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Given a directed, acyclic graph, a source and a sink node, and a set of forbidden pairs of arcs, the path avoiding forbidden pairs (PAFP) problem is to find a path that connects the source and sink nodes and contains at most one arc from each forbidden pair. The general version of the problem is NP-hard, but it becomes polynomially solvable for certain topological configurations of the pairs. We present the first polyhedral study of the PAFP problem. We introduce a new family of valid inequalities for the PAFP polytope and show that they are sufficient to provide a complete linear description in the special case where the forbidden pairs satisfy a disjointness property. Furthermore, we show that the number of facets of the PAFP polytope is exponential in the size of the graph, even for the case of a single forbidden pair.
Planning and operating railway transportation systems is an extremely hard task due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. Because of that, however, mathematical models and optimization techniques can result in large gains for both railway cus- tomers and operators, e.g., in terms of cost reductions or service quality improvements. In the last years a large and growing group of researchers in the OR community have devoted their attention to this domain devel- oping mathematical models and optimization approaches to tackle many of the relevant problems in the railway planning process. However, there is still a gap to bridge between theory and practice, with a few notable exceptions. In this paper we address three success stories, namely, long-term freight train routing (part I), mid-term rolling stock rotation planning (part II), and real-time train dispatching (part III). In each case, we describe real-life, successful implementations. We will dis- cuss the individual problem setting, survey the optimization literature, and focus on particular aspects addressed by the mathematical models. We demonstrate on concrete applications how mathematical optimization can support railway planning and operations. This gives proof that math- ematical optimization can support the planning of rolling stock resources. Thus, mathematical models and optimization can lead to a greater effi- ciency of railway operations and will serve as a powerful and innovative tool to meet recent challenges of the railway industry.
Duty rostering problems occur in different application contexts and come in different flavors. They give rise to very large scale integer programs which ypically have lots of solutions and extremely fractional LP relaxations. In such a situation, heuristics can be a viable algorithmic choice. We propose an mprovement method of the Lin-Kernighan type for the solution of duty rostering problems. We illustrate its versatility and solution quality on three different applications in public transit, vehicle routing, and airline rostering with a focus on the management of preferences, fairness, and fatigue, respectively.
Railway transportation and in particular train timetabling is one of the basic and source application areas of combinatorial optimization and integer programming. We will discuss two well established modeling techniques for the train timetabling problem. In this paper we focus on one major ingredient - the bounding by dual relaxations. We compare two classical dual relaxations of large scale time expanded train timetabling problems - the Lagrangean Dual and Lagrangean Decomposition. We discuss the convergence behavior and show limitations of the Lagrangean Decomposition approach for a configuration based model. We introduce a third dualization approach to overcome those limitations. Finally, we present promising preliminary computational experiments that show that our new approach indeed has superior convergence properties.
Rolling stock, i.e., rail vehicles, are among the most expensive and limited assets of a railway company. They must be used efficiently applying optimization techniques. One important aspect is re-optimization, which is the topic that we consider in this paper. We propose a template concept that allows to compute cost minimal rolling stock rotations under a large variety of re-optimization requirements. Two examples, involving a connection template and a rotation template, are discussed. An implementation within the rolling stock rotation optimizer rotor and computational results for scenarios provided by DB Fernverkehr AG, one of the leading railway operators in Europe, are presented.
Railway transportation and in particular train timetabling is one of the basic and source application areas of combinatorial optimization and integer programming. We will discuss two well established modeling techniques for the train timetabling problem. In this paper we focus on one major ingredient - the bounding by dual relaxations. We compare two classical dual relaxations of large scale time expanded train timetabling problems - the Lagrangean Dual and Lagrangean Decomposition. We discuss the convergence behavior and show limitations of the Lagrangean Decomposition approach for a configuration based model. We introduce a third dualization approach to overcome those limitations. Finally, we present promising preliminary computational experiments that show that our new approach indeed has superior convergence properties.
