Foundations of Dynamic Traffic Assignment: The Past, the Present and the Future

Abstract

Dynamic Traffic Assignment (DTA) has evolved substantially since the pioneering work of Merchant and Nemhauser. Numerous formulations and solutions approaches have been introduced ranging from mathematical programming, to variational inequality, optimal control, and simulation-based. The aim of this special issue is to document the main existing DTA approaches for future reference. This opening paper will summarize the current understanding of DTA, review the existing literature, make the connection to the approaches presented in this special issue, and attempt to hypothesize about the future.

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Networks and Spatial Economics, 1: 2001 233±265
#2001 Kluwer Academic Publishers, Manufactured in the Netherlands.
Foundations of Dynamic Traf®c Assignment:
The Past, the Present and the Future
SRINIVAS PEETA
School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA
ATHANASIOS K. ZILIASKOPOULOS
Department of Civil Engineering, Northwestern University, Evanston, IL 60208, USA
Abstract
Dynamic Traf®c Assignment (DTA) has evolved substantially since the pioneering work of Merchant and
Nemhauser. Numerous formulations and solutions approaches have been introduced ranging from mathematical
programming, to variational inequality, optimal control, and simulation-based. The aim of this special issue is to
document the main existing DTA approaches for future reference. This opening paper will summarize the current
understanding of DTA, review the existing literature, make the connection to the approaches presented in this
special issue, and attempt to hypothesize about the future.
Keywords: Dynamic traf®c assignment, real-time deployment, planning
1. Introduction
Dynamic Traf®c Assignment (DTA), though still in a state of ¯ux, has evolved
substantially since the seminal work of Merchant and Nemhauser (1978a, 1978b). There
is currently heightened interest in DTA, particularly in the development of approaches that
can be deployed for large-scale real-time and planning applications. In addition, researchers
have become increasingly aware that the theory of DTA is still relatively undeveloped,
which necessitates new approaches that account for challenges from the application
domains as well as for the fundamental questions related to tractability and realism.
Agencies and practitioners are also increasingly realizing the potential of DTA to address
longstanding problems with the unrealistic assumptions of existing static planning
methods, as well as the potential of DTA to both evaluate Intelligent Transportation
Systems (ITS) technologies, as well as be the main operational engine for deployment.
DTA refers to a broad spectrum of problems, each corresponding to different sets of
decision variables and underlying behavioral and system assumptions, and possessing
varying data requirements and capabilities in terms of representing the traf®c system or
control actions. One common feature of these models is that they depart from the standard
static assignment assumptions to deal with time-varying ¯ows. Another feature shared by
these models is that none presently provides a universal solution for general networks.
Perhaps the one aspect that fosters unanimity among researchers is that the general DTA

problem is inherently characterized by ill-behaved system properties that are imposed by
the need to adequately represent traf®c realism and human behavior. This is further
exacerbated by the time-dependency and randomness in system inputs. A fundamental
consequence of this reality is that a theoretical guarantee of properties such as existence,
uniqueness, and stability can be tenable only through compromises in depicting traf®c
theoretic phenomena and potentially restrictive assumptions on driver behavior. Viewed
from the complementary perspective, an ability to adequately capture traf®c dynamics and
driver behavioral tendencies precludes the guarantee of the standard mathematical proper-
ties. This inherent complexity of DTA has spawned a clear dichotomy of approaches that
range from the the analytical to the simulation-based. A fundamental practical con-
sequence of the theoretical intractability is the focus of DTA researchers on developing
deployable solution procedures that seek close-to-optimal solutions with a clear under-
standing that claims of uniqueness and/or global optimality are neither essential nor
particularly meaningful in the real-world. This has manifested as the development of
mostly heuristic implementation procedures that seek effectiveness, robustness, and
deployment ef®ciency. Another outcome is the notion of commensurability of features
among different approaches so that trade-offs among desirable features allow different
degrees of responsiveness to different DTA problems given the broad scope of objectives
and functional needs addressed under the general umbrella of DTA.
A reassuring practical aspect of the general DTA problem is that its mathematical
intractability is not an all-encompassing barrier to the real-world utility of the associated
solution approaches. Substantial research over the past decade suggests that effective and
ef®cient solutions can be obtained for several realistic scenarios characterized by incon-
sequential mild violations and/or infrequent physical manifestations of the mathematically
intractable aspects. There are several actual situations where ill-behaved problem char-
acteristics do not arise, and even if they do, it is with minimal practical temporal and spatial
consequences vis-a
Á-vis the modeling assumptions. Hence, different approaches may address
different functional needs with different degrees of robustness, precluding the notion of
sweeping generalizations on mathematical intractability by focusing on pathological
scenarios, especially if they are rare in practice. The consensus is that a deployable DTA
approach should adequately represent traf®c realism in the context of the problem objective.
This paper is organized as follows. The next section reviews past efforts in DTA by
classifying the major approaches and putting them in perspective with the current
understanding of DTA. Section 3 discusses some future research issues and directions,
currently in their incipient stages, that are motivated by application domains or problem
characteristics. Section 4 presents some concluding comments.
2. Review of the Past
This section reviews past DTA literature by classifying the various approaches into four
broad methodological groups: mathematical programming, optimal control, variational
inequality, and simulation-based. Of these, the ®rst three groups are further labeled as
analytical approaches. As in the static case (for comprehensive coverage, see Shef® (1985)
234 PEETA AND ZILIASKOPOULOS

and Patricksson (1994)), most formulations tend to focus on the user equilibrium (UE) and
system optimal (SO) objectives, or some variants of them. While a vast body of literature
has been developed in this area over the past two decades given the wide range of problems
addressed under the DTA label, this section focuses on some of the efforts that highlight
the basic problem dimensions. Peeta (1994) provides a review of the literature on
mathematical programming and optimal control based DTA formulations, with particular
focus on real-time information provision. Ran and Boyce (1994) discuss various optimal
control models. Recent efforts that focus on variational inequality based formulations are
discussed in Ran and Boyce (1996a) and Chen (1999). Mahmassani et al. (1998a) review
various simulation-based DTA models.
2.1. Mathematical programming formulations
Mathematical programming DTA models formulate the problem in a discretized time-
setting. Merchant and Nemhauser (1978a, 1978b) represent the ®rst attempt to formulate
the DTA problem as a mathematical program. The formulation was limited to the deter-
ministic, ®xed-demand, single-destination, single-commodity, SO case. The model (here-
after referred to as the M±N model) uses a link exit function to propagate traf®c and a
static link performance function to represent the travel cost as a function of link volume. It
results in a ¯ow-based, discrete time, non-convex non-linear programming formulation.
The model is shown to provide a proper generalization of the conventional static SO
assignment problem, and the global solution is obtained by solving a piecewise linear
version of the model. Ho (1980) shows that such a global optimum can be obtained by
solving a sequence of at most N+1 linear programs, where N is the number of periods.
Carey (1986) proved that the M±N model satis®es the linear independence constraint
quali®cation since the proposed exit function is continuously differentiable, validating the
optimality analysis in Merchant±Nemhauser (1978a, 1978b). Carey (1987) reformulates
the Merchant±Nemhauser problem as a well-behaved convex nonlinear program through
the manipulation of exit functions, which offers mathematical and algorithmic advantages
over the original formulation. The basic formulation is very similar to the M±N model. For
the formulation to be convex, an exit function is used to bound the link out¯ow rather than
being the out¯ow itself as in the M±N model. Since it is convex, standard mathematical
programming software can be used to solve the problem. However, the paper indicates that
special structures such as the staircase structure in the constraints should be explored to
develop algorithms that are more ef®cient. Extensions of the basic model are introduced to
handle multiple destinations and commodities. However, the resulting formulations remain
problematic because of non-convexity issues arising from a ``®rst-in, ®rst-out'' (FIFO)
requirement. These dif®culties appear to be inherent in all mathematical programming
approaches to the time-dependent assignment problem, for both the UE and SO cases.
Multiple destinations require the models to explicitly satisfy a ®rst-in, ®rst-out requirement
that is essential from a traf®c realism viewpoint. FIFO has two dimensions, physical and
algorithmic. Since individual vehicles may at times violate FIFO in actual traf®c networks
(overtaking), it is satis®ed only in an average sense from a traf®c propagation standpoint.
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 235

The problematic FIFO violation implied here is algorithm-induced, one where some
commodity (for example, traf®c between one origin-destination (O-D) pair) physically
jumps over another to reduce system costs. This is inconsistent with traf®c realism. The
FIFO requirement is easily satis®ed in single-destination formulations and for certain
special network structures. In general networks, this requirement would introduce addi-
tional constraints that yield a non-convex constraint set, destroying many of the nice
properties of the formulation, and severely increasing the computational requirements of
any eventual solution algorithm (Carey, 1992).
Another well-known phenomenon related to the SO mathematical programming models
that impinges on traf®c realism is the ``holding-back'' of vehicles on links. It arises when
link exit constraints are satis®ed as strict inequalities. In a traf®c network, it may often be
advantageous to favor certain traf®c streams or movements over others to minimize
system-wide travel delays (e.g. holding back traf®c at the minor approach of an
intersection in favor of the major approach). Unless otherwise speci®ed, the solution of
a SO assignment formulation may entail holding of traf®c on one path in favor of traf®c on
other paths for some signi®cant amount of time at points where the paths overlap or
intersect. In other words, vehicles may be arti®cially delayed on a link for a time that
exceeds what may be considered ``fair'' or ``reasonable''. Such a solution is probably not
acceptable socially nor realistic operationally. From a modeling standpoint this would
entail explicit constraints to preclude holding-back of traf®c. Carey and Subrahmanian
(2000) illustrate some aspects of the FIFO and holding-back issues.
Janson (1991a, 1991b) represents one of the earliest attempts to model the UE DTA
problem as a mathematical program. One feature of his approach is that it seeks an
equilibrium described in terms of experienced path travel times, instead of the instantaneous
travel times assumed in several prior studies. Non-linear mixed integer constraints are
proposed in the formulation to ensure temporal continuity of O-D ¯ows, though they may
be violated in the solution procedure speci®ed which is a straightforward extension of the
well-known incremental assignment heuristic for static formulations. The properties of this
procedure are not suf®ciently well-established, and it may lead to possible unrealistic traf®c
behavior. In addition, it relies on static link performance functions for traf®c modeling.
Birge and Ho (1993) extend the M±N problem to the stochastic case by relaxing the
assumption that O-D desires are known for the entire planning horizon. They develop a
multistage stochastic mathematical programming formulation that is non-linear and non-
convex as in the deterministic case. The model assumes a ®nite number of scenarios of
random variable realizations, where a scenario is de®ned as a possible combination of past
O-D desires in every period. However, the formulation assumes current assignment
decisions to be independent of future O-D desires. The resulting model determines the
O-D assignment pattern that minimizes the expected piecewise linear convex costs over
several time periods through a sequence of linear optimizations.
Ziliaskopoulos (2000) introduces a linear programming formulation for the single
destination SO DTA problem based on the cell transmission model (Daganzo, 1994) for
traf®c propagation. It circumvents the need for link performance functions as the ¯ow
propagates according to the cell transmission model, thereby being more sensitive to traf®c
realities. While not an operational model for real-world applications, it provides some
236 PEETA AND ZILIASKOPOULOS

insights on the DTA problem properties. Carey and Subrahmanian (2000) also propose a
linear single destination SO DTA formulation, again with the intent to derive insights on
the model properties. As stated earlier, they study the FIFO and holding-back issues in the
context of mathematical programming formulations.
The substantial research in mathematical programming based DTA approaches high-
lights its current limitations for developing deployable models for general networks. A
persistent issue is the need to trade-off mathematical tractability with traf®c realism. This
is primarily manifest in terms of the inadequate modeling ability and/or inconvenient
constraints arising vis-a
Á-vis the representation of traf®c dynamics. One such aspect is the
non-convex constraints needed to explicitly address the FIFO property. While there is a
whole body of optimization literature that deals with non-convex programming, the non-
convexity in the DTA context leads to a loss of analytical and computational tractability for
deployment in general networks. In addition, in general, mathematical programming DTA
formulations tend to have dif®culties related to: (i) the use of link performance and/or link
exit functions, (ii) holding-back of traf®c, (iii) ef®cient solutions for real-time deployment
in large-scale traf®c networks, and (iv) a clear understanding of solution properties for
realistic problem scenarios.
2.2. Optimal control formulations
In constrained optimal control theory DTA formulations, the O-D trip rates are assumed as
known continuous functions of time, and the link ¯ows are sought as continuous functions
of time. The constraints are analogous to those for the mathematical programming
formulations, but are de®ned in a continuous-time setting. This results in a continuous-
time optimal control formulation, rather than a discrete-time mathematical program.
Friesz et al. (1989) discuss link-based optimal control formulations for both the SO and
UE objectives for the single destination case. The models assume that adjustments from one
system state to another may occur concurrently as the network conditions change; that is, the
routing decisions are made based on current network conditions, but can be continuously
modi®ed as conditions change. The SO model is a temporal extension of the static SO model,
and proves that at the optimal solution the instantaneous ¯ow marginal costs on the used
paths for an O-D pair are identical and less than or equal to the ones on the unused paths. It
should be noted that a discrete-time version of this SO formulation would be identical to the
M±N model. They also propose a time-dependent generalization of Beckmann's equivalent
optimization problem (Beckmann et al., 1956) for static UE traf®c assignment in the form of
an optimal control problem by the equilibration of instantaneous user path costs. The models
use exit functions to propagate traf®c, and link performance functions to determine travel
costs. Key issues include the lack of meaningful link performance and exit functions, and the
dif®culties in developing ef®cient solution algorithms. Also, the formulations treat the link
in¯ow as a control variable, though the out¯ow is a function. This is problematic as the non-
linearity of the exit function makes it dif®cult to establish the generalization of Wardrop's
®rst principle (Wardrop, 1952) to the time-dependent case for a network with multiple
origins and destinations. Wie (1990) extends the UE model to include elastic time-varying
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 237

travel demand, which leads to the implicit consideration of departure time choices. Wie also
enumerates several limitations of this approach.
Ran and Shimazaki (1989a) use the optimal control approach to develop a link-based
SO model for an urban transportation network with multiple origins and destinations. They
also de®ne exit ¯ow to be a function, precluding the generalization of optimality
conditions. They use linear exit functions and quadratic link performance functions so
as to reduce the computational burden for a time-space decomposition solution procedure
that can only handle very small network problems. In addition to the unrealistic modeling
of congestion, they do not consider the FIFO issue that arises for multiple destinations.
Ran and Shimazaki (1989b) present an optimal control theory based instantaneous UE
DTA model. The ¯ows exiting links are treated as a set of control variables rather than
functions to circumvent the generalization issue. No ef®cient algorithms are available to
solve these formulations.
Ran et al. (1993) use the optimal control approach to obtain a convex model for the
instantaneous UE DTA problem by de®ning link in¯ows and out¯ows to be control
variables. They recognize the inability of the usual cost functions to account for dynamic
queuing and congestion costs, and propose splitting the link travel cost into moving and
queuing components. However, these functions are assumed to be non-negative, increasing
and differentiable, and hence may not re¯ect traf®c realism. Also, no speci®c instances of
the functions are provided or tested. Boyce et al. (1995) discuss a methodology to solve the
discretized version of the problem using the Frank±Wolfe algorithm and an expanded time-
space network representation. However, they do not implement the procedure or illustrate
it through examples. Also, the use of static link performance functions is a limitation of
this model.
Though an attractive approach for describing dynamic systems, an optimal control type
of DTA formulation suffers from many limitations, such as the lack of explicit constraints
to ensure FIFO and preclude holding of vehicles at nodes, the inadequate and possibly
unrealistic modeling of traf®c congestion, and more crucially, the lack of a solution
procedure for general networks. In addition, the substantive interpretation of the UE
formulations based on instantaneous travel times requires rather strong and possibly
unrealistic assumptions about the nature of the behavioral processes underlying the
particular equilibrium conditions postulated by the researchers. Because of the limitations
of optimal control theory in the DTA context and the advantages offered by variational
inequality (VI), analytical DTA models have in recent years migrated towards the VI
approach, discussed hereafter.
2.3. Variational inequality formulations
Variational inequality provides a general formulation platform for several classes of
problems in the DTA context such as optimization, ®xed point, and complementarity. It
fosters a uni®ed mechanism to address equilibrium and equivalent optimization problems.
Also, its mathematical properties such as uniqueness can be illustrated in a simple manner.
Dafermos (1980) introduced the VI approach in the static traf®c equilibrium context.
238 PEETA AND ZILIASKOPOULOS

Nagurney (1998) provides a comprehensive summary of VI and addresses various
equilibrium problems. VI circumvents analytical tractability issues arising in constrained
optimization formulations due to asymmetric link interactions. In that sense, it can handle
more realistic traf®c scenarios. Also, extensions and sensitivity analyses can be conveni-
ently performed. While it is more general, and better equipped than the other analytical
approaches to address several aspects of DTA problems, the broader limitations associated
with analytical models, discussed earlier, remain.
Friesz et al. (1993) formulate a continuous time VI model to solve for the departure time/
route choice by equilibrating the experienced travel times. The model uses link performance
functions, penalty functions for early/late arrivals, travel demands, desired arrival times, and
all possible paths between origins and destinations. The path cost is a combination of the
travel cost determined by the link performance function and the penalty for early/late arrival
caused by traveling along the path. The formulation incorporates relatively more realism in
terms of traveler behavior, but has unresolved issues. For instance, there is no proof of
solution existence or uniqueness. Also, since the formulation is a continuous-time in®nite-
dimensional VI problem, its solution requires solving a complex system of simultaneous
integral equations and there is no ef®cient algorithm for this purpose.
Wie et al. (1995) introduce a discretized VI formulation for the simultaneous route-
departure equilibrium problem to enable computational tractability, and propose a heuristic
algorithm to approximately solve it. They show solution existence under certain regularity
conditions, and use exit ¯ow functions instead of exit time functions used in the Friesz et
al. (1993) formulation. The use of exit ¯ow functions raises the usual traf®c ¯ow realism
issues. Since the formulation is path-based, and complete path enumeration is computa-
tionally burdensome, there is a need for an ef®cient method to identify a subset of relevant
paths. However, that may entail some limiting assumptions on the traf®c modeling (Ran
and Boyce, 1996b).
To circumvent the problems with path-based VI models, Ran and Boyce (1996b)
propose a link-based discretized VI formulation with ®xed departure times. Akin to Friesz
et al. (1993), they also equilibrate the experienced travel times. They include a queuing
delay component to partly alleviate the traf®c realism issues arising in the context of
analytical models. However, the capacity and oversaturation constraints signi®cantly
increase the computational burden, leading to computational feasibility issues for realistic
networks. Also, path-based formulations are inherently convenient in the context of route
guidance. Ran et al. (1996) extend the proposed model to a link-based VI model for the
simultaneous departure time/route choice problem.
Chen and Hsueh (1998) also propose a link-based VI formulation for the UE DTA
problem. They show that without loss of generality, travel time on a link can be represented
as a function of link in¯ow only (instead of function of link in¯ow, exit ¯ow, and number
of vehicles on the link). Through this simpli®cation, they illustrate that the Jacobian matrix
of the travel time function is asymmetric, and that any dynamic travel choice problem with
such travel time function characteristics does not have an equivalent optimization program.
A solution algorithm based on nested diagonalization procedure is proposed; however, it is
still prohibitively expensive to implement on real networks.
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 239

