Pareto- -improving Congestion improving Congestion Pareto Pricing - - PowerPoint PPT Presentation

SLIDE 1

Pareto Pareto-

• improving Congestion

improving Congestion Pricing on Multimodal Pricing on Multimodal Transportation Networks Transportation Networks

Wu, Di Wu, Di

Civil & Coastal Engineering, Univ. of Florida Civil & Coastal Engineering, Univ. of Florida

Yin, Yin, Yafeng Yafeng

Civil & Coastal Engineering, Univ. of Florida Civil & Coastal Engineering, Univ. of Florida

Lawphongpanich Lawphongpanich, , Siriphong Siriphong ( ( Toi Toi) )

Industrial & System Engineering, Univ. of Florida Industrial & System Engineering, Univ. of Florida

CMS Annual Student Conference March 2009

SLIDE 2

Outline Outline

• Background

Background

• Pareto

Pareto-

• Improving Pricing Scheme

Improving Pricing Scheme

• Pareto

Pareto-

• Improving Pricing Problem on

Improving Pricing Problem on Multi Multi-

• modal Network

modal Network

• Mathematical Model

Mathematical Model

• Numerical Examples

Numerical Examples

• Conclusion

Conclusion

SLIDE 3

Background Background

• Congestion Pricing

Congestion Pricing

• Alleviate traffic congestion by charging tolls.

Alleviate traffic congestion by charging tolls.

• Current Practice:

Current Practice:

• London, UK

London, UK

• Singapore

Singapore

• Stockholm, Sweden

Stockholm, Sweden

• Can successfully reduce congestion, but is still

Can successfully reduce congestion, but is still facing strong objection from the general facing strong objection from the general public. public.

SLIDE 4

Example Example

2 4 1 3 10v13 50+ v12 2+ 25v34 10v24 10+ v32

There are 3.6 travelers for OD pair (1, 3)

SLIDE 5

Example (Contd.) Example (Contd.)

2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3 2 4 1 3

1.3223 2.2772 1.1685 0.8956 1.5359 1.79 1.36 0.45

User Equilibrium System Optimal Under Marginal Cost Toll Pareto-I mproving

36.00 50.00 12.28 22.78 35.06 20.64+ 20.64 51.53+ 1.54 10.89+ 0.90 31.21+ 29.21 24.32+ 24.32 22.40 51.36 18.10 10.45+ 18.51 46.75+ 0.31

Total travel time: 255.82 Total travel time: 227.11 Total travel time: 241.17 1-2-3: 71.06 1-2-4-3: 71.06 1-2-3: 101.70 1-2-4-3: 101.70 1-4-3: 101.70 1-2-3: 69.46 1-2-4-3: 69.46 1-4-3: 69.46

SLIDE 6

Pareto Pareto-

• Improving Approach

Improving Approach

• By Italian economist

By Italian economist Vilfredo Vilfredo Pareto Pareto

• make at least one individual better off

make at least one individual better off without making any other individual worse without making any other individual worse

• ff
• ff

SLIDE 7

Single Modal Pareto Single Modal Pareto-

• Improving

Improving Scheme Scheme

• Studied by

Studied by Lawphongpanich Lawphongpanich and Yin and Yin (2008) (2008)

• Pareto

Pareto-

• improving condition may be

improving condition may be relatively prevalent relatively prevalent

• Exact Pareto

Exact Pareto-

• improvement may not lead

improvement may not lead to significant travel time reduction. to significant travel time reduction.

SLIDE 8

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improvement

Improvement

• Develop a Pareto

Develop a Pareto-

• improving model

improving model for for multimodal networks. multimodal networks.

• Allow cross

Allow cross-

• subsidy of different travel

subsidy of different travel modes to encourage travelers switch to modes to encourage travelers switch to higher occupancy travel modes in order to higher occupancy travel modes in order to further increase the system efficiency. further increase the system efficiency.

SLIDE 9

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improvement

Improvement (Contd.) (Contd.)

