The Evolution of Transport...

Innovations in Travel Modeling, Baltimore, MD, April 27-30, 2014

The Evolution of Transport Planning


Michael FlorianCalin D. Morosan

On Proportionality and Uniqueness inEquilibrium Assignment


Several bush-based algorithms for computingequilibrium assignments can obtain very finelyconverged flows Bar Gera (2002) Dial (2006) Gentile (2012)

All require post processing to obtain unique pathflows by maximizing the entropy of path flowsTAPAS method (Bar-Gera, 2010) exhibitsproportionality and hence uniqueness of path flows

Background


Bush-based methods are not particularly efficientfor solving multi-class network equilibrium models usually require cycling over the classes cannot be efficiently parallelized

The motivation of this investigation is the need fora more efficient multi-class traffic assignment thatcan obtain unique path and class flows

Motivation


The bi-conjugate variant of the linearapproximation method solves multi-class networkequilibrium models efficientlyThe path and class flows exhibit proportionalityand hence uniqueness as the relative gapdecreasesThe linear approximation method is not as efficientbut also shares the near uniqueness properties

Findings


Two classes of traffic: 1->8 100 trips; 2->8 60 tripsTotal link flow indicated on the links

Path Flows and Proportionality

1

2

3

5

6

84 7

100

60

160

40

120

40

120

160


These path flows are proportional25/75=15/45=40/120=1/3


1

2

3

5

6

84 7

100

60

100;60

25;15

75;45

25;15

75;45

100;60


These path flows are not proportional40/60 is not equal to 0/60


1

2

3

5

6

84 7

100

60

100;60

40;0

60;60

40;0

60;60

100;60


A two-class network equilibrium problem on theChicago test networkThree sets of origin-destination matrices cars and trucks three different levels of congestion

Flows obtained with the TAPAS algorithm run to arelative gap of 10-12

Available Data for Comparison


1,790 zones11,192 nodes39,018 linkstwo classes cars and trucks

truck prohibition on563 links

The Chicago Test Network


1 A multi-threaded implementation of the classicallinear approximation method, as implemented inEmme 4.1 Standard traffic assignment

2 A multi-threaded, conjugate direction linearapproximation method* as implemented in Emme4.1 SOLA traffic assignment The convergence of the second-order method is one

order of magnitude better with reasonable computationtimes for relative gaps of 10-5 and 10-6

Algorithms Used in the Computations

*As described in Mitradjieva, M. and Lindberg, P.O. (2013)


TAPAS (10-12 relativegap) and SOLA (10-6

relative gap) havenearly identical carflowsSOLA trafficassignment runs in12.7 minutes on a 16-core, 2.9 GHz Xeonprocessor (32 threads)

Car Flows Comparison


TAPAS (10-12 relativegap) and SOLA (10-6

relative gap) havenearly identical truckflowsSOLA trafficassignment runs in12.7 minutes on a 16-core, 2.9 GHz Xeonprocessor (32 threads)

Truck Flows Comparison


Paired Alternative SegmentsSOLA - Relative Gap 10-6

200

233

Ratio of flow onpaired alternativesegments is~200/233 = 0.8584


O-D pairs contributingflow to each leg of thepair of alternativesegmentsO-D demand areobtained with a ‘select-link’ analysisSlope is~200/233 =.8584

Paired Alternative SegmentsSOLA Car flows


O-D pairs contributingflow to each leg of thepair of alternativesegmentsO-D demand areobtained with a ‘select-link’ analysisSlope is~200/233 =.8584

Paired Alternative SegmentsSOLA Truck flows


A comparison of the flow differences between SOLA flows at different levels of convergence TAPAS flows at 10-12 relative gap

Comparison using a log-log scale


Car Flows Comparison: SOLA vs TAPASSOLA 10-3 Relative Gap



RMSE=0.89


Truck Flows Comparison: SOLA vs TAPASSOLA 10-3 Relative Gap


TAPAS (10-12 relative gap)and SOLA (10-6 relative gap)have nearly identical carflowsSOLA traffic assignmentruns in 17.7 minuteson a 16-core, 2.9 GHz Xeonprocessor (32 threads)

Car Flows ComparisonMore congestion


TAPAS (10-12 relative gap)and SOLA (10-6 relative gap)have nearly identical truckflowsSOLA traffic assignmentruns in 17.7 minuteson a 16-core, 2.9 GHz Xeonprocessor (32 threads)

Truck Flows ComparisonMore congestion


TAPAS (10-12 relative gap)and SOLA (10-6 relative gap)have nearly identical carflowsSOLA traffic assignmentruns in 32 minuteson a 16-core, 2.9 GHz Xeonprocessor (32 threads)

Car Flows ComparisonEven more congestion


TAPAS (10-12 relative gap)and SOLA (10-6 relative gap)have nearly identical truckflowsSOLA traffic assignmentruns in 32 minuteson a 16-core, 2.9 GHz Xeonprocessor (32 threads)

Truck Flows ComparisonEven more congestion


Interesting to see how the linear approximationmethod performsStandard traffic assignment was run for 15,000iterations to achieve a relative gap of 2.3*10 -6 (notrecommended for use in practice…)3.5 hours with the linear approximation method ona 16-core, 2.9 GHz Xeon processor (32 threads)

Back to Linear Approximation


Car Flows ComparisonTAPAS vs Linear Approximation

RMSE=1.35


Truck Flows ComparisonTAPAS vs Linear Approximation


Results obtained with the linear approximationmethod and the second-order method (SOLA)exhibit near class uniqueness and proportionalityThe SOLA traffic assignment, a multi-threaded bi-conjugate variant of the linear approximationmethod, provides an attractive and computationallyefficient method for solving multi-classassignments to fine convergence

Conclusions

The Evolution of Transport...

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