Production, Manufacturing and Logistics
Incorporating transfer reliability into equilibrium analysis of railway passenger flow

https://doi.org/10.1016/j.ejor.2012.02.012Get rights and content

Abstract

Passenger’s transfer route choice behavior is one of the prominent research topics in the field of railway transportation. Existing traffic assignment approaches do not properly account for passenger’s expectation for transfer reliability. In this study, the transfer reliability is explicitly defined and a multi-class user equilibrium model is established, given which passengers choose the minimal-cost path based on their expected reliability thresholds. In particular, a path-based traffic assignment algorithm which combines a k-shortest path algorithm and the method of successive averages is proposed. The validity of the proposed approach is verified by an illustrative example. Using the proposed modeling approach, it is possible to determine the passenger’s collective route choice behavior based on the user equilibrium pattern. Moreover, the railway timetables can be evaluated and optimized based on the cost-based level of service estimation.

Highlights

► Railway transfer reliability is subject to train delays. ► We defined transfer reliability for multi-class passengers. ► Transfer reliability was incorporated into user equilibrium model. ► Railway timetables can be optimized by the proposed method.

Introduction

Railway is the major transport mode for medium-long distance journey in many countries of vast territory (e.g. China and Russia). In 2010, the total rail ridership in China was 1.6 billion trips and the rail passenger distance travelled was 876.2 billion kilometers. It is also reported that China has the world’s longest high-speed rail (HSR) network with about 9,676 kilometer of routes in service as of June 2011, including 3,515 kilometer of rail lines with top speeds of 300 kilometer/hour.

Due to the constraints of network structure and operation plans, passengers would have to arrive at the destination by transferring if there is no direct train available between the origin and destination stations. In reality, missing connections may occur and result in great inconvenience for passengers. It is natural to postulate that passengers would respond against the transfer uncertainties by rationally choosing routes with reasonable transfer time. Thereby, special attention should be paid to this risk-aversion psychology in modeling railway passenger’s route choice behavior, which serves as a useful planning tool at both strategic and operational levels.

Equilibrium approach is wildly used in modeling passenger’s transfer route choice behavior. In traditional equilibrium analysis, of vital concern is to minimize the travel costs in relation to transfer trips in terms of travel time, fare and congestion disutility. Passengers would choose the route with the minimum cost, and finally form a long-term habitual equilibrium pattern (Sheffi, 1985, De Cea and Ferández, 1993, Wu et al., 1994, Shi et al., 2007.

While most of the previous studies ignored the reliability of transportation systems, a number of recent studies have highlighted the effect of travel reliability on passenger’s route choice behavior. Abdel-Aty et al. (1995) found that travel time reliability was one of the most important factors for route choice considerations. Rietveld et al. (2001) simulated the arrival delays in multi-modal public transport services, and found that connection failure usually resulted from serious arrival delays. In their survey, up to 85% of the surveyed passengers prefer routes that are a little longer but more reliable over a short but unreliable route. Some empirical studies reviewed by Li et al. (2010) also revealed that travelers are willing to pay for reduction in travel time variability in addition to travel time saving. Moreover, some researchers studied the route choice behavior considering travel time uncertainties by using analytic equilibrium approaches. Watling (2006) proposed a late arrival penalized user equilibrium model under fixed departure time and penalized arrival delay. Lo et al. (2006) modeled travelers’ route choice behavior under stochastic link capacity degradation by the notion of “travel time budget”. Shao et al. (2006) proposed a demand driven travel time reliability-based stochastic user equilibrium model, considering both travel time variability from demand fluctuation and travelers’ perception error on travel time budget. Siu and Lo, 2008, Lam et al., 2008 further extended the approach of “travel time budget equilibrium” to the doubly uncertain transportation network with both stochastic supply and demand.

In particular for rail transport reliability, recently, Goverde (2007) modeled the regularity of train delays and delay propagation based on the theory of max-plus algebra, and evaluated the stability and robustness of railway timetables. Vansteenwegen and Van Oudheusden, 2006, Vansteenwegen and Van Oudheusden, 2007 estimated the effect of section buffer times on the transfer cost and waiting cost when train delays, and constructed improved timetables with ideal buffer time using a linear programming approach. Liebchen et al. (2010) proposed delay resistant timetables, in which the objective was to optimize the transfer time between any two adjacent trains so as to guarantee successful transfer on the one hand and to minimize the total travel cost on the other hand. Since the previous studies are limited to operational stability of rail transportation systems, there is a lack of research on the effect of transfer reliability on passenger’s route choice behavior.

