In this paper, we propose a novel approach to model stochastic route choice in a tolled road network. First of all, we assume that all users have two objectives: (1) minimise travel time; and (2) minimise toll cost. Users are all rational in the sense that given a choice set, they will only choose one of the efficient paths. This will result in a bi-objective user equilibrium (BUE) condition such that traffic arranges itself in such a way that no individual trip maker can improve either his/her toll or travel time or both without worsening the other objective by unilaterally switching routes.

We assume further that users have different preferences in the sense that for any given path with a specific toll, there is a limit on the time that an individual would be willing to spend. Each individual can have his/her own preference represented by this indifference function between toll and time. Time surplus is defined as the maximum time minus the actual time. Given a set of paths, the one with the highest (or least negative) time surplus will be the preferred path for the individual. As a result, for a specific origin-destination (O-D) pair, each individual can have a different preferred path, even though all individuals are considering the same choice set.

In this way, based on the distribution of individual indifference curves, we can deduce the bi-objective equilibrium solution satisfying the time surplus maximisation bi-objective user equilibrium (TSMaxBUE) condition. That is, for each O-D pair, all individuals are travelling on the path with the highest time surplus value among all the efficient paths between this O-D pair.

The philosophy behind the TSMaxBUE model is to overcome the restrictions that came with the two most commonly applied methods in tolling analysis, namely, user equilibrium (UE) and stochastic user equilibrium (SUE). UE assumes that all individuals behave the same way, i.e. to minimise generalised cost. The difference in individual preferences is modelled by creating user classes with different values of time (VOT). As a result, UE is restricted by the assumption that all users with the same VOT behave in exactly the same way, which can be dictated by a generalised cost function. SUE assumes that drivers choose their preferred routes based on perceived costs on different routes. Some formulations of this model come with the well known independence of irrelevant alternatives (IIA) property and assumptions on the probability density function of the error term of the utility function. Both the IIA property and assumptions on the error term have imposed limitations on the capability to replicate travel behaviour in reality. The model proposed here can overcome all these difficulties by introducing bi-objective traffic assignment and an indifference function between toll and time which can vary between individuals with no restrictions.