A hierarchical optimization problem: Estimating traffic flow using Gamma random variables in a Bayesian context

In this paper a hierarchical optimization problem generated by a Bayesian method to estimate origin-destination matrices, based on Gamma models, is given. The problem can be considered as a system of equations in which three of them are optimization problems: (1) a Wardrop minimum variance (WMV) assignment model, which is used to derive the route choice probabilities, (2) a least squares problem, used to obtain the OD sample data, and (3) a maximum likelihood problem to estimate the posterior modes. A multi-level iterative approach is proposed to solve the multi-objective problem that converges in a few iterations. Finally, two examples of applications are used to illustrate the proposed methods and procedures, a simple and the medium size Ciudad Real networks. A comparison with existing techniques, which provide similar flows, seems to validate the proposed methods.