# Passenger Flow-Oriented Metro Operation without Timetables
## 3 Mathematical Description
### System Constraints  
The dwelling time could be adjusted in operation, and, in doing so, the time interval between trains would be changed. Therefore, a lower bound for the time interval should be provided to guarantee safety in operation. \[h_{i,\ i-1,\ s}\geq\ h_{min} \tag{1}\]  
Meanwhile, the departure time interval between two trains in the first station should be included in the collection of optional intervals. \[h_{lh_{i,\ i-1}}\in H_{i} \tag{2}\]  
The dwelling time for each train should be chosen from the optional dwelling times set. \[\omega_{ld_{i,s}}\in\Omega_{s}^{i} \tag{3}\]  
The number of passengers on a train after its departure from a station should be less than or equal to the train's maximum capacity. \[p_{i,s}^{train}\leq\ C_{i}^{max} \tag{4}\]  
The arrival time of train \(i\) at the first station equals the departure time of train \(i-1\) plus the departure time interval according to level choice for train \(i\). \[a_{i,\ 0}=d_{i-1,\ 0}+h_{lh_{i,\ i-1}} \tag{5}\] \begin{table}
\begin{tabular}{c l} \hline
**Notation** & **Definition** \\ \hline \(\overline{S}\) & Set of stations in the metro system \\ \(\overline{I}\) & Set of train services \\ \(T\) & Set of times in the research time horizon \\ \(i\) & Index of train service numbers \\ \(s\) & Index of station numbers \\ \(t\) & Index of times \\ \(t_{D}\) & Detecting time \\ \(p_{i,(s+1)}^{pre}\) & Predetermined running time for train \(i\) to travel from station \(s\) to station \(s+1\) \\ \(p_{i,(s+1)}^{pre}\) & Operational running time for train \(i\) to travel from station \(s\) to station \(s+1\) \\ \(\Omega_{s}^{i}\) & Set of optional dwelling times for train \(i\) at station \(s\) \\ \(\omega_{ld_{i,s}}\) & Dwelling time for train \(i\) at station s, according to the choice of dwelling time level \\ \(H_{i}\) & Set of optional original time interval for train \(i\) \\ \(h_{lh_{i,\ i-1}}\) & Original time interval between train \(i\) and train \(i-1\) at station 0, according to choice of departure time level \\ \(h_{i,i-1,s}\) & Time interval between train \(i\) and train \(i-1\) at station \(s\), \(s>0\) \\ \(h_{min}\) & Minimum time interval between two trains \\ \(a_{i,s}\) & Arrival time of train \(i\) at station \(s\) \\ \(d_{i,s}\) & Departure time of train \(i\) at station \(s\) \\ \(\lambda_{s}^{s}(t)\) & Average passenger entering speed at station \(s\) to station s\({}^{\prime}\) \\ \(p_{i,s}^{station}\) & Passengers at station \(s\) waiting for train \(i\) \\ \(p_{i,s}^{left}\) & Passengers left by train \(i\) at station \(s\) \\ \(C_{i}^{max}\) & Maximum capacity of train \(i\) \\ \(c_{i,s}\) & Capacity of train \(i\) after passengers get off at station \(s\) \\ \(p_{i,s}^{off}\) & Number of passengers who want to get off at station \(s\) on train \(i\) \\ \(p_{i,s}^{on}\) & Number of passengers who can get on train \(i\) at station \(s\) \\ \(p_{i,s}^{train}\) & Number of passengers on train \(i\) when it is departing from station \(s\) \\ \hline \end{tabular}
\end{table} 
Table 3: Notations and Parameters in this Problem. The operational running time between different stations is predetermined if the time interval between two trains is larger than the minimum interval; otherwise, it should be increased to guarantee the minimum interval. \[r_{i,(s,s+1)}^{ope}=\left\{ \begin{array}{cc}r_{i,(s,s+1)}^{pre}\,&r_{i,(s,s+1)}^{ pre}+d_{i,\ s-1}-d_{i-1,\ s}\geq h_{min}\\ h_{min}-d_{i,\ s-1}+d_{i-1,\ s},&r_{i,(s,s+1)}^{pre}+d_{i,\ s-1}-d_{i-1,\ s}<h_{ min}\end{array} \right. \tag{6}\]  
The arrival time \(a_{i,\ s}\) of train \(i\) at station \(s\) equals the sum of the departure time at station \(s-1\) and the operational running time between these two stations. \[a_{i,\ s}=d_{i,\ s-1}+r_{i,(s,s+1)}^{ope},\ \ 0<s,\ s\in\overline{S} \tag{7}\]  
The departure time \(d_{i,\ s}\) of train \(i\) at station \(s\) is equal to the sum of the arrival time \(a_{i,\ s}\) and the dwell time \(\omega_{Id_{i,s}}\).