Assigning User Class Link and Route Flows
Uniquely to Urban Road Networks
______________________________________________________________________________
David Boyce and Jun Xie
Northwestern University, Evanston, Illinois
Tongji University, Shanghai, China
__________________________________________________________________________________
Research in collaboration with
Hillel Bar-Gera and Yu (Marco) Nie
___________________________________________________________________________________
14th TRB Transportation Planning
Applications Conference
May 8, 2013
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.
2
What is the Traffic Assignment Problem?
• Traditionally, a method for “loading” origindestination (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.
3
Brief History of TAP
1952
1956
Wardrop proposed criteria for route choice
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
4
Quick-precision methods
1994
2002
2006
2009
2010
2011
Gradient Projection (Jayakrishnan et al.)
Origin-Based Assignment (Bar-Gera)
Algorithm B (Dial)
Projected Gradient (Florian et al.)
TAPAS (Bar-Gera)
Applications of TAPAS (Bar-Gera et al.)
5
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.
6
Multiple user-equilibrium route flow solutions
Car
2
100
40
40
1
Truck
From
Car
Truck
4
120
60
3
To
D
D
160
D
120
Route
A→1→2→4→D
h*
25
Flow
h1
40
h2
0
A→1→3→4→D
75
60
100
B→1→2→4→D
15
0
40
B→1→3→4→D
45
60
20 7
How to choose a unique route flow solution?
Proportionality Condition
The same proportions apply to all travelers facing a
choice between a pair of alternative segments (PAS).
Consider the pair of segments [1,2,4] and [1,3,4].
First segment’s proportion is 40/(40+120)=1/4.
C
100
2
40
40
4
1
60
D
120
120
T
160
3
8
How to choose a unique route flow solution?
Proportionality Condition
Same proportions apply to all travelers facing a
choice between a pair of alternative segments (PAS).
Consider the pair of segments [1,2,4] and [1,3,4].
First segment proportion is 40/(40+120)=1/4.
C
2
25
100
40
40
4
1
120
T
60
160
75
D
120
3
9
For trips by Car to D, the proportion is 25/(25+75)=1/4.
How to choose a unique route flow solution?
Proportionality Condition
The same proportions apply to all travelers facing a
choice between a pair of alternative segments (PAS).
Consider the pair of segments [1,2,4] and [1,3,4].
First segment’s proportion is 40/(40+120)=1/4.
C
2
100
40
15
40
160
4
1
120
T
60
D
120
3
45
10
For trips by Truck to D the proportion is 15/(15+45)=1/4.
Basis for the proportionality condition
Reasons:
• Simple, reasonable, consistent, stable, and useful.
• Proportionality is testable, but not yet verified.
• Are there any other practical suggestions?
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.
11
Chicago regional zone system and road network
12
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:
m
m
d pq  Rp Sq exp   u pq


• 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.
13
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
14
Segments 1 and 2 connecting node 8032 to node 10344
8032
Segment 2
Scale in miles
|_______________|_______________|
0.0
0.25
0.5
10344
6380
Segment 1
6389
North Avenue
15
Total Segment Flows without and with Proportionality
status
segment
cars
trucks
segment
total flow
flows without proportionality vph
1
2
1/total
338
25
0.93
56
155
0.26
394
181
0.69
flows with proportionality vph
1
2
1/total
249
114
0.69
145
66
0.69
394
181
class
total flow
363
211
0.69
574
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.
