ASYMMETRICALLYDISTRIBUTED VARIATIONS IN
TRAVELERPERCEIVED TRAVEL TIMES IN
STOCHASTIC USEREQUILIBRIUM TRAFFIC ASSIGNMENT
By
MINTANG LI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
ACKNOWLEDGMENTS
I wish to express my gratitude to certain individuals for contributions in the
preparation of this dissertation. Extreme gratitude is expressed to Dr. Gary Long for his
inspiration in this endeavor, along with his excellent review of the original manuscript. I also
wish to thank Dr. Charles Wallace and Dr. ChengTin Gan for their continued encouragement
and strong support during my doctoral research study.
Extreme gratitude is expressed to members of my supervisory committeeDr. Gary
Long (chairman), Professor Kenneth G. Courage, Dr. Bon A. Dewitt, Dr. Sherman X. Bai,
and Dr. Mark C. K. Yangfor their review of the original manuscript and many useful
suggestions.
Special thanks is given to my parents for their patience and for supporting me in many
ways during my studies.
Finally, a very special thanks to my wife, LeeFang, and my son, Andrew, for their
love and support.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ....
LIST OF TABLES .........
LIST OF FIGURES .........
ABSTRACT ..............
CHAPTERS
1 INTRODUCTION ......
Background ......
Problem Statement .
Goal and Objectives
Study Area .......
Organization ......
2 BACKGROUND AND STATISTICAL TOOLS ...
2.1 Discrete Choice M odels ...................
2.1.1 Introduction ...................
2.1.2 Gumbel Distribution .............
2.1.2.1 Derivation of multinomial logit
2.2 Stochastic
2.2.1
2.2.2
2.2.3
2.2.4
2.1.2.2 Derivation of binary asymmetrical Gumbel
distribution .................
2.1.2.3 Derivation of multiasymmetrical Gumbel
distribution .................
2.1.2.4 Method of probabilityweighted moments
U ser Equilibrium .........................
Introduction ............................
Stochastic Network Loading Procedures .......
SUE Solution Algorithm ................. .
Define Route Choice SetCompetitive Paths . .
. . . . . . 1 9
. . . . . . 2 0
. . . . . . 2 1
. . . . . . 2 1
. . . . . . 2 1
. . . . . 24
. . . . . 2 6
. . . . . . 2 8
Paeg
2.2.5 Variation in CORSIM ................................ 30
2.2.6 Variation in Perceived Travel Time .................. 37
2.3 N onparam etric Statistics ................................ 45
2.3.1 W ilcoxon SignedRank Test ........................... 47
2 .3.2 Sign T est .............................. ......... 48
2.4 Percent Root M ean Square Error ................................ 49
3 NETW ORK DESCRIPTION ..................... ...... .......... 51
3.1 G ainesville N etw ork ................................ . 51
3.2 FTO W N N etw ork ........................................ 55
4 SUE ALGORITHM AND MODEL INITIALIZATION ................. 57
4.1 SUE Algorithm .............
4.2 SUE Initialization Procedures .
4.2.1 Read Profile .........
4.2.2 Read Speed/Capacity ..
4.2.3 Read Vfactors ........
4.2.4 Read Assigned Volumes
4.2.5 Make Batch .........
4.2.6 Create Links File ......
4.2.7 Change HNET Controls
4.2.8 Read Mode ..........
5 COMPETITIVE PATH ENUMERATION ........................... 73
5.1 Introduction ....... .. .. .................. .......... 73
5.2 SUE Competitive Path Enumeration Procedures .............. . 74
5.2.1 Change HPATH Controls ....................... 76
5.2.2 Line .................... . . ............. ..... 76
5.2.3 P ath L ist ....................... ....... ......... 77
5.2.4 Check Path Duplication ............................... 78
5.2.5 Change LINK S File ........ ............. ......... 79
5.3 Competitive Path Enumeration Results for the Gainesville
N etw ork ............. .. ............... ......... 79
5.3.1 Overestimated Travel Times from the BPR Formula ......... 79
5.3.2 Roadway System Detail .............................. 82
5.3.3 Tendency to Convergence ....................... 87
6 TRAVEL TIME VARIATION .................................... 89
6.1 Introduction ............................................. 89
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
6.2 SUE Travel Time Variation Procedures .................... 89
6.2.1 Read Random Seed .................................. 91
6.2.2 M ake CORSIM Input File ....................... 92
6.2.2.1 Check need to simulate ..................... 92
6.2.2.2 Get simulated travel times from memory ............ 93
6.2.2.3 Create record types . . . . . . . 94
6.2.2.4 Run CORSIM .......... ......... ............ 99
6.2.2.5 Get simulated travel times from CORSIM output ..... 99
6.2.2.6 Get travel times from incomplete CORSIM simulation 101
6.3 Travel Time Variation Results for the Gainesville Network ......... 101
7 ROUTE CHOICE M ODEL ...................................... 106
7.1 Introduction ......................... . ... ..... ....... 106
7.2 SUE Route Choice Implementation Procedures ............... 106
7.2.1 Get Gumbel Parameters ............................. 106
7.2.2 G et Path Probability ................................ 108
7.2.3 Trip Allocation ...... ..... .... ....... ......... 110
7.3 Route Choice Implementation Results for the Gainesville
N etw ork ............................... ......... 112
8 CONVERGENCE TEST .............................. ........ 116
8.1 SUE Test Convergence Procedures .......................... 116
8.1.1 Update LINK File .................. ............... 116
8.1.2 Check Convergence ............................ 117
8.2 Test Convergence Results for the Gainesville Network ............ 117
8.3 Test Convergence Results for the FTOWN Network ............. 118
8.4 Examine Differences in Assigned Trips From Successive Iterations ... 119
9 M ODEL VALIDATION ........................................ 122
9.1 Introduction . . . . ..................... . 122
9.2 Percent Root M ean Square Error ............................ 122
9 .3 Sign T est .................................... ........ 124
9.4 Computer Running Time .................................. 126
10 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .......... 130
10.1 Sum m ary ............................................. 130
10.2 C onclu sions ............................................ 13 1
10.3 R ecom m endations ....................................... 133
APPENDIX GAMMA FUNCTION .................................... 136
R E FE R E N C E S ..................................................... 138
BIOGRAPHICAL SKETCH .................................... . 141
LIST OF TABLES
Table page
21 Variance Equality Test Results for Simulated Travel Times from CORSIM ... 33
22 KolmogorovSmirnov Test Results for 3,000 vph Mainline Entry Volume .... 34
23 KolmogorovSmirnov Test Results for 5,000 vph Mainline Entry Volume .... 36
24 Perceived Travel Times for Links under Different Traffic Flows ............ 38
25 Test Results for Homogeneity of Variances in Errors of Perceived
T ravel T im es ............................................ 40
26 Test Results for Link (2204, 2012) Under Daily Traffic .............. 41
27 Test Results for Link (2204, 2012) Under Peak15 Minute Traffic .......... 42
28 Test Results for Link (1453, 1556) Under Peak15 Minute Traffic .......... 43
29 Test Results for Link (1453, 1556) Under Daily Traffic .............. 44
31 Socioeconomic Data Summary from ZDATA1.85V ................. 53
32 Socioeconomic Data Summary from ZDATA2.85V ................. 54
33 Special Trip Generator Data Summary from ZDATA3.85V ............... 54
51 Number of Enumerated Paths in Each SUE Iteration ................ . 80
52 Number of Zones with Lower, Equal, and Higher Numbers of Paths
Compared to the Previous Iteration ........................... 81
53 Origin Zones Enumerated with Lowest and Highest Numbers of Paths
Per Zone Pair in Each SUE Iteration ..................... 86
54 Sum of Absolute Differences in Number of Enumerated Paths for All
Zones in the Current and Previous Iteration ....................
61 Random Number Seeds for CORSIM Simulation ......................
62 Median, Mean, and Variance Travel Times for Link (2204, 2012) .........
63 Median, Mean, and Variance Travel Times for Link (1453, 1556) .........
71 Parameters for Example in Route Choice Implementation Step ............
81 Wilcoxon SignedRank Test Variables for Assigned Trips from
Successive SUE Iterations in the Gainesville Network ............
82 Wilcoxon SignedRank Test Variables for Assigned Trips from
Successive SUE Iterations in the FTOWN Network ..............
91 Mean and Median for Paired Observations from UE and Each
SU E Iterations ..........................................
92 Sign Test Variables during Each SUE Iteration ..................... .
93 Computer Running Times from Executing an SUE Assignment on
the Gainesville Network (in minutes) .........................
94 Computer Running Times from Executing an SUE Assignment on
the FTOWN Network (in minutes) ...........................
118
124
125
126
126
LIST OF FIGURES
Figure
21
31
32
41
42
43
44
51
52
53
54
55
56
57
61
62
63
. . . . 32
. . . . 52
. . . . 56
. . . . 5 8
....... 62
....... 67
. . . . 7 1
....... 74
. . . . 7 5
....... 77
. . . . 7 8
Average Number of Paths Per ZonePair Enumerated in the SUE1 Iteration
Average Number of Paths Per ZonePair Enumerated in the SUE4 Iteration
Average Number of Paths Per ZonePair Enumerated in the SUE5 Iteration
Flow Chart for the Travel Time Variation Step ..................... .
O overlapping P ath .............................................
LinkNode Diagram for Path Example .............................
. 83
. 84
. 85
. 90
. 93
. 95
LinkNode Diagram for Example Freeway System ............
1985 Gainesville Highway Network .......................
FTOW N Highway Network .............................
Flowchart for the SUE Algorithm ........................
Modules Processed at the Initialization Step .................
Example Batch File Created by the MakeRunBatch Module . . .
Parameters in the HNET.ALL Control File ..................
Flow Chart for the Path Enumeration Step ..................
Link Contained within Multiple Paths ......................
Example Output File of PATH.PRN from Line Module ........
Illustration of Complete Linked Path List ...................
64 Numbers of CORSIM Simulations in the SUE1 Iteration ............... 103
71 Flow Chart for the Route Choice Model Implementation Step ............ 107
72 Numbers of Assigned Trips from UE and the SUE1 Iteration ............ 114
81 Pattern for Difference in Assigned Trips from SUE4 and SUE5 Iterations . 120
91 % RM SE for Each Iteration ...................................... 123
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ASYMMETRICALLYDISTRIBUTED VARIATIONS IN
TRAVELERPERCEIVED TRAVEL TIMES IN
STOCHASTIC USEREQUILIBRIUM TRAFFIC ASSIGNMENT
By
MINTANG LI
December 1999
Chairman: Gary Long
Major Department: Civil Engineering
Most of the current approaches for applying a stochastic user equilibrium (SUE)
traffic assignment model have been based on iterative applications of the Dial multipath
assignment algorithm. The errors of perceived travel times by motorists in the route choice
model are assumed to be symmetrically Gumbeldistributed. Yet, observations suggest the
errors are asymmetrical. This study describes an alternative for solving the SUE optimization
problem by treating errors of perception as being asymmetricallydistributed for competitive
paths that are enumerated. Paths are generated by a modified shortestpath algorithm which
imposes a penalty to the links along an associated shortest path. The competitive paths are
then simulated to obtain expected travel time variations, due to different perceptions by
motorists, which allows more refined travel time estimation. The pathbased variations of
travel time are implemented in a random disutility model for route choice. The assumptions
of this approach are more reasonable than those of the other popular methodologies and can
be integrated into the Florida Standard Urban Transportation Model Structure (FSUTMS)
with minor modification to the existing structure. The model was applied and tested with a
network from a typical urban transportation study and found to converge on assigned
volumes that provided better agreement with traffic counts than the results from a standard
userequilibrium assignment. Although it provided both a better conceptual foundation and
more accurate numerical results, the model is currently not yet costeffective for realworld
applications due to exorbitant computer running times.
CHAPTER 1
INTRODUCTION
1.1 Background
A decade usually elapses from the time that a new transportation facility is planned
until it can be constructed and opened for traffic service. This makes it important to plan
transportation facilities today that will be needed 10 or 20 years or more in the future. For
this purpose, an urban transportation planning process which involves the use of a sequence
of forecasting models, e.g., traffic assignment models, has been developed.
Traffic assignment is essential and fundamental in the transportation planning process
since it attempts to predict the vehicular flows on the network by allocating travel demands
to routes in a traffic network according to associated mathematical models [DAM96]. In
these mathematical models, an urban transportation system is usually represented by a
network which is composed of nodes and directed links where each link is associated with a
travel cost function.
The linkbased travel cost function, also known as a link congestion function, is
usually dependent on link flows, e.g., the BPR (Bureau of Public Roads) formula or
Davidson's link capacity function [BOY81]. The values obtained from the link congestion
functions mainly reflect the mean travel times corresponding to a certain flow traveling
through the corresponding link. The traffic demands, specified by an origindestination (OD)
2
matrix, are allocated to the links to reflect the expected or forecasted traffic conditions in the
urban area through executing mathematical models in the traffic assignment process.
Because of different assumptions, various mathematical models take different
approaches in performing the traffic assignment process. Basically, they can be divided into
two categories: deterministic models and stochastic models. A wellknown deterministic
model for traffic assignment is the user equilibrium (UE) model. In the UE assignment, the
path chosen by each traveler is generally assumed to be the path that minimizes his or her
journey time, or some combination of time and cost. Although the UE assignment is efficient
computationally in allocating trips on a network, it assumes that all travelers have identical
perceptions of travel time and cost [SHE85].
Stochastic models differ from deterministic models by recognizing that travelers do
not have perfect knowledge of the traffic system and that there is some variation in how
systemwide travel disutility is perceived by travelers. This assumption allows partitioning
origindestination trips between several alternative paths, even if travel time on the paths
differ from each other. Thus, stochastic models are suggested as a more realistic approach
[CHE91]. At stochastic user equilibrium, no motorist can improve his or her perceived travel
time by unilaterally changing routes. This follows directly from the interpretation of the
choice probability as the probability that the perceived travel time on the chosen route is the
smallest of all the routes connecting the origindestination under consideration [SHE85].
In stochastic models, discrete choice models are applied in the path selection process
by including a random component (known as random error) in travelers' perceptions of travel
disutility, and a path's perceived travel disutility is usually assumed as "link additive", i.e.,
3
obtained as the sum of corresponding perceived link travel disutilities [CAS96]. Thus, the
routechoice can be formulated as a process of selection among alternative paths, on which
the travel disutility is random. Motorists are then assumed to choose the paths which have
the minimum perceived travel disutilities from the finite number of paths in a traffic network.
