Periodic Dynamic Traffic Assignment Problem

Transcription

1 Periodic Dynamic Traffic Assignment Problem Artyom Nahapetyan Center for Applied Optimization, ISE Department University of Florida Joint work with: Siriphong Lawphongpanich and Donald W. Hearn Periodic Dynamic Traffic Assignment Problem p. 1/?

2 Outline Dynamic traffic assignment problem. Traditional optimal control formulation. Basics of the periodic DTA. Continuous formulation of PDTA. Discrete formulation of PDTA. Approximation procedure. Computational results. Periodic Dynamic Traffic Assignment Problem p. 2/?

3 DTA Problem Data: Network - G(A, N). Set of OD pairs - C. Travel time - η a ( ). Demand - h k (t). Problem: Find traffic flows which minimize the total travel time of all vehicles during time period [0, T]. Periodic Dynamic Traffic Assignment Problem p. 3/?

4 OC Formulation x p a(t) - number of cars on arc a in path p at time t, u p a(t) - inflow into arc a in path p at time t, v p a(t) - outflow from arc a in path p at time t. Assumption: If a car enters an arc a at time t then it must leave the arc at t + η a (x a (t)). Typical constraints: ẋ p a(t) = u p a(t) v p a(t), u p a(t) = v p a(t + η a (x a (t))), v p (i,j) (t) = up (j,k) (t) Periodic Dynamic Traffic Assignment Problem p. 4/?

5 OC Formulation Disadvantage: Typically OC formulation requires the network to be empty at time 0 and T. If x a (0) > 0 then the times that these cars enter arc a unknown, thus the time that these cars leave the arc cannot be computed. Periodic Dynamic Traffic Assignment Problem p. 5/?

6 OC Formulation It is not necessarily valid to ignore x a (0) even when it is small. Periodic Dynamic Traffic Assignment Problem p. 6/?

7 Periodic DTA problem How to model DTA problem with positive x a (0) and x a (T)? versus Periodic Dynamic Traffic Assignment Problem p. 7/?

8 Periodic DTA problem How to model DTA problem with positive x a (0) and x a (T)? versus Periodic Dynamic Traffic Assignment Problem p. 7/?

9 Periodic DTA problem How to model DTA problem with positive x a (0) and x a (T)? versus Periodic Dynamic Traffic Assignment Problem p. 7/?

10 Set of Enter-Leave Times for Computing v a (t) L a (t) = {θ [0, T) θ + η a (x a (θ)) = t} {θ [0, T) θ + η a (x a (θ)) T = t} Periodic Dynamic Traffic Assignment Problem p. 8/?

11 Set of Enter-Remain Times for Computing x a (t) R a (t) = {ω [0, t) ω + η a (x a (ω)) t} {ω [t, T) ω + η a (x a (ω)) T t} Periodic Dynamic Traffic Assignment Problem p. 9/?

12 Periodic DTA problem (PDTA) In the model, time 0 and T are the same, planing horizon is [0, T), if an event occurs at time t, the same event must occur at the time t + αt, for all positive integer α, h k (0) = h k (T), and 0 < η a ( ) < T. Periodic Dynamic Traffic Assignment Problem p. 10/?

13 Periodic DTA Problem T min u,v 0 s.t. ([ ] a A p P δp au p a(t) ) η a (x a (t)) x p a(t) = R a (t) up a(z)dz valid (t, a, p) v p a(t) = θ L a (t) up a(θ) valid (t, a, p) v p (i,j) (t) = up (j,q)(t) valid (t, a, p) x a (t) = p P δp ax p a(t) valid (t, a, p) h k (t) = p P k δ p a 1 u p a 1 (t) valid (t, a, p) u p a(t) 0 valid (t, a, p) dt Periodic Dynamic Traffic Assignment Problem p. 11/?

14 PDTA: Theoretical Results 1. Any solution (u p a(t), v p a(t), x p a(t)) feasible to PDTA satisfies the periodicity condition. 2. Under mild condition, PDTA has a solution. Let p k P k and G( N, Â) be a subgraph of G(N, A) induced by the set of paths Φ = {p k : k C}. If there is a set Φ s.t. G( N, Â) is acyclic then PDTA has a feasible solution. Periodic Dynamic Traffic Assignment Problem p. 12/?

