# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 4 Exemplary Application to a Sample Network
### Results  
The optimization model described in the previous sections is implemented in Python 3.9 and solved for the sample network specified in Section 4.1 on the compute cluster of the RWTH Aachen University using Gurobi 9.5. For technical details of the compute cluster see [30]. For the application, a time period of three hours (\(T=180\)) is considered. In the following, the results are first analyzed for the entire network as depicted in Figure 3.  
For the entire network (see Figure 3), 38 trains of 32 different route numbers are initially scheduled per hour, 6 of which are freight trains. This means that, within the whole investigation period, 114 trains are initially scheduled (cf. Table 3). All freight train route numbers are then considered for the saturation of the network, and in each iteration, it is checked whether another train of each route number can be incorporated. This procedure leads to a total number of 132 trains in the network. Hence, eighteen additional trains that were added in seven iterations could be scheduled.  
The runtime of the entire optimization process was approximately 477 min. To achieve this, further accelerating constraints were implemented in the optimization model. Most of these constraints helped the solver make irrelevant but necessary decisions that do not change the objective value of the best possible solution. An example of these constraints is the specification that no trains use holdover arcs of the time-expanded network outside of the core network. One example of constraints that can have a minor impact on the results but contribute significantly to the acceleration of the optimization process, is the specification for trains of passenger route numbers in the set \(K_{sch}\) to maintain a minimum time interval of half their frequency at their origin vertices.  
Apart from the evaluation regarding the number of totally planned trains, the train numbers can also be evaluated with respect to the single line sections of the considered network. The resulting train numbers can then be compared to the line capacities that were used as input parameters for the model as an upper bound. Figure 5 shows this comparison for the entire network, where the first number indicates the actual number of trains on each line section and the second number indicates the maximum number of trains that could theoretically run on that section according to the calculated line section capacity.  
Figure 5: Entire sample network after saturation by the additional freight train route numbers F1, F2 and F3 (in green, dark red and blue color, respectively). Red numbers on the arcs highlight saturated line sections after optimization [27].  
To allow for the derivation of network effects, the network is divided into two subnetworks, each of which the analysis is repeated for. The division into the partial networks is illustrated in Figure 6.  
In total, 84 trains were initially planned for the \"upper\" part of the network and 21 trains were added during the following seven iterations. The runtime of the complete optimization process was approximately 17 min. The subnetwork, consisting of the \"lower\" part of the entire network, is already saturated after six iterations, and 12 additional trains could be added to the originally planned 78 trains. The runtime of the entire optimization process was approximately 91 min. The load of the respective subnetworks after saturation can also be seen in Figure 6.  
In comparison to the analysis of the entire network (cf. Figure 5), it can be seen that in these partial analyses at least as many (and sometimes even more) trains can be scheduled on the individual line sections.  
The optimization thus allows the capacities that would be available when looking at a single element to be compared with the actually usable capacity when considering the interaction with surrounding elements. As expected, the usable capacity tends to decrease the larger the network under consideration becomes, as the influences between the elements increase.