90C27 Combinatorial optimization
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- Passenger routing (2)
- Periodic timetabling (2)
- Public transport (2)
- combinatorial optimization (2)
- direct connection (2)
- line planning (2)
- metric inequalities (2)
- Degree Property (1)
- Hypergraph (1)
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(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.
The task of periodic timetabling is to schedule the trips in a public transport system by determining arrival and departure times at every station such that travel and transfer times are minimized. To date, the optimization literature generally assumes that passengers do not respond to changes in the timetable, i.e., the passenger routes are fixed. This is unrealistic and ignores potentially valuable degrees of freedom. We investigate in this paper periodic timetabling models with integrated passenger routing. We show that different routing models have a huge influence on the quality of the entire system: Whatever metric is applied, the performance ratios of timetables w.r.t. to different routing models can be arbitrarily large. Computations on a real-world instance for the city of Wuppertal substantiate the theoretical findings. These results indicate the existence of untapped optimization potentials that can be used to improve the efficiency of public transport systems.
The task of timetabling is to schedule the trips in a public transport system by determining periodic arrival and departure times at every station. The goal is to provide a service that is both attractive for passengers and can be operated economically. To date, timetable optimization is generally done with respect to fixed passenger routes, i.e., it is assumed that passengers do not respond to changes in the timetable. This is unrealistic and ignores potentially valuable degrees of freedom. We investigate in this paper periodic timetabling models with integrated passenger routing. We propose several models that differ in the allowed passenger paths and the objectives. We compare these models theoretically and report on computations on real-world instances for the city of Wuppertal.
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.
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 show that the concept of metric inequalities to characterize capacities that
support a multi-commodity flow can be generalized to deal with direct connections.
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.
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.