How much convergence is enough for traffic assignments used in feedback?

John Gibb

DKS Associates

For the 14th TRB National Transportation Planning Applications Conference

*Annotated version* - see notes.

Partners, Sources, Assistance• Sacramento Area Council of Governments• Puget Sound Regional Council• John Bowman, Mark Bradley, RSG• Carson Area Metropolitan Planning Organization• Spokane Regional Transportation Council• Citilabs• PTV America• Caliper• Inro Consultants

Convergence of traffic assignments: How much is enough?

David Boyce

Biljana Ralevic-Dekic

Hillel Bar-Gera

• Link flow stability – avoid random noise• Relative gap → 0.0001

ASCE Journal of Transportation Engineering 130, 49-55 (2004)

Skims

Assignment

Link volumes

Demand model

Skims

Assignment

Demand model

Skims

Assignment

Demand model

Skims

Assignment

Demand model

Link timesLink timesLink times

The rest of the assignments

Skim comparison statistics

Iteration vs. PreceedingIteration vs. Equilibrium

Used paths vs. Shortest path = Relative Gap

Convergence Progress of a Trip-Based Model – AM Skim Change

2 3 4 5 6 7 8 9 10 11 12 13 14 150.0001

0.001

0.01

0.1

1

RG≈0.05

RG≈0.01

RG≈0.003

RG≈0.001

RG≈0.0004

RG≈0.0002

Iteration (demand model)

Re

lati

ve

Av

era

ge

Sk

im C

ha

ng

e

Convergence Progress of a Trip-Based Model – AM Displaced Trips

2 3 4 5 6 7 8 9 10 11 12 13 14 150.0001

0.001

0.01

0.1

1

RG≈0.05

RG≈0.01

RG≈0.003

RG≈0.001

RG≈0.0004

RG≈0.0002

Iteration (demand model)

Re

lati

ve

Dis

pla

ce

d T

rip

s

Convergence Progress of draft Sacramento Activity-Based Model

2 3 4 5 6 7 8 9 100.0001

0.001

0.01

0.1

1

ev2h05h06h07h08h09h14h15h16h17md4ni9Max RelGapRel Gap Criterion

Demand Model Iteration (during newest skim)

Re

lati

ve

Av

g-A

bs

olu

te S

kim

De

lta

fro

m

pre

vio

us

ite

rati

on

;R

ela

tiv

e G

ap

Period of Day

Skim error study

Well-converged assignment

Less-converged assignment

Skim Skim

Poorly-converged assignment

Skim

Comparison statistics

Comparison statistics

Trip Table & Network

Skim comparison:unconverged vs. best equilibrium

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

RG 0.016RG 0.0012

Best Equilibrium Skim (min)

Sk

im a

t in

dic

ate

d R

ela

tiv

e G

ap

Skim Error v. Relative Gap: Sacramento AM (F-W)

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

r

Average-Absolute

Extreme

RMS

convergence

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

rSkim Error v. Relative Gap: Sacramento Mid-Day

Extreme

RMS

Average-Absoluteconvergence

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

rSkim Error v. Relative Gap: Reduced congestion (AM)

Extreme

RMS

Average-Absoluteconvergence

Skim Error v. Relative Gap: Increased congestion (AM)

Extreme

RMS

Average-Absolute

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

r

convergence

Skim Error v. Relative Gap: BPR^8(AM)

Extreme

RMS

Average-Absolute

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

r

convergence

Skim Error v. Relative Gap: Visum (Spokane)

Extreme

RMS

Average-Absolute

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

r

convergence

Skim Error v. Relative Gap: TransCAD (Carson City)

Extreme

RMS

Average-Absolute

0.000001 0.0001 0.01 10.000001

0.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

r

convergence

Convergence Progress of a Trip-Based Model – AM Average Skim Change

2 3 4 5 6 7 8 9 10 11 12 13 14 150.0001

0.001

0.01

0.1

1

RG≈0.05

RG≈0.01

RG≈0.003

RG≈0.001

RG≈0.0004

RG≈0.0002

Iteration (demand model)

Re

lati

ve

Sk

im C

ha

ng

e

0.0001 0.001 0.01 0.1 10.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

rSkim Error, AM changes v. Relative Gap

Extreme

RMS

Average-Absolute

Convergence Progress of a Trip-Based Model – Mid-Day Avg. Skim Change

2 3 4 5 6 7 8 9 10 11 12 13 14 150.00001

0.0001

0.001

0.01

0.1

1

AM 0.05AM 0.01AM 0.003AM 0.001AM 0.0004AM 0.0002RG≈0.05RG≈0.01RG≈0.003RG≈0.001RG≈0.0004RG≈0.0002

Iteration

Re

lati

ve

Sk

im C

ha

ng

e

AM

Mid-Day

0.00001 0.0001 0.001 0.01 0.1 10.00001

0.0001

0.001

0.01

0.1

1

Relative Gap (posterior)

Re

lati

ve

Sk

im E

rro

rSkim Error, MD changes v. Relative Gap

Extreme

RMS

Average-Absolute

Conclusions• Relative Gap seems to imply relative skim errors.• Successive skim differences can be misleading – less than skim error implied by RG – esp. in low congestion.

• Choose Relative Gaps to create small skim errors compared to skim changes. • Low RGs don’t seem to accelerate demand convergence,

but high RGs limit it.• Doing so should avoid the misleading-differences problem.

• This is a small study of a few models. Test your own!

Convergence Progress of draft Sacramento Activity-Based Model

2 3 4 5 6 7 8 9 100.0001

0.001

0.01

0.1

1 ev2

h05

h06

h07

h08

h09

h14

h15

h16

h17

md4

ni9

Max RelGap

Rel Gap Criterion

Expected ErrorDemand Model Iteration (during newest skim)

Re

lati

ve

Ch

an

ge

, E

rro

r, G

ap

Contact

John Gibb

jag (at) dksassociates (dot) com

Extra slides

Example of spurious flow change

Skim comparison statisticsRelative Gap

(links)

(skims)

= link cost = link flow = virtual shortest-path flow

= demand (trip table) = used-paths average travel time = shortest-path skim travel time

Average Absolute(trip-weighted)

= equilibrium travel time

RMS(trip-weighted)

Max Absolute