Aroon Aungsuyanon, David Boyce, and Bin Ran
1
Assessment of Adherence to the Condition of Proportionality in
User Equilibrium Traffic Assignments with Uniquely Determined Route Flows
Aroon Aungsuyanon
Department of Civil and Environmental Engineering
University of Wisconsin
1415 Engineering Drive, Madison, WI 53706-1691
Tel.: 608-770-9426 Fax: 608-262-5199
Email: [email protected]
David Boyce
Department of Civil and Environmental Engineering
Northwestern University
2145 Sheridan Road, Evanston, IL 60208-3109
Tel.: 847-570-9501 Fax: 847-491-4011
Email: [email protected]
Bin Ran
Department of Civil and Environmental Engineering
University of Wisconsin
1415 Engineering Drive, Madison, WI 53706-1691
Tel.: 608-262-0052 Fax: 608-262-5199
Email: [email protected]
Word count : 6,192
Figure count: 5
Table count: 0
Total equivalent word count: 7,442
Corresponding author: Aroon Aungsuyanon
Submission date: November 8, 2012
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
2
ABSTRACT
The standard formulation of static deterministic user equilibrium (UE) traffic assignment
problem based on the criterion of Wardrop provides a unique solution in terms of link flows;
however, route flows are not determined uniquely. Analyses based on an arbitrary choice among
the infinite number of possible route flow solutions could cause inconsistencies or even
controversies in applications. In 2010, a computationally efficient algorithm called Traffic
Assignment by Paired Alternative Segments (TAPAS) was successfully implemented to identify
UE route flows uniquely. So far, no effort has been made to assess adherence to the condition of
proportionality in UE traffic assignment with uniquely determined route flows. In this paper,
TAPAS was solved to obtain proportional UE route flows for the Chicago regional network in
the closest proximity to uniqueness of the solution. Various assessments of adherence to
proportionality are performed for a selected pair of alternative segments. The results show that
route flows determined by TAPAS correspond closely to exact proportionality. Only minor
differences occur between computed and exactly proportional UE route flows. Systematic
characteristics of the plots for the two alternative segments show that TAPAS behaves properly
according to the condition of proportionality. Insights from these empirical results may help
transportation planning professionals to be aware of the magnitude of differences in UE route
flows based on proportionality and to be able to differentiate uniqueness from non-uniqueness of
route flows in UE traffic assignment. The results may also be useful to software developers in
seeking improved adherence to proportionality of route flow solutions.
Keywords: traffic assignment, the condition of proportionality, pairs of alternative segments
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
3
INTRODUCTION
The standard formulation of the static deterministic user equilibrium (UE) traffic assignment
problem based on the criterion of Wardrop provides a unique solution in terms of link flows;
however, route flows are not determined uniquely. Analyses based on an arbitrary choice among
the infinite number of possible UE route flow solutions could cause inconsistencies or even
controversies in applications. In 1999, a behaviorally justifiable condition of proportionality on
paired alternative segments was first proposed by Bar-Gera to determine route flows consistently
(1). Unfortunately, proportionality is only a necessary, but not a sufficient condition to identify
route flows uniquely. In 2006, Bar-Gera proposed the condition of route set consistency to obtain
a set of routes that is likely to be similar to the exact set of UE routes and showed that this
condition suffices for any solution that aims to satisfy uniqueness of route flows (2). In 2010, a
computationally efficient UE solution algorithm called Traffic Assignment by Paired Alternative
Segments (TAPAS) was successfully implemented to identify UE route flow solutions that
concurrently satisfy the conditions of proportionality and route set consistency (3).
Due to its ability to satisfy the condition of proportionality within used routes, TAPAS
was widely applied as reference solutions in evaluating consistency, or adherence to the
condition of proportionality, of various UE route flow solutions (4, 5, 6). Results generated with
TAPAS available thus far have only considered relative gap and no effort has been made to
assess adherence to the condition of proportionality in UE traffic assignment with uniquely
determined route flows. In this paper, proportionality is precisely enforced under the most
reasonable conditions of UE and route set consistency to obtain proportional route flows in the
closest proximity to uniqueness of the solution. Various assessments of adherence to the
condition of proportionality are performed for a selected pair of alternative segments over three
single-class congestion scenarios for the Chicago regional network. Selected results are
presented in a way that transportation planning professionals may find helpful in understanding
underlying solution characteristics of unique UE route flows and in differentiating uniqueness
from non-uniqueness of route flows in UE traffic assignment.
In the remainder of this paper, the issue of non-uniqueness of UE route flow solutions as
well as consistent treatments for uniqueness of the solutions are offered, followed by brief
descriptions of the model and overall algorithm solutions. Principal findings regarding
proportionality assessments, effects of proportionality on individual link flows, and differences
between computed and exactly proportional UE route flows are then presented. Finally,
conclusions, usefulness, and future research directions complete the paper.
CONSISTENT CHOICES OF UE ROUTE FLOWS
This section aims to address the issue of non-uniqueness of route flows under the UE condition
as well as to introduce two equivalent approaches in choosing route flows uniquely. Figure 1(a)
shows a UE link flow solution for a simple road network, which consists of three O-D pairs
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
4
connecting three origins to one destination. Assume that link flows shown beside each link in
Figure 1(a) represent a perfect UE solution. In principle, the number of possible UE route flow
solutions that correspond to the same link flows of any road network can be infinite; four
solutions that correspond to the same link flows are shown in Figures 1(b)-1(e) by O-D pair.
Each O-D pair is connected by two alternative routes. Assume further that all equilibrium routes
shown in each solution represent an exact set of UE routes. Route flows are denoted along the
links that comprise the route. Below each route flow solution is a scatter plot of vehicle flows
along the routes; each point represents flows for one O-D pair passing through the upper
alternative route (x-axis) and the lower alternative route (y-axis) respectively. The trend line and
linear regression equation are shown on each scatter plot.
The question of how many vehicles from each origin use each of the two alternative
segments arises naturally. Clearly, the results from the four route flow solutions are different. For
example, flows on the upper alternative segment from origin A are substantially different,
varying from 1.25 in solution 1(c) to 5 in solution 1(d). Without any specific mathematical
criterion or behavioral assumption to support decisions, the choice among the infinite number of
possible UE route flow solutions is arbitrary and may cause inconsistencies or even controversies
in applications. The early development for consistent choices of UE route flows is based on the
criterion of entropy maximization (7, 8). The entropy function gives the probability of route
choices made by individual travelers within a specific route flow pattern. Since entropy presumes
an equal probability of occurrences for each route choice, any pattern that occurs most frequently
is regarded as the most likely route flows (8). In principle, maximizing entropy subject to the
constraints that the total link flows are UE leads to the identification of the most likely route flow
pattern. Solution 1(e) is the one and only one result that maximizes entropy. Entropy values,
which are provided on top of each scatter plot, show that route flows in solution 1(c) are farthest
away from being the most likely route flows, followed by those in solution 1(b) and 1(d)
respectively.
