• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Background and statistical...
 Network description
 SUE algorithm and model initia...
 Competitive path enumeration
 Travel time variation
 Route choice model
 Convergence test
 Model validation
 Summary, conclusions and recom...
 Gamma function
 References
 Biographical sketch














Title: Asymmetrically-distributed variations in traveler-perceived travel times in stochastic user-equilibrium traffic assignment
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Title: Asymmetrically-distributed variations in traveler-perceived travel times in stochastic user-equilibrium traffic assignment
Physical Description: Book
Language: English
Creator: Li, Min-Tang, 1966-
Publisher: State University System of Florida
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Publication Date: 1999
Copyright Date: 1999
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Subject: Civil Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Civil Engineering -- UF   ( lcsh )
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Summary: ABSTRACT: Most of the current approaches for applying a stochastic user equilibrium (SUE) traffic assignment model have been based on iterative applications of the Dial multi-path assignment algorithm. The errors of perceived travel times by motorists in the route choice model are assumed to be symmetrically Gumbel-distributed. 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 asymmetrically-distributed for competitive paths that are enumerated. Paths are generated by a modified shortest-path 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 path-based 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 user-equilibrium assignment. Although it provided both a better conceptual foundation and more accurate numerical results, the model is currently not yet cost-effective for real-world applications due to exorbitant computer running times.
Thesis: Thesis (Ph. D.)--University of Florida, 1999.
Bibliography: Includes bibliographical references (p. 138-140).
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Statement of Responsibility: by Min-Tang Li.
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General Note: Document formatted into pages; contains xii, 141 p.; also contains graphics.
General Note: Vita.
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
        Page viii
    List of Figures
        Page ix
        Page x
    Abstract
        Page xi
        Page xii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    Background and statistical tools
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
    Network description
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
    SUE algorithm and model initialization
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Competitive path enumeration
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
    Travel time variation
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
    Route choice model
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
    Convergence test
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
    Model validation
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
    Summary, conclusions and recommendations
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
    Gamma function
        Page 136
        Page 137
    References
        Page 138
        Page 139
        Page 140
    Biographical sketch
        Page 141
Full Text











ASYMMETRICALLY-DISTRIBUTED VARIATIONS IN
TRAVELER-PERCEIVED TRAVEL TIMES IN
STOCHASTIC USER-EQUILIBRIUM TRAFFIC ASSIGNMENT













By

MIN-TANG 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. Cheng-Tin Gan for their continued encouragement

and strong support during my doctoral research study.

Extreme gratitude is expressed to members of my supervisory committee-Dr. Gary

Long (chairman), Professor Kenneth G. Courage, Dr. Bon A. Dewitt, Dr. Sherman X. Bai,

and Dr. Mark C. K. Yang-for 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, Lee-Fang, 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 multi-asymmetrical Gumbel
distribution .................
2.1.2.4 Method of probability-weighted moments
U ser Equilibrium .........................
Introduction ............................
Stochastic Network Loading Procedures .......
SUE Solution Algorithm ................. .
Define Route Choice Set-Competitive 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 Signed-Rank 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

2-1 Variance Equality Test Results for Simulated Travel Times from CORSIM ... 33

2-2 Kolmogorov-Smirnov Test Results for 3,000 vph Mainline Entry Volume .... 34

2-3 Kolmogorov-Smirnov Test Results for 5,000 vph Mainline Entry Volume .... 36

2-4 Perceived Travel Times for Links under Different Traffic Flows ............ 38

2-5 Test Results for Homogeneity of Variances in Errors of Perceived
T ravel T im es ............................................ 40

2-6 Test Results for Link (2204, 2012) Under Daily Traffic .............. 41

2-7 Test Results for Link (2204, 2012) Under Peak-15 Minute Traffic .......... 42

2-8 Test Results for Link (1453, 1556) Under Peak-15 Minute Traffic .......... 43

2-9 Test Results for Link (1453, 1556) Under Daily Traffic .............. 44

3-1 Socioeconomic Data Summary from ZDATA1.85V ................. 53

3-2 Socioeconomic Data Summary from ZDATA2.85V ................. 54

3-3 Special Trip Generator Data Summary from ZDATA3.85V ............... 54

5-1 Number of Enumerated Paths in Each SUE Iteration ................ . 80

5-2 Number of Zones with Lower, Equal, and Higher Numbers of Paths
Compared to the Previous Iteration ........................... 81

5-3 Origin Zones Enumerated with Lowest and Highest Numbers of Paths
Per Zone Pair in Each SUE Iteration ..................... 86









5-4 Sum of Absolute Differences in Number of Enumerated Paths for All
Zones in the Current and Previous Iteration ....................

