Networks and Spatial Economics

, Volume 10, Issue 1, pp 147–172 | Cite as

A Relaxation Approach for Estimating Origin–Destination Trip Tables

  • Yu Marco Nie
  • H. M. Zhang


The problem of estimating origin-destination travel demands from partial observations of traffic conditions has often been formulated as a network design problem (NDP) with a bi-level structure. The upper level problem in such a formulation minimizes a distance metric between measured and estimated traffic conditions, and the lower level enforces user-equilibrium traffic conditions in the network. Since bi-level problems are usually challenging to solve numerically, especially for large-scale networks, we proposed, in an earlier effort (Nie et al., Transp Res, 39B:497–518, 2005), a decoupling scheme that transforms the O–D estimation problem into a single-level optimization problem. In this paper, a novel formulation is proposed to relax the user equilibrium conditions while taking users’ route choice behavior into account. This relaxation approach allows the development of efficient solution procedures that can handle large-scale problems, and makes the integration of other inputs, such as path travel times and historical O–Ds rather straightforward. An algorithm based on column generation is devised to solve the relaxed formulation and its convergence is proved. Using a benchmark example, we compare the estimation results obtained from bi-level, decoupled and relaxed formulations, and conduct various sensitivity analysis. A large example is also provided to illustrate the efficiency of the relaxation method.


Static O–D estimation problem Bi-level program Relaxation Column generation 



The authors would like to thank the area editor, and two anonymous referees for their helpful comments and suggestion. This research is supported in part by a grant from the National Science Foundation under the number CMS#9984239 and by a Caltrans grant under the Task Orders 5502 and 5300. The views are those of the authors alone.


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringNorthwestern UniversityEvanstonUSA
  2. 2.Department of Civil and Environmental EngineeringUniversity of CaliforniaDavisUSA

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