Assigning User Class Link and Route Flows Uniquely to Urban Road Networks______________________________________________________________________________

David Boyce and Jun XieNorthwestern University, Evanston, Illinois

Tongji University, Shanghai, China__________________________________________________________________________________

Research in collaboration withHillel Bar-Gera and Yu (Marco) Nie

___________________________________________________________________________________

14th TRB Transportation Planning Applications Conference

May 8, 2013

22

Objectives of this talk

• Review the status of solving the traffic assignment problem (TAP);

• Describe the condition of proportionality to solve for unique class link flows and route flows;

• Analyze class link flows on the Chicago regional network with regard to:– order of the assignment of the trip matrices;– flows over a pair of links and alternative segments;

• Explore the prospects for the validation of proportionality in a real network.

3

What is the Traffic Assignment Problem?

• Traditionally, a method for “loading” origin-destination (O-D) flows onto a road network;

• A simple behavioral model of route choice in a congested urban road network;

• A method for determining link flows and travel times and costs for: – computing origin-destination-mode flows;– computing emissions from cars and trucks;

• Inputs for analyses using route flows and multi-class link flows.

4

Brief History of TAP

1952 Wardrop proposed criteria for route choice

1956 Beckmann formulated the user-equilibrium route choice problem with variable

demand

1958 Algorithms for the shortest route problem

1958-75 Heuristic methods for assigning traffic to road networks

1973 Solution of Beckmann’s formulation with the link-based linear approximation

method

1978 First link-based practitioner code

1990~ Research on quick-precision methods

5

Quick-precision methods

1994 Gradient Projection (Jayakrishnan et al.)

2002 Origin-Based Assignment (Bar-Gera)

2006 Algorithm B (Dial)

2009 Projected Gradient (Florian et al.)

2010 TAPAS (Bar-Gera)

2011 Applications of TAPAS (Bar-Gera et al.)

6

Who needs route flows or user class link flows?

• Solution of TAP finds unique total link flows, but not route flows or user class links flows;

However, transportation planners require:

• Link flows by user class for project evaluation;

• Route flows for select link analyses to find the source of link flows by origin and destination;

• This talk examines the idea of applying a condition of proportionality to determine user class link and route flows uniquely.

7

Multiple user-equilibrium route flow solutions

From To Route Flow

h* h1 h2

Car DA→1→2→4→D 25 40 0

A→1→3→4→D 75 60 100

Truck DB→1→2→4→D 15 0 40

B→1→3→4→D 45 60 20

Car

Truck

1

2

3

4 D

100

60 120

40 40

120

160

8

How to choose a unique route flow solution?

C

T

1

2

3

4 D

100

60120

40 40

120

160

Consider the pair of segments [1,2,4] and [1,3,4].

First segment’s proportion is 40/(40+120)=1/4.

Proportionality ConditionThe same proportions apply to all travelers facing a choice between a pair of alternative segments (PAS).

9

How to choose a unique route flow solution?

C

T

1

2

3

4 D

100

60120

40 40

120

160

Consider the pair of segments [1,2,4] and [1,3,4].

First segment proportion is 40/(40+120)=1/4.

Proportionality ConditionSame proportions apply to all travelers facing a

choice between a pair of alternative segments (PAS).

25

75

For trips by Car to D, the proportion is 25/(25+75)=1/4.

10

For trips by Truck to D the proportion is 15/(15+45)=1/4.

How to choose a unique route flow solution?

C

T

1

2

3

4 D

100

60120

40 40

120

160

Consider the pair of segments [1,2,4] and [1,3,4].

First segment’s proportion is 40/(40+120)=1/4.

Proportionality ConditionThe same proportions apply to all travelers facing a

choice between a pair of alternative segments (PAS).

15

45

11

Basis for the proportionality condition

Implications:Any user-equilibrium (UE) route that can be used, keeping the same total link flows, should be used

Traffic Assignment by Paired Alternative Segments (TAPAS), as well as two commercial software systems, now implement the condition of proportionality.

