The continuous dynamic network loading problem: A mathematical formulation and solution method

The continuous dynamic network loading problem aims to find, on a congested network, temporal arc volumes, arc travel times, and path travel times given time-dependent path flow rates for a given time period. This problem may be considered as a subproblem of a temporal (dynamic) traffic assignment problem. In this paper, we study this problem and formulate it as a system of functional equations. Based on this system, we show that the FIFO (First In, First Out) condition is respected under a reasonable assumption. For computational purposes, we develop a polynomial approximation, which is almost equivalent to the original formulation on a set of finite discrete points. The approximation formulation is a finite dimensional system of equations which is solved as an optimization problem. The optimization problem may or may not be smooth depending on the structure of arc travel times. Several numerical examples are given to illustrate this approach.