The VI approach is more general than the other analytical approaches discussed earlier,
providing greater analytical ¯exibility and convenience in addressing various DTA
problems. It has been used to illustrate, with relative ease, the notion of experienced
travel times for the so-called ``simultaneous'' and ``ideal'' UE DTA problems. Also, it
highlights the inability of the mathematical programming approach in addressing scenarios
with asymmetric Jacobian matrices for the travel cost functions. However, VI approaches
are more computationally intensive than optimization models, raising issues of computa-
tional tractability for real-time deployment. These issues are further magni®ed for path-
based VI formulations requiring complete path enumeration. Also, despite capabilities to
better represent link interactions, traf®c realism issues arising in the context analytical
models persist.
2.4. Simulation-based models
Simulation-based DTA models use a traf®c simulator to replicate the complex traf®c ¯ow
dynamics, critical for developing meaningful operational strategies for real-time deploy-
ment. The terminology ``simulation-based models'' may be a misnomer. This is because
the mathematical abstraction of the problem is a typical analytical formulation, mostly of
the mathematical programming variety in the current literature. However, the critical
constraints that describe the traf®c ¯ow propagation and the spatio-temporal interactions,
such as the link-path incidence relationships, ¯ow conservation, and vehicular movements,
are addressed through simulation instead of analytical evaluation while solving the
problem. This is because analytical representations of traf®c ¯ow that adequately replicate
traf®c theoretic relationships and yield well-behaved mathematical formulations are
currently unavailable. Hence, the term ``simulation-based'' primarily connotes the solution
methodology rather than the problem formulation. A key issue with simulation-based
models is that theoretical insights cannot be analytically derived as the complex traf®c
interactions are modeled using simulation. On the other hand, due to the inherently ill-
behaved nature of the DTA problem, notions of convergence and uniqueness of the
associated solution may not be particularly meaningful from a practical standpoint. In
addition, due to their better ®delity vis-a
Á-vis realistic traf®c modeling, simulation-based
models have gained greater acceptability in the context of real-world deployment
(Mahmassani et al., 1998b; Ben-Akiva et al., 1998).
In addition to the use of a simulator in a descriptive mode to determine the traf®c ¯ow
propagation, most existing simulation-based models also use it as part of the search
process to determine the optimal solution. Labeled the predictive-iterative mode, the
simulator is used in each iteration to project the future traf®c conditions as part of the
direction-®nding mechanism for the search process. Given the substantial computational
burden associated with the use of a simulator, the choice of granularity (macroscopic,
microscopic or mesoscopic) has signi®cant implications for the real-time computational
tractability of simulation-based models.
Mahmassani and Peeta (1992, 1993, 1995), and Peeta and Mahmassani
(1995a), develop DTA models that use a mesoscopic traf®c simulator, DYNASMART
240 PEETA AND ZILIASKOPOULOS

(Jayakrishnan et al., 1994), as part of an iterative algorithm to solve for the SO and UE
solutions for O-D demand with ®xed departure times. Labeled as deterministic DTA
models in the context of the assumptions on information availability for Advanced
Traveler Information Systems (ATIS) operations, the problem assumes complete a priori
knowledge of O-D demands for the entire planning horizon of interest. Both descriptive
and normative objectives are considered. Also, akin to the analytical formulations, the
models assume identical users, manifest as a single class of users in terms of information
availability, information supply strategy, and driver response to the information provided.
The adopted simulation logic combines a microscopic level of representation of
individual tripmakers and drivers, with a macroscopic description of some of the
interactions taking place in the traf®c stream. This enables an acceptable solution
accuracy level at a fraction of the computational cost required for a microscopic
representation of traf®c maneuvers. The use of a traf®c simulator circumvents the traf®c
realism issues of analytical formulations and provides a solution procedure for general
networks. Mahmassani et al. (1993) extend the single user class models to the more
realistic multiple user classes scenarios where multiple user classes in terms of informa-
tion availability, information supply strategy, and driver response to the information
provided, are assumed. However, as with the analytical formulations, they are inadequate
operationally for providing optimal real-time path information and/or instructions to
network users under ATIS in response to unpredicted variations in network conditions. In
addition to the conceptual and algorithmic aspects of the models, the real-time and large-
scale nature of these problems make computational issues associated with the imple-
mentation of the models an integral part of the problem. In particular, the development of
algorithmic procedures must re¯ect the issue of computational ef®ciency. As these
deterministic DTA solution methodologies incorporate a simulator as part of the iterative
search process, they are not feasible for real-time deployment without further modi®ca-
tions vis-a
Á-vis implementation.
Ghali and Smith (1992a) propose a single user class formulation for the deterministic
SO DTA problem in which congestion arises exclusively at speci®ed bottlenecks modeled
as deterministic queues. A simulation-based solution procedure is proposed, whereby
vehicles are routed individually on paths determined using link marginal travel costs.
Although the approach does not ensure system optimality, and has limitations due to
certain assumptions on queuing, it addresses several traf®c modeling issues that limit the
realism and validity of analytical formulations. Ghali and Smith (1992b) discuss different
levels of computing approximate marginal travel times, in addition to the computation of
the global marginal travel times. However, their methodology for the computation of
marginal travel times involves the brute force simulation of alternative scenarios, which is
neither feasible for real-time application in the ATIS context, nor ef®cient in large-scale
real-world networks. Ghali and Smith (1992c), and Smith (1994), address the SO and UE
DTA problems and implement their solution procedures using the CONTRAM (Leonard
et al., 1989; Taylor, 1990, 1996) simulation model. Ghali and Smith (1993) construct
examples to illustrate the possibility of non-convexity and non-differentiability in DTA
problems, which preclude the guarantee of convergence and of obtaining globally optimal
solutions for general networks.
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 241

With the objective of generating a real-time deployment capability, Peeta and Mahmas-
sani (1995b) develop rolling horizon DTA models to explicitly incorporate real-time
variations in network conditions and foster computational ef®ciency to enable real-time
tractability. The rolling horizon approach provides a practical method for addressing
problems which ideally require future demand information for the entire planning horizon,
a de®ning characteristic of DTA problems (Peeta, 1994). Its advantage is the ability to use
currently available information, and near-term forecasts (of the order of 5 to 15 minutes)
with some degree of reliability, to solve a problem in quasi real-time while preserving the
effectiveness of the computational procedure in determining ``good'' control strategies.
From an operational perspective, these information requirements are more realistic
compared to the perfect knowledge assumptions of deterministic models. Since it is
stage-based, the rolling horizon approach ensures that unpredicted variations in on-line
traf®c conditions can be adequately accounted for in subsequent stages. However, if the
actual O-D desires in a stage are signi®cantly different from the forecasts, the solution is
sub-optimal. Also, despite being a stage-based approach and more ef®cient than determin-
istic DTA solution procedures, it can be computationally intensive in a centralized
architecture.
Ben-Akiva et al. (1997a, 1997b) propose DynaMIT as a dynamic traf®c assignment
system to estimate and predict in real-time current and future traf®c conditions. It consists
of a demand and a supply simulator that interact to generate UE route guidance under the
rolling horizon framework. No underlying formulation is proposed. The demand simulator
estimates and predicts O-D demand using a Kalman Filtering methodology. It considers
both historical information and the driver response to information. The supply simulator is
used to determine the ¯ow pattern based on the demand. It is a mesoscopic traf®c
simulator, where vehicles are moved in packets and links are divided into segments that
include a moving part and a queuing part to model traf®c ¯ow.
Ziliaskopoulos and Waller (2000) introduce an internet-based GIS system that integrates
data and models into one framework. The simulation-based DTA model in this system uses
RouteSim, which is a mesoscopic model based on cell transmission (Daganzo, 1994) for
traf®c propagation, as the traf®c simulator. The model captures many realities on the network,
such as traf®c signals, by using time-dependent cell capacities and saturation ¯ow rates.
Simulation-based DTA models address, with relative ease, several modeling issues that
are troublesome in analytical formulations. The use of a simulation model that incorpor-
ates traf®c theoretic relationships to model traf®c ¯ow circumvents the limitations of
analytical link performance and exit functions in replicating dynamic traf®c phenomena
and obtains ¯ows consistent with those relationships. A traf®c simulator also captures the
complex vehicle interactions, thereby evaluating the non-linear objective function satis-
factorily compared to idealized cost functions. In addition, a simulation model can
implicitly satisfy the FIFO constraint and circumvent the problem due to holding-back
of vehicles. All these factors, and the ability to keep track of paths of individual vehicles in
the ATIS context, represent the advantages of simulation-based approaches in the DTA
context. Hence, while analytical models have focused primarily on deriving theoretical
insights, simulation-based models have concentrated on enabling practical deployment for
realistic networks. Therefore, issues such as multiple user classes, information availability,
242 PEETA AND ZILIASKOPOULOS

