## 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING

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#### Keywords

- Periodic timetabling (2)
- Bipartite matching (1)
- Change-cycle inequality (1)
- Controls in transportation networks (1)
- Cycle inequality (1)
- Linienplanung (1)
- Mixed-Integer Nonlinear Programming (1)
- Polymatrix game (1)
- Rolling Stock Planning, Hypergraph Modeling, Integer Programming, Column Generation, Rapid Branching (1)
- Stackelberg game (1)

#### Institute

Cycle inequalities play an important role in the polyhedral study of the periodic
timetabling problem. We give the first pseudo-polynomial time separation algo-
rithm for cycle inequalities, and we give a rigorous proof for the pseudo-polynomial
time separability of the change-cycle inequalities. Moreover, we provide several
NP-completeness results, indicating that pseudo-polynomial time is best possible.
The efficiency of these cutting planes is demonstrated on real-world instances of the
periodic timetabling problem.

A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable
(2018)

We consider the following planning problem in public transportation: Given a
periodic timetable, how many vehicles are required to operate it?
In [9], for this sequential approach, it is proposed to first expand the periodic
timetable over time, and then answer the above question by solving a flow-based
aperiodic optimization problem.
In this contribution we propose to keep the compact periodic representation of
the timetable and simply solve a particular perfect matching problem. For practical
networks, it is very much likely that the matching problem decomposes into several
connected components. Our key observation is that there is no need to change any
turnaround decision for the vehicles of a line during the day, as long as the timetable
stays exactly the same.

Real world routing problems, e.g., in the airline industry or in public and rail transit, can feature complex non-linear cost functions. An important case are costs for crossing regions, such as countries or fare zones. We introduce the shortest path problem with crossing costs (SPPCC) to address such situations; it generalizes the classical shortest path problem and variants such as the resource constrained shortest path problem and the minimum label path problem. Motivated by an application in flight trajectory optimization with overflight costs, we focus on the case in which the crossing costs of a region depend only on the nodes used to enter or exit it. We propose an exact Two-Layer-Dijkstra Algorithm as well as a novel cost-projection linearization technique that approximates crossing costs by shadow costs on individual arcs, thus reducing the SPPCC to a standard shortest path problem. We evaluate all algorithms’ performance on real-world flight trajectory optimization instances, obtaining very good à posteriori error bounds.

Bus rapid transit systems in developing and newly industrialized countries are often operated at the limits of passenger capacity. In particular, demand during morning and afternoon peaks is hardly or even not covered with available line plans. In order to develop demand-driven line plans, we use two mathematical models in the form of integer programming problem formulations. While the actual demand data is specified with origin-destination pairs, the arc-based model considers the demand over the arcs derived from the origin-destination demand. In order to test the accuracy of the models in terms of demand satisfaction, we simulate the optimal solutions and compare number of transfers and travel times. We also question the effect of a selfish route choice behavior which in theory results in a Braess-like paradox by increasing the number of transfers when system capacity is increased with additional lines.

The rolling stock, i.e., railway vehicles, are one of the key ingredients of a running railway system. As it is well known, the offer of a railway company to their customers, i.e., the railway timetable, changes from time to time. Typical reasons for that are different timetables associated with different seasons, maintenance periods or holidays. Therefore, the regular lifetime of a timetable is split into (more or less) irregular periods where parts of the
timetable are changed. In order to operate a railway timetable most railway companies set up sequences that define the operation of timetabled trips by a single physical railway vehicle called (rolling stock) rotations. Not surprisingly, the individual parts of a timetable also affect the rotations. More precisely, each of the parts brings up an acyclic rolling stock rotation problem with start and end conditions associated with the beginning and ending of the corresponding period. In this paper, we propose a propagation approach to deal with large planning horizons that are composed of many timetables with shorter individual lifetimes. The approach is based on an integer linear programming formulation that propagates rolling stock rotations through the irregular parts of the timetable while taking a large variety of operational requirements into account. This approach is implemented within the rolling stock rotation optimization framework ROTOR used by DB Fernverkehr AG, one of the leading railway operators in Europe. Computational results for real world scenarios are presented to evaluate the approach.

We consider railway timetables of our industrial partner DB Fernverkehr AG that operates the ICE high speed trains in the long-distance passenger railway network of Germany. Such a timetable covers a whole year with 364 days and, typically, includes more than 45,000 trips. A rolling stock rotation plan is not created for the whole timetable at once. Instead the timetable is divided into regular invariant sections and irregular deviations (e.g. for public holidays). A separate rotation plan with a weekly period can then be provided for each of the different sections of the timetable. We present an algorithmic approach to automatically recognize these sections. Together with the supplementing visualisation of the timetable this method has shown to be very relevant for our industrial partner.

A railway operator creates (rolling stock) rotations in order to have a precise master plan for the
operation of a timetable by railway vehicles. A rotation is considered as a cycle that multiply
traverses a set of operational days while covering trips of the timetable. As it is well known,
the proper creation of rolling stock rotations by, e.g., optimization algorithms is challenging
and still a topical research subject. Nevertheless, we study a completely different but strongly
related question in this paper, i.e.: How to visualize a rotation? For this purpose, we introduce
a basic handout concept, which directly leads to the visualization, i.e., handout of a rotation. In
our industrial application at DB Fernverkehr AG, the handout is exactly as important as the
rotation itself. Moreover, it turns out that also other European railway operators use exactly the
same methodology (but not terminology). Since a rotation can have many handouts of different
quality, we show how to compute optimal ones through an integer program (IP) by standard
software. In addition, a construction as well as an improvement heuristic are presented. Our
computational results show that the heuristics are a very reliable standalone approach to quickly
find near-optimal and even optimal handouts. The efficiency of the heuristics is shown via a
computational comparison to the IP approach.

We introduce the shortest path problem with crossing costs (SPPCC), a shortest path problem in a directed graph, in which the objective function is the sum of arc weights and crossing costs. The former are independently paid for each arc used by the path, the latter need to be paid every time the path intersects certain sets of arcs, which we call regions.
The SPPCC generalizes not only the classical shortest path problem but also variants such as the resource constrained shortest path problem and the minimum label path problem. We use the SPPCC to model the flight trajectory optimization problem with overflight costs.
In this paper, we provide a comprehensive analysis of the problem. In particular,
we identify efficient exact and approximation algorithms for the cases that are most relevant in practice.

We investigate a graph theoretical problem arising in the automatic billing of a network toll. Given a network and a family of user paths, we study the graph segmentation problem (GSP) to cover parts of the user paths by a set of disjoint segments. The GSP is shown to be NP-hard but for special cases it can be solved in polynomial time. We also show that the marginal utility of a segment is bounded. Computational results for real-world instances show that in practice the problem is more amenable than the theoretic bounds suggest.