# Delay Management in Regional Railway Transport
## 2 Materials and Methods
### The Maximum Transfer Waiting Times Principle Based on Modified CPM
If \(m[x,y]\) is an oriented path from vertex \(x\) to vertex \(y\) and \(ed\in m\) is the total of oriented edges contained in the path, for all vertices \(v\
otin I_{w}\) of operational/technological processes graph: 1. if there is at least one oriented path \(m[v,d_{t^{*},i^{*}}]\), where \(d_{t^{*},i^{*}}\in I_{w}\), the latest necessary arrival, departure, or passage time \(l(v)\) is assigned by Equation (4); \[l(v)=\min_{\forall d_{t^{*},i^{*}}\in I_{w}}\left(l_{IN}(d_{t^{*},i^{*}})-\max _{\forall m[v,d_{t^{*},i^{*}}]}\left(\sum_{d\in[v,d_{t^{*},i^{*}}]}mnt[ed] \right)\right)\] (4) 2. if there is no oriented path \(m[v,d_{t^{*},i^{*}}]\), where \(d_{t^{*},i^{*}}\in I_{w}\), the evaluation \(l(v)\) stays undetermined. 3. For all vertices \(v\) of the operational/technological processes graph, evaluation \(e(v)\) was determined (i.e., the earliest possible arrival, departure, and passage times at traffic points). The following factors must be considered: \(\circ\) the scheduled arrival, departure, and passage times at traffic points; \(\circ\) the possibility of scheduled technological times shortening in light of timetable reserves and spaces utilization (however, they cannot be shortened below their minimum value, therefore \(e(y)\geq e(x)+nmt[x,y]\) is valid for each oriented edge \([x,y]\) of the operational/technological processes graph); \(\circ\) the unacceptability of a train's departure before the scheduled departure time (e.g., departure at arrival time, departure immediately after passenger boarding, or departure if the train stops for traffic reasons). 4. If at least one vertex \(v\) exists for which \(e(v)>l(v)\) is valid, it was not possible to ensure at least one transfer binding with used input waiting times; this situation should not occur when no disposal measure is used. Let \(m_{c>l}[x,d_{t^{*},i^{*}}]\), where \(d_{t^{*},i^{*}}\in I_{w}\), be an oriented path with unfeasible transfer binding if for all edges \([f,g]\in\ m_{c>l}\) both evaluations \(l(f)\) and \(l(g)\) are determined, \(e(f)>l(f)\) and \(e(g)>l(g)\) are valid, and there concurrently exists no oriented edge \([y,x]\) where \(e(y)>l(y)\) and \(e(d_{t^{*},i^{*}})>l_{IN}(d_{t^{*},i^{*}})\). For each oriented path with unfeasible transfer binding \(m_{e>l}[x,d_{t^{*},i^{*}}]\), it is necessary to drop at least one transfer binding edge \(t_{u1-u2}\in m_{c>l}[x,d_{t^{*},i^{*}}]\); this step means resignation from the corresponding transfer binding. If at least one edge was dropped, the algorithm must be performed again from the second step; otherwise, the algorithm should be continued to the fifth step. 5. For all vertices \(d_{t^{*},i^{*}}\in I_{w}\), the evaluation \(l(d_{t^{*},i^{*}})\) was adapted by Formula (5): \[l(d_{t^{*},i^{*}})=\min\{l(d_{t^{*},i^{*}});l_{IN}(d_{t^{*},i^{*}})\}\] (5) 6. For all vertices \(d_{t,i}\), which are entered by at least one oriented edge \(t_{u-t,i}\), the transfer waiting time \(w_{t,i}\) of train \(t\) at traffic point \(i\) was determined: \(\circ\) if evaluation \(l(d_{t,i})\) was determined, the transfer waiting time was calculated by Formula (6): \[w_{t,i}=l(d_{t,i})-s(d_{t,i})\] (6) \(\circ\) if evaluation \(l(d_{t,i})\) was not determined, the transfer waiting time \(w_{t,i}\) was not restricted by the input transfer waiting times and it was necessary to determine the transfer waiting time by another method or by an expert estimate, e.g., according to the transportation effects on passengers in train \(t\) and in trains that directly or indirectly interact with train \(t\). 7. If \(m_{lim}[x,y]\) is a limiting (oriented) path if for all edges \([f,g]\in m_{lim}\) both evaluations \(l(f)\) and \(l(g)\) are determined and \(l(g)-l(f)=nmt[f,g]\) is valid. Such a path limits the maximum value of at least one transfer waiting time.