Solving time-dependent multimodal transport problems using a transfer graph model H. Ayed a,b, * , C. Galvez-Fernandez a , Z. Habbas b , D. Khadraoui a a CRP Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg, Luxembourg b LITA, University Paul Verlaine – Metz, Idle du Saulcy 57045, Metz Cedex 1, France article info Article history: Available online 10 June 2010 Keywords: Time-dependent multimodal transport problem Shortest path problem Dijkstra ACO abstract In this paper we present an hybrid approach for solving the time-dependent multimodal transport prob- lem. This approach has been tested on realistic instances of the problem providing an adequate balance between computation time and memory space. This solution can be applied to real transport networks in order to reduce the impact of traffic congestion on pollution, economy, and citizen’s welfare. A compar- ison with two previous approaches are given from theoretical point of view as well as experimental performance. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Nowadays, the daily mobility of passenger and goods has be- come a very important problem in our society. Traffic congestion produces a direct impact on the economy, causes an increase of pollution, and reduces citizens’ welfare. According to the recent data available, in Beijing (China) the road transport sector gener- ates 23% of the total air pollution (Zhao, 2009), close behind the industrial sector, while in the European Union the transport emis- sions are accounted for around 20% of total greenhouse gas emis- sions (Bart, in press). In US, urban traffic congestion caused during 2007 a waste of fuel equal to $87.2 billion as well as 4.2 bil- lion hours of transport delay (Schrank & Lomax, 2009). There are different policy options for dealing with this men- tioned problem. In order to avoid that urban development becomes exclusively car-oriented, one of the measures consists in improving the quality of public transport and encouraging its use (Bart, in press). However, greater effectiveness can be reached through combining the different private and public transport means, spe- cially in big cities and in interregional scenarios. Multimodal transport, the combination of public and private transport modes, has been addressed by several authors in the re- search community. For solving the multimodal transport problem (MTP) different abstractions are proposed, generally based on the concept of graph theory: hypergraphs (Lozano & Storchi, 2001), hierarchical graph (Bielli, Boulmakoul, & Mouncif, 2006), and clas- sical multigraph (Lo, Yip, & Wan, 2003). Besides, several algorithms to compute the shortest path for the MTP are given in the litera- ture. The Dijkstra algorithm is the most used approach (Kamoun, Uster, & Hammadi, 2005; Zidi, 2006), while other algorithms like the label correcting algorithm (Lozano & Storchi, 2002; Ziliaskopo- ulos & Wardell, 2000), Breadth-First search (Fragouli & Delis, 2002) and heuristic algorithms (Chang, 2008; Chiu, Lee, Fung Leung, Au, & Wong, 2005; Li & Kurt, 2000) are also investigated. Despite the great effort done in this field, the complexity of the MTP has not been fully addressed. In realistic scenarios, traveling cost (e.g., time, price, comfort) depends on time, and thus the opti- mal solution. This time-dependent multimodal transport problem (TMTP) is more complex for solving, since it contemplates different transport modes available and their schedules. In fact, there exist few works that take into account this constraint (Bielli et al., 2006; Galvez-Fernandez, Khadraoui, Ayed, Habbas, & Alba, 2009; Ziliaskopoulos & Wardell, 2000). In our previous works we have developed a solution for solving TMTP. In Galvez-Fernandez et al. (2009) an alternative abstraction to model time-dependent multimodal networks called transfer graph was presented as well as an approach for this abstraction. Two implementations of this approach were proposed. A variant of Dijkstra algorithm was developed in Galvez-Fernandez et al. (2009). It provides better performance in terms of computation time than other algorithms in the literature. Nevertheless, the re- quired memory space makes it unfeasible to apply on big-sized transport networks. In Ayed, Habbas, and Khadraoui (2009), we present a second solution that uses Ant Colony Optimization (ACO) metaheuristic (Dorigo, Birattari, & Stntzle, 2006). It requires less memory space but increases the computation time. 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.05.018 * Corresponding author at: CRP Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg, Luxembourg. Tel.: +352 42 59 91 803; fax: +352 42 59 91 333. E-mail addresses: hedi.ayed@tudor.lu (H. Ayed), carlos.galvez@tudor.lu (C. Galvez-Fernandez), zineb@univ-metz.fr (Z. Habbas), djamel.khadraoui@tudor.lu (D. Khadraoui). Computers & Industrial Engineering 61 (2011) 391–401 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie