A modified late arrival penalized user equilibrium model and robustness in data perturbation

In a seminal paper (Watling, 2006), Watling proposes a stochastic variational inequality approach to model traffic flow equilibrium over a network where the transportation time is random and a path is selected by to transport if the user’s expected utility of the transportation of the path is maximized over their paths. A key feature of Watling’s model is that the user’s utility function incorporates a penalty term for lateness and the resulting equilibrium is known as Late Arrival Penalized User Equilibrium (LAPUE). In this paper, we revisit the LAPUE model with a different focus: we begin by adopting a new penalty function which gives a smooth transition of the boundary between lateness and no lateness and demonstrate the LAPUE model based on the new penalty function has a unique equilibrium and is stable with respect to (w.r.t.) small perturbation of probability distribution under moderate conditions. We then move on to discuss statistical robustness of the modified LAPUE (MLAPUE) model by considering the case that the data to be used for fitting the density function may be perturbed in practice or there is a discrepancy between the probability distribution of the underlying uncertainty constructed with empirical data and the true probability distribution in future, we investigate how the data perturbation may affect the equilibrium. We undertake the analysis from two perspectives: (a) a few data are perturbed by outliers and (b) all data are potentially perturbed. In case (a), we use the well-known influence function to quantify the sensitivity of the equilibrium by the outliers and in case (b) we examine the difference between empirical distributions of the equilibrium based on perturbed data and the equilibrium based on unperturbed data. Moreover, we extend the discussions in case (b) to the LAPUE model and the other UE models. To examine the performance of the MLAPUE model and our theoretical analysis of statistical robustness, we carry out some numerical experiments, the preliminary results confirm the statistical robustness as desired.