driver response characteristics, deployable solution procedures, and deployment aspects
such as consistency, robustness, calibration, and computational ef®ciency, have been the
focus of simulation-based models. The deployment aspects are discussed further in Section
3. Simulation is an especially convenient option to realistically capture the complex
interactions among multiple user classes that arise in actual traf®c networks. Given the
traf®c modeling issues that arise for analytical models in the single user class context itself,
introducing the notion of multiple user classes substantially adds to their intractability.
The key limitation of simulation-based DTA models, in general, is the inability to derive
the associated mathematical properties. As discussed earlier, this is not a signi®cant issue
for general traf®c networks. This is because the DTA problem for general traf®c networks
is inherently ill-behaved and rather intractable, restricting the ability of analytical models
as well to analyzing problems with simpli®ed assumptions that reduce realism. However,
an ability to analyze the system properties even under simpli®ed assumptions can be
insightful in generating future directions to address the problems. Another limitation of
using a simulator is in the deployment context. The computational burden associated with
the use of a simulator as part of an iterative mechanism to project the future can be
operationally restrictive. Hence, many recent simulation-based deployment frameworks,
discussed further in Section 3, trade-off solution accuracy with computational ef®ciency.
3. Current and Future Directions: Challenges and Opportunities
The review of past DTA efforts provides pointers to the challenges and opportunities in the
general DTA domain, both in terms of research directions and practical applications. DTA
appeals to a broad audience of researchers and practitioners, and promises the potential for
a wide range of applications. Recent trends suggest that it can impact a broad range of
problems in transportation operations, planning, and engineering. Each, in turn, introduces
a unique set of challenges as well as opportunities for DTA research. This section brie¯y
discusses some of the current and/or future research directions and potential challenges. It
also identi®es some of the opportunities they present for advancing the DTA ®eld. The
challenges and opportunities are grouped into three categories: two in the application
domain (real-time deployment and planning) and one related to the fundamental issues.
3.1. Real-time deployment
Till recently, the primary focus of research efforts in the DTA arena has been the
development of methodological and algorithmic constructs that address the fundamental
concepts, architecture, and logic, representing mostly an off-line developmental perspec-
tive. Current efforts, while continuing to address the fundamental issues, are increasingly
focusing on real-time operational issues, to develop and re®ne capabilities for the ef®cient
and effective real-time deployment of these methodologies and algorithms. These issues
are: (i) computational tractability, (ii) robustness of the solution methodologies and
stability of the associated solutions given the inherent system randomness, (iii) fault
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 243

tolerance and system reliability, (iv) operational consistency and model calibration/
validation, and (v) demand estimation and prediction. The associated objective is a
practical one: how to implement these algorithms and strategies to achieve a feasible
degree of practical effectiveness in large-scale traf®c networks? Currently, the research on
most of these issues is rather sparse.
While perfect a priori knowledge (on O-D demand and incidents) and system-wide
coordination are highly desirable, they remain the obstacles to real-time operability in
general networks because they introduce unrealistic expectations and computational
intractability, respectively. This motivates the development of methodologies that are
both robust and computationally ef®cient on-line.
3.1.1. Deployment frameworks: computational tractability The real-time deployment of
DTA requires computationally tractable solution procedures for general networks. Real-
time computational tractability can be addressed at different levels. At the computing
hardware level, it strongly depends on the environment used (sequential, parallel,
distributed, etc.) and the technological progress of computational processing and commu-
nication capabilities. At the computing software level, it depends on the ef®cient
algorithmic coding and the overall system integration architecture used, for example the
CORBA based implementation of a DTA system (Hawas et al., 1997). At the level of
algorithmic logic, it depends on the control architecture (e.g., centralized, decentralized, or
hybrid). As stated earlier, here, the deployment frameworks related to algorithmic logic are
addressed. Most of these frameworks either circumvent the use of a simulator on-line or
use it in a computationally ef®cient mode that avoids a predictive-iterative process.
To address the real-time computational burden, Hawas and Mahmassani (1995) propose
a non-cooperative reactive decentralized architecture where spatially distributed controllers
make routing decisions independent of each other. Local control rules using arti®cial
intelligence techniques and currently available partial traf®c information in the local area
are used to determine user routes within the control territory. An attractive feature of the
approach is the ¯exibility in de®ning the territorial size of each controller based on its
processing capabilities, thereby circumventing issues of computational burden. Another
signi®cant aspect is that the approach does not require any O-D demand predictions unlike
centralized frameworks. However, since controllers act independently based only on local
rules, there is no coordination between them. If the proportion of inter-territory vehicles is
large, the solution under this scheme deviates substantially from the optimal solution
(Hawas and Mahmassani, 1995).
Hawas (1995) extends the above approach to develop a cooperative scheme that enables
exchanging non-local information between neighboring controllers. While the decentral-
ized architecture is more robust under incident situations because the local rule heuristics
are more responsive to current network conditions, there can be substantial degradation in
performance under non-incident conditions compared to the benchmark centralized
deterministic DTA solution (Hawas and Mahmassani, 1997). Since the approach is
reactive, it does not exploit available historical data, especially on time-dependent O-D
demand and incidents.
244 PEETA AND ZILIASKOPOULOS

Pavlis and Papageorgiou (1999) develop a reactive decentralized feedback control
strategy based DTA model for meshed networks. The essential components of the strategy
are simple decentralized control laws of the bang-bang, P, or PI type that could be designed
based on trial-and-error. It reacts solely to the real-time measurements. The controllers
independently calculate splitting rates for each path based on the traf®c information on
alternative paths. The control strategy is to equalize the instantaneous path travel times of
alternative paths for each O-D pair. Since it is decentralized and circumvents forecasts of
future traf®c conditions, signi®cant computational ef®ciency is achieved to promote real-
time tractability. However, the associated framework is limited to speci®c network
topologies and is opaque to systematic correctable errors in the prediction process.
Peeta and Zhou (1999a, 1999b) propose a hybrid predictive-reactive framework that
combines off-line and real-time strategies to solve the real-time DTA problem. It ensures
that the computationally intensive components are executed off-line while circumventing
the need for very accurate real-time O-D demand forecast models. The primary concept of
this approach involves the development of an a priori solution using an iterative
centralized deterministic DTA-based mechanism off-line that exploits the available
historical data to develop a robust initial solution for real-time use. The historical O-D
demand and incident distributions are used to incorporate randomness. The a priori
solution is based on the computationally intensive Monte Carlo simulation of various
realizations. The initial solution is updated on-line based on the unfolding conditions
through ef®cient real-time reactive strategies obtained using the ®eld traf®c measurements.
Hence, no O-D demand prediction is required in real-time.
Peeta and Yang (2000a) develop an ef®cient dynamical systems based search process to
solve the DTA problem in real-time. It is used in conjunction with a simulator in the
descriptive mode as part of a feedback control strategy that circumvents the need for real-
time O-D demand predictions and uses the ®eld traf®c measurements. A truncated decision
horizon, called the assignment interval, is used to implement the strategy over the planning
horizon of interest, akin to a rolling horizon scheme. The control strategy is extended
(Yang, 2001) to incorporate an anticipatory capability that uses historical traf®c data within
a hybrid framework. Here as well, the simulator is used in a descriptive mode for a
truncated horizon, ensuring real-time tractability.
3.1.2. Consistency checking The operational consistency problem, representing the
potential divergence of the predicted system state from the actual conditions unfolding
on-line, arises because of several factors that can signi®cantly in¯uence the performance of
dynamic traf®c networks. They include: (i) incorrect prediction of the time-dependent
origin-destination (O-D) trip demand, (ii) unpredicted incidents, (iii) incorrect path
predictions, (iv) incorrect traf®c modeling, (v) incorrect assumptions on driver behavior
and/or response to information provided, (vi) incorrect assumptions on system related
parameters, (vii) noise and/or sparsity in measurements, and (viii) failure of the ATIS
system components. Ensuring consistency is critical to the effectiveness of any DTA
procedure. Additionally, from the perspective of traf®c networks with advanced informa-
tion systems, not all network users are likely to be equipped with in-vehicle route guidance
systems for two-way communication with traf®c control centers. Hence, it is imperative to
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 245

robustly predict the travel actions and decisions of unequipped users so as to provide more
accurate routing information to equipped users, with the broader objective of enhancing
system performance. In a control context, the consistency aspect raises issues of system
observability and controllability. A related problem that needs to be addressed in an
operational context is the calibration of the various static and dynamic modeling and
system parameters for the speci®c network. This aspect is also related to the model
validation issue.
Mahmassani et al. (1998a) propose an integrated real-time and off-line adjustment
module for consistency using a proportional-integral-derivative (PID) feedback control
strategy. They develop a framework for real-time monitoring that reacts to any observed
on-line deviations in traf®c conditions. The real-time corrections are not necessarily the
most accurate adjustments because some information could be missing. Thus, the system
is further updated off-line where full information on past conditions is used to perform
optimal adjustments to address observed real-time errors. The real-time scheme is used to
correct the parameters of the travel time function and the ¯ow propagation equations using
real-time data on traf®c measures such as average speed, in¯ow, and out¯ow. However, the
associated adjustment process does not address the underlying fundamental phenomena
that cause the inconsistencies, and the procedure can also be computationally expensive
on-line. The approach is extended to an online traf®c monitoring system in Ziliaskopoulos
et al. (1998).
Peeta and Bulusu (1999) propose a framework for ensuring operational consistency vis-
a
Á-vis real-time dynamic traf®c assignment in networks with ATIS. Formulated within a
stage-based rolling horizon framework, the model ®rst solves a deterministic DTA problem
to predict the system state for the near future while optimizing certain system-wide
objectives for the controller, and later seeks consistency between the predicted system state
and the actual conditions unfolding on-line. The approach ensures that future state
predictions and path assignments are consistent with the current actual system state
rather than a presumed estimate of it, which synergistically reduces the propagation of
consistency errors over time. The consistency problem is formulated as a constrained least
squares model. It is under-determined, rank de®cient, and potentially ill-conditioned for
general networks. In addition, it lacks well-behaved properties and has a ®xed-point
element, characteristics inherited from the DTA problem. It is solved using generalized
singular value decomposition based orthogonal transformations. However, the approach is
computationally intensive.
3.1.3. Robustness: incorporating randomness The issue of on-line robustness of the
DTA solution procedure arises primarily because of inherent stochastic system inputs and/
or factors in general traf®c networks. As discussed earlier, there are several sources of
randomness that signi®cantly in¯uence the performance of dynamic traf®c networks. They
include the randomness in demand, supply conditions (for example, incidents), and the
fractions of users belonging to the different user classes. Solution procedures that
explicitly incorporate uncertainties so as to minimize or limit the deterioration in system
performance are referred to as robust. The associated problem is typically labeled as the
stochastic DTA problem.
246 PEETA AND ZILIASKOPOULOS

