Projection and Fukushima’s Gap Based Methods for the Asymmetric Traffic Assignment Problem

Traffic Assignment is the problem of assigning an Origin to Destination, OD matrix onto a network, under given conditions of using the links of the network, to determine the resulting traffic flows in the network. The underlying hypothesis is that travelers travel from origins to destinations in the network along the available routes connecting them. The characteristics of a traffic assignment procedure are determined by the hypothesis on how travelers use the routes. The main modelling hypothesis is based on the concept of user equilibrium which assumes that travelers try to minimize their individual travel times, that is, travelers chose the routes that they perceive as the shortest under the prevailing traffic conditions. The translation of these modelling hypotheses in terms of a mathematical model leads in the general case to a formulation in terms of a system of variational inequalities that has an equivalent convex optimization model when volume-delay functions are separable. However, the separability assumptions on the volume delay functions may lead quite frequently to modelling inaccuracies due to the over simplifications that they represent when dealing with generalized cost in complex multiclass-multimode planning models, or accounting for priorities at intersections, then the problems become asymmetric in terms of the Jacobian of the cost functions and the associated system of variational inequalities must be solved. Projection and Gap Function methods are among the most computationally efficient algorithms to solve the models. This paper explores a combination of a variant of Fukushima’s projection algorithm and gap Functions. The new algorithm is computationally tested for several large networks and the computational results are presented and discussed. 1 Codina, Ibáñez, Barceló STATIC TRAFFIC ASSIGNMENT: EQUILIBRIUM ASSIGNMENT Traffic Assignment is the problem of assigning an Origin to Destination, OD matrix onto a network, under given conditions of using the links of the network, to determine the resulting traffic flows in the network. The underlying hypothesis is that travelers travel from origins to destinations in the network along the available routes connecting them. The characteristics of a traffic assignment procedure are determined by the hypothesis on how travelers use the routes. The main modelling hypothesis is based on the concept of user equilibrium which assumes that travelers try to minimize their individual travel times, that is, travelers chose the routes that they perceive as the shortest under the prevailing traffic conditions. User equilibrium modeling hypothesis: the routes chosen by the travellers are those individually perceived as being the shortest under the prevailing traffic conditions. This hypothesis assumes that travellers try to minimize their individual travel times. It was formulated by Wardrop (1) in terms of what is now known as Wardrop’s First Principle, or Wardrop’s User Equilibrium: The journey times on all the routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route. Consider a network defined in terms of a graph G=(N,A) with a set of nodes N representing either intersections or centroids, dummy nodes associated with the transportation zones, and a set A of arcs modeling the infrastructure and the connectors linking centroids to the networks. Consider also an OD matrix modeling the demand between transportation zones. The notation used through this paper is the following: • ( ) { } q p w q p w W n destinatio and origin for pair OD the is , = = = Set of all OD pairs • w W w Γ ⊗ = Γ ∈ ; { } W w r r w ∈ = Γ pair OD the joining path • ( ) { } 1 , , = ∈ Γ ∈ ∃ ∈ ∃ ∈ = ar w r a r W w A a A δ   • ( ) { } a a x A a A link over link priority ˆ ∃ ∈ =   We say x(a) is a priority link over link a when x(a) and a are two incident links to the same intersection node, and flow arriving to the intersection from link x(a) has priority for passing through it. • : a v link flows, A a∈ • : ) (v sa cost on link A a∈ • : w r h path flows through the path Γ ∈ r , joining the OD pair W w∈ • r s : cost on path Γ ∈ r • ( ) : h C path costs • w W w H H ∈ ⊗ = ; ( )       ≥ = R ∈ =  Γ ∈ 0 , w r w r w r n w w h g h h H