6-1 Random Number Seeds for CORSIM Simulation ......................

6-2 Median, Mean, and Variance Travel Times for Link (2204, 2012) .........

6-3 Median, Mean, and Variance Travel Times for Link (1453, 1556) .........

7-1 Parameters for Example in Route Choice Implementation Step ............

8-1 Wilcoxon Signed-Rank Test Variables for Assigned Trips from
Successive SUE Iterations in the Gainesville Network ............

8-2 Wilcoxon Signed-Rank Test Variables for Assigned Trips from
Successive SUE Iterations in the FTOWN Network ..............

9-1 Mean and Median for Paired Observations from UE and Each
SU E Iterations ..........................................

9-2 Sign Test Variables during Each SUE Iteration ..................... .

9-3 Computer Running Times from Executing an SUE Assignment on
the Gainesville Network (in minutes) .........................

9-4 Computer Running Times from Executing an SUE Assignment on
the FTOWN Network (in minutes) ...........................


118


124

125


126


126


















LIST OF FIGURES


Figure


2-1


3-1


3-2


4-1


4-2


4-3


4-4


5-1


5-2


5-3


5-4


5-5


5-6


5-7


6-1


6-2


6-3


. . . . 32


. . . . 52


. . . . 56


. . . . 5 8


....... 62


....... 67


. . . . 7 1


....... 74


. . . . 7 5


....... 77


. . . . 7 8


Average Number of Paths Per Zone-Pair Enumerated in the SUE-1 Iteration


Average Number of Paths Per Zone-Pair Enumerated in the SUE-4 Iteration


Average Number of Paths Per Zone-Pair Enumerated in the SUE-5 Iteration


Flow Chart for the Travel Time Variation Step ..................... .


O overlapping P ath .............................................


Link-Node Diagram for Path Example .............................


. 83


. 84


. 85


. 90


. 93


. 95


Link-Node 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 ...................









6-4 Numbers of CORSIM Simulations in the SUE-1 Iteration ............... 103

7-1 Flow Chart for the Route Choice Model Implementation Step ............ 107

7-2 Numbers of Assigned Trips from UE and the SUE-1 Iteration ............ 114

8-1 Pattern for Difference in Assigned Trips from SUE-4 and SUE-5 Iterations . 120

9-1 % 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

ASYMMETRICALLY-DISTRIBUTED VARIATIONS IN
TRAVELER-PERCEIVED TRAVEL TIMES IN
STOCHASTIC USER-EQUILIBRIUM TRAFFIC ASSIGNMENT

By

MIN-TANG 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 multi-path

assignment algorithm. The errors of perceived travel times by motorists in the route choice

model are assumed to be symmetrically Gumbel-distributed. 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 asymmetrically-distributed for competitive

paths that are enumerated. Paths are generated by a modified shortest-path 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 path-based 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

user-equilibrium assignment. Although it provided both a better conceptual foundation and

more accurate numerical results, the model is currently not yet cost-effective for real-world

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 link-based 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 origin-destination (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 well-known 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

system-wide travel disutility is perceived by travelers. This assumption allows partitioning

origin-destination 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 origin-destination 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

route-choice 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 user-perceived 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 well-known 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 commercially-available 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 assignment-simulation framework.

However, dynamic traffic assignment is designed to analyze the traffic state over a

specific time interval such as a peak hour in a short-term time horizon, whereas traditional

traffic assignment is designed to estimate or forecast traffic flows in the more distant,

long-term 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 lane-changing 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 Gumbel-distributed 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 "non-standard" 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 Gumbel-distributed, 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 Gumbel-distributed 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 origin-destination zone-pair. 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 choice-makers 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 Gumbel-distributed error is

replaced by asymmetric Gumbel-distributed error. The concept is to allow the travel time on

a path to be treated as a random variable which is Gumbel-distributed 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 origin-destination

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 path-based 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 non-stochastic 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 Gumbel-distributed 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 non-stochastic 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 exp-e()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(-A-e t)dt = e -nlf e texp(-A-e t)dt


-e nlf e texp(-A-e -)d(-t)


e T0
r1 0
A f exp(-A-e )d(-Ae t)

e e

A e-i -V + V 2
e + e


e T+ e2 12


-e nlf exp(-A-e t)d(e )
0
e I d(exp- Aet)
A
0


As indicated earlier, the difference of two independent Gumbel-distributed 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 Gumbel-distributed 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 Gumbel-distributed is addressed as follows.