Peeta and Zhou (1999a, 1999b) explicitly model the inherent stochasticity in O-D
demand and/or network supply conditions. Thereby, the O-D desires and/or non-recurrent
congestion characteristics are treated as random variables with known distributions (based
on a historical database updated on-line or on a day-to-day basis). The associated real-time
stochastic DTA problem is solved using the hybrid solution framework discussed in
Section 3.1.1, which addresses the computationally intensive procedures off-line to reduce
the processing time on-line. It involves the determination of an off-line solution that serves
as a ``good'' initial solution for real-time application, to which adjustments are made on-
line as warranted by unfolding traf®c conditions. A robust solution is de®ned here as one
that has minimal deviation, in terms of the expected system-wide travel time, from the
corresponding deterministic DTA solution, on average, across the range of likely O-D
demand matrices. The approach models only the time-dependent O-D demand explicitly as
random variables while determining the off-line solution. While incidents can also be
treated as random variables in this framework, they can be more robustly managed on-line
using reactive strategies (Hawas and Mahmassani, 1997; Peeta and Zhou, 1999b). Also,
the marginal effect of the occurrence or non-occurrence of a predicted incident on the
degradation of the robustness of the off-line solution is, in general, relatively much higher
compared to that of an O-D desire, as individual incidents can impact network perfor-
mance signi®cantly.
3.1.4. Stability Stability of the DTA solution is an important operational issue for the
control of dynamic traf®c networks. This is because inappropriate assignment proportions
may lead to increased unpredictability and volatility, and/or catastrophic consequences for
the system. There are several key factors that can affect the stability of the traf®c system.
They include incidents, randomness in time-dependent demand, driver behavior, informa-
tion provision, driver response to information, user class fractions in the traf®c stream, and
external traf®c controls. Conceptually, the notion of stability implies that all solutions are
bounded and converge to the time-dependent desirable states. The practical implication is
that a stable solution minimizes or limits the deterioration of system performance.
There is a substantial body of literature that addresses stability in the context of static
traf®c assignment. Yang (2001) provides a detailed summary of the various approaches.
Smith (1979, 1984) addresses the stability of traf®c network equilibrium for the static
assignment problem. A dynamical system is used to model the route choice behavior. The
Lyapunov function approach is used to study the stability of the equilibrium solutions.
Horowitz (1984) proposes three models for the route choice decision-making process in a
two-link network based on three weighted average measures. He de®nes the network
equilibrium to be stable if the equilibrium point is unique and the convergence of link
volumes to the equilibrium state from arbitrary initial points is guaranteed. Friesz et al.
(1994) apply tatonnement adjustment processes from classical microeconomic equilibrium
models to predict day-to-day changes in response to changes in demand. They analyze the
behavior of day-to-day trajectories from disequilibrium under complete or incomplete
information provision and discuss the stability properties. Cantarella and Cascetta (1995)
consider the stability of stochastic equilibrium for general networks. Regularity conditions
are proposed to ensure the existence and uniqueness of a stationary probability distribution
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 247

of system states. Zhang and Nagurney (1995, 1996), and Nagurney and Zhang (1996,
1997), introduce the projected dynamical system concept to study the route choice
adjustment process in elastic and ®xed demand networks. The stationary point of such a
dynamical system coincides with the UE ¯ow pattern. They propose two distinct
approaches: the monotonicity approach to analyze global stability and the regularity
approach for local stability analysis. Watling (1999) extended Horowitz's results (1984) to
general networks. A dynamical adjustment process is proposed for studying the stability of
the general asymmetric stochastic equilibrium assignment problem. The stability analysis
concentrates on the linear approximation of the original non-linear model.
Peeta and Yang (2000a, 2000b) analyze stability in dynamic traf®c networks using a
dynamical systems approach and propose stable real-time deployable route guidance
control strategy models for different assignment principles. They show that the Lyapunov
functions for the SO and UE objectives are their corresponding objective functions under
DTA. The problem is formulated as a non-linear dynamical system. Incorporation of time-
dependence enables the consideration of time variance in the O-D demand and link travel
times. Rather than seeking the time-dependent desirable stable states themselves, the
approach moves the system towards these states based on the current or predicted future
network conditions (Yang, 2001). This is because the desirable states themselves may
never be reached in practice given the complexities inherent to dynamic traf®c networks.
3.1.5. Demand estimation and prediction As discussed in Section 2, the assumptions on
the amount of information available on the O-D trip demand lead to different DTA
formulations based on information availability. In this context, the real-time estimation and
prediction of dynamic O-D demand has attracted a lot of research attention. Chang and Tao
(1999) classify the associated models into two broad categories: DTA based and non-DTA
based. Cremer and Keller (1987), Wu and Chang (1996), Sherali et al. (1997), Ashok and
Ben-Akiva (2000), and Hu et al. (2001) belong to the non-DTA based category, while
Cascetta et al. (1993) and Chang and Tao (1999) are representative of DTA based O-D
demand models.
Cremer and Keller (1987) link the in¯ows into the network and the out¯ows from it
through split parameters. The problem is then transformed into one where an estimate for
the split parameters is sought. The formulation assumes that the travel time between the
various origins and destinations is shorter than the sampling interval, an unrealistic
assumption for general networks. Four different approaches are proposed for estimating
the time-dependent splits. The ®rst one uses the cross-correlation matrices to estimate the
splits. This method cannot guarantee that the conditions on the error distribution of the
splits are satis®ed. The second method includes those conditions in the problem formula-
tion as constraints and minimizes the squared error between the estimated exit ¯ows and
the observed values. These two methods seek the time-dependent splits in one step, while
the other two approaches proposed are recursive and result in shorter computational times.
Wu and Chang (1996) propose a formulation that uses traf®c ¯ows through screenlines
in addition to link volumes to estimate the dynamic O-D trip matrices. They consider the
time-dependent travel times between O-D pairs, thereby relaxing the assumption in Cremer
and Keller (1987) on travel times. The system observability is enhanced through the
248 PEETA AND ZILIASKOPOULOS

consideration of these two aspects. However, as no traf®c assignment component is
included in the formulation, the path choices are approximated using a logit model. The
use of a logit model implies independence among paths, a restrictive assumption. Sherali
et al. (1997) develop solution algorithms based on dynamic link volumes. The two
formulations addressed are optimization problems: one is a constrained least-squares
problem, the other attempts to minimize the sum of the absolute deviation between the
estimated and observed ¯ows. The solution algorithms perform better than the Kalman
®lter based recursive approach in Cremer and Keller (1987). However, static travel times
are used and no assignment component is included.
Cascetta et al. (1993) and Chang and Tao (1999) incorporate a network loading process
into the O-D demand estimation problem. Cascetta et al. (1993) propose a dynamic
network loading approach that uses the link-path incidence matrix to describe how traf®c
propagates. However, it uses average travel times to compute this matrix and is, hence, not
a robust assignment model. Two estimators are proposed for the demand matrix. One
simultaneously estimates the O-D demand matrices for all time intervals; the other
estimates an O-D demand matrix for each interval sequentially, using the current one as
an initial estimate for the next time interval. Chang and Tao (1999) extend the screenline
approach (Wu and Chang, 1996) to decompose the network into smaller sub-networks. A
two-stage O-D demand estimation framework is proposed for a large network based on the
network decomposition. A non-DTA based approach is used to estimate an initial O-D
demand matrix. A DTA model assigns this initial demand matrix to the network and
calibrates the demand matrix using the resulting link volumes.
Ashok and Ben-Akiva (2000) present two state-space models for the real-time estima-
tion and prediction of time-dependent O-D demand. Instead of de®ning the state-vector as
the O-D ¯ows themselves, the ®rst model focuses on the deviations in O-D demand. The
second model de®nes the state-vector as the deviations of departure rates from each origin
and the shares headed to each destination. Preliminary test results indicate that such
formulations make the real-time estimation process computationally tractable. The second
model yields better predictions with some loss of accuracy in the ®ltered estimates.
Hu et al. (2001) propose an adaptive Kalman Filtering algorithm for the dynamic
estimation of O-D demands using the time-varying link traf®c count information. The
study does not address the problem of estimation of O-D demand matrices in networks
with multiple routes, but can be extended to that scenario. A key aspect of the adaptive
estimator is the use of a traf®c simulator to predict the time-varying travel times used to
compute time-varying assignment fractions.
Estimating and predicting the dynamic O-D demand from link volumes (and possibly,
other information such as the historical O-D demand matrix) is a relatively under-explored
research topic, with no approach currently capable of fully dealing with general networks.
Models that have a DTA component can robustly address the various time-dependent
network issues; however they can entail signi®cant computational burden.
3.1.6. Error and fault tolerance The real-time deployment of DTA under ATIS involves
the transmission of ®eld data to the Traf®c Control Center (TCC) for real-time processing.
To enable reliable and uninterrupted operation, these real-time systems should be fault
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 249

tolerant to critical hardware failure modes such as malfunctioning detectors and failed
transmission/communications links. Transportation systems possess signi®cant random
elements, such as human behavior, weather conditions, demand variability, and incidents,
which complicate fault detection in vehicular traf®c networks.
Anastassopoulos (2000) proposes a Fourier transform based fault tolerant framework for
a deployable DTA control architecture where both data faults (erroneous data) and
incidents are treated as abnormalities in the monitored network. The approach ®rst detects
an abnormality and then distinguishes data faults from incidents. Data faults are corrected
using a Fourier transform based data correction heuristic. The approach uses data directly
without any predictive modeling, circumventing likely modeling errors and enabling
adaptability to future demand/supply changes. It also predicts near-term traf®c conditions
ef®ciently.
3.2. Planning
3.2.1. Estimating and forecasting time-dependent demand Probably the single most
challenging obstacle to overcome, before deploying DTA for planning applications, is that
of estimating and predicting the time-dependent origin-destination demand. Most research-
ers assume that such a time-dependent O-D matrix is readily available. However, the
current practice for demand modeling does not provide methodologies for estimating time-
dependent O-D demand. While substantial work has been done in this area, the issue is far
from closed. Planning agencies currently spend signi®cant resources to obtain a 24-hour
trip table; but no resources, to the best of our knowledge, are invested for collecting time-
dependent data. Demand modeling in the traditional four-step procedure uses data from
travel surveys to calibrate demand models and forecast future demand. Trip generation
usually takes the form of either regression or category analysis, both of which use socio-
economic characteristics of the travelers in a zone as independent variables. Trip
distribution mainly relies on gravity models (which can be derived through entropy
maximization) to estimate the linking of trip ends predicted by the trip generation model
and to form an O-D matrix. Discrete choice models are then applied to compute the mode
splits that estimate the percentage of travelers using each mode. The results from these
steps are the O-D matrices for each travel mode. While this process has been questioned as
arbitrary and various combined models have been suggested, the four-step process tends to
be the one mostly adopted by practitioners.
Surprisingly, the problem of estimating the temporal distribution of demand has been
addressed by only a few studies. The approaches appearing in this special issue discuss
demand estimation, though the attention is mostly directed towards real-time applications
(Mahmassani, 2001; Ben-Akiva, et al. 2001). Demand estimation and prediction are
typically treated outside the simulation based DTA modules aiming to provide robust
estimates of historical time-dependent demand. None of the papers, however, presents
compelling evidence that such an approach would work for an actual network. Mahmas-
sani (2001) was the ®rst to adopt the rolling-horizon approach for real-time DTA
applications, which can account for demand uncertainties by updating the ¯ow patterns
250 PEETA AND ZILIASKOPOULOS

produced by those demands as they become known. Ben-Akiva et al. (2001) recognize the
need for historical time-dependent data, but they mostly focus on short-term predictions
for real-time applications. The Kalman ®ltering approach, however, aims to use all existing
information to produce real-time O-D estimates, and could lead to the development of
meaningful O-D matrices even for planning applications. A fairly elaborate discussion of
the Ben-Akiva et al. approach based on a Kalman ®ltering framework is included in the
paper. Carey (2001) introduces an elastic demand model in a similar manner. Speci®cally,
for each O-D pair and time period the demand is treated as a price elastic variable, where
travel demands may be subject to costs or penalties incurred if traf®c arrives at the
destination earlier or later than some desired times.
Another promising approach is suggested, where dynamic demand data are generated
from two sets of information: the ®rst set is the static demand data and the second includes
a set of behavioral rules that describe the travelers' choice of departure time. A process is
proposed for calibrating the set of behavioral rules with increasingly detailed and
disaggregate data. The synthetic dynamic demand module is then incorporated into an
iterative process with DTA within a simulation based model. Li (2001) suggested a time-
dependent trip-distribution approach that extends the static trip generation and distribution
gravity models. While it stops short of proposing a framework that would produce time-
dependent demand data, it makes an important contribution by identifying temporal
elements in existing data sets that could certainly be used to further enhance the temporal
demand estimates.
While the time-varying nature of the demand in DTA models presents insurmountable
dif®culties, it also presents some unprecedented opportunities, namely accounting for the
departure time choice and the uncertain nature of demand. The opportunity to account for
people desiring to arrive at a certain time (®xed arrival time demand) is dealt with
extensively in Friesz et al. (2001). Many of the references in the paper by Friesz provide
promising formulations and state necessary and suf®cient conditions. Li et al. (1999)
introduced a linear programming formulation that can solve reasonable size networks
accounting for ®xed arrival time as well as mixed demand.
Waller (2000), Waller and Ziliaskopoulos (1998) and Ziliaskopoulos and Waller (2000)
proposed approaches for accounting for demand uncertainty. Note that by relaxing the
determinism of the demand, one can potentially afford to make more mistakes in
estimating it, since the solutions of stochastic DTA models are by de®nition more
robust. The speci®cs of these approaches are brie¯y discussed in section 3.3.3.
In summary, the estimation of time-dependent demand is a seemingly intractable
problem that will occupy researchers in the years to come. However, accounting for
additional complexities such as demand uncertainty and ®xed arrival time users, coupled
with data from a real-time system, may provide opportunities to overcome some of the
dif®culties in estimating time-dependent demand.
3.2.2. Modeling multiple user classes The simulation based approaches are convenient
for capturing different user classes, as well as other realities of street networks. All
approaches in this special issue deal with the issue of accounting for many user classes,
while recognizing that the issue is complex and many assumptions need to be made.
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 251

Mahmassani et al. (2001), Boyce et al. (2001) and Ben-Akiva et al. (2001) report on
approaches that account for many types of users. However, a key question that will occupy
researchers for the years to come is: what are the fundamentally different user classes?
Mahmassani et al. (2001) focus on user classes in terms of information availability and
supply strategies (UE, SO, following local rules, etc.). Ben-Akiva et al. (2001) classify
drivers according to the information level they receive (®xed-route vehicles, unguided
drivers, drivers receiving descriptive or prescriptive information). Boyce et al. (2001) deal
with the problem in a more abstract fashion, by considering classes of users, where each
class is associated with a disutility or generalized cost function. The ideal UE DTA route
choice conditions are de®ned for each class on the basis of travel disutilities instead of
travel time only. Implicitly, Friesz et al. (2001) and Li et al. (1999) distinguish drivers
based on whether they are departure time or arrival time based. Finally, Li (2001) and Li et
al. (1999) classify users for planning applications by trip purpose and propose an
analytical approach to model the interaction between ®xed arrival time (FAT) and ®xed
departure time (FDT) based users.
The range of the classes modeled indicates the complexity and signi®cance of the
problem. At the disaggregate level, every user can be viewed as representative of a unique
class; however, this would be impractical to describe in a model. A classi®cation based on
the type of on-board device, information compliance level, trip purpose and vehicle type
should provide reasonable realism for modeling purposes. There is a great deal that needs
to be done to account for all these classes; the dif®culties are mostly related to representing
user behavior and equilibrium conditions.
3.2.3. Modeling many modes: person assignment Very little has been done on assigning
routes to people instead of drivers. As information technology matures in the transporta-
tion arena, ultimately, mode split will be more of a daily or real-time decision, instead of a
long term planning decision assumed by the third planning step. The main dif®culties for
implementing a simulation based person-assignment approach would be representing the
many modes of transportation and their perceived costs, de®ning equilibrium conditions,
modeling the mode choice and computing paths on intermodal/multimodal networks. The
intermodal optimum path algorithm by Ziliaskopoulos and Wardell (2000) provides a
promising framework to both represent and compute paths that account for both highway
and transit modes. Constructing analytical dynamic assignment approaches on networks
with many modes would be even more challenging, especially since some of the ®xed
schedule lines need integer variables to be properly captured, potentially leading to
intractable formulations.
3.2.4. Accounting for network controls and other realities Capturing network realities
will be an ongoing research endeavor. While, in recent years, the computational power of
commercially available machines has doubled approximately every other year, the need to
capture more network realities will always stretch the hardware capabilities to the limit.
Currently, the simulation based approaches (Mahmassani, 2001; Ben-Akiva et al., 2001;
and Ziliaskopoulos and Waller, 2000) seem to be able to account for signal control, ramp
metering, variable message signs, intersection movement delays, detector logic, different
252 PEETA AND ZILIASKOPOULOS

vehicle types and their interactions (trucks, buses), as well as information provision
technologies. As DTA moves closer to deployment, it will become more critical to
correctly account for control and the associated responses of drivers. It may not be
suf®cient to simply obtain the approximate impact of a traf®c signal timing plan; a very
precise representation may be required. This requires either microscopic or mesoscopic
simulation models with time steps of a few seconds. Otherwise, the evolution of the queues
behind traf®c signals and ramp meters cannot be appropriately captured. However, the use
of a very small time step (about 2 seconds) to capture the evolution of traf®c on a fairly
uniform long freeway segment is wasteful and unnecessary. Using multiple simulation
time intervals is a promising alternative to ®ner simulation, though the traf®c model should
provide for such a capability. An extensive discussion on the models used to propagate
traf®c is included in the next section.
Another issue related to the realism of the network representation is that of capturing the
interactions between some classes of vehicles and the infrastructure; such as modeling a
truck negotiating a right turn at an intersection with inadequate geometry. For example, a
stated preference survey by Peeta et al. (2000a) on driver response to variable message sign
based messages suggests perceptible differences in the responses of truck drivers
compared to those of non-truck drivers. This is because not all alternative diversion
routes are viewed as feasible by truck drivers. Existing models only tentatively address this
issue, especially on how infrastructure in¯uences vehicle routing behavior. In a mixed
traf®c case, even computing the likely routes of vehicles is a dif®cult problem, as the travel
delay experienced by a non-truck vehicle at an intersection with inadequate geometry will
depend on whether a truck is present or not.
Finally, capturing the impact of information technology devices and the response of
drivers to information can only be guessed at this point. Different devices and service
providers enter the market every year; some have technology that needs to be simulated
such as electronic toll collection readers, transit signal pre-emption technologies, real-time
traf®c adaptive responsive control (RT-TRAC) controllers and data collected from wireless
phones.
Capturing some of the above realities with analytical closed form models will probably
remain an open problem for a long time. Striving however to address these realities using
any type of approach will be increasingly relevant as DTA is adopted by operating
agencies.
3.2.5. Simultaneously optimizing controls and routing As discussed above, representing
controls is an important issue that adds to the realism of the approach, but it is
considerably simpler than actually optimizing the controls. To the best of our knowledge,
no approach can currently simultaneously optimize routes and controls. The simulation
DTA models can fairly easily represent them but they do not provide an appropriate
structure to actually optimize them. The analytical approaches would become intractable
even if all signalized intersections were treated in isolation. Again, this will possibly be an
open issue for a very long time, which of course should not discourage researchers from
striving to address it. Including signal optimization, capacity analysis and evaluation
capabilities in a simulation assignment framework that can do user-speci®ed hybrid
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 253

