An investigation into some aspects of Braess

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
Volume delay functions Volume delay functions
t1 50 x1
t2 50 x2
t3 10x3
t4 10x4
Total tavel time 498
4
Braess paradox 2
Additional link t5 10 x5
Total tavel time 552
5
Other Paradoxes
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Downs - Thomson
A
B
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Downs - Thomson
A
B
8
Downs - Thomson
A
B
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Mechanical Analogue of Braess Paradox
Cohen Horowitz
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Mechanical Analogue of Braess Paradox
Cohen Horowitz
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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
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From PWV Update Study
Year No. of Projects No. of Projects with Braess P Largest Paradox
1989 31 1 13
1992 64 28 62
1995 107 9 20
1996 123 71 70
1998 158 52 65
2000 186 79 84
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From PWV Update Study
Stopping criterion 15 iterations
Year No. of Projects No. of Braess Projects Largest Paradox
1989 31 1 13
1992 64 28 62
1995 107 9 20
1996 123 71 70
1998 158 52 65
2000 186 79 84
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Occurrence of Braess Paradox with Different
Stopping Criteria
1996 123 projects
Stopping Criterion No. of Braess Projects Largest Paradox
15 Iterations 71 70
Rel Gap 0.20 29 16
Rel Gap 0.10 0 -
Rel Gap 0.05 4 5
Rel Gap 0.03 2 4
Rel Gap 0.01 1 1
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Pas Principio
Paradox occurs when 2.58 lt Q lt 8.89
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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?

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Braess paradox type network with BPR functions
Link Length (km) FF Speed Capacity
1 2 1.56 60 830
3 4 0.75 70 920
5 1.56 110 1110
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Paradox occurs when 508.25 lt Q lt 873.99
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Difference in costs original - augmented
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Flows on BPR Network where Braess Paradox Occurs
Link Capacity Flow where Paradox begins V/C Flow where Paradox ends V/C
1 2 830 0 0 245.15 0.30
3 4 920 508.25 0.55 628.85 0.68
5 1110 508.25 0.46 383.71 0.35
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Effect on Differences with Matrix Factor
Project 57 26 18 21 28 76
0.5 3 10 -2 -1 9 -9
0.6 8 24 -1 4 14 -12
0.7 9 24 2 8 18 -16
0.8 6 16 4 12 20 -21
0.9 4 14 4 22 21 -28
1.0 11 16 10 35 26 -40
1.1 11 18 11 45 28 -41
1.2 12 28 11 56 34 -54
1.3 14 45 22 72 39 -57
1.4 19 33 21 101 49 -80
1.5 26 26 20 137 53 -93
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Effect on Differences with Select Link Matrix
Factor
Project 34 38 58 69 76
0.5 -1 0 -1 -1 49
0.7 0 0 0 0 42
0.9 -2 -1 -1 0 41
1.1 0 0 1 0 39
1.3 2 2 1 2 23
1.5 4 0 0 0 23
1.7 7 -1 -2 1 30
1.9 6 4 1 3 26
2.1 10 4 3 3 15
2.3 15 4 6 5 4
2.5 17 4 12 7 7
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Analysis of Flows on Links
Demand Q Original network flow on all links
0.5Q
Augmented network flow on new link P Then
flows on links 1 2 0.5P 0.5Q on
links 3 4 0.5P 0.5Q
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Difference in costs original - augmented
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Difference in Costs Original Augmented with
Different Costs on Links 3 4
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Replacing Link Results in Braess Paradox
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Change in V/C Ratios After Adding New Link
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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

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Acknowledgements

• Gautrans
• Vela VKE
• Prof P H Potgieter