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Keywords
- column generation (3)
- line planning (3)
- public transport (3)
- Optimization (2)
- Rapid Branching (2)
- computational complexity (2)
- demand function (2)
- fare planning (2)
- integer programming (2)
- rapid branching (2)
Application Area
- B (26)
The fare planning problem for public transport is to design a system of fares that maximize the revenue. We introduce a nonlinear optimization model to approach this problem. It is based on a discrete choice logit model that expresses demand as a function of the fares. We illustrate our approach by computing and comparing two different fare systems for the intercity network of the Netherlands.
The line planning problem is one of the fundamental problems in strategic planning of public and rail transport. It consists in finding lines and corresponding frequencies in a network such that a giv en demand can be satisfied. There are two objectives. passengers want to minimize travel times, the transport company wishes to minimize operating costs. We investigate three variants of a multi-commo dity flow model for line planning that differ with respect to passenger routings. The first model allows arbitrary routings, the second only unsplittable routings, and the third only shortest path rou tings with respect to the network. We compare these models theoretically and computationally on data for the city of Potsdam.
Wir stellen in dieser Arbeit ein mathematisches Optimierungsmodell zur Bestimmung eines optimalen Linienplans vor, das sowohl die Fahrzeiten und die Anzahl der Umstiege berücksichtigt als auch die Kosten des Liniennetzes. Dieses Modell deckt wichtige praktische Anforderungen ab, die in einem gemeinsamen Projekt mit den Verkehrsbetrieben in Potsdam (ViP) formuliert wurden. In diesem Projekt wurde der Linienplan 2010 für Potsdam entwickelt. Unsere Berechnungen zeigen, dass die mathematische Optimierung in nichts einer „Handplanung“ des Liniennetzes nachsteht. Im Gegenteil, mit Hilfe des Optimierungsprogramms ist es möglich, durch Veränderung der Parameter mehrere verschiedene Szenarien zu berechnen, miteinander zu vergleichen und Aussagen über minimale Kosten und Fahrzeiten zu machen.
The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.
In this paper a bottom-up approach of automatic simplification of a railway network is presented. Starting from a very detailed, microscopic level, as it is used in railway simulation, the network is transformed by an algorithm to a less detailed level (macroscopic network), that is sufficient for long-term planning and optimization. In addition running and headway times are rounded to a pre-chosen time discretization by a special cumulative method, which we will present and analyse in this paper. After the transformation we fill the network with given train requests to compute an optimal slot allocation. Then the optimized schedule is re-transformed into the microscopic level and can be simulated without any conflicts occuring between the slots. The algorithm is used to transform the network of the very dense Simplon corridor between Swiss and Italy. With our aggregation it is possible for the first time to generate a profit maximal and conflict free timetable for the corridor across a day by a simultaneously optimization run.
Rapid Branching
(2012)
We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning.
This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers.
The hypergraph assignment problem (HAP) is the generalization of assignments
from directed graphs to directed hypergraphs. It serves, in particular,
as a universal tool to model several train composition rules in vehicle rotation
planning for long distance passenger railways. We prove that even for problems
with a small hyperarc size and hypergraphs with a special partitioned structure
the HAP is NP-hard and APX-hard. Further, we present an extended integer
linear programming formulation which implies, e. g., all clique inequalities.
Vehicle rotation planning is a fundamental problem in rail transport. It decides how the railcars, locomotives, and carriages are operated in order to implement the trips of the timetable. One important planning requirement is operational regularity, i.e., using the rolling stock in the same way on every day of operation. We propose to take regularity into account by modeling the vehicle rotation planning problem as a minimum cost hyperassignment problem (HAP). Hyperassignments are generalizations of assignments from directed graphs to directed hypergraphs. Finding a minimum cost hyperassignment is NP-hard. Most instances arising from regular vehicle rotation planning, however, can be solved well in practice. We show that, in particular, clique inequalities strengthen the canonical LP relaxation substantially.
We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the "regularity" of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved.
Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.
The line planning problem is one of the fundamental problems in strategic
planning of public and rail transport. It consists of finding lines
and corresponding frequencies in a public transport network such that
a given travel demand can be satisfied. There are (at least) two objectives.
The transport company wishes to minimize its operating cost;
the passengers request short travel times. We propose two new multicommodity
ow models for line planning. Their main features, in comparison
to existing models, are that the passenger paths can be freely
routed and that the lines are generated dynamically.
The line planning problem is one of the fundamental problems in strategic
planning of public and rail transport. It consists in finding lines
and corresponding frequencies in a transport network such that a given
travel demand can be satisfied. There are (at least) two objectives. The
transport company wishes to minimize operating costs, the passengers
want to minimize travel times. We propose a new multi-commodity
ow model for line planning. Its main features, in comparison to existing
models, are that the passenger paths can be freely routed and
that the lines are generated dynamically. We discuss properties of this
model and investigate its complexity. Results with data for the city of
Potsdam, Germany, are reported.