macroscopic, mesoscopic and microscopic simulation and optimize routes will probably be
the model that can ®nally integrate transportation planning, engineering and operations
modeling and software. The bene®ts from the enhanced communication and database
integration among these traditionally fragmented ®elds will be immense.
3.3. Other fundamental modeling issues
3.3.1. Describing equilibrium and behavioral issues De®ning equilibrium on networks
with time-dependent demand has consumed signi®cant research energies, leading to
various de®nitions and proofs. Most of them are simple extrapolations of Wardrop's
static conditions. User equilibrium (UE) conditions mean that drivers follow time
dependent least travel time paths, while system optimum (SO) conditions result from
drivers following time dependent least marginal travel time paths. The simulation based
approaches are ¯exible to perform UE, SO or multiple user classes assignment, though the
equilibrium conditions are only heuristically approximated. Mahmassani (2001) assumes
that users follow some user optimal routing behavior that could be in¯uenced by
information, leading to various classes of users. The boundedly rational drivers are
probably the most ¯exible class that has the potential to capture a wide range of behavioral
patterns. Ben-Akiva et al. (2001) assume user optimal behavior which implies that users
cannot ®nd a route they would prefer compared to the one they chose based on the
provided information. Analytical approaches can guarantee equilibrium (UE or SO) and
they can typically prove theoretically that a given formulation meets such conditions.
Friesz et al. (1993) introduced the path integral equilibrium model, which is the ®rst to
generalize the Wardropian principle of equilibrium and account for the simultaneous route
and departure-time choice on general networks. The generalization of the Wardropian
principle leads to a variational inequalities (VI) formulation for the problem, where the
dynamic equilibrium is expressed as a function of the integrals of path delays.
The question of whether equilibrium actually takes place or is a mathematical construct
is a very old issue and probably precedes even the de®nition of traf®c equilibrium itself.
The context of time-dependent networks provides opportunities to relax the restrictive
steady-state equilibrium assumptions and model phenomena of evolving disequilibria
(Friesz et al., 1994). Probably the only experimental evidence of user decision-making
behavior is an experiment, that involved 100 travelers over a 24-day period, performed by
Chang and Mahmassani (1988). Two heuristic rules were proposed and calibrated for the
adjustment of departure time. Friesz et al. (1994) addressed Mahmassani and Chang's
design theoretically by introducing a tatonnement model for modeling transition of
disequilibria from one state to another. The model led to the same conclusions as in
Chang and Mahmassani (1988), that a day-to-day adjustment process could lead to an
equilibrium state. The fact that there are always changes in supply, demand and traf®c
propagation, in combination with the stochasticity of all the involved parameters, makes
the notion of equilibrium highly questionable.
254 PEETA AND ZILIASKOPOULOS

3.3.2. Modeling traf®c ¯ow propagation Modeling traf®c ¯ow propagation is one of the
fundamental DTA issues that concerned most researchers in the ®eld, in their effort to
balance realism and computational tractability. Merchant and Nemhauser (1978a, 1978b)
were the ®rst to realize that link performance functions (similar to the static BPR
relationships) are not appropriate for dynamic networks, and they devised what they
called exit functions to regulate the link out¯ow. Other formulations have been proposed
since, including link performance functions used by some simulation based models, or
approaches based on the cell transmission model and microscopic models. Next, we brie¯y
discuss the main existing approaches to properly capture ¯ow propagation on the links and
nodes, and speculate on future developments.
Exit function Merchant and Nemhauser (1978a, 1978b) suggested the use of an exit
function to capture congestion on a link. The exit function determines the out¯ow from a
link given the number of vehicles on it. To express this function symbolically, let x
a
(t)
denote the volume of traf®c on arc a at time t, u
a
(t) the rate at which traf®c enters arc a, and
g
a
(x
a
(t)) the rate at which traf®c exits the link. The M±N model, as stated earlier, is non-
convex due to the exit functions considered. Carey (1986) showed that the model satis®ed
constraint quali®cations, which ensured that the optimality conditions would hold at an
optimum. Carey (1987) also showed that if the exit function constraint was re-written as an
inequality, then the model is convex and, if certain weak restrictions hold, then in any
solution of the model these inequalities will be binding, that is, satis®ed as equalities. As
discussed in Friesz et al.'s (2001) paper in this focus issue, the M±N ¯ow balance equation
based on the exit function can be written as follows:
dxa
dt uagaxafor all arcs:
Since ®rst introduced, exit functions have been used extensively as a means to move traf®c
in DTA models (Carey, 1987; Friesz et al., 1989; Wie et al., 1995).
Since an exit function computes the out¯ow as a function of the number of vehicles on the
link, it implicitly assumes that changes in density propagate instantaneously across the link.
However, in a congested network, this shortcoming may not be a serious problem since links
are not empty at either the beginning or the end of the time period of interest. The purpose of
the model is to re¯ect the aggregate behavior of the vehicles on the network, not to track
individual vehicle's movement. Exit ¯ow functions have been criticized as dif®cult to specify
and measure. In addition, they typically violate the ®rst-in, ®rst-out (FIFO) condition on the
link as demonstrated in Carey (1986, 1987). This has led to the abandonment of exit functions
for capturing traf®c propagation on links. In the Friesz et al. (2001) paper, there is an
extensive discussion of extensions to exit functions that can provide reasonably consistent
propagation such as the exit time functions and their inverses. They also provide a way of
expressing exit-time function based model of arc dynamics to obtain an alternative
formulation involving constrained differential equations, state-dependent lags, and arc
entrance and exit ¯ows that are control variables rather than operators.
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 255

Link performance function Link performance functions are used in Janson (1991a), Ran
et al. (1996), and Chen and Hsueh (1998) to decide the time-dependent link-path matrix.
Most link performance functional forms appear to be straightforward temporal extensions
of static BPR functions, which suffer from the well-documented drawbacks of realism and
consistency. These are further exacerbated by the need to capture dynamics; i.e., to account
for the spatio-temporal distribution of traf®c on the link while propagating traf®c. Some
simulation models (Ben-Akiva et al., 2001) use similar functions but only on part of the
link. Speci®cally, they divide links into moving and queuing sections; the BPR type
function is applied only to the moving part which is characterized by more uniform
conditions than the rest of the link. Mahmassani (2001) moves vehicles on links or sub-
links according to a modi®ed Greenshield type speed-density relationship, noting that
other traf®c stream models could also be incorporated based on ®eld investigation.
Link performance functions tend to be convex, which makes them convenient for closed
form formulations, but they suffer from serious drawbacks of realism, mainly because they
cannot capture the link dynamics in propagating traf®c. Dividing the link into small sub-
sections can approximate the link queue evolution but it is not clear how accurate this
representation is. Field calibration experiments will ultimately determine the suitability of
these models.
The cell transmission model The cell transmission model was proposed by Daganzo
(1994) to simulate the evolution of traf®c on a single highway link. It was shown that if the
relationship between traf®c ¯ow (q) and density (k) is of the form:
qminfnk;qmax;wkjkg;for 0 kkj
then Lighthill and Whitham's (1955) and Richards' (1956) equations for a single highway
link can be approximated by a set of difference equations, with current conditions (the
state of the system) being updated at every time interval. In the relationship above, n,q
max
,
w, and k
j
are constants that denote the free ¯ow speed, the maximum ¯ow (or capacity), the
backward propagation speed, and the jam density, respectively. The model discretizes the
time period of interest (assignment period) into small intervals; based on that, it divides
every link of the street network into small homogeneous segments, called cells, so that the
length of each cell is equal to the distance traveled by the free-¯ow moving vehicles in one
time interval.
Ziliaskopoulos and Lee (1997) extended the model to represent freeway and arterial
street networks. Ziliaskopoulos (2000) suggested that the traf®c ¯ow propagation relation-
ships could be captured by a set of linear constraints that results in linear programming
DTA models. The main advantages of the cell transmission model are its simplicity,
versatility and linearity; its major drawbacks are the determinism of the traf®c propagation
on highly congested links and the precision errors introduced when the discretization time
interval is smaller than 2 seconds. While both concerns can be addressed, they could result
in less tractable formulations. Embedding the cell transmission model in analytical
formulations, however, presents unprecedented advantages, since it provides superior
realism in a linear program that allows the vast existing literature on linear programming to
256 PEETA AND ZILIASKOPOULOS

