A Link-Node Discrete-Time Dynamic Second Best Toll Pricing Model with a Relaxation Solution Algorithm

Dynamic congestion pricing has become an important research topic because of its practical implications. In this paper, we formulate dynamic second-best toll pricing (DSBTP) on general networks as a bilevel problem: the upper level is to minimize the total weighted system travel time and the lower level is to capture motorists' route choice behavior. Different from most of existing DSBTP models, our formulation is in discrete-time, which has very distinct properties comparing with its continuous-time counterpart. Solution existence condition of the proposed model is established independent of the actual formulation of the underlying dynamic user equilibrium (DUE). To solve the bilevel DSBTP model, we adopt a relaxation scheme. For this purpose, we convert the bilevel formulation into a single level nonlinear programming problem by applying a link-node based nonlinear complementarity formulation for DUE. The single level problem is solved iteratively by first relaxing the strick complementarity by a relaxation parameter, which is then progressively reduced. Numerical results are also provided in this paper to illustrate the proposed model and algorithm. In particular, we show that by varying travel time weights on different links, DSBTP can help traffic management agencies better achieve certain system objectives. Examples are given on how changes of the weights impact the optimal tolls and associated objective function values.