An alternative approach to choosing route flow solutions consistently is based on route
choice behavior of individual travelers. This approach implicitly assumes that all individual
travelers, no matter where they are from or going, are rational in their behavior; therefore, if
facing choices between the same two alternative route segments, they should distribute
themselves over the two alternative route segments in the same proportions. This assumption is
formally known as the condition of proportionality, which was first introduced by Bar-Gera and
Boyce in 1999 as an intuitive behavioral interpretation for necessary conditions that characterize
entropy maximizing route flows (1). To help examine individual route choice behavior in each
route flow solution, the ratio of travelers traversing the lower to upper alternative route segment,
namely the slope of a straight line that passes through the origin of a scatter plot, is denoted in
the middle of each solution subfigure. Inconsistent ratios of travelers across three origins in
solutions 1(b)-1(d) suggest that the assumption of proportionality is violated in all three solutions
in that not all individual travelers behave the same across origins; they only behave similarly
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
5
10
27
56
17
12
46
(a) A simple road network with a perfect UE solution on total link flows
A
3
1.25
3.75
3.75
D
D
D
3.00
9
C
Entropy value=25.442
20
10
7
C
y = -5x + 32
10
2
4
Upper route flows
6
0
2
4
Upper route flows
6
(c) A disproportionately
assigned UE route flow solution
(b) An arbitrarily
assigned UE route flow solution
C
D
D
9.857
10
10
C
9.857
9.857
Entropy value=26.276
30
y = 4x + 2
20
10
0
0
0
D
4.60
Entropy value=26.266
30
20
0
2.143
2.143
D
10
Entropy value=22.182
30
Lower route flows
Lower route flows
30
7
2.143
2
7
9
13.964
C
D
13.964
13.964
5.00
C
D
D
B
2
2
D
9
C
D
1.40
C
D
14
5
5
3.036
4.60
14
14
5
3
3.036
3.036
B
D
13.25
3
22.179
3
B
13.25
13.25
13
22.179
4.67
B
13
3
3
3.53
B
D
22.179
2
B
D
3.25
D
4.60
3
B
D
13
22
3.75
4.821
A
22
25.75
4
B
4.821
D
22
25.75
4
4
A
D
25.75
2
4.821
4.40
D
24
D
20.60
24
A
5
5
D
A
8.00
24
5
1.25
D
A
A
1.25
3
Lower route flows
3
Lower route flows
A
y = 4.6x
20
10
0
0
2
4
Upper route flows
6
(d) A disproportionately
assigned UE route flow solution
0
2
4
Upper route flows
6
(e) A proportionately
assigned UE route flow solution
Note that entropy is defined by ;
where is the flow on route connecting O-D pair and is total O-D demand flow between origin and destination .
FIGURE 1 Consistent choices of UE route flows.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
6
within the same origin but differently across origins. Without any supportive rationale, solutions
1(b)-1(d) should not be considered for use in any meaningful analysis. Only all individual
travelers in solution 1(e) behave intuitively and distribute themselves proportionately over the
two alternative route segments across all origins; that is, within-each-origin proportionality is the
same as between-origin proportions in a fashion that the ratios of travelers are identical across all
three origins and equal to the slope of the regression line. Basically, the two alternative segments
of the equal cost routes are referred as a pair of alternative segments or “a PAS”. By definition,
every PAS consists of one diverge node, one merge node, and two equal cost segments; each
segment is defined by a distinct sequence of one or more directed links and an identical set of
relevant origins.
MODEL DESCRIPTION AND OVERALL ALGORITHM SOLUTIONS
TAPAS, which is used to prepare the results presented in this paper, is an iterative traffic
assignment algorithm designed to identify UE route flow solutions that satisfy conditions of
proportionality and route set consistency (3). TAPAS was utilized to distribute O-D flows from
three single-class trip matrices to the Chicago regional network, which consists of 1,790 zones,
12,982 nodes, and 39,018 links. Each trip matrix was constructed according to a mode-origindestination-trip distribution model with doubly-constrained logit form, d mpq = A p Bq exp(− µ ⋅ c mpq ) ,
where (A p , Bq ) are balancing factors, µ is a cost sensitivity (CS) parameter, and cmpq is modeorigin-destination generalized travel cost. Three trip matrices only differ by the value of CS
parameter: 0.20, 0.10, and 0.05. The largest value has the highest sensitivity to cost, the lowest
generalized travel cost, and the least congestion. The generalized link travel cost is assumed to
equal link travel times given by the conventional BPR function. Details regarding the three trip
matrices are available in Bar-Gera and Boyce (9).
To assure that proportionality is precisely converged under the most reasonable conditions
of UE and route set consistency, TAPAS was terminated at 200 iterations for the 0.20 and 0.10
solutions and 1,500 iterations for the 0.05 solution. Additional refinements did not show to
improve convergence of the solutions. In this paper, TAPAS achieved a sub-consistency ratio of
3.02E-4 for the 0.20 and 0.10 solutions and a super-consistency ratio of 81 for the 0.05 solution.
The first two solutions fail to reach super-consistency because they require the precise level of
convergence that is well beyond the precision limits of present computer technology. Unlike past
studies, all solutions in this paper were computed to the maximum relative gap of 4.9E-16 and
the maximum flow deviation from proportionality of 8.45E-9 (10). Proportional route flows in
this paper are, therefore, considered as the closest proximity to uniquely determined UE route
flows. Formal definitions of three convergence measures are found in Bar-Gera (3).
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
7
PROPORTIONALITY ASSESSMENTS
In Chicago regional network, the number of PASs identified in forming uniquely determined UE
routes for the 0.20, 0.10, and 0.05 solutions are 5,617, 11,702, and 22,500 respectively (10).
Comprehensive analysis of individual PASs does not seem to be a reasonable basis for
proportionality assessing. In this paper, only one PAS with the maximum number of relevant
origins is selected from each solution to display the principal findings. Although this approach is
not statistically representative, there is a good reason to believe that it does provide enough
evidence to see what might occur with other PASs. A basis for this selection is that PASs with
fewer numbers of relevant origins will adhere to proportionality in the same way as that with
maximum number of relevant origins. Figure 2 shows a map of a selected PAS for three
solutions. Past studies experimenting on the same road network with the same values of CS
showed that a simple PAS formed by four links occurs most frequently (10). The selected PAS
of four links is typical of all PASs, assuring that findings and conclusions obtained from this
paper are representative. In the following, adherences to the condition of proportionality are
assessed through selected paired segments analyses, which are performed at two levels: by
aggregate O-D pairs and by disaggregate O-D pairs.
Selected Paired Segments Analysis at Aggregate Levels
Key attributes for the three solutions summarized beneath a map of the selected PAS in Figure 2
are helpful in interpreting analyses in subsequent sections. Information regarding the number of
relevant origins and destinations suggests that flows traversing the two segments come from
almost everywhere in the region and the destination zone is the same for all origins. It seems
surprising that a very small PAS is used by almost all origins. However, this is most favorable
for TAPAS. In principle, PASs with shorter total link lengths and fewer links are more
computationally desirable for they are likely to be relevant to more origins, and a shift of flows
between pairs of routes can be computed faster (2). Recall that one O-D pair may consist of one
or more disaggregate O-D pairs; each disaggregate O-D pair is defined by the two routes that
only distinguish at alternative segments. For an O-D pair with more than one disaggregate O-D
pairs, route segments taken from the origin to the diverge node and/or from the merge node to
the destination for one disaggregate O-D pair must be physically different from others.
Information regarding the number of O-D pairs and unique UE routes suggests that most O-D
pairs have more than one pair of UE routes or more than one disaggregate O-D pair. For
example, there are 7,926 different UE routes connecting 1,781 O-D pairs or equivalently 3,963
disaggregate O-D pairs in the 0.20 solution.
In each solution, between-origin proportionality is observed on each of the two equal cost
segments. Comparisons between solutions reveal that congestion and proportionality are not
necessarily related. In fact, the effects of congestion on proportionality are PAS-specific,
meaning that among the solutions the proportions of flows over the two segments for a given
PAS respond to congestion differently. For example, the proportions of flows on one segment
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
8
7058
12184
N
10126
1668
11928
Node
Zone centroid (destination zone)
Link that is part of segment 1
Link that is part of segment 2
Link that is not part of a selected PAS
Direction of flows
12310
12389
7057
6185
5830
Cost sensitivity
Segment
Total number of links in a segment
Total link lengths in a segment (mi)
Total flows on a segment (vph)
Total costs on a segment (auto in-vehicle min)
Exact proportion of flows on a segment
Total origins relevant to a segment
Total destinations relevant to a segment
Total O-D pairs relevant to a segment
Total disaggregate O-D pairs relevant to a segment
Total unique UE routes relevant to a segment
Total number of links relevant to a segment
12430
11929
CS-0.20
1
2
1.49
284.249
2.676
0.343
1,781
1
1,781
3,963
3,963
8,207
2
2
1.07
543.627
2.676
0.657
1,781
1
1,781
3,963
3,963
8,207
CS-0.10
1
2
1.49
255.189
2.673
0.320
1,781
1
1,781
10,331
10,331
8,495
2
2
1.07
542.283
2.673
0.680
1,781
1
1,781
10,331
10,331
8,495
CS-0.05
1
2
1.49
108.502
2.667
0.156
1,775
1
1,775
21,471
21,471
8,916
2
2
1.07
585.433
2.667
0.844
1,775
1
1,775
21,471
21,471
8,916
FIGURE 2 Selected paired segments analysis at aggregate levels.
could increase or decrease as congestion grows. For this particular selected PAS, the proportions
of flows over the two segments are substantially influenced by congestion levels on the network.
The proportion of flows in each solution is higher on segment 2 than segment 1. As seen, the
difference between segment proportions in the 0.20 and 0.10 solutions is about half that of the
0.05 solutions. The decreases in total flows over the two segments from the 0.20 to 0.10 to 0.05
solution pertain to two intertwined factors: a decrease in CS values, and an increase in link and
route costs resulting from increased traffic on the network. Since the segment proportions shown
in Figure 2 are computed from aggregations of flows on all routes traversing each of the two
alternative segments, they are regarded as the exact proportionality with which every
disaggregate O-D pair must agree.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
9
Selected Paired Segments Analysis at Disaggregate Levels
A way to assess adherence to proportionality of (disaggregate) O-D pairs is to make scatter plots
of the data. Each data point shown in Figure 3(a) represents one O-D pair plotted on the linear
scale. The top row shows plots of total O-D flows (x-axis) against O-D flows for routes
traversing segment 1 (y-axis) for three solutions, arranged by decreasing values of CS from the
left to the right. The middle row shows the same plots as with the first row, with the y-axis
represents O-D flows for routes traversing segment 2. Plots in the bottom row shows flows of a
particular O-D pair split between the two segments. Above each trend line is a linear regression
equation in conjunction with a statistical measure of R2 to assist assessments of adherence to
proportionality. To enhance interpretation of the plots, the total number of O-D pairs are also
shown above each plot. As seen in each plot, all the points fall along one straight line through the
origin (0,0), indicating that for this pair of segments the same proportion of flows is perfectly
applied to all O-D pairs. The extremely high degree of adherence to proportionality indicated by
R2 = 1 is not unexpected for TAPAS assignments in which the solutions are highly converged
and the condition of proportionality is precisely enforced for every pair of alternative segments
on the network. The observed differences in the proportions from solution to solution result from
sensitivity to travel cost and congestion on the network, and not from imprecise solutions or
arbitrary choices of used routes over a selected pair of alternative segments. The slopes of the
trend lines on the first two rows correspond precisely to the exact proportions, indicating that
every disaggregate O-D pair agrees with the exact proportion and that TAPAS works perfectly in
assigning the O-D flows to the two segments in the same proportion. The three different slopes
of the trend lines on the bottom row reflect three different ratios between the flows on segment 1
and 2. Assuming that each slope is represented by a constant m, the plots suggest that the vertical
coordinate of each point in the middle row is m times those in the top rows. To put it another
way, for every disaggregate O-D pair the flow on the route traversing segment 2 is m times that
of segment 1. Without the imposition of proportionality, the ratio of the flows on segment 1 to
the flows on segment 2 in each solution could be rather arbitrary, resulting from non-uniqueness
of route flows.
One may be interested to know whether the proportions on segment 1 and 2 are actually
the same for every O-D pair. Plots shown in Figure 3(a) do not permit visual comparisons for
every O-D pair adheres to proportionality so perfectly that the differences in the proportions
among O-D pairs are not visible in any plot. To amplify the differences, plots in Figure 3(a) are
slightly modified as shown in Figure 3(b). The similar layouts of the plots are maintained for
ease of comparison. For the plots of the first two rows, the O-D flows on the x-axis are plotted
on the log scale to spread out the points by different orders of magnitude, ranging from -14 to 4.
As an example, an x-axis value of -14 corresponds to an O-D flow equal to 1E-14 vph. To obtain
the proportions of flow on segment 1 and 2, each value on the y-axis is divided by its total O-D
flows. Since segment proportions shown in Figure 3(b) are computed from the flows of only one
disaggregate pair of routes that traverse each segment, the y-axis value of each data point is
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
30
30
O-D-segment 2 flows (vph)
3,963 disaggregate O-D pairs
120
y = 0.657x
R² = 1.000
90
60
30
0
0
30
60
90
120
150
180
3,963 disaggregate O-D pairs
90
y = 1.913x
R² = 1.000
30
0
20
40
O-D-segment 1 flows (vph)
0
y =0.680x
R² = 1.000
40
20
0
0
60
O-D-segment 1- flows vs. O-D-segment 2-flows : linear : CS-0.20
20
40
60
80
y = 2.125x
R² = 1.000
20
0
5
10
15
O-D-segment 1 flows (vph)
y = 0.156x
R² = 1.000
20
10
0
0
20
O-D-segment 1- flows vs. O-D-segment 2-flows : linear : CS-0.10
10
21,471 disaggregate O-D pairs
40
30
y = 0.844x
R² = 1.000
20
10
0
0
10
21,471 disaggregate O-D pairs
40
dssdsdsd
0
30
20
30
40
Total O-D flows (vph)
Total O-D flows vs. O-D-segment 2-flows : linear : CS-0.05
10,331 disaggregate O-D pairs
40
21,471 disaggregate O-D pairs
40
20
30
40
Total O-D flows (vph)
Total O-D flows vs. O-D-segment 1-flows : linear : CS-0.05
10,331 disaggregate O-D pairs
60
60
60
0
0
Total O-D flows (vph)
Total O-D flows vs. O-D-segment 2-flows : linear : CS-0.10
O-D-segment 2 flows (vph)
O-D-segment 2 flows (vph)
Total O-D flows (vph)
Total O-D flows vs. O-D-segment 2-flows : linear : CS-0.20
120
20
20
40
60
80
Total O-D flows (vph)
Total O-D flows vs. O-D-segment 1-flows : linear : CS-0.10
60
90
120
150
180
Total O-D flows (vph)
Total O-D flows vs. O-D-segment 1-flows : linear : CS-0.20
O-D-segment 2 flows (vph)
0
O-D-segment 1 flows (vph)
60
y = 0.320x
R² = 1.000
40
O-D-segment 2 flows (vph)
y = 0.343x
R² = 1.000
10,331 disaggregate O-D pairs
60
O-D-segment 2 flows (vph)
O-D-segment 1 flows (vph)
90
O-D-segment 1 flows (vph)
3,963 disaggregate O-D pairs
120
0
10
30
y = 5.396x
R² = 1.000
20
10
0
0
2
4
6
8
O-D-segment 1 flows (vph)
O-D-segment 1- flows vs. O-D-segment 2-flows : linear : CS-0.05
FIGURE 3(a) Selected paired segments analysis at disaggregate levels:- assessed by O-D-segment flows.
