Transit users' route-choice modelling in transit assignment: a review

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Transit users' route‐choice modelling in transit assignment : a 

review. Transport Reviews , 30(6). pp. 753‐769. 

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Transit Users’ Route-Choice Modelling in Transit

Assignment: a Review

Yulin Liu*, Jonathan Bunker, Luis Ferreira

School of Urban Development, Queensland University of Technology, 2 George Street, GPO BOX 2434, Brisbane, QLD 4001, Australia

* Corresponding author’s e-mail address: y68.liu@qut.edu.au

Abstract

This paper reviews the main studies on transit users’ route choice in the context of transit assignment. The studies are categorised into three groups: static transit assignment; within-day dynamic transit assignment; and emerging approaches. The motivations and behavioural assumptions of these approaches are re-examined. The first group includes shortest-path heuristics in all-or-nothing assignment, random utility maximisation route-choice models in stochastic assignment, and user equilibrium based assignment. The second group covers within-day dynamics in transit users’ route choice, transit network formulations, and dynamic transit assignment. The third group introduces the emerging studies on behavioural complexities, day-to-day and real-time dynamics in transit users’ route choice. Future research directions are also discussed.

Keywords: public transport, transit, demand modelling, transit assignment, route choice

1 Introduction

Transit assignment refers to a manner in which a given aggregate origin-destination (O-D) passenger traffic demand is assigned to the transit route(s) of that O-D pair. As an important part of transit demand analysis, transit assignment plays an indispensable role in transit modelling. There is a substantial reservoir of studies for transit assignment, covering facets of this complicated problem. Particularly, the studies on modelling transit users’ route choice form one of the cornerstones. This modelling work is not only significant but also challenging, for it can involve lots of complexities from both transit system attributes and human decision-making process. More sophisticated models have been proposed, and this will keep its momentum since developing and studying transit systems are gaining increasing attention. This paper thus presents a review of the studies on transit users’ route choice within the context of transit assignment. It is intended as a guidepost along the way, in an area that is still very much under active development.

This review classifies existing transit assignment approaches into three major categories: static transit assignment, within-day dynamic transit assignment and emerging approaches. This classification represents three typical perspectives of

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understanding transit users’ route-choice behaviour. The first category includes ‘shortest-path’ based all-or-nothing assignment, stochastic assignment based on random utility maximisation (RUM) choice models, and user equilibrium (UE) paradigm based assignment. The common trait of these approaches is that they consider no time dimensions in transit users’ route choice while having different assumptions about the final state of assigned transit network. The second category distinguishes itself from the first one in that the within-day time dimension in users’ route choice is accommodated through introducing time-dependent O-D demand and/or transit-schedule based departure-time choice modelling. If the approaches in the first two categories are considered as traditional and more commonly found in engineering practice, the approaches grouped in the third category are relatively miscellaneous and theoretical. They include the studies on the complexities in transit users’ behavioural mechanisms of decision making and information integration, in a day-to-day and/or real-time dynamic modelling framework. These emerging approaches may challenge traditional concepts and methodologies and are drawing increasing attention.

The remainder of this paper is organised as follows. The approaches in the aforementioned three categories are respectively introduced in the following three sections. Their motivations and behavioural assumptions are re-examined. Because this paper is neither an exhaustive list of the past studies nor a detailed reference text, only representative studies are reviewed here to exhibit a panorama of the research on transit users’ route-choice behaviour. Finally, future research directions are envisioned in a summary.

2 Static Transit Assignment

2.1 Shortest-path heuristics in all-or-nothing assignment

In all-or-nothing transit assignment, all the passengers of an O-D pair are assumed to choose that O-D pair’s ‘shortest path’, which, from the analyst’s perspective, is usually the fastest (or least cost, or shortest in distance); the inferior paths are assigned nothing. Notably, as pointed out Nguyen and Pallottino (1988), a major obstacle in transit assignment stems from the ‘common lines’ problem, that is, passengers at a transit stop shared by several competitive transit lines can choose between boarding the arriving vehicle and waiting for the vehicle of another express line to minimise their total travel time. This makes shortest-path based assignment in transit networks with common lines different from that in the auto traffic context. A shortest transit path could be a set of multiple routes/lines, each with a certain probability of being used by passengers. And those probabilities could depend on such factors as vehicle-arrival pattern, service frequency, passenger-arrival pattern, and schedule coordination. Thus many shortest-path heuristics have been proposed based on different assumptions about those network attributes.