(2.21)








19

2.1.2.2 Derivation of binary asymmetrical Gumbel distribution

If C1 and c2 are independent Gumbel-distributed but not identical, the following

equations can be derived. That is:

F(c 1, 82) = F(c,) F2(C2)

SF (,) = expl-e -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(t-1 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(t-ndexp[-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 multi-asymmetrical Gumbel

distribution when there are more than two variables that are independently but not identically

Gumbel-distributed.

2.1.2.3 Derivation of multi-asymmetrical 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 Gumbel-distributed.

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+V-V,, ..., t+V-VJ) dt

(2.30)
f i,e -(t exp[-e -(t-n exp[- i(t +V-VI- 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 probability-weighted moments

Landwehr, Matalas, and Wallis [LAN79] proposed a simple method for estimation of

the parameters ir and [ based on probability-weighted 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 first-order 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 first-order 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 Gumbel-distributed, 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 well-known 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 trip-makers 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 multi-path assignment algorithm, the errors of perceived travel times by motorists in the

route choice model are assumed to be symmetrically Gumbel-distributed. 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 Gumbel-distributed 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 Gumbel-distributed 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 Set-Competitive Paths

A route choice set which contains the competitive paths for road users to select for

traversing between a given zone-pair 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 origin-destination (O-D) 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 paths-the set of resulting paths must be those of least

generalized cost, assuming that users are economically rational to a certain degree-by

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 re-emerge 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 zone-pair 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 2-1, 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 2-1 Link-Node 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 2-1. 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 2-1 illustrates the related variables and their corresponding

values that were calculated in this variance equality test. The results in Table 2-1 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 2-1 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

Gumbel-distributed variates. Goodness of fit for the errors in simulated travel time was tested

using the Kolmogorov-Smirnov (K-S) procedure which is generally preferable to chi-square

when dealing with continuous distributions [ZAR84]. Table 2-2 illustrates the variables and

their corresponding values in the K-S 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 2-2 Kolmogorov-Smirnov 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 -0.8--0.7 2 2 0.01 0.00937 0.00064 0.00937
3 -0.7--0.6 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 -0.4--0.3 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 -0.2--0.1 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 0-0.1 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 0.3-0.4 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 0.7-0.8 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 2-2, 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 Kolmogorov-Smirnov test procedure was applied again to test the errors of

simulated travel times using a mainline entry volume of 5,000 vph. Table 2-3 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 Gumbel-distributed.

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

Gumbel-distributed.












Table 2-3 Kolmogorov-Smirnov 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 -14--13 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 -9--8 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 -6--5 11 50 0.25 0.24114 0.00887 0.04614
11 -5 ~-4 8 58 0.29 0.29876 0.00876 0.04876
12 -4--3 15 73 0.365 0.35840 0.00660 0.06840
13 -3--2 13 86 0.43 0.41831 0.00117 0.05331
14 -2--1 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 2-3 8 129 0.645 0.68029 0.03529 0.07529
19 3 4 14 143 0.715 0.72094 0.00564 0.07594
20 4-5 8 151 0.755 0.75737 0.00237 0.04237
21 5 6 5 156 0.78 0.78975 0.00975 0.03475
22 6-7 5 161 0.805 0.81834 0.01334 0.03834
23 7 8 6 167 0.835 0.84343 0.00843 0.03843
24 8-9 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 2-3 (Cont.) Kolmogorov-Smirnov 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 19-20 1 197 0.985 0.97629 0.00871 0.00371
36 20-21 1 198 0.99 0.97982 0.01018 0.00518
37 21 -22 1 199 0.995 0.98284 0.01217 0.00717
38 22-23 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 2-4, 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 2-4 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 chi-square distribution with k 1 degrees

of freedom, but a more accurate chi-square approximation is obtained by computing a

correction factor [ZAR84],









40


C= + 1 1--1
3 x (k 1) v k (2.50)




with the corrected test statistic being B, = B/C.

Table 2-5 illustrates the related variables and their corresponding values in Bartlett's

test for homogeneity of variances.

Table 2-5 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 2-4 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

Gumbel-distributed. The Kolmogorov-Smirnov 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 2-6 through 2-9.





Table 2-6 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 -1.5--1 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 0-0.5 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 1.5-2 1 21 0.80769 0.73289 0.07480 0.03634
9 2-2.5 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 2-7 Test Results for Link (2204, 2012) Under Peak-15 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 -4--3 1 1 0.03846 0.11327 0.07481 0.11327
3 -3--2 2 3 0.11539 0.18676 0.07138 0.14830
4 -2--1 7 10 0.38462 0.27454 0.11008 0.15916

5 -1-0 3 13 0.50000 0.36941 0.13059 0.01521
6 0 1 3 16 0.61539 0.46431 0.15108 0.03569
7 1-2 2 18 0.69231 0.55375 0.13856 0.06164
8 2-3 0 18 0.69231 0.63423 0.05808 0.05808
9 3 4 0 18 0.69231 0.70413 0.01182 0.01182
10 4-5 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 7-8 0 22 0.84615 0.88376 0.03761 0.03761
14 8-9 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 2-8 Test Results for Link (1453, 1556) Under Peak-15 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 0-0.25 1 13 0.52 0.38289 0.13711 0.09711
8 0.25-0.5 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 2-2.25 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 2-9 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 0-0.1 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 0.2-0.3 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 0.4-0.5 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 15-minute traffic were Gumbel-distributed.

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 Gumbel-distributed 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 Gumbel-distributed under moderate traffic, errors of perceived travel times

are likely to be Gumbel-distributed 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

distribution-free methods.

The sign test and the Wilcoxon Signed-Rank test procedures are two of the most

commonly used distribution-free 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 Signed-Rank 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 t-test 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 t-test procedure to be undesirable [RA098].

Equation (2.51) illustrates the statistical model associated with the sign, signed-rank

and t-test for inference about a population median [RA098].

Y = + e. (2.51)


where Y, = the i-th observation in a sample;
p = population median; and
C, = error in the i-th 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 t-test 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 Signed-Rank 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

symmetrically-distributed 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 symmetrically-distributed.











2.3.1 Wilcoxon Signed-Rank 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 Signed-Rank 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 Signed-Rank 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 root-mean-square 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 root-mean-square 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 path-based 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 path-based 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 two-way and 17 one-way 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 3-1

illustrates the 1985 Gainesville highway network.











imm G %%5 ighwa Netork7
1 -177,


Figure 3-1 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 path-based 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 3-1

summarizes the socioeconomic data for trip production-oriented 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 3-1 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 3-2 summarizes trip attraction socioeconomic data listed by traffic analysis zone

from the zone-based 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 3-2 Socioeconomic Data Summary from ZDATA2.85V


Category Industrial Commercial Service Total

Number of Employees 4,462 12,195 39,788 56,445


Table 3-3 summarizes special generator data listed by traffic analysis zone in

ZDATA3.85V, where HBW, HBS, HBSR, HBO and NHB stand for home-based work,

home-based shopping, home-based social/recreation, home-based other and nonhome-based

trips. The total number of external-internal (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 3-3 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 "1-260" 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 3-2 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 3-2 FTOWN Highway Network


1-istFTOWN 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 4-1 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 4-1, the steps in the SUE algorithm are:

Step 0-Initialization 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 1-Competitive 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 2-Travel 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 4-1 Flowchart for the SUE Algorithm




Step 3-Route 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 origin-destination pair based on the simulated path travel

times from Step 2. This yields auxiliary link flow volumes { yj }.

Step 4-Test 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 non-stochastic 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 Gumbel-distributed

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 non-stochastic 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 4-2 illustrates the flow chart for the modules at the

initialization step.









































Figure 4-2 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: NAME-study area name,

ZONESA-total number of zones including internal and external, NODES-highest node

number permitted, and UNITS-coordinate 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, non-transit 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: TR1-Single-path transit,

TR2-Multi-path/single-period transit, and TR3-Multi-period/multi-path transit [FSU97].

