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Hyunsoo Noh and Mark Hickman 2011 INFORMS Annual Meeting 11 / 14 / 2011 The University of Arizona PowerPoint Presentation
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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. Contents Background UE and SUE problem

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slide1

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

slide2
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
slide4

User Equilibrium

    • Beckman (1956) introduced the formulation for solving traffic UE problem by Wardrop (1952)
    • A representative solution method is Frank-Wolfe (1956) algorithm.
slide5

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.
slide6

Path-based Assignment

    • Newton Approximation
    • Iterative Solution Update
    • Matrix from
slide7

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)
slide8

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

slide9

Stochastic Path-based Method

    • Bekhor and Toledo (2005) introduced a stochastic version of path-based model
    • Hessian (diagonal)
slide11

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

slide12

Capacity Constraint: ceaeb

    • Congestion level is determined by the residual capacity of forward link
    • Soft capacity form but working hard capacity
slide13

With capacity constraint

    • Objective
    • Lagrangian multiplier
slide15

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

slide16

DGPM Example

    • Priority is on e2 → e3
    • What is the estimated UE solution?
    • What is the optimal objective cost?
slide17

Result

    • α = 3.0
slide18

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:

slide19

SGPM model

    • Formulation
    • Hessian (diagonal)
slide20

Proposed SGPM Algorithm

    • Same to DGPM except path cost: Entropy term is included
slide21

SGPM Example

    • Priority is on e2 → e3
    • What is the estimated SUE solution?
    • What is the optimal objective cost?
slide22

Result

    • α = 1.0
slide23

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.
slide24

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.
slide25

?

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)