TRB 2013 Annual Meeting
Paper revised from original submittal.
0.2900
0.2700
-14 1E-11
-11 1E-08
-8
-5
-2 1E+01
1 1E+04
4
1E-14
1E-05
1E-02
Total O-D flows (vph)
3,963 disaggregate O-D pairs
0.7000
0.6800
0.6600
1
-2
-5
-8
y = 1.000x + 0.279
R² = 1.000
-11
-14
-11
-8
-5
-2
1
4
O-D-segment1 flows (vph)
O-D-segment1- flows vs. O-D-segment 2-flows : log : CS-0.20
10,331 disaggregate O-D pairs
0.6820
0.6810
0.6800
0.6790
0.6780
1E-14
1E-05
1E-02
-14 1E-11
-11 1E-08
-8
-5
-2 1E+01
1 1E+04
4
Total O-D flows (vph)
Total O-D flows vs. proportion of flows on segment 2 : semi-log : CS-0.10
3,963
disaggregate
O-Ds
3,963
disaggregate
O-D
pairs
4
-14
0.3180
1E-14
1E-08
1E-05
1E-02
-14 1E-11
-11
-8
-5
-2 1E+01
1 1E+04
4
Total O-D flows (vph)
O-D-segment2 flows (vph)
O-D-segment2 flows (vph)
0.6400
1E-14
1E-05
-14 1E-11
-11 1E-08
-8
-5 1E-02
-2 1E+01
1 1E+04
4
Total O-D flows (vph)
Total O-D flows vs. proportion of flows on segment 2 : semi-log : CS-0.20
0.3190
Total O-D flows vs. proportion of flows on segment 1 : semi-log : CS-0.10
O-D-segment 2 flows/O-D flows
O-D-segment 2 flows/O-D flows
Total O-D flows vs. proportion of flows on segment 1 : semi-log : CS-0.20
0.3200
1
-2
-5
-8
y = 1.000x + 0.327
R² = 1.000
-11
-14
-11
-8
-5
-2
1
4
0.1570
0.1565
0.1560
0.1555
0.1550
1E-14
1E-05
1E-02
-14 1E-11
-11 1E-08
-8
-5
-2 1E+01
1 1E+04
4
Total O-D flows (vph)
Total O-D flows vs. proportion of flows on segment 1 : semi-log : CS-0.05
21,471 disaggregate O-D pairs
0.8450
0.8445
0.8440
0.8435
0.8430
0.8425
1E-14
1E-05
-14 1E-11
-11 1E-08
-8
-5 1E-02
-2 1E+01
1 1E+04
4
Total O-D flows (vph)
Total O-D flows vs. proportion of flows on segment 2 : semi-log : CS-0.05
21,471
disaggregate
O-Ds
21,471
disaggregate
O-D
pairs
4
1
-2
-5
-8
y =1.000x + 0.732
R² = 1.000
-11
-14
O-D-segment1 flows (vph)
O-D-segment1- flows vs. O-D-segment 2-flows : log : CS-0.10
21,471 disaggregate O-D pairs
0.1575
10,331
disaggregate
O-Ds
10,331
disaggregate
O-D
pairs
4
-14
O-D-segment 1 flows/O-D flows
0.3100
0.3210
O-D-segment 2 flows/O-D flows
0.3300
10,331 disaggregate O-D pairs
0.3220
O-D-segment2 flows (vph)
3,963 disaggregate O-D pairs
0.3500
11
O-D-segment 1 flows/O-D flows
O-D-segment 1 flows/O-D flows
Aroon Aungsuyanon, David Boyce, and Bin Ran
-14
-11
-8
-5
-2
1
4
O-D-segment1 flows (vph)
O-D-segment1- flows vs. O-D-segment 2-flows : log : CS-0.05
FIGURE 3(b) Selected paired segments analysis at disaggregate levels:- assessed by proportions of flows.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
12
regarded as the computed proportionality. Since the proportions must lie between 0 and 1, a
linear scale is suitable for used to display the magnitude of the differences in the proportions
among O-D pairs
The semi-log plots on the first two rows exhibit notable patterns of reverse symmetry in
which a plot in the top row is a mirror image of that in the middle row. These reverse symmetry
patterns result from the complementary effects of flow shift operations made between any pair of
routes. Shifting the same flows from a higher cost route to a lower cost route does not result in
any change in the total O-D flow. As a result, the proportions of flow for routes traversing
segment 1 and 2 are complementary in the sense that two proportions, when added together, will
equal unity. The presence of a reverse symmetry pattern ensures that TAPAS behaves properly
and the properties of the PASs are as expected. As can be seen, the variations in the proportions
of flows only occur with regard to O-D pairs with flows less than 1E-3 vph, whereas O-D pairs
with flows greater than 1E-3 vph seem to correspond best to the exact proportions as indicated
by the horizontal alignment of the data points. Since these variations are only shown to be in the
very narrow range of very small values, they are most likely caused by rounding errors, which
could not be eliminated perfectly. These extremely small variations can be largely attributed to
the high precision of TAPAS in computing the proportional UE route flow solution. The very
small O-D flows are a characteristic of how the trip matrices were computed.
The plots in the bottom rows of Figure 3(b) show exactly the same points as those in
Figure 3(a), but they are shown in log scale to allow a closer look at O-D pairs with small
segment flows. The values shown on the axes are the orders of magnitude of O-D flows
traversing the two segments. As indicated by the coefficient of linear regression equations and
corresponding values of R2, all the points in each solution perfectly fit the 45-degree line, but
they belong to different intercepts on the y-axis. The y-intercept of the log plots corresponds to
the slope of the linear plots. Positive signs of the y-intercepts simply indicate there are more O-D
flows on route traversing segment 2 than segment 1. An agreement of all data points with the 45degree line ensures that O-D flows are split proportionately between the two alternative
segments.
