188 European Journal of Operational Research 71 (1993) 188-204 North-Holland A variational inequality formulation of the Dynamic Traffic Assignment Problem * Omar Drissi-Ka'itouni Centre de Recherche sur les Transports, Universit~ de Montreal, 3535 Queen Mary Road, Montreal H3C 3J7, Canada Received November 1990; revised June 1992 Abstract: This paper deals with the modelling of the Dynamic Traffic Assignment Problem (DTAP) for analyzing the day time varying flows of urban transportation networks. During the past two decades, many models have been proposed in the literature, but some of them are based on heuristic concepts, and most of them incorporate important limitations. In this paper, we propose a Dynamic Traffic Assignment Model which is mainly based on the following assumption: the time spent by a vehicle on a link may be decomposed into a fixed travel time and a waiting time. The fixed travel time is the free flow travel time, after which vehicles are put in the link exit queue, until there is room for them to proceed their trip. We show that this model leads to a network structure (a temporal expansion of the base network, including the queues), and then formulate the DTAP as a network equilibrium problem over the expanded network. The mathematical formulation is achieved through a variational inequality where the variables are the path (or link) flows over the space-time expanded network. Numerical results show that the model may handle large networks, so it may be used in practice to analyze the traffic congestion moves in real cities, as well in space (physical links) as in day time. Keywords: Dynamic; Traffic; Assignment; Network; Equilibrium; Variational inequality I. Introduction This paper deals with the modelling of the traffic flows on urban transportation networks, mainly during the peak periods of a day. Most of the research done in the past has been concentrated in the Static Traffic Assignment Problem (STAP) and its variants, where a static demand is assigned on the network based on the descriptive assignment rules of Wardrop (1952) (see, among many others, Dafermos and Sparrow, 1969). Altough the demand is not constant over the day time in the urban context, the static models have been extensively used in practice. During the two past decades, research has been more and more concentrated in developing new tools regarding the Dynamic Traffic Assignment Problem (DTAP) that may help to analyze the traffic flow variations over the time, in a city transportation network. The early DTAP models were often based on heuristic concepts (Yagar, 1970; Leonard et al., 1978) where the traffic flows are assigned on instanta- neous minimum cost paths. Other models were too restrictive: some do not apply to capacity constrained links (Robillard, 1974) and others consider only simple networks with only one destination (Hendrickson and Kocur, 1981) or having only parallel paths between the (only) origin-destination pair (Arnott et al., * This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Correspondence to: O. Drissi-Ka'itouni,D6partement de genie industriel, Ecole Mohammadia d'Ing6nieurs, Avenue Ibn Sina, BP 765, Rabat, Maroc. email drissi%emi@corton.inria.fr 0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved