# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 3 Optimization Model for Determining a Network-Wide Capacity
### Mathematical Formulation of the Model  
#### 3.3.1 Decision Variables  
For every train \(k\in K\) and each of its relevant arcs \(a\in A^{T,k}\) in the time-expanded network, we define a binary routing variable \(x_{a}^{k}\in\{0,1\}\) that indicates whether train \(k\) uses arc \(a\) (\(x_{a}^{k}=1\)) or not (\(x_{a}^{k}=0\)). The decision \(x_{a}^{k}=1\) for a travel arc of the form \(a=\left((i,t),\left(j,t+\tau_{ij}^{k}\right)\right)\) means that train \(k\) leaves station (or junction) \(i\) at time \(t\) and arrives in \(j\) exactly \(\tau_{ij}^{k}\) time steps later.  
Furthermore, for every train \(k\in K\), we define a binary variable \(y^{k}\in\{0,1\}\) that states whether the network's capacity suffices to schedule train \(k\) under the valid restrictions within the given time horizon (\(y^{k}=1\)) or not (\(y^{k}=0\)).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(T\) & Time horizon of the examination \\ \hline \(\delta^{+}((i,t))\), & Set of outgoing/ incoming arcs of vertex \((i,t)\) in \(G^{T}\) \\ \hline \((i,\alpha)\) & Vertex in \(G^{T}\) called source of vertex \(i\in V\) (\(\delta^{-}((i,\alpha))=\varnothing\)) \\ \hline \((i,\omega)\) & Vertex in \(G^{T}\) called sink of vertex \(i\in V\) (\(\delta^{+}((i,\omega))=\varnothing\)) \\ \hline \(\kappa_{i}\) & Capacity of vertex \(i\in V_{c}\) (number of available tracks for stopping trains) \\ \hline \multirow{3}{*}{\(C_{i}\)} & Set of (potentially occurring) route conflicts between trains arriving and/or departing in vertex \(i\in V_{c}\); \\  & One conflict \(c\in C_{i}\) is given as tuple \(((j,\ n,\ \mu),\ (j^{\prime},\ n^{\prime},\ \mu^{\prime}))\). \\ \hline \(\mu\in(^{\prime}d^{\prime},^{\prime}a^{\prime})\) & Parameter to indicate station departures (\(d^{\prime}\)) and arrivals (\({}^{\prime}a^{\prime}\)) \\ \hline \(K\) & Set of trains (\(K=K_{sch}\cup K_{pot}\)) \\ \hline \(K_{sch}\) & Set of trains that must be scheduled by the model in any case \\ \hline \(K_{pot}\) & Set of trains to be potentially planned in addition \\ \hline \(n(k)\) & Route number of train \(k\) \\ \hline \(N\) & Set of all occurring route numbers (\(N:=\{n(k)\mid k\in K\}\)) \\ \hline \(K_{n}\) & Set of all trains that have route number \(n\) (\(K_{n}:=\{k\in K\mid n(k)=n\}\)) \\ \hline \(o(k)\), \(d(k)\) & Origin/ destination vertex of train \(k\) (\(o(k),\ d(k)\in V\)) \\ \hline \(f_{n}\) & Departure frequency (trains per hour) of route number \(n\) \\ \hline \(V^{k}\), \(A^{k}\) & Set of vertices/ arcs that train \(k\) must pass throughout its journey iff scheduled (\(V^{k}\subset V\), \(A^{k}\subset A\)) \\ \hline \(A^{T,k}\) & Set of relevant arcs for train \(k\) in \(G^{T}\) (\(A^{T,k}\subset A^{T}\)), i.e., arcs corresponding to its assigned route (derived from \(V^{k}\) and \(A^{k}\)) \\ \hline \(w_{min,i^{\prime}}^{k}\) & Minimum and maximum dwell times of train \(k\) in station \(i\in V^{k}\) \\ \hline \(\tau_{ij}^{k}\) & Travel time of train \(k\) on arc \((i,j)\in A^{k}\) \\ \hline \(K_{ij}\) & Set of trains whose routes contain arc \((i,j)\) (\(K_{ij}:=\left\{k\in K\mid(i,j)\in A^{k}\right\}\)) \\ \hline \(s_{ij}\in\{0,1\}\) & Information if arc \((i,j)\in A_{c}\) corresponds to a single-track (\(s_{ij}=1\)) or double-track (\(s_{ij}=0\)) line section \\ \hline \(\overline{h}_{ij}\) & Average minimum headway time on arc \((i,j)\in A_{c}\) (only in case \(s_{ij}=0\)) \\ \hline \(b_{ij}\) & Minimum required buffer time on arc \((i,j)\in A_{c}\) (only in case \(s_{ij}=0\)) \\ \hline \(u_{ij}\) & Hourly capacity of arc \((i,j)\in A_{c}\) (in case \(s_{ij}=1\): sum of both directions) \\ \hline \end{tabular}
\end{table}
Table 1: Cont.