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The early shortest-path heuristics usually assume deterministic vehicle-running times and a waiting time depending on the frequencies of routes serving the O-D pair. Dial (1967) and le Clercq (1972) probably perform the earliest studies on transit assignment. Both studies assume exponentially distributed headways, random passenger arrivals, and as such one-half of the headways as the waiting or transfer time. In the case of common lines, the expected waiting or transfer time is set to be equal to one-half of the inverse of the sum of the routes’ frequencies. The assignment to the equally fast routes of common lines is proportional to their relative line frequencies.

Chriqui and Robillard (1975) explicitly define the optimal choice set of transit lines between two consecutive points as a subset of the common lines. Passengers are assumed to choose their optimal choice set such that when they take the first bus to arrive on any routes of the set, their expected travel time is minimal. Under the same assumptions as Dial (1967) and le Clercq (1972), Chriqui and Robillard design a greedy heuristic to find optimal choice set, which always produces the optimal solution when all routes have independently and exponentially distributed headways. Although this heuristic is proved by Marguier (1981) not necessarily optimal for all headway distributions, the idea of optimal choice set inspires quite a few researchers. Spiess (1984) introduces a notion of strategy, which is a choice of an attractive set of lines at each boarding-decision point. An optimal strategy minimises the passenger's travel time. Strategy is later expressed in graph-theoretic language by Nguyen and Pallottino (1988) under the denomination of hyperpath. Spiess and Florian (1989) formulate a linear programming in the space of link flows to determine the probability of choosing different lines. Also, de Cea et al. (1988) present a network model that is partially different from Spiess’s strategy approach, and propose a linear programming to determine the optimal choice. de Cea and Fernandez (1989) further propose a non-linear programming solution.

With the conceptual appeal and modelling convenience, the shortest-path paradigm enjoys great popularity in engineering practice, especially for sparse transit networks. However, some assumptions of the above mentioned models are not applicable in the cases of schedule-based reliable transit systems. For example, the assumption of exponentially distributed headways is inconsistent either with the calculation of waiting/transfer time as being one-half of the prevailing headway, or with the frequency-share approach to assigning passengers to common lines (Marguier and Ceder, 1984; Jansson and Ridderstolpe, 1992). More importantly, the shortest-path heuristics implicate the perspective towards transit service from an analyst rather than transit users. Although these models employ passenger-oriented performance measures, they do not specifically focus on analysing passengers’ route-choice behaviour. A more passenger-oriented approach is needed.

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2.2 RUM choice models in stochastic assignment

Random utility maximisation choice modelling was introduced into travel demand analysis in the 1970s and has now become the backbone of travel behaviour studies (McFadden, 2000). As choice determinants, both attributes of choice alternatives and decision-maker social-demographic characteristics can be incorporated into utility functions. The RUM choice models assume that a decision-maker has a perfect discrimination capability to choose the alternative with the maximum utility perceived, but the analyst has incomplete information. The uncertainties are represented by random term. In the case of transit users’ route choice, Nielsen (2000) summarises four uncertain factors which the random term can encompass: first, passengers do not have full knowledge of transit network and they only choose according to their perceived utilities; second, travel times along different routes may vary from day-to-day; third, different routes are often chosen for the sake of variation; and fourth, different passengers may have different preferences. According to different assumptions about its statistical distribution, types of RUM choice models have been proposed. Hess (2005) comprehensively examines the advanced stochastic discrete choice models with applications to transport demand analysis.

The RUM choice models explain choice behaviour of individuals. However, an important objective of discrete choice analysis is to predict the group behaviour of individuals, just as traffic assignment to route-choice modelling. This prediction can be achieved through aggregating a choice model over its target population. Ben-Akiva and Lerman (1985) compare five aggregation approaches. Traffic assignment based on aggregation of route-choice models is known as stochastic assignment. The passenger flow of one O-D pair is split over the alternative routes rather than be all assigned to the shortest one.

Most RUM choice models are applied in the context of analysing travellers’ mode choice rather than transit users’ route choice. In addition, policy measures can be obtained from parameter estimates of utility functions in a RUM choice model, such as value of travel time savings, elasticities, and willingness-to-pay. Thus transit studies using RUM choice models often pay more attention to the impacts of level-of-service (LOS) factors on passengers’ choice. van der Ward (1988) examines relative importance of transit trip-time attributes in route choice. Hunt (1990) proposes a Logit model for transit route choice. Pursula and Weurlander (1999) investigate LOS factors in transit route choice by building a structured-tree-type Logit model with combined revealed preference (RP) and stated preference (SP) data from Helsinki. Lam and Xie (2002) study LOS factors impacting transit route choice by calibrating a path-size Logit model using mixed RP and SP data from Singapore. Douglas and Karpouzis (2005, 2006) estimate values of overcrowding in rail station and aboard train, respectively, by calibrating simple Logit models with SP data collected from urban rail users in Sydney. Lo et al. (2004) develop a three-level Nested Logit choice model to deal with travel choices in a multi-modal transit

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network: the first level focuses on combined-mode choice, the second on transfer location choice, and the third on route choice.

