Solving time-dependent multimodal transport problems using a transfer graph model
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.
Section snippets
The transfer graph model
In this section we present the transfer graph, a graphical structure that abstracts the time-dependent multimodal transport network. The main advantage of this model is that it adapts to the distributed nature of real-world transport information providers since it separates and keeps all transport modes in different unimodal networks. Another benefit of this abstraction is that each unimodal network can be easily and independently updated without requiring any further recalculation (
The relevant graph approach
So far, the transfer graph to abstract time-dependent multimodal networks is defined. In this section we present an approach to compute the shortest path (SP) in this model.
The relevant graph approach is divided into three main steps (see Fig. 2). The first one is called precalculation and consists in computing and storing in advance part of the calculations. After, based on this information and additional computation a more compact structure is built, called relevant graph. Finally the SP is
The hybrid approach
In Section 3 we have formally presented the relevant graph approach for computing the SPP in transfer graph as well as two implementations of this approach. We have demonstrated that these previous solutions are limited by time or space. These results are partially presented in Galvez-Fernandez et al. (2009). In this section we propose an alternative solution, the hybrid approach.
In Fig. 4 a general scheme of relevant graph and hybrid approach is shown. As it can be seen, the hybrid approach
Theoretical comparison
In this section we demonstrate that solving the SPP by hybrid approach is similar to solving it by the relevant graph approach.
In order to validate the hybrid approach from theoretical point of view, we must show that any SP obtained by the relevant graph approach can be found by the hybrid approach as well, i.e., the hybrid approach is correct in respect to the relevant graph approach. In fact, the correctness is possible only if we use exact method in both approaches.
Before establishing the
Experimental results
In this section, an implementation of the hybrid approach is proposed. Also, several tests in order to select their appropriated parameters are analyzed. Finally, the performance of the algorithm on different instances of the problem is compared with the other two approaches presented in previous sections.
Related work
The multimodal transport problem under the time-dependent constraint is not often studied in literature. As far as we know there are two main works related to time-dependent multimodal transport problem that present the performance of their approach on different instances of the problem (Bielli et al., 2006, Ziliaskopoulos and Wardell, 2000).
Authors in Ziliaskopoulos and Wardell (2000) abstract the problem with non-hierarchical multimodal graph and propose several data structures and tables,
Conclusion
In this work we proposed an hybrid approach for optimizing the time-dependent multimodal transport problems.
The hybrid approach is compared with two previous approaches, developed by the authors. The experimental results show that abstract graph is a better intermediary model than precalculation for solving the SP in TDMG. Besides, abstract graph does not consider time or cost. For this reason, this graph is independent of changes in transport conditions (e.g., traffic jams, delays, roadworks),
Acknowledgments
This work is partially supported by the National Research Fund of Luxembourg, under contract TR-PDR BFR08-086 and TR-PHD BFR08-005.
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