A percentile system optimization approach with and without path enumeration

In this paper we deal with the travel time reliability PUE (probabilistic user equilibrium) problem studied by Lo et al. (2006) [12] and Nie (2011) [15] and we propose an alternative model that assumes a location-scale family for the path travel times, whose means and variances are evaluated in terms of link travel times. This avoids the use of the central limit theorem and convolutions providing a flexible and simple alternative. Contrary to the most existing models that require path enumeration or an iterative method to add paths sequentially, we present a percentile system optimization in its two versions: with and without path enumeration. Two examples of applications, one of them real, are used to illustrate the power of the proposed method. The cpu times required to solve the problem seem reasonable. In addition, we answer an open question raised by Nie (2011) [15] about the permutability of percentiles and partial derivatives of route travel times with respect to route flows. A family of counterexamples is given to demonstrate that the two operations: (a) obtain percentiles and (b) partial derivation of route travel times do not commute. Finally, to reproduce the trial-and-error sequence followed by users when selecting paths, we also present an algorithm that simulates this iterative process and shows that the final long-term user behavior coincides with PUE (probabilistic user equilibrium) problem resulting from some existing models.