Grounded in well established microeconomic theories, RUM choice models are the prevailing analysis tools in travel demand modelling. But the utility maximisation assumption is also being challenged by a line of research originating in cognitive psychology. This research shows that individual is less organized, and more adaptive and imitative, than the RUM approach requires. The economists’ calculus of utility assessment and maximization is reduced to one of many factors in the decision-making environment, with an influence that may be overridden by context effects, emotion, and errors in perception and judgment (McFadden, 2003). The potentially important roles of information processing, perception formation and cognitive illusions deserve more exploration. Therefore, some studies in this direction are emerging in route-choice modelling, which are introduced in the fourth section of this paper.

2.3 User equilibrium based assignment

Considering the negative externalities of road traffic congestion upon travel time, Wardrop (1952) conceptualises the user equilibrium (UE) of users’ route choice in auto traffic network, an analogue of the ideal market equilibrium in neoclassical economics. Wardrop assumes that travellers know the precise route travel time and choose the fastest route; when the UE is reached, all the used route(s) of an O-D pair have the same and shortest travel time, no traveller can reduce his/her travel time solely by changing route, and the equilibrium will stay stable. Traffic assignment is to seek the equilibrium state. Daganzo and Sheffi (1977) extend Wardrop UE to stochastic user equilibrium (SUE) to overcome the unrealistic assumption of precise perception of route travel time across users. SUE is obtained where no traveller’s perceived cost can be reduced solely by changing route. That is underpinned by RUM route-choice modelling.

Similarly, overcrowding in transit systems also exerts negative externalities upon route cost, such as discomfort in vehicle, extended vehicle dwell time, and failure to board due to vehicle capacity, which can impact passengers’ route choice. The UE concept should also be applicable for transit assignment. However, UE assignment is not directly comparable between transit and auto networks. In auto networks, link travel time is usually the only variable affected by congestion. But in transit networks, such variables can include waiting time and in-vehicle discomfort. Plus the common lines problem, overcrowding impacts on transit users’ route choice have more complexities. A few early studies explore the flow split over common lines in overcrowded transit systems, by considering vehicle capacity, line capacity, boarding queue, and the like (Last and Leak, 1976; Gendreau, 1984; Nguyen and Pallottino, 1988; Spiess and Florian, 1989; de Cea and Fernandez, 1993). But much more efforts are taken in developing equilibrium-based transit assignment.

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Based on the works of Spiess and Florian (1989) and de Cea and Fernandez (1993), Wu et al. (1994) present a UE transit assignment. The overcrowding at a transit link is described by the nonlinearity of the link-cost function, an unbounded increasing convex-volume function of overcrowding, in an empirical formula. Bouzaiene-Ayari et al. (1995) reform the model of Wu et al. (1994) as a variational inequality fixed-point problem in the space of hyperpath flows. Cominetti and Correa (2001) treat the waiting-time function of flow as a simplified bulk queue model of Gendreau (1984). They give an UE transit assignment in the space of line flows.

Lam et al. (1999b) present a mathematical programming for SUE transit assignment based on a multinominal Logit route-choice model. They prove that, when the link-capacity constraints are reached, the Lagrangian multipliers of the mathematical programming are equivalent to the equilibrium passenger-overload delays in overcrowded transit network. Nielsen (2000) gives a SUE transit assignment based on a multinominal Probit route-choice model. Lam and Xie (2002) incorporate the dwell-time model from Lam et al. (1999a) into a SUE transit assignment model. Nielsen and Frederiksen (2006) propose a SUE transit assignment model using the Nested Logit route-choice model. Yang and Lam (2006) put forward a Probit-type reliability-based SUE transit assignment model, in which in-vehicle travel times are normally distributed. A disutility function is used for considering passengers’ risk-averse behaviour under unreliable transit conditions, which is associated with passenger travel time and its variation.