Lo et al. (2006) proposed a multi-class mixed-equilibrium mathematical program to capture the route choice behavior of travelers with heterogeneous risk aversions given that travelers valued travel time variability in degradable network as a form of travel time budget. The study provides a promising solution to incorporate transfer reliability into route choice behavior analysis. Nevertheless, the difference between the road and rail transport generates a further research need of an adapted method to study transfer reliability in rail transport.

Specifically, the road travel cost, which is depicted as route travel time (including congestion disutility), increases in a continuous pattern with the increase of traffic flow. But the railway travel cost is reflected by the in-vehicle discomfort and travel time. While the in-vehicle discomfort would increase in a continuous pattern with the increase of seat occupancy, the travel time, which includes in-vehicle time and transfer time, would be predetermined by railway timetables. In addition, the road traffic reliability is represented by the probability of on-time arrival in travelers’ expectation, and as such the travel time budget increases continuously with the increased expectation for travel reliability. But railway reliability is reflected by the probability of successful connection for the whole trip, and the passenger’s travel time would increase discretely with the increase of transfer reliability expectation.

This study proposes an equilibrium-based rail passenger flow model into which the transfer reliability is incorporated. In particular, the transfer reliability is revealed in term of the expected probability of successful connection in Section 2. In Section 3, back-boned with an actual rail transfer timetable, a multi-class user equilibrium model is developed to explicitly depict the passenger’s route choice behavior considering transfer reliability. A path-based traffic assignment algorithm which combines a k-shortest path algorithm and the method of successive averages is developed, as reported in Section 4. Furthermore, in Section 5, an illustrative example is presented to demonstrate the validity of the model and the feasibility of solution algorithm. Finally, Section 6 concludes the study and presents recommendations for future research.

Section snippets

The transfer reliability

Let (i, s, j) denote the scheme that passengers transfer from the train i to j at the station s. The train i arrives at the station s at the time taris and the train j departs from the station s at the time tdejs. The minimum time needed for transfer at the station s is denoted as tmins. Clearly, we have 0taris,tdejs,tminsT, where T is the timetable cycle (T = 1440 minutes or 86400 seconds). The connection time in the transfer scheme (i, s, j) can be expressed asttrisj=tdejstaris,tdejstaris>tminstde

Schedule-based transfer network design

An essential step in equilibrium analysis is transfer network design based on train arrival and departure times in accordance to railway timetables (Shi and Deng, 2004). Mathematically, a transfer network (V, A) is constructed by three kinds of nodes and five kinds of arcs, where V and A are the set of nodes and the set of arcs respectively. It shows mutual relationships among trains in train set in operation. Define the set of nodes V=SVarVde, where S, Var and Vde represent the set of

Solution algorithm

A path-based traffic assignment algorithm which combines a k-shortest path algorithm and the method of successive averages is developed for solution. In the transfer network (V, A), we utilize the k-shortest path algorithm to obtain the shortest paths kwu (i.e. the uth shortest path between O–D pair w), where uu¯ and u¯ is a predefined upper bound of the shortest path number, and hence the aggregate set of transfer paths Kw is obtained. For passenger class l, we solve Pk(τl) and Ekl(τl) for

Numerical example

The proposed approach is illustrated with a small railway network as shown in Fig. 4. Three trains run among stations s1, s2, s3, s4, and s5 their specific operation lines are denoted as t1 = (s1, s2, s3, s5), t2 = (s1, s2, s3), t3 = (s2, s3, s4, s5) respectively. The section fares are labeled on the sections (in Chinese yuan, a USD is equivalent to 6.5 RMB yuans). Train timetable is shown in Table 1. Note that two train grades are defined, i.e. medium (train 1 and train 3) and high (train 2), and the

Conclusion

Railway transport is an increasingly popular transportation mode for medium-long distance journey in many countries in the past decades. However, railway passengers are often suffered from the train delays. An undesirable consequence of train delays is connection failure, for passengers who have to make transfer(s) during the journey. Passengers desire the certainty for successful connection when they make travel plan. Existing traffic assignment approaches do not properly consider passenger’s

Acknowledgements

This study was supported by the National Nature Science Foundation of China (Grant No.71171200, No.71101155). The authors thank the two anonymous referees for their helpful comments, which greatly improve the quality of this paper.

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