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We propose a novel extended formulation for the line planning problem in public transport. It is based on a new concept of frequency configurations that account for all possible options to provide a required transportation capacity on an infrastructure edge. We show that this model yields a strong LP relaxation. It implies, in particular, general classes of facet defining inequalities for the standard model.
The treatment of transfers is a major challenge in line planning. Existing models either route passengers and lines sequentially, and hence disregard essential degrees of freedom, or they are of
extremely large scale, and seem to be computationally intractable. We propose a novel direct connection approach that allows an integrated optimization of line and passenger routing, including accurate estimates of the number of direct travelers, for large-scale real-world instances.
We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree property for terminal nodes in hypergraphs which generalizes the well-known graph property that the average degree of terminal nodes in Steiner trees is at most 2.
A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable
(2018)
We consider the following planning problem in public transportation: Given a periodic timetable, how many vehicles are required to operate it? In [9], for this sequential approach, it is proposed to first expand the periodic timetable over time, and then answer the above question by solving a flow-based aperiodic optimization problem. In this contribution we propose to keep the compact periodic representation of the timetable and simply solve a particular perfect matching problem. For practical networks, it is very much likely that the matching problem decomposes into several connected components. Our key observation is that there is no need to change any turnaround decision for the vehicles of a line during the day, as long as the timetable stays exactly the same.
A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable
(2018)
We consider the following planning problem in public transportation: Given a
periodic timetable, how many vehicles are required to operate it?
In [9], for this sequential approach, it is proposed to first expand the periodic
timetable over time, and then answer the above question by solving a flow-based
aperiodic optimization problem.
In this contribution we propose to keep the compact periodic representation of
the timetable and simply solve a particular perfect matching problem. For practical
networks, it is very much likely that the matching problem decomposes into several
connected components. Our key observation is that there is no need to change any
turnaround decision for the vehicles of a line during the day, as long as the timetable
stays exactly the same.
In dieser Arbeit wird ein graphenbasiertes Modell zur Einbindung von Preissystemen des öffentlichen Nahverkehrs in Routing-Algorithmen vorgestellt. Jeder Knoten des Graphen repräsentiert einen abstrakten Preiszustand einer Route und ist an einen tatsächlichen Preis gekoppelt. Damit sind sehr einfache und konzise Beschreibungen von Tarifstrukturen möglich, diesich algorithmisch behandeln lassen. Durch das zeitgleiche Tracken eines Pfades im Routinggraphen im Ticketgraphen kann schon während einer Routenberechnung der Preis bestimmt werden. Dies ermöglicht die Berechnung von preisoptimalen Routen. An den Tarifsystemen der Verkehrsverbünde MDV (Mitteldeutscher Verkehrsverbund) und VBB (Verkehrsverbund Berlin-Brandenburg) wird die Konstruktion des Modells detailliert erläutert.
Dieses Dokument fasst den Stand der mathematischen Modellierung von
Preissystemen des öV mittels eines am ZIB entwickelten Tarifgraphenmodells zusammen. Damit sind sehr einfache und konzise
Beschreibungen von Tarifstrukturen möglich, die sich algorithmisch
behandeln lassen: Durch das zeitgleiche Tracken eines Pfades im
Routinggraphen im Tarifgraphen kann schon während einer Routenberechnung der Preis bestimmt werden. Wir beschreiben
zunächst das Konzept. Die konkrete Realisierung wird im Folgenden
beispielhaft an den Tarifsystemen der Verkehrsverbünde Warnow,
MDV, Vogtland, Bremen/Niedersachsen, Berlin/Brandenburg und Mittelsachsen erläutert. Anschließend folgen Überlegungen zur konkreten Implementierung von Kurzstrecken-Tarifen und zur Behandlung des Verbundübergriffs.