Seeking the equilibrium state for assignment solution, both researchers and practitioners are quite concerned about the existence and uniqueness of equilibrium state and its solution algorithms. Wu et al. (1994) give the sufficient conditions for equilibrium’s existence and uniqueness and a solution algorithm. Bouzaiene-Ayari et al. (1995) prove that the existence and uniqueness properties require unrealistic assumptions about waiting-time and travel-time functions of flow. Cominetti and Correa (2001) prove that these equilibrium properties can be obtained under weak conditions (that is, the travel-time functions of flow are not necessarily to be strongly monotonic) without giving a solution algorithm. Babazadeh and Aashtiani (2005) prove that strategy-based transit assignment in overcrowded networks is equivalent to a nonlinear complementarity problem (NCP) in terms of ordinary path flows. They propose a solution approach for the NCP model. The convergence of the algorithm is not proven, but the computational results for a real network are presented. Cepeda et al. (2006) extend the work of Cominetti and Correa (2001) to obtain a new characterization of the equilibriums to formulate an equivalent optimization problem in terms of a computable gap function that vanishes at equilibrium. This model formulation can deal with flow-dependent travel times and is a generalization of the strategy (hyperpath) based transit network equilibrium models. The approach leads to an algorithm which has been applied on large scale networks.

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Because of solid theoretical foundations, the equilibrium concept is currently the dominant paradigm in transport network analysis. The advancing mathematics knowledge and computing techniques are making the high-cost equilibrium approaches practical to large-scale networks. However, the equilibrium transit assignment is no perfection. Its static character is criticised as inapplicable to the networks with daily time-dependent demand. More significantly, the classical equilibrium paradigm is being challenged by the emerging research in experimental economics and behaviour science, where the process of reaching possible equilibriums is paid specific attention. These criticisms and challenges are reviewed in the next two sections, respectively.

3 Within-Day Dynamic Transit Assignment

Transit systems usually experience significant peak loading, with peak-period demand exceeding capacity and affecting passenger satisfaction (see, e.g., Lam et al., 1999a). This fluctuation of the time-dependent demand within one day leads to dynamics in passengers’ route choice and network assignment. Another factor contributing to the within-day dynamics is passengers’ departure-time choice, which is strongly correlated with passengers’ choice of run in the transit systems with schedules. Conventionally, passenger inter-arrival times at origin stops are assumed to follow random distributions. This is justified when service is at sufficiently short headways, or is so unreliable that passengers cannot effectively reduce the expected waiting time by means of clever arrival strategies. However, studies find that, for headways of over 10 to 12 minutes, there are significant reductions in expected waiting time over that expected if passengers arrive randomly (Jollife and Hutchinson, 1975; Turnquist, 1978). In addition, the within-day variations in transit supply could also impact the dynamics in demand.

But static assignment framework, assuming that the demand and supply are constant within the specified period of analysis, cannot reveal this bottleneck induced overcrowding problem. While the static models are applicable for the strategic and long-range planning of proposed transit systems, they can be problematic in the cases of advanced transit management, which requires short-term and more precise prediction. For example, using a constant average rate of passenger arrivals at stops can lead to severe errors in vehicle-load calculations, when this rate changes significantly during the reference period. Therefore, dynamic transit assignment is proposed to accommodate the within-day dynamics in passengers’ route choice. It is also notable that schedule-based network models are more commonly used in the dynamic transit assignment studies, while the static assignments usually employ frequency-based network formulations.

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3.2 Transit network formulation

Transit network formulation can be broadly divided into frequency-based and schedule-based types. In frequency-based approach, each transit line is assumed to operate on a constant frequency and the travel time along a link is determined by a volume/delay function. When passengers are assumed to board the first arriving vehicle in their choice set, the waiting time for boarding a line at a stop is a probabilistic function of the passenger-arrival pattern at the stop and the line frequency. In the case of common lines, there may be multiple optimal paths for one O-D pair.

In schedule-based approach, schedules are used to describe the clock-dependent movement of transit vehicles. The passenger waiting time for boarding a line is a deterministic function of line schedules and the passenger-arrival pattern at the stop. Given an O-D pair and departure time, the optimal path is deterministic and can be described as an itinerary which specifies precisely the lines used and the clock time of boarding at each line. The period of analysis could be one hour, several hours or the whole day and multi-interval trip matrices can be used to represent time varying passenger demand. The assignment results would then show the number of passengers assigned to each scheduled vehicle run during the period of analysis.