EFFECTS OF PROPORTIONALITY ON INDIVIDUAL LINK FLOWS
Proportionality affects not only the flows on routes traversing the two segments, but also the
flows on individual links of routes traversing segments. Figure 4(a) shows scatter plots for flows
on routes traversing a selected segment aggregated by links. Each data point represents one link.
For the plots in the first two rows, the x-axis represents the link number; whereas the y-axis
represents the total flows on all routes traversing both that link and the segment on the log scale.
For the plots in the bottom row, the x and y axes simply correspond to the vertical axes of the top
and middle rows but plotted on the linear scale. The slope at each point corresponds to the ratio
of total flows on a link for routes traversing segment 2 to total flows on that link for routes
traversing segment 1. Parameters of linear regression are estimated and shown in the body of the
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
13
plots. Note that total number of links shown on the top of each plot does not include links that
are part of the selected PAS nor zone connectors. As compared within the solution, the patterns
of total link flows given by the plot in the top row are exactly identical to those in the middle
row. Identical total link flow patterns are a direct consequence of proportionality, which can be
observed by a single alignment of points in the bottom row. Notice that a linear regression
equation in the bottom row of Figure 4(a) is the same as that in Figure 3(a), reaffirming
proportionality on link flows. Without the inclusion of proportionality, these link flow patterns
could be arbitrarily different. As compared across the solutions, the wider ranges of total link
flows found in the 0.20 solution become narrower and are shifted upwards in response to
growing congestion in the 0.10 and 0.05 solutions. However, flows on links for routes traversing
segment 2 are higher than those traversing segment 1 and the relative total flows on each link are
equal to the slope of the line shown in the bottom row.
To allow one to observe spatial distributions of individual link flows over the entire road
network for each solution, each data point of the plots in the top and middle row of Figure 4(a) is
slightly modified and translated onto a map of the physical road network. In the map, the total
flow on a given link is shown as a percentage of segment flows. Displaying all possible values of
total link flows relative to segment flows on map is rather cumbersome and not necessary.
Therefore, the values shown are categorized into eight different scales, ranging from 0 percent
for links that are unused by the routes traversing a selected segment to 100 percent for links that
are part of a selected segment. Maps corresponding to each solution are, for ease of comparisons,
placed side by side and shown in Figure 4(b). Supplementing each map, two sets of solution
attributes are summarized on the upper right of the map: attributes for a selected segment shown
on the top half are reiterated to enhance interpretations of proportionality, and attributes for the
routes traversing a selected segment shown on the bottom half are summarized to provide
fundamental insights into effects of proportionality on route flow solutions. In each solution, the
black lines represent a set of PASs that comprises initial route segments taken from any origin to
diverge node 12389. Since the initial route segments are commonly used by paired routes, both
sets of PASs are common regardless of which selected segments are used. As a result, unique UE
route segments taken from any origin to diverge node 12389 are identical. The area marked by a
small circle represents the geographical location of the selected PAS. Notice that the two
previously excluded links that are part of a selected segment are currently included in the total
number of links. As seen in each map, total route flows equal segment flows. Minimum route
flows indicate that no route flow over the two selected segments is zero, confirming the
fundamental principle of a PAS. Identical link flow patterns may be observed between the two
maps. Since proportionality is applied in each solution, the percentage of flows on any specific
link relative to flows on segment 1 must be identical to the percentage of flows on the same link
relative to flows on segment 2. Without the imposition of proportionality, arbitrary differences in
flows on links for routes traversing segment 1 and 2 would be expected. Equal mean travel costs
for routes traversing segment 1 and 2 are the results of imposed proportionality as well.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
8,207 Links
O-D-link-segment 1 flows
(vph)
O-D-link-segment 1 flows
(vph)
1E-07
-7
1E-10
-10
1E-10
-10
1E-10
-10
0
10000
20000
30000
40000
1E-13
-13
8,207 Links
10000
40000
10000
20000
30000
40000
Link Number
Total flows on links for routes traversing segment 2 : CS-0.20
400
y = 1.913x
R² = 1.000
300
200
100
0
100
200
300
400
500
600
Flows
O-D-link-segment
on routes traversing
1 flows
segment 1 aggregated
by link
(vph)
Relative flows on links for routes(vph)
traversing segment 2 to 1: CS-0.20
10000
20000
30000
40000
1E-13
-13
20000
30000
40000
Link Number
Total flows on links for routes traversing segment 2 : CS-0.05
8,495 Links
600
O-D-link-segment 2 flows
(vph)
500
0
Link Number
Total flows on links for routes traversing segment 2 : CS-0.10
8,207 Links
600
400
y = 2.125x
R² = 1.000
300
200
100
0
100
200
300
0
10000
8,916 Links
600
500
0
8,916 Links
1E-10
-10
O-D-link-segment 2 flows
(vph)
0
Total flows on links for routes traversing segment 1 : CS-0.05
Link Number
1E-07
-7
1E-10
-10
1E-13
-13
40000
1E-04
-4
1E-07
-7
1E-10
-10
20000
Link
Number30000
Link Number
1E-01
-1
1E-04
-4
1E-07
-7
10000
1E+02
2
1E-01
-1
1E-04
-4
0
O-D-link-segment 2 flows
(vph)
1E+02
2
1E-01
-1
1E-13
-13
1E+05
5
O-D-link-segment 2 flows
(vph)
1E+02
2
Link20000
Number 30000
8,495 Links
1E+05
5
O-D-link-segment 2 flows
(vph)
5
1E+05
0
Link Number
Total flows on links for Link
routesNumber
traversing segment 1 : CS-0.10
Link Number
Total flows on links for routes traversing segment 1 : CS-0.20
O-D-link-segment 2 flows
(vph)
1E-04
-4
1E-07
-7
1E-07
-7
0
1E-01
-1
-4
1E-04
1E-04
-4
1E-13
-13
1E+02
2
-1
1E-01
1E-01
-1
8,916 Links
1E+05
5
2
1E+02
1E+02
2
1E-13
-13
8,495 Links
5
1E+05
O-D-link-segment 1 flows
(vph)
1E+05
5
14
400
500
600
O-D-link-segment 1 flows
(vph)
Relative flows on links for routes traversing segment 2 to 1: CS-0.10
500
400
y = 5.396x
R² = 1.000
300
200
100
0
0
100
200
300
400
500
O-D-link-segment 1 flows
(vph)
600
Relative flows on links for routes traversing segment 2 to 1: CS-0.05
FIGURE 4(a) Plots showing the effects of proportionality on individual link flows for the three solutions.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
CS-0.20
15
Attributes of a selected segment:
No. of links: 2
Link lengths: 1.49 mi
