Norm approximation method for handling traffic count inconsistencies in path flow estimator Anthony Chen a, * , Piya Chootinan b , Will Recker c a Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110, USA b Bureau of Planning, Department of Highways, Bangkok 10400, Thailand c Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697-3600, USA article info Article history: Received 28 April 2007 Received in revised form 26 February 2009 Accepted 27 February 2009 Keywords: Origin–destination estimation Path flow estimator Stochastic user equilibrium Norm approximation Partial linearization method abstract Path flow estimator (PFE) is a one-stage network observer proposed to estimate path flows and hence origin–destination (O–D) flows from traffic counts in a transportation network. Although PFE does not require traffic counts to be collected on all network links when inferring unmeasured traffic conditions, it does require all available counts to be reason- ably consistent. This requirement is difficult to fulfill in practice due to errors inherited in data collection and processing. The original PFE model handles this issue by relaxing the requirement of perfect replication of traffic counts through the specification of error bounds. This method enhances the flexibility of PFE by allowing the incorporation of local knowledge, regarding the traffic conditions and the nature of traffic data, into the estima- tion process. However, specifying appropriate error bounds for all observed links in real networks turns out to be a difficult and time-consuming task. In addition, improper spec- ification of the error bounds could lead to a biased estimation of total travel demand in the network. This paper therefore proposes the norm approximation method capable of inter- nally handling inconsistent traffic counts in PFE. Specifically, three norm approximation criteria are adopted to formulate three L p -PFE models for estimating consistent path flows and O–D flows that simultaneously minimize the deviation between the estimated and observed link volumes. A partial linearization algorithm embedded with an iterative bal- ancing scheme and a column generation procedure is developed to solve the three L p - PFE models. In addition, the proposed L p -PFE models are illustrated with numerical exam- ples and the characteristics of solutions obtained by these models are discussed. Published by Elsevier Ltd. 1. Introduction Path flow estimator (PFE), originally developed by Bell and Shield (1995), is one of the efficient methods for estimating path flows (hence origin–destination (O–D) flows) from traffic counts. The attractiveness of PFE lies on the fact that it is a single level mathematical program in which the interdependency between O–D demands and route choice behavior (con- gestion effect) is taken into account without the need to employ the bi-level mathematical program (one level estimates the O–D trip table while the other level represents the behavioral responses of network users). Network users are assumed to follow the stochastic user equilibrium (SUE) principle, which allows the selection of non-equal travel time paths due to the imperfect knowledge of network travel times and yields unique path flow estimates. Besides, PFE can perform the estimation using traffic counts collected only on a subset of network links. Nevertheless, these available counts must be 0191-2615/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.trb.2009.02.007 * Corresponding author. Tel.: +1 435 797 7109; fax: +1 435 797 1185. E-mail address: achen@engineering.usu.edu (A. Chen). Transportation Research Part B 43 (2009) 852–872 Contents lists available at ScienceDirect Transportation Research Part B journal homepage: www.elsevier.com/locate/trb