A demand model with departure time choice for within-day dynamic traffic assignment

Abstract

A within-day dynamic demand model is formulated, embodying, in addition to the classic generation, distribution and modal split stages, an actual demand model taking into account departure time choice. The work focuses on this last stage, represented through an extension of the discrete choice framework to a continuous choice set. The dynamic multimodal supply and equilibrium model based on implicit path enumeration, which have been developed in previous work are outlined here, to define within-day dynamic elastic demand stochastic multimodal equilibrium as a fixed point problem on users flows and transit line frequencies. A MSA algorithm capable, in the case of Logit route choice models, of supplying equilibrium flows and frequencies on real dimension networks, is presented, as well as the specific procedures implementing the departure time choice and actual demand models. Finally, the results obtained on a test network are presented and conclusions are drawn.

Introduction

The dynamic analysis of transportation networks gathered increasing attention through the past years. The static models, traditionally used in this field, are in fact unable to represent relevant phenomena, such as demand variations over time and temporary over saturation of network elements. On the other hand, dynamic models are more complex than static ones: on the supply side, they require to ensure temporal consistency, besides spatial consistency, among system variables (for instance, see Cascetta, 2001); on the demand side, choice of the time when to travel has to be modelled explicitly in order to achieve a correct representation of users’ travel behaviour (Mahmassani and Chang, 1986, Van Vuren et al., 1998, Mahmassani and Liu, 1999).

Different approaches to the latter problem can be found in the literature. One of them is to regard departure time choice as a discrete choice among temporal intervals and use static assignment to characterize the utility of each interval, as in Daly et al. (1990); models based on this approach yield a rough representation of travel demand during day time as a sequence of static equilibriums.

An alternative approach, relying on a more realistic representation of dynamic travel times, can be found in De Palma et al. (1983) and Ben-Akiva et al. (1986) where a within-day dynamic stochastic equilibrium model and a doubly dynamic stochastic model, respectively, are presented. Travel, departure time and path choices are addressed through a mixed discrete/continuous nested Logit model and travel times are determined by means of a deterministic queuing model; however, both models are applied only to single origin destination pair idealized networks.

Another approach is based on the hypothesis that departure time and path choices are made jointly, so that, for each origin–destination pair, a discrete choice set is defined whose finite number of alternatives is equal to the number of departure intervals multiplied by the number of paths (Arnott et al., 1990, Cascetta et al., 1992); models based on this approach require explicit path enumeration and the introduction of a diachronic graph, as in Van der Zijpp and Lindveld (2000), so they can be hardly applied to congested urban networks.

An extensive analysis of departure time choice for shopping trips is presented in Bhat (1998) and Bhat and Steed (2002). In the first paper, a discrete choice model is proposed, able to represent correlation among adjacent departure time periods. In the second paper, a continuous-time model of departure time is proposed, accommodating time-varying coefficients. Both papers however does not investigate path choice, since travel times and cost are assumed exogenously.

In a recent paper by Bellei et al. (2005) the within-day road dynamic traffic assignment (DTA) was regarded as a stochastic dynamic user equilibrium. A new fixed point formulation of the problem in terms of time-continuous real valued temporal profiles of arc flows and arc performances (travel times and generalized costs) was presented, where the concept of network loading map, yielding arc flows for given demand flows consistently with certain arc performances, was extended to the dynamic case, thus avoiding the introduction of both the dynamic network loading as a sub-problem, and the explicit path enumeration. OD flows temporal profiles were taken as given. An implicit path enumeration algorithm was thus proposed for the Logit case.

Based on the above modelling framework, two more papers, namely Gentile et al. (2002) and Gentile et al. (2003), focused on a new dynamic transit supply model relying on a frequency based approach; the introduction of a diachronic graph is thus avoided and implicit path enumeration is allowed. This model is able to represent both intra and inter modal congestion effects (i.e. interaction among cars and bus flows). In order to define a multimodal within-day DTA, a dynamic mode choice model was introduced in these papers, while demand flows were assumed to be rigid with respect to any other choice dimension.

In this paper a mixed discrete/continuous nested Logit dynamic demand model with five choice levels is presented, where, besides the usual to travel or not to travel (generation), destination, mode and path choices, the departure time choice is introduced. The model is conceived to extend previous work to the most general case of elastic demand multimodal within-day DTA. With reference to departure time choice, the proposed demand model adopts a continuous approach, thus not requiring to enumerate explicitly the desired departure time intervals. The resulting within-day DTA model is then capable of representing both supply and demand dynamic phenomena concerning congested multimodal urban networks, and leads to a fixed point formulation that can be solved by an efficient implicit path MSA algorithm applicable to real networks.