# An Integrated Optimization Method of High-Speed Railway Rescheduling Problem at the Network Level
## 3 Multi-Objective Integrated Optimization Model
### Objectives
\tag{7}\]  
In Constraint (7), binary variables \(tos_{j;j+1}^{c,c^{\prime},t,t^{\prime}}\) indicate the sequence of train \(t\) and train \(t^{\prime}\) in section \(j,j+1\), \(tos_{j;j+1}^{c,c^{\prime},t,t^{\prime}}=1\) represents that train \(t^{\prime}\) is in front of train \(t\); otherwise, \(tos_{j;j+1}^{c,c^{\prime},t,t^{\prime}}=0\). \(TDH\) is the minimize train delivery headway of two trains. \(M\) is a sufficiently large numerical value. Cross-line trains should meet the constraint of station operation time at boundary stations, as shown in Constraint (8):  
\[ \begin{array}{l}aat_{j}^{c,t}+aws_{j}^{c,t}\times(EST_{j}+TST_{j}+EDT_{j}) \leq adI_{j}^{c,t},\\ \forall c\in C,\forall t\in T^{cross},\forall j\in J^{cross}\end{array}  \tag{8}\]  
In which, \(T^{cross}\) is the set of cross-line trains. \(J^{cross}\) is the set of stations connecting different lines. If two trains are coupled, they must meet the following Constraints (9)-(16):  
\[wtc_{c,t}^{c^{\prime},t^{\prime}}=0,t\
otin T^{\circ}\lor t^{\prime}\
otin T^{\circ} \tag{9}\]  
The arrival time and the departure time of the two coupled trains are the same at all the stations during the process of coupling, as Constraints (10) and (11) show:  
\[aat_{j}^{c,t}=aat_{j}^{c^{\prime},t^{\prime}},\forall j\in CS_{c,t}^{c^{\prime},t^{\prime}} \tag{10}\]  
\[adI_{j}^{c,t}=adI_{j}^{c^{\prime},t^{\prime}},\forall j\in CS_{c,t}^{c^{\prime},t^{\prime}} \tag{11}\]  
Moreover, the stop stations of the two coupled trains are the same, as shown in Constraint (12):  
\[aws_{j}^{c,t}=aws_{j}^{c^{\prime},t^{\prime}},\forall j\in CS_{c,t}^{c^{\prime},t^{\prime}} \tag{12}\]  
In addition, the two coupled trains should meet the time constraints at coupled station and the decoupled station, as Constraints (13) and (14) show:  
\[ \begin{array}{l}adI_{j}^{c^{\prime},t^{\prime}}-\max\left\{aat_{j}^{c^{ \prime},t^{\prime}},aat_{j}^{c,t}\right\}+(1-wtc_{c,t}^{c^{\prime},t^{\prime} })\times M\geq COT,\\ \forall c\in C,c^{\prime}\in C,t\in T^{s},t^{\prime}\in T^{s},j=STCS_{c,t}^{c^ {\prime},t^{\prime}}\\ \max\left\{adI_{j}^{c,t},adI_{j}^{c^{\prime},t^{\prime}}\right\}-aat_{j}^{c,t }+(1-wtc_{c,t}^{c^{\prime},t^{\prime}})\times M\geq COT,\\ \forall c\in C,c^{\prime}\in C,t\in T^{s},t^{\prime}\in T^{s},j=STCS_{c,t}^{c^ {\prime},t^{\prime}}\end{array}  \tag{14}\]  
These constraints can be linearized as follows:  
\[ \begin{array}{l}adI_{j}^{c^{\prime},t^{\prime}}-aat_{j}^{c^{\prime},t^{ \prime}}+(1-wtc_{c,t}^{c^{\prime},t^{\prime}})\times M\geq COT,\\ adI_{j}^{c^{\prime},t^{\prime}}-aat_{j}^{c,t}+(1-wtc_{c,t}^{c^{\prime},t^{ \prime}})\times M\geq COT\\ \forall c\in C,c^{\prime}\in C,t\in T^{s},t^{\prime}\in T^{s},j=STCS_{c,t}^{c^ {\prime},t^{\prime}}\end{array}  \tag{15}\]\[ \begin{array}{l}adt_{j}^{c,t}-aat_{j}^{c,t}+(1-wtc_{c,t}^{c^{\prime},t^{\prime}}) \times M\geq COT,\\ adt_{j}^{c^{\prime},t^{\prime}}-aat_{j}^{c,t}+(1-wtc_{c,t}^{c^{\prime},t^{ \prime}})\times M\geq COT\\ \forall c\in C,c^{\prime}\in C,t\in T^{s},t^{\prime}\in T^{s},j=STDS_{c,t}^{c^ {\prime},t^{\prime}}\end{array}  \tag{16}\]  
In Constraints (9)-(16), \(T^{s}\) represents the set of short trains. \(CS_{c,t}^{c^{\prime},t^{\prime}}\) are the common sections of train \(t\) in circulation \(c\) and train \(t^{\prime}\) in circulation \(c^{\prime}\). Binary variables \(wtc_{c,t}^{c^{\prime},t^{\prime}}\) indicate train \(t\) in circulation \(c\) is coupled with train \(t^{\prime}\) in circulation \(c^{\prime}\) (\(wtc_{c,t}^{c^{\prime},t^{\prime}}=1\)) or not (\(wtc_{c,t}^{c^{\prime},t^{\prime}}=0\)). \(COT\) is the operation time of train coupling. \(DOT\) is the operation time of train decoupling. \(STCS_{c,t}^{c^{\prime},t^{\prime}}\) is the station where short train \(t\) in circulation \(c\) and short train \(t^{\prime}\) in circulation \(c^{\prime}\) are coupled. \(STDS_{c,t}^{c^{\prime},t^{\prime}}\) is the station where short train \(t\) in circulation \(c\) and short train \(t^{\prime}\) in circulation \(c^{\prime}\) are decoupled.