Our main result is that every n-dimensional polytope can be
described by at most 2n ? 1 polynomial inequalities and, moreover, these
polynomials can explicitly be constructed. For an n-dimensional pointed
polyhedral cone we prove the bound 2n ? 2 and for arbitrary polyhedra we
get a constructible representation by 2n polynomial inequalities.
Ein gemischt-ganzzahliges lineares Optimierungsmodell für ein Laserschweißproblem im Karosseriebau
(2006)
Wir betrachten das Problem der Betriebsplanung von Laserschweißrobotern im Karosseriebau. Gegeben ist eine Menge von Schweißnähten, die innerhalb einer Fertigungszelle an einem Karosserieteil gefertigt werden müssen. Die Schweißnähte werden durch mehrere parallel betriebene Roboter bearbeitet. Die Aufgabe besteht darin, für jeden Roboter eine Reihenfolge und eine zeitliche Koordinierung seiner Bewegungen zu finden, so dass alle Schweißnähte innerhalb der Taktzeit der Fertigungszelle bearbeitet werden und so wenig Laserquellen wie möglich eingesetzt werden. Dabei müssen einige Nebenbedingungen berücksichtigt werden. Für dieses spezielle Schweißproblem haben wir eine Formulierung als gemischt-ganzzahliges lineares Programm entwickelt, welches sich für die untersuchten praktischen Fälle sehr schnell lösen lässt.
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.
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.
In "classical optimization" it is assumed that full information about the problem to be solved is given. This, in particular, includes that all data are at hand. The real world may not be so "nice" to optimizers. Some problem constraints may not be known, the data may be corrupted, or some data may not be available at the moments when decisions have to be made. The last issue is the subject of "online optimization" which will be addressed here. We explain some theory that has been developed to cope with such situations and provide examples from practice where unavailable information is not
the result of bad data handling but an inevitable phenomenon.
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).