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The Equilibrium Assignment – What is a “Correct” Solution?

The Equilibrium Assignment – What is a “Correct” Solution?. Keith Bloy. Contents. Brief history of trip assignment Stopping criteria Results of analysis Conclusions & Recommendations. Assignment: A Brief History. All-or-Nothing Assignments. Assignment: A Brief History.

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The Equilibrium Assignment – What is a “Correct” Solution?

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  1. The Equilibrium Assignment – What is a “Correct” Solution? Keith Bloy

  2. Contents • Brief history of trip assignment • Stopping criteria • Results of analysis • Conclusions & Recommendations

  3. Assignment: A Brief History • All-or-Nothing Assignments

  4. Assignment: A Brief History • All-or-Nothing Assignments

  5. Assignment: A Brief History • Wardrop (1952): “The costs on all routes used between any given pair of end points are equal and not greater than the cost experienced by a single vehicle on any unused route between them.”

  6. Assignment: A Brief History t1 A B t2 Initially: t1 < t2 At Equilibrium: t1 = t2

  7. Assignment: A Brief History Beckmann, McGuire & Winsten: Studies in the Economics of Transportation (1956)

  8. Assignment: A Brief History • Incremental Assignment • Capacity Restraint • Approximate Methods:

  9. Equilibrium Assignment: Done using the Franke-Wolfe algorithm Consists of successive all-or-nothing assignments which are combined mathematically so as to minimize an objective function. The travel times are recalculated using the combined assignments and the process repeated.

  10. From Urban Transportation Networks by Sheffi

  11. From Gautrans Model

  12. Stopping Criteria in EMME/2 • Number of Iterations (15) • Relative Gap (0.50) An estimate of the difference between the current assignment and a perfect equilibrium assignment • Normalized Gap (0.50) The difference between the mean trip time of the current assignment and the mean minimal trip time. The mean trip time is the average trip times used in the previous iteration while the mean minimal trip time is the average time computed using the shortest paths of the current iteration.

  13. Convergence of TrafficAssignments: How Much Is Enough? The Delaware Valley Case Study by David Boyce, Biljana Ralevic-Dekic & Hillel Bar-Gera EMME/2 User Group Meeting Albuquerque 2002

  14. Reasons For Doing Assignments • For economic evaluation • For geometric and pavement design

  15. Projects Not Ranked in the Top 12

  16. Regression Coefficients

  17. Cumulative Percentage Within Given Percentage of “Equilibrium” Link Volumes

  18. Cumulative Percentage Within Given Percentage of “Equilibrium” Link Volumes

  19. Computational Effort

  20. Conclusions • Be wary of using default values for stopping criteria • An objective measure of convergence is preferable to number of iterations • Different stopping criteria can be used depending on purpose of assignment

  21. Recommendations • For Economic Evaluation: R Gap = 0.01 • For Link Volumes: Use Table 4 to provide a “level of confidence” • Do not use number of iterations

  22. Final Remarks • Thanks to Gautrans • My own views

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