# An Integrated Optimization Method of High-Speed Railway Rescheduling Problem at the Network Level
## 3 Multi-Objective Integrated Optimization Model
### Objectives
\begin{table}
\begin{tabular}{c c} \hline
**Symbol** & **Definition** \\ \hline \(a_{i}^{c,t}\) & Whether the train-set \(i\) executes train \(t\) of circulation \(c\), \(a_{i}^{c,t}=1\) represents that train-set \(i\) executes train \(t\); \\  & otherwise, \(a_{i}^{c,t}=0\). \\ \(h_{b}^{c,t}\) & Whether the standby train-set \(b\) executes train \(t\) of circulation \(c\), \(h_{b}^{c,t}=1\) represents that train-set \(b\) executes train \(t\); otherwise, \(h_{b}^{c,t}=0\). \\ \(h_{b}\) & Whether the standby train-set \(b\) is operated, \(h_{b}=1\) represents that train-set \(b\) is operated; otherwise, \(h_{b}=0\). \\ \(can^{c,t}\) & Whether the train \(t\) of circulation \(c\) is cancelled, \(can^{c,t}=1\) represents that train \(t\) is cancelled; otherwise, \(can^{c,t}=0\). \\ \(tr_{c^{\prime},t^{\prime}}^{c,t}\) & Whether the train \(t\) in circulation \(c\) is executed by train \(t^{\prime}\) in circulation \(c^{\prime}\), \(tr_{c^{\prime},t^{\prime}}^{c,t}=1\) represents that train \(t\) is executed by the train \(t^{\prime}\); otherwise, \(tr_{c^{\prime},t^{\prime}}^{c,t}=0\). \\ \(tr_{c^{\prime},t^{\prime}}^{c,t,x}\) & Whether the passenger demand \(x\) of train \(t\) in circulation \(c\) is served by train \(t^{\prime}\) in circulation \(c^{\prime}\), \(tr_{c^{\prime},t^{\prime}}^{c,t,x}=1\) represents that demand \(x\) is served by train \(t^{\prime}\); otherwise, \(tr_{c^{\prime},t^{\prime}}^{c,t,x}=0\). \\ \(tr_{b}^{c,t,x}\) & Whether the train \(t\) in circulation \(c\) is executed by standby train-set \(b\), \(tr_{b}^{c,t,x}=1\) represents that train \(t\) is executed by the train \(t^{\prime}\); otherwise, \(tr_{b}^{c,t,x}=0\). \\ \(aat_{j}^{b}\) & Actual arrival time of standby train-set \(b\) at station \(j\) after rescheduling. \\ \(aat_{j}^{b}\) & Actual departure time of standby train-set \(b\) at station \(j\) after rescheduling. \\ \(aat_{j}^{c,t}\) & Actual arrival time of train \(t\) in circulation \(c\) at station \(j\) after rescheduling. \\ \(aat_{j}^{c,t}\) & Actual departure time of train \(t\) in circulation \(c\) at station \(j\) after rescheduling. \\ \(aat_{j}^{c,t}\) & Whether train \(t\) in circulation \(c\) actually stops at station \(j\) after rescheduling, \(aws_{j}^{c,t}=1\) represents that train \(t\) stops at station \(j\); otherwise, \(aws_{j}^{c,t}=0\). \\ \(tos_{s}^{c,t}\) & The sequence of train \(t\) in circulation \(c\) at section \(s\), \(tos_{s}^{c,t}\in Z\wedge tos_{s}^{c,t}\geq 0\) \\ \(tos_{j,t}^{c,t^{\prime},t^{\prime}}\) & The sequence of train \(t\) and train \(t^{\prime}\) in section \(j,j+1\), \(tos_{j,t+1}^{c,t^{\prime},t^{\prime}}=1\) represents that train \(t^{\prime}\) is in front of train \(t\); \\ \(tos_{j,t+1}^{c^{\prime},t^{\prime}}\) & otherwise, \(tos_{j,t+1}^{c^{\prime},t,t^{\prime}}=0\). \\ \(wt_{c^{\prime}_{c,t}}^{c^{\prime},t^{\prime}}\) & Whether train \(t\) in circulation \(c\) is coupled with train \(t^{\prime}\) in circulation \(c^{\prime}\), \(wtc_{c,t}^{c^{\prime},t^{\prime}}=1\) represents that train \(t\) is coupled with train \(t^{\prime}\); otherwise, \(wt_{c^{\prime}_{c,t}}^{c^{\prime},t^{\prime}}=0\). \\ \hline \end{tabular}
\end{table} 
Table 3: Decision variables. In which, \(ET_{itr}\) is the end time of the interruption. \(SS_{itr}\) is the start station of the section where the interruption occurs. In order to ensure the safety of the trains, the headway time between any two trains should be greater than or equal to the minimum headway time, as shown in Constraint (7). \[\left\{ \begin{array}{l}aaI_{j}^{c,t}+TDH\leq aat_{j}^{c,t^{\prime}}+(1-tos_{j ;j+1}^{c,c^{\prime},t,t^{\prime}})\times M\\ adI_{j}^{c,t}+TDH\leq adI_{j}^{c^{\prime},t^{\prime}}+(1-tos_{j;j+1}^{c,c^{ \prime},t,t^{\prime}})\times M\\ \forall c\in C,\forall c^{\prime}\in C,\forall t\in T,\forall t^{\prime}\in T,t \
eq t^{\prime},\forall j\in J\end{array} \right.