16
By reversing the assignment order of the trip
matrices, the class flows without proportionality
were changed dramatically.
status
segment
cars
trucks
segment
total flow
status
segment
cars
trucks
segment
total flow
order of trip matrices in the assignment: car/truck
flows without proportionality vph flows with proportionality vph
1
2
1/total
1
2
1/total
338
25
0.93
249
114
0.69
56
155
0.26
145
66
0.69
394
181
0.69
394
181
0.69
order of trip matrices in the assignment: truck/car
flows without proportionality vph flows with proportionality vph
1
2
1/total
1
2
1/total
199
164
0.55
249
114
0.69
195
16
0.92
145
66
0.69
394
181
0.69
394
181
0.69
class
total flow
363
211
574
class
total flow
363
211
574
17
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
18
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
19
Car and truck link flows: without proportionality
Order of initial assignment: truck/car versus car/truck
Car flows on 30,147 links
Root mean square difference = 33.2
Truck flows on 29,810 links
Root mean square difference = 33.4
20
Car and truck link flows: with proportionality
Order of initial assignment: truck/car versus car/truck
Car flows on 30,149 links
Root mean square difference = 4.5
Truck flows on 29,831 links
Root mean square difference = 4.5
21
Two Links and a Pair of Alternative Segments
B
B
Edens
Exp′way
A
edens
Left – Edens Exp′way
link and segment
Right – Cicero Avenue
link and segment
Among 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.
Cicero
Avenue
A
edens
22
Flows on two links without and with proportionality
roadway link
car flow
(solid arrow):
(vph)
proportionality: without with
Edens Expressway 1,942 1,739
Cicero Avenue
946 1,145
truck flow
(pceph)
without with
4,747 4,950
1,292 1,093
delta =
delta/
delta/
total
|w – wo| with-car with-trk flow
vph
%
%
vph
203
12
4
6,689
199
17
18
2,238
Flows on two segments with proportionality (TAPAS)
segment:
Edens Expressway
Cicero Avenue
car flow
(vph)
173
28
truck flow
(pceph)
216
35
proportion by segment (%)
car
truck
total
86
86
86
14
14
14
segment
flow (vph)
389
62
23
Difference in CAR link flows: with minus without
proportionality versus car link flow with proportionality
Car link flow differences: with minus
without proportionality (vph)
200
150
100
50
0
-50
-100
-150
-200
0
1,000
2,000
3,000
4,000
5,000
Car link flow with proportionality (vph)
6,000
7,000
24
Difference in TRUCK link flows: with minus without
proportionality versus truck link flow with proportionality
Truck link flow differences: with minus
without proportionality (pceph)
200
150
100
50
0
-50
-100
-150
-200
0
1,000
2,000
3,000
4,000
5,000
Truck link flow with proportionality (pceph)
6,000
7,000
25
Histogram of 30,365 used links by car link flow differences
100,000
15,234
10,000
5,034
4,862
Number of links
2,491
2,448
1,000
164
132
100
10
1
< –100
–100 ~ –10
–10 ~ 0
0
0 ~ 10
10 ~ 100
> 100
26
Car link flow differences: with minus without proportionality
Histogram of 30,365 used links by car link flow differences
Average total link flow (vph)
2,000
1,670
1,645
1,500
1,320
1,181
983
916
1,000
500
269
0
< –100
–100 ~ –10
–10 ~ 0
0
0 ~ 10
10 ~ 100
> 100
27
Car link flow differences: with minus without proportionality
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.
28
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.
29
30
Test for proportionality of observed segment flows
1000
origins 1
origins 2-6
origins 7-9
800
origins 10
zero intercept, origins 10
Segment 2
non-zero intercept, origins 10
600
y = 0.5023x
R2 = 0.9569
400
y = 0.481x + 12.654
R2 = 0.9614
200
0
0
200
400
600
Segment 1
800
1000
31
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.
32
33
Number of O-D pairs without and with
proportionality by order of assignment
order of trip matrices in the assignment: car/truck
segment
cars
trucks
total
O-D pairs without proportionality
1
2
either/both
668
392
670/390
204
245
256/193
678
483
680/481
O-D pairs with proportionality
1
2
either/both
670
670
670/670
256
256
256/256
680
680
680/680
order of trip matrices in the assignment: truck/car
segment
cars
trucks
total
O-D pairs without proportionality
1
2
either/both
624
641
670/595
256
131
256/131
649
655
680/624
O-D pairs with proportionality
1
2
either/both
670
670
670/670
256
256
256/256
680
680
680/680
34