Fixed point approaches to the estimation of O/D matrices using traffic counts on congested networks

Over recent years, increasing attention has been devoted to the problem of estimating Origin/Destination (O/D) matrices by using traffic counts, named in the following O/D Count Based Estimation (ODCBE) problem. These methods combine traffic flow measures with other available information to ‘correct’ and improve an initial estimate of the O/D trip matrix. Virtually all models and procedures proposed in the literature are formulated as mathematical programming problems. Most ODCBE models use a relationship relating traffic counts to the unknown O/D matrix; this relationship is often in the form of an explicit estimate of the assignment matrix, i.e., of the fractions of O/D flows using each link of the network for which traffic counts are available. The computation of the assignment matrix is not trivial for congested networks, where travel costs and path choice fractions depend on flows. This problem has been dealt with in relatively few papers in the literature, usually formulated as a bilevel optimisation model under the assumption of a Deterministic User Equilibrium (DUE) assignment model. In this paper, the general within-day static ODCBE problem for congested networks is formulated as a fixed-point problem of an implicit function which results from the solution of a mathematical programming problem. In other words, the solution of the ODCBE problem is an O/D matrix that, once assigned to the network, reproduces flows and costs consistent with the values used to compute the assignment matrix. Fixed-point theorems and algorithms have been extended to the problem at hand. Different fixed-point algorithms, namely, Functional Iteration, Method of Successive Averages, and Method of Successive Averages with Decreasing Reinitialisation are proposed, and their performances are compared on a small test network. It has been verified that all algorithms converge to the same solution, though with different speeds. Furthermore, the fixed-point solution outperforms the initial solution, both in terms of bias and capability to reproduce both counted and noncounted flows.