1. The Impact of Convergence Criteria on Equilibrium Assignment Yongqiang Wu, Huiwei Shen, and Terry Corkery Florida Department of Transportation 11 th Conference on Transportation Applications May 7, 2007

2. Florida Department of Transportation 2 Content  Background  Methodology  Preliminary Findings  Conclusions and Recommendations

3. Florida Department of Transportation 3 Motives for the Study  New Florida Standard Model framework  Increasing computational efforts with more complicated models  FTA New Starts analysis guidelines  Implications of link flow stability on different types of transportation engineering and planning projects

4. Florida Department of Transportation 4 Objectives of the Study  Evaluate how link volumes vary with closing criteria for user equilibrium traffic assignment  Provide some empirical evidence on how to choose closing criteria for assignment  Establish standards and guidelines for new Florida Standard Model

5. Florida Department of Transportation 5 Study Methodology  Use Florida Statewide Model (FLSWM)  4,059 zones  98,098 directional links  Run the model on two computers with different configurations  Test two scenarios with different congestion levels  Base Year 2000 – validation  Future Year 2030 – forecast

6. Florida Department of Transportation 6 Study Methodology (Continued)  Perform traffic assignment using different closing criteria Number of Iterations – 1,5,10,15,20,25,30,35,40,45,50, 75,100,150, 200, and 300.  Analyze assignment results  Compare volumes with traffic counts  Compare volumes from iteration to iteration

7. Florida Department of Transportation 7 Florida Statewide Model Structure Passenger Model Trip Generation Trip Distribution Auto Occupancy Freight Model Trip Generation Trip Distribution Mode Choice SE Data/Network Joint Assignment

8. Florida Department of Transportation 8 User Equilibrium (UE) Assignment  Wardrop's Condition For a given origin-destination pair, travel costs (travel times) are equal on all paths actually utilized, and are less than or equal to the travel time on any other paths. No traveler can improve his or her travel time by unilaterally switching to a different route.

9. Florida Department of Transportation 9 User Equilibrium Assignment  Franke-Wolfe Algorithm  A series of all-or-nothing assignments are performed  Link flows are combined mathematically using weights derived from a line search to minimize the objective function  Travel times are recalculated using the combined assignment.  The process is repeated for a specified number of iterations or until certain stopping criterion is met.

10. Florida Department of Transportation 10 Gap and Relative Gap = the current iteration Where = link volume from an AON assignment based on CE n-1 = cost based on equilibrium volume VE n = equilibrium weighted volume for iteration n = summation over all network links  n n n l VA CE VE n   l 11 11       nn l nn-1 l nn CEVE CEVECEVA RGAP 11 11       l nn l nn l nn CEVE CEVECEVE GAP

11. Florida Department of Transportation 11 Preliminary Findings  Computational efforts  Computer Specifications  Congestion level  Gap and relative gap  Validation - aggregate statistics  V/C, VMT, and VHT  Root Mean Square Error (RMSE)  Forecast - link volume Changes  Percentages  Absolute differences

12. Florida Department of Transportation 12 Computer Configurations  Computer 1  Dell Mobile Station M90  CPU – Intel® Core TM 2 T7600 2.33 GHz Duo Mobile Processors  RAM – 4GB  Computer 2  Dell Desktop Optiplex GX620  CPU – Intel® Pentium®4 3.20GHz Duo Processors with Hyper-Treading Technology  RAM – 2GB

13. Florida Department of Transportation 13 CPU Time vs. No. of Iterations

14. Florida Department of Transportation 14 GAP/RGAP vs. No. of Iterations

15. Florida Department of Transportation 15 Observation 1  Computer run time is proportional to number of iterations.  Level of congestion has direct impact on computer run time.  To achieve the same level of gap/relative gap, more iterations are needed for more congested conditions  Relative gap seems to be higher and more stable than gap.

16. Florida Department of Transportation 16 Modeled vs. Observed - Volumes

17. Florida Department of Transportation 17 Modeled vs. Observed - VMT

18. Florida Department of Transportation 18 Vehicle Hours Traveled - VHT

19. Florida Department of Transportation 19 Observation 2  Aggregate statistics used for validation purposes, such as V/C, VMT, and VHT, tend to stabilize with less computational efforts (10-15 iterations).  The overall good match between modeled volumes and observed traffic counts might obscure the large differences in individual links.