Note that all of the control files with extensions ".HWY", except MODE.HWY", and ".TR?"

remain unchanged during the process of path-based 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 look-up 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 level-of-service (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 path-based 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 4-3. 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 4-3 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 sub-steps 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 sub-steps.

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 sub-step. 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 field-measured 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 path-building 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, free-flow 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 path-based 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 4-4 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 4-4 Parameters in the HNET.ALL Control File




4.2.8 Read Mode

The ReadMode module retrieves the origin-destination daily vehicle-trip 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

person-trips 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: Non-Transit (also known

as "highway-only"), Single-Path Transit, Multi-Path Transit, and Multi-Path/Multi-Period

Transit. The highway-only 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

origin-destination vehicle-trip 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 zone-pair 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 non-duplicated 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 5-1 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 5-1 Flow Chart for the Path Enumeration Step









75

In each path enumeration iteration, the batch file listed in Figure 4-2 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 5-2 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 5-2 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

sub-modules 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 5-3 illustrates a partial example from the output file (PATH.PRN) of the Line module.

Figure 5-3 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 repeat-printing

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 5-3 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 5-3, 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 5-4 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 5-4 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 non-duplicated 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 5-1 shows a summary for the total number of distinct competitive paths, along

with the average number of paths per origin zone and per zone-pair, 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 zone-pair.









80

Table 5-1 Number of Enumerated Paths in Each SUE Iteration


The results in Table 5-1 indicate that the number of paths enumerated in each

subsequent iteration was close to that obtained in the SUE-1 iteration. Also, even though the

maximum number of allowable path enumerations was ten, the average number of paths per

zone-pair was less than 6 in each model iteration. The number of paths per zone-pair is

limited by a maximum value because additional enumerations could cause unreasonably long

computer execution times.

In addition, the results in Table 5-1 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 SUE-1 iteration.

Table 5-2 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 5-2 also illustrate the effects of

overestimated BPR travel times on the path enumeration results. The paths enumerated


Iteration SUE-1 SUE-2 SUE-3 SUE-4 SUE-5

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 Zone-Pair 5.44 5.64 5.51 5.50 5.61









81

during the SUE-1 iteration tend to include the links with lower v/c ratios calculated during

the initialization step. The paths enumerated during the SUE-2 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 iteration-1 to produce lower numbers of

paths during iteration-2 and vice versa.



Table 5-2 Number of Zones with Lower, Equal, and Higher Numbers of Paths Compared
to the Previous Iteration


Iteration SUE-2 SUE-3 SUE-4 SUE-5

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 SUE-1 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 SUE-1 iteration. Thus, the path

enumeration process in the SUE-1 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 5-5 illustrates the numbers of paths that were enumerated in the SUE-1 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
SUE-1 -Linear (SUE-1)

Figure 5-5 Average Number of Paths Per Zone-Pair Enumerated in the SUE-1 Iteration





By plotting a linear trend line for the number of enumerated paths along with the

position from the Gainesville downtown area in Figure 5-5, 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 5-6 and 5-7 for the SUE-4 and SUE-5 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




6-o







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
SUE-4 -Linear (SUE-4)

Figure 5-6 Average Number of Paths Per Zone-Pair Enumerated in the SUE-4 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
SUE-5 -Linear (SUE-5)

Figure 5-7 Average Number of Paths Per Zone-Pair Enumerated in the SUE-5 Iteration





Table 5-3 illustrates the lowest and highest numbers of competitive paths per zone pair

that were enumerated in each SUE iteration. Centroid number in Table 5-3 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 zone-pair was obtained by dividing the number of paths

enumerated for an origin zone by 259. As with Figure 5-5, the results in Table 5-3 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 5-3 Origin Zones Enumerated with Lowest and Highest Numbers of Paths Per Zone
Pair in Each SUE Iteration

Iteration SUE-1 SUE-2 SUE-3 SUE-4 SUE-5

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 5-3 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 SUE-1 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 SUE-1 iteration.

5.3.3 Tendency to Convergence

Table 5-4 shows the sum of the absolute differences in numbers of enumerated paths

for all zones between the specified and previous iteration. Table 5-4 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 5-4 Sum of Absolute Differences in Number of Enumerated Paths for All Zones in
the Current and Previous Iterations


Iteration SUE-2 SUE-3 SUE-4 SUE-5

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.




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