Static traffic assignment with side constraints in a dense orthotropic network

AbstractThis paper proposes an investigation of bidimensional modeling for traffic flows in a large and dense network of an urban area. The network is viewed as a continuum of orthotropic roads. This (and the side-constraints) constitutes the main research contributions of the paper. Each point of the area is characterized by a travel cost and a side constraint that depend on the privileged directions. A side constraint is a means to obtain a more precise description of traffic flow as far as it does not allow it to exceed the capacity of the roads (Larsson & Patriksson, 1995; Taguchi & Iri, 1982). Each destination is characterized both by a monotonic strictly decreasing function to reflect the customers’ elastic demand with respect to the total cost (Yang & Wong, 2000) and a satisfaction function to reflect the customers’ gain as entering the desired destination. The study deals with a multi-commodity traffic assignment. We suppose that there are a few destinations for the commuters in the area. All the commuters are uniformly distributed in the area and try to reach their destination. A commodity is defined as the traffic flows from the area to one of the destinations (Yang & Wong, 2000). We express equilibrium as a primal problem which is the minimization of a mathematical convex program (Yang, Yagar & Iida, 1994) adapted to the orthotropic case. This program is inspired from the Beckmann objective function (Beckmann, 1952). To solve this program, we use a Lagrangian scheme and a dual method. This technique provides a potential function that explains the flows of traffic over the city, and the over costs generated by the saturated flows. We prove that the solution fits with the User Equilibrium principle (Wardrop, 1952). We apply the model to the case of Paris urban area, where we put an orthotropic network. The commuters are uniformly distributed on the city, and try to reach some destinations located on the ring on-ramps