11 th trb transportation planning applications conference, daytona beach, fl achieving planning...

11th TRB Transportation Planning Applications Conference, Daytona Beach, FL

Achieving Planning Model Convergence

Howard SlavinJonathan BrandonAndres RabinowiczSrinivasan Sundaram

Caliper CorporationMay 2007


Feedback Convergence Motivation

• Required for model consistency

• Use of the proper congested travel times

• Fuller evaluation of project impacts


Feedback Convergence Requirements

• Requires traffic assignment convergence• Distribution and Mode choice models

must have suitable properties so that the overall model is convergent and a fixed point solution exists

• Skim or trip table convergence needed• Absence of divergent elements• Must be computable in reasonable time


Feedback Convergence Questions

• How should we measure feedback convergence?

• How much is enough?

• What is the best way to compute it?


Traffic Assignment Convergence

• Most traffic assignments not sufficiently converged

• Tighter convergence is needed for calibration

• Tighter convergence is needed to achieve feedback convergence

• Much tighter convergence is needed for impact assessment

GAPs of the same name are not all Rose’s

Change in the objective function (LeBlanc, Sheffi)

z = Objective function

Maximum flow change x flow, i link, n iteration

Ortuzar / Willumsen x flow, t time

(Average Excess Cost)Bar-Gera / Boyce

x flow, OD = demand from i to j

Relative Gap (Rose et al.) x flow, t travel time

%100z 1n

n

n

z

z

ni

ni

ni

x

xx 1

max

*

*

ijij

ijrijijrijr

Cx

ttx

tx

txtx

UE

aonUE

OD

txtx aonUE

Many different measures of convergence usedMany are poor indicators (Rose et al. 1988)


Rel % Diff in VHT is a poor measureConvergence Pattern: Rel Gap vs. % VHT


Two approaches now proven for faster traffic assignment convergence

• Multi-threaded Traditional Frank-Wolfe UE

• Origin User Equilibrium OUE


Multi-threaded Frank-Wolfe

• Speedups proportional to the number of cores

• Must be done carefully or the results will be different on different machines

• Most new hardware has multiple cores• Leads to significant improvement in TA

convergence in the same computing time.• Universally applicable to the largest

networks• Less costly and more effective than

distributed processing


Origin UE Assignment(proposed by Bar Gera & Boyce, Dial)• Can be faster to small gaps

• Requires more memory

• Order dependent so not readily amenable to multi-threading

• Has excellent warm start properties

• Implemented for multi-mode assignment with turn penalties for TransCAD 5


Test Environment

• Well-calibrated regional model for Washington DC

• 2500 zones, 6 purposes, 3 time periods, 5 assignment classes

• Feedback through distribution, mode choice, & assignment

• Calibrated to Relative Gap of .001, Skim matrix root mean square error < 1%

• 80 – 170 Assignment iterations and 4 feedback loops

• TransCAD 4.8/5.0 environment• Initial congested times from 5+ loop runs

Washington Regional NetworkNodes 20343

Links 57374

OD Pairs

6365529

Trips 2977171

Extent 92 x 109 mi

TA Convergence versus CPU Time (min.) for Multi-threaded FW & OUE

Gap 4 Core 8 Core

FW OUE FW OUE

0.001 62 12 36 12

0.0001 230 86 130 86


Feedback Tests

• Skim matrix stability used as feedback convergence stability

• RMSE of 1% and 0.1%

• Feedback convergence easily achievable with a good starting point

Time in min to reach Traffic Assignment Convergence – Washington Regional Net- 5 User Classes

Gap 0.001 0.0001

Multi-threaded FW3 GHZ, 4 core Woodcrest

60 215

Multi-threaded FW2.66 GHz 8 core Clovertown

36 129

OUE-Woodcrest(warm start)

12 82

OUE-Clovertown(warm start)

14 87

(Starting from congested link times from 5 feedback loops)

Feedback Convergence-Washington D.C. Regional Model-Minutes to Skim RMSE of 1 % (#Loops)

Gap 0.001 0.0001 .00001


154(2)

466(2)

n/a


112(2)

303(2)

n/a


67(2)

147(2)


64(2)

145(2)

Gap 0.001 .0001


369(5)

917(4)


263(5)

594(4)


67(2)

185(3)


64(2)

193(3)

Feedback Convergence-Washington D.C. Regional Model-Minutes to Skim RMSE of .1 % (#Loops)


Comparison of Feedback Calculation Approaches

• MSA AVERAGING OF FLOWS

• TRIP TABLE AVERAGING

• FLOW AND TRIP TABLE AVERAGING


FEEDBACK APPROACH COMPARISON GRAPH


Feedback Conclusions

• Feedback convergence in sequential models easily achievable with multi-threaded UE or OUE

• Beginning with congested travel times greatly reduces the computational cost.

• Tighter traffic assignment convergence reduces the number of feedback loops required for FW

• OUE with a warm start is fastest for assignment to low gaps and takes fewer loops for feedback

• MSA Flow Averaging is effective• Trip Table Averaging may help MSA Flow Averaging• More research on measures, methods and solution

characteristics is needed


Acknowledgements

We would like to thank Robert Dial, David Boyce and Hillel Bar-Gera for their research and many helpful discussions.


ReferencesH. Bar-Gera 1999 Origin-based algorithms for transportation network modeling,

Technical Report #103, NISS, Research Triangle Park, NC

H. Bar-Gera 2002 Origin-based algorithm for the traffic assignment problem, Transportation Science 36 (4), pp. 398-417

H. Bar-Gera and Amos Luzon 2007 Non-unique Solutions of User-Equilibrium Assignments and their Practical Implications, Paper presented at the 86th Annual Meeting of the Transportation Research Board

D. Boyce, B. Ralevic-Dekic, and H. Bar-Gera 2004 Convergence of Traffic Assignments: How much is enough, Journal of Transportation Engineering, ASCE, Jan./Feb. 2004

R. Dial 1999 Algorithm B: Accurate Traffic Equilibrium (and How to Bobtail Frank-Wolfe, Volpe National Transportation Systems Center, Cambridge, MA July 25, 1999

R. Dial 2006 A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration, Transportation Research B, December 2006

G. Rose, M. Daskin, F. Koppleman 1988. An Examination of Convergence Error in Equilibrium Traffic Assignment Models:, Transportation Research Vol 22B (4)

H. Slavin, J. Brandon, and Andres Rabinowicz 2006 An Empirical Comparison of Alternative User Equilibrium Traffic Assignment Methods, Proceedings of the European Transport Conference 2006, Strasbourg, France

11 th trb transportation planning applications conference, daytona beach, fl achieving planning...

Documents

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