1.225J
1.225J (ESD 205) Transportation Flow Systems
Lecture 5
Assignment on Traffic Networks
Prof. Ismail Chabini and Prof. Amedeo R. Odoni
Lecture 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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Lecture 5, Page 2
1
NonNon-Congested Road Network and OO-D Matrix
O-D Matrix
Mixed network
From 1
2
2
4
1
1
1
1
2
4
8
[t (i , j )]
5
4
3
4
2
3
5
6
3
Directed network
ª
«10
«
«50
«
« 25
«¬ 45
To
3
2
4
5
30 35 40 15 º
15 12 10 »»
40 35 20 »
»
30 35 40 »
30 35 40 »¼
• In R7, t(i, j) means demand for
O-D pair (i, j). (t(i, j) does not mean
travel time of link (i,j)
• Other common notation: q(i, j) or qij
2
2
4
1
1
1
4
8
2
3
5
4
6
3
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Lecture 5, Page 3
Assign OO-D Flows Originating from Node 1
(6,4)+ 2
2
1
Shortest path tree
from node 1
(0,*)+ 1
5 (8,2)+
(5,3)+ 4
3
2
(3,1)+
30
3
2
15
30+35+40+15
40
Assign flows originating
from node 1
1
30+15
30+35+
40+15
4
5
15
30+40+15
3
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35
Lecture 5, Page 4
2
Assign OO-D Flows Originating from Node 2
(0,*)+ 2
2
4
Shortest path tree
from node 2
1
(2,4)+ 1
5 (2,2)+
(1,2)+ 4
3
(7,1)+
3
10+15+12+10
2
10
10+15
Assign flows originating
from node 2
12
1
12
4
5
10
15
10
3
15
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Lecture 5, Page 5
Assign OO-D Flows Originating from Node 3
(3,4)+ 2
2
1
Shortest path tree
from node 3
(3,3)+ 1
5 (5,2)+
(2,3)+ 4
3
2
(0,*)+
3
40
2
20
50
35
Assign flows originating
from node 3
40+20
1
4
5
20
40+35+20
50
3
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50+40+35+20
Lecture 5, Page 6
3
Assign OO-D Flows Originating from Node 4
(1,4)+ 2
2
4
Shortest path tree
from node 4
1
(5,2)+ 1
5 (3,2)+
(0,*)+ 4
3
(8,1)+
3
30
2
25
Assign flows originating
from node 4
40
25+35
25+30+
35+40
1
25+30+35+40
4
5
40
35
3
35
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Lecture 5, Page 7
Assign OO-D Flows Originating from Node 5
(1,5)+ 2
4
Shortest path tree
from node 5
1
(5,2)+ 1
1
5 (0,*)+
(2,2)+ 4
6
+
(6,5)
30
2
45
45
Assign flows originating
from node 5
3
45+30+40
40
1
40
4
5
45+30+35+40
35
3
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35
Lecture 5, Page 8
4
Results of AllAll-oror-Nothing Assignment
0
2
85
0+25+0+60+45
=130
Add all flows on each link
52
115
235
1
0
4
170
5
0
0
180
50
0
35
3
‰ All-or-nothing (AON) assignment does 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?
Ÿ Equilibrium traffic assignment
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Lecture 5, Page 9
Derived Diagrams from the Fundamental Diagram
(flow, travel time) diagram
(flow, speed) diagram
u
t
true relationship
“unstable”
“stable”
classical
volume-delay
function
uc
“unstable”
“stable”
qmax
q
qmax
q
‰ 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
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Lecture 5, Page 10
5
Framework for Static Traffic Assignment Models
‰ Conceptual framework:
(input)
Supply
• Network representation of
the structure of a physical
road network
• Link performance functions
Supply/Demand
Interaction
(input)
Demand
• Origin-destination flows
• Centroid nodes and links
output
Flows and Travel Times
‰ Principles of assignment to represent the supply/demand interaction
• User Optimal (U.O.): O-D flows are assigned to paths with
minimum travel time
• System Optimal (S.O.): O-D flows are assigned such that total
travel time on the network is minimum
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Lecture 5, Page 11
Example
qac Ÿ
a
Small On-Ramp (Link 3)
t3 x3 0
Highway (Link 1)
t1 x1 10 x1
c
Ÿ qac
Ÿ qbc
Arterial Street (Link 2)
t 2 x2 90 x2
qbc Ÿ
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b
Lecture 5, Page 12
6
Traffic Assignment Concepts
‰ Conceptual Network
x O - Ds : (a, c) and (b, c)
1
a
‰ Demand
c
x (a, c) : q ac vehicles / hr
x (b, c) : q bc vehicles / hr
3
2
b
‰ O-D flows and path flows
‰ Path flow variables
x O - D (a, c) : one path
1 : Link 1 ( f1ac )
x O - D (b, c) : two paths
qac
f1ac
qbc
f1bc f 2bc
1 : Link 2 ( f1bc )
2 : Link 3, Link 1 ( f 2bc )
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Lecture 5, Page 13
Traffic Assignment Concepts (cont.)(cont.)-1
‰ Link (arc) flows and path flows
x1
f1ac f 2bc
x2
f1bc
x3
f 2bc
bc
‰ Assume that f 2
x1
qac pqbc
x2
1 p qbc
x3
pqbc
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‰ Arc-path incidence matrix
O-D
Path
b-c
a -- c 1
2
1
1 ª1
«
Link 2 « 0
3 «¬ 0
pqbc
0
1
0
1º
0 »»
1 »¼
‰ Assignment matrix
ª x1 º
«x »
« 2»
«¬ x3 »¼
p º
ª1
«0 1 p » ªqac º
«
» «q »
«¬0
p »¼ ¬ bc ¼
Lecture 5, Page 14
7
Traffic Assignment Concepts (cont.)(cont.)-2
‰ t1 , t2 , t3 are the travel times of links 1, 2, 3.
