Solving network equilibrium problems on multimodal urban transportation networks with multiple user classes

A modelling approach for solving quite general network equilibrium problems ( with fixed trip productions and attractions) intrinsic to the urban transport planning process is presented. The framework can consider a variety of demand models and route choice behaviours within the same implementation, including multiple user classes and combined travel modes that interact on the same physical network. The demand choices are assumed to have a hierarchical structure. When trip distribution is variable, a doubly constrained entropy-maximizing model is considered at the first level of choice and a hierarchical logit model is used for the remaining demand choices ( time of departure, travel mode, transfer point for combined modes, etc.). If trip distribution is considered to be exogenous, the demand choices are modelled as a hierarchical logit. One of the main features of the model is that it considers the effects of congestion on the road network as well as congestion and capacity constraints effects in each public transport service network. The problem is formulated mathematically as a variational inequality, with asymmetric cost functions, and solved following the diagonalization procedure. Each iteration of the aforementioned procedure solves an optimization problem using the Evans algorithm. Sufficient conditions for the existence and uniqueness of the solution to the diagonalized problem are obtained. The main results of a simple example ( solved with an academic version of the proposed algorithm) are presented to show the consistency of the equilibrium flows and levels of services obtained using the model. Finally, a real scale implementation of the model is briefly described to show the feasibility of its application.