The Impact of The Impact of ConvergenceConvergence Criteria Criteriaon Equilibrium Assignmenton Equilibrium Assignment

Yongqiang Wu, Huiwei Shen, and Terry CorkeryFlorida Department of Transportation

11th Conference on Transportation ApplicationsMay 7, 2007

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ContentContent

Background Methodology Preliminary Findings Conclusions and Recommendations

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Motives for the StudyMotives for the Study

New Florida Standard Model framework Increasing computational efforts with

more complicated models FTA New Starts analysis guidelines Implications of link flow stability on

different types of transportation engineering and planning projects

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Objectives of the StudyObjectives of the Study

Evaluate how link volumes vary with closing criteria for user equilibrium traffic assignment

Provide some empirical evidence on how to choose closing criteria for assignment

Establish standards and guidelines for new Florida Standard Model

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Study MethodologyStudy Methodology

Use Florida Statewide Model (FLSWM)4,059 zones98,098 directional links

Run the model on two computers with different configurations

Test two scenarios with different congestion levelsBase Year 2000 – validationFuture Year 2030 – forecast

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Study Methodology (Continued)Study Methodology (Continued)

Perform traffic assignment using different closing criteria

Number of Iterations – 1,5,10,15,20,25,30,35,40,45,50,

75,100,150, 200, and 300. Analyze assignment results

Compare volumes with traffic countsCompare volumes from iteration to iteration

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Florida Statewide Model StructureFlorida Statewide Model Structure

Passenger Model

• Trip Generation• Trip Distribution• Auto Occupancy

Freight Model

• Trip Generation• Trip Distribution• Mode Choice

SE Data/Network

Joint Assignment

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User Equilibrium (UE) AssignmentUser Equilibrium (UE) Assignment

Wardrop's ConditionFor a given origin-destination pair, travel costs (travel times) are equal on all paths actually utilized, and are less than or equal to the travel time on any other paths. No traveler can improve his or her travel time by unilaterally switching to a different route.

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User Equilibrium AssignmentUser Equilibrium Assignment

Franke-Wolfe AlgorithmA series of all-or-nothing assignments are

performedLink flows are combined mathematically

using weights derived from a line search to minimize the objective function

Travel times are recalculated using the combined assignment.

The process is repeated for a specified number of iterations or until certain stopping criterion is met.

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Gap and Relative GapGap and Relative Gap

= the current iteration

Where

= link volume from an AON assignment based on CEn-1

= cost based on equilibrium volume VEn

= equilibrium weighted volume for iteration n

= summation over all network links

n

n

n

l

VA

CE

VE

n

l11

11

nn

lnn-1

lnn

CEVE

CEVECEVA

RGAP

11

11

lnn

lnn

lnn

CEVE

CEVECEVE

GAP

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Preliminary FindingsPreliminary Findings

Computational effortsComputer SpecificationsCongestion level

Gap and relative gap Validation - aggregate statistics

V/C, VMT, and VHTRoot Mean Square Error (RMSE)

Forecast - link volume ChangesPercentagesAbsolute differences

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Computer ConfigurationsComputer Configurations

Computer 1Dell Mobile Station M90CPU – Intel® CoreTM 2 T7600 2.33 GHz Duo

Mobile ProcessorsRAM – 4GB

Computer 2Dell Desktop Optiplex GX620CPU – Intel® Pentium®4 3.20GHz Duo

Processors with Hyper-Treading TechnologyRAM – 2GB

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CPU Time vs. No. of IterationsCPU Time vs. No. of Iterations

0

180

360

540

720

900

1,080

1,260

1,440

1,620

1,800

1,980

2,160

0 50 100 150 200 250 300

No. of Iterations

Ru

n T

ime,

Min

ute

s

Y2000-PC1

Y2030-PC1

Y2000-PC2

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Base Year 2000

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 50 100 150 200 250 300

No. of Iterations

GA

P/R

GA

P, L

og

Y2000 GAP Y 2000 RGAP

Future Year 2030

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 50 100 150 200 250 300

No. of Iterations

GA

P/R

GA

P, L

og

Y2030 GAP Y2030 RGAP

GAP/RGAP vs. No. of IterationsGAP/RGAP vs. No. of Iterations

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Observation 1Observation 1

Computer run time is proportional to number of iterations.

