## 90B20 Traffic problems

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- ZIB-Report (22)
- Doctoral Thesis (2)

#### Keywords

- Optimierung (7)
- integer programming (4)
- Linienplanung (3)
- Nahverkehr (3)
- combinatorial optimization (3)
- duty scheduling (3)
- line planning (3)
- transfers (3)
- vehicle scheduling (3)
- Dienstplanung (2)

#### Institute

This thesis describes the algorithm IS-OPT that integrates scheduling of vehicles and duties in public bus transit. IS-OPT is the first algorithm which solves integrated vehicle and duty scheduling problems arising in medium sized carriers such that its solutions can be used in daily operations without further adaptions. This thesis is structured as follows: The first chapter highlights mathematical models of the planning process of public transit companies and examines their potential for integrating them with other planning steps. It also introduces descriptions of the vehicle and the duty scheduling problem. Chapter 2 motivates why it can be useful to integrate vehicle and duty scheduling, explains approaches of the literature, and gives an outline of our algorithm IS-OPT. The following chapters go into the details of the most important techniques and methods of IS-OPT: In Chapter 3 we describe how we use Lagrangean relaxation in a column generation framework. Next, in Chapter 4, we describe a variant of the proximal bundle method (PBM) that is used to approximate linear programs occurring in the solution process. We introduce here a new variant of the PBM which is able to utilize inexact function evaluation and the use of epsilon-subgradients. We also show the convergence of this method under certain assumptions. Chapter 5 treats the generation of duties for the duty scheduling problem. This problem is modeled as a resourceconstraint- shortest-path-problem with non-linear side constraints and nearly linear objective function. It is solved in a two-stage approach. At first we calculate lower bounds on the reduced costs of duties using certain nodes by a new inexact label-setting algorithm. Then we use these bounds to speed up a depth-first-search algorithm that finds feasible duties. In Chapter 6 we present the primal heuristic of IS-OPT that solves the integrated problem to integrality. We introduce a new branch-and-bound based heuristic which we call rapid branching. Rapid branching uses the proximal bundle method to compute lower bounds, it introduces a heuristic node selection scheme, and it utilizes a new branching rule that fixes sets of many variables at once. The common approach to solve the problems occurring in IS-OPT is to trade inexactness of the solutions for speed of the algorithms. This enables, as we show in Chapter 7, to solve large real world integrated problems by IS-OPT. The scheduled produced by IS-OPT save up to 5% of the vehicle and duty cost of existing schedules of regional and urban public transport companies.

In this paper, we study the influence of technology, traffic properties and price trends on optimized
design of a reference IP-over-WDM network with rich underlying fiber topology. In each network node,
we investigate the optimal degree of traffic switching in an optical (lambda) domain versus an electrical
(packet) domain, also known as measure of \emph{node transparency}. This measure is studied in connection to changes in
traffic volume,
demand affinity, optical circuit speeds and equipment cost. By applying variable design constraints,
we assess the relative roles of the two distinct equipment groups, IP routers and optical
cross-connects, with respect to resulting changes in cost-sensitive network architectures.

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 present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, the crews of the tours have to be scheduled. Thus, we developed a personalized crew rostering model. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The paper focuses first on different requirements for the rostering and how they can be modeled in detail. The second focus is on a bicriterion analysis of the planning problem to find the balance between the control quality and the roster acceptance. On the one hand the tour planning is a profit maximization problem and on the other hand the rostering should be made in a employee friendly way. Finally, computational results on real-world instances show the practicability of our method.

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 i s 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 numb er of trips per month. We demonstrate the examples by computational experiments.

The \emph{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 d iscrete 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.

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.

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.

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.