Some network flow problems in urban road networks

Michael ZhangCivil 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 % SHAREAir 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 0.001363Highway 42013 0.00993Railroad 602 0.022856Transit 275 0.001898Waterborne 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

NAIJR i j

( ), : link travel cost function: Traffic demand from origin to destination

: Capacity on link

a a a

ij

a

t v Cq 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∈ ∈ ∈

, ,

0ij

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

( ), , Tav 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

where1, if path using link

0, otherwiseijij

ar

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 rr R

c 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{ }, , ,ij

r 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

211

1:functionscost Link

3

122

11

=

++=

+=

vt

vvvt

vvt

( ) ( ) ( )

0,,12

min

321

321

332211

≥=++

++

vvvvvv

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

10,67,6

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 dvdtvvtvt +=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

kxk k

+

= +

Stop Yes

No

Algorithms for Solving the UE Problem (Cont’d)

Start

Stop

( )

Find a descent direction

kd

( ) 0?kd =

( )( )

Finδ a step size

0,1ka ∈

( 1) ( ) ( ) ( )

Upδate the link floωs by

set 1

k k k kv v δk k

+ = +a= +

(0) anδ 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)