1. Application of Accelerated User Equilibrium Traffic Assignments Howard Slavin Jonathan Brandon Andres Rabinowicz Srinivasan Sundaram Caliper Corporation May 2009

2. Traffic Assignment Convergence Most traffic assignments not sufficiently converged and give semi-random results At low convergence-Counter-intuitive or error-prone forecasts and noise all over the network for even minor local changes More rapid convergence is now readily available and very helpful in modeling. Congested speeds are key model inputs as well as a primary benefit measure.

3. 11 th TRB Transportation Planning Applications Conference, Daytona Beach, FL Empirical Testing of Faster Algorithms & Convergence Impacts Examination of algorithmic sales claims from the transportation science literature Speed enhancements through distributed processing and multi-threading Benefits of tighter convergence Use of more realistic and appropriate test cases

4. Two approaches now proven for faster traffic assignment convergence Multi-threaded (and/or distributed) FW Dial’s Algorithm B (“OUE” in TransCAD) B is a significant innovation Both improvements provided in TransCAD 5

5. Test Case Well-calibrated regional model for Washington DC that Caliper developed for MNCPPC-Prince George’s County 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%, Close match to ground counts. 80 – 170 Assignment iterations and 4 feedback loops HCM planning BPR coefficients that vary by road class Subsequent more accurate traffic assignments performed Primary test computer-3 year old 3GHz dual Xeons

6. Washington Regional Network Nodes20343 Links57374 OD Pairs 6365529 Trips2977171 Extent92 x 109 mi

7. Cold Start Convergence Rates PM Assignment

8. Cold Start Convergence Times (Min) PM Assignment Gap FW – 4 core Woodcrest FW -8 core Ocho OUE – 4 core Woodcrest OUE – 8 core Ocho 1.00E-028.34.316.415.5 1.00E-0328.51533.729.6 1.00E-04108.156.160.453.5 1.00E-05715.1369.793.782.8 1.00E-06191.5172.5 1.00E-07431.2406.7

9. Warm Start Convergence with Random Trip Table Perturbations RG=10 -5 Time to converge Cold Start01:28:02 +/-5% perturbation run 100:08:51 +/-5% perturbation run 200:08:53 +/-5% perturbation run 300:08:45 +/-10% perturbation run 100:11:10 +/-10% perturbation run 200:11:18 +/-10% perturbation run 300:10:00

10. Cold Start and Warm Start Convergence Rate Comparison

11. Feedback loop run times with OUE assignment (0.001 RG) Model StepsLoop 1Loop 2Loop 3Loop 4 All other Steps25 min16 min AM Assn18 min5 min6 min5 min PM Assn34 min8 min6 min5 min MD Assn20 min7 min6 min5 min Loop Time1 hr 37 min36 min34 min31 min

12. Comparison of OUE and FW solutions at the same Relative Gap GAP%RMSE between UE and OUE Max Link Flow Difference Objective Function Value UE Objective Function Value OUE 0.017.871663.5552,329,996.952,196,755.6 0.0012.95 776.1551,959,104.151,946,295.8 0.00010.90 378.351,922,622.251,916,312.2 0.000010.36 225.951,918,464.751,914,194.1

13. Model run times with OUE assignment (0.0001 RG) Model StepsLoop 1Loop 2Loop 3Loop 4 All other Steps25 min16 min AM Assn31 min6 min 5 min PM Assn1 hr 1 min8 min 7 min6 min MD Assn35 min9 min 6 min Loop Time2 hr 32 min39 min 34 min33 min

14. How much error is there in the link flows in an unconverged assignment? Easy to quantify with these tools Using the lens of OUE, we compare less converged solutions with more highly converged ones.

15. Average and Maximum Link Flow Differences between the OUE equilibrium solution and the solutions at lower relative gaps

16. Differences from the equilibrium solution (RG 10 -15 ) Gap Number of Links with Abs_Flow_Diff>200Avg Abs DiffMax Abs DiffRMSE % 1e-2 6,339106.31,879152.6 1e-3 96035.71,14860.6 1e-4 4910.346519.2 1e-5 13.12787.3 1e-6 00.7922.3 1e-7 00.13260.6 1e-8 00.01430.07 1e-9 00.001450.310.007 1e-10 00.0001480.0330.001

17. Flow Differences of OUE assignments at different relative gaps with the equilibrium OUE solution computed to a RG of 10-15

18. Convergence Levels & Project Impacts Three Examples Examined An Irrelevant network change-doubling the capacity of 2 links in rural VA New MD-5 and Beltway Interchange- Addition of a flyover ramp in PG County Woodrow Wilson Bridge Improvement- from 6 to 10 lanes.

19. Links with flow differences greater than 200 vehicles – Irrelevant Change Example

20. Links with flow changes greater than 200 vehicles – Interchange Project

21. Comparison of Base and Scenario – Interchange Project GapNumber of Links with Abs_Flow_Diff > 200 VHT Base Case VHT Scenario VHT Saving (Veh-Hrs) 1e-21621,105,7261,106,134-408 1e-3561,091,1531,091,247-94 1e-4441,090,1361,090,116+20 1e-5451,090,0901,090,088+2 1e-6471,090,0841,090,079+5 1e-7451,090,0871,090,085+2 1e-8451,090,0891,090,086+3

23. Links with flow differences greater than 200 vehicles – Bridge Project

24. Comparison of Base and Scenario – Bridge Project GapNumber of Links with Abs_Flow_Diff > 200 VHT Base Case VHT Scenario VHT Saving (Veh-Hrs) 1e-213631,105,7261,092,979+ 12,747 1e-311261,091,1531,079,974+ 11,179 1e-411971,090,1361,078,699+ 11,437 1e-512651,090,0901,078,631+ 11,459 1e-612601,090,0841,078,686+ 11,398 1e-712591,090,0871,078,688+ 11,399 1e-812601,090,0891,078,689+ 11,400

25. Other Findings Benefits estimated from FW were similar Our real problem was much tougher computationally than problems reported in the literature In these examples, a relative gap of 10 -4 seems sufficient for impact analysis. Convergence levels should be tested for other assignment models

26. Conclusions Orders of magnitude greater convergence can be achieved with low computing times Greater convergence can reduce errors in models and estimated project impacts There is little risk in taking advantage of these developments