# An Algorithm for Rescheduling of Trains under Planned Track Closures
## 4 Joint Problem of Train Rescheduling and Track Closure Planning
### Mathematical Model
#### 4.5.2 Train Rescheduling
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline \multicolumn{1}{c}{**Symbol**} & \multicolumn{1}{c}{**Description**} & \multicolumn{1}{c}{**Symbol**} & \multicolumn{1}{c}{**Description**} \\ \hline \(\mathbf{y}\) & binary matrix of closure planning & \(\Delta t^{\text{ARR}}_{i,p}\in\mathbb{N}\) & element of \(\mathbf{\Delta t}^{\text{ARR}}\) representing time shift of train \(p\) arrival to station \(i\) & element of \(\mathbf{\Delta t}^{\text{DEP}}\) & element of \(\mathbf{\Delta t}^{\text{ARR}}\) representing time shift of train \(p\) arrival to station \(i\) \\ \(y_{n}\in\{0,1\}\) & element of \(\mathbf{y}\) indicating the status of closure \(n\) & \(\mathbf{\Delta t}^{\text{DEP}}\) & matrix with time shifts of train departures from stations \\ \(\mathbf{\Delta t}^{\text{C}}\) & matrix with time shifts of closure start moments & \(\mathbf{S}\) & matrix representing the plan of track closures \\ \(\Delta t^{\text{C}}_{n}\in\mathbb{N}\) & element of \(\mathbf{\Delta t}^{\text{C}}\) representing the start time shift of closure \(n\) & \(S_{n}\) & element of \(S\) representing plan for closure \(n\) \\ \(\mathbf{r}^{\text{T}}_{p}\) & matrix describing routes of all trains & \(r^{\text{C}}_{n}\) & tracks to be closed during closure \(n\) \\ \(\mathbf{r}^{\text{T}}_{p}\) & matrix describing the route of train \(p\) being the element of \(\mathbf{r}^{\text{T}}\) & vector with timetable of trains \\ \(r^{\text{T}}_{p,i,j,k}\in\{0,1\}\) & element of \(\mathbf{r}^{\text{T}}_{p}\) describing the connection between stations \(i\) and \(j\) is used in the route of train \(p\) & element of \(\mathbf{TT}\) given by a tuple representing the timetable of train \(p\) \\ \(\mathbf{\Delta t}^{\text{ARR}}\) & matrix with time shifts of train arrivals to stations & \(\mathbf{x}\) & binary matrix describing inclusion of trains into timetable \(\mathbf{TT}\) \\ \(\Delta t^{\text{DEP}}_{i,p}\in\mathbb{N}\) & element of \(\mathbf{\Delta t}^{\text{DEP}}\) representing time shift of train \(p\) departure from station \(i\) & \(x_{p}\in\{0,1\}\) & element of \(\mathbf{x}\) indicating the possible inclusion of train \(p\) into timetable \(\mathbf{TT}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Decision variables. if the \(p\)th train run is present in the new timetable (otherwise). It is important to point out that some train runs denoted by \(\mathbf{TR}^{\text{OB}}\subseteq\mathbf{TR}\) may exist and have to be obligatorily inserted into \(\boldsymbol{TT}\). The train rescheduling has to obey some regulations on any train travel time extensions concerning timetable \(\boldsymbol{TT}^{\text{G}}\). They assure that the total travel time extension cannot exceed the given limit \(T^{\text{E}}\), and the travel time extension on a given distance \(D^{\text{L}}\) cannot exceed the given limit \(T^{\text{DE}}\) (see Formulas (10) and (11)). The matrices \(\boldsymbol{r}^{\text{T}}\), \(\boldsymbol{\Delta t}^{\text{ARR}}\), \(\boldsymbol{\Delta t}^{\text{DEF}}\), and \(\boldsymbol{x}\), or equivalently \(\boldsymbol{TT}\), comprise all decisions made while rescheduling of trains.