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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 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
Braess’ paradox 2 Total tavel time = 552
Downs - Thomson A B
Downs - Thomson A B
Downs - Thomson A B
Mechanical Analogue of Braess’ Paradox Cohen & Horowitz
Mechanical Analogue of Braess’ Paradox Cohen & Horowitz
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”
From PWV Update Study Stopping criterion: 15 iterations
Occurrence of Braess’ Paradox with Different Stopping Criteria 1996 – 123 projects
Pas & Principio Paradox occurs when 2.58 < Q < 8.89
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?
Paradox occurs when 508.25 < Q < 873.99
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
Difference in Costs: Original – Augmented with Different Costs on Links 3 & 4
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
Acknowledgements • Gautrans • Vela VKE • Prof P H Potgieter