Level of congestion has direct impact on computer run time.

To achieve the same level of gap/relative gap, more iterations are needed for more congested conditions

Relative gap seems to be higher and more stable than gap.

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Modeled vs. Observed - VolumesModeled vs. Observed - Volumes

0.50

0.60

0.70

0.80

0.90

1.00

1.10

Vo

lum

e/C

ou

nt

(V/C

) R

ati

o

1 5 10 15 20 25 30 35 40 45 50 75 100 150 200 300

No. of Iterations

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Modeled vs. Observed - VMTModeled vs. Observed - VMT

0.50

0.60

0.70

0.80

0.90

1.00

1.10

Mo

de

led

VM

T/C

ou

nts

VM

T

1 5 10 15 20 25 30 35 40 45 50 75 100 150 200 300

No. of Iterations

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Vehicle Hours Traveled - VHTVehicle Hours Traveled - VHT

0.60

0.80

1.00

1.20

1.40

1.60

1.80

Mo

de

led

VH

T/C

ou

nts

VH

T

1 5 10 15 20 25 30 35 40 45 50 75 100 150 200 300

No. of Iterations

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Observation 2Observation 2

Aggregate statistics used for validation purposes, such as V/C, VMT, and VHT, tend to stabilize with less computational efforts (10-15 iterations).

The overall good match between modeled volumes and observed traffic counts might obscure the large differences in individual links.

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Link Volumes vs. Counts (RMSE)Link Volumes vs. Counts (RMSE)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

Vo

lum

e/C

ou

nt

(V/C

) R

ati

o

1 5 10 15 20 25 30 35 40 45 50 75 100 150 200 300

No. of Iterations

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RMSE by Volume Groups (25 ITER)RMSE by Volume Groups (25 ITER)

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

RM

SE

0-5 5-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 100-500

Volume Group, 1,000 vpd

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Y2000 Link Volume Changes Y2000 Link Volume Changes

IterationsPercentage of Links with Volume Changes within, %

0→10% 10→20% 20→30% 30→40% 40→50% 50→100% 100%→

5 → 10 68.2 15.0 7.5 3.6 1.9 2.8 1.1

10 → 15 86.9 8.3 2.3 0.9 1.4 0.1 0.1

15 → 20 92.8 4.5 1.1 0.4 1.2 0.0 0.0

20 → 25 95.9 2.3 0.5 1.3 0.0 0.0 0.0

25 → 30 97.0 1.3 0.3 1.3 0.0 0.0 0.0

30 → 35 97.6 0.9 1.4 0.0 0.0 0.0 0.0

35 → 40 97.7 0.8 1.4 0.0 0.0 0.0 0.0

40 → 45 98.0 0.6 1.3 0.0 0.0 0.0 0.0

45 → 50 98.2 0.5 1.3 0.0 0.0 0.0 0.0

50 → 75 97.6 0.8 0.3 0.1 0.1 1.1 0.0

75 → 100 98.3 0.4 0.1 0.1 1.1 0.0 0.0

100 → 150 98.3 0.4 0.1 0.0 0.0 1.1 0.0

150 → 200 98.6 0.2 0.1 1.2 0.0 0.0 0.0

200 → 300 98.6 0.2 0.1 0.0 1.1 0.0 0.0

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Y2000 Max & Avg Volume ChangesY2000 Max & Avg Volume Changes

IterationsPercent Difference, % Absolute Difference, vpd

Maximum Average Maximum Average

5 → 10 500.00+ 17.18 72,394 1,310

10 → 15 500.00+ 5.75 17,095 497

15 → 20 500.00+ 3.71 9,069 301

20 → 25 500.00+ 2.51 10,080 190

25 → 30 500.00+ 1.94 5,626 135

30 → 35 500.00+ 1.59 5,888 103

35 → 40 394.64 1.43 4,539 93

40 → 45 500.00+ 1.17 4,297 73

45 → 50 298.12 0.95 3,489 55

50 → 75 500.00+ 1.87 4,210 71

75 → 100 166.75 1.14 3,252 38

100 → 150 633.91 1.22 3,125 31

150 → 200 193.25 0.77 1,273 16

200 → 300 99.99 0.85 1,526 14

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Y2030 Link Volume Changes Y2030 Link Volume Changes