While frequency-based models are commonly used in static transit assignments, schedule-based models have been largely neglected. This is mainly because the latter type usually requires a larger database, a faster computer, and more detailed information on demand. But these problems can be overcome with today’s information and communication technologies. More importantly, frequency-based models can be inappropriate for today’s reality. The highly regular and information-rich properties of advanced transit systems may justify passengers’ clever arrival strategies; and advanced transit management requires network formulation that is suitable for modelling the within-day dynamics in passengers’ route choice. Hence, schedule-based models appear more in recent studies on dynamic transit assignment. Nuzzolo and Crisalli (2004) review the representative scheduled-based approaches in dynamic transit modelling. A natural and well established approach represents transit supply, which is intrinsically discrete in time, as a diachronic graph (Nuzzolo et al., 2001), where each run is modelled through a specific sub-graph whose nodes have space and time coordinates according to timetable. Alternatively, it is possible to define a dual graph (Anez et al., 1996; Nielsen and Jovicic, 1999), where each run section is a node, while arcs represent the connections at stops satisfying temporal consistency. A third approach, referred to as mixed line-database, is to describe the topology of the transit network through a graph analogous to that used in the static assignment, and to characterize its arcs with the information relative to timetable (Tong and Wong, 1999; Hickman and Bernstein, 1997; Nielsen, 2000).

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Nevertheless frequency-based models are not doomed. Teklu et al. (2007) present a composite frequency-based and schedule-based approach to transit assignment. The proposed model uses aggregate line frequencies to parameterise bus headway and a micro-simulator to enforce capacity constraints on individual vehicles. Schmöcker et al. (2008) compare the advantages of schedule-based versus frequency-based transit assignment tools and discuss when these models are most applicable. They also propose afrequency-based dynamic transit assignment model. This study aims to fill exactly the niche by introducing fairly large headways, thus network flows and platform overcrowding could be assumed to be relatively constant. Nökel and Wekeck (2009) also compare several models of route choice in frequency-based transit assignment in terms of the underlying assumptions on service regularity, passenger information, and choice set structure.

3.3 Dynamic transit assignment

The terminology in the literature of dynamic transit assignment is inconsistent across different researchers. Dynamic transit assignment can refer to the problems that may, or may not involve passengers’ departure-time choice explicitly as part of the formulation. The bulk of contributions to date have been concerned with finding an assignment in the network given a known time-dependent O-D trip pattern. Some recent efforts have also attempted to explicitly incorporate passengers’ trip timing. Tong and Wong (1999) present a dynamic stochastic transit assignment, using a schedule-based model with given time-dependent O-D matrix. The time-dependent optimal path is generated by a branch-and-bound algorithm based on the work of Tong and Richardson (1984). A Monte Carlo approach is employed to solve the stochastic assignment problem. A case study is conducted based on the data from Mass Transit Railway System in Hong Kong.

Poon et al. (2004) give a dynamic UE transit assignment, also using a schedule-based network with given time-dependent O-D demand. It is assumed that transit vehicles with capacity constraint operate precisely as scheduled; passengers queue according to the single channel FIFO rule. Each vehicle’s available capacity is updated dynamically as demand is loaded onto network through time-increment simulation. Passenger arrival-departure profiles at all stations are recorded after each simulation run to predict dynamic queuing delays. With such delays, optimal paths are updated by the algorithm of Tong and Wong (1999), and used in the next simulation run. The UE assignment is solved by the method of successive averages (MSA).

Hamdouch and Lawphongpanich (2008) present a schedule-based UE transit assignment model with vehicle capacity constraint. Passengers’ travel strategies are adaptive over time and represented as sub-graphs. When loading a vehicle, on-board passengers have priority, and waiting passengers are loaded by a FIFO rule or in a random manner. Passengers unable to board must wait for the next vehicle. UE conditions are formulated as a variational inequality involving a vector-valued

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function of expected strategy costs. They propose a method that takes successive averages as it iterates and generates strategies during each time of iteration by solving a dynamic program. Numerical examples demonstrate that the algorithm converges to an equilibrium solution.

Schmöcker et al. (2008) propose the first frequency-based dynamic transit assignment model for overcrowded high-frequency transit networks, where passengers might not be able to board vehicle and hence remain on platform. A fail-to-board probability is introduced to reflect passenger’s attitude of risk-averse. The expected delay along a hyperpath by a fail-to-board probability is incorporated into a generalised cost function. The hyperpath search algorithm follows the one suggested by Nguyen and Pallottino (1988). A time-dependent O-D demand is loaded by time intervals, and those passengers who failed to board are added to the demand in the subsequent time interval. But the route-choice model only partially considers the dynamics effects. It assumes that passengers expected current level of overcrowding remains constant in future time intervals; thus the chance of failing to board in future time intervals remains the same as in current interval. The UE assignment problem is solved through the MSA. The approach is illustrated with an application in London underground. When timetables are reasonably reliable and service frequencies are low, the departure-time and route choice can be equally important to passengers. Nguyen et al. (2001) propose a transit assignment that encompasses simultaneously departure-time and route choice. This approach is built upon the concept of path available capacity, which captures the flow priority induced by the FIFO rule that binds passengers at every access point to transit network. A boarding penalty cost function is introduced to impose capacity constraint. That function is dependent on the difference between the queue length before the boarding passenger and the capacity of the vehicle being boarded. A UE flow model is mathematically formulated, and a solution algorithm is developed based on the asymmetric boarding penalty cost functions.