In this paper we introduce the fare planning problem for public
transport which consists in designing a system of fares maximizing
revenue. We propose a new simple general model for this problem.
It is based on a demand function and constraints for the different
fares. The constraints define the structure of the fare system, e.g.,
distance dependent fares or zone fares. We discuss a simple example
with a quadratic demand function and distance dependent fares. Then
we introduce a more realistic discrete choice model in which passengers
choose between different alternatives depending on the number
of trips per month. We demonstrate the examples by computational
experiments.
Can OR methods help the public transport industry to break even?
How would you build a public transport system? For example, have a look at
Berlin. The BVG, Berlin's public transport company, maintains a network
of 2,423 km, operates 197 lines with 3,286 stops, using 1,554 busses, 1,391
subway cars, and 599 trams from 12 depots, and has 13,409 employees [7].
The BVG currently transports about 800 million passengers per year and
covers about 40% of the total non-pedestrian traffic volume of the city [18].
Does Berlin have a "reasonable" public transportation network? Does
the BVG run a "good" transportation system? Is it "efficient"?
These are difficult questions. In fact, politicians, transportation managers,
customers, taxpayers, etc. frequently employ judgments such as "good"
and "efficient", but nobody can give a defiition what this exactly means.
Since almost every public transportation system in the world is in the red,
the cheapest system is no public transportation at all. On the other hand,
the most convenient system for the passenger - a stop in front of every house
with direct connections to everywhere - is much too expensive. What is the
right compromise? Operations Research has no good answer either - so far.
But OR can improve aspects of public transportation significantly, as we
want to demonstrate in the following.
Millionen von Menschen werden allein in Deutschland täglich von
Bussen, Bahnen und Flugzeugen transportiert. Der öffentliche
Personenverkehr (ÖV) ist von großer Bedeutung für die
Lebensqualität einzelner aber auch für die Leistungsfähigkeit ganzer
Regionen. Qualität und Effizienz von ÖV-Systemen hängen ab von
politischen Rahmenbedingungen (staatlich geplant,
wettbewerblich organisiert) und der Eignung der Infrastruktur
(Schienensysteme, Flughafenstandorte), vom vorhandenen
Verkehrsangebot (Fahr- und Flugplan), von der Verwendung
angemessener Technologien (Informations-, Kontroll- und
Buchungssysteme) und dem bestmöglichen Einsatz der
Betriebsmittel (Energie, Fahrzeuge und Personal). Die hierbei
auftretenden Entscheidungs-, Planungs- und
Optimierungsprobleme sind z.T. gigantisch und "schreien"
aufgrund ihrer hohen Komplexität nach Unterstützung durch Mathematik.
Dieser Artikel skizziert den Stand und die Bedeutung des Einsatzes von
Mathematik bei der Planung und Durchführung von öffentlichem
Personenverkehr, beschreibt die bestehenden Herausforderungen und
regt zukunftsweisende Maßnahmen an.
Every day, millions of people are transported by buses, trains, and airplanes
in Germany. Public transit (PT) is of major importance for the quality of
life of individuals as well as the productivity of entire regions. Quality and
efficiency of PT systems depend on the political framework (state-run, market
oriented) and the suitability of the infrastructure (railway tracks, airport
locations), the existing level of service (timetable, flight schedule), the use
of adequate technologies (information, control, and booking systems), and
the best possible deployment of equipment and resources (energy, vehicles,
crews). The decision, planning, and optimization problems arising in this
context are often gigantic and “scream” for mathematical support because of
their complexity.
This article sketches the state and the relevance of mathematics in planning
and operating public transit, describes today’s challenges, and suggests a
number of innovative actions.
The current contribution of mathematics to public transit is — depending
on the transportation mode — of varying depth. Air traffic is already well
supported by mathematics. Bus traffic made significant advances in recent
years, while rail traffic still bears significant opportunities for improvements.
In all areas of public transit, the existing potentials are far from being exhausted.
For some PT problems, such as vehicle and crew scheduling in bus and
air traffic, excellent mathematical tools are not only available, but used in
many places. In other areas, such as rolling stock rostering in rail traffic,
the performance of the existing mathematical algorithms is not yet sufficient.
Some topics are essentially untouched from a mathematical point
of view; e.g., there are (except for air traffic) no network design or fare
planning models of practical relevance. PT infrastructure construction is
essentially devoid of mathematics, even though enormous capital investments
are made in this area. These problems lead to questions that can only be
tackled by engineers, economists, politicians, and mathematicians in a joint
effort.
Among other things, the authors propose to investigate two specific topics,
which can be addressed at short notice, are of fundamental importance not
only for the area of traffic planning, should lead to a significant improvement
in the collaboration of all involved parties, and, if successful, will be of real
value for companies and customers:
• discrete optimal control: real-time re-planning of traffic systems in case
of disruptions,
• model integration: service design in bus and rail traffic.