20. Florida Department of Transportation 20 Link Volumes vs. Counts (RMSE)

21. Florida Department of Transportation 21 RMSE by Volume Groups (25 ITER)

22. Florida Department of Transportation 22 Y2000 Link Volume Changes Iterations Percentage of Links with Volume Changes within, % 0→10%10→20%20→30%30→40%40→50%50→100%100%→ 5 → 1068.215.07.53.61.92.81.1 10 → 1586.98.32.30.91.40.1 15 → 2092.84.51.10.41.20.0 20 → 2595.92.30.51.30.0 25 → 3097.01.30.31.30.0 30 → 3597.60.91.40.0 35 → 4097.70.81.40.0 40 → 4598.00.61.30.0 45 → 5098.20.51.30.0 50 → 7597.60.80.30.1 1.10.0 75 → 10098.30.40.1 1.10.0 100 → 15098.30.40.10.0 1.10.0 150 → 20098.60.20.11.20.0 200 → 30098.60.20.10.01.10.0

23. Florida Department of Transportation 23 Y2000 Max & Avg Volume Changes Iterations Percent Difference, %Absolute Difference, vpd MaximumAverageMaximumAverage 5 → 10 500.00+17.1872,3941,310 10 → 15 500.00+5.7517,095497 15 → 20 500.00+3.719,069301 20 → 25 500.00+2.5110,080190 25 → 30 500.00+1.945,626135 30 → 35 500.00+1.595,888103 35 → 40 394.641.434,53993 40 → 45 500.00+1.174,29773 45 → 50 298.120.953,48955 50 → 75 500.00+1.874,21071 75 → 100 166.751.143,25238 100 → 150 633.911.223,12531 150 → 200 193.250.771,27316 200 → 300 99.990.851,52614

24. Florida Department of Transportation 24 Y2030 Link Volume Changes Iterations Percentage of Links with Volume Changes within, % 0→10%10→20%20→30%30→40%40→50%50→100%100%→ 5 → 1053.118.510.55.83.75.52.9 10 → 1575.314.94.82.12.00.50.4 15 → 2086.58.82.20.91.30.20.1 20 → 2591.85.31.21.50.1 0.0 25 → 3095.02.91.90.10.0 30 → 3596.12.21.70.0 35 → 4097.01.51.40.0 40 → 4597.31.31.40.0 45 → 5097.62.30.10.0 50 → 7595.82.10.50.2 1.20.0 75 → 10097.50.90.2 1.20.0 100 → 15097.40.90.20.1 1.20.0 150 → 20098.20.40.1 1.20.0 200 → 30098.20.40.1 0.01.20.0

25. Florida Department of Transportation 25 Y2030 Max & Avg Volume Changes Iterations Percent Difference, %Absolute Difference, vpd MaximumAverageMaximumAverage 5 → 10 500.00+29.4092,9083,430 10 → 15 500.00+9.9344,4301,426 15 → 20 500.00+5.8820,978813 20 → 25 500.00+4.1018,304540 25 → 30 500.00+2.9112,778352 30 → 35 500.00+2.4011,842275 35 → 40 320.161.888,498215 40 → 45 363.301.7712,164193 45 → 50 331.981.537,722167 50 → 75 212.132.717,414225 75 → 100 167.141.704,210122 100 → 150 276.321.794,269106 150 → 200 158.451.162,64757 200 → 300 52.041.252,35349

26. Florida Department of Transportation 26 Links with Highest Percent Changes

27. Florida Department of Transportation 27 Links with Highest Volume Changes

28. Florida Department of Transportation 28 Observation 3  Large variations in link volumes do exist  Variations might cause a lane change in some cases  Largest percentage changes occur on low volume links located near centroid connectors  Largest volume changes occur on mid- to high-volume links located in congested areas

29. Florida Department of Transportation 29 Conclusions and Recommendations  Use gap/relative gap as stopping criterion and set the values sufficiently small  Set number of iterations large enough  Use number of iterations only when  Checking grammatical or functional errors  Preventing model being run indefintely  Consider different stopping criterion for different applications

30. Florida Department of Transportation 30 Conclusions and Recommendations  Aggregate statistics can obscure the large variations in individual links  Large percentage changes occur to links with low volume links and located near centroid connectors  Large volume changes happen to links with mid- to high volume links located in congested areas  More studies considering other factors using different models are needed.

31. Florida Department of Transportation 31 Questions?