Mathematical Problems in Engineering

Volume 2015, Article ID 289072, 12 pages

http://dx.doi.org/10.1155/2015/289072

## A Branch and Bound Algorithm and Iterative Reordering Strategies for Inserting Additional Trains in Real Time: A Case Study in Germany

^{1}Institute of Railway Systems Engineering and Traffic Safety, Technical University of Braunschweig, Pockelsstrasse 3, 38106 Braunschweig, Germany^{2}School of Transportation Engineering, Tongji University, 4800 Caoan Road, Shanghai 201804, China

Received 6 June 2014; Revised 1 September 2014; Accepted 2 September 2014

Academic Editor: Huimin Niu

Copyright © 2015 Yuyan Tan and Zhibin Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the aim of supporting the process of adapting railway infrastructure and future traffic needs, we have developed a method to insert additional trains efficiently to an existing timetable without introducing large consecutive delays to scheduled trains. In this work, the problem is characterized as a job-shop scheduling problem. In order to meet the limited time requirement and minimize deviations to the existing timetable, the modification that consists of retiming or reordering trains is implemented if and only if it potentially leads to a better solution. With these issues in mind, the problem of adding train paths is decomposed into two subproblems. One is finding the optimal insertion for a fixed order timetable and the other is reordering trains. The two subproblems are solved iteratively until no improvement is possible within a time limit of computation. An innovative branch and bound algorithm and iterative reordering strategy are proposed to solve this problem in real time. Unoccupied capacities are utilized as primary resources for additional trains and the transfer connections for passengers can be guaranteed in the new timetable. From numerical investigations, the proposed framework and associated techniques are tested and shown to be effective.

#### 1. Introduction and Literature Review

This paper will give an account of how to reconstruct an existing train schedule by inserting additional train services. This timetable-based extra train paths inserting (TETPI) problem is an integration of railway dispatching and scheduling. Train dispatcher both modifies the given timetable to manage delay in the running operation and establishes schedules for extra trains.

##### 1.1. Background

The primary motivation of this research occurs as a result of the following application areas.

* (1) The Demands of Extra Trains for Train Operating Companies in European Railways.* Pachl [1] describes the open access networks in European railways. In the train paths management, the train operating companies order train paths from the infrastructure operator, and then the infrastructure operator allocates these train paths on its lines in accordance with the orders of the train operating companies. From the view of scheduling, there is a distinction between regular trains and extra trains. Regular trains are all trains that have a schedule in the yearly timetable. Train paths for regular trains have to be ordered several months before the yearly timetable comes into effect. Extra trains are trains that have no schedule in the yearly timetable. Train paths for extra trains can be ordered at any time. Thus, when establishing the yearly timetable, a lot of traffic is not yet known. It contains only a part of the real traffic, mainly passenger trains and freight trains that run on a regular basis. Nowadays, in order to meet the needs of the shippers, an increasing share of freight trains runs as extra trains. Train paths for many freight trains, which do often not even appear in the scheduling systems, are ordered in a very short time in advance, sometimes just a few hours [1]. For this reason, the train operating companies have to rise to the challenge that establishing conflict-free train paths for these trains in a short time and keeping the disruptions to regular trains minimised or similarly within acceptable levels.

* (2) A key Concept for Noncyclic Timetable in China Railway System.* The cyclic timetable has been widely adopted on high-speed railways because of its obvious advantages in transport marketing and train operation. However, it is not wise to apply the complete cyclic timetable model on Chinese railway due to its own features. Recently, researchers have shown an increased interest in an incomplete cyclic timetable model, an integration model of cyclic and noncyclic timetables, on China high-speed railways, such as [2, 3]. When constructing incomplete cyclic timetables, the trains of short (or medium long) distances and high frequencies are scheduled as cyclic trains, and then trains of long distances or low frequencies need to be inserted into the planned cyclic timetable as extra acyclic trains. Nowadays, the cyclic timetable models have been well developed, and the time-based train insertion technique without breaking the original cyclic structure is still a significant demand for research.

In this paper, the TETPI problem via a job shop scheduling approach is considered. More specifically the purpose of the TETPI problem in this paper is to establish a new conflict-free timetable for additional trains in a short time, guarantee the necessary transfer connections between passenger trains, and aim to minimize the delays incurred by additional trains, where delay is defined as deviation of actual service schedules and the initial schedules.