• Three travel modes:

Three travel modes:

• Single Occupancy Vehicle

Single Occupancy Vehicle

• High Occupancy Vehicle

High Occupancy Vehicle

• Transit (transit shares highway lanes with auto modes)

Transit (transit shares highway lanes with auto modes)

• Three types of transportation facilities:

Three types of transportation facilities:

• Regular (Toll) Lanes

Regular (Toll) Lanes

• High Occupancy Toll (HOT) Lanes

High Occupancy Toll (HOT) Lanes

• Transit services (fixed frequency and capacity)

Transit services (fixed frequency and capacity)

SLIDE 10

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improvement

Improvement (Contd.) (Contd.)

• Assumptions

Assumptions

• One class of homogenous users

One class of homogenous users

• Total O

Total O-

• D demand is fixed and known.

D demand is fixed and known.

• Users

Users’ ’ decision on travel decision on travel-

• mode choosing

mode choosing follows logit model based on travel cost. follows logit model based on travel cost.

• Traffic flow distribution follows user

Traffic flow distribution follows user equilibrium condition within a chosen mode. equilibrium condition within a chosen mode.

SLIDE 11

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improvement

Improvement (Contd.) (Contd.)

• Objectives

Objectives

• The user utility will not decrease for any

The user utility will not decrease for any traveler. traveler.

• Remain in the same travel mode and the travel

Remain in the same travel mode and the travel cost will not increase. cost will not increase.

• Switch to another more preferred travel mode.

Switch to another more preferred travel mode.

• The total collected toll will be enough to cover

The total collected toll will be enough to cover all the transit subsidy. all the transit subsidy.

• The total social benefits will increase.

The total social benefits will increase.

SLIDE 12

Modeling Transit User Behavior Modeling Transit User Behavior

• Waiting time

Waiting time

• Transfers

Transfers

• Strategies

Strategies

• Boarding the first arrived vehicle within a

Boarding the first arrived vehicle within a selected subset in order to achieve shortest selected subset in order to achieve shortest expected travel time. expected travel time.

SLIDE 13

Network Structure Network Structure

• Original Network:

Original Network:

2 Line a: Line b: Line c:

Transit

1 3 Line a: Line b: Line c:

Transit

SLIDE 14

Network Structure (Contd.) Network Structure (Contd.)

• Modified Network:

Modified Network:

1 3 1a 1b 2b 2a 3a 2c 3c Regular link Transit link Embarking link Alighting link HOT link (travel time, inf, 0) (travel time, inf, 0) (travel time, 0, transit capacity) (boarding time, waiting time, inf) (alighting time, 0, inf) 2 Regular link Transit link Embarking link Alighting link HOT link (travel time, inf, 0) (travel time, inf, 0) (travel time, 0, transit capacity) (boarding time, waiting time, inf) (alighting time, 0, inf) 1 3 1a 1b 2b 2a 3a 2c 3c Regular link Transit link Embarking link Alighting link HOT link (travel time, inf, 0) (travel time, inf, 0) (travel time, 0, transit capacity) (boarding time, waiting time, inf) (alighting time, 0, inf) 2 Regular link Transit link Embarking link Alighting link HOT link (travel time, inf, 0) (travel time, inf, 0) (travel time, 0, transit capacity) (boarding time, waiting time, inf) (alighting time, 0, inf)

SLIDE 15

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improving

Improving Model Model

, , , , , , ,

1 max ln exp( ) . . Tolled User Equilibrium Condition (1 10), , , , , 0, ( ) , , , , ,

w m w m m j i w m w m w m U w m w m l l E m w m l l w m w m m l w l H S S l l H l

E t w W m M x w W m D x s t l l L L M θ ρ τ ρ β ρ τ θ τ τ τ − − − − ≤ ∀ ∈ ⎛ ⎞ ⋅ ⋅ + ⎜ ⎟ ⎝ ⎠ − ∈ ≥ ∀ ∈ ∈ ∈ = ∀ ∈ =

∑ ∑ ∑∑∑ ∑∑ ∑

, , 0, , ( , , ) .