The hypothesis underlying discrete choice model is that when faced with a choice
situation, an individual's preferences toward each alternative can be described by an
"attractiveness" or "utility" measure associated with each alternative [SHE85]. However,
utility cannot be measured directly. In addition, many of the attributes that influence
individuals' utilities cannot be observed and thus must be treated as random. Consequently,
the utilities themselves are modeled as random, meaning that choice models can give only the
probability with which alternatives are chosen, not the choice itself [SHE85]. Equation (1.1)
illustrates the utility function that is typically used in stochastic traffic assignments.
U (v) = (v) + (v) (1.1)
where:
v = volume on path k;
U,(v) = perceived utility function with respect to the volume v for the
0kh path between origin r and destination s;
V,(v) = measured utility with respect to the volume v; and
er(v) = random error term in perceived travel disutility that is
associated with the kh path under consideration.
Since the userperceived travel disutility of each path is a random variable, it is
associated with some probability density function which gives the probability that a driver
randomly drawn from the population will associate a given travel disutility with that path
[SHE85]. Once the distribution of the error term is specified, the distribution of the utilities
can be determined, and the choice function can be calculated explicitly.
4
Most of the current stochastic models [SHE85] assume that the random errors in
perceived travel disutility for various paths are independently and identically distributed (IID)
Gumbel variates. This assumption is derived for operational convenience since the widely
used multinomial logit (MNL) model will be derived when random errors in perceived travel
times are assumed to be IID Gumbel variates.
There are several packages of computer programs for performing traffic assignment
by either deterministic or stochastic models that are applied by transportation planners.
Among these computer packages, the Florida Standard Urban Transportation Model
Structure (FSUTMS) is a wellknown tool for transportation planning studies and has been
developed and maintained by the Systems Planning Office of the Florida Department of
Transportation (FDOT) since 1977. FSUTMS is used to represent a formal set of modeling
steps, procedures, software, file formats, and guidelines established by FDOT for use in travel
demand forecasting throughout the State [FSU97].
The microcomputer version of FSUTMS consists of specific FSUTMS modules
converted to interface with the commerciallyavailable TRANPLAN travel demand software
system. TRANPLAN is an acronym for TRANsportation PLANning, which is an integrated
suite of programs for forecasting the impacts of alternative land use scenarios and/or
transportation networks on highway and public transit systems [URB98].
Along with the existing computer programs for travel demand forecasting, more and
more transportation research has been undertaken with the implementation of computer
simulation tools in order to predict a transportation system's operational performance as
expressed in terms of travel time and other related measures of effectiveness (MOEs). A
5
good example of the application of computer simulation is the dynamic traffic assignment
methods under development by the Massachusetts Institution of Technology (MIT) and
University of Texas based on the implementation of an assignmentsimulation framework.
However, dynamic traffic assignment is designed to analyze the traffic state over a
specific time interval such as a peak hour in a shortterm time horizon, whereas traditional
traffic assignment is designed to estimate or forecast traffic flows in the more distant,
longterm future. In addition, dynamic traffic assignment is an ongoing development and the
feasibility of dynamic traffic assignment models for real world applications has not been
investigated thoroughly. Nevertheless, the traditional traffic assignment process has been
developed and implemented for decades and its performance has been well recognized. Thus,
it is important to preserve and improve the traditional traffic assignment process.
One of the most important and powerful computer simulation tools is the corridor
microscopic simulation (CORSIM) package. CORSIM is a combination of NETSIM
(NETwork SIMulation) and FRESIM (FREeway SIMulation) models. The NETSIM and
FRESIM models are microscopic stochastic simulation models for urban traffic and freeway
traffic, respectively [COR98]. The CORSIM program has been applied extensively to a wide
variety of problem areas by both practitioners and researchers and is the most widely used
traffic simulation model [COR98].
1.2 Problem Statement
Although stochastic traffic assignments are developed based on solid mathematical
proof with a more reasonable assumption of random travel disutility than the assumption in
the UE model that all travelers have identical perceptions of travel time and cost, several
6
problems are recognized in stochastic traffic assignment which may cause these models not
to be able to predict vehicular flows on a network accurately.
First, there is currently no simulation tool available for estimating perceived travel
times (or disutilities). Perceived travel times by travelers cannot be properly measured or
estimated unless a tremendous effort is made in surveying large samples of motorists who are
known to be road users of a given traffic network. It could be difficult for each road user to
respond with travel time estimates for all possible links contained in alternative paths. This
is even more difficult when traffic assignment for forecasting future traffic conditions is
performed. The perception of future travel times is likely to vary. Also, it would be difficult
to accurately define or predict the population of future road users. Since perceived travel
times cannot currently be approximated by a model, the distribution and its parameters for the
error term in perceived travel times can only be assumed.
Second, link congestion functions are applied in stochastic assignment to obtain
measured travel disutility. However, these functions are usually not calibrated from field data
obtained for a given traffic network. In addition, the link congestion functions, e.g., the BPR
formula or Davidson's link capacity function, have their own weaknesses. For example, the
BPR formula is known to have the limitation of ambiguous definition of capacity in the
function [BOY81]. Also, although limiting traffic volumes to link capacities is a realistic
portrayal of traffic operations, Davidson's link capacity function is based on a simplified
queuing theory formulation in which no lanechanging maneuvers are allowed [BOY81].
In addition, the assumption that the variations of errors in perceived path travel times
are identically distributed is merely for reducing the computational complexity [BEN85]. A
7
more general assumption is to presume that the errors in perceived path travel times are not
identically distributed. The results from a field experiment, addressed in Section 2.2.6, show
that the variations of the errors in perceived travel times increase with congestion and are not
identically distributed. Thus, the assumption that errors of perceived travel disutility in SUE
traffic assignment are IID Gumbeldistributed is not realistic.
Besides the above problems, stochastic traffic assignment is currently not the standard
procedure in FSUTMS. When a user wants to perform a "nonstandard" analysis, knowledge
of TRANPLAN and its syntax is required. If the application of SUE traffic assignment in
FSUTMS is going to be developed, there should be minimum changes to the current program
structure to conserve the effort of modification.
To deal with the condition that errors in perceived travel times are not IID but
asymmetrically Gumbeldistributed, the utility model for SUE traffic assignment is revised and
is illustrated in Equation (1.2).
U v) = C = (c(v) + e(v)') (1.2)
where
Crs(v) = perceived travel disutility with respect to the volume v for the
kth path between origin r and destination s;
ck (v) = the measured or actual travel time with respect to the volume
v for the kth path between origin r and destination s; and
e (v)' = the variation in travel times that is Gumbeldistributed with
parameters rs (v) and gj(v).
The error in perceived travel time is revised to be asymmetrically distributed, instead
of identically distributed, among various paths for a given origindestination zonepair. The
utility function in Equation (1.2) is assumed to consider the disutility of travel time only,
8
i.e., ci (v), and the other unknown attributes are assumed to not have any impacts on the
utility of path selection. Thus, travel time is assumed as the only determinant of route choice,
and perception errors will be a random variable distributed across the population of motorists.
If other attributes of the alternative paths, e.g., toll roads, as well as the decision maker's
characteristics, are believed to have impacts on route choicemakers in selecting a path for
traveling, the utility function can be updated by including these travel disutility terms in
Cks(v) in Equation (1.2) as long as these disutility terms can be properly estimated.
For example, the perceived travel disutility can be formulated as follows:
C(v) c(v) + (v)' = a x TY(v) + x Tollksv+ (v)' (1.3)
where:
T'7(v) = the measured or actual travel time with respect to the volume
v for the kth path between origin r and destination s;
Tollks(v) = the measured or actual toll cost with respect to the volume v
for the kth path between origin r and destination s; and
qa, aol = the corresponding weight coefficients.
If toll cost as well as travel time can be measured precisely and the distribution of the
error term is specified, the distribution of Cr'(v) can be determined, and the choice function
for road users to select a path for traversing between a given zone pair can be calculated
explicitly. However, for simplification purposes, this study focuses on travel time disutility
only.
In the revised utility function, the assumption of IID Gumbeldistributed error is
replaced by asymmetric Gumbeldistributed error. The concept is to allow the travel time on
a path to be treated as a random variable which is Gumbeldistributed with parameters
rC(v) and kg(v). This requires a new algorithm for solving the stochastic user
9
equilibrium traffic assignment problem based on the assumed utility function in Equation
(1.2).
1.3 Goal and Objectives
The goal of this study is to develop a model to perform stochastic user equilibrium
traffic assignment with a new assumption that the errors in perceived travel times are Gumbel
distributed but not identical. This allows a set of distinct Gumbel variables, rlk and tk to be
associated with path k of the enumerated paths. Since only a subset of all possible routes is
selected by the explicit path enumeration algorithm, the solution obtained is an approximate
stochastic user equilibrium. However, the SUE algorithm gives the optimal route flows for
the subset of the routes that were selected.
The location parameter, rl, and the scale parameter, t, are considered to be flow
dependent and are not presumed to be constant. The assumption that errors in perceived
travel times, for those paths that are enumerated by mathematical models and included in the
choice sets for road users, are independent of each other is still applied for simplification
purposes. This implies that errors for every enumerated path are independent of each other
although overlapping links may exist on multiple paths. In addition, the travel demand from
origin to destination is assumed to be constant over time, and traffic assignment is performed
by allocating the demand to each enumerated competitive path between an origindestination
pair according to the proposed random utility function in Equation (1.2).
The objectives of this research are to:
1. develop a new SUE algorithm to perform a stochastic traffic assignment that complies
with the situation where errors in perceived travel times are flow dependent;
10
2. provide mathematical proof to support the developed algorithm based on the
assumptions;
3. establish a viable method to estimate traveler perceived travel times; and
4. implement the algorithm on a typical traffic network and examine the results.
The product of this research will be an enhanced algorithm to perform stochastic user
equilibrium traffic assignment when variations in travel times perceived by users increase with
congestion. A computer program is produced in order to implement this algorithm on a
transportation network. This process is expected to contribute to stochastic user equilibrium
traffic assignment modeling in the following ways:
1. allow a stochastic user equilibrium traffic assignment under the assumption that errors
in perceived travel times for each path are independently but not identically Gumbel
distributed;
2. allow pathbased travel times to be implemented in SUE traffic assignment;
3. allow more realistic estimation of perceived travel times; and
4. enhance the capability of FSUTMS by allowing SUE traffic assignment to be
implemented with minimum modifications.
1.4 Study Area
The study area selected to evaluate the SUE traffic assignment model is the 1985
Gainesville highway network. The highway network was defined using data provided by
FDOT. This research also applies the 1985 socioeconomic data for Gainesville obtained from
FDOT to perform the analysis. Since the Gainesville data had been validated by FDOT, it is
assumed that the zonal socioeconomic data of FDOT were accurate and no extra
modifications or adjustments needed to be performed. Traffic counts from 1985 contained
in the network data are used for assignment validation.
11
1.5 Organization
In Chapter 1, an introduction to the research topic and the needs for the research have
been presented. Improvements needed in the previous research have been identified and the
specific objectives of the research have been stated.
A literature review on the development of the SUE model and the statistical
evaluation tools for model performance is presented in Chapter 2. The discrete choice model
is first reviewed, followed by the Gumbel distribution. The traditional SUE model and its
solution algorithm are also addressed. The results of an experiment are addressed to illustrate
the distribution pattern for the errors in perceived travel times.
Chapter 3 briefly describes the networks that are used to evaluate the SUE model's
performance. Chapter 4 addresses the new SUE traffic assignment algorithm and the
methodology for the Initialization step of this SUE model. Chapter 5 presents the
methodology and results for the Competitive Path Enumeration step of the SUE model.
Chapter 6 addresses the methodology and results for the Travel Time Variation step of SUE
model. Chapter 7 describes the methodology and summarized results for the Route Choice
Model Implementation step of the SUE model. Chapter 8 presents the methodology and
summarized results for the Test Convergence step of SUE model.
Chapter 9 validates the assigned results from the SUE model to evaluate the model's
performance. The final chapter presents a summary and conclusions and recommends areas
for further research.
CHAPTER 2
BACKGROUND AND STATISTICAL TOOLS
2.1 Discrete Choice Models
2.1.1 Introduction
Discrete choice models assume that when an individual is faced with a choice
situation, his or her preferences toward each alternative can be described by an
"attractiveness" or "utility" measure associated with each alternative [SHE85]. This utility
is a function of the attributes of the alternatives as well as the decision maker's characteristics.
The decision maker is assumed to select the alternative with the highest utility at the time a
choice is made.
Utilities, however, cannot be observed or measured directly. Furthermore, many of
the attributes that influence individuals' utilities cannot be observed and must therefore be
treated as random. Consequently, the utilities themselves are modeled as random, meaning
that choice models can give only the probability with which an alternative is chosen, not the
choice itself. That is,
P, = Prob [U1 > U for j ] i, j = 1, ..., J]. (2.1)
where P, is the probability that the ith alternative is chosen, U, is the stochastic utility
associated with thejth alternative and J is the joint set of all possible alternatives.
13
The stochastic utility function can be written as the following:
U = + e (2.2)
where V, is nonstochastic and reflects the representative tastes of the population and e, is the
stochastic component for alternative i, respectively. If the stochastic term in Equation (2.2)
is assumed to be IID Gumbel distributed, a multinomial logit model can be derived [DOM75].
The following section introduces the Gumbel distribution and how it is implemented in
developing the multinomial logit model.
2.1.2 Gumbel Distribution
A random variable e is Gumbel distributed if [BEN85]
F(e) = exp [ e (E ) ], p > 0
(2.3)
f(e) = pe (n) exp [ e (c n)]
where F(e) = the cumulative distribution function of random variable e;
f(e) = probability density function of random variable e;
11, p = location parameter and scale parameter, respectively.
Both "exp" and "e" in the above equations represent the exponential function.
The mean and variance ofe can be derived by the first (p'i) and second (G'2) moments
about the origin via the moment generating function method since E(e) = i'1 and Var(e) = i'2
 (p' 1)2. The moment generating function for e is [MEN90]
m(t) = E[ et ]= f ete e n)exp[e 9(c )] de (2.4)
Let p(e rl) = y, then d/ = [ip dy, and e = ['y + ir. Hence
14
my(t) = f t( ly + T) e y exp[e Y] dy
(2.5)
Let z = e Y, then dz
e y dy. Hence
mz(t) = etf z'it e (dz)
etnf zr g t e
(2.6)
= etrnP(1 A p't) = E(eAt)
It is shown in Appendix A that
ae7' Y (1 +
n=]
1
+ ... + 
n
1
where y = lim ( 1 + 
n 2
1 0
F(1 't)
Inn ) 0.57722 is known as Euler's constant. Thus,
 (1 lt)
(1 I t)e (1 10 I (1 + (1 )e
(2.8)
Consequently,
!'
1 1 t Lt
[ it n=1 n (n +1 t)
!t 11 + Y 
P'(1 't)
F(1 i t)
(2.9)
n = n(n + 1)
1
F(a)
a)e
n
(2.7)
a[ In(1/r(1 tt))
a8t
a1n r(1 't)
at
Hence
r'(1 p t) = F(1 p t) (2.10)
As a result, the mean value of c, i.e., E(c), and the variance of c, i.e., var(c), are
E) y 0.57722
E(e) = ri + ~ +  (2.11)
1 2
var(c) =  (2.12)
6 2
The Gumbel frequency distribution has the same general bell shape as the normal
frequency distribution, but is skewed, with a thinner left tail than the normal distribution and
a thicker right tail. The right tail behaves like the tail of an exponential distribution. The
parameter ir determines the mode of the Gumbel distribution; hence changing 1r shifts the
location of the mode and mean, but not the shape of the distribution as long as i remains
constant. For a Gumbel distribution with parameters ir and equal to 0 and 1, F(0.366513)
is approximately equal to 0.5 whereas the mean of the corresponding distribution is 0.57722.
There are two significant properties of the Gumbel distribution [DOM75]. First, the
distribution is stable under maximization, in the sense that the maximum of two independent
Gumbel random variables is again a Gumbel random variable. This property can be compared
with the property that the sum of two normal variables is again normal, so the normal family
is stable under addition. When maximization of utility is the critical operation, this stability
property of the Gumbel distribution makes it a natural distribution with which to work, just
as the normal distribution is natural for problems involving addition of random variables.
16
Second, the difference of two Gumbel distribution variables has a binary logistic
distribution. This special feature leads to the development of multinomial logit model. The
logit model is computationally tractable and in many applications corresponds to a plausible
stochastic specification. The key to the assumption of IID Gumbeldistributed random errors
in perceived travel disutility is the specification of a statistical distribution with the property
that the difference of two independent random variables having this distribution is a
logistically distributed random variable [DOM75]. The development is illustrated in the
following section.
2.1.2.1 Derivation of multinomial logit
Let F(t, ..., tj) andf(el, ..., ej) denote the joint distribution function and joint density
function of (el, ..., ey), respectively, i.e.,
t t, tj
F(t ..., t, ..., t,) = f ... f ... f f( ..., ..., ,e) dej ... de, ... de (2.13)
Let F, denote the derivative of F with respect to its ith argument, i.e.,
t1 tj
F,(t ..., t, ..., t,) = f ... f( ..., e ..., ej) de, ... de, (2.14)
Equation (2.1) can be written as
P = Prob [c e < V, V forj i, j = 1, ..., J]. (2.15)
Equation (2.15) becomes the following, which is known as the convolution formula:
P f FI(t + v, V1, ..., t + V V) dt
(2.16)
St + W, 1 t J V V
= f f ... f f(e, ..., e, ..., ej) dej ... de, dt
Any specified joint probability distribution will yield a family of probabilities depending on the
unknown parameters of the distribution and the function V,, the nonstochastic component of
disutility. If random variables e, have independent Gumbel distributions with parameters il,
and common i (assumed equal to 1 for convenience) for i = 1, ..., n, then [DOM75]:
Prob[V1 + C, > V2 + C2] = e V (2.17)
To establish Equation (2.17), the convolution formula in Equation (2.16) can be written as
Prob [V, + 1, > V2 + 2] =f F'1(t) F2(t + V1 V2) dt, (2.18)
since F(el, C2) = F(e)F2(C2) where F, is the cumulative distribution function of e,. Thus,
Fl(t) = exp[ e (t + Ti)
(2.19)
F'(t) = e (t+ T) exp e (t + ) (2.9)
Then Equation (2.18) becomes
ProbE[V +c, > V2 +2] = f expe()exp(e (t+))exp(e e(t+v v2+nT2)) dt
(2.20)
f e (tT+n)exp(e t(ei 1 + e )) dt
Let A = e +e + 2 2
18
then Equation (2.20) becomes:
f e (t + )exp(Ae t)dt = e nlf e texp(Ae t)dt
e nlf e texp(Ae )d(t)
e T0
r1 0
A f exp(Ae )d(Ae t)
e e
A ei V + V 2
e + e
e T+ e2 12
e nlf exp(Ae t)d(e )
0
e I d(exp Aet)
A
0
As indicated earlier, the difference of two independent Gumbeldistributed random
variables has a binary logit distribution
P1 = Prob[e2 e< V1 V2 = G(V1 2)
(2.22)
e Tb, + Q e 2 12
Hence, independent Gumbeldistributed stochastic components of utility lead to the
logit model. In general, the parameters il, can be absorbed into v, and interpreted as an effect
specific to each alternative i. Thus, an assumption that all 11, are zero involves no loss of
generality [DOM75]. Although the logit model can be developed by assuming the stochastic
components to be lID Gumbel distributed, the assumption is used only for reasons of
operational convenience in order to obtain a simplified model [BEN85].
The development of a binary asymmetrical Gumbel distribution for two variables that
are independently but not identically Gumbeldistributed is addressed as follows.
(2.21)
19
2.1.2.2 Derivation of binary asymmetrical Gumbel distribution
If C1 and c2 are independent Gumbeldistributed but not identical, the following
equations can be derived. That is:
F(c 1, 82) = F(c,) F2(C2)
SF (,) = exple gi(1 1 T and F2c) = exp[e 92(C2 ~ 2 (2.23)
.. F(8, C82) = exp[e g(1i d exp[e 2 2 2
Thus
P, f Fl(t, t+V,V2) dt
(2.24)
= f e 1(t1 exp[e g1 11) exp[e 2 12) dt
The above equation is identical to the general binary logit model if il, and t, are equal
to 0 and 1, respectively. It is obvious that P1 cannot be displayed in as simple form as the
logit model due to different t, in the equation. Hence, let
Pi(t) = gLe g(tndexp[e g(t T) exp[e 2 (+ V2 T2) (2.25)
and divide t into infinite intervals each with length 1, then Equation (2.24) can be written as
P1 = i Pl(t). It can be verified that Pl(t) becomes very small when t in Equation (2.25)
is close to o or +o. Thus, if the values at the two extremes are ignored, Equation (2.24) can
be approximated as follows when the range for integration is specified, i.e.,
b
P1 = (t) < a < 0, 0 < b < + (2.26)
a
20
Thus, P, can be determined by numerical approximation and hence can be implemented in the
SUE traffic assignment as the probability that a path is selected by road users.
The following section addresses the development of the multiasymmetrical Gumbel
distribution when there are more than two variables that are independently but not identically
Gumbeldistributed.
2.1.2.3 Derivation of multiasymmetrical Gumbel distribution
Let el, c2, ..., ej be the errors of perceived travel times for the J paths between a given
zone pair and assume these variables are independently but not identically Gumbeldistributed.
The joint distribution function for these variables can be derived as follows:
F(ce, ..., c,) = F,( e) ... F/(e) (2.27)
F,(e) = exp[e 9101 l T]
:(2.28)
F (c,) = exp e ,gi( (2.28
Thus
F(yc, ..., ,) = exp[ e gi(cl ) ] ... exp[ e c TV ] (2.29)
SP, = f F(t+VV,, ..., t+VVJ) dt
(2.30)
f i,e (t exp[e (tn exp[ i(t +VVI I...exp[e (t + V, V dt
P, then becomes the probability that path i is selected by road users for traversing
between a given zone pair. Since the above equation cannot be expressed analytically and
21
evaluated in closed form, the same numerical approximation method presented in Section
2.1.2.2 is applied to obtain the value of P,. The following section introduces an approach to
estimate the Gumbel parameters, i.e., pi and ri for path i, in Equation (2.30).
2.1.2.4 Method of probabilityweighted moments
Landwehr, Matalas, and Wallis [LAN79] proposed a simple method for estimation of
the parameters ir and [ based on probabilityweighted moments (PWM) given by
M(k) k = 0, 1, 2,... (2.31)
n =1 n
k
where k is the moment number and n is the number of data points. X, 's are the data values
sorted in ascending order. The PWM estimators are given in Equations (2.32).
In2
(0) (1)
(2.32)
Sy 0.57722
S (0) (0)
where , and f1, are the estimates of 9, and rl,, respectively. The method of PWM is
simple and also highly efficient even for a sample size as small as 5 [LAN79].
2.2 Stochastic User Equilibrium
2.2.1 Introduction
A stochastic user equilibrium (SUE) model assumes that motorists perceive different
travel times (disutilities) and choose the path with the least perceived disutility from origin to
22
destination. The stochastic equilibrium conditions can be characterized by the following
equation [SHE85]:
f" = q P" V k, r, s (2.33)
where Ps is the probability that route k between r and s is chosen given a set of measured
travel times, t, i.e., P[ = P s(t) = Pr(Cs < Crs, V / k e K It), where Ck is the
random variable representing the perceived travel time on route k between r and s. In
addition, the following network constraints have to hold, i.e.,
t t (x ) V a
(2.34)
fkrs rs V r, s
k
where t = E[T] and x = fkrs 8rs V a. Ta denotes the perceived travel
time on link a. Thus, the perceived travel times are modeled as random variables as well as
flow dependent since the choice probability is conditional on the values of the mean link travel
times at equilibrium. This dependence is accounted for by assuming that the mean travel time
for each link is a function of the flow on that link [SHE85]. Consequently, at equilibrium, no
motorist can unilaterally changing routes to improve his or her perceived travel time.
Sheffi and Powell developed the following minimization program of which the
solution is the desired set of SUE flows [SHE85].
x
min z(x) = q E min { C  c rs(x) } + E xat (x) t(w) dw (2.35)
rs keK, a a 0
23
where Cs7 represents the perceived travel time on the kh path between origin r and
destination s. Cs is a random variable that is equal to the mean travel time cs,
rS
measured at a given flow level x, plus its associated random error term, ek. The program
in Equation (2.35) does not have any intuitive economic or behavioral interpretation. It is
only a mathematical model that is utilized to solve the SUE problem.
Sheffi and Powell show that the flow pattern that minimizes the Equation (2.35) also
satisfies the SUE conditions. To prove the equivalence between the minimization program
in Equation (2.35) and the SUE conditions, the firstorder conditions of this program have
to coincide with the SUE conditions illustrated in Equation (2.33).
The partial derivative of z(x) with respect to a typical path flow, f", is
9z[x(f)] rs dt(xa) rs
a rs a k (2.36)
fk a dx
Since the firstorder condition for unconstrained minimizations requires only the
gradient vector of the objective function be equal to 0, the above derivatives have to equal
zero for all k, r, and s, meaning that: fk = q,, Pkrs, V k, r, s, which is the SUE condition.
Moreover, the flow conservation constraints (>kf 'k = q) are automatically satisfied at
equilibrium since Yk Pkrs = 1.
The expected perceived travel time function, i.e., E[ min { Cf cr(x) } ], is
included as the first term of the objective function from origin r to destination s. The partial
derivative of the cost function with respect to c k is the probability of choosing path k
between r and s, P which is the probability that Ck is less than the cost of any other
route, C's between rands. If ers is identically and independently Gumbeldistributed, ers
24
can be replaced by er and Ps can then be expressed by a logit model where J is the choice
set of paths that can be enumerated:
prs e
k e c (2.37)
where 0 is a positive parameter.
2.2.2 Stochastic Network Loading Procedures
The wellknown STOCH algorithm [SHE85] or Dial's algorithm [DIA71] is a
procedure which effectively implements a logit route choice model at the network level. This
algorithm uses the logit formula with parameter 0 to assign choice probabilities and thus,
flows, to a set of reasonable paths connecting each OD pair. In the algorithm, a link (i,j) is
included in a reasonable path only when r(i) < r(j) and s(i) > s(j) where r(i) denotes the travel
time from origin node r to node i along the minimum travel time path and s(i) denotes the
travel time from node i to destination node s along the minimum path.
The steps of the STOCH algorithm for one OD pair are addressed below. These steps
need to be repeated for every OD pair in the network.
Step 0: (Preliminaries.)
(1) Compute the minimum travel time from node r to all other nodes. Determine
r(i) for each node i;
(2) Compute the minimum travel time from node i to node s. Determine s(i) for
each node i;
(3) Define X, as the set of downstream nodes of all links leaving node i.
(4) Define i, as the set of upstream nodes of all links arriving at node i.
25
(5) For each link (i,j) compute the "link likelihood", L(i,j), where
eO[rL() r( t(,)] if r(i) < r() and s(i) > s(j)
L(i, j) = (2.38)
0 otherwise
In the above expression, t(i,j) is the measured travel time on link (i,j).
Step 1: (Forward Pass.) Consider nodes in ascending order of r(i), starting with the origin,
r. For each node, i, calculate the "link weight," w(ij), for each e X,, i.e., for each
link emanating from i, where
L(i, j) if i = r, i.e., if node i is the origin
wQJ) (2.39)
S L(i, j) Y w(m, i) otherwise
This step is completed when the destination node, s, is reached.
Step 2: (Backward Pass.) Consider nodes in ascending values of s(i) starting with the
destination, s. When each node, j, is considered, compute the link flow x(i, j) for each
i e ,, i.e., for each link entering, by following the assignment:
q w(i, j) for j = s, i.e., if j is the destination
s w(mj)
x(i, j) = (2.40)
[Zx(j, mn) w(i', j for all other links (i, j)
This step is applied repeatedly until the origin node is reached. The sum in the
denominator of the quotient includes all links arriving at the downstream node of the
link under consideration. The sum of the flow variables encompasses all links
emanating at the downstream node of the link under consideration.
26
Since the STOCH algorithm leads to heavy computational demands, Dial introduced
a second definition where an efficient path would be one where every link extends further
from the origin [DIA71]. Thus, rather than a pass for each OD pair, only a single pass of the
assignment algorithm is required to assign all of the trips for each origin to all of the
destinations. However, this definition is illogical in the sense that most tripmakers are more
concerned about reaching their destinations than leaving their origin [BEL93].
2.2.3 SUE Solution Algorithm
Most algorithms for solving SUE programs search for the minimum of the objective
function along the descent direction at each iteration. The iterative process to find the move
size along the descent direction is difficult because the partial derivative of the objective
function at a particular iteration could be random in some cases.
The method of successive averages (MSA) is based on predetermined move sizes
along the descent directions [SHE85]. In other words, a, is not determined on the basis of
some characteristics of the current solution. Instead, the sequence of move sizes a,, a2, ...
is determined a priori. The MSA algorithm can be summarized as follows [SHE85]:
Step 0: Initialization. Perform a stochastic network loading based on a set of initial travel
times { to }. This generates a set of link flows { xa }. Set n:= 1.
Step 1: Update. Set tan = ta(x ), V a.
Step 2: Direction finding. Perform a stochastic network loading procedure based on the
current set of link travel times, { t a }. This yields an auxiliary link flow pattern
{ ya 1.
