Consistent objectives and solution of dynamic user equilibrium models

The aim of this paper is to establish that there exist suitable objective functions for mathematical programming formulations of dynamic user equilibrium assignment in the sense of Wardrop (1952). Recently, Lin and Lo (2000) showed that we cannot necessarily calculate a dynamic user equilibrium assignment by minimising the integral over time of Beckmann's objective function that is widely used to solve for static user equilibrium assignment. In the present paper, two distinct objective functions are presented that result from formulations of assignment either as a variational inequality or as an infinitesimal optimisation: each of these formulations can be solved as a forward dynamic programme. With these formulations and solution approaches, we solve an example based upon Lin and Lo's one correctly.