5_assignment on traffic network

8/9/2019 5_Assignment on Traffic Network

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Lecture 5Lecture 5

Assignment on Traffic NetworksAssignment on Traffic Networks

Prof.Prof. Ismail ChabiniIsmail Chabini and Prof.and Prof. AmedeoAmedeo R.R. OdoniOdoni

1.2251.225J (ESD 205) Transportation Flow SystemsJ (ESD 205) Transportation Flow Systems

1.225, 11/14/02 Lecture 5, Page 2

Lecture 5 OutlineLecture 5 Outline

Summary from previous lectures:

Assignment on non-congested networks: All-or-nothing

assignment

Volume-delay functions for congested networks

Framework for static traffic assignment models

Static traffic assignment: concepts

Static traffic assignment: principles

User Optimal (UO) and System Optimal (SO) static traffic

assignment

Summary


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1.225, 11/14/02 Lecture 5, Page 5

Assign OAssign O--D Flows Originating from Node 2D Flows Originating from Node 2

Shortest path tree

from node 2

1

2

3

4 5Assign flows originating

from node 215

12

10

1

2

3

4 5

4

3

1

2

(2,4)+

(7,1)+

(1,2)+ (2,2)+

(0,*)+

10+15

10+15+12+10

12

15

10

10

1.225, 11/14/02 Lecture 5, Page 6


1

2

3


from node 3

50

40+35+20

40+20

20

1

2

3

4 5

3 2

2

(3,3)+

(0,*)+

(2,3)+ (5,2)+

(3,4)+

Shortest path tree

from node 3

1

50+40+35+20

50

35

20

40


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1.225, 11/14/02 Lecture 5, Page 7


1

2

3


from node 435

25+35

25+30+

35+40

40

Shortest path tree

from node 41

2

3

4 5

4

3

2

(5,2)+

(8,1)+

(0,*)+ (3,2)+

(1,4)+

1

25+30+35+40

30

40

25

35

1.225, 11/14/02 Lecture 5, Page 8


1

2

3


from node 5

4540

45+30+40

Shortest path tree

from node 51

2

3

4 5

4 1 1

6

(5,2)+

(6,5)+

(2,2)+ (0,*)+

(1,5)+

35

45+30+35+40

40

35

30

45


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1.225, 11/14/02 Lecture 5, Page 9

Results of AllResults of All--oror--Nothing AssignmentNothing Assignment

1

2

3

4 5Add all flows on each link

170

0+25+0+60+45

=130235

85

115

50 180

35

52

0

0

0

0

0

All-or-nothing (AON) assignmentdoes not consider congestion

The solution may not be unique (Why? Is unique solution important?

AON assignment does not make sense if certain links are congested

To account for congestion, travel time must depend on link flow

How to change the AON assignment method to make it work?

Equilibrium traffic assignment

1.225, 11/14/02 Lecture 5, Page 10

Derived Diagrams from the Fundamental DiagramDerived Diagrams from the Fundamental Diagram

qmax

uc

unstable

q

u

(flow, speed) diagram

stable

qmaxq

t

(flow, travel time) diagram

In general, q cannot be used as a variable (why?)

In the traffic planning area: q is also called volume

travel time is also called travel delay

In the case of volume-delay functions, q is used as a variable

classical

volume-delay

function

true relationship

unstablestable


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1.225, 11/14/02 Lecture 5, Page 11

Framework for Static Traffic Assignment ModelsFramework for Static Traffic Assignment Models

Conceptual framework:

Principles of assignment to represent the supply/demand interaction

User Optimal (U.O.): O-D flows are assigned to paths withminimum travel time

System Optimal (S.O.): O-D flows are assigned such that total

travel time on the network is minimum

Supply/Demand

Interaction

Flows and Travel Times

(input)

output

Supply

Network representation of

the structure of a physical

road network

Link performance functions

Demand

Origin-destination flows

Centroid nodes and links

(input)

1.225, 11/14/02 Lecture 5, Page 12

ExampleExample

a c

b

Highway (Link 1)

Small On-Ramp (Link 3)

Arterial Street (Link 2)

111 10 xxt

222 90 xxt

033 xt

acq

bcq

bc

ac

q

q


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1.225, 11/14/02 Lecture 5, Page 13

