# An Algorithm for Rescheduling of Trains under Planned Track Closures
## 5 Heuristic Solution Algorithm
### Managing of Closures
\begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{4}{c}{**Notation Used in Algorithm 1**} \\ \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(r_{n}^{\text{C}}\) & tracks to be closed during closure \(n\) & \(\mathbf{C}^{\text{OB}}\) & set of obligatory closures \\ \(t_{n}^{\text{CE}}\) & actual completion time of closure \(n\) & \(\mathbf{C}^{\text{F}}\) & set of facultative closures \\ \(z_{i}\) & sequence of obligatory closures & \(\mathbf{C}^{\text{S}}\) & set of selected closures to be passed to \\  & & & Algorithm 2 \\ \hline \multicolumn{4}{c}{**Notation Used in Algorithm 2**} \\ \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline TR\({}^{\text{S}}\) & selected train runs to be passed to Algorithm 3 & \(W_{p,k}\) & variable indicating the run of train \(p\) by track \(k\) between given stations \\ \(\Delta\) & parameter of Algorithm 2 representing the limit of time shifts of closures or trains & \(U_{p}\) & conflict index for the train run \(p\) \\ \hline \hline \multicolumn{4}{c}{**Notation Used in Algorithm 3**} \\ \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(\mathbf{S^{\prime}}\) & temporary closure plan & \(\mathbf{V}_{n}\) & set of stations of the decomposed sub-network \(n\) \\ \(\boldsymbol{TT^{\prime}}\) & temporary train timetable & \(\eta\) & number of sub-networks in the decomposition \\ \(\mathbf{GR}\) & set of track assignments to groups used for the decomposition & \(\pi_{n,p}\) & string of stations representing the path of train \(p\) \\ E & set of tuples describing each arc that exists in network \(\mathbf{G}\) & \(\mathbf{C}^{\text{S}}_{n}\) & set of closures in the decomposed sub-network \(n\) \\ \(\mathbf{G}_{n}\) & graph representing the decomposed railway sub-network \(n\). & \(\mathbf{TR}^{\text{S}}_{n}\) & set of trains in decomposed sub-network \(n\) \\ \(\boldsymbol{E}_{n}\) & matrix of tracks (matrix extended by virtual tracks) for the decomposed sub-network \(n\) & \(q\) & train run marked for rerouting \\ \hline \multicolumn{4}{c}{**Notation Used in Algorithm 4**} \\ \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(\boldsymbol{\Psi}^{i,j,k}\) & series of elements indicating tracks that have trains or track closures and their order & \(\mathbf{C}_{i,j,k}\) & set of closures on the given track \\ \(\boldsymbol{\Psi}^{i,j,k}_{m}\) & element of \(\boldsymbol{\Psi}^{i,j,k}\) indicating if tracks have trains or track closures at position \(m\) & \(\mathbf{TR}_{i,j,k}\) & set of trains going through track \((i,j,k)\) \\ \(\boldsymbol{\theta}^{i,j,k}\) & matrix representing the order of track usage by either trains or closures & \(\mathbf{TR}^{\text{S}}_{n}\) & sub-set of \(\mathbf{TR}^{\text{S}}_{n}\), \\ \(\boldsymbol{\theta}^{i,j,k}_{m}\) & element of \(\boldsymbol{\theta}^{i,j,k}\) & \(Q^{i,j,k}_{1}\) & total time shift for trains limited to a single track at a time \\ \(\boldsymbol{\Theta}^{i,j,k}\) & number of trains and closures on track \(k\) & \(\mathbf{TR}_{i}\) & set of trains on tracks adjacent to station \(i\). \\  & connecting stations \(i\) & \(\mathbf{K}^{\text{ADJ}}_{i}\) & set of tracks adjacent to station \(i\). \\ \(\boldsymbol{\lambda}^{i}_{m}\) & element of \(\boldsymbol{\lambda}^{i}\) indicating number of train entering station \(i\) in the order \(m\) & \(Q^{i}_{2}\) & total time shift of trains for station \(i\). \\ \(\boldsymbol{\Lambda}^{i}\) & number of trains entering station \(i\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Decision variables.