Work on these topics in interdisciplinary research projects could be funded
by the German ministry of research and education (BMBF), the German
ministry of economics (BMWi), or the German science foundation (DFG).
The optimization of fare systems in public transit allows to pursue
objectives such as the maximization of demand, revenue, profit, or
social welfare. We propose a non-linear optimization approach to fare
planning that is based on a detailed discrete choice model of user
behavior. The approach allows to analyze different fare structures,
optimization objectives, and operational scenarios involving, e.g.,
subsidies. We use the resulting models to compute optimized fare
systems for the city of Potsdam, Germany.
We introduce an optimization model for the line planning problem in a public transportation system that aims at minimizing operational costs while ensuring a given level of quality of service in terms of available transport capacity. We discuss the computational complexity of the model for tree network topologies and line structures that arise in a real-world application at the Trolebus Integrated System in Quito. Computational results for this system are reported.
The Steiner connectivity problem is a generalization of
the Steiner tree problem. It consists in finding a minimum cost set of
simple paths to connect a subset of nodes in an undirected graph.
We show that important polyhedral and algorithmic results on the
Steiner tree problem carry over to the Steiner connectivity problem,
namely, the Steiner cut and the Steiner partition inequalities, as
well as the associated polynomial time separation algorithms, can be
generalized. Similar to the Steiner tree case, a certain directed
formulation, which is stronger than the natural undirected one,
plays a central role.
This paper introduces the line connectivity
problem, a generalization of the Steiner tree problem and a
special case of the line planning problem. We study its complexity and
give an IP formulation in terms of an exponential number of
constraints associated with "line cut constraints". These inequalities
can be separated in polynomial time. We also generalize the Steiner
partition inequalities.
Die Angebotsplanung im öffentlichen Nahverkehr umfasst die Aufgaben der Netz-, Linien-,
Fahr- und Preisplanung. Wir stellen zwei mathematische Optimierungsmodelle zur Linien- und
Preisplanung vor. Wir zeigen anhand von Berechnungen für die Verkehrsbetriebe in Potsdam
(ViP), dass sich damit komplexe Zusammenhänge quantitativ analysieren lassen. Auf diese
Weise untersuchen wir die Auswirkungen von Freiheitsgraden auf die Konstruktion von Linien
und die Wahl von Reisewegen der Passagiere, Abhängigkeiten zwischen Kosten und Reisezeiten
sowie den Einfluss verschiedener Preissysteme auf Nachfrage und Kostendeckung.
Line planning is an important step in the strategic planning
process of a public transportation system. In this paper, we discuss an
optimization model for this problem in order to minimize operation costs
while guaranteeing a certain level of quality of service, in terms of
available transport capacity. We analyze the problem for path and tree
network topologies as well as several categories of line operation that
are important for the Quito Trolebus system. It turns out that, from a
computational complexity worst case point of view, the problem is hard in
all but the most simple variants. In practice, however, instances based
on real data from the Trolebus System in Quito can be solved quite well,
and significant optimization potentials can be demonstrated.
Line planning is an important step in the strategic planning process of a public transportation system. In this paper, we discuss an optimization model for this problem in order to minimize operation costs while guaranteeing a certain level of quality of service, in terms of available transport capacity. We analyze the problem for path and tree network topologies as well as several categories of line operation that are important for the Quito Trolebus system. It turns out that, from a computational complexity worst case point of view, the problem is hard in all but the most simple variants. In practice, however, instances based on real data from the Trolebus System in Quito can be solved quite well, and significant optimization potentials can be demonstrated.
The mathematical treatment of planning problems in public transit has made significant advances in the last decade. Among others, the classical problems of vehicle and crew scheduling can nowadays be solved on a routine basis using combinatorial optimization methods. This is not yet the case for problems that pertain to the design of public transit networks, and for the problems of operations control that address the implementation of a schedule in the presence of disturbances. The article gives a sketch of the state and important developments in these areas, and it addresses important challenges. The vision is that mathematical tools of computer aided scheduling (CAS) will soon play a similar role in the design and operation of public transport systems as CAD systems in manufacturing.
The timetable is the essence of the service offered by any provider
of public transport'' (Jonathan Tyler, CASPT 2006). Indeed, the
timetable has a major impact on both operating costs and on passenger
comfort. Most European agglomerations and railways use periodic timetables in which operation repeats in regular intervals. In contrast, many North and South American municipalities use trip timetables in which the vehicle trips are scheduled individually subject to frequency constraints. We compare these two
strategies with respect to vehicle operation costs. It turns out that
for short time horizons, periodic timetabling can be suboptimal; for
sufficiently long time horizons, however, periodic timetabling can
always be done in an optimal way'.
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