By extending the equilibrium concept into passengers’ departure-time choice, Huang et al. (2004) give an equilibrium formulation of bus riding in morning peak for a network with single origin and single destination. With an overcrowding-cost function, passengers are assumed to make a trade-off between overcrowding cost (that increases as closer to the peak of rush hour) and early/late arrival penalty at destinations in determining their optimal departure times. Tian et al. (2007) further analyse the equilibrium properties of the morning peak-period commuting on a many-to-one mass transit system. Commuters are assumed to choose their optimal departure times from various origins to a single destination by trading off the travel time and overcrowding cost against the schedule-delay cost.

In general, these formulations of dynamic transit assignment are based on the mainstream behavioural assumptions of passengers’ route choice as in the static approaches. Further probing into the behavioural mechanisms underlying passengers’

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trip routing reasonably considers this choice process in a day-to-day framework. Some emerging studies investigate the alternative transit route-choice models that explicitly recognise users’ learning and adaptation mechanisms in a dynamically varying system.

4 The Emerging Approaches

The above reviewed transit users’ route-choice models all assume that passengers make rational decisions to maximise expected utility. However, assuming the strict rationality in individual decision making is criticised by some behavioural scientists. For example, Simon (1955) is among the first to criticise the assumption of rational and fully informed ‘economic man’. He argues that people have bounded rationality and tend to seek satisfying choices rather than the best. Kahneman and Tversky (1979) find that people do not follow the rules of objective expected utility maximisation in economic choices, but have a non-linear perception of the probability and the value of a certain outcome. This is described in their Prospect Theory as the ‘framing’ of decision under risk. Recently, transport researchers also pass similar criticisms on travel choice models (e.g., Gärling, 1998; Avineri and Prashker, 2003). Behavioural assumptions in travel choice modelling are almost always made without reference to the existing theories in the behavioural sciences (Gärling, 1998). Hence, there is an emerging stream of research on the behavioural complexities in travel choices, such as bounded rationality, risk attitudes, habit effects, learning and adaptation.

Avineri (2004) finds the bus waiting time paradox when studying the effect of information format on passengers’ preferences of bus lines. Questionnaires present two formats of alternative bus lines’ information: headways and waiting time. The results show the evidence to size-biased sampling when subjects are provided with information about bus headways. By formulating and interpreting the subjects' preferences with the framework of Cumulative Prospect Theory (Tversky and Kahneman, 1992), Avineri finds that the value of subjects' reference point is much lower than their experienced waiting times. Schmöcker et al. (2009) discover the equivalencebetween the risk-averse path set created through a multi-agent, zero-sum game and the hyperpath (Spiess and Florian, 1989) created through a linear program to describe the behaviour of transit users who aim to minimise their expected travel times. The congestion effects are not considered.

Wardrop’s UE paradigm in traffic assignment is also challenged, since no scientific evidence has been reported to confirm the existence of such a state in reality (Mahmassani, 1997). For the more plausible SUE, Hazelton (1998) argues that SUE is by nature a deterministic state, because it relates to a specific isolated state which is guaranteed by the weak Law of Large Number.The equilibrium paradigmiscriticised for just pursuing a presumed isolated and self-consistent equilibrium in market, while

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neglecting the history of system’s evolution. Thus, there are novel studies analysing the route-choice and assignment problems from a dynamical system’s perspective. Considered as a particular state in dynamical system’s evolution, equilibrium is not necessarily unique but dependent on the evolutionary path and individual user’s behavioural mechanisms of information integration and decision making over day-to-day repeated choices.

4.2 Real-time transit information

Another new ingredient in transit users’ route choice is the provision of real-time information. Emergent Intelligent Transportation Systems (ITS) and Advanced Public Transport Systems (APTS) are able to provide timely information to transit passengers on the conditions of network, such as lines, schedule, arrival time, departure time, occupancy and transfer. This information can be available before trip departure (pre-trip) or during the trip (en-route) and be delivered via a wide variety of media, such as audible or visual messages through at-stop or in-vehicle information devices, telephone, Internet, and mobile phone, to individual travellers or traveller groups. The information provided can be either descriptive to improve travellers’ knowledge of the actual state of network, or prescriptive to advise on travel choices which travellers can follow or not.

