Ordering Constraints in Time Expanded Networks for Train Timetabling Problems

The task of the train timetabling problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. This kind of problem has proven to be very challenging and numerous solution approaches have been proposed. One of the most successful approaches is based on time discretized network models. However, one of the major weaknesses of these models is that fractional solutions tend to change the order of trains along some track, which is not allowed for integer solutions, leading to poor relaxations. In this paper, we present an extension for these kind of models, which aims at overcoming these problems. By exploiting a configuration based formulation, we propose to extend the model with additional ordering constraints. These constraints enforce compatibility of orderings along a sequence of tracks and greatly improve the quality of the relaxations. We show in some promising preliminary computational experiments that our approach indeed helps to resolve many of the invalid overtaking problems of relaxations for the standard models

Strong Relaxations for the Train Timetabling Problem Using Connected Configurations

The task of the train timetabling problem or track allocation problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. Especially for non-periodic instances models based on time expanded networks are often used. Unfortunately, the linear programming relaxation of these models is often extremely weak because these models do not describe combinatorial relations like overtaking possibilities very well. In this paper we extend the model by so called connected configuration subproblems. These subproblems perfectly describe feasible schedules of a small subset of trains (2-3) on consecutive track segments. In a Lagrangian relaxation approach we solve several of these subproblems together in order to produce solutions which consist of combinatorially compatible schedules along the track segments. The computational results on a mostly single track corridor taken from the INFORMS RAS Problem Solving Competition 2012 data indicate that our new solution approach is rather strong. Indeed, for this instance the solution of the Lagrangian relaxation is already integral

Phase Synchronization in Railway Timetables

Timetable construction belongs to the most important optimization problems in public transport. Finding optimal or near-optimal timetables under the subsidiary conditions of minimizing travel times and other criteria is a targeted contribution to the functioning of public transport. In addition to efficiency (given, e.g., by minimal average travel times), a significant feature of a timetable is its robustness against delay propagation. Here we study the balance of efficiency and robustness in long-distance railway timetables (in particular the current long-distance railway timetable in Germany) from the perspective of synchronization, exploiting the fact that a major part of the trains run nearly periodically. We find that synchronization is highest at intermediate-sized stations. We argue that this synchronization perspective opens a new avenue towards an understanding of railway timetables by representing them as spatio-temporal phase patterns. Robustness and efficiency can then be viewed as properties of this phase pattern

A review of key planning and scheduling in the rail industry in Europe and UK

Planning and scheduling activities within the rail industry have benefited from developments in computer-based simulation and modelling techniques over the last 25 years. Increasingly, the use of computational intelligence in such tasks is featuring more heavily in research publications. This paper examines a number of common rail-based planning and scheduling activities and how they benefit from five broad technology approaches. Summary tables of papers are provided relating to rail planning and scheduling activities and to the use of expert and decision systems in the rail industry.EPSR

Engineering Graph-Based Models for Dynamic Timetable Information Systems

Many efforts have been done in the last years to model public transport timetables in order to find optimal routes. The proposed models can be classified into two types: those representing the timetable as an array, and those representing it as a graph. The array-based models have been shown to be very effective in terms of query time, while the graph-based models usually answer queries by computing shortest paths, and hence they are suitable to be used in combination with speed-up techniques developed for road networks. In this paper, we focus on the dynamic behavior of graph-based models considering the case where transportation systems are subject to delays with respect to the given timetable. We make three contributions: (i) we give a simplified and optimized update routine for the well-known time-expanded model along with an engineered query algorithm; (ii) we propose a new graph-based model tailored for handling dynamic updates; (iii) we assess the effectiveness of the proposed models and algorithms by an experimental study, which shows that both models require negligible update time and a query time which is comparable to that required by some array-based models

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes