{0})=t_{0} \tag{8}\]  
\[travelTime(R)\leq b \tag{9}\]  
\[dt(n_{j})=\sum_{i=0}^{k}\left(dt(n_{i})+travelTime(e_{ij})\right)\times x_{i,j} \tag{10}\]  
where \(x_{i,j}\) represents whether the edge from node \(n_{i}\) to node \(n_{j}\) is included in the path, and \(x_{i,j}=1\) represents that the edge from node \(n_{i}\) to node \(n_{j}\) is included in the path; otherwise, the edges from node \(n_{i}\) to node \(n_{j}\) are not included in the path. \(dt\big{(}n_{j}\big{)}\) is the departure time from node \(n_{j}\), and \(travelTime\big{(}e_{ij}\big{)}\) is the travel time on edge \(e_{ij}\), which is determined by the length of edge \(e_{ij}\) and the travel speed when starting from \(n_{i}\).  
Formula (3) is an objective function, which means minimizing the travel time of the path, maximizing the quality ratio of the path, or maximizing the scenic score of the path according to the difference values of urgency. Formula (5) ensures that the planned path starts from the starting point and ends at the end point. Formula (6) ensures that no edge enters the starting point and no edge leaves the end point. Formula (7) guarantees that any node connected with the edge on the path can pass through once at most. Formula (8) guarantees that the user starts at \(t_{0}\). Formula (9) guarantees that the path travel time does not exceed the time budget specified by the user. Formula (10) calculates the departure time of node \(n_{j}\).