Allocation of Transportation Resources. Presented by: Anteneh Yohannes

Transcription

1 Allocation of Transportation Resources Presented by: Anteneh Yohannes

2 Problem State DOTs must allocate a budget to given projects Budget is often limited Social Welfare Benefits Different Viewpoints (Two Players) Planners Road Users

3 Purpose of Project Produce a method of prioritizing projects Consider Total System Travel Time (TSTT) performance measure Allocate the budget to priority links Provide State DOTs with a decision making tool

4 Methodology Bi-Level Optimization Planners Upper Level Problem (ULP) Minimize Total System Travel Time (TSTT) Users Lower Level Problem (LLP) Traffic Assignment (UE)

5 Data Required Number of Links in the network Capacity Length Free Flow Travel Time Alpha and Beta parameters Connecting Nodes O/D Matrix Budget

6 Formulation Upper Level problem (ULP) Objective Function : Minimize TSTT = a x a t a (x a, y a ) Subject to: a g a (y a ) B y a 0: a A Where: TSTT : Total System Travel Time x a : Flow for link a y a : Capacity expansion for link a (nonnegative real value) t a : Travel time for link a t a (x a, y a ) : Travel cost on link a as a function of flow and capacity expansion g a y a : improvement cost function for link a B : Budget (nonnegative real value)

7 Formulation Lower Level problem (LLP) Minimize TT = a A x a t a (x a, y a )dx 0 (11) Subject to: x a = f k ij q ij = k k ij f k ij (i,j) IJ k K ij (i, j) IJ ij δ ak f ij k, a A (12) (13) 0, k k ij, (i, j) IJ, (14) q ij 0, (i, j) IJ (15)

8 Notations C a : The capacity for link a r f ij : Flow on path r, connecting each Origin-Destination (O-D) pair (i-j) q ij : Demand between each Origin-Destination (O-D) pair (i-j) t a : Travel time for link a t a (x a, y a ) : Travel cost on link a as a function of flow and capacity expansion x a : Flow for link a α a : Constant, varying by facility type (BPR function) β a : Constant, varying by facility type (BPR function) r δ a,ij : binary variable 0,1 {1,if link a A is on path k k^ij:0,otherwise} t o : Free flow time on link a y a : Capacity expansion for link a (nonnegative real value)

9 Flow Chart Initial Traffic Assignment Base traffic flow X Network planner s problem Minimize TSTT = (x a t a (x a, y a )) a A Design constraints No Microsoft Solver Foundation Generalized Reduced Gradient (GRG2) algorithm Did users stop responding to improvements? Y Yes X' User equilibrium problem Minimize TT x a = t a (x a, y a )dx a A 0 Frank Wolfe Algorithm (FW) Definitional constraint Demand conservation constraint Non-negativity constraint Stop

10 LLP Frank Wolfe Algorithm (FW) SOURCE: YOSEF SHEFFI, Massachusetts Institute of Technology Step 0: Initialization Perform all-or-nothing assignment based on ta =ta (0) a. A new flow vector {xa} will be generated. Set counter n = 1. Step 1: Update Set ta=ta (xa) a Step 2: Finding Direction Perform all-or-nothing assignment based on {ta}. A new auxiliary flow vector {x a} will be generated. Step 3: Line search Find αn (0 α 1) that solves equation : x a + (x n a x n a ) 0 min z x = t a w dw Step 4: Move x a n+1 = x a n + n x a n x a n, a Step 5: Convergence test a a x a n+1 x a n 2 a x a n k

11 Test Network 1 t a x a, y a = A a + B a x a C a + y a 4 TSTT y = (t a x a, y a. x a + 1.5d a y2 a a Arc a A a B a C a d a

12 Comparison of Results Hooke- Jeeves (H-J) EDO GA Current Study Case MINOS 1 Demand =100 y y y y y Z Demand =150 y y y y y Z Demand =200 y y y y y Z Demand =300 y y y y y Z GA H-J EDO MINOS Names of heuristics Genetic Algorithm Hooke-Jeeves algorithm Equilibrium Decomposed Optimization (Bolzano search) Modular In-core Non linear System Sources Mathew (2009) Abdulaal and LeBlanc (1979) Suwansirikul et al. (1987) Suwansirikul et al. (1987)

13 Test Network 2 O/D t a x a, y a = A a + B a x a C a + y a 4 Source: Chiou et al (2005) TSTT y = (t a x a, y a. x a + θy a a Link a Aa Ba Ca θa

14 Comparison of Results Case SAB GP CG QNEW PT Current Study y y y y y Zy GP CG QNEW PT SAB Gradient Projection method Conjugate Gradient projection method Quasi-NEWton projection method PARTAN version of gradient projection method Sensitivity Analysis Based Source: Chiou et al (2005)

15 Comparison of Results Comparison of results on 9-node grid network with scaling factors Scalar SAB GP CG QNEW PT Current Study GP CG QNEW PT SAB Gradient Projection method Conjugate Gradient projection method Quasi-NEWton projection method PARTAN version of gradient projection method Sensitivity Analysis Based

16 Test Network 3 O/D

17 Comparison of Results Case 1 Demand (1,6)=5.0 Demand (6,1)=10.0 Names of heuristics Sources y 1 y 2 MINOS H-J EDO IOA Current Study y y 4 y 5 y y 7 y 8 y 9 IOA H-J EDO Iterative Optimization- Assignment algorithm Hooke-Jeeves algorithm Equilibrium Decomposed Optimization (Bolzano search) Allsop (1974) Abdulaal and LeBlanc (1979) Suwansirikul et al. (1987) y 10 y 11 y 12 MINOS Modular In-core Non linear System Suwansirikul et al. (1987) y 13 y 14 y y Z

18 Test Network 4 candidate links are marked by red arrow Sioux Falls Network

19 Comparison of Results Comparison of Results for Sioux Falls Network Case H-J H-J EDO SA SAB GP CG QNew PT GA Current Study y y y y y y y y y y Zy GP CG QNEW PT Gradient Projection method Conjugate Gradient projection method Quasi-NEWton projection method PARTAN version of gradient projection method

20 Comparison of Results Comparison of Results for Sioux Falls Network for different demand level Scalar SAB GP CG QNew PT EDO IOA GA Current Study FW Itr FW Itr FW Itr FW Itr FW Itr GA GP CG QNEW PT Genetic Algorithm Gradient Projection method Conjugate Gradient projection method Quasi-NEWton projection method PARTAN version of gradient projection method

21 Test Network 5 O/D t a x a, y a = A a + B a x a C a + y a TSTT y = (t a x a, y a. x a + θ y a a Linka Aa Ba Ka θa Linka Aa Ba Ka θa

22 Comparison of Results Comparison of results on 25-node grid network with scaling factors Scalar SAB GP CG QNEW PT Current Study GP CG QNEW PT Gradient Projection method Conjugate Gradient projection method Quasi-NEWton projection method PARTAN version of gradient projection method

23 Questions