Networks and Spatial Economics

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

A Relaxation Approach for Estimating Origin–Destination Trip Tables



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.


  1. Abdulaal M, LeBlanc LJ (1979) Continuous equilibrium network design models. Transp Res 13B:19–32CrossRefGoogle Scholar
  2. Allsop RE (1974) Some possibilities for using traffic control to influence trip destinations and route choice. In: Buckley DJ (ed) Proceedings of the Sixth International Symposium on Transportation and Traffic Theory. Elsevier, Amsterdam, pp 345–374Google Scholar
  3. Beckmann M, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven, CTGoogle Scholar
  4. Cascetta E (1984) Estimation of trip matrices from traffic counts and survey data: a generalized least squares estimator. Transp Res 18B:289–299CrossRefGoogle Scholar
  5. Fisk CS (1980) Some developments in equilibrium traffic assignment. Transp Res 14B:243–255CrossRefGoogle Scholar
  6. Fisk CS (1984) Game theory and transportation systems modeling. Transp Res 18B:301–313CrossRefGoogle Scholar
  7. Fisk CS (1988) On combining maximum entropy trip matrix estimation with user optimal assignment. Transp Res 22B:66–79Google Scholar
  8. Friesz TL, Anandalingam G, Mehta NJ, Nam K, Shah S, Tobin RL (1993) The multiobjective equilibrium network design problem revisited: a simulated annealing approach. Eur J Oper Res 65:44–57CrossRefGoogle Scholar
  9. Friesz TL, Cho H-J, Mehta NJ, Tobin RL, Anandalingam G (1992) A simulated annealing approach to the network design problem with variational inequality constraints. Transp Sci 28:18–26CrossRefGoogle Scholar
  10. Gartner NH, Gershwin SB, Little JDC, Ross P (1980) Pilot study of computer based urban traffic management. Transp Res 14B:203–217CrossRefGoogle Scholar
  11. Gur Y, Turnquist M, Schneider M, Leblanc L (1980) Estimation of an O–D trip table based on observed O-D volumes and turning movements, vols 1–3. Technical report FHWA/RD-80/035, FHWA, Washington, DCGoogle Scholar
  12. LeBlanc L, Farhangian K (1982) Selection of a trip table which reproduces observed link flows. Transp Res 22B:83–88CrossRefGoogle Scholar
  13. Lim AC (2002) Transportation network design problems: an MPEC approach. PhD thesis, Johns Hopkins University, USAGoogle Scholar
  14. Low D (1972) A new approach to transportation system modelling. Traffic Q 26:391–404Google Scholar
  15. Nguyen S (1977) Estimating an OD matrix from network data: a network equilibrium approach. Technical report 60, University of MontrealGoogle Scholar
  16. Nguyen S (1984) Estimating origin-destination matrices from observed flows. In: Florian M (ed) Transportation planning models. Elsevier Science, Amsterdam, pp 363–380Google Scholar
  17. Nie Y (2006) A variational inequality approach for inferring dynamic origin-destination travel demands. PhD thesis, University of California, Davis, USAGoogle Scholar
  18. Nie Y, Zhang HM, Recker WW (2005) Inferring origin-destination trip matrices with a decoupled GLS path flow estimator. Transp Res 39B:497–518CrossRefGoogle Scholar
  19. Ortuzar J, Willumsen L (2001) Modeling transport, 3rd edn. Wiley, New YorkGoogle Scholar
  20. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice Hall, Englewood Cliffs, NJGoogle Scholar
  21. Sherali HD, Park T (2001) Estimation of dynamic origin-destination trip tables for a general network. Transp Res 35B:217–235CrossRefGoogle Scholar
  22. Sherali HD, Sivanandan R, Hobeika AG (1994) A linear programming approach for synthesizing origin-destination trip tables from link traffic volumes. Transp Res 28B:213–233CrossRefGoogle Scholar
  23. Tan H, Gershwin SB, Athans M (1979) Hybrid optimization in urban traffic networks. Report DOT-TSC-RSP-79-7, Laboratory for Information and Decision Systems, MIT, CambridgeGoogle Scholar
  24. Van-Zuylen JH, Willumsen LG (1980) The most likely trip matrix estimated from traffic counts. Transp Res 14B:281–293CrossRefGoogle Scholar
  25. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proceedings of the Institute of Civil Engineers, Part II 1:325–378Google Scholar
  26. Willumsen LG (1981) Simplified transport models based traffic counts. Transportation 10:257–278CrossRefGoogle Scholar
  27. Yang H (1995) Heuristic algorithms for the bi-level origin-destination matrix estimation problem. Transp Res 29B:231–242CrossRefGoogle Scholar
  28. Yang H, Sasaki T, Iida Y, Asakura Y (1992) Estimation of origin-destination matrices from traffic counts on congested networks. Transp Res 26B:417–433CrossRefGoogle Scholar

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

Personalised recommendations