Hyunsoo Noh and Mark Hickman 2011 INFORMS Annual Meeting 11 / 14 / 2011 The University of Arizona

1. A Logit-based Transit Assignment Using Gradient Projection with the Priority of Boarding on a Transit Schedule Network Hyunsoo Noh and Mark Hickman 2011 INFORMS Annual Meeting 11 / 14 / 2011 The University of Arizona

2. Contents Background • UE and SUE problem • Path-Based Assignment Using Gradient Projection Proposed Model • Transit Behavior : Priority and Congestion • UE with Priority on Congested Transit Schedule Network • SUE with Priority on Congested Transit Schedule Network

3. Background

4. User Equilibrium • Beckman (1956) introduced the formulation for solving traffic UE problem by Wardrop (1952) • A representative solution method is Frank-Wolfe (1956) algorithm.

5. Stochastic User Equilibrium • Fisk (1981) introduced a path-based stochastic model equivalent to Logit model based on the gravity model of Evans (1973) • For the solution algorithm, fixed demand incremental assignment algorithm (a kind of MSA) was introduced.

6. Path-based Assignment • Newton Approximation • Iterative Solution Update • Matrix from

7. Deterministic Path-based Method • Restate the Beckman’s objective and constraints for non-negative non-shortest paths based on the Goldstein-Levitin-Poljak gradient projection by Bertsekas (1976) (Jayakrishnan et al., 1999) • Model • Flow update: Hessian approximation (diagonal)

8. Path-based Assignment Algorithm • Step 0: (Initialization) - All-or-Nothing assignment and initialize a set of paths K • Step 1: (update) - Update first derivative length d of all paths in K • Step 2: (direction) - Search direction and set d’ for the direction - If direction is different from an alternative in K, add it in K • Step 3: (move) - Flow update by gradient projection model • Step 4: (convergence test) - If converged, stop - Else, go to Step 1

9. Stochastic Path-based Method • Bekhor and Toledo (2005) introduced a stochastic version of path-based model • Hessian (diagonal)

10. Proposed Model

11. Priority on a Congested Transit Schedule Network • FIFO on board and waiting (Poon et al., 2004; Hamdouch et al. 2008) • FIFO 1: On vehicle, on-board passengers vs. boarding passengers • FIFO 2: At stop, early arrival passengers vs. late arrival passengers • On-board passenger < early arrival passenger < late arrival passengers • Priority to access link e4 : e2 < e1 < e3 t1arr < t2arr < t3arr e1 t1arr e4 e2 t2arr r i s t3arr e3

12. Capacity Constraint: ceaeb • Congestion level is determined by the residual capacity of forward link • Soft capacity form but working hard capacity

13. With capacity constraint • Objective • Lagrangian multiplier

14. Deterministic Gradient Projection Method (DGPM) • Formulation • Hessian (diagonal)

15. Proposed DGPM Algorithm • Step 0 (initialization) - Search the least cost path - Load flows on the searched path • Step 1 (Cost Update) - If sub-loop (from Step 2), then flows are fixed - Else (from Step 3), then flows are changed • Step 2 (Diagonalization) - Update the cost path - Step 2.1 (Direction) Search the least cost path - Step 2.2 (Move) Update new flows - Step 2.3 (Convergence Test) - If satisfied, then go to Step 3; Else then go to Step 1 • Step 3 (Convergence Test) - If Satisfied, then Stop; Else then go to Step 1

16. DGPM Example • Priority is on e2 → e3 • What is the estimated UE solution? • What is the optimal objective cost?

17. Result • α = 3.0

18. Stochastic Gradient Projection Method (SGPM) • Objective function for capacity constraint • Stochastic path cost (Chen, 1999) • If flow f is small enough? E.g., almost 0 Solution:

19. SGPM model • Formulation • Hessian (diagonal)

20. Proposed SGPM Algorithm • Same to DGPM except path cost: Entropy term is included

21. SGPM Example • Priority is on e2 → e3 • What is the estimated SUE solution? • What is the optimal objective cost?

22. Result • α = 1.0

23. Conclusion • Stochastic path-based assignment is developed using gradient projection method with priority, including deterministic model. • As the proposed algorithm, diagonalization methodology is utilized. • Ongoing Work • Computation efficiency will be considered to get the solution including accuracy • Stochastic solution on the capacity constraint will be analyzed in detail. • Large network will be tested.

24. References • Beckman MJ, McGuire CB, and Winston CB (1956) Studies in the Economics of Transportation. Yale University Press, Connecticut. • Bekhor S, Toledo T (2005) Investigating path-based solution algorithms to the stochastic user equilibrium problem. Transportation Research Part B: Methodological 39(3):279-295. • Bertsekas D (1976) On the Goldstein-Levitin-Polyak Gradient Projection Method. Automatic Control, IEEE Transactions 21(2):174-184. • Chen H- (1999) Dynamic travel choice model: A variational inequality approach. Springer. • Evans SP (1973) A relationship between the gravity model for trip distribution and the transportation problem in linear programming. Transportation Research 7(1):39-61. • Fisk C (1980) Some developments in equilibrium traffic assignment. Transportation Research Part B: Methodological 14(3):243-255. • Frank M and Wolfe P (1956). An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly 3(1-2):95-110. • Hamdouch Y, Lawphongpanich S (2008) Schedule-based transit assignment model with travel strategies and capacity constraints. Transportation Research Part B: Methodological 42(7-8):663-684. • Jayakrishnan R, Tsai WK, Prashker JN, Rajadhyaksha S (1994) Faster path-based algorithm for traffic assignment. Transportation Research Record: Journal of the Transportation Research Board 1443:75-83. • Poon MH, Wong SC, Tong CO (2004) A dynamic schedule-based model for congested transit networks. Transportation Research Part B: Methodological 38(4):343-368. • Wardrop JG (1952). Some theoretical aspects of road traffic research, Proceedings, Institute of Civil Engineers, PART II(1): 325–378.

25. ? Thank you. • - Contact Information - • Hyunsoo Noh (hsnoh@email.arizona.edu) • Mark Hickman (mhickman@email.arizona.edu) • UATRU(University of Arizona Transit Research Unit) website (www.transit.arizona.edu)