Reasons:• Simple, reasonable, consistent, stable, and useful.• Proportionality is testable, but not yet verified.• Are there any other practical suggestions?

12

Chicago regional zone system and road network

13

Construction of trip matrices for our experiments• Person trips by mode are directly related to the

number of origins and destinations and inversely related to UE travel time, distance, tolls and transit fares:

• This function results in many small OD flows << 1;• Cost sensitivity parameter β = 0.20 determines

mode and trip length with a moderate level of endogenous congestion;

• Truck trips are exogenous, and represented in car equivalent units.

mpqqp

mpq uSRd exp

14

Solution procedure with VISUM-LUCE

• The car and truck trip matrices were assigned to UE travel time routes as precisely as possible assuming that both consider only travel time in choosing their routes (Relative Gap = 1E-8).

• Class link and route flows were adjusted for proportionality with a post-processing procedure.

• The result is two link flow arrays for cars and trucks:– one solution for class link flows without proportionality– uniquely determined class link flow solution with

proportionality

15

8032

10344

6380

6389

Segment 1

Segment 2

Scale in miles |_______________|_______________| 0.0 0.25 0.5

Segments 1 and 2 connecting node 8032 to node 10344

North Avenue

16

In performing the assignment, we noticed an unexpected result in the class flows without proportionality: cars are assigned mainly on segment 1, and trucks mainly on segment 2. The order of the initial assignment was cars followed by trucks.

Total Segment Flows without and with Proportionality

status flows without proportionality vph flows with proportionality vph segment 1 2 1/total 1 2 1/total

class total flow

cars 338 25 0.93 249 114 0.69 363 trucks 56 155 0.26 145 66 0.69 211 segment total flow

394 181 0.69 394 181 0.69 574

17

order of trip matrices in the assignment: car/truck status flows without proportionality vph flows with proportionality vph segment 1 2 1/total 1 2 1/total

class total flow

cars 338 25 0.93 249 114 0.69 363 trucks 56 155 0.26 145 66 0.69 211 segment total flow

394 181 0.69 394 181 0.69 574

order of trip matrices in the assignment: truck/car status flows without proportionality vph flows with proportionality vph segment 1 2 1/total 1 2 1/total

class total flow

cars 199 164 0.55 249 114 0.69 363 trucks 195 16 0.92 145 66 0.69 211 segment total flow

394 181 0.69 394 181 0.69 574

By reversing the assignment order of the trip matrices, the class flows without proportionality were changed dramatically.

18

North Avenue PAS: Car and truck OD flows with and without proportionality

Car: With: Segt 1 - 249; Segt 2 - 114 vph Without: 338; 25 vph

Truck: With: Segt 1 - 145; Segt 2 - 66 vph Without: 55; 155 vph

19

North Avenue EB select link analysis: car and truck OD flows

Car: With - 1,052 vph; Without - 1,171 vph

Truck: With - 696 vph; Without - 577 vph

20

Car and truck link flows: without proportionality

Order of initial assignment: truck/car versus car/truck

Car flows on 30,147 linksRoot mean square difference = 33.2

Truck flows on 29,810 linksRoot mean square difference = 33.4

21

Car and truck link flows: with proportionalityOrder of initial assignment: truck/car versus car/truck

Car flows on 30,149 linksRoot mean square difference = 4.5

Truck flows on 29,831 linksRoot mean square difference = 4.5

22

Two Links and a Pair of Alternative SegmentsLeft – Edens Exp′way

link and segmentRight – Cicero Avenue

link and segmentAmong 7,266 pairs of

alternative segments:

• expressway link lies in 101 segments;

• arterial link lies in 45 segments;

• both lie in 11 pairs of alternative segments.

