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Some network flow problems in urban road networks
Michael Zhang
Civil and Environmental Engineering
University of California Davis
Outline of Lecture
• Transportation modes, and some basic statistics
• Characteristics of transportation networks
• Flows and costs
• Distribution of flows – Behavioral assumptions– Mathematical formulation and solution– Applications
Vehicle Miles of Travel: by mode
Mode Vehicle-miles (millions)
Air Carriers 4,911 0.3%
General Aviation 3,877 0.2%
Passenger Cars 1,502,000 88%
Trucks 11%
Single Unit 66,800 4%
Combination 124,500 7%
Amtrak(RAIL) 288 0.0%
((U.S., 1997, Pocket Guide to Transp.)U.S., 1997, Pocket Guide to Transp.)
(U.S., 1997, Pocket Guide to Transp.)
ModeMode Passenger-miles (millions) Passenger-miles (millions) % SHARE % SHARE
Air CarriersAir Carriers 450,600 450,600 9.75% 9.75%
General AviationGeneral Aviation 12,500 12,500 0.27% 0.27%
Passenger CarsPassenger Cars 2,388,000 2,388,000 51.67%51.67%
Other vehicles 1,843,100 Other vehicles 1,843,100 34.56% 34.56%
BusesBuses 144,900 144,900 3.14% 3.14%
Rail Rail 26,339 26,339 0.56% 0.56%
Other Other 1,627 1,627 0.04% 0.04%
Passenger miles by modePassenger miles by mode
Fatalities by mode (1997, US)
Air 631 0.001363
Highway 42013 0.00993
Railroad 602 0.022856
Transit 275 0.001898
Waterborne 959 N/A
Mode # of fatalities # per million pas.-mile
How to “grow” a transportation system:
Population & Economic Growth
Land Use Change
Mobility/AccessibilityChange
NETWORK“Growth”
CHANGES IN ACTIVITYTravel Demand
pop. & economic growth, land use and demand/supply balance
An example: Beijing, China
Population: 5.6 million (1986) -> 10.8 million (2000)GDP: ~9-10% annual growth
Changes in land use
Changes in the highway network
Growing the highway network (Beijing)
The four step planning process
Activity pattern andforecast
Trip Generation
Trip Distribution
Modal Split
Trip Assignment
Link Flow
4step process
NEED FEEDBACK
HIGHWAY TRANSPORTATION
RAIL (SUBWAY) TRANSPORATION
London Stockholm
AIR TRANSPORTATION
TRANSPORATION NETWORKS AND THEIR REPRESENATIONS
• Nodes (vertices) for connecting points– Flow conservation, capacity and delay
• Links (arcs, edges) for routes– Capacity, cost (travel time), flow propagation
• Degree of a node, path and connectedness
• A node-node adjacency or node-link incidence matrix for network structure
Characteristics of transportation networks
• Highway networks– Nodes rarely have degrees higher than 4– Many node pairs are connected by multiple paths– Usually the number of nodes < number of links <
number of paths in a highway network
• Air route networks– Some nodes have much higher degrees than others
(most nodes have degree one)– Many node pairs are connected by a unique path
• Urban rail networks– Falls between highway and air networks
Flows in a Highway Network
: set of nodes
: set of links
: set of origins
: set of destinations
: set of paths from origin to destination ij
N
A
I
J
R i j
( ), : link travel cost function
: Traffic demand from origin to destination
: Capacity on link
a a a
ij
a
t v C
q i j
C a
i
j
Flows in a Highway Network (Cont’d)
• Path flows:– Flow conservation equations
– Set of feasible path flows
{ }, , ,ijr ijf r R i I j J∈ ∈ ∈
, ,
0
ij
ijr ij
r R
ijr
f q i I j J
f
∈
= ∈ ∈
≥
∑
1
2
5
6
3 4
3
4
5
1
26
7
( ), , ; 0; , ,ij
Tij ij ijr r ij r ij
r R
S f f f q f r R i I j J∈
⎧ ⎫⎪ ⎪= = = ≥ ∈ ∈ ∈⎨ ⎬⎪ ⎪⎩ ⎭
∑L L
Flows in a Highway Network (Cont’d)
• Origin based link flows:– Flow conservation equations
– Set of feasible origin based link flows
{ }, ,iav a A i I∈ ∈
{ }
,
,
0, \
0, ,
i
j
n n
ia ij
j Ja A
ia ij
a A
i ia a
a A a A
ia
v q i I
v q j J
v v n N I J
v i I a A
−
+
+ −
∈∈
∈
∈ ∈
= ∈
= ∈
− = ∈
≥ ∈ ∈
∑ ∑
∑
∑ ∑ U
{ }{ }all links entering node ,
all links leaving node ,
n
n
A n n N
A n n N
−
+
= ∈
= ∈
( ) { }{ }, , , , satisfies the above equationsTI I i i
a aS v v v i I a A= = ∈ ∈L L
Flows in a Highway Network (Cont’d)
• Link flows: – Set of feasible link flows
( ), ,T
av v= L L
, ; ; 0; , ,ij ij
ij ij ij ija r ar r ij r ij
i I j J r R r R
v v f a A f q f r R i I j J∈ ∈ ∈ ∈
⎧ ⎫⎪ ⎪Ω= = ∈ = ≥ ∈ ∈ ∈⎨ ⎬⎪ ⎪⎩ ⎭
∑∑∑ ∑δ
where
1, if path using link
0, otherwise
ijijar
r R a∈⎧δ =⎨
⎩
It is a convex, closed and bounded set
Costs in a Highway Network (Cont’d)
• Travel cost on a path
• The shortest path from origin i to destination j
• Total system travel cost
, ,ijr R i I j J∈ ∈ ∈
( ) , , ,ij ijr a a ar ij
a A
c t v r R i I j J∈
= δ ∈ ∈ ∈∑
{ }min , ,ij
ijij r
r Rc i i j J
∈μ = ∈ ∈
( )a a aa A
t v v∈∑
Behavioral Assumptions
• Travelers have full knowledge of the network and its traffic conditions
• Each traveler minimizes his/her own travel cost (time)
• Travelers choose routes to make the total travel time of all travelers minimal (which can be achieved through choosing the routes with minimal marginal travel cost)
Act on self interests (User Equilibrium):
Act on public interests (System Optimal):
( )a a aa A
t v v∈∑min
THE USER EQUILIBRIUM CONDITION
• At UE, no traveler can unilaterally change his/her route to shorten his/her travel time (Wardrop, 1952). It’s a Nash Equilibrium. Or
• At UE, all paths connecting an origin-destination pair that carry flow must have minimal and equal travel time for that O-D pair
• However, the total travel time for all travelers may not be the minimum possible under UE.
