New Empirical Study of Alternative Traffic Equilibrium Algorithms

1. New Empirical Study of Alternative Traffic Equilibrium Algorithms Zhong Zhou & Matthew Martimo Citilabs

2. Outline Background & Motivation New Assignment Algorithms in Cube Voyager Empirical Studies Conclusions

3. Background & Motivation

4. Traffic Assignment Problems Traffic Assignment is a process of allocating the given origin-destination (OD) trip table to the transportation network under certain rules User Equilibrium (UE) Principle: “No traveler can improve his or her travel cost by unilaterally changing routes” (Wardrop, 1952) (All of the used paths have equal and minimum travel times; all of the unused paths have equal or higher travel times)

5. Frank-Wolfe Algorithm Basic Idea: Solve a linearized subproblem to get a decent direction Find a new solution by using line search Advantage: The linearized subproblem can be efficiently solved by AON assignment Memory efficiency (only link variables need to be stored) Disadvantage: Slow convergence near the optimal point, take long time to reach high precision Zig-zagging effect that means the flow may not stable

6. Basic Idea: Solve a linearized subproblem to get a decent direction Find a new solution by using line search Frank-Wolfe Algorithm

7. High Level of Convergence is Important A Relative Gap of 0.01 % (0.0001) is required to assure that the assignment is sufficiently converged to achieve stable link flows. (Boyce, et al., 2004) Traditional FW suffers from slow converge speed to desired precision level ( e.g., relative gap < 0.0001)

8. Overview of Different Traffic Assignment Algorithms

9. New Assignment Algorithms in CUBE Voyager

10. Improved Frank-Wolfe algorithms

11. New Link-based Assignment Algorithms in Voyager Basic Idea: Conjugate FW & Bi-Conjugate FW (Daneva, 2003) • Advantages • Only one line search step has to be performed in order to find the new solution (same as FW) • At each iteration, only three (four) vectors in memory to find a new conjugate search direction

12. Good Features Able to keep consistence with existing practice Fullfunctionalityas that provided by the traditional FW assignment procedure without need to modify anything (network, input data etc.) Multiple user classes, Turning penalties, Junction Modeling Select link analysis and similar analysis Distributed computing, Etc. Maintain ‘proportionality’ very well in select link analysis based on our preliminary tests New Link-based Assignment Algorithms in Voyager (Cont.)

13. New Path-based Assignment Algorithms in Voyager Firstly introduced to transportation field by Jayakrishnan et al. (1994). Based on Goldstein-Levitin-Polyak gradient projection method formulated by Bertsekas (1976) for general nonlinear multi-commodity problem Extensively used in computer communication networks for optimal flow routing Basic Idea In contrast to FW, which finds auxiliary solutions that are vertices (extreme points) of the feasible region, GP uses a transformed objective function and makes successive moves in the direction of negative gradient, scaled by the approximation of the second derivative Hessian

14. New Path-based Assignment Algorithms in Voyager Feasibility The memory restriction for tracking the paths has been relaxed considerably in recent years due to rapid advances in computing environment Advantages Quickly converge to desired level of accuracy Unique Link Flow Solution Disadvantages Does not maintain proportionality assumptions “Funny” results on detailed inspection Select Link, Select Zone, … Turning Movements

15. New Empirical Studies

16. Testing Environments Computing Platform 64 bit Intel Platform with Vista 64 Two Xeon E5335 2GHz Quad Core Processors and 8GB of RAM Chicago Regional Network 1790 Zones 12982 Nodes 39018 Links 1429901.19 Total OD Flow

17. Convergence Analysis on the Enhanced Link-based Algorithms

19. Run Time to Reach Relative Gap 0.01 • Run Time to Reach Relative Gap 0.001 • Run Time to Reach Relative Gap 0.0001 • Run Time to Reach Relative Gap 0.00001

20. Effect of Distributed Computing • Run Time to Reach Relative Gap 0.01 • Run Time to Reach Relative Gap 0.001

21. Effect of Distributed Computing • Run Time to Reach Relative Gap 0.0001 • Run Time to Reach Relative Gap 0.00001

22. Convergence Analysis on the New Path Based Algorithm

25. Run Time to Reach Relative Gap 0.01 • Run Time to Reach Relative Gap 0.001 • Run Time to Reach Relative Gap 0.0001 • Run Time to Reach Relative Gap 0.00001

26. Select Link Analysis Alternative segments: Using North Ave Bridge L=8032-8037 && L=8037-8752 && L=8752-8753 && L=8753-6380 && L=6380-6389 && L=6389-10344 Not using North Ave Bridge L=8032-8749 && L=8749-8750 && L=8750-8751 && L=8751-8994 && L=8994-8993 && L=8993-10344

27. Conjugate FW with Relative Gap = 1e-4 OD Flow Proportion Number of OD

28. Flow Distribution

29. Proportionality under Relative Gap 1e-4

30. Conclusions

31. Conclusions Two new link-based algorithms (CFW & BiFW) have been implemented Converge faster to small relative gap than traditional FW algorithm Memory efficiency (require similar memory as FW algorithm) Consistent with existing practice Keep all available abilities as that provided by FW algorithm (select link analysis, distributed computing, etc.) Maintain ‘proportionality’ in select link analysis based on our preliminary tests

32. Conclusions (Cont.) A new path-based algorithms (GP) are introduced Converge much quickly to desired precision level than FW algorithms Loss of detail and proportionality in results More research and enhancement are undergoing, and more tests are needed Will be available soon in new release of Cube Voyager

33. Acknowledgements We would like to congratulate and thank Professor David Boyce, Hillel Bar-Gera and Yu Nie for their research and helpful discussions!