Combined distribution and assignment model for a continuum traffic equilibrium problem with multiple user classes

Consider a city with several facilities competing for multi-class users that are distributed continuously over space. Within the city region, the road network is relatively dense and is considered as a continuum. A user equilibrium condition results when users freely make their choice of facilities and their optimal routes within this continuum based on the criterion of minimizing the total facility and travel cost. A logit-type demand distribution function is specified to model the probabilistic choice behavior of destinations for the multi-class users. Furthermore, facility externality is introduced to capture the effect of congestion and economies of scale, as the usage of the facility varies. A mathematical program is formulated for the above complex problem, and a promising solution algorithm is developed. Finally, a numerical example is given to show the feasibility of the mathematical program and the effectiveness of the solution algorithm.