Traffic Assignment ConceptsTraffic Assignment Concepts

Conceptual Network Demand

Path flow variables

)(1Link3,Link:2

)(2Link:1

pathstwo:c)(b,D-O

)(1Link:1

pathone:c)(a,D-O

2

1

1

bc

bc

ac

f

f

f

O-D flows and path flows

hrvehicles

hrvehicles

/q:c)(b,

/q:c)(a,

c)(b,andc)(a,:Ds-O

bc

ac

a c

b

1

32

bcbcbc

ac

ac

ffq

fq

21

1

1.225, 11/14/02 Lecture 5, Page 14

Traffic Assignment Concepts (cont.)Traffic Assignment Concepts (cont.)--11

Link (arc) flows and path flows Arc-path incidence matrix

bc

bc

bcac

pqx

qpx

pqqx

3

2

1

1

Assignment matrix

bc

ac

q

q

p

pp

x

xx

0

101

3

2

1

bc

bc

bcac

fx

fx

ffx

23

12

211

100

010

101

3

2

1

Link

c-a

1

c-b

21O-D

Path

Assume that bcbc pqf 2


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1.225, 11/14/02 Lecture 5, Page 15

Traffic Assignment Concepts (cont.)Traffic Assignment Concepts (cont.)--22

are the travel times of links 1, 2, 3.

Congested networks:

Link travel times depend on link flows

Example:

Path travel-times as a function of link travel-times:

Total travel times (What is the unit of this quantity?):

321 ,, ttt

0,90,10 33222111 xtxxtxxt

3122111 ,, ttCtCtCbcbcac

0*90*10* 32211 xxxxx

1.225, 11/14/02 Lecture 5, Page 16

Assignment PrinciplesAssignment Principles

If OD flows are infinitely divisible:

There is an infinite number of assignments

Which assignment should we choose?

We need additional assumptions to define a less ambiguous

assignment

Assignment principle: a principle used to determine an assignment

Examples of assignment principles:

User-optimal: between each O-D pair, all used paths have equal

and minimum travel times

System-optimal: the total travel times are minimum


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1.225, 11/14/02 Lecture 5, Page 17

Mathematical Expressions of Assignment PrinciplesMathematical Expressions of Assignment Principles

User-optimal traffic assignment principle: findp such that:

O-D (b, c):

O-D (a, c): the question is not posed as there is only one path

System optimal: findp that minimizes:

231

231

231

,10

,1

,0

tttpIf

tttpIf

tttpIf

332211221111 txtxtxCfCfCfacacbcbcacac

1.225, 11/14/02 Lecture 5, Page 18

Solution of the U.O. AssignmentSolution of the U.O. Assignment

We want to solve for a such that:

and

1,0p

223311

223311

223311

,10

,1

,0

xtxtxtpIf

xtxtxtpIf

xtxtxtpIf

)10(

)80(

bc

ac

q

q

090

10

33

222

111

xtxxt

xxt

bc

ac

q

q

p

p

p

x

x

x

0

10

1

3

2

1


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1.225, 11/14/02 Lecture 5, Page 21

Building More Roads Is Not Always BetterBuilding More Roads Is Not Always Better

Without Link 3, there is only one possible assignment

Total travel times in one hour: 80*(10+80)+10*(90+10)=8200 veh-hr

(in one hour)

U.O. with Link 3:

Total travel times (in one hour): 8550 veh-hr (in one hour)

System travel times are worse if one adds Link 3! (Is this intuitive?)

This phenomenon is known as Braess Paradox, and is not an

isolated phenomenon.

10,80,, *2*1 bcac qqxx

1.225, 11/14/02 Lecture 5, Page 22

System Optimal AssignmentSystem Optimal Assignment

S.O. solution:

bc

bc

bcac

bcbcac

bc

bcbc

ac

ac

fx

fx

ffx

fff

qff

qfts

xtxxtxxtx

23

12

211

211

21

1

333222111

flowslinkofDefinition

negativity-Non0,0,0

Demand..

min

10,80,if,0,10,80,, *3*2*1 bcac qqxxx

3

33333

2

22222

1

11111

0

333

0

222

0

111333222111

,,:

minmin321

dx

xtxdxm

dx

xtxdxm

dx

xtxdxmWhere

dxxmdxxmdxxmxtxxtxxtx

xxx


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1.225, 11/14/02 Lecture 5, Page 23

Lecture 5 SummaryLecture 5 Summary

Assignment on non-congested networks: All-or-nothing assignment

Volume-delay functions for congested networks and static demand

Framework for static traffic assignment models

Static traffic assignment concepts and principles

User Optimal and System Optimal static traffic assignment

Summary

5_assignment on traffic network

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