Segment flows: 284.249 vph
Segment cost: 2.676 min
Segment proportion: 0.343
No. of relevant origins: 1781
No. of relevant destinations: 1
No. of O-D pairs: 1,781
No. of disaggregate O-D pairs: 3,963
Attributes of relevant routes:
No. of links: 8,209
No. of PASs: 449
No. of unique UE routes: 3,963
Flows on all routes: 284.249 vph
Maximum route flows: 56.422 vph
Minimum route flows: 9.870E-15 vph
Mean route travel distance: 65.745 mi
Mean route travel cost: 73.784 min
Mean route travel speed: 51.804 mph
Legend:
CS-0.20
Attributes of a selected segment:
No. of links: 2
Link lengths: 1.07 mi
Segment flows: 543.627 vph
Segment cost: 2.676 min
Segment proportion: 0.657
No. of relevant origins: 1,781
No. of relevant destinations: 1
No. of O-D pairs: 1,781
No. of disaggregate O-D pairs: 3,963
Attributes of relevant routes:
No. of links: 8,209
No. of PASs: 449
No. of unique UE routes: 3,963
Flows on all routes: 543.627 vph
Maximum route flows: 107.907 vph
Minimum route flows: 1.910E-14 vph
Mean route travel distance: 65.325 mi
Mean route travel cost: 73.784 min
Mean route travel speed: 50.400 mph
Legend:
Link flows relative to segment flows
Link flows relative to segment flows
Segment 1
CS-0.10
0%
(0,1]%
(1,2]%
0%
(0,1]%
(1,2]%
(2,5]%
(5,10]%
(10,20]%
(2,5]%
(5,10]%
(10,20]%
(20,50]%
(50,100]%
(20,50]%
(50,100]%
Links that are part of a considered PAS
Links that are part of a considered PAS
Segment 2
Links that are part of other PASs
Attributes of a selected segment:
No. of links: 2
Link lengths: 1.49 mi
Segment flows: 255.189 vph
Segment cost: 12.673 min
Segment proportion: 0.320
No. of relevant origins: 1,781
No. of relevant destinations: 1
No. of O-D pairs: 1,781
No. of disaggregate O-D pairs: 10,331
Attributes of relevant routes:
No. of links: 8,497
No. of PASs: 651
No. of unique UE routes: 10,331
Flows on all routes: 255.189 vph
Maximum route flows: 19.351 vph
Minimum route flows: 2.180E-12 vph
Mean route travel distance: 61.006 mi
Mean route travel cost: 76.235 min
Mean route travel speed: 48.523 mph
Legend:
Links that are part of other PASs
CS-0.10
Attributes of a selected segment:
No. of links: 2
Link lengths: 1.07 mi
Segment flows: 542.283 vph
Segment cost: 2.673 min
Segment proportion: 0.680
No. of relevant origins: 1,781
No. of relevant destinations: 1
No. of O-D pairs: 1,781
No. of disaggregate O-D pairs: 10,331
Attributes of relevant routes:
No. of links: 8,497
No. of PASs: 651
No. of unique UE routes: 10,331
Flows on all routes: 542.283 vph
Maximum route flows: 41.120 vph
Minimum route flows: 5.980E-12 vph
Mean route travel distance: 60.586 mi
Mean route travel cost: 76.235 min
Mean route travel speed: 47.488 mph
Legend:
Link flows relative to segment flows
Link flows relative to segment flows
Segment 1
CS-0.05
0%
(0,1]%
(1,2]%
0%
(0,1]%
(1,2]%
(2,5]%
(5,10]%
(10,20]%
(2,5]%
(5,10]%
(10,20]%
(20,50]%
(50,100]%
(20,50]%
(50,100]%
Links that are part of a considered PAS
Links that are part of a considered PAS
Segment 2
Links that are part of other PASs
Attributes of a selected segment:
No. of links: 2
Lengths: 1.49 mi
Segment flows: 108.502 vph
Segment cost: 2.667 min
Segment proportion: 0.156
No. of relevant origins: 1,775
No. of relevant destinations: 1
No. of O-D pairs: 1,775
No. of disaggregate O-D pairs: 21,471
Attributes of relevant routes:
No. of links: 8,918
No. of PASs: 1,122
No. of unique UE routes: 21,471
Flows on all routes: 108.502 vph
Maximum route flows: 5.813 vph
Minimum route flows: 1.350E-11 vph
Mean route travel distance: 63.229 mi
Mean route travel cost: 102.104 min
Mean route travel speed: 42.757 mph
Legend:
Links that are part of other PASs
CS-0.05
Attributes of a selected segment:
No. of links 2
Link lengths: 1.07 mi
Segment flows: 585.433 vph
Segment cost: 2.667 min
Segment proportion: 0.884
No. of relevant origins: 1,775
No. of relevant destinations: 1
No. of O-D pairs: 1,775
No. of disaggregate O-D pairs: 21,471
Attributes of relevant routes:
No. of links: 8,918
No. of PASs: 1,122
Total (consistent UE) routes: 21,471
Flows on all routes: 585.433 vph
Maximum route flows: 31.367 vph
Minimum route flows: 7.270E-12 vph
Mean route travel distance: 62.809 mi
Mean route travel cost: 102.104 min
Mean route travel speed: 42.014 mph
Legend:
Link flows relative to segment flows
Link flows relative to segment flows
0%
(0,1]%
(1,2]%
0%
(0,1]%
(1,2]%
(2,5]%
(5,10]%
(10,20]%
(2,5]%
(5,10]%
(10,20]%
(20,50]%
(50,100]%
(20,50]%
Segment 1
(50,100]%
Links that are part of a considered PAS
Links that are part of other PASs
Links that are part of a considered PAS
Segment 2
Links that are part of other PASs
FIGURE 4(b) Illustrations for the effect of proportionality on individual link flows for the three solutions.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
16
By comparing between solutions, one may observe that the number of PASs and the
number of uniquely determined UE routes depend to a substantial extent on the congestion levels
in the network. Nonetheless, the number of PASs in each solution is relatively small in
comparison to the number of uniquely determined UE routes. Substantially higher values of
mean route travel costs in the 0.05 solution indicate to what extent the travelers in this solution
are insensitive to travel costs, as compared with those in the 0.20 solution, which is considered to
be somewhat realistic. For uncongested networks higher route travel costs primarily result from
longer route travel distances; however, these characteristics are not necessarily valid under
congested networks. Therefore, the levels of congestion between the solutions can be
alternatively measured by mean route travel speed, which unifies both route travel distances and
costs.
In each solution, there are three major corridors in which flows on links are distinct from
each other. Each corridor in each solution has a fairly close correspondence to its geographical
location. However, the corridors from the heavily congested network appear to be longer than
those from the moderately or less congested network. The general impressions regarding spatial
arrangements of PASs in the network are that PASs with long total link lengths are mostly
scattered at outer suburbs of the region where flows are initiated, while those with short total link
lengths are predominantly located at inner suburbs and the central city where the main corridors
are formed. As congestion increases, PASs with long or extremely long total link lengths tend to
occur less frequently, whereas those with short total link lengths prevail throughout the network.
The reason is that the routes with long or extremely long total link lengths could no longer be
least cost as congestion increases. In addition, with increasing congestion there is an increasing
number of PASs along the corridors; most of them are relatively short, thereby providing
travelers with more alternatives to avoid congested segments of the corridors or to alleviate
congestion over the corridors.
DIFFERENCES BETWEEN COMPUTED AND EXACT PROPORTIONALITY
Selected paired segments analysis at disaggregate levels in the previous section shows that there
exist minimal variations in computed and exact proportionality for O-D pairs with small flows
and there seem to be very perfect adherences to exact proportionality for O-D pairs with large
flows. When multiplying exact proportions of flows for each O-D pair with corresponding O-D
flows, O-D pairs with small flows do not seem to cause any major difference between computed
and exactly proportional route flows. However, the differences may be substantial for O-D pairs
with large flows and these could be consequential in applications. Therefore, investigating the
magnitude of differences between the two proportional route flows is essential in determining
whether the extremely small variations of computed proportionality have any significant
practical implications in interpreting uniquely determined UE route flows or whether they can
simply be neglected from further considerations. Recall that the computed proportionality
involves the flows from only one disaggregate O-D pair traversing each segment, whereas the
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
17
exact proportionality involves the flows from all disaggregate O-D pairs traversing the same
segment.
Figures 5(a)-5(c) show the magnitude of differences between computed and exactly
proportional route flows for the 0.20, 0.10, and 0.05 solutions respectively. Each data point
represents one O-D pair. In each plot, the x-axis represents total O-D flows. The y-axis
represents the absolute difference between computed and exactly proportional route flows for
routes traversing a selected segment. Plots on the left and right columns correspond to the routes
traversing segment 1 and 2 respectively. Those on the top row include all routes traversing a
selected segment and having either positive or negative differences of flows. In order to help
explore fundamental characteristics of the differences, positive and negative values are plotted
separately and shown in the middle and bottom rows of each figure. Each point is plotted on log
scale to enhance the visualization of extremely small differences for O-D pairs with extremely
small flows. All plots are on the same scales of x and y axes for easy comparison. Absolute
values on the y-axis are simply needed to enable negative differences to be displayed on log
scale. Figures 5(a)-5(c) should be interpreted in conjunction with Figure 3(b).
The plots in each solution show that when computed O-D flows for routes traversing one
segment exceeds the exactly proportional route flows in the positive direction, those traversing
the other segment will evenly exceed the exactly proportional route flows in the negative
direction. This remarkable characteristic is seen much more clearly by comparing plots side by
side as shown in the middle and bottom rows of each solution. Precise similarity of the visual
appearances of the patterns and their magnitude of differences are naturally governed by
complementarity of total O-D flows over the two selected segments. Failure to comply with
complementarity will lead to violations of proportionality in the solution, and arbitrary
differences in the patterns would occur. Therefore, similarity of the differences between two
proportional route flows over the same selected segments is an essentially desirable characteristic
for any solution that aims to satisfy proportionality.
As seen in the plots of each figure, computed and exactly proportional route flows differ
marginally in the range of very small values between -18 and -3 orders of magnitude. The
observed magnitude of differences is influenced more heavily by increasing O-D flows. In light
of these results, this magnitude of differences appears to be insufficiently significant to influence
the entire analysis. It is of interesting to note that the notable differences between the two
proportional UE route flows for O-D pairs with total flows greater than 1E-3 veh/h, which seem
to correspond best to the exact proportionality as shown by the precise single horizontal
alignment of data points for plots in the first two rows of Figure 3(b), reveal that there seems to
be no perfect fit to the exact proportionality. This characteristic was not expected at the outset of
this study.
TRB 2013 Annual Meeting
Paper revised from original submittal.
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| -2.524E-04 |
3,963 disaggregate O-D pairs
1E+000
18
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| +2.524E-04 |
Aroon Aungsuyanon, David Boyce, and Bin Ran
1E-03
-3
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-15
-15
1E-18
-18
-14
1E-14
-11
1E-11
-8
-5
-2
1E-08
1E-05
1E-02
Total O-D flows (vph)
(284.249)
1
1E+01
4
1E+04
1E-18
-18
1E-14
-14
1E-03
-3
1E-11
-11
1,763 disaggregate O-D pairs
1E+000
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| -2.545E-04 |
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(543.627)
All routes traversing segment 2 with either positive or negative
differences of flows
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| -2.060E-06 |
1,763 disaggregate O-D pairs
1E+000
1E-15
-15
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(9.154E-03)
Only the routes traversing segment 1 with positive
differences of flows
2,200 disaggregate O-D pairs
1E+00
0
1E-03
-3
1E-18
-18
1E-14
-14
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| +2.545E-04 |
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| +2.060E-06 |
All routes traversing segment 1 with either positive or negative
differences of flows
1E-18
-18
1E-14
-14
3,963 disaggregate O-D pairs
1E+00
0
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(1.751E-02)
Only the routes traversing segment 2 with negative
differences of flows
2,200 disaggregate O-D pairs
1E+000
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-15
-15
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(284.240)
Only the routes traversing segment 1 with negative
differences of flows
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(543.610)
Only the routes traversing segment 2 with positive
differences of flows
FIGURE 5(a) Differences between computed and exactly proportional route flows for the 0.20 solution.
TRB 2013 Annual Meeting
Paper revised from original submittal.
1E+00
0
1E-03
-3
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-18
-18
1E-14
-14
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(255.189)
All routes traversing segment 1 with either positive or negative
differences of flows
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| +1.345E-03 |
1E-11
-11
5,009 disaggregate O-D pairs
1E+000
1E-08
-8
1E-05
-5
1E-02
-2
1E+01
1
1E+04
4
Total O-D flows (vph)
(542.283)
All routes traversing segment 2 with either positive or negative
differences of flows
5,009 disaggregate O-D pairs
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| -1.271E-03 |
1E-11
-11
1E+00
0
1E-03
-3
1E-15
-15
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(117.065)
Only the routes traversing segment 1 with positive
differences of flows
5,322 disaggregate O-D pairs
1E+000
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(248.769)
Only the routes traversing segment 2 with negative
differences of flows
5,322 disaggregate O-D pairs
1E+00
0
1E-03
-3
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-15
-15
1E-18
-18
1E-14
-14
1E-18
-18
1E-14
-14
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| -1.345E-03 |
1E-15
-15
1E-18
-18
1E-14
-14
10,331 disaggregate O-D pairs
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| -7.426E-05 |
10,331 disaggregate O-D pairs
1E+00
0
19
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| +1.271E-03 |
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| +7.426E-05 |
Aroon Aungsuyanon, David Boyce, and Bin Ran
1E-11
-11
1E-08
1E-05
1E-02
-8
-5
-2
Total O-D flows (vph)
(138.124)
1E+01
1
1E+04
4
Only the routes traversing segment 1 with negative
differences of flows
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(293.514)
Only the routes traversing segment 2 with positive
differences of flows
FIGURE 5(b) Differences between computed and exactly proportional route flows for the 0.10 solution.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
1E+00
0
1E-03
-3
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-15
-15
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
-8
-5
-2
Total O-D flows (vph)
(108.502)
1E+01
1
1E+04
4
1E-18
-18
1E-14
-14
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| +3.325E-05 |
7,033 disaggregate O-D pairs
1E+000
1E-11
-11
1E+01
1
1E+04
4
7,033 disaggregate O-D pairs
1E+000
1E-03
-3
1E-08
1E-05
1E-02
-8
-5
-2
Total O-D flows (vph)
(585.433)
All routes traversing segment 2 with either positive or negative
differences of flows
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| -3.325E-05 |
All routes traversing segment 1 with either positive or negative
differences of flows
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-15
-15
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
-8
-5
-2
Total O-D flows (vph)
(8.870E-02)
1E+01
1
1E+04
4
1E-18
-18
1E-14
-14
1E-08
1E-05
1E-02
-8
-5
-2
Total O-D flows (vph)
(4.788E-01)
1E+01
1
1E+04
4
14,438 disaggregate O-D pairs
1E+000
1E-03
-3
1E-03
-3
1E-06
-6
1E-06
-6
1E-09
-9
1E-09
-9
1E-12
-12
1E-12
-12
1E-15
-15
1E-18
-18
1E-14
-14
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| +2.213E-04 |
14,438 disaggregate O-D pairs
1E+000
1E-11
-11
Only the routes traversing segment 2 with negative
differences of flows
Only the routes traversing segment 1 with positive
differences of flows
|Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| -2.213E-04 |
21,471 disaggregate O-D pairs
|Difference of O-D-segment 2 flows from
exactly proportional route flows (vph)|
| +1.880E-04 |
| Difference of O-D-segment 1 flows from
exactly proportional route flows (vph)|
| -1.880E-04 |
21,471 disaggregate O-D pairs
1E+000
20
1E-15
-15
1E-11
-11
1E-08
1E-05
1E-02
-8
-5
-2
Total O-D flows (vph)
(108.413)
1E+01
1
Only the routes traversing segment 1 with negative
differences of flows
1E+04
4
1E-18
-18
1E-14
-14
1E-11
-11
1E-08
1E-05
1E-02
1E+01
1E+04
-8
-5
-2
1
4
Total O-D flows (vph)
(584.955)
Only the routes traversing segment 2 with positive
differences of flows
FIGURE 5(c) Differences between computed and exactly proportional route flows for the 0.05 solution.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
21
In order to determine overall significances of the magnitude of differences between the
two proportional UE route flows in each solution, the number in parentheses on the x-axis label
shows the total O-D flows on all the routes traversing one of the two alternative segments; the
number in absolute value symbol on the y-axis label shows either net or total differences of the
two proportional UE flows on all routes traversing a corresponding segment. Absolute aggregate
differences of the two proportional UE route flows with respect to total O-D flows, which is the
ratio of the number on the y-axis label to the number on the x-axis label, fall in the very narrow
range of 1.369E-7 to 1.733E-6 for the net differences and in the somewhat wider range of
3.782E-7 to 3.749E-4 for the total differences. Such relatively small ratios appear to justify the
use of uniquely determined UE route flows yielded by TAPAS in practical applications. Notice
that the numbers on the x-axis labels are actually the total flows on a segment found previously
in Figure 2. Since the plots in the top row are a composite of plots in the middle and bottom
rows, the sum of the numbers on the x and y axis labels for plots in the middle and bottom rows
is equal to the number found on the corresponding x and y axis labels for plots in the top row.
CONCLUSIONS, USEFULNESS, AND FUTURE RESEARCH
This empirical research aims to advance understandings of UE traffic assignments with uniquely
determined route flows through various assessments of adherence to the condition of
proportionality, which are performed for one selected pair of alternative segments over three
congestion scenarios. The results show that route flows over the two segments determined by
TAPAS nearly perfectly adhere to exact proportionality. Due to numerical errors, only minor
differences occur between computed and exactly proportional UE route flows. Systematic
solution characteristics between routes traversing each of the two alternative segments assure
that TAPAS behaves properly according to the condition of proportionality. Insights from these
empirical results may help transportation planning professionals to be aware of the magnitude of
differences in UE route flows based on the proportionality condition, and to decide whether such
differences are important for their analyses. The results may also be useful to software
developers in seeking improved adherence to proportionality in route flow solutions. Essentially
desirable solution characteristics of TAPAS can be used as a basis for transportation planning
professionals in differentiating uniqueness from non-uniqueness of route flows in UE traffic
assignment.
Since the results presented in this paper pertain only to one PAS, studies with more PASs
are warranted to establish mere definite conclusions. Perhaps, a set of PASs with common merge
and diverge nodes may merit future explorations. Another research direction is to determine an
acceptable level of proportionality in a solution by seeking how large the relative differences
between the computed and exactly proportional UE route flows should be tolerated in practical
applications without changing the results of aggregate benefits. Experimenting with placing a
lower bound on the O-D flows, such as 1E-4 vph, would be also useful to know whether such a
bound would improve the convergence of proportionality. Assessing whether the condition of
proportionality is observed in reality is a good subject for future research as well.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Aroon Aungsuyanon, David Boyce, and Bin Ran
22
ACKNOWLEDGEMENTS
The authors are grateful to Dr. Hillel Bar-Gera for the opportunity to use a research tool of
TAPAS and for his ongoing technical advice. It is a pleasure to acknowledge the Chicago Area
Transportation Study for the provision of the Chicago regional road network and origindestination trip tables. Comments and advices of three anonymous reviewers are also gratefully
acknowledged.
REFERENCES
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University of Illinois at Chicago, Chicago, IL., 1999.
2. Bar-Gera, H. Primal Method for Determining the Most Likely Route Flows in Large Road
Networks. Transportation Science, Vol. 40, 2006, pp. 269-286.
3. Bar-Gera, H. Traffic Assignment by Paired Alternative Segments. Transportation Research
Part B, Vol. 44, 2010, pp. 1022-1046.
4. Boyce, D., Y. Nie, H. Bar-Gera, Y. Liu, and Y. Hu. Field Test of a Method for Finding Consistent
Route Flows and Multiple-Class Link Flows in Road Traffic Assignments. Final Report to the
Federal Highway Administration, Washington, DC, March
2010. http://www.transportation.northwestern.edu/docs/research/Boyce_FieldTestConsistentR
outeFlows.pdf (Accessed July 25, 2012).
5. Bar-Gera, H., Nie, Y., Boyce, D., Hu, Y., and Liu, Y. Consistent Route Flows and the
Condition of Proportionality. In TRB 90th Annual Meeting Compendium of Paper. DVDROM. Transportation Board of the National Academies, Washington, D.C., 2010, paper #101526.
6. Bar-Gera, H., Boyce, D., and Nie, Y. User-equilibrium Route Flows and the Condition of
Proportionality. Transportation Research Part B, Vol. 46, 2012, pp. 440-462.
7. Rossi, T.F., McNeil, S., Hendrickson, C. Entropy Model for Consistent Impact Fee
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8. Janson, B.N. Most Likely Origin-destination Link Uses from
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9. Bar-Gera, H., and Boyce, D. Some Amazing Properties of Road Traffic Network Equilibria.
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Berlin, 2007, pp. 305-335.
10.Aungsuyanon, A., Boyce, D., and Ran, B. Solution Attributes of the Static Deterministic
Traffic Assignment Problem with Unique Route Flows Determined by the Condition of
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Transportation Board of the National Academies, Washington, D.C., 2012, paper #12-1434.
TRB 2013 Annual Meeting
Paper revised from original submittal.