11 th TRB Transportation Planning Applications Conference, Daytona Beach, FL Achieving Planning Model Convergence Howard Slavin Jonathan Brandon Andres Rabinowicz Srinivasan Sundaram Caliper Corporation May 2007

11 th 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

11 th 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

11 th 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?

11 th 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 changex flow, i link, n iteration Ortuzar / Willumsenx 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 Many different measures of convergence used Many are poor indicators (Rose et al. 1988)

11 th TRB Transportation Planning Applications Conference, Daytona Beach, FL Rel % Diff in VHT is a poor measure Convergence Pattern: Rel Gap vs. % VHT

11 th 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

11 th 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

11 th 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

11 th 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 Network Nodes20343 Links57374 OD Pairs Trips Extent92 x 109 mi

TA Convergence versus CPU Time (min.) for Multi-threaded FW & OUE Gap4 Core8 Core FWOUEFWOUE

11 th 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 Multi-threaded FW 3 GHZ, 4 core Woodcrest Multi-threaded FW 2.66 GHz 8 core Clovertown OUE-Woodcrest (warm start) 1282 OUE-Clovertown (warm start) 1487 (Starting from congested link times from 5 feedback loops)

Feedback Convergence-Washington D.C. Regional Model-Minutes to Skim RMSE of 1 % (#Loops) Gap Multi-threaded FW 3 GHZ, 4 core Woodcrest 154 (2) 466 (2) n/a Multi-threaded FW 2.66 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 Multi-threaded FW 3 GHZ, 4 core Woodcrest 369 (5) 917 (4) Multi-threaded FW 2.66 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)

11 th 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

11 th TRB Transportation Planning Applications Conference, Daytona Beach, FL FEEDBACK APPROACH COMPARISON GRAPH

11 th 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

11 th 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.

11 th TRB Transportation Planning Applications Conference, Daytona Beach, FL References H. 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 H. Bar-Gera and Amos Luzon 2007 Non-unique Solutions of User-Equilibrium Assignments and their Practical Implications, Paper presented at the 86 th 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 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 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