A percentile system optimization approach with and without path enumeration Enrique Castillo a,n , Aida Calviño a , Santos Sánchez-Cambronero b , María Nogal a , Ana Rivas b a Department of Applied Mathematics and Computational Sciences, University of Cantabria, 39005 Santander, Spain b Department of Civil Engineering, University of Castilla La Mancha, 13071 Ciudad Real, Spain article info Available online 21 May 2013 Keywords: Percentile user equilibrium Percentiles Percentile system optimization abstract 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. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The problem of user interaction in traffic networks where travelers compete for space has been dealt with in the existing literature for several decades. The most common approach assumes that users behave in a homogeneous way (see [18]). However, if perfect online information of continuous real-time link travel time is available (see [3]) users react in a different form, depending on costs (see [21]), leading to heterogeneity. Thus, recently, some heteroge- neous cases have arisen as is the case of the travel time reliability problem, in which different users perceive the problem from a different perspective. This occurs when travelers are concerned about reaching the destination on time because of possible consequences in terms of prestige, money losses, etc. but the perception of these consequences is not the same for all of them, so that they can be grouped in different classes. It is common to measure travel time reliability as the probability that trip can be completed on time (see [4]). Consequently, users reduce the risk of late arrival and decide to start the trip with sufficient time to guarantee a high probability of success (arrive at the target on time) according to their classes. However, making this decision is not easy because travel times are random in nature and its statistical properties are not well known. As it has been shown in Refs. [10,11,1], travelers' decisions are known to be largely influenced by travel time variability and reliability. More precisely, travel time reliability has been recog- nized by users as one of the two main criteria for route choice. In the existing literature there have been some approaches to solve the travel time reliability problem. For example, Mirchandani and Soroush [14] present a model in which each traveller chooses a “perceived optimal route” (as opposed to a shortest route) which minimizes the perceived expected disutility of traveling from his origin node to his destination node. This allows one to incorporate travellers' risk taking behavior when some travel conditions are uncertain. Lo and Tung [13] postulate that drivers select routes to lower their travel time variabilities. Based on the experience on travel time variations of different routes, users learn their variability and choose routes accordingly, so that a long-term equilibrium pattern is reached after this learning process. Lo and Tung [13] solve the reliability problem by introducing the probabilistic user equili- brium (PUE) models. Lo et al. [12] indicate that travel time reliability plays an important role in travelers' route choice behavior and present an approach to relate the travel time variability due to stochastic Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2013.05.004 n Corresponding author. Tel.: +34 942201722; fax: +34 942201703. E-mail address: castie@unican.es (E. Castillo). Computers & Operations Research 40 (2013) 2711–2723