90B20 Traffic problems
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Umsteigen ohne Warten
(2017)
Wir stellen einen mathematischen Optimierungsansatz zur Berechnung von periodischen Taktfahrplänen vor, bei dem die Umsteigezeiten unter
Berücksichtigung des Passagierverhaltens minimiert werden. Wir untersuchen damit den Einfluss wichtiger Systemparameter und Verhaltensmuster auf die Beförderungsqualität.
Wir illustrieren anhand des Liniennetzes der Stadt Potsdam das Potenzial mathematischer Methoden der Angebotsplanung. Wir zeigen,
dass das "bestmögliche" Verkehrsangebot stark von planerischen Vorgaben beeinflusst wird, mit denen man die Erreichung unterschiedlicher und teilweise gegenläufiger Ziele steuern kann. Die Komplexität des Systems führt zum Auftreten von Rückkoppelungseffekten, die man nicht mit Hilfe von Daumenregeln beherrschen kann. Vielmehr ist der Einsatz moderner Planungsverfahren in einer interdisziplinären Zusammenarbeit von politischen Entscheidungsträgern, Verkehrsingenieuren und Mathematikern notwendig, um die aktuellen Herausforderungen in der Verkehrsplanung zu meistern. Der Artikel dokumentiert einen Beitrag zum 7. ÖPNV Innovationskongress des Ministeriums für Verkehr und Infrastruktur des Landes Baden-Württemberg, der vom 9.-11. März 2015 in Freiburg stattfand.
We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We derive a feasibility condition for path capacities supporting such direct connection flows similar to the feasibility condition for arc capacities in ordinary multi-commodity flows.
The concept allows to solve large-scale real-world line planning problems in public transport including a novel passenger routing model that favors direct connections over connections with transfers.
The integrated line planning and passenger routing problem is an important planning problem in service design of public transport. A major challenge is the treatment of transfers. A main property of a line system is its connectivity.
In this paper we show that analysing the connecvitiy aspect of a line plan gives a new idea to handle the transfer aspect of the line planning problem.
This thesis introduces the Steiner connectivity problem. It is a generalization of the well
known Steiner tree problem. Given a graph G = (V, E) and a subset T ⊆ V of the nodes,
the Steiner tree problem consists in finding a cost minimal set of edges connecting all
nodes in T . The Steiner connectivity problem chooses, instead of edges, from a given set
of paths a subset to connect all nodes in T . We show in the first part of this thesis that
main results about complexity, approximation, integer programming formulations, and
polyhedra can be generalized from the Steiner tree problem to the Steiner connectivity
problem.
An example for a straightforward generalization are the Steiner partition inequalities, a
fundamental class of facet defining inequalities for the Steiner tree problem. They can be
defined for the Steiner connectivity problem in an analogous way as for the Steiner tree
problem. An example for a generalization that needs more effort is the definition of a
directed cut formulation and the proof that this dominates the canonical undirected cut
formulation enriched by all Steiner partition inequalities. For the Steiner connectivity
problem this directed cut formulation leads to extended formulations, a concept that is
not necessary for the Steiner tree problem. There are also major differences between
both problems. For instance, the case T = V for the Steiner connectivity problem is
equivalent to a set covering problem and, hence, not a polynomial solvable case as in the
Steiner tree problem.
The Steiner connectivity problem is not only an interesting generalization of the Steiner
tree problem but also the underlying connectivity problem in line planning with inte-
grated passenger routing. The integrated line planning and passenger routing problem
is an important planning problem in service design of public transport and the topic of
the second part. Given is the infrastructure network of a public transport system where
the edges correspond to streets and tracks and the nodes correspond to stations/stops
of lines. The task is to find paths in the infrastructure network for lines and passengers
such that the capacities of the lines suffice to transport all passengers. Existing models
in the literature that integrate a passenger routing in line planning either treat transfers
in a rudimentary way and, hence, neglect an important aspect for the choice of the pas-
senger routes, or they treat transfers in a too comprehensive way and cannot be solved
for large scale real world problems. We propose a new model that focuses on direct
connections. The attractiveness of transfer free connections is increased by introducing a transfer penalty for each non-direct connection. In this way, a passenger routing is
computed that favors direct connections.
For the computation of this model we also implemented algorithms influenced by the
results for the Steiner connectivity problem. We can compute with our model good
solutions that minimize a weighted sum of line operating costs and passengers travel
times. These solutions improve the solutions of an existing approach, that does not
consider direct connections, by up to 17%. In contrast to a comprehensive approach,
that considers every transfer and for which we could not even solve the root LP within
10 hours for large instances, the solutions of the new model, computed in the same time,
are close to optimality (<1%) or even optimal for real world instances. In a project with
the Verkehr in Potsdam GmbH to compute the line plan for 2010 we showed that our
approach is applicable in practice and can be used to solve real world problems.
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.
Im Zuge der Übernahme von 6 Linien der Havelbus Verkehrsgesellschaft
mbH durch die ViP Verkehr in Potsdam GmbH ergab sich 2009 die Notwendigkeit der
Entwicklung eines neuen Linien- und Taktplans für das Jahr 2010.
Das Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
entwickelt in einem Projekt des DFG-Forschungszentrums Matheon ein Verfahren zur mathematischen Linienoptimierung. Dieses Tool wurde bei der Optimierung des ViP Linienplans 2010 in
einer projektbegleitenden Studie eingesetzt, um Alternativen bei
verschiedenen Planungs- und Zielvorgaben auszuloten.
In dem Artikel wird eine Auswertung der Ergebnisse mit dem Verkehrsanalysesystem Visum der PTV AG beschrieben. Die Auswertungen bestätigen, dass mit Hilfe von mathematischer Optimierung eine weitere Verkürzung der Reisezeit um 1%, eine
als um 6% verkürzt empfundene Reisezeit, 10% weniger Fahrzeit im
Fahrzeug und eine gleichzeitige Kostenreduktion um 5% möglich sind.