Step 3: Move. Find the new flow pattern by setting
27
n+l n 1 n n
Xa = xa y+ (y xa ). (2.41)
Step 4: Convergence criterion. If convergence is attained, stop. If not, set n = n + 1 and go
to step 1.
The results from the method of successive averages were proved to converge to the
minimum [SHE85]. Dial's stochastic loading algorithm was typically used as the inner part
of the SUE assignment as addressed in Step 2 of the MSA algorithm because of its efficiency
[PRA98]. However, Dial's algorithm produces a flow that is deterministic, not random.
Thus, when link weights and OD demands are deterministic, the only way to get truly
probabilistic behavior is via behavioral models that postulate a probabilistic individual choice
mechanism [FLO76].
Since the solution from the MSA algorithm is based on iterative applications of the
Dial multipath assignment algorithm, the errors of perceived travel times by motorists in the
route choice model are assumed to be symmetrically Gumbeldistributed. In addition, the Dial
algorithm implicitly defines the route choice set, i.e., the competitive paths, for each origin
destination pair [PRA98]. The assumption of IID Gumbeldistributed errors is only for
operational convenience in order to obtain a route choice model as simplified as a logit model.
Under the assumption of asymmetrically Gumbeldistributed errors in perceived travel times,
the traditional mathematical models for solving the stochastic user equilibrium problem are
not applicable. A more general approach in solving stochastic traffic assignment is needed.
One of the important steps in the SUE algorithm is to enumerate competitive paths.
28
2.2.4 Define Route Choice SetCompetitive Paths
A route choice set which contains the competitive paths for road users to select for
traversing between a given zonepair is defined either implicitly or explicitly. A good example
of implicitly defining a path choice set is the STOCH method, also known as Dial's algorithm
that is addressed in the previous section. This algorithm implements the logit route choice
model to assign choice probabilities and flows to 'efficient' paths only instead of all paths
connecting each origindestination (OD) pair.
An efficient path is defined as one which does not backtrack, i.e., a path is efficient
if every link in it has its initial node closer in travel time to the origin than to its final node, and
has its final node closer in travel time to the destination than to its initial node [DIA71].
Although efficient computationally, the stochastic traffic assignment models that implement
Dial's algorithm are known to assign excess flows on overlapping paths [PRA98]. Also, it
is difficult to browse and demonstrate the assignment results from stochastic models on
implicit paths.
Explicit path enumeration in stochastic models is generally performed by some
heuristic rules in order to restrict the number of available routes. De La Barra et al. derive
a set of n distinct reasonable pathsthe set of resulting paths must be those of least
generalized cost, assuming that users are economically rational to a certain degreeby
repeatedly applying an overlapping zealot (Oz) factor to the current shortest path [BAR93].
The De La Barra method proceeds repeatedly for a given origin i and destination
with the following steps:
29
Step 1: search for the minimum path from i toj and store it;
Step 2: penalize the cost (travel time) of all links in the network that form part of the
minimum path by an Oz factor; and
Step 3: continue the search for more paths; stop if the current minimum path is identical to
any of the previous paths that were stored, else return to Step 2.
The key element of this method is the Oz factor used in Step 2. The factor controls
the degree of overlapping between the paths that succeed in getting stored. If the Oz factor
is greater than 1.0, the first minimum path will take more iterations to reemerge and other
distinct paths will succeed in getting stored. In addition, since overlapping with previous
paths is penalized, all intermediate paths will be different from each other as much as possible.
It follows that the cost of all paths without overlapping penalization will be greater than or
equal to the cost of the first minimum path.
The De La Barra method can require long computer execution times since it performs
the path enumeration on a zonepair basis. However, the shortest path algorithm in FSUTMS
performs the shortest path searching on a zonal basis. Thus, a new method is derived by
slightly revising the original De La Barra method in this research, i.e.,
Step 1: search for the minimum paths from origin zone i to all destination zonesj and store
them; set n = 1;
Step 2: penalize the cost (travel time) for all links in the network that form part of the
minimum paths to all destination zones by an Oz factor; and
Step 3: continue the search for more paths; store the new enumerated paths if they are not
identical to any of the previous paths to each destination zone that were stored; stop
30
if n is greater than or equal to the desired maximum number of path enumerations,
otherwise increment n by 1 and return to Step 2.
The purpose of revising the original De La Barra method is mainly to improve its
efficiency in the path enumeration process. In this study, travel times for paths enumerated
by the revised De La Barra method will be estimated and used as an input in the
aforementioned route choice model. If a proportion of these enumerated paths from the
revised method are not competitive enough, the probabilities that they will be chosen by
travelers for traversing between zone pairs will be low.
2.2.5 Variation in CORSIM
CORSIM employs a linear recursive procedure to generate a sequence of numbers,
known as a random number stream, in order to simulate the stochastic elements of traffic flow
[COR98]. Hence, unlike the case with truly random numbers, the random values generated
in each simulation can be controlled by specifying the initial seed value.
In CORSIM, record type (RT) 2 is a run control record and is required for setting up
a CORSIM simulation run. The random number seed in entry 17 is used as the base seed to
start the stochastic process affecting the traffic stream, such as the characteristics of each
driver/vehicle combination [COR98]. The random number seed in entry 18 is used for all
stochastic processes, such as the decision for accepting available gaps in the oncoming traffic
for left turns [COR98].
CORSIM employs Equation (2.42) to generate the random number stream for Entry
17 [RAT90]. The default value for Entry 17 is 7781 [COR98].
31
Seed = Mod Seed, x 9 + Mod ( Seed 104 x 104 10 1
Seed (2.42)
RND 1 + +1
106
where Seed,1, = random number seed for the (i + 1)th driver/vehicle;
Seed, = random number seed for the (i)th driver/vehicle; and
RND,+, = the random number for the (i + 1)th driver/vehicle.
Equation (2.43), for which all of its variables are previously defined, illustrates the
formula that is used to generate the random number seeds and the random number stream for
Entry 18 [RAT90]. The default value for Entry 18 is 7581 [COR98].
Seed+1 = Mod [Seed, x 3 + Mod (Seeda 104 x 104 108s
(2.43)
Seed11
106
Thus, by simply specifying different random number seeds in a CORSIM input file, variations
in the simulated results can be obtained.
A freeway system, as illustrated in Figure 21, was used as a template example for
further examining the variation in simulated travel times from CORSIM. This experiment was
performed by entering two different traffic flow rates, i.e., 3,000 and 5,000 vph, on the
mainline of the freeway system. Both systems, one with moderate traffic and the other with
congested traffic, were then respectively simulated by CORSIM with 200 sets of randomly
selected random number seeds. The total simulated travel times on the mainline links, i.e.,
the sum of the travel times from links (11, 1), (1, 2), and (2, 14), were collected to examine
the variation in simulated travel times from CORSIM. The first thing to examine was the
equality of variances under different levels of traffic flow.
Figure 21 LinkNode Diagram for Example Freeway System
The following test statistic was applied to test the standard deviations S, and S2 for
errors in simulated travel time from the 200 samples obtained by specifying 3,000 and 5,000
vph mainline entry volumes, respectively, for the example freeway system in Figure 21. The
errors in simulated travel times were obtained by subtracting the median value in each sample
group with the simulated travel times from CORSIM.
ZO 1 S2
I 1 1 (2.44)
S2n1 2n2
where
S ( 1)2 2 S 2 (2.45)
p nn + n2 2
The null and alternative hypotheses are illustrated as follows:
H0: o2 = o2
33
Because sample sizes for the two different levels of traffic flow were large, i.e, 200,
this test procedure for the equality of two variances does not require the normality
assumption [MON94]. Table 21 illustrates the related variables and their corresponding
values that were calculated in this variance equality test. The results in Table 21 suggest that
the errors in simulated travel times were likely to be flow dependent. This is because the
variances in errors of simulated travel times under different traffic conditions were
significantly different.
Table 21 Variance Equality Test Results for Simulated Travel Times from CORSIM
Variable S, S2 S, Zo
Value 0.445 7.562 5.356 18.790
The next task was to investigate whether the errors in simulated travel times were
Gumbeldistributed variates. Goodness of fit for the errors in simulated travel time was tested
using the KolmogorovSmirnov (KS) procedure which is generally preferable to chisquare
when dealing with continuous distributions [ZAR84]. Table 22 illustrates the variables and
their corresponding values in the KS goodness of fit procedure for an entry flow rate of
3,000 vph, where is the observed frequency, F, is the cumulative observed frequency, rel
F, is the cumulative relative observed frequency, and rel F, is the cumulative relative
expected frequency. The PWM method was applied to estimate 1 and if for the Gumbel
distribution, in order to obtain the cumulative relative expected frequency, i.e., rel F,.
Table 22 KolmogorovSmirnov Test Results for 3,000 vph Mainline Entry Volume
i Error Class (sec) f F, rel F, rel F D, D '1
1 <0.8 0 0 0 0.00213 0.00213 0.00213
2 0.80.7 2 2 0.01 0.00937 0.00064 0.00937
3 0.70.6 6 8 0.04 0.02884 0.01116 0.01884
4 0.6 0.5 7 15 0.075 0.06774 0.00726 0.02774
5 0.5 0.4 12 27 0.135 0.12953 0.00547 0.05453
6 0.40.3 8 35 0.175 0.21189 0.03689 0.07689
7 0.3 0.2 13 48 0.24 0.30788 0.06788 0.13288
8 0.20.1 19 67 0.335 0.40887 0.07387 0.16887
9 0.1 0 15 82 0.41 0.50712 0.09712 0.17212
10 00.1 23 105 0.525 0.59720 0.07220 0.18720
11 0.1 0.2 19 124 0.62 0.67613 0.05613 0.15113
12 0.2 0.3 11 135 0.675 0.74295 0.06795 0.12295
13 0.30.4 11 146 0.73 0.79806 0.06806 0.12306
14 0.4 0.5 13 159 0.795 0.84261 0.04761 0.11261
15 0.5 0.6 13 172 0.86 0.87808 0.01808 0.08308
16 0.6 0.7 7 179 0.895 0.90601 0.01101 0.04601
17 0.70.8 6 185 0.925 0.92780 0.00280 0.03280
18 0.8 0.9 4 189 0.945 0.94469 0.00031 0.01969
19 0.9 1.0 4 193 0.965 0.95773 0.00727 0.01273
20 1.0 1.1 3 196 0.98 0.96774 0.01226 0.00274
21 1.1 1.2 1 197 0.985 0.97541 0.00959 0.00459
22 1.2 1.3 1 198 0.99 0.98128 0.00872 0.00372
23 > 1.3 2 200 1 1 0 0.01
0.14052; maxD, = 0.09712; maxD ', = 0.18720; D
t = 2.755; i1
0.18720
35
Variable D, is the absolute difference between variables rel F, and rel Fj and variable
D ', is the absolute difference between variables rel F,_1 and relF,. The test statistic is the
largest D, or the largest D ',, whichever is larger, and is illustrated as follows:
D = max[(max D), (max D')] (2.46)
For large n, critical values of D,, can be approximated by Equation (2.47) [ZAR84].
In()
2n (2.47)
a, ~ \ 2n
From Table 22, D was larger than Do.05,200, i.e., 0.096032 calculated from Equation
(2.47), with 95% confidence. Therefore, the sample population for errors in simulated travel
times from CORSIM, using a mainline entry volume of 3,000 vph, was not Gumbel
distributed.
The KolmogorovSmirnov test procedure was applied again to test the errors of
simulated travel times using a mainline entry volume of 5,000 vph. Table 23 illustrates the
related variables and their corresponding values with all variables previously defined. Unlike
the results under moderate traffic flow, the errors under congested traffic passed the goodness
of fit test and were Gumbeldistributed.
Results from the tasks of examining variations in CORSIM simulated travel times
suggest that the variances for errors in simulated travel times are likely to be flow dependent,
and that the sample population for errors of simulated travel times may not always be
Gumbeldistributed.
Table 23 KolmogorovSmirnov Test Results for 5,000 vph Mainline Entry Volume
i Error Class (sec) f F, rel F, rel F, D, D '1
1 <14 0 0 0 0.00206 0.00206 0.00206
2 1413 2 2 0.01 0.00524 0.00476 0.00524
3 13 12 1 3 0.015 0.01155 0.00345 0.00155
4 12~11 1 4 0.02 0.02262 0.00262 0.00762
5 11 10 6 10 0.05 0.04004 0.00996 0.02004
6 10 9 1 11 0.055 0.06501 0.01001 0.01501
7 98 10 21 0.105 0.09813 0.00687 0.04313
8 8 7 11 32 0.16 0.13922 0.02078 0.03422
9 7 6 7 39 0.195 0.18737 0.00763 0.02737
10 65 11 50 0.25 0.24114 0.00887 0.04614
11 5 ~4 8 58 0.29 0.29876 0.00876 0.04876
12 43 15 73 0.365 0.35840 0.00660 0.06840
13 32 13 86 0.43 0.41831 0.00117 0.05331
14 21 4 90 0.45 0.47700 0.02700 0.04700
15 1 0 9 99 0.495 0.53327 0.03827 0.08327
16 0 1 12 111 0.555 0.58625 0.03125 0.09125
17 1 2 10 121 0.605 0.63536 0.03036 0.08036
18 23 8 129 0.645 0.68029 0.03529 0.07529
19 3 4 14 143 0.715 0.72094 0.00564 0.07594
20 45 8 151 0.755 0.75737 0.00237 0.04237
21 5 6 5 156 0.78 0.78975 0.00975 0.03475
22 67 5 161 0.805 0.81834 0.01334 0.03834
23 7 8 6 167 0.835 0.84343 0.00843 0.03843
24 89 5 172 0.86 0.86535 0.00535 0.03035
25 9 10 5 177 0.885 0.88441 0.00059 0.02441
26 10 11 1 178 0.89 0.90093 0.01093 0.01593
27 11 12 2 180 0.9 0.91520 0.01520 0.02520
28 12 13 1 181 0.905 0.92750 0.02250 0.02750
29 13 14 7 188 0.94 0.93807 0.00193 0.03307
30 14 15 1 189 0.945 0.94715 0.00215 0.00715
37
Table 23 (Cont.) KolmogorovSmirnov Test Results for 5,000 vph Mainline Entry
Volume
i Error Class (sec) f F, rel F, rel F D, D '
31 15 16 3 192 0.96 0.95493 0.00507 0.00993
32 16 17 2 194 0.97 0.96159 0.00841 0.00159
33 17 18 1 195 0.975 0.96728 0.00772 0.00272
34 18 19 1 196 0.98 0.97214 0.00786 0.00286
35 1920 1 197 0.985 0.97629 0.00871 0.00371
36 2021 1 198 0.99 0.97982 0.01018 0.00518
37 21 22 1 199 0.995 0.98284 0.01217 0.00717
38 2223 1 200 1 0.98540 0.01460 0.00960
39 > 23 0 200 1 1 0 0
I = 0.1633; if
2.8421; maxD,
0.03827; max D ', = 0.09125; D
The following section addresses the experimental results from interviewing a small
sample of individuals about the travel times they perceive under different levels of traffic
conditions to test if the errors in perceived travel times are asymmetrically distributed and are
flow dependent.
2.2.6 Variation in Perceived Travel Time
An experiment was performed to examine the patterns of errors in travel times
perceived by road users. Data for perceived travel times, as illustrated in Table 24, for two
links under two different levels of traffic conditions from a sample size of 26 persons were
collected. The free flow travel times for links (2204, 2012) and (1453, 1556) are estimated
as 2.256 and 0.213 minutes, respectively, according to their corresponding free flow speeds.
0.09125
Table 24 Perceived Travel Times for Links under Different Traffic Flows
Link (2204, 2012) (min) Link (1453, 1556) (min)
Sample # Peak 15 min Daily Average Peak 15 min Daily Average
1 0.45 0.50 NA NA
2 1.50 1.60 0.33 0.07
3 1.67 1.67 0.50 0.08
4 2.00 1.70 0.50 0.08
5 2.25 1.75 0.50 0.20
6 2.50 1.75 0.75 0.25
7 2.50 1.85 0.75 0.30
8 2.50 1.85 0.75 0.33
9 2.50 1.90 1.00 0.40
10 2.65 2.00 1.00 0.40
11 3.00 2.00 1.00 0.40
12 3.50 2.00 1.15 0.50
13 3.53 2.00 1.40 0.50
14 4.00 2.00 1.50 0.50
15 4.00 2.00 2.00 0.50
16 4.00 2.20 2.00 0.52
17 5.25 2.50 2.52 0.55
18 5.50 3.00 2.60 0.56
19 8.00 3.00 3.00 0.75
20 9.56 3.25 3.00 0.75
21 10.00 3.50 3.50 0.75
22 10.00 4.00 5.00 1.00
23 14.20 5.00 5.00 4.00
24 15.00 5.00 8.15 5.30
25 15.00 10.00 10.00 7.00
26 20.00 12.00 20.00 15.00
Mean 5.96 3.08 3.12 1.63
Median 3.77 2.00 1.50 0.50
Variance 26.73 6.59 18.17 10.72
39
Since there were less than or equal to 26 observations available for the perceived
travel times, the large sample test procedure for the equality of two variances is not
appropriate for testing the perceived travel times data. However, since the traditional
assumption for errors of perceived travel times is that they are identically distributed, the test
can be performed under the hypothesis that all samples for different links under different
traffic conditions came from a population with identical variances. The hypotheses of the
equality, or homogeneity, of variances can be written as follows [ZAR84]:
HO: C2 = C2 = C 2 = C2
Ho 1 2 03 04
Ha: The four population variances are heterogeneous (i.e., are not all equal)
The most common method employed to test for homogeneity of variances is Bartlett's
test [ZAR84]. The test statistic for this procedure is illustrated in Equation (2.48).
B = 2.30259 x (log S ) ( v) v logSi2 (2.48)
where v, is equal to n, 1 and n, is the size of sample i. Variable S2 is calculated as follows:
k k n
2 (2.49)
p k k
1=1 i=1
The distribution of B is approximated by the chisquare distribution with k 1 degrees
of freedom, but a more accurate chisquare approximation is obtained by computing a
correction factor [ZAR84],
40
C= + 1 11
3 x (k 1) v k (2.50)
with the corrected test statistic being B, = B/C.
Table 25 illustrates the related variables and their corresponding values in Bartlett's
test for homogeneity of variances.
Table 25 Test Results for Homogeneity of Variances in Errors of Perceived Travel Times
Link (2204, 2012) Link (1453, 1556)
Variable Sum
Peak 15 min Daily Average Peak 15 min Daily Average
SS, 668.13 164.69 436.13 257.24 1526.185
v, 25 25 24 24 98
S2 26.725 6.588 18.172 10.718 62.203
log S,2 1.427 0.819 1.259 1.030 4.535
v, log S,2 35.673 20.468 31.485 25.753 113.38
1/v, 0.04 0.04 0.0417 0.0417 0.1633
S2= 1526.185/98= 15.573, log S2 = 1.1924
B = 2.30259 [1.1924 98 113.38] = 8.00196
C= 1 + 1/(3*3) (0.1633 1/98) = 1.017
B, = B/C= 7.8682
Since 0o.o5, 3 is equal to 7.815, with 95% confidence, the errors in perceived travel
times are not all equal. Although the data in Table 24 were collected from a small sample
and were only for two roadway links, the results indicate that errors in perceived travel times
are not likely to be identically distributed. In addition, errors in perceived travel times are
likely to be flow dependent since variances tend to increase along with higher traffic.
The next task was to investigate whether the errors in perceived travel times are
Gumbeldistributed. The KolmogorovSmirnov goodness of fit procedure that was
performed in Section 2.2.5 was repeated four times to test the data from the perceived travel
time experiment. The results are illustrated in Tables 26 through 29.
Table 26 Test Results for Link (2204, 2012) Under Daily Traffic
i Error Class (sec) f F, rel F, rel F D, D '
1 <1.5 0 0 0 0.05354 0.05354 0.05354
2 1.51 1 1 0.03846 0.11945 0.08099 0.11945
3 1 0.5 0 1 0.03846 0.21388 0.17542 0.17542
4 0.5 0 8 9 0.34615 0.32644 0.01971 0.28798
5 00.5 7 16 0.61539 0.44371 0.17168 0.09755
6 0.5 1 1 17 0.65385 0.55443 0.09942 0.06096
7 1 1.5 3 20 0.76923 0.65173 0.11750 0.00212
8 1.52 1 21 0.80769 0.73289 0.07480 0.03634
9 22.5 1 22 0.84615 0.79807 0.04809 0.00963
10 2.5 3 0 22 0.84615 0.84898 0.00282 0.00282
11 3 3.5 2 24 0.92308 0.88795 0.03513 0.04180
12 >3.5 2 26 1 1 0 0.07692
0.17615; maxD,= 0.17542; maxD',= 0.28798; D :
 = 0.6408; fl
0.28798
Table 27 Test Results for Link (2204, 2012) Under Peak15 Minute Traffic
i Error Class (sec) f F, rel F, rel F D, D '
1 <4 0 0 0 0.05918 0.05918 0.05918
2 43 1 1 0.03846 0.11327 0.07481 0.11327
3 32 2 3 0.11539 0.18676 0.07138 0.14830
4 21 7 10 0.38462 0.27454 0.11008 0.15916
5 10 3 13 0.50000 0.36941 0.13059 0.01521
6 0 1 3 16 0.61539 0.46431 0.15108 0.03569
7 12 2 18 0.69231 0.55375 0.13856 0.06164
8 23 0 18 0.69231 0.63423 0.05808 0.05808
9 3 4 0 18 0.69231 0.70413 0.01182 0.01182
10 45 1 19 0.73077 0.76319 0.03242 0.07088
11 5 6 1 20 0.76923 0.81205 0.04281 0.08128
12 6 7 2 22 0.84615 0.85181 0.00565 0.08258
13 78 0 22 0.84615 0.88376 0.03761 0.03761
14 89 0 22 0.84615 0.90919 0.06304 0.06304
15 9 10 0 22 0.84615 0.92929 0.08313 0.08313
16 10 11 1 23 0.88462 0.94507 0.06045 0.09891
17 11 12 2 25 0.96154 0.95741 0.00413 0.07279
18 > 12 1 26 1 0 0 0.03846
0.01591; maxD, = 0.15107; maxD ', = 0.15916; D = 0.15916
t = 0.26085; il
Table 28 Test Results for Link (1453, 1556) Under Peak15 Minute Traffic
i Error Class (sec) f F, rel F, rel F D, D '
1 <1.25 0 0 0 0.17709 0.17709 0.17709
2 1.25 1 1 1 0.04 0.20823 0.168233 0.168233
3 1 0.75 3 4 0.16 0.24117 0.08117 0.20117
4 0.75 0.5 3 7 0.28 0.27551 0.00449 0.11551
5 0.5 0.25 4 11 0.44 0.31084 0.12916 0.03084
6 0.25 0 1 12 0.48 0.34676 0.13324 0.09324
7 00.25 1 13 0.52 0.38289 0.13711 0.09711
8 0.250.5 0 13 0.52 0.41888 0.10112 0.10112
9 0.5 0.75 2 15 0.6 0.45442 0.14558 0.06558
10 0.75 1 0 15 0.6 0.48924 0.11076 0.11076
11 1 1.25 2 17 0.68 0.52309 0.15691 0.07691
12 1.25 1.5 0 17 0.68 0.55580 0.12420 0.12420
13 1.5 1.75 2 19 0.76 0.58720 0.17280 0.09280
14 1.75 2 0 19 0.76 0.61720 0.14280 0.14280
15 22.25 1 20 0.8 0.64572 0.15428 0.11428
16 > 2.25 5 25 1 1 0 0.2
1 = 0.39305; f~ = 0.14616; maxD,
0.17709; max D ', = 0.20117; D
0.20117
44
Table 29 Test Results for Link (1453, 1556)
Under Daily Traffic
i Error Class (sec) f F, rel F, rel F D, D ',
1 <0.5 0 0 0 0.22072 0.22072 0.22072
2 0.5 0.4 3 3 0.12 0.24131 0.12131 0.24131
3 0.4 0.3 0 0 0.12 0.26244 0.14244 0.14244
4 0.3 0.2 2 5 0.2 0.28401 0.08401 0.16401
5 0.2 0.1 2 7 0.28 0.30582 0.02592 0.10592
6 0.1 0 3 10 0.4 0.32807 0.07193 0.04807
7 00.1 7 17 0.68 0.35038 0.32962 0.04962
8 0.1 0.2 0 17 0.68 0.37276 0.30724 0.30724
9 0.20.3 3 20 0.8 0.39512 0.40488 0.28488
10 0.3 0.4 0 20 0.8 0.41739 0.38261 0.38261
11 0.40.5 0 20 0.8 0.43948 0.36052 0.36052
12 > 0.5 5 25 1 1 0 0.2
0.6085; f
0.178187;
max D,
0.40488;
max D',= 0.38261; D
Since DO.05 26 and D0.05 25 are equal to 0.25908 and 0.26404, respectively, sample
populations for the two links under peak 15minute traffic were Gumbeldistributed.
However, as with the results for simulated travel times from CORSIM, the goodness of fit
test did not show that the errors in perceived travel times were Gumbeldistributed under
moderate traffic volume conditions.
Thus, the experiment results in this section indicate that the errors in perceived travel
times are not likely to be identically distributed. The errors are likely to be flow dependent
since higher variances were obtained at higher traffic volume conditions, although this was
not tested using any statistical tools. In addition, although the perceived travel times were
0.40488
45
not likely to be Gumbeldistributed under moderate traffic, errors of perceived travel times
are likely to be Gumbeldistributed under congested traffic.
2.3 Nonparametric Statistics
Since the SUE algorithm introduced later in Section 4.1 is an iteration process, a
strategy is required to terminate the model iteration. Statistical tests are applied to determine
whether the process can be terminated. The term nonparametric statistics has no standard
definition but nonparametric statistical methods are known to work well under fairly general
assumptions about the nature of all probability distributions or parameters that are involved
in an inferential problem [MEN90]. The validity of nonparametric statistical methods does
not depend upon the actual mathematical form, e.g., normal distribution, of the population
distributions. Often, assumptions such as "the population distribution is continuous," or "the
population distribution is continuous and symmetric" is all that is needed to validate the use
of nonparametric statistics. Thus, nonparametric statistics methods are also known as
distributionfree methods.
The sign test and the Wilcoxon SignedRank test procedures are two of the most
commonly used distributionfree procedures for making inferences about the location
parameter of a population distribution [RA098]. Both of these procedures are designed for
inferences about the median of a continuous, but not necessarily normal, population.
Since the mean and median are the same for a normal distribution, the sign test and
Wilcoxon SignedRank test procedures can also be used for making inferences about the
mean of a normal population. Therefore, these two procedures are alternatives to the
procedures based on the one sample ttest and are useful when there is reason to believe that
46
violations of the assumption of a normal distribution is serious enough to render the use of
the one sample ttest procedure to be undesirable [RA098].
Equation (2.51) illustrates the statistical model associated with the sign, signedrank
and ttest for inference about a population median [RA098].
Y = + e. (2.51)
where Y, = the ith observation in a sample;
p = population median; and
C, = error in the ith observation.
The sign test requires the least restrictive assumption that the errors, i.e., E,, have a
continuous distribution with a median of zero. The assumption for the Wilcoxon Signed
Rank test is more restrictive because it not only requires the errors to be continuously
distributed with medians equal to zero, but also requires the error distributions to be
symmetric. The ttest requires the most restrictive assumption that the errors have a normal
distribution with median of zero since the assumption of a normal distribution with mean of
zero implies that the error distribution is continuous symmetric about a median of zero. In
this study, the Wilcoxon SignedRank test is used to terminate the iteration of an SUE traffic
assignment by assuming a median of zero for differences in assigned trips between successive
SUE iterations. However, prior to implementing this test, the differences in assigned trips
between successive SUE iterations need to be observed by examining the assumption of
symmetricallydistributed errors. The sign test can be applied to validate results from the
SUE model by comparing the assigned trips from a UE traffic assignment and each SUE
iteration with the counted volumes if the error term is not symmetricallydistributed.
2.3.1 Wilcoxon SignedRank Test
Let the assigned trips from iterations n 1 and n for link i be x,"' and x,', respectively.
In addition, let the relative change between iterations n 1 and n for link i be D, and be equal
to x," x,"'. The null and alternative hypotheses are illustrated as follows [VIMEN90]:
Ho: The population relative frequency distributions for the assigned trips from the
two successive iterations are identical.
Ha: The relative frequency distributions for the two populations differ.
Under the null hypothesis of no difference in the distributions of assigned trips from
two successive iterations, half of the differences in pairs would be expected to be negative and
the other half positive. In other words, the expected number of negative differences between
pairs would be n/2 where n is the number of links in a transportation network. In addition,
it would be expected that positive and negative differences of equal absolute magnitude
should occur with equal probability. That is, by ordering the differences according to their
absolute values and by ranking them from smallest to largest, the expected rank sums for the
negative and positive differences would be equal.
To carry out the convergence test, the differences for assigned trips from the previous
and current iterations for each link in the network are calculated. Differences equal to zero
are eliminated, and the number of links, n, is reduced accordingly. The absolute values of the
differences, i.e., Dj, are then ranked by assigning a 1 to the smallest, a 2 to the second
smallest, and so on. If two or more absolute differences are tied for the same rank, the
average of the ranks that would have been assigned to these differences is assigned to each
member of the tied group. The rank sums for the negative differences and the positive
48
differences are then calculated and denoted by T and T respectively. For a large sample
size, i.e., n > 25, the test statistic for the Wilcoxon SignedRank test becomes the following
[MEN90]:
T n x (n + 1)
Z = (2.52)
n x (n + 1) x (2n + 1)
24
At a equal to 0.05, the hypothesis of identical population distributions would be rejected
when Z > 1.96.
2.3.2 Sign Test
Let the absolute difference between the counted volume and assigned trips from a UE
traffic assignment for link i in a given transportation network be A, for experiment A. Let the
absolute difference between the counted volume and assigned trips for link i from a specific
SUE traffic assignment be B, for experiment B. There are numerous pairs of observations of
the form (A,, B,) when listing the data from groups A and B for the same link in a pair.
The sign test is based on the signs ofD,, i.e., the differences between A, and B,. Under
the null hypothesis that A, and B, come from the same continuous probability distributions,
the probability that D, is positive is equal to 0.5 and so is the probability that D, is negative.
The median ofD, is assumed to be zero. Unlike the Wilcoxon SignedRank test applied for
terminating the SUE iteration, in the sign test the distribution for the error term between D,
and its median is unrestricted. The null and alternative hypotheses for the sign test can be
written as:
49
Ho: p = 0.5 (neither the UE or SUE model is preferred, i.e., the results from these
two models are identical)
Ha: p > 0.5 (Experiment B produces preferred assigned trips since the differences
between the UE trips and counted volumes are greater than the ones
between the SUE trips and counted volumes)
Equation (2.53) illustrates the test statistic for the sign test when the sample size is
larger than 25 [MEN90].
n
M 
2
Z (2.53)
where M denotes the number of positive D, values and n is the number of samples. The
observations associated with one or more pairs may be equal and result in ties, i.e., D, equal
to zero. These links are deleted and n is reduced accordingly. The sign test rejects Ho if Z
> Z,. With a significance level of 0.05, Z, is equal to 1.645.
2.4 Percent Root Mean Square Error
The percent rootmeansquare error (%RMSE) is a statistic for comparison between
the model generated volume and the actual ground count on the link and is adopted by the
Florida Department of Transportation (FDOT) as a criterion for calibrating trip assignment
results [ORL92].
Equation (2.54) is the formula for calculating the rootmeansquare error (RMSE).
The %RMSE, illustrated in Equation (2.55), is then derived by dividing the RMSE by the
average of traffic counts.
50
N
RE AISE = X )2 (2.54)
oR/ ISE RAISE
Xjc (2.55)
i = i
N
where X, = traffic count of link i;
X,' = assigned volume to link i; and
N = total number of links in the network with counted volumes.
RMSE is used to measure the deviation between the counted volumes and assigned
trips from two successive iterations for every link with traffic count data available.
CHAPTER 3
NETWORK DESCRIPTION
3.1 Gainesville Network
As addressed in Chapter 1, the pathbased SUE traffic assignment model is applied
to the 1985 Gainesville highway network. This allows SUE model performance to be
evaluated by comparing its assigned link volumes with (1) the counted link volumes collected
from the roadways and with (2) the assigned link volumes from the UE model in FSUTMS.
Before applying the pathbased SUE model to the 1985 Gainesville network,
FSUTMS needs to be executed first. Subsequently, it generates files containing the
information required at the SUE model initialization step described in Chapter 4. The files
required specifically for the SUE model are PROFILE.MAS, LINKS.85V, XY.85V,
SPDCAP.85V, HASSIGN.OUT, and MODE.OUT.
According to LINKS.85V and XY.85V, the network contained 1,027 nodes (with
5018 as the maximum node number), 1,404 twoway and 17 oneway roadway links, i.e.,
2,825 directional links in total. Of these, 1,144 are centroid connection links leaving 1,681
major roadway links. Among the major roadway links in the network, there are 549 links
with traffic volume counts. Traffic volume counts are not available for the rest of the links
in the 1985 network, but the counts available exceed what are usually available. Figure 31
illustrates the 1985 Gainesville highway network.
imm G %%5 ighwa Netork7
1 177,
Figure 31 1985 Gainesville Highway Network
53
The 1985 Gainesville network has 246 internal zones, 33 dummy zones held in reserve
for future use, and 14 external stations, i.e., 293 centroids in total. Due to the intensive
computational nature of the pathbased SUE traffic assignment model, the dummy zones were
taken out manually from the Gainesville network, reducing the total centroids to 260, to
reduce the model execution times. In addition, since the regional public transit system for the
Gainesville urban area served low ridership and data for the 1985 Gainesville network show
virtually no other transportation modes except private vehicles, the "highway only"
assignment option of FSUTMS was selected.
In addition to link and node data, socioeconomic characteristics data are necessary to
represent the basis for estimating trip generations, which ultimately govern the results of
traffic assignment. The following are summaries of the socioeconomic characteristics from
the zonal data files or "ZDATA" files for the 1985 Gainesville network. Table 31
summarizes the socioeconomic data for trip productionoriented zonal data from the
ZDATA1.85V file. In addition, a total of 1,635 units were occupied among the 2,420
hotel/motel units by 2,478 people.
Table 31 Socioeconomic Data Summary from ZDATA1.85V
Socioeconomic Characteristics Single Family (SF) Multi Family (MF)
Total Occupied Dwelling Units (DUs) 25,399 28,162
Population of Permanent & Seasonal Residents 63,788 59,600
DUs with 0 Cars 1,270 3,661
DUs with 1 Cars 8,382 14,363
DUs with 2+ Cars 15,747 10,138
54
Table 32 summarizes trip attraction socioeconomic data listed by traffic analysis zone
from the zonebased dataset, ZDATA2.85V. In addition, there was a total student enrollment
in schools of 18,327 in the 1985 Gainesville data. Note that colleges and universities with
enrollments of 2,000 or more are treated separately as special traffic generators.
Table 32 Socioeconomic Data Summary from ZDATA2.85V
Category Industrial Commercial Service Total
Number of Employees 4,462 12,195 39,788 56,445
Table 33 summarizes special generator data listed by traffic analysis zone in
ZDATA3.85V, where HBW, HBS, HBSR, HBO and NHB stand for homebased work,
homebased shopping, homebased social/recreation, homebased other and nonhomebased
trips. The total number of externalinternal (EI) trips for the 14 external zones in the 1985
network that are listed in ZDATA4.85V is 115,667. For the purpose of simplification, the
individual El zonal trips are not illustrated in a table.
Table 33 Special Trip Generator Data Summary from ZDATA3.85V
Trip Trip Purpose
Type
HBW HBS HBSR HBO NHB Total
Productions 0 0 0 58,118 0 58,118
Attractions 16,557 9,032 593 130,949 0 157,132
55
There is no VFACTORS.85V file for the 1985 Gainesville network. This is because
the previous versions of FSUTMS did not allow the information in that file to vary by facility
type. Since this file is now a required input, the file with recommended default values for
FACTORS in FSUTMS was applied. The &SELDEST in the PROFILE.MAS was
specified as "1260" to include every centroid in the 1985 Gainesville network.
3.2 FTOWN Network
In order to further examine the convergence likelihood of the SUE traffic assignment
model, the FTOWN example network in FSUTMS was used to execute the traffic assignment
process for five model iterations. Figure 32 illustrates the FTOWN highway network.
The FTOWN network has 15 internal zones and 9 external stations, i.e., 24 centroids
in total. The network contained 140 nodes (with 4100 as the maximum node number) and
328 directional links. The FTOWN network is a hypothetical network which was created
merely as an example in FSUTMS.
Figure 32 FTOWN Highway Network
1istFTOWN Highway Network F3
CHAPTER 4
SUE ALGORITHM AND MODEL INITIALIZATION
4.1 SUE Algorithm
There are five steps in the SUE traffic assignment algorithm. Figure 41 illustrates the
flowchart of the SUE algorithm. Excluding the initialization step, the other four steps in the
SUE algorithm form an iterative process which requires a convergence test to terminate the
model iteration process. The overall algorithm contains the underlying assumptions that
errors of perceived travel times are flow dependent and are independently distributed for each
enumerated competitive path.
As illustrated in Figure 41, the steps in the SUE algorithm are:
Step 0Initialization Step. Perform a standard UE assignment and obtain the initial
geometric and general traffic information from the input and output files of FSUTMS. Set
n = 1 where n is the SUE model iteration number.
Step 1Competitive Path Enumeration Step. Apply an explicit path enumeration
approach to enumerate multiple competitive paths for each centroid of travel origin. Repeat
for all centroids of origin. To avoid confusion with the model iteration, "path enumeration
iteration" will be used to indicate the calculations described in this step.
Step 2Travel Time Variation Step. Obtain and store the simulated path travel times
and determine their variation for each path enumerated in the path enumeration step by
58
applying assigned volumes from the (n 1)th model iteration for component links along the
path. Store the simulated travel times for the path and its component links.
Figure 41 Flowchart for the SUE Algorithm
Step 3Route Choice Model Implementation Step. Obtain the Gumbel distribution
parameters for each path. Perform a route choice assignment procedure which allocates trips
to each enumerated path for each origindestination pair based on the simulated path travel
times from Step 2. This yields auxiliary link flow volumes { yj }.
Step 4Test Convergence Step. Compare the related change in assigned trips from
previous and current iterations for every link in a transportation network. If convergence is
attained, stop. If not, set n = n + 1 and update the link travel times. Go to Step 1.
59
The SUE condition is reached when the model iterations are terminated at
convergence. This is because perceived travel times are assumed to be flow dependent.
When assigned trips from successive iterations are not found to be significantly different,
travelers cannot find a path that can improve their travel times.
Since perceived travel times cannot be measured directly, most stochastic traffic
assignment models apply measured travel times to represent the nonstochastic component
of perceived travel times for paths between a given zone pair. Also, they assume the error
term of perceived path travel times as independently and identically Gumbeldistributed
variates. The error term is expected to take into consideration the unknown and/or
unobserved attributes that influence individuals' disutilities for choosing routes.
Because CORSIM is a well recognized simulation tool for real world applications in
estimating travel times, the median (or average) simulated travel times from CORSIM can be
used as the measured travel times for the nonstochastic component of perceived travel times
under a given flow rate. As addressed in Section 2.2.5, the variation in simulated CORSIM
travel times are obtained under different driver/vehicle combinations and traffic environments.
Thus, when using the median value of simulated travel times from a finite number of
CORSIM simulations (performed using various random number seeds) as the measured travel
time for a path, the variations among the multiple simulated travel times become a realization
of a proportion of true errors in perceived travel times by travelers.
It is evident that the proportion of errors among simulated travel times from CORSIM
cannot be a full realization of errors in perceived travel times by travelers since there are
factors, e.g., weather, lighting, and numerous unknown factors that influence individuals'
60
perceptions of travel disutilities, which are excluded from the CORSIM simulation model.
However, the known factors that produce variations in CORSIM simulated travel times can
contribute to distinguishing the difference between various alternative paths. This is preferable
to using a simplified assumption about the errors of perceived travel times.
This study applies CORSIM to emulate the perceived path travel times, i.e., Cs(v),
in Equation (1.2), for each enumerated competitive path. The median value of the simulated
travel times is used to approximate the measured travel time, i.e., c~ (v), in Equation (1.2).
The variations in simulated travel times from multiple CORSIM simulations are then treated
as the errors in perceived travel times by road users, i.e., er(v)', in Equation (1.2). As
addressed earlier, the simulated travel times from CORSIM may not be exact realizations of
path travel times perceived by road users. However, the implementation of CORSIM is more
practical than a field experiment because it is less costly and the results are obtained more
quickly.
The FRESIM model in CORSIM is used in the SUE traffic assignment model to
obtain the simulated path travel times. Although a better approximation for perceived path
travel times could be to implement the NETSIM model with the necessary signal timing
control information entered, FRESIM is used in order to reduce the effort of creating the
CORSIM input files. In reality, however, road users have to stop and wait for traffic signals
at intersections when traveling through an arterial network. This should inflict considerably
more delay and cause much longer travel times than traveling at uninterrupted freeway
speeds.
61
Consequently, the use of FRESIM is expected to grossly underestimate travel times
and errors, i.e., variables c, (v) and e, (v)' in Equation (1.2). If the new SUE method can
provide better agreement with traffic counts, future improvements should apply NETSIM,
which models intersection delay to obtain more realistic travel times and errors.
4.2 SUE Initialization Procedures
The initialization step is designed mainly to obtain the initial geometric and general
traffic information, either from the input or the output files of FSUTMS, that will be used in
the subsequent steps of the SUE traffic assignment algorithm. For example, the initialization
step processes the retrieval of network variables in the PROFILE.MAS file, e.g., the total
number of traffic analysis zones, and the highest node number in the network. The network
information in the input files ofFSUTMS, e.g., XY, LINKS and SPDCAP files, etc., are also
retrieved. In addition, the traffic flows generated by a user equilibrium traffic assignment
model in FSUTMS are retrieved from the FSUTMS output file and are stored as the initial
link flows. The initialization step then provides an initial estimate of link travel times under
the UE assigned volumes by the BPR formula for all links in the system.
There are eight sequential modules in the initialization step: ReadProfile,
ReadSpdCapFile, ReadVfactorsFile, ReadHassign, MakeRunBatch, CreateLinksFile,
ChangeHnetAll and ReadMode. Each module prepared in this study is basically named after
a FSUTMS input or output file that is processed within the module and is described in detail
in the following sections. Figure 42 illustrates the flow chart for the modules at the
initialization step.
Figure 42 Modules Processed at the Initialization Step
4.2.1 Read Profile
The ReadProfile module retrieves the information that reflects characteristics of an
urban area specified in the PROFILE.MAS file. They are: NAMEstudy area name,
ZONESAtotal number of zones including internal and external, NODEShighest node
number permitted, and UNITScoordinate units per mile. "NAME" is retrieved only for
reference purpose during the execution of the SUE computer program and has no impacts on
63
the analysis. "ZONESA" and "NODES" are required in the program for initial memory
allocations. "UNITS" is a scale factor for calculating link distances by using the coordinate
node data.
If the link distance values are specified in the LINKS file, the value for the "UNITS"
variable will have no impacts on the analysis. Otherwise, the link distance is calculated by the
formula illustrated in Equation (4.1) with the coordinate data contained in the XY file.
x2 + /5280
distance = (x1 x2 )2 + (l Y2 x 5280 (4.1)
UNITS
where distance = link length in feet
x1, x2 = x coordinates for nodes 1 and 2; and
y, y, = y coordinates for nodes 1 and 2.
The "PROFILE.MAS" file is referenced during the execution of FSUTMS to identify
parameter settings used in each step of the model that are specified in the FSUTMS control
files. FSUTMS control files (also called "script" files) are used to define the modeling sub
steps, configurations, and options used during a given FSUTMS model run.
Control files ending with the extension ".ALL" are used during FSUTMS execution
for all transit, nontransit and alternative modeling approaches. Control files ending with the
extension ".HWY" are used during highway modeling only. ".TR?" control files are used for
the processing of FSUTMS transit modeling steps. The last character in the filename
extension refers to the specific type of transit approach, as follows: TR1Singlepath transit,
TR2Multipath/singleperiod transit, and TR3Multiperiod/multipath transit [FSU97].
Note that all of the control files with extensions ".HWY", except MODE.HWY", and ".TR?"
remain unchanged during the process of pathbased SUE model.
64
A review of any FSUTMS control file would indicate a number of "&" parameter
references. The PROFILE.MAS file contains replacement values for each of these parameter
references. The format for this file is one line which references the parameter name and is
followed by the parameter value on the following line.
4.2.2 Read Speed/Capacity
The ReadSpdCapFile module retrieves the information that is specified in the
SPDCAP.yya file, where "yy" and "a" stand for the simulation year and the alternative
identification, respectively. Simulation year defines the last two digits of the model
application year and part of the input (yya) and output (ayy) file extensions. Alternative
Identification defines the alternative (usually "A," "B," "C," etc.) and completes the
input/output file extension definition [FSU97].
Instead of individually coding directional speed and capacity for each network link,
appropriate speed and capacity values can be designated in FSUTMS for each link according
to different combinations of area type, facility type, and number of lanes. Area type codes are
an indication of the land use characteristics of areas traversed by links. Facility type codes
are an indication of the physical type of facility for highway links. For the SUE computer
program, the speed and capacity lookup table is stored in memory and is used in subsequent
modules for calculating link travel times and link capacities as well. Note that the capacities
specified in SPDCAP.yya are usually based on levelofservice (LOS) "E." The capacities are
converted to LOS "C" for subsequent traffic assignment procedures performed in the process
by multiplying by a factor designated as UROADF.
4.2.3 Read Vfactors
The ReadVfactorsFile module retrieves the values for the variables of CONFAC,
UROADF, BPR LOS and BPR EXP that are specified in the VFACTORS.yya file for each
facility type. The CONFAC parameter is used to factor daily capacities to hourly capacities.
A CONFAC value of 10 means that approximately 10 percent of daily trips occur during the
peak hour. The UROADF factor is used to adjust LOS "E" capacities to represent practical
capacities, usually at LOS "C." In FSUTMS, the BPR LOS and BPR EXP parameters are
used within the Bureau of Public Roads (BPR) formula to calculate the iterative link travel
times, as illustrated in Equation (4.2).
Tn = T x (1.0 + BPR LOS x (v/c)BPR EP) (4.2)
where: T = Travel time for current iteration;
T = Travel time for previous iteration (or optionally the free flow travel
time, To); and
v/c Ratio of assigned volume to practical capacity.
The SUE computer program applies the BPR formula with the parameters specified
in the "VFACTORS.yya" file to calculate the initial link travel times on each network link by
substituting v and T,1 in Equation (4.2) with the UE assigned volumes and free flow time,
respectively.
The program is designed to automatically apply the default values recommended in
FSUTMS Release 5 if no "VFACTORS.yya" can be found. In this case, 0.10 and 4.00 are
used for CONFAC and BPR EXP, respectively, for all facility types and 0.15 is applied for
66
BPR LOS except facility types 80 through 89 where 0.30 is used. UROADF has varying
values from 0.57 to 1.0 with the most frequent being 0.66 in about 25% of the different
facility types, and the second most frequent being 1.0 in almost as many instances.
4.2.4 Read Assigned Volumes
The ReadHassign module retrieves the assigned traffic volumes listed in the
HASSIGN.OUT file for each network link. The assigned volumes are assigned to the
network links according to the UE traffic assignment model and are used in the subsequent
modules for calculating the initial link travel times. The UE assigned volumes are also used
for evaluating the performance of the pathbased SUE traffic assignment procedure.
A linked list is created in this module of the SUE computer program to store the
necessary information for each highway link in objects called cells. Each cell includes a
pointer to the next cell in the list. The information includes: 'from' and 'to' nodes of a link,
link distance, assigned link volume, cumulative simulated link travel time, the number of
simulations for this specific link during a model iteration, and the NextCell variable that
indicates the next cell in the list.
This linked list is updated accordingly in the subsequent module of the SUE computer
program each time the information for a certain highway link changes during the assignment
process. In the remaining text, the term "highway linked list" is used when this linked list of
highway links is referred to.
4.2.5 Make Batch
The MakeRunBatch module creates the batch file illustrated in Figure 43. The batch
file is designed to perform the highway network building and then to report the shortest paths.
The batch file is responsible for launching the following four FSUTMS executable files:
TRNPLNXT, NETPRO, HWYNET, and RPTPAT. Iterative executions of the batch file are
performed in the path enumeration step to search for the feasible paths between zone pairs.
Rem Start HNET
cd c:\student\li\cgn7980\program2\path
if exist disk*.* del disk*.*
if exist *.err del *.err
copy C:\Student\li\CGN7980\GNV01\hnet.in tmpln.in
trnplnxt
del tmpln.In
del tmplnx.bat
del tmpl002.ins
copy C:\Student\li\CGN7980\GNV01\Profile.mas Profile.mas
copy C:\Student\li\CGN7980\GNV01\xy.85v xy.xxx
copy C:\Student\li\CGN7980\GNV0 1\links.tem links.xxx
copy C:\Student\li\CGN7980\GNV01\spdcap.85v spdcap.xxx
netpro
del xy.xxx
del links.xxx
del spdcap.xxx
del hnetaux.xxx
hwynet
del tmplOO1.ins
copy hnet.out hnet.v85
del hnet.out
del netpro.inf
del tmpl*.out
del tmplnx.con
Rem Start HPATH
copy C:\Student\li\CGN7980\GNV01\hpath.in tmpln.in
trnplnxt
del tmpln.In
del tmplnx.bat
rptpat
del tmplnx.con
del tmplOO1.ins
copy tmpl001.out C:\Student\li\CGN7980\GNV01\hpath.out
del tmplOO 1.out
del tcards.xxx
del Profile.mas
del hnet.v85
cd C:\Student\li\CGN7980\GNV01
Figure 43 Example Batch File Created by the MakeRunBatch Module
68
The highway network building model of FSUTMS is called "HNET." The execution
of the HNET module in FSUTMS requires completion of the following four substeps by
default: preparation of data, NETPRO conversion, build highway network, and report
highway network [HNE97]. In this study, however, the highway network building model is
simplified to require only the completion of the first three substeps.
The first step in the batch file is to execute the "TRNPLNXT" module which reads
the HNET.ALL file and then creates the following output files: TRNPL001.INS and
TRNPLNX.CON. The ".CON" file simply contains the first control function file name, i.e.,
TRNPL001.INS, and its output file name, i.e., TRNPL001.OUT, after the function controls
have been processed by a TRANPLAN module. The ".INS" file contains the function
controls which are input to the TRANPLAN programs in building the highway network. The
control file contains the control codes for execution of the second and third steps of the
HNET module in FSUTMS.
The second step in the batch file is to execute an FSUTMS conversion program called
"NETPRO." NETPRO converts the data in the files required for the HNET model in
FSUTMS, i.e., LINKS.yya, XY.yya and SPDCAP.yya, into a temporary ASCII formatted file
called "HWYNET.TEM". This ".TEM" file is the input file for the next substep. The
extension name is changed to XXX prior executing NETPRO.
The LINKS file contains highway characteristics data for each link in the highway
network. The following information is specified in LINKS.yya for each highway network
link: link endpoint nodes, facility type, area type, number of lanes, and fieldmeasured traffic
volume. The XY file contains X and Y coordinates for each node in the highway network.
69
If link lengths are specified in the LINKS file, the coordinate data are only used to graphically
display the paths that are enumerated.
The third step in the batch file is to execute "HWYNET" in order to generate a
highway network database in a binary code compatible with other TRANPLAN programs.
This binary file was originally called HNET.OUT and is renamed to HNET.ayy. This finishes
the highway network building process. The rest of the commands in the batch file are used
to search and report the minimum paths for each traffic analysis zone.
The highway network pathbuilding model in FSUTMS is called "HPATH." The
HPATH module uses a database of highway network information (from the HNET module)
to calculate matrices of travel times (and distances, if desired) between each pair of traffic
analysis zones in the network. These matrices (sometimes called "skims") are accumulated
over the shortest highway paths [HPA97].
The path building process in the HPATH module of FSUTMS is implemented in the
batch file. Similar to the highway network building process described earlier, the batch file
first executes the "TRNPLNXT" file which reads the HPATH.ALL file and creates the
following output files: TRNPL001.INS and TRNPLNX.CON. The next step is to execute
the RPTPAT (report highway paths) file to construct the minimum travel paths by listing
nodes traversed from a selected origin zone, specified in the HPATH.ALL file, to all other
zones in the network.
4.2.6 Create Links File
The CreateLinksFile module is designed to create the LINKS file which contains the
link travel times calculated by the BPR formula. The travel time values are obtained by using
70
assigned volumes from the UE model in FSUTMS and also by using the default BPR
parameters specified in the VFACTORS file for different facility types. In addition, the
attributes for each highway link that are specified in the original LINKS file are stored in the
linked list created in the ReadHassign module, e.g., distance, freeflow speed and capacity.
Travel times for the connector links, i.e., connections between the centroid nodes and
the major roadway system, are virtually exempted from the BPR formula by nearly unlimited
link capacities in the first model iteration and are not updated thereafter in the remaining
iterations. This is because centroid connectors do not necessarily correspond to specific
roadways on the ground and are specified with nearly unlimited link capacities in
transportation planning studies. Thus, the impacts on travel times by various volumes
traversing these links are ignored in the pathbased SUE model.
4.2.7 Change HNET Controls
As described in the previous sections, the HNET.ALL control file contains the
controls for "HWYNET." The parameters for number of zones, maximum node and
coordinate distance factor that are specified in the PARAMETERS section of the HNET.ALL
file are obtained from the values of ZONESA, NODES, and UNITS that are contained within
the PROFILE.MAS file. Section 4.2.1 discusses the process of retrieving this global network
information from the PROFILE.MAS file.
Because these parameters are network specific, the HNET.ALL file must be modified
during the initialization step for an SUE traffic assignment in order to insert these variables
that are required by the highway network building procedure in the batch file addressed in
Section 4.2.5. FSUTMS performs the same task automatically each time it is executed.
71
Figure 44 illustrates the parameters inserted into the PARAMETER section of the
HNET.ALL file.
$PARAMETERS
NUMBER OF ZONES = 24
MAXIMUM NODE = 4003
ERROR LIMIT = 50
COORDINATE DISTANCE FACTOR = 100
Figure 44 Parameters in the HNET.ALL Control File
4.2.8 Read Mode
The ReadMode module retrieves the origindestination daily vehicletrip table from
the output file, MODE.OUT, which is generated by the FSUTMS Mode Choice Models.
Mode choice models, also known as mode split or mode usage models, estimate the
proportion of total travel to be carried by public transit and private automobiles. The mode
choice model in FSUTMS is called "MODE." Mode choice models generally apportion
persontrips between origin and destination zone to several categories of automobile use (e.g.,
drive alone, carpool) and public transportation (e.g., bus, rail, etc.) [MOD97]. The person
trips obtained from mode choice models are then converted to vehicle trips via auto
occupancy factors.
There are four alternative mode choice models in FSUTMS: NonTransit (also known
as "highwayonly"), SinglePath Transit, MultiPath Transit, and MultiPath/MultiPeriod
Transit. The highwayonly model, directed by the MODE.HWY control file, differs from all
other mode choice options in that transit trips are not considered. In order to retrieve the
72
origindestination vehicletrip table for every zone in the urban network, the MODE.HWY
file must be manually edited to contain the commands "PRINT TABLE" and "SELECTED
ZONES=&SELDEST" in the $OPTION and $PARAMETERS sections, respectively.
After these initialization tasks are completed for obtaining the necessary information
from the input and output files of FSUTMS, the next task in the SUE traffic assignment
model is to enumerate competitive paths.
CHAPTER 5
COMPETITIVE PATH ENUMERATION
5.1 Introduction
The competitive path enumeration step applies the revised De La Barra approach
repeatedly to enumerate feasible paths from each origin zone to all destination zones in the
network. In order to increase the efficiency of the path enumeration process, the maximum
number of path enumeration iterations is specified as ten internally in the SUE computer
program. Thus, the maximum number of paths for a zonepair that can be enumerated in this
step is ten. This upper limit is expected to be sufficient for enumerating paths in medium and
small urban transportation networks because it is expected that seldom will there be more
than 10 different paths between an origin and a destination centroid that are used by a
significant proportion of the interchanging trips.
In addition, the enumerated shortest paths (for a specified origin zone to its
destination zones) are evaluated to assure that they are not duplicated. As soon as at least
one link on a new enumerated path is not part of the previous enumerated paths, the new path
is defined to be nonduplicated and is added to a linked path list that is created by dynamically
allocating memory. If it has been duplicated, the path must not be added to the linked list to
avoid "double counting."
74
5.2 SUE Competitive Path Enumeration Procedures
Figure 51 illustrates a flow chart for the modules executed in the competitive path
enumeration step with ten as the maximum number of path enumerations allowed between
an origin zone and each destination zone.
Figure 51 Flow Chart for the Path Enumeration Step
75
In each path enumeration iteration, the batch file listed in Figure 42 is executed in the
Shell(runbat) module to obtain the links that compose the shortest path. These links are
imposed with a penalty weighting factor, i.e., overlapping zealot (Oz) factor, in order to
increase link travel times for the next path enumeration iteration. This causes the paths
enumerated in the subsequent path enumeration iterations to be unlikely overlapped.
Also, while applying the Oz factor to the links along a path, if a link is detected that
is also contained in the shortest paths to other destination zones, e.g., link (100, 111) in
Figure 52 is contained in the path from zone 1 to zone 2 as well as from zone 1 to zone 3,
the link is imposed with a penalty only once.
Figure 52 Link Contained within Multiple Paths
The SUE computer program creates multiple output files, PATHxy.OUT, at the end
of the path enumeration step and releases memory for the linked path list of the current origin
zone before processing the next origin zone in the sequence. These output files store a
complete list of all enumerated shortest paths from a specific origin zone y to all other
destination zones at the current SUE model iteration x. The following sections describe the
submodules that are processed in the path enumeration step.