Duty rostering problems occur in different application contexts and come in different flavors. They give rise to very large scale integer programs which ypically have lots of solutions and extremely fractional LP relaxations. In such a situation, heuristics
can be a viable algorithmic choice. We propose an mprovement method of the Lin-Kernighan type for the solution of duty rostering problems. We illustrate its versatility and solution quality on three different applications in public transit, vehicle routing, and
airline rostering with a focus on the management of preferences, fairness, and fatigue, respectively.
The Graduate-Level Research in Industrial Projects (G-RIPS) Program provides an
opportunity for high-achieving graduate-level students to work in teams on a
real-world research project proposed by a sponsor from industry or the public
sector. Each G-RIPS team consists of four international students (two from
the US and two from European universities), an academic mentor, and an industrial sponsor.
This is the report of the Rail-Lab project on the definition and integration of
robustness aspects into optimizing rolling stock schedules. In general, there is
a trade-off for complex systems between robustness and efficiency. The ambitious
goal was to explore this trade-off by implementing numerical simulations and
developing analytic models.
In rolling stock planning a very large set of industrial railway requirements,
such as vehicle composition, maintenance constraints, infrastructure capacity,
and regularity aspects, have to be considered in an integrated model. General
hypergraphs provide the modeling power to tackle those requirements.
Furthermore, integer programming approaches are able to produce high quality
solutions for the deterministic problem.
When stochastic time delays are considered, the mathematical programming problem
is much more complex and presents additional challenges. Thus, we started with a
basic variant of the deterministic case, i.e., we are only considering
hypergraphs representing vehicle composition and regularity.
We transfered solution approaches for robust optimization
from the airline industry to the setting of railways and attained a
reasonable measure of robustness. Finally, we present and discuss different
methods to optimize this robustness measure.
Planning and operating railway transportation systems is an extremely
hard task due to the combinatorial complexity of the underlying discrete
optimization problems, the technical intricacies, and the immense size of
the problem instances. Because of that, however, mathematical models
and optimization techniques can result in large gains for both railway cus-
tomers and operators, e.g., in terms of cost reductions or service quality
improvements. In the last years a large and growing group of researchers
in the OR community have devoted their attention to this domain devel-
oping mathematical models and optimization approaches to tackle many
of the relevant problems in the railway planning process. However, there
is still a gap to bridge between theory and practice, with
a few notable exceptions. In this paper we address three success stories,
namely, long-term freight train routing (part I), mid-term rolling stock
rotation planning (part II), and real-time train dispatching (part III). In
each case, we describe real-life, successful implementations. We will dis-
cuss the individual problem setting, survey the optimization literature,
and focus on particular aspects addressed by the mathematical models.
We demonstrate on concrete applications how mathematical optimization
can support railway planning and operations. This gives proof that math-
ematical optimization can support the planning of rolling stock resources.
Thus, mathematical models and optimization can lead to a greater effi-
ciency of railway operations and will serve as a powerful and innovative
tool to meet recent challenges of the railway industry.
We consider problems concerning the scheduling of a set of trains on a single track. For every pair of trains there is a minimum headway, which every train must wait before it enters the track after another train. The speed of each train is also given. Hence for every schedule - a sequence of trains - we may compute the time that is at least needed for all trains to travel along the track in the given order. We give the solution to three problems: the fastest schedule, the average schedule, and the problem of quantile schedules. The last problem is a question about the smallest upper bound on the time of a given fraction of all possible schedules. We show how these problems are related to the travelling salesman problem. We prove NP-completeness of the fastest schedule problem, NP-hardness of quantile of schedules problem, and polynomiality of the average schedule problem. We also describe some algorithms for all three problems. In the solution of the quantile problem we give an algorithm, based on a reverse search method, generating with polynomial delay all Eulerian multigraphs with the given degree sequence and a bound on the number of such multigraphs. A better bound is left as an open question.