The benefits that real-time information offers in terms of reducing uncertainty in waiting, increasing public transport patronage, reducing waiting time, increasing willingness-to-pay, and so on are well documented by different studies reporting the results of introduction of APTIS in specific case-studies (see, e.g., Atkins, 1994; Nuzzolo and Coppola, 2002; Dziekan and Kottenhoff , 2007; Caulfield and O’ Mahnoy, 2009). These types of benefits could add considerable support for a transit agency’s decision to install real-time passenger information systems. Moreover, the real-time information of transit network could impact passengers’ route and/or departure-time choice, which would change the spatial and/or temporal patterns of flow assignment over the network.

Some early studies have already discussed the effect of real-time information on transit route choice. Chriqui and Robillard (1975) and Marguier (1981) introduce the concept of the clever passenger, who is assumed to be able to revise his/her optimal choice set of routes while waiting, based on headway distribution and the time already spent in waiting. Hall (1982) proposes an adaptive choice-decision rule to find the optimal path in the transit networks of both stochastic and time-dependent travel times, and examines the shortest-path heuristic under scenarios of information availability to investigate information’s impacts. Hall’s experiments with transit users show that the actual travel time is significantly longer than the optimal according to the heuristics; real-world passengers seem to employ simpler route-choice rules and not take full advantage of the available information. To evaluate the quantitative benefits of real-time information for transit passengers’ route choice, Hickman (1993) develops a dynamic path-choice algorithm of the passenger receiving real-time information as to

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whether to board a departing vehicle. By assuming that passengers make the best possible use of real-time information to shorten their travel times, Hickman’s algorithm’s results suggest the highest level of benefits. But through the computer simulation of a transit corridor, Hickman and Wilson (1995) suggest that real-time information yields only very modest improvements in travel time savings.

As noted by Hall (1982), a transportation engineer can be very successful at reducing the travel time for ideal travellers, yet fail at improving the actual travel time seen by real travellers. The behavioural mechanisms of passengers’ information integration and decision making are worthy of more exploration. The traditional transportation planning methods have serious limitations in evaluating the effects of information technologies, since they are neither sensitive to the types of information that may be provided to travellers nor to the traveller’s response to that information (Wahba and Shalaby, 2009).

4.3 Day-to-day and real-time dynamics

The importance of passengers’ mechanism of information integration is embodied by the recent studies on day-to-day and real-time dynamics in transit users’ route choice. In terms of the day-to-day dynamics, Nuzzolo et al. (2001) propose a schedule-based transit assignment model, which takes into account explicitly the within-day and day-to-day variations of services and user choices. The utility-based choice model proposed integrates both real-time information (including waiting time, time already spent at stop, and on-board comfort) and experience of network attributes (including on-board time and comfort, and transfer time) through a day-to-day learning process that is specified using an exponential filter. Teklu et al. (2007) also employs in a transit assignment model a weighted average formula to simulate passengers updating route cost expectations as a day-to-day learning process.

Wahba and Shalaby (2009) introduce a Micro-simulation Learning-based Approach for Transit Assignment (MILATRAS). Accommodating the time-dependent and stochastic transit service characteristics, MILATRAS also models adaptive trip choices by passengers and captures the complex interactions between passengers’ decisions and transit network performance. As a multi-agent simulation environment, this approach models each passenger as a microscopic entity and models that entity’s reaction to system directly, while simultaneously modelling the dynamic system performance as a response to passengers’ behaviour, without presetting an equilibrium state. MILATRAS is based on representing passengers and both their learning and planning activities explicitly. In the short run, learning activities include updating passenger’s perceptions of transit network conditions and perceptions of the accuracy of real-time information provided. In the long run, learning activities include updating passenger’s mechanism for making predictions about network conditions and the strategy for making trip choices. Planning activities include making trip choices of departure-time, stop and run. The learning process is concerned with the specification of the cost of different transit trip components; the planning process

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considers how experience and information about those components on previous days influence the choice on the current day, like a Markovian Decision Process. The learning and decision-making processes of passengers are assumed to follow Reinforcement Learning principles for experience updating and choice techniques. More studies are found in literature on evaluating the effects of real-time information on passengers’ route choice and transit assignment. Gentile et al (2005) propose a framework for determining the probability of boarding each line available at a stop when online information on bus waiting times is provided to passengers. This study shows that the classical assumption of boarding the first-arriving vehicle in attractive set without online information may be interpreted as a particular instance of the proposed framework. Shimamoto et al. (2005) compare the effects of arrival time information with that of hardware improvements in terms of congestion mitigation. The researchers employ a static capacity-constraint transit assignment model, where common lines are dealt with hyperpath approach (Nguyen and Pallotino, 1988) and passengers are assumed to know the accurate train arrival time at platform. This comparison shows that arrival time information improve the probability of arriving at destination without failing to board at any stations for the whole network; it also encourages passengers to choose shorter in-vehicle time lines, and increases expected waiting time and platform congestion. In terms of hyperpath cost (including expected travel time and risk of failing to board), arrival time information leads to the same effects as hardware improvements, such as increasing the frequency of the congested line. However, the effect is different between OD pairs, that is, arrival time provision can expect more effect than hardware improvements at the origin stations of the lines whereas less effect at the interim stations.