‰ “Congested” networks:
• Link travel times depend on link flows
• Example:
t1 x1 10 x1 , t2 x2 90 x2 , t3 x3 0
‰ Path travel-times as a function of link travel-times:
C1ac
t1 , C1bc
t 2 , C2bc
t1 t3
‰ Total travel times (What is the unit of this quantity?):
x1 * 10 x1 x2 * 90 x2 x3 * 0 1.225, 11/14/02
Lecture 5, Page 15
Assignment Principles
‰ If OD flows are infinitely divisible:
• There is an infinite number of assignments
• Which assignment should we choose?
Ј 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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Lecture 5, Page 16
8
Mathematical Expressions of Assignment Principles
‰ User-optimal traffic assignment principle: find p such that:
• O-D (b, c):
If p 1,
t1 t3 !
t1 t3 If 0 p 1,
t1 t3
If p
0,
t2
t2
t2
• O-D (a, c): the question is not posed as there is only one path
‰ System optimal: find p that minimizes:
f1ac C1ac f1bc C1bc f 2ac C2ac x1t1 x2t 2 x3t3 1.225, 11/14/02
Lecture 5, Page 17
Solution of the U.O. Assignment
‰ We want to solve for a p  >0 ,1@ such that:
If p
0,
If p 1,
If 0 p 1,
t1x1 t3 x3 ! t2 x2 t1x1 t3 x3 t2 x2 t1 x1 t3 x3 t2 x2 and
qac ( 80)
qbc ( 10)
1.225, 11/14/02
t1 x1 10 x1
t2 x2 90 x2
t3 x3 0
ª x1 º
«x »
« 2»
«¬ x3 »¼
p º
ª1
«0 1 p » ªqac º
» «q »
«
«¬0
p »¼ ¬ bc ¼
Lecture 5, Page 18
9
Solution of the U.O. Assignment (cont.)
‰
t 2 x 2 t1 x1 t 3 x3 ‰
x2 x1
‰
t 2 x2 t1 x1 t3 x3 80 qbc qac 2 x3
If
90 x 2 10 x1 qbc x3 qac x3 qbc qac 2x3
80 qbc qac
t 0, then x3
2
‰ Example: qac ,q bc 80 x 2 x1
80 qbc qac
2
80,10 o x1 , x2 , x3 85,5,5
‰ The travel time on any path is: 95
‰ Total travel time (per hour): (80+10)*95=8550 veh-hrs (per hour)
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Lecture 5, Page 19
(UO) Assignment May Depend on the Demand
‰
80 qbc qac
t 0, then
2
§ 80 qbc qac qbc 80 qac 80 qbc qac ·
x1* , x2* , x3* ¨
,
,
¸
2
2
2
©
¹
80 qbc qac
*
*
*
0, then x1 , x2 , x3 qac , qbc ,0 If
2
80 qbc qac
Proof : p * 0 and t 2 qbc t1 qac t3 0 0
2
If
‰
‰ The U.O. assignment is a function of the demand, and the function
may be non-linear. Example:
qac , qbc 80,10 o x1* , x2* , x3* 85,5,5
qac , qbc 160,20 o x1* , x2* , x3* 160,20,0
160,20 2 u 80,10 But 160,20,0 z 2 u 85,5,5
‰ Remark: An increase in demand may lead to a decrease in the flow on
a link
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Lecture 5, Page 20
10
Building More Roads Is Not Always Better
‰ Without Link 3, there is only one possible assignment
x , x q
, qbc 80,10
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.
*
1
*
2
ac
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‰
Lecture 5, Page 21
min
System Optimal Assignment
x1t1 x1 x2t 2 x2 x3t3 x3 s.t .
f1ac
qac
bc
1
bc
2
f
f
Demand
qbc
f1ac t 0, f1bc t 0, f 2bc t 0
x1
f1ac f 2bc
x2
f1bc
x3
bc
2
f
*
*
*
‰ S.O. solution: x1 , x 2 , x 3 Non - negativity
Definition of link flows
80 ,10 ,0 , xif q ac , q bc x 80 ,10 1
2
0
0
x3
‰ min x1t1 x1 x2t 2 x2 x3t3 x3 min ³ m1 x1 dx1 ³ m2 x2 dx2 ³ m3 x3 dx3
Where : m1 x1 1.225, 11/14/02
d x1t1 x1 , m2 x2 dx1
d x2t 2 x2 , m3 x3 dx2
0
d x3t3 x3 dx3
Lecture 5, Page 22
11
Lecture 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
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Lecture 5, Page 23
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