Congestion and environmental toll schemes for the morning commute with heterogeneous users and parallel routes

Abstract

We design a congestion and environmental toll (CET) scheme for the morning commute with heterogeneous users in a single OD network with parallel routes. The designed toll scheme relies upon the concept of marginal-cost pricing and is anonymous. The Henderson approach is used to model road congestion and the tolling problem to examine commuter's arrival time and route choice at the CET equilibrium (CETE). Linear interpolation is applied to approximate the emission cost function and the resulting CETE problem is formulated as an unconstrained optimization problem, which is solved by the modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. Unlike the existing approach, this novel approach does not require that the arrival of each group of commuters at the destination at the equilibrium follows a predetermined order, and can handle non-monotone (emission) cost function. As two special cases, no-toll equilibrium (NTE) and the congestion toll equilibrium (CTE) are also examined, and the two resultant equilibrium problems are formulated and solved by the same approach. This approach is shown to be more efficient than the existing approach. Bi-level programming models are proposed to formulate the optimal congestion toll and CET design problems, in which the CTE and CETE problems are the corresponding lower level problem. These models are solved by the double BFGS method, which uses a classical BFGS method to solve the upper level model and the proposed BFGS method to solve the lower level model. Finally, numerical examples are provided to illustrate the properties of the models and the efficiency of the proposed solution algorithms.