Production, Manufacturing and Logistics A disjunctive graph model and framework for constructing new train schedules R.L. Burdett, E. Kozan * School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434, Brisbane, QLD 4001, Australia article info Article history: Received 20 April 2007 Accepted 6 December 2008 Available online 16 December 2008 Keywords: Train scheduling Job shop scheduling abstract Train scheduling is a complex and time consuming task of vital importance in many countries. To create completely new train schedules that are more accurate and efficient than permitted by current tech- niques, a novel ‘‘hybrid” job shop approach is proposed and implemented in this paper. Unique charac- teristics of train scheduling are firstly incorporated into a disjunctive graph representation of the solution. Dedicated ‘‘stand-alone” constructive algorithms that utilise this representation are then devel- oped. The modelling approach and the constructive algorithms are essential as they provide the basis for which meta-heuristics and other iterative refinement algorithms can be applied. A numerical investiga- tion and case study is provided and demonstrates the viability of the modelling approach. Furthermore it is demonstrated that good quality solutions are provided with reasonable computational effort. Crown Copyright Ó 2008 Published by Elsevier B.V. All rights reserved. 1. Introduction Trains provide a relatively clean and cheap method of transpor- tation for passengers and freight, and compare favourably if not better than alternative modes of transportation such as road, air and sea in many circumstances. Furthermore the utilisation of railway systems can only increase in the future as roads become even more congested, trains become faster and infrastructure is extended. Due to the size, weight and speed of trains the coordina- tion of train movements (by train scheduling) is vital in order to utilise these systems safely and effectively. However train schedul- ing on current systems is still a relatively difficult and time consuming task as the size and complexity is prohibitive. Train scheduling problems have unique properties and pose a number of unique difficulties that distinguish it from other related schedul- ing problems. These will be discussed in a later section. The manual construction of a schedule by a human expert with the help of computer software is the most common first and last resort in practice. In practice there are a variety of different scheduling problems that must be solved, though in principle two main variants exist. The first considers the development of a new timetable that is typ- ically but not necessarily to be applied at regular intervals such as daily, weekly or monthly. The second scheduling problem concerns the re-development of an existing timetable. For example, an exist- ing timetable may become undesirable and or infeasible after unforseen delays have caused significant deviations to the original plan. In the first variant there is usually no limitation on when trains may enter the system, i.e. they may enter at any time. How- ever in the second variant trains have to enter at predefined time and some trains may already be within the system at the start of the schedule. In recent years the majority of papers in the literature have ad- dressed the second ‘‘rescheduling” problem, and examples include Carey (1994a,b) and Higgins et al. (1996) for exact approaches and Cai and Goh (1994), Higgins et al. (1997), Cai et al. (1998), Chiang et al. (1998), Sahin (1999), Adenso-Diaz et al. (1999), and Dorfman and Medanic (2004) for heuristic approaches . The first problem has been addressed more recently by Odijk (1996), Brannlund et al. (1998), Goverde (1999), Lindner (2000), Kroon and Peeters (2003), Ghoseiri et al. (2004) and Zhou and Zhong (2005). Train platforming and pathing is another aspect that has received atten- tion recently by Carey and Lockwood (1992), Romeijn et al. (1996), Zwaneveld et al. (1996), Kroon et al. (1997), Cordeau et al. (1998), Zwaneveld et al. (2001), Billionnet (2003), Carey and Carville (2003). In this paper the development of completely new train sched- ules via a new ‘‘hybrid” job shop scheduling approach is consid- ered. More specifically this paper considers the most efficient way for a specified number (mix) of trains with predefined routes to traverse a railway system (network) between their predefined origin and destination location subject to a variety of technical constraints. A railway system is a single track or a complex net- work of interconnected tracks. A makespan objective criterion is used in this paper to measure the relative merits of a new timeta- ble though other criterions could easily be used as our approaches are quite independent. The makespan objective is a well known scheduling measure and provides a good benchmark for comparing the efficiency of the techniques proposed in this paper. In our expe- rience train scheduling criteria varies from one region and operator to the next and when constructing a new timetable the best 0377-2217/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.12.005 * Corresponding author. Tel.: +61 7 3864 1069; fax: +61 7 3864 2310. E-mail address: e.kozan@qut.edu.au (E. Kozan). European Journal of Operational Research 200 (2010) 85–98 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor