Housing allocation problem in a continuum transportation system

We consider a city with a central business district (CBD) with a road network outside of the CBD that is relatively dense and is considered to be a continuum. In this transportation system, several classes of users with different perceptions and behavior are considered. Their demands are continuously distributed over the city, and their travel patterns to the CBD satisfy the user equilibrium conditions under which each individual user chooses the least costly route in the continuum to the CBD. A logit-type demand distribution function that incorporates housing rent and travel cost is specified to model the housing location choice behavior of the commuters. A bi-level model is set up for modeling the housing allocation problem in the continuum transportation system. At the lower level, a set of differential equations is constructed to describe this housing location and traffic equilibrium problem. We present a promising solution algorithm that applies the finite element method (FEM) to solve this set of differential equations. At the upper level, a constrained minimization problem is set up to find the optimal housing provision pattern that maximizes the total utility of the system. The FEM and convex combination method are proposed to solve the minimization problem with the sensitivity information from the lower level. A numerical example is given to show the workability of the proposed bi-level model and the effectiveness of the solution algorithm.