H H S l l

l L x l L d ω τ τ ∀ ∈ ≥ ∀ ∈ ∈ Φ

SLIDE 16

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improving

Improving Model (Contd.) Model (Contd.)

• Tolled User Equilibrium Condition

Tolled User Equilibrium Condition

( ) { }

, , ,

0, , , ( ( )) , , ,

m w m w m l i w l i m l j j

t x l L w W x m S H l L l L τ ρ ρ

+ −

= ∀ + − − ∈ ∪ ∈ ∈ ∈ ∈ (1)

( )

( )

, , ,

( ) 0, , , , ,

T w w T w T l l l l j w i j T l i

x t x l L w W l L l L τ μ γ ρ ρ

+ −

+ + + − − = ∀ ∈ ∈ ∈ ∈ (2)

, , ,

1 (ln ) , , ,

w m m w w m w m

d E w W m M β λ ρ θ + + − = ∀ ∈ ∈ (3) , 1 , ,

i

w l l l L

f i I w W μ

+

∀ ∈

= ∀ ∈ ∈

∑

(4)

,

( ) , ,

w T T l l l w W T

x l L c γ

∈

= ∀ ∈ −

∑

(5)

,

) 0, ( , ,

w T w T l l i w l

x f l L w W ω μ − = ∀ ∈ ∈ (6)

( ) { }

, ,

, ( ) , , , , ,

m w m w m l l j i i j

l L t x m S H l L l L w W τ ρ ρ

+ −

+ − − ≥ ∈ ∪ ∈ ∈ ∀ ∈ ∈ (7)

( )

, ,

, ( ) , , , ,

T w w T w T l l l l j i i j

t x l L w W l L l L τ μ γ ρ ρ

+ −

+ + + − − ≥ ∀ ∈ ∈ ∈ ∈ (8) , ,

l T

l L γ ∀ ∈ ≥ (9) , , ,

w l T

l L w W μ ∀ ∈ ∈ ≥ (10)

SLIDE 17

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improving

Improving Model (Contd.) Model (Contd.)

, , , , , , ,

1 max ln exp( ) . . Tolled User Equilibrium Condition (1 10), , , , , 0, ( ) , , , , ,

w m w m m j i w m w m w m U w m w m l l E m w m l l w m w m m l w l H S S l l H l

E t w W m M x w W m D x s t l l L L M θ ρ τ ρ β ρ τ θ τ τ τ − − − − ≤ ∀ ∈ ⎛ ⎞ ⋅ ⋅ + ⎜ ⎟ ⎝ ⎠ − ∈ ≥ ∀ ∈ ∈ ∈ = ∀ ∈ =

∑ ∑ ∑∑∑ ∑∑ ∑

, , 0, , ( , , ) .

H H S l l

l L x l L d ω τ τ ∀ ∈ ≥ ∀ ∈ ∈ Φ

SLIDE 18

Multi Multi-

• Modal Pareto

Modal Pareto-

• Improving

Improving Model (Contd.) Model (Contd.)

• This problem is a mathematical

This problem is a mathematical programming with programming with complementarity complementarity constraints (MPCC). constraints (MPCC).

• Solved by adapting manifold

Solved by adapting manifold suboptimization suboptimization algorithm proposed by algorithm proposed by Lawphongpanich Lawphongpanich and Yin (2008) and Yin (2008)

SLIDE 19

Numerical Example I Numerical Example I

(5, 12) 1 2 5 6 Regular link HOT link 7 8 3 4 9 (5, 12) (2, 11) (2, 11) (3, 25) (3, 12) (9, 35) (9, 12) (6, 33) (6, 11) (6, 43) (6, 11) (6, 18) (3, 35) (9, 20) (4, 11) (2, 19) (4, 36) (8, 39) (6, 24) (8, 26) (4, 26) (7, 32) (8, 30) O-D demand: 1-3: 30 1-4: 30 2-3: 30 2-4: 40 (5, 12) 1 2 5 6 Regular link HOT link 7 8 3 4 9 (5, 12) (2, 11) (2, 11) (3, 25) (3, 12) (9, 35) (9, 12) (6, 33) (6, 11) (6, 43) (6, 11) (6, 18) (3, 35) (9, 20) (4, 11) (2, 19) (4, 36) (8, 39) (6, 24) (8, 26) (4, 26) (7, 32) (8, 30) O-D demand: 1-3: 30 1-4: 30 2-3: 30 2-4: 40

SLIDE 20

Numerical Example I (Contd.) Numerical Example I (Contd.)

