achieving planning model convergence
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11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Achieving Planning Model Convergence
Howard SlavinJonathan BrandonAndres RabinowiczSrinivasan Sundaram
Caliper CorporationMay 2007
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Feedback Convergence Motivation
• Required for model consistency
• Use of the proper congested travel times
• Fuller evaluation of project impacts
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Feedback Convergence Questions
• How should we measure feedback convergence?
• How much is enough?
• What is the best way to compute it?
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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)
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Rel % Diff in VHT is a poor measureConvergence Pattern: Rel Gap vs. % VHT
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Two approaches now proven for faster traffic assignment convergence
• Multi-threaded Traditional Frank-Wolfe UE
• Origin User Equilibrium OUE
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
Multi-threaded FW3 GHZ, 4 core Woodcrest
154(2)
466(2)
n/a
Multi-threaded FW2.66 GHz 8 core Clovertown
112(2)
303(2)
n/a
OUE-Woodcrest(warm start)
67(2)
147(2)
OUE-Clovertown(warm start)
64(2)
145(2)
Gap 0.001 .0001
Multi-threaded FW3 GHZ, 4 core Woodcrest
369(5)
917(4)
Multi-threaded FW2.66 GHz 8 core Clovertown
263(5)
594(4)
OUE-Woodcrest(warm start)
67(2)
185(3)
OUE-Clovertown(warm start)
64(2)
193(3)
Feedback Convergence-Washington D.C. Regional Model-Minutes to Skim RMSE of .1 % (#Loops)
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Comparison of Feedback Calculation Approaches
• MSA AVERAGING OF FLOWS
• TRIP TABLE AVERAGING
• FLOW AND TRIP TABLE AVERAGING
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
FEEDBACK APPROACH COMPARISON GRAPH
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Acknowledgements
We would like to thank Robert Dial, David Boyce and Hillel Bar-Gera for their research and many helpful discussions.
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
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
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