15 Discrete PDTA 1. Replace [0, T) by a set of discrete points {0, 1, 2,..., T 1}. 2. Assume that cars enter or leave the arc at discrete points {0, 1, 2,..., T 1}. 3. Assign h m k = m+1 m h k (z)dz. 4. Construct time-expanded network. Periodic Dynamic Traffic Assignment Problem p. 13/?

16 Time-Expanded (TE) Network Periodic Dynamic Traffic Assignment Problem p. 14/?

17 Discrete PDTA: Variables Arc flow on TE network: y j at = k C ykj at { z j 1 if the arc is used at = 0 o/w x a = Q a y a Periodic Dynamic Traffic Assignment Problem p. 15/?

18 Discrete PDTA (D-PDTA): Formulation s.t. min y,z t a A [ 4 j=2 yj at ] η a (x t a) Wy k = d k y j at = k C ykj at x a = Q a y a 4 j=2 (j 1)zj at < η a (x t a) 4 j=2 jzj at 4 j=2 zj at = 1 y j at Mz j at, y kj at 0, z j at {0, 1} Periodic Dynamic Traffic Assignment Problem p. 16/?

19 Discrete PDTA: Linearization Objective function t a A [ 4 j=2 yj at ] η a (x t a) t = t a A [ 4 j=2 yj at ] [ 4 a A 4 j=2 jyj at = c T y j=2 jzj at ] Periodic Dynamic Traffic Assignment Problem p. 17/?

20 Discrete PDTA: Linearization Constraint 4 j=2 (j 1)zj at < η a (x t a) 4 j=2 jzj at Denote τ j a = η 1 a (j), j 1 and τ 1 a = 0 4 j=2 τj 1 a z j at < x t a 4 j=2 τj az j at Periodic Dynamic Traffic Assignment Problem p. 18/?

21 Linear Approx. of PDTA (LA-PDTA) min y,z c T y s.t. Wy k = d k y j at = k C ykj at x a = Q a y a 4 j=2 τj 1 a z j at x t a 4 j=2 τj az j at 4 j=2 zj at = 1 y j at Mz j at, y kj at 0, z j at {0, 1} Periodic Dynamic Traffic Assignment Problem p. 19/?

22 Approximation Procedure Let (ŷ, ẑ) solves LA-PDTA y solves D-PDTA with z = ẑ The approximate solution is (y, ẑ). Periodic Dynamic Traffic Assignment Problem p. 20/?

23 Convergence Lets [0, T) divide into N intervals with equal length. (y (N), ẑ(n)) is the solution of the approximation procedure. (ȳ, z) is the solution of the continuous PDTA. Then lim N (y (N), ẑ(n)) = (ȳ, z) Periodic Dynamic Traffic Assignment Problem p. 21/?

24 Computational Results: Data [0, T) = [0, 10) Travel time functions: η a (x a (t)) = x a (t) η a (x a (t)) = x 2 a(t): Periodic Dynamic Traffic Assignment Problem p. 22/?

25 Computational Results: Data Three travel demand functions for the interval [0, 10): Total demands: 600, 800, and Periodic Dynamic Traffic Assignment Problem p. 23/?

26 Computational Results: Solvers All problems were solved on NEOS server using the following solvers D-PDTA - SBB solver LA-PDTA - Xpress-MP solver D-PDTA with z = ẑ - CONOPT Periodic Dynamic Traffic Assignment Problem p. 24/?

27 Computational Results: Linear Travel Time CPU time. Demand D-PDTA LA-PDTA Ratio Solution quality of LA-PDTA Demand D-PDTA LA-PDTA Diff (%) Periodic Dynamic Traffic Assignment Problem p. 25/?

28 Computational Results: Quadratic Travel Time CPU time. Demand D-PDTA LA-PDTA Ratio Solution quality of LA-PDTA Demand D-PDTA LA-PDTA Diff (%) Periodic Dynamic Traffic Assignment Problem p. 26/?

29 Conclusion Periodic Dynamic Traffic Assignment Problem. Does not require the network to be empty at time 0 and T. Discrete Dynamic Traffic Assignment Problem Constructed based on time-expanded network. It is Mixed Integer NLP. Linear Approximation Procedure Can be solved by MIP and NLP solvers. lim N (y (N),ẑ(N)) = (ȳ, z). Computational results. Periodic Dynamic Traffic Assignment Problem p. 27/?

30 Questions? Periodic Dynamic Traffic Assignment Problem p. 28/?