SOV HOV Transit SOV HOV Transit SOV HOV Transit 1-3 7.79 13.56 8.65 20.80 7.65 1.55 10.97 12.60 6.42 1-4 14.67 11.56 3.77 21.51 7.91 0.58 16.38 7.30 6.32 2-3 10.95 17.85 1.20 21.49 7.91 0.61 13.29 15.26 1.44 2-4 16.58 21.59 1.83 27.74 10.20 2.06 16.85 19.35 3.80 1-3 24.59 16.82 14.07 22.40 22.40 25.40 21.22 15.53 13.90 1-4 23.22 19.41 20.02 22.22 22.22 30.27 19.90 18.94 14.66 2-3 21.91 14.47 22.95 20.34 20.34 28.19 18.65 12.96 19.75 2-4 23.38 17.06 24.40 22.07 22.07 25.07 22.07 16.37 19.52 1-3 9.8

• 24.9
• 44.6
• 5.3
• 30.7
• 45.3

1-4 4.5

• 12.6
• 33.9
• 10.4
• 14.8
• 51.6

2-3 7.7

• 28.9
• 18.6
• 8.3
• 36.3
• 29.9

2-4 5.9

• 22.7
• 2.7
• 0.0
• 25.8
• 22.1

SO UE Pareto-improving Social benefits

• 1875.90
• 2603.27
• 2068.95

System travel time 1994.13 2851.63 2241.59 Revenue 513.97 0.00 70.22 Travel cost increase (%) Demand Travel cost

SLIDE 21

Numerical Example I (Contd.) Numerical Example I (Contd.)

SOV HOV Transit SOV HOV Transit SOV HOV Transit 1-3 7.79 13.56 8.65 20.80 7.65 1.55 10.97 12.60 6.42 1-4 14.67 11.56 3.77 21.51 7.91 0.58 16.38 7.30 6.32 2-3 10.95 17.85 1.20 21.49 7.91 0.61 13.29 15.26 1.44 2-4 16.58 21.59 1.83 27.74 10.20 2.06 16.85 19.35 3.80 1-3 24.59 16.82 14.07 22.40 22.40 25.40 21.22 15.53 13.90 1-4 23.22 19.41 20.02 22.22 22.22 30.27 19.90 18.94 14.66 2-3 21.91 14.47 22.95 20.34 20.34 28.19 18.65 12.96 19.75 2-4 23.38 17.06 24.40 22.07 22.07 25.07 22.07 16.37 19.52 1-3 9.8

• 24.9
• 44.6
• 5.3
• 30.7
• 45.3

1-4 4.5

• 12.6
• 33.9
• 10.4
• 14.8
• 51.6

2-3 7.7

• 28.9
• 18.6
• 8.3
• 36.3
• 29.9

2-4 5.9

• 22.7
• 2.7
• 0.0
• 25.8
• 22.1

SO UE Pareto-improving Social benefits

• 1875.90
• 2603.27
• 2068.95

System travel time 1994.13 2851.63 2241.59 Revenue 513.97 0.00 70.22 Travel cost increase (%) Demand Travel cost

SLIDE 22

Numerical Example I (Contd.) Numerical Example I (Contd.)