( ) ( ) 0,0,0 ≥≥−=− ijrij
ijrij
ijr
ijr fccf μμ
A special case: no congestion, infinite capacity
• Travel time is independent of flow intensity• UE & SO both predict that all travelers will
travel on the shortest path(s)• The UE and SO flow patterns are the same
{ }, , ,ijr ijf r R i I j J∈ ∈ ∈
1 2
1
2
3
i=1, j=2, q12=12t1=1t2=10t3=40
V1=12, f1=12, =1V2=0, f2=0V3=0, f3=0
We can check the UE and SO conditions
A case with congestion
1 2
1
2
3
( )
( )( ) 40
2
11
1
:functionscost Link
3
122
11
=
++=
+=
vt
vvvt
vvt
( ) ( ) ( )
0,,
12
min
321
321
332211
≥
=++
++
vvv
vvv
vtvvtvvtv
12
O-D demand:
12q =
* * *1 2 3
UE link flow pattern:
8, 4, 0v v v= = =
12
UE O-D travel cost:
9 =
12* 12* 12*1 2 3
UE path flow pattern:
8, 4, 0f f f= = =
12* 12* 12*1 2 3
Path travel cost pattern:
9, 9, 40c c c= = =
1* 1* 1*1 2 3
UE origin based link flow pattern
8.0, 4.0, 0.0v v v= = =
SO:
10240,0
10,6
7,6
33
22
11
===
====
∑ ii
itvtv
tvtv
108=∑ ii
itv
The Braess’ Paradox
The Braess’ Paradox-Cont.
The Braess’ Paradox-Cont.
General Cases for UE:
( ) ( )0
min
subject to
, ,
,
0, , ,
a
ij
ij
v
av
a A
ijr ij
r R
ij ija r ar
i I j J r R
ijr ij
h v t d
f q i I j J
v f a A
f r R i I j J
∈
∈
∈ ∈ ∈
= ω ω
= ∈ ∈
= δ ∈
≥ ∈ ∈ ∈
∑∫
∑
∑∑ ∑
With an increasing travel time function, this isa strictly (nonlinear) convex minimization problem.
It can be shown that the KKT condition of the aboveproblem gives precisely the UE condition
The relation between UE and SO
( ) ( ) ωω dtvhAa
v
av
a
∑ ∫∈
Ω∈ =0
min ( ) ( )aaAa
av vtvvH ∑∈
Ω∈ =min
( ) ( )a
aaaaaa dv
dtvvtvt +=ˆ
UE SO
( ) ( ) ωω dtvhAa
v
av
a
∑ ∫∈
Ω∈ =0
ˆˆmin
( ) ( ) ωω dtv
vtav
aa
aa ∫=0
1(
( ) ( )aaAa
av vtvvH((
∑∈
Ω∈ =min
Algorithms for for Solving the UE Problem
• Generic numerical iterative algorithmic framework for a minimization problem
Start
(0) and set 0x k =
Is the stop criterionsatisfied?
( 1)
By a strategy, choose
the next point
set 1
kx
k k
+
= +
StopYe
s
No
Algorithms for Solving the UE Problem (Cont’d)
Start
Stop
( )
Find a descent direction
kd
( ) 0?kd =
( )( )
Find a step size
0,1kα ∈
( 1) ( ) ( ) ( )
Update the link flows by
set 1
k k k kv v dk k
+ = +α= +
(0) and set 0v k=
Yes
NO
Applications
• Problems with thousands of nodes and links can be routinely solved
• A wide variety of applications for the UE problem– Traffic impact study – Development of future transportation plans– Emission and air quality studies
Intelligent Transportation System (ITS) Technologies
• On the road
• Inside the vehicle
• In the control room
“EYES” OF THE ROAD
- Ultrasonic detector- Infrared detector
- Loop detector - Video detector
SMART ROADS
SMART VEHICLESSAFETY, TRAVEL SMART GAGETS, MOBILE OFFICE(?)
SMART PUBLIC TRANSIT
• GPS + COMMUNICATIONS FOR– BETTER SCHEDULING & ON-TIME
SERVICE– INCREASED RELIABILITY
• COLLISION AVOIDANCE FOR– INCREASED SAFETY
SMART CONTROL ROOM
If you wish to learn more about urban traffic problems
• ECI 256: Urban Congestion and Control (every Fall)
• ECI 257: Flows in Transportation Networks (Winter)