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  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

Background

 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

 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

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

 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

 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

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

Proposed Model

 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

 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

 With capacity constraint Objective Lagrangian multiplier 13

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

 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

 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∞

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

 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:

 SGPM model Formulation Hessian (diagonal) 19

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

 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∞

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

 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

 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–

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