IterationsPercentage of Links with Volume Changes within, %

0→10% 10→20% 20→30% 30→40% 40→50% 50→100% 100%→

5 → 10 53.1 18.5 10.5 5.8 3.7 5.5 2.9

10 → 15 75.3 14.9 4.8 2.1 2.0 0.5 0.4

15 → 20 86.5 8.8 2.2 0.9 1.3 0.2 0.1

20 → 25 91.8 5.3 1.2 1.5 0.1 0.1 0.0

25 → 30 95.0 2.9 1.9 0.1 0.0 0.0 0.0

30 → 35 96.1 2.2 1.7 0.0 0.0 0.0 0.0

35 → 40 97.0 1.5 1.4 0.0 0.0 0.0 0.0

40 → 45 97.3 1.3 1.4 0.0 0.0 0.0 0.0

45 → 50 97.6 2.3 0.1 0.0 0.0 0.0 0.0

50 → 75 95.8 2.1 0.5 0.2 0.2 1.2 0.0

75 → 100 97.5 0.9 0.2 0.2 1.2 0.0 0.0

100 → 150 97.4 0.9 0.2 0.1 0.1 1.2 0.0

150 → 200 98.2 0.4 0.1 0.1 1.2 0.0 0.0

200 → 300 98.2 0.4 0.1 0.1 0.0 1.2 0.0

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Y2030 Max & Avg Volume ChangesY2030 Max & Avg Volume Changes

IterationsPercent Difference, % Absolute Difference, vpd

Maximum Average Maximum Average

5 → 10 500.00+ 29.40 92,908 3,430

10 → 15 500.00+ 9.93 44,430 1,426

15 → 20 500.00+ 5.88 20,978 813

20 → 25 500.00+ 4.10 18,304 540

25 → 30 500.00+ 2.91 12,778 352

30 → 35 500.00+ 2.40 11,842 275

35 → 40 320.16 1.88 8,498 215

40 → 45 363.30 1.77 12,164 193

45 → 50 331.98 1.53 7,722 167

50 → 75 212.13 2.71 7,414 225

75 → 100 167.14 1.70 4,210 122

100 → 150 276.32 1.79 4,269 106

150 → 200 158.45 1.16 2,647 57

200 → 300 52.04 1.25 2,353 49

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Links with Highest Percent ChangesLinks with Highest Percent Changes

0

2000

4000

6000

8000

10000

12000

14000

0 10 20 30 40 50 60 70 80 90 100

No. of Iterations

Lin

k V

olu

me

97755-132116

131461-131411

107325-107329

120507-127697

107220-107507

132160-132161

112465-114758

112465-112466

21571-104914

120762-119446

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Links with Highest Volume ChangesLinks with Highest Volume Changes

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

180,000

0 10 20 30 40 50 60 70 80 90 100

No. of Iterations

Lin

k V

olu

me

97478-99281

111672-132033

109791-132161

97756-97757

132107-99429

132004-131995

132066-132062

97631-97632

111521-132053

131994-131996

132080-132084

110754-132177

132158-110949

123680-128775

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Observation 3Observation 3

Large variations in link volumes do exist Variations might cause a lane change in

some cases Largest percentage changes occur on low

volume links located near centroid connectors

Largest volume changes occur on mid- to high-volume links located in congested areas

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Conclusions and RecommendationsConclusions and Recommendations

Use gap/relative gap as stopping criterion and set the values sufficiently small

Set number of iterations large enough Use number of iterations only when

Checking grammatical or functional errorsPreventing model being run indefintely

Consider different stopping criterion for different applications

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Conclusions and RecommendationsConclusions and Recommendations

Aggregate statistics can obscure the large variations in individual links

Large percentage changes occur to links with low volume links and located near centroid connectors

Large volume changes happen to links with mid- to high volume links located in congested areas

More studies considering other factors using different models are needed.

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Questions?Questions?