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An investigation into some aspects of Braess’ Paradox

An investigation into some aspects of Braess’ Paradox. Keith Bloy. Vela VKE Consulting Engineers. Contents. Classical example Other paradoxes Literature survey Comparison between literature and modelled results Eliminating Braess’ paradox. Braess’ paradox. Total tavel time = 498.

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An investigation into some aspects of Braess’ Paradox

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  1. An investigation into some aspects of Braess’ Paradox Keith Bloy Vela VKE Consulting Engineers

  2. Contents • Classical example • Other paradoxes • Literature survey • Comparison between literature and modelled results • Eliminating Braess’ paradox

  3. Braess’ paradox Total tavel time = 498

  4. Braess’ paradox 2 Total tavel time = 552

  5. Other Paradoxes

  6. Downs - Thomson A B

  7. Downs - Thomson A B

  8. Downs - Thomson A B

  9. Mechanical Analogue of Braess’ Paradox Cohen & Horowitz

  10. Mechanical Analogue of Braess’ Paradox Cohen & Horowitz

  11. How Prevelant is Braess’ Paradox? LeBlanc: “When dealing with a network with many origins and destinations, it is not clear whether adding an arc will increase or decrease the congestion at equilibrium.” Steinberg and Zangwill (1983): “Braess’ paradox is as likely to occur as not”

  12. From PWV Update Study

  13. From PWV Update Study Stopping criterion: 15 iterations

  14. Occurrence of Braess’ Paradox with Different Stopping Criteria 1996 – 123 projects

  15. Pas & Principio Paradox occurs when 2.58 < Q < 8.89

  16. Questions arising from Pas & Principio • What does figure look like for non-linear functions (eg BPR)? • What is the level of congestion where paradox occurs • Does paradox occur only over a range in real networks?

  17. Braess’ paradox type network with BPR functions

  18. Paradox occurs when 508.25 < Q < 873.99

  19. Difference in costs: original - augmented

  20. Flows on BPR Network where Braess’ Paradox Occurs

  21. Effect on Differences with Matrix * Factor

  22. Effect on Differences with Select Link Matrix * Factor

  23. Analysis of Flows on Links Demand = Q Original network: flow on all links = 0.5*Q Augmented network: flow on new link = P Then flows: on links 1 & 2 = 0.5*P – 0.5*Q on links 3 & 4 = 0.5*P + 0.5*Q

  24. Difference in costs: original - augmented

  25. Difference in Costs: Original – Augmented with Different Costs on Links 3 & 4

  26. Replacing Link Results in Braess’ Paradox

  27. Change in V/C Ratios After Adding New Link

  28. Conclusions • Braess’ paradox less likely to occur at high level of convergence • Braess’ paradox less likely to occur at high volumes • A possible methodology to eliminate Braess’ Paradox was suggested

  29. Acknowledgements • Gautrans • Vela VKE • Prof P H Potgieter

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