# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 3 Optimization Model for Determining a Network-Wide Capacity
### Input Sets and Parameters of the Model  
#### 3.2.1 Different Trains and Train-Related Parameters  
In the railway context of this study, we consider different trains denoted by \(k\in K\). \(K\) is the total set of trains, which is split into the disjoint subsets \(K_{sch}\) and \(K_{pot}\). \(K_{sch}\) is defined as the set of trains that are scheduled by the model in any case, while \(K_{pot}\) describes the set of trains that can potentially be routed in addition. This division is made to have a fixed number of scheduled trains on the one hand and to allow for the possibility of saturation by additional trains for capacity calculation on the other hand. To determine the capacity of the core network, the model developed in this paper therefore aims to schedule as many additional trains from the set \(K_{pot}\) as possible. Note that the presented optimization model only leads to a feasible solution if \(K_{sch}\) does not yet contain too many trains, i.e., if residual capacities are available.  
Each train \(k\in K\) is assigned a route number \(n(k)\), an origin vertex \(o(k)\in V\), a destination vertex \(d(k)\in V\), and its route through the network given as set of arcs \(A^{k}\subset A\) and set of vertices \(V^{k}\subset V\). Each train is assigned a route through the network that it is permitted to travel, for example with regard to the permissible clearance gauge as well as length and weight restrictions. We also introduce the set \(N\) of occurring route numbers (\(N:=\{n(k)\mid k\in K\}\)), and, for each route number \(n\in N\), the set \(K_{n}\), which comprises all trains that share route number \(n\) (\(K_{n}:=\{k\in K\mid n(k)=n\}\)). Trains that share the identical route number are trains of the same type that are assigned the same origin and destination vertices, and the same route through the network--they only differ in their given index \(k\) (which can also be interpreted as train identification number). To allow the result of the optimization model to contain timetable-like structures, each route number \(n\in N\) (for which trains are defined in the set \(K_{sch}\)) is assigned a number \(f_{n}\) that specifies how often trains of this route number should run per hour. Note that the total number of trains of route number \(n\) defined in the set \(K_{sch}\) must equal the number \(f_{n}\) multiplied by the number of hours covered by the time horizon of the investigation.  
Furthermore, for each train \(k\in K\), minimum and maximum dwell times (\(w^{k}_{min,\,i},w^{k}_{max,\,i}\)) are specified at each station \(i\in V^{k}\) of its route. By setting the maximum dwell time to zero, a train can be forced not to stop at a station. In contrast, a train can be forced to stop at a station by setting the minimum dwell time to at least one.  
In addition, each train \(k\in K\) is assigned an individual travel time \(\tau^{k}_{ij}\in\mathbb{N}\) on each arc \((i,j)\in A^{k}\) of its route. Since each train can have its own travel time on each arc of the network, the model can consider trains with different characteristics. Travel times on linking lines \((i,j)\in A_{l}\) are zero for all trains \(k\in K\).