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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

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2 ContentsBackground 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

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Background

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

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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. 5

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

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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) 7

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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 8

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

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Proposed Model

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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 e 4 : e 2 < e 1 < e 3 11 rsi e1e1 e2e2 e3e3 e4e4 t 1 arr t 2 arr t 3 arr t 1 arr < t 2 arr < t 3 arr

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Capacity Constraint: c e a e b Congestion level is determined by the residual capacity of forward link Soft capacity form but working hard capacity 12

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With capacity constraint Objective Lagrangian multiplier 13

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

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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 15

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DGPM Example Priority is on e 2 → e 3 What is the estimated UE solution? What is the optimal objective cost? 16 linkcostcapacity e13∞ e24∞ e3310 e49∞

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Result α = pathFlow(α = 3.0)flow(α = 7.0) e e1,e300 e2,e

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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 18 Solution:

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SGPM model Formulation Hessian (diagonal) 19

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

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SGPM Example Priority is on e 2 → e 3 What is the estimated SUE solution? What is the optimal objective cost? 21 linkcostcapacity e13∞ e24∞ e3310 e49∞

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Result α = pathflow*new flow*Logit e e1,e e2,e

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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. 23

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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): Bertsekas D (1976) On the Goldstein-Levitin-Polyak Gradient Projection Method. Automatic Control, IEEE Transactions 21(2): 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): Fisk C (1980) Some developments in equilibrium traffic assignment. Transportation Research Part B: Methodological 14(3): Frank M and Wolfe P (1956). An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly 3(1-2): Hamdouch Y, Lawphongpanich S (2008) Schedule-based transit assignment model with travel strategies and capacity constraints. Transportation Research Part B: Methodological 42(7-8): 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: Poon MH, Wong SC, Tong CO (2004) A dynamic schedule-based model for congested transit networks. Transportation Research Part B: Methodological 38(4): Wardrop JG (1952). Some theoretical aspects of road traffic research, Proceedings, Institute of Civil Engineers, PART II(1): 325–

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

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