Ren et al. (2009) propose a model for assessing the effects of the integrated implementation of en-route transit information systems and time-varying transit pricing systems. The proposed model reveals the interaction between the two types of systems, and the potential benefits of the joint implementation. Passengers are classified into two groups: equipped and unequipped with en-route information. It is assumed that unequipped passengers make their travel choices according to dynamic stochastic user optimal principles, with equipped passengers having a lower perception variation of travel cost due to the availability of better information. A bi-level program is formulated to model the integrated effects on passengers’ departure-time choice, route choice, transit network performance, and transit operators’ revenue. The lower level is a multi-class dynamic stochastic transit assignment model. The combined system total cost and operators’ benefits under varied transit conditions are investigated with a numerical example.

Coppla and Rosati (2009) provide a system simulation approach to evaluate the impacts of the en-route descriptive shared information to transit passengers on network performance. The transit simulation system consists of a schedule-based network model, an information provider and the passengers. The waiting time and the

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on-board comfort are the only information provided to passengers. A Kalman filter based model and a schedule-based dynamic transit assignment model are used for the prediction of link travel times and bus occupancy, respectively, within the simulation period. By means of passenger interviews in Naples, a binomial Logit run choice model is specified and calibrated, that is, the passengers are assumed to consider only two alternatives: the run arriving at the stop and the next arriving run. An exponential smoothing filter is employed to simulate how passengers, based on their experience, react to the forecasted information received at stops. Thus the attributes in the systematic utility function for which information is provided are a result of a process of information acquisition and knowledge updating. A simulation case study shows that the impact of the waiting-time information is an increase of passengers’ average waiting time and a decrease of the average total travel time, similar to the findings from Shimamoto et al. (2005); the impact bus-occupancy information is null and negligible in the cases of regular and irregular service in terms of average waiting and on-board time; the impact of bus-occupancy information is significant in terms of travellers’ utility, the higher utility increase corresponding to the higher level of congestion on the network.

5 Summary

The above tour of main developments of modelling transit users’ route choice shows researchers’ deepening understanding of this significant and challenging problem. All three research streams, classified in this paper, with different levels of complexities continue to be active and fertile areas of investigation. This highlights the wealth of opportunities that remain for both methodological and practical contributions to this important aspect of travel behaviour. Referring to an agenda proposed by Mahmassani (1997), areas of opportunity for future work may include:

• developing theoretical constructs for representing passenger behaviour, especially with regard to (i) integrating departure-time choice with route choice, and (ii) capturing day-to-day learning and travel-cost prediction processes of passengers in response to actual experience and exogenous information;

• integrating passenger decisions in the context of transit system simulation, thereby incorporating passengers’ travel choices firmly in network-flow assignment, and manifesting the interaction between individual behaviour and network performance through passenger’s behavioural mechanisms of information integration and decision making;

• developing novel measurement techniques that yield the desired level of temporal and spatial richness, for large number of users. In addition to advances in longitudinal survey techniques, and growing acceptance of laboratory-like experiments, it is particularly fascinating to explore potentials

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of the emergent automated data collection systems, which include automatic vehicle location, passenger counting, and fare collection (see, e.g., Zhao, 2004; Wilson et al., 2009; Slavin et al., 2009);

• understanding the differences and link between traditional equilibrium and dynamical system approaches, and developing econometric and psychometric modelling frameworks that recognise the complex nature of the dynamic processes of interest and the resulting challenges for interpreting data generated from observation of these processes;

• enriching demand-forecasting framework, such that the evaluation of transportation options can shift from being exclusively an exercise in comparative statics to one where the evolution of system adjustment in response to control actions and policies may be of equal, if not greater, concern than the eventual final state. Transit planners and demand specialists might have to recognise, accept, manage, and eventually take advantage of the possible presence of chaotic behaviour in transit systems.

Acknowledgements

This paper’s authors are grateful to the anonymous referees for their constructive comments.

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