Christian Liebchen

The Concept of Recoverable Robustness, Linear Programming Recovery, and Railway Applications

We present a new concept for optimization under uncertainty: recoverable robustness. A solution is recovery robust if it can be recovered by limited means in all likely scenarios. Specializing the general concept to linear programming we can show that recoverable robustness combines the flexibility of stochastic programming with the tractability and performances guarantee of the classical robust approach. We exemplify recoverable robustness in delay resistant, periodic and aperiodic timetabling problems, and train platforming.

The First Optimized Railway Timetable in Practice

A short time ago, decision support by operations research methods in railway companies was limited to operations planning (e.g., vehicle scheduling, duty scheduling, crew rostering). In effect since December 12, 2004, the 2005 timetable of the Berlin subway is based on the results of mathematical programming techniques. It is the first such service concept that has been put into daily operation. Profiting from these techniques, compared with the previous timetable, the Berlin subway today operates with a timetable that offers shorter passenger waiting times---both at stops and at transfers---and even saves one train. The work is based on a well-established graph model, the periodic event-scheduling problem (Pesp). This model was introduced as early as 1989. Besides describing in detail its first success story in practice, in this paper we also deepen a result on the asymptotic complexity of the Pesp: we provide MAXSNP-hardness proofs of two natural optimization variants.

Periodic Timetable Optimization in Public Transport

„The timetable is the essence of the service offered by any provider of public transport.“ (Jonothan Tyler, CASPT 2006)

The modeling power of the periodic event scheduling problem: railway timetables-and beyond

In the planning process of railway companies, we propose to integrate important decisions of network planning, line planning, and vehicle scheduling into the task of periodic timetabling. From such an integration, we expect to achieve an additional potential for optimization. Models for periodic timetabling are commonly based on the Periodic Event Scheduling Problem (PESP). We show that, for our purpose of this integration, the PESP has to be extended by only two features, namely a linear objective function and a symmetry requirement. These extensions of the PESP do not really impose new types of constraints. Indeed, practitioners have already required them even when only planning timetables autonomously without interaction with other planning steps. Even more important, we only suggest extensions that can be formulated by mixed integer linear programs. Moreover, in a selfcontained presentation we summarize the traditional PESP modeling capabilities for railway timetabling. For the first time, also special practical requirements are considered that we proove not being expressible in terms of the PESP.

Survey: Cycle bases in graphs characterization, algorithms, complexity, and applications

Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and a priori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX-hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

A Case Study in Periodic Timetabling

Abstract In the overwhelming majority of public transportation companies, designing a periodic timetable is even nowadays largely performed manually. Software tools only support the planners in evaluating a periodic timetable, or by letting them comfortably shift sets of trips by some minutes, but they rarely use optimization methods. One of the main arguments against optimization is that there is no clear objective in practice, but that many criteria such as amount of rolling stock required, average passenger changing time, average speed of the trains, and the number of cross-wise correspondences have to be considered. This case study will demonstrate on the example of the Berlin underground (BVG) that all these goals can be met if carefully modeled, and that timetables constructed by optimization lead to considerable improvements. Our approach uses the Periodic Event Scheduling Problem (PESP) with several add-ons concerning problem reduction and strengthening. The resulting integer linear programs are solved with the CPLEX MIP-Solver. We have been able to construct periodic timetables that improve the current timetable considerably. For any of the above criteria, we have been able to identify global lower and upper bounds. Our favorite timetable improves the current BVG timetable in each of these criteria.

Classes of cycle bases

In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiries. At present, the complexity status of the MCB problem has been settled only for undirected, directed, and strictly fundamental cycle bases. In this paper, we offer an unitary classification accommodating these three classes and further including the following four relevant classes: 2-bases (or planar bases), weakly fundamental cycle bases, totally unimodular cycle bases, and integral cycle bases. The classification is complete in that, for each ordered pair (A,B) of classes considered, we either prove that A@?B holds for every graph or provide a counterexample graph for which A@?B. The seven notions of cycle bases are distinct (either A@?B or B@?A is exhibited for each pair (A,B)). All counterexamples proposed have been designed to be ultimately effective in separating the various algorithmic variants of the MCB problem naturally associated to each one of these seven classes. Furthermore, we provide a linear time algorithm for computing a minimum 2-basis of a graph. Finally, notice that the resolution of the complexity status of some of the remaining three classes would have an immediate impact on practical applications, as for instance in periodic railway timetabling, only integral cycle bases are of direct use.

Performance of Algorithms for Periodic Timetable Optimization

During the last 15 years, many solution methods for the important task of constructing periodic timetables for public transportation companies have been proposed. We first point out the importance of an objective function, where we observe that in particular a linear objective function turns out to be a good compromise between essential practical requirements and computational tractability. Then, we enter into a detailed empirical analysis of various Mixed Integer Programming (MIP) procedures — those using node variables and those using arc variables — genetic algorithms, simulated annealing and constraint programming. To our knowledge, this is the first comparison of five conceptually different solution approaches for periodic timetable optimization.

Delay resistant timetabling

In public transport punctuality has prominent influence on the customers’ satisfaction. Our task is to support a management decision to optimally invest passengers’ nominal travel time to secure the nominal schedule against delay. For aperiodic scheduling we clarify the notion and use of a fixed amount of time supplements, so-called buffers, both theoretically and by realistic examples. The general tool to solve such optimization problems is a sampling approach. We show how this approach is mathematically justified. As its applicability to large networks is limited, we show an efficient alternative for the case of series-parallel graphs. For periodic timetabling we propose two heuristic approaches to ensure a certain level of delay resistance at the least expense of slightly increased nominal passengers travel time, and analyze in detail their advantages and drawbacks.

Finding Short Integral Cycle Bases for Cyclic Timetabling

Cyclic timetabling for public transportation companies is usually modeled by the periodic event scheduling problem. To obtain a mixed-integer programming formulation, artificial integer variables have to be introduced. There are many ways to define these integer variables.