In order to meet the limited time requirement and minimize deviations to the existing timetable, in practice it is not necessary to take all of the scheduled trains into consideration. The modification which usually consists of retiming or reordering trains is implemented if and only if it potentially leads to a better solution. With these issues in mind the following process is possible for solving the TETPI problem.

*Phase 1 (FX strategy). *Fix all previously scheduled services.

*Phase 2 (RM strategy). *The scheduled services can be retimed, but the relative order between trains is kept.

*Phase 3 (RO strategy). *The scheduled services can be reordered.

Note that the TETPI process does not necessarily have to be applied in the order shown above since not every phase is required. For instance when track infrastructure utilisation is light then Phase 1 would be applied. When track capacity is already heavily utilised, additional trains can only be inserted by disrupting existing services. Phase 1 will be inappropriate and Phase 2 should be applied. In both Phase 1 and Phase 2, additional train services may be added using a constructive algorithm. In Phase 3, a decision is taken in this phase to restate a priority order between trains, competing for the same single track section, in order to get a better solution.

Based on the above analysis, the contributions of this paper are as follows.(1)An iterative procedure for inserting extra trains to an existing timetable is defined in this paper. The TETPI problem is decomposed into two subproblems; one is finding the optimal insertion for a fixed order timetable (Phase 1 and Phase 2) and the other is reordering trains (Phase 3). The two subproblems are solved iteratively until no improvement is possible within a time limit of computation.(2)An innovative branch and bound search algorithm is introduced to solve the first subproblem. In this step, the additional trains mainly utilize the unoccupied capacities to be inserted in order to minimise the deviations to existing trains.(3)The order between trains is restated based on dispatching rules to get a better solution. Note that only the change between train services which occur in the* critical path* is propitious to decrease the makespan (i.e., objective in the model) of the alternative graph.

##### 1.2. Review of the Related Literature

###### 1.2.1. Literature Review on Scheduling and Rescheduling Problem

The TETPI problem is related to a variety of topics in the literature. The first and foremost is rail timetable optimization. Recently, train scheduling and rescheduling problems have a great deal of attention. Corman et al. [4], for example, present a list of foremost papers published on this area between 1999 and 2007. There are varied models for formulating train operation problem. Since then we have observed the following papers based on* graph theory* to solve these problems efficiently: D’Ariano et al. [5] propose a fixed speed model and variable speed model to find a conflict-free timetable in real time after train operations are perturbed. D’Ariano et al. [6] consider the problem of managing disturbance in real time. In this paper, a real-time traffic management system called Railway traffic Optimization by means of alternative graph (ROMA) is introduced. They model the railway traffic optimization based no-store alternative graph, including constraints of rolling stock and passenger connections. This problem is decomposed into two subproblems, one is reordering which is solved by branch and bound algorithm and the other is rerouting which is solved by a local search algorithm. The two subproblems are then solved iteratively. Cacchiani et al. [7] deal with the problem of timetabling noncyclic trains. A mixed-integer programming model (MIP) is proposed to look for the maximum-weight path in a comparability graph. Schachtebeck [8] considers the delay management in public transportation. Based on graph theory he uses integer programming model (IP) formulation and suggests various heuristic solution approaches to solve large-scale real-world instances to optimality. In addition, papers such as [9–13] also present innovative models which are of certain reference value in the field of railway operation optimization.

Furthermore, there are indications that some of the previous models and techniques in job shop scheduling could be modified and adapted to solve the TETPI problem. For instance, Kis and Hertz [14] give for the classical job shop a polyhedral description of the feasible job insertions and use it to derive a lower bound for the minimum makespan job insertion problem. Gröflin et al. investigate [15, 16] insertion problems in a general disjunctive scheduling framework capturing a variety of job shop scheduling problems and inserting types. They propose a short cycle property for job insertion problem. A polyhedral description of all the feasible job insertions is derived to find the lower and upper bound for the minimum makespan.

###### 1.2.2. Literature Review on Problem of Adding Train Paths

Although the adding extra train paths technology is very important, there has been few direct related discussion about adding paths problem. The only papers to our knowledge are presented in Table 1 which summarizes the studies, like ours, dealing with inserting passenger or freight trains into an exiting timetable.