Solving time-dependent multimodal transport problems using a transfer graph model

Abstract

In this paper we present an hybrid approach for solving the time-dependent multimodal transport problem. 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 comparison with two previous approaches are given from theoretical point of view as well as experimental performance.

Introduction

Nowadays, the daily mobility of passenger and goods has become 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 generates 23% of the total air pollution (Zhao, 2009), close behind the industrial sector, while in the European Union the transport emissions are accounted for around 20% of total greenhouse gas emissions (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 billion hours of transport delay (Schrank & Lomax, 2009).

There are different policy options for dealing with this mentioned 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, specially 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 research 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 classical multigraph (Lo, Yip, & Wan, 2003). Besides, several algorithms to compute the shortest path for the MTP are given in the literature. The Dijkstra algorithm is the most used approach (Kamoun et al., 2005, Zidi, 2006), while other algorithms like the label correcting algorithm (Lozano and Storchi, 2002, Ziliaskopoulos and Wardell, 2000), Breadth-First search (Fragouli & Delis, 2002) and heuristic algorithms (Chang, 2008, Chiu et al., 2005, Li and 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 optimal 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 et al., 2009, Ziliaskopoulos and 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 required 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.

In this paper a new approach for the TMTP is proposed, which is based on Dijkstra and ACO. Its main contribution is to provide an adequate balance between computation time and space. Therefore, this solution can be scalable and applied to realistic scenarios involving several cities, regions or countries.

The outline of this paper is the following. Section 2 gives some definitions and presents the transfer graph model. Next, in Section 3 we introduce the relevant graph approach for computing the shortest path in transfer graph and two implementations based on Dijkstra and ACO, concluding with the benefits and drawbacks of both strategies. Then, we present the new algorithm for TMTP in Section 4 and we prove that is correct with respect to the relevant graph approach in Section 5. Next, in Section 6 some experiments outline the performances of this new approach. In Section 7 we compare the hybrid approach with other existing approaches proposed in the literature. This paper finishes with conclusions in Section 8.