5.2.1 Change HPATH Controls
The ChangeHpathAll module prepared in this study is designed mainly to change the
"SELECTED ZONE" value in the $PARAMETER section of HPATH.ALL to obtain the
shortest paths from the specific origin zone to all other destination zones. In addition, the
input file name specified as the "USER ID" variable in the $FILE section is changed to
"$HNET.ayy$" as well. This file contains the highway network in a binary format and serves
as the input file into the "RPRPAT" program.
5.2.2 Line
In this study, the Line module is designed to organize the shortest path listed in the
output file of program RPTPAT, HPATH.OUT, into a single line for each destination zone.
Figure 53 illustrates a partial example from the output file (PATH.PRN) of the Line module.
Figure 53 depicts a path that traverses nodes 1462 and 1461, from origin zone 1 to
destination zone 2, with a total of 2.5 minutes of path travel time.
The complete path list cannot be retrieved directly for the zone pair from 1 to 4,
because the output associated with zone pair 1 to 4 is incomplete. This is because, in general,
the program tries to save space when it prints the output file. In the case of zone pair 1 to
4, when node 1461 is reached, the program recognizes that the remainder of the path list is
a duplicate of a path that has already been printed. Therefore, instead of just repeatprinting
the duplicated portion of the path, it simply prints the zone number, in this case "2". Thus,
if the program were not concerned with saving space and instead attempted to print out the
entire list of nodes every time, they would be listed 1449, 1457, 1461, and then 1462, for
zone pair 1 to 4.
1 HOME NODE
2 2.50 1461 2.00 1462 1.00 1
3 2.40 1463 2.00 1464 1.60 1466 1.20 1
4 3.00 1449 2.80 1457 2.40 1461 2.00 2
5 4.50 1446 4.20 1447 4.00 1448 3.80 1449 2.80 4
Figure 53 Example Output File of PATH.PRN from Line Module
5.2.3 Path List
The PathList module is designed in this study to obtain the complete node list for the
shortest path enumerated at the current path iteration (< 10). The module first retrieves the
path information from the output file of the Line module and then creates a temporary path
list. As shown in Figure 53, the paths (in terms of nodes traversed) might not end with the
origin zone shown as HOME NODE in the output. In order to obtain the complete path list,
the technique of linked lists is applied again to retrieve path information in the current path
iteration in the Link Node module.
The Link_Node subroutine is designed to dynamically allocate memory for the linked
list and also to simultaneously search for the list in order to get the complete node list for each
shortest path that is retrieved from the PATH.PRN file. In the Link_Node module, the paths
that terminate with the origin node are directly stored in the linked path list. Otherwise, a
search is performed to get the path information that was stored previously. The search is
performed by linking the incomplete path to its referral node number until it reaches the origin
zone.
Figure 54 illustrates the linked list created for the example described in Section 5.2.2.
The listO object is the top sentinel of the linked list and is referred to and accessed whenever
78
the path for a certain zone pair is requested. Note that the memory allocated dynamically for
the current path enumeration iteration is not released until the process reaches the end of the
CheckDone module which follows.
list(l, 2) 2 1461 1462 1
list(l, 4) 4 1449 1457 1461
list(l, 5) 5 1446 1447 1448 1449
Figure 54 Illustration of Complete Linked Path List
5.2.4 Check Path Duplication
In this study, the CheckDone module checks to see if the shortest paths enumerated
at the current iteration are duplicates of those in the previous path iterations. If a path
contains at least one link that is not part of any previously enumerated path, this new path is
added to the linked path list. This dynamically allocated linked path list is created at the
beginning of the PathSkim module to store all of the nonduplicated paths. Its memory is not
released until the overall process reaches the end of PathSkim module. Otherwise, memory
for the path list at the current path iteration is released at the end of this module.
5.2.5 Change LINKS File
The ChangeLinksFile module is designed in this study to increase the travel times by
the revised De La Barra approach for those links that form part of the enumerated paths at
the current path iteration. Consequently, the shortest paths in the next iteration favor the
links that are not selected in the current and previous iterations.
The program logic of this module is designed to increase link travel time only once,
even though the link may be part of multiple shortest paths between a specific origin zone and
several destination zones. In addition, the NETPRO program of FSUTMS restricts link travel
time to be less than 40.95 minutes. Thus, no additional path enumeration iterations are
performed whenever a weighted link travel time is greater than 40.9 minutes.
5.3 Competitive Path Enumeration Results for the Gainesville Network
This section describes the path enumeration results obtained from the 1985 Gainesville
network. The effects of overestimated travel times from the BPR formula, and the effects of
roadway system detail on path enumeration results are addressed, followed by a discussion
of the tendency of the SUE model to converge.
5.3.1 Overestimated Travel Times from the BPR Formula
Table 51 shows a summary for the total number of distinct competitive paths, along
with the average number of paths per origin zone and per zonepair, that were enumerated
in each iteration of SUE traffic assignment. The number of paths per zone was derived by
dividing the total number of enumerated paths for the network by 260, i.e., the total number
of centroids in the 1985 Gainesville network. The number of paths per zone was then divided
by 259 to obtain the number of paths per zonepair.
80
Table 51 Number of Enumerated Paths in Each SUE Iteration
The results in Table 51 indicate that the number of paths enumerated in each
subsequent iteration was close to that obtained in the SUE1 iteration. Also, even though the
maximum number of allowable path enumerations was ten, the average number of paths per
zonepair was less than 6 in each model iteration. The number of paths per zonepair is
limited by a maximum value because additional enumerations could cause unreasonably long
computer execution times.
In addition, the results in Table 51 show that travel times may be overestimated by
the BPR formula during the initialization step on links with high v/c ratios, where variables
v and c represent the number of assigned trips from a UE traffic assignment and the link
capacities, respectively. The overestimation effect prevents the path enumeration algorithm
from being able to locate additional competitive paths and hence results in the lowest total
number of enumerated competitive paths in the SUE1 iteration.
Table 52 shows the numbers of zones with lower, equal and higher numbers of
enumerated paths, respectively, during the specified iteration compared with those obtained
during the previous iteration. The results in Table 52 also illustrate the effects of
overestimated BPR travel times on the path enumeration results. The paths enumerated
Iteration SUE1 SUE2 SUE3 SUE4 SUE5
Total Number of Paths 366,391 379,557 371,362 370,352 377,770
Number of Paths per Zone 1,409.2 1,459.8 1,428.3 1424.4 1453.0
Number of Paths per ZonePair 5.44 5.64 5.51 5.50 5.61
81
during the SUE1 iteration tend to include the links with lower v/c ratios calculated during
the initialization step. The paths enumerated during the SUE2 iteration then tend to include
the links with higher v/c ratios during the initialization step. This causes the zones that are
enumerated with higher numbers of paths during iteration1 to produce lower numbers of
paths during iteration2 and vice versa.
Table 52 Number of Zones with Lower, Equal, and Higher Numbers of Paths Compared
to the Previous Iteration
Iteration SUE2 SUE3 SUE4 SUE5
Number Low Equal High Low Equal High Low Equal High Low Equal High
of Zones
85 1 174 153 1 106 138 1 121 109 1 150
The possible overestimation effects from the BPR formula on the path enumeration
results are addressed as follows.
First, a relatively small Oz factor cannot increase travel times on the current
enumerated paths enough to allow the enumeration algorithm to find other paths with larger
travel times and thus prevents the modified De La Barra method from being able to function
well.
Second, this overestimation caused the upper limit for a link's travel time to be
reached too often during the path enumeration process of the SUE1 iteration. Due to the
requirement of TRANPLAN that limits a link's travel time to be less than 40.95 minutes, the
SUE traffic assignment model restricts a weighted link travel time to less than or equal to
82
40.95 minutes. The path enumeration iteration is terminated as soon as the travel times on
one or more links along a path exceeds that limit.
The purpose of implementing the BPR formula, to estimate the initial link travel times
for the SUE using UE assigned trips, was to allow the links that were allocated with low UE
trips to be included in the enumerated paths during the SUE1 iteration. Thus, the path
enumeration process in the SUE1 iteration was to favor the inclusion of minor street links
in the enumerated paths, which should allow the maximum number of alternative paths to be
enumerated. In other words, if UE traffic assignment does allocate vehicle trips reasonably
on the links in the network, the paths that are enumerated during the first iteration will likely
be those with lower v/c ratios since the travel times for the links with higher v/c ratios are
likely to be overestimated. This suggests that fewer road users may truly traverse some of
the paths enumerated during the first iteration. After these paths are simulated by CORSIM,
and trips are allocated to each path, including minor street links, more realistic path travel
times should result. Since the BPR formula was only implemented to estimate initial link
travel times, variable BPR EXP in Equation (4.2) may be reduced to 2 or lower, from its
default value of 4, in order to mitigate the overestimation effect in future applications.
5.3.2 Roadway System Detail
Figure 55 illustrates the numbers of paths that were enumerated in the SUE1 model
iteration for every origin centroid in the 1985 Gainesville network. The figure was prepared
by sorting the distance between each zone's centroid and the central core of the Gainesville
downtown area in ascending order. This presumed center is located at the crossroad of
University Avenue and Main Street, i.e, node 1457 in the 1985 Gainesville network. The
83
distance between each centroid to node 1457 is obtained from Equation (4.1) by substituting
coordinates for nodes 1 and 2 with the ones for each centroid and node 1457, respectively.
8
6 
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260
Zones Ordered in Ascending Sequence from Gainesville Downtown
SUE1 Linear (SUE1)
Figure 55 Average Number of Paths Per ZonePair Enumerated in the SUE1 Iteration
By plotting a linear trend line for the number of enumerated paths along with the
position from the Gainesville downtown area in Figure 55, it shows that zones further from
the Downtown area tend to have lower numbers of enumerated paths. Zones located mostly
in the downtown area of Gainesville tend to have larger numbers of enumerated paths.
The linear trend lines for the path enumeration results from the other four SUE model
iterations, such as the ones shown in Figures 56 and 57 for the SUE4 and SUE5 iterations,
84
reveal a similar tendency of fewer and more enumerated paths for zones that are located
further from and close to the Gainesville downtown area, respectively. This is because the
periphery area for zones far away from Downtown were coded with less roadway system
detail and were likely connected directly to a primary arterial. With an Oz factor of 1.25, the
penalty magnitude may not be significant enough for the path enumeration model to search
for additional competitive paths. If a larger Oz factor were applied, more paths for the zones
that are far away from Downtown might have been enumerated. However, these paths may
not be reasonable since they would take much longer for road users traversing between
centroids.
8
0
6o
4 
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260
Zones Ordered in Ascending Sequence from Gainesville Downtown
SUE4 Linear (SUE4)
Figure 56 Average Number of Paths Per ZonePair Enumerated in the SUE4 Iteration
8
6 
o *
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260
Zones Ordered in Ascending Sequence from Gainesville Downtown
SUE5 Linear (SUE5)
Figure 57 Average Number of Paths Per ZonePair Enumerated in the SUE5 Iteration
Table 53 illustrates the lowest and highest numbers of competitive paths per zone pair
that were enumerated in each SUE iteration. Centroid number in Table 53 refers to the
number specified for a zone centroid in the Gainesville network after the dummy zones were
taken out. The number of paths per zonepair was obtained by dividing the number of paths
enumerated for an origin zone by 259. As with Figure 55, the results in Table 53 show that
the origin zones with lowest numbers of enumerated paths per zone pair in each SUE iteration
were located relatively far away from the Gainesville downtown area.
For example, Centroid 220 is located in the area that is surrounded by N.W. 23rd
Avenue, Newberry Road, NW 98th street and Interstate Road 75; Centroid 244 is located
86
near the crossroad of N.W. 24th Blvd and N.W. 45th Avenue; Centroid 182 is located near
the crossroad of N.W. 34th street and W. University Avenue; and Centroid 260 is an external
station for traffic towards the High Springs area.
Table 53 Origin Zones Enumerated with Lowest and Highest Numbers of Paths Per Zone
Pair in Each SUE Iteration
Iteration SUE1 SUE2 SUE3 SUE4 SUE5
Magnitude Low High Low High Low High Low High Low High
Paths/Zone Pair 4.01 7.10 3.99 7.42 4.00 7.29 4.00 6.94 4.17 7.31
Centroid Number 220 98 244 162 182 98 182 98 260 156
Zone centroids with the highest numbers of enumerated paths per zone pair were
generally located relatively close to the downtown area. For example, Centroid 98 is located
near the crossroad of Waldo Road and Hawthorne Road; Centroid 162 is located close to
Newnans Lake and near the crossroad of County Road 329B and State Road 20; and
Centroid 156 is located near the crossroad of E. University Avenue and State Road 26.
The results in Table 53 also show that the minimum and maximum average number
of paths from a zone centroid to the other 259 zones were about 4 and 7, respectively, for the
Gainesville network. The maximum number of path enumerations could be increased in order
to enumerate more paths if the resulting average values are not sufficient for a given
transportation network. However, increasing the limit above 10 may not yield much increase
in the average results because there may not be more competing paths that exist.
87
Links with travel times greater than 40.9 minutes are unlikely to exist in a
transportation network since links are usually relatively short. However, the path
enumeration process sets the upper limit of weighted link travel time to be 40.9 minutes. This
is because a weighted link travel time can be significantly large after imposing an overlapping
penalty. For example, by imposing a 1.25 overlapping penalty ten times on a link with a free
flow time of 5 minutes, the weighted travel time for the link becomes 46.6 minutes which is
much higher than its free flow time.
For vehicles from zones that are specified with less roadway system detail, it is
expected that these vehicles will utilize the major roadways to reach their destinations. Thus,
the overlapping penalty will be imposed most on some of the arterial links. For an extreme
case, a zone centroid may have a connection to only one arterial link. Consequently, this
arterial link is a part of every path enumerated. If a reduced limit below 40.95 is applied,
there will be fewer paths enumerated for this specific zone. In other words, with the
maximum allowable link travel time as the upper limit for path enumeration, more paths can
be enumerated for the centroids that are coded with less roadway system detail. In addition,
since the BPR formula overestimated the initial travel time for the SUE1 iteration, the upper
limit for a weighted link travel time should absorb the overestimated travel time by the BPR
formula in order to allow more paths to be enumerated in the SUE1 iteration.
5.3.3 Tendency to Convergence
Table 54 shows the sum of the absolute differences in numbers of enumerated paths
for all zones between the specified and previous iteration. Table 54 shows that the number
of enumerated paths for every zone between successive iterations becomes less significant in
88
later iterations. This is an indication that the SUE traffic assignment is converging for the
1985 Gainesville network.
Table 54 Sum of Absolute Differences in Number of Enumerated Paths for All Zones in
the Current and Previous Iterations
Iteration SUE2 SUE3 SUE4 SUE5
Sum of Different Paths 28,724 23,673 21,968 21,256
After the competitive paths are determined, the next task is to simulate the variations
in travel times associated with each path that are expected to be perceived by travelers.
