## 2 Problem Description
### Symbol Definition  
For ease of description, Line A represents the origin line of a large transfer passenger flow, and Line B represents the target line of the flow. Meanwhile, we define the following sets, parameters, and variables. **Set:**  
\(\mathbf{N}^{A}/\mathbf{N}^{B}\): Set of trains in the planned timetable of Line A/Line B, and index is \(i\);  
\(\mathbf{S}^{A}/\mathbf{S}^{B}\): Set of stations at Line A/Line B, and index is \(j\) and \(k\);  
\(\mathbf{S}_{i}\): Set of stations that train \(i\) passes through on Line A or Line B;  
\(\{l\}\): Set of cross-line backup trains, including train \(l\);  
\(\mathbf{S}_{i}^{A}/\mathbf{S}_{l}^{B}\): Set of stations that train \(l\) passes through on Line A/Line B;  
**T**: Set of time units, and index is \(m\). **Parameter:**  
\(A_{i,j}/D_{i,j}\): The planned arrival/departure time of train \(i\) at station \(j\);  
\(r_{j-1,j}^{\min,\text{A}}/r_{j-1,j}^{\min,\text{B}}\): The minimum running time between station \(j\) and \(j-1\) on Line A/Line B;  
\(\frac{l_{a}^{th}/l_{b}^{th}}{l_{a}^{th}}\): The minimum tracking interval time for train operation on Line A/Line B;  
\(l_{a}^{p}/l_{b}^{p}\): The minimum departure-arrival interval for train operation on Line A/Line B;  
\(l_{b}^{front}\): The minimum interval between train \(l\) and its preceding train at the cross-line station;  
\(l_{b}^{back}\): The minimum interval between train \(l\) and its following train at the cross-line station;  
\(w_{j}^{\min,\text{A}}/w_{j}^{\min,\text{B}}\): The minimum dwelling time of trains at station \(j\) of Line A/Line B;  
\(w^{0}\): The additional dwelling time provided by train \(l\) for non-transfer passengers to alight at the previous station of the cross-line station;  
\(T^{\min,\text{dev}}\): The minimum deviation value of the planned timetables for Line A and B due to the addition of the cross-line backup train;  
\(e^{d}\): The deviation tolerance coefficient of the planned timetables, and the larger the tolerance coefficient, the smaller the limit on the degree of deviation;  
\(\psi_{d}\): Adjustment step for the deviation tolerance coefficient of the planned timetables;  
\(T^{\min,\text{r}}\): The fastest time for train \(l\) to reach the cross-line station;  
\(e^{\ell}\): The running efficiency tolerance coefficient set for train \(l\);  
\(\psi_{e\ell}\): Adjustment step for the running efficiency tolerance coefficient of train \(l\);  
\(t_{m}\): The \(m\)th timestamp;  
\(o_{j,m}\): The number of passengers arriving at station \(j\) in the \(m\)th time unit;  
\(p_{k,j}\): The proportion of passengers boarding at station \(k\) with the purpose of station \(j\);  
\(q_{j}\): The proportion of passengers transferring out to the corresponding direction of Line B in the passengers who boarding at station \(j\) of Line A with cross-line stations as their target stations for transferring out or exiting. \(L_{\max}\): The maximum passenger capacity of a train;  
M: A reasonably large number. **Decision variable:**  
\(a_{ij}/d_{i,j}\): The arrival/departure time of train \(i\) at station \(j\);  
\(a_{lj}/d_{l,j}\): The arrival/departure time of train \(l\) at station \(j\);  
\(\frac{a_{l,i-1}^{d}/z_{j,i-1}^{b}}{\text{: The decision variable of whether train }l\) is located after train \(i-1\) and before train \(i\) when it is on Line A/Line B.