A

B B

A

Edens Exp′way edens

Cicero Avenue edens

23

Flows on two links without and with proportionality

roadway link (solid arrow):

car flow (vph)

truck flow (pceph)

delta = |w – wo|

delta/ with-car

delta/ with-trk

total flow

proportionality: without with without with vph % % vph Edens Expressway 1,942 1,739 4,747 4,950 203 12 4 6,689 Cicero Avenue 946 1,145 1,292 1,093 199 17 18 2,238

proportion by segment (%) segment:

car flow (vph)

truck flow (pceph) car truck total

segment flow (vph)

Edens Expressway 173 216 86 86 86 389 Cicero Avenue 28 35 14 14 14 62

Flows on two segments with proportionality (TAPAS)

24

Difference in CAR link flows: with minus without proportionality versus car link flow with proportionality

-200

-150

-100

-50

0

50

100

150

200

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000

Car link flow with proportionality (vph)

Car

link

flow

diff

ere

nce

s: w

ith m

inu

s w

itho

ut

pro

po

rtio

na

lity

(vp

h)

25

Difference in TRUCK link flows: with minus without proportionality versus truck link flow with proportionality

-200

-150

-100

-50

0

50

100

150

200

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000

Truck link flow with proportionality (pceph)

Tru

ck li

nk

flow

diff

ere

nce

s: w

ith m

inus

with

ou

t p

rop

ort

ion

alit

y (p

ceph

)

26

Histogram of 30,365 used links by car link flow differences

132

2,4915,034

15,234

4,8622,448

164

1

10

100

1,000

10,000

100,000

< –100 –100 ~ –10 –10 ~ 0 0 0 ~ 10 10 ~ 100 > 100

Car link flow differences: with minus without proportionality

Num

ber

of li

nks

27

Histogram of 30,365 used links by car link flow differences

1,645

1,181

916

269

983

1,320

1,670

0

500

1,000

1,500

2,000

< –100 –100 ~ –10 –10 ~ 0 0 0 ~ 10 10 ~ 100 > 100

Car link flow differences: with minus without proportionality

Ave

rage

tota

l lin

k flo

w (

vph)

28

Conclusions so far• Differences in car and truck flows with and without

proportionality are primarily less than +/–100 vph;• Differences mainly occur on link flows < 2,000 vph;• Adjustments can occur only on pairs of segments;• From related research, we know that links found in

pairs of segments are primarily links with flows under 2,000 vph: if links are ranked ordered by flow, links with flow < 2,000 vph comprise 77% of links in PASs;

• Additional studies are required to reach more specific conclusions.

29

Progress on validation of proportionality

• The static traffic assignment model is very simple, but it has proven to be useful for plan evaluation.

“Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.” George Box and Norman Draper, Empirical Model-Building, Wiley, 1987, p. 74.

• How to validate the condition of proportionality?

– locate where alternative routes occur;

– identify a high volume pair of segments;

– observe segment flows over many days to identify whether the pattern is stable.

30

31

y = 0. 5023xR2 = 0. 9569

y = 0. 481x + 12. 654R2 = 0. 9614

0

200

400

600

800

1000

0 200 400 600 800 1000

Segment 1

Segm

ent

2

ori gi ns 1

ori gi ns 2-6

ori gi ns 7-9

ori gi ns 10

zero i ntercept, ori gi ns 10

non-zero i ntercept, ori gi ns 10

Test for proportionality of observed segment flows

32

Conclusions• The charts show substantial differences with respect

to class link flows by order of assignment, and with and without proportionality;

• The proportionality condition can be applied to determine class link flows and route flows uniquely. Without proportionality, these flows are not comparable among scenarios.

• Initial validation tests of proportionality appear promising; next we need to examine whether these patterns repeat daily or weekly.

33

34

Number of O-D pairs without and with proportionality by order of assignment

order of trip matrices in the assignment: car/truck

O-D pairs without proportionality O-D pairs with proportionality segment 1 2 either/both 1 2 either/both cars 668 392 670/390 670 670 670/670 trucks 204 245 256/193 256 256 256/256 total 678 483 680/481 680 680 680/680

order of trip matrices in the assignment: truck/car

O-D pairs without proportionality O-D pairs with proportionality segment 1 2 either/both 1 2 either/both cars 624 641 670/595 670 670 670/670 trucks 256 131 256/131 256 256 256/256 total 649 655 680/624 680 680 680/680