# Delay Management in Regional Railway Transport
## 2 Materials and Methods
### The Maximum Transfer Waiting Times Principle Based on Modified CPM  
In the Czech praxis, one of the normal methods for estimating transfer waiting times is maximization, so that a certain subset of transfer bindings set out in a timetable can be maintained and further marked (as crucial transfer bindings), while considering the mutual interactions between trains (i.e., crossing, post-rides, vehicle reversions, etc.) and using disposal measures (see Section 2.2.1). By that method, it is common to progress from the highest segment trains to the lowest segment trains. Transfer waiting times for higher segment trains serve as input for assigning the transfer waiting times for lower segment trains. When choosing crucial transfer bindings, it is necessary to consider the volume of passengers of a corresponding transfer binding and the frequency of continuing trains (i.e., how long passengers will wait for the following connection when the continuing train is missed). It is also possible to reduce transfer waiting times based on the assessment of these factors. Net-planning methods may be used to calculate transfer waiting times, rather than simply estimating the times, via a modified critical path method (CPM). In doing so, the task may be expressed as an \"operational/technological processes graph\" that represents train arrivals, departures, and passages, as well as the bindings between the trains (transfers, operational bindings, circulation bindings, etc.). The graph contains:  
* vertices \(a_{t,i}\), which express the arrival of train \(t\) to traffic point \(i\);
* vertices \(d_{t,i}\), which express the departure or passage of train \(t\) from/trough traffic point \(i\);
* oriented edges \(r_{t,i-j}=\left[d_{t,i};a_{t,j}\right]\) or \(r_{t,i-j}=\left[d_{t,i};d_{t,j}\right]\), which introduce the run of the train \(t\) from traffic point \(i\) to traffic point \(j\) and potential stops at intermediate traffic points between traffic points \(i\) and \(j\) (if the edge goes into a vertex \(a_{t,j}\), train \(t\) stops at traffic point \(j\); if the edge goes into a vertex \(d_{i,j}\), train \(t\) passes through traffic point \(j\));
* oriented edges \(dw_{t,i}=\left[a_{t,i};d_{t,i}\right]\), which introduce the dwell of train \(t\) at traffic point \(i\);
* oriented edges \(t_{t-u,i}=\left[a_{t,i};a_{u,i}\right]\), which introduce the transfer binding from connecting train \(t\) to continuing train \(u\) at traffic point \(i\) (while \(t\
eq u\));
* oriented edges \(h_{t,i-u,j}=\left[a_{t,i};a_{u,j}\right]\) or \(h_{t,i-u,j}=\left[a_{t,i};d_{u,j}\right]\) or \(h_{t,i-u,j}=\left[d_{t,i};a_{u,j}\right]\) or \(h_{t,i-u,j}=\left[d_{t,i};d_{u,j}\right]\), which introduce operational interactions between the trains (while \(t\
eq u\)); and
* oriented edges \(c_{t-u,i}=\left[a_{t,i};d_{u,i}\right]\), which introduce the circulation binding between trains \(t\) and \(u\) (while \(t\
eq u\)) at traffic point \(i\) (e.g., the train set reversion). Vertices \(a_{t,i}\) and \(d_{t,i}\) each have three evaluations:
* \(s(a_{t,i})\) or \(s(d_{t,i})\), which means the scheduled arrival, departure, or passage time of train \(t\) at traffic point \(i\);
* \(e(a_{t,i})\) or \(e(d_{t,i})\), which means the earliest possible arrival, departure, or passage time of train \(t\) at traffic point \(i\); and
* \(l(a_{t,i})\) or \(l(d_{t,i})\), which means the latest possible arrival, departure, or passage time of train \(t\) at traffic point \(i\). The vertices \(d_{t,i}\), whose highest tolerable waiting times are known for a certain combination of train \(t\) and traffic point \(i\) and which are used as an input for assigning the other trains' waiting times, have one more evaluation \(l_{IN}(d_{t,i})\), which means the latest possible departure time of train \(t\) at traffic point \(i\) while waiting for delayed connecting trains.