SOV HOV Transit SOV HOV Transit

1-5 1.71 0.68 0.00 0.72 0.72

• 4.71

1-6 2.39 0.96 0.00 0.00 0.00

• 1.74

2-5 1.15 0.46 0.00 0.00 0.00

• 0.77

2-6 0.12 0.05 0.00 0.00 0.00 0.00 5-6 0.00 0.00 0.00 0.00 0.00

• 3.25

5-7 9.84 3.94 0.00 7.92 7.92 6.54 5-9* 0.50 0.20 0.00 0.00 0.00 0.00 6-5 0.00 0.00 0.00 0.00 0.00

• 1.30

6-8 1.28 0.51 0.00 0.39 0.39 2.37 6-9* 0.00 0.00 0.00 0.00 0.00

• 2.29

7-3 1.41 0.56 0.00 0.00 0.00

• 3.52

7-4 0.84 0.34 0.00 0.79 0.79

• 0.76

7-8 0.00 0.00 0.00 0.00 0.00

• 0.64

8-3 0.00 0.00 0.00 0.69 0.69

• 11.40

8-4 0.50 0.20 0.00 0.00 0.00

• 7.20

8-7 0.00 0.00 0.00 0.13 0.13

• 6.36

9-7* 0.25 0.10 0.00 0.00 0.00

• 1.44

9-8* 0.00 0.00 0.00 0.00 0.00

• 2.05

1-5* 2.07 0.83 0.00 0.00 0.00

• 5.42

2-6* 0.12 0.05 0.00 0.00 0.00

• 0.06

5-7* 10.56 4.22 0.00 5.69 0.00 4.31 6-8* 1.31 0.52 0.00 0.78 0.00 3.18 7-3* 1.74 0.70 0.00 1.60 0.00

• 3.52

8-4* 0.55 0.22 0.00 0.18 0.00

• 7.71

Toll SO Pareto-improving

SLIDE 23

Numerical Examples II Numerical Examples II

• 90 links with 14

90 links with 14 HOT links. HOT links.

• 1 transit line on

1 transit line on each link. each link.

• 528 OD pairs.

528 OD pairs.

10 12 11 16 18 17 14 15 19 23 22 13 24 21 20 77 4 80 13 79 78 84 16 81 8 3 22 89 82 86 21 18 88 37 34 85 48 90 87 56 41 45 42 46 59 44 73 65 63 39 66 62 1 2 3 4 5 6 9 8 7 2 5 3 6 7 8 9 10 11 12 14 15 17 19 20 23 24 25 26 27 28 29 30 31 32 33 35 36 38 40 43 47 49 50 51 52 53 5 4 55 57 58 60 61 64 67 68 69 70 71 72 74 75 76

Regular link HOT link

10 12 11 16 18 17 14 15 19 23 22 13 24 21 20 77 4 80 13 79 78 84 16 81 8 3 22 89 82 86 21 18 88 37 34 85 48 90 87 56 41 45 42 46 59 44 73 65 63 39 66 62 1 2 3 4 5 6 9 8 7 2 5 3 6 7 8 9 10 11 12 14 15 17 19 20 23 24 25 26 27 28 29 30 31 32 33 35 36 38 40 43 47 49 50 51 52 53 5 4 55 57 58 60 61 64 67 68 69 70 71 72 74 75 76

Regular link HOT link

SLIDE 24

Numerical Examples II (Contd.) Numerical Examples II (Contd.)

SOV HOV Transit SOV HOV Transit SOV HOV Transit 3.21 0.66 0.00

• 0.00

0.00 0.00 27.51 27.51 49.08

• 0.68

0.68 39.78 UE Pareto-improving SO Social benefit

• 4147.4
• 4084.6
• 3890.3

Total vehicle travel time 3859.2 3691.0 2907.8 Total passenger travel time 4406.3 4397.0 4173.5 Revenue 0.0 30.4 463.8 Max travel cost increase (%) Max travel cost decrease (%)

SLIDE 25

Conclusion Conclusion

• Pareto improvement can be achieved in

Pareto improvement can be achieved in multimodal networks by adjusting the transit multimodal networks by adjusting the transit fares and charging tolls on highway links to fares and charging tolls on highway links to redistribute traffic flows among travel modes redistribute traffic flows among travel modes and among the network. and among the network.

• The multimodal Pareto

The multimodal Pareto-

• improving tolls can be

improving tolls can be

• btained by solving an MPCC problem using a
• btained by solving an MPCC problem using a

manifold manifold suboptimization suboptimization technique. technique.

SLIDE 26

Thank you! Thank you!

SLIDE 27

Mathematical Model Mathematical Model

• Feasible Region

Feasible Region

• Flow balance constraint:

Flow balance constraint:

• Total OD demand:

Total OD demand:

• Transit capacity:

Transit capacity:

, , , ,

, ,

w m w m w m

Ax E d w W m M = ∀ ∈ ∈

,

, ,

w m w m M

d D w W

∈

= ∀ ∈

∑

,

, ,

w T T T l l w W

x c l L

∈

≤ ∀ ∈

∑

SLIDE 28

Mathematical Model (Contd.) Mathematical Model (Contd.)

• Feasible Region

Feasible Region

• Common

Common-

• line:

line:

,

, , , ,

w T w l l i i

x f i N l L w W ω

+

≤ ∀ ∈ ∈ ∈

SLIDE 29

Mathematical Model (Contd.) Mathematical Model (Contd.)

• Link Travel Time

Link Travel Time

• Transit link travel times are the same with the auto

Transit link travel times are the same with the auto link travel times that they share. link travel times that they share.

4

1 0.15 , .

H S l l l l

v t fft l L L cap ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = ⋅ + ∀ ∈ ∪ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

SLIDE 30

Mathematical Model (Contd.) Mathematical Model (Contd.)

• Total Link Vehicle Flow

Total Link Vehicle Flow

,

' ' , ,

∑ ∑ ∑

∈

+ + =

T l

L l l H w H w l w S w l l

f

• x

x v

SLIDE 31

Mathematical Model (Contd.) Mathematical Model (Contd.)

• User Equilibrium (UE) Condition within the

User Equilibrium (UE) Condition within the Same Travel Mode Same Travel Mode

, , , , ,

0 if , , , . 0 if

w m p w m w m w m p w m p

b C t p P w W m M b ⎧= > ⎪ − ∀ ∈ ∈ ∈ ⎨≥ = ⎪ ⎩

. , ,

: path travel time on path for OD pair by mode , : the smallest travel time for OD pair by mode , : flow on path for OD pair by mode .

w m p w m w m p

C p w m t w m b p w m

SLIDE 32

Mathematical Model (Contd.) Mathematical Model (Contd.)

• Modal Split

Modal Split

• The users

The users’ ’ mode choice behavior follow logit mode choice behavior follow logit model. model.

W w t t D d

M m m m w m m w w m w

∈ ∀ − ⋅ − − ⋅ − = ∑

∈

) exp( ) exp(

, , ,

β θ β θ

, : parameters to be calibrated.

m

θ β

SLIDE 33

Multi Multi-

• Modal User Equilibrium

Modal User Equilibrium

• Theorem.
• Theorem. The solution of the following

The solution of the following variational inequality (VI) problem (or MUE variational inequality (VI) problem (or MUE-

• VI)

VI) satisfies the multi satisfies the multi-

• modal user equilibrium

modal user equilibrium conditions. conditions.

( ) ( )( ) ( )

* , , * , * , , * *

( ) 1 ln 0, ( , , ) , : the previously defined feasible region.

w m w m l l l w m l w m m w m w m w m w w i i w i

t x x x d d d x d β θ ω ω ω − + + − + − ≥ ∀ ∈Φ Φ

∑∑∑ ∑∑ ∑∑

SLIDE 34

Multi Multi-

• Modal System Optimum

Modal System Optimum

• Objective: Maximize the total social

Objective: Maximize the total social benefits. benefits.

• Multi

Multi-

• modal system optimal (MSO)

modal system optimal (MSO)

w M m m m w w

D t ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ − ⋅

∑ ∑

∈

) exp( ln 1

,

β θ θ

, , , ,

1 ( ) ln 1 . . ( , in , ) . m

w m w m w m l l w m l w m w m m w i w m w i

t x x d d d s t x d θ β ω θ ω + + + ∈Φ

∑∑∑ ∑∑ ∑∑ ∑∑