be used to better understand and compute DTA. It can also provide for extensions to
formulate network design and stochastic programming formulations as discussed in the
following section.
In conclusion, it needs to be emphasized again that all the representations above are
simply models and as such abstractions of reality; the ultimate model suitability will
be determined when enough data is available to calibrate and evaluate them with real data.
3.3.3. Demand uncertainty DTA models typically assume known point estimates of
time-dependent O-D demand. The main objective of the DTA research is to relax the
assumption of time-invariance of demand. The next natural question is whether the
assumption of the determinism of demand can be relaxed. In general, for both planning
and operational applications, it is unclear what action should be taken or what the cost is,
when the forecast demand is not realized. It becomes necessary, therefore, to develop a
systematic way of accounting for demand uncertainty and develop a more robust solution,
which is more likely to withstand extreme events. One such possibility is to run the
deterministic model with a large number of randomly generated demand patterns, and to
infer some rules and principles from the results. Such an approach has been proposed by
Peeta and Zhou (1999b) to produce robust DTA solutions. The only drawback with this
approach appears to be the computational requirements of a single DTA run and the
enormous amount of scenarios that need to run to produce meaningful results and
inferences. However, as discussed in Peeta and Zhou (1999a), the computational aspect
can be addressed by using a hybrid deployment framework consisting of off-line and real-
time components. Chance Constrained Programming (CCP) introduced by Waller and
Ziliaskopoulos (1998) is an alternative approach. The CCP model is introduced as an
extension of a deterministic linear programming model. The insights from the analytical
model suggest that the uncertain demand needs to be in¯ated depending on its variance,
which can also be applied to simulation based approaches. Waller (2000) formulates a two-
stage stochastic programming approach with recourse for planning DTA approaches, with
promising results. However, the issue remains open and would probably be one of the most
active areas in DTA and traf®c control/optimization in the years to come, as stochasticity
seems to be the rule rather than the exception in traf®c systems. In fact, one of the
emerging questions is whether accounting for the stochastic nature of traf®c systems is
more important than accounting for dynamics. This is an issue that will probably be
debated by the research community in the years to come.
3.3.4. User behavior Mahmassani (2001) reports extensively on the effort to account for
behavioral elements. One of the principal features of this approach's interface with activity
based behavioral models is its explicit representation of individual trip-making decisions,
particularly for path selection decisions, both at the trip origin and en-route. Behavioral
rules governing route-choice decisions are incorporated, including the special case in
which drivers are assumed to follow speci®c route guidance instructions. Experimental
evidence presented by Chang and Mahmassani (1988) and Mahmassani (2001) suggests
that commuter route choice behavior exhibits a boundedly-rational character. This means
that drivers look for gains only outside a threshold, within which the results are satisfying
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 257

and suf®cing for them. This can be translated to the following route switching model
(Mahmassani and Jayakrishnan, 1991):
djk 1 if TTCjkTTBjk>maxZjTTCjk;tj
0 otherwise

where d
j
(k) is a binary indicator variable equal to 1 when user j switches from the current
path to the best alternate, and 0 if the current path is maintained; TTC
j
(k) and TTB
j
(k) are
the trip times along the current path and along the best path from node k to the destination
on current path, respectively; Z
j
is a relative indifference threshold, and t
j
is an absolute
minimum travel time improvement needed for a switch. The threshold level may re¯ect
perceptual factors, preferential indifference, or persistence and aversion to switching. The
quantity Z
j
governs users' responses to the supplied information and their propensity to
switch. The minimum improvement t
j
is currently taken to be identical across users
according to user-de®ned values. Further discussion on this issue can be found in
Mahmassani's (2001) paper in this special issue. Very little seems to have been done,
besides the above-mentioned work, in this important ®eld.
3.3.5. Path processing Path processing lies at the core of DTA development; routes need to
be computed for every modeled user class accounting for intersection movement delays,
general link cost functions, many modes of transportation, ®xed arrival times and other
network realities. Thework by Ziliaskopoulos and Mahmassani (1993, 1994, 1996) addresses
many of these issues, but given that in the existing operational models path processing
seems to be a key computational bottleneck, more needs to be done. Ziliaskopoulos et al.
(1997) proposed a parallel implementation that can potentially reduce the required
computational time, but it requires rather elaborate hardware settings which discourages
researchers from applying it. The evolution of the Internet and information technology
state-of-practice presents many opportunities for distribution and parallelism not available
only a few years ago. More research needs to be done to take advantage of these
technologies. For example, Beowulf clusters (Peeta et al., 2000b) represent an economical
and ¯exible high-performance computing environment in this context. Another area related
to path processing is that of computing routes when the link travel times are not only time-
dependent but also stochastic, which necessitates on-line shortest path approaches.
3.3.6. Tractable analytical approaches, FIFO violations and ef®cient decomposition
approaches While there are various analytical approaches and solution algorithms, as
discussed in the review section of this paper, there is no analytical formulation that can
ef®ciently solve actual size networks. In other words, there is no equivalent to the Frank-
Wolfe solution algorithm we have for static approaches that could conveniently decompose
on of the analytical approaches to some sort of a path processing algorithm. Waller (2000)
suggested a decomposition approach for the cell transmission based linear programming
formulation that solves min-cost ¯ow sub-problems on a modi®ed network, but the
approach is yet to be applied on actual size networks. In addition the analytical approaches
seem to still suffer from the limitations of single destination (unless the ®rst-in, ®rst-out
258 PEETA AND ZILIASKOPOULOS

condition on the links is violated), especially for user optimum behavior. For system
optimum solutions satisfying the FIFO condition may not be as critical, especially if they
simply aim to provide a lower bound on the system performance. Constraining for non-
FIFO conditions would at best provide a tighter bound which makes the solution
questionable. It is the belief of the authors that the FIFO violation for SO solutions is
the temporal equivalent of unreasonable paths produced by static approaches; the SO
approach should hold traf®c at various places in the network, since this provides a better
system optimum solution.
3.3.7. Departure time choice modeling or FAT/FDT According to Li (2001), a recent
travel survey in the Puget Sound region in the United States revealed that almost 70% of
the trips during the morning commute are work or school related trips that tend to ®x their
arrival rather than their departure time (see Table 1). This presents challenges for
researchers to develop models that account for this class of users, but also opportunities,
since arrival time based demand is arguably easier to estimate than departure time based
one. Accounting for ®xed arrival time based demand leads to the simultaneous route
choice and departure time assignment problem. The existing approaches in the literature
for this problem can be grouped into empirical approaches, which are based on
experimental or survey data, and analytical ones, which are typically limited to formula-
tions with no ef®cient solution algorithms.
Chang and Mahmassani (1988), Mahmassani and Herman (1984), Friesz et al. (1993),
De Palma et al. (1983), Hendrickson and Kocur (1981), and Friesz et al. (1994) dealt with
the simultaneous route and departure-time choice on urban networks. Equilibrium
conditions for simultaneous departure time and route were ®rst de®ned by Friesz et al.
(1993), and can be loosely stated as follows:
For any h (hkb
rs t:k2P, b2B), the pair (h,m)is designated as a simultaneous route-
departure time equilibrium, if and only if the following two conditions are satis®ed:
hkb
rs t>0)Tk
rsbt;hftTk
rsbt;h  mrsbh;all r;s;b:
Tk
rsbt;hftTk
rsbt;h  mrsbh;all r;s;b:
Where, hkb
rs tis the ¯ow on path kthat connects nodes rand sat time tand arrival time b
measured at the entrance of the ®rst link on the path, ftTk
rsbt;h is a cost function of
Tab le 1. Temporal distribution of work and school trips
Time of day
0:01±5:00 5:01±9:00 9:01±14:00 14:01±19:00 19:01±24:00
Work and school trips 49 2036 1115 493 55
Total trips 154 2998 4491 5986 1867
Percentage of work
and school trips 31.5% 67.9% 24.8% 8.2% 2.9%
FOUNDATIONS OF DYNAMIC TRAFFIC ASSIGNMENT 259

early or late schedule delay fb0and mrsbhminfTk
rsbt;hftTk
rsbt;hg
over all paths kin Pthat connect rand s. These conditions simply state that at equilibrium
a user cannot switch either departure time and/or path and reduce his/her personal gain.
Based on this de®nition Friesz et al. (1993) proposed a variational inequality formulation
and proved that it is equivalent to the de®nition stated above. Ziliaskopoulos and Rao
(1999) introduced a simulation based ®xed arrival based heuristic solution methodology
that approximates the above equilibrium condition. A similar heuristic solution aimed to
emulate day-to-day dynamics and individual drivers' decisions as discussed in Chang and
Mahmassani (1988).
Finally, Li et al. (1999) and Li (2001) introduced a linear programming (LP) formulation
for the simultaneous departure time and route choice problem under SO conditions for
networks with multiple origins and destinations. The LP is formulated using the cell
transmission model. The necessary and suf®cient conditions for SO DTA ®xed arrival time
were proved to be the following:
The necessary and suf®cient condition for SO FAT-DTA is that all used paths to a node
at a certain arrival time from an origin have cost equal to the marginal cost of an
additional unit of demand at that node and time interval from that origin, while all
unused paths have cost higher than or equal to the marginal cost.
Extensions to the model that account for penalties associated with schedule delays, and
for simultaneous loading of arrival time based and departure time based demand were also
proposed.
Despite the efforts outlined above, estimation of the departure time choices is not well
understood. It is potentially one of the main advantages of DTA over steady state models,
where departure time is meaningless. In closing this section, we venture to state that the
potential of DTA to account for this important decision variable makes it not only superior
to static approaches, but also necessary for planning applications. Ignoring the departure
time when the common sense and experimental data suggest that it is the main decision
variable most users are likely to consider adjusting would be unfortunate, especially since
time-dependent demand for this user class can potentially be estimated more easily than
for other user classes. If the arguably obsolete four planning steps were to be followed,
estimating departure time would have been the ®fth planning step.
4. Closing Comments
Dynamic traf®c assignment has evolved rapidly over the past two decades, fueled by the
needs of applications domains ranging from real-time traf®c operations to long-term
planning. Characterized by inherent mathematical intractability and challenging complex-
ities, it has nevertheless spawned a vast body of literature that encompasses a broad gamut
of problems with different underlying assumptions and functional objectives. While
focused research has led to rapid strides in the understanding of the problem character-
istics, it has also highlighted the dif®culties involved in developing a universally applicable
260 PEETA AND ZILIASKOPOULOS

approach for general networks in the operational domain. For example, a mathematically
tractable analytical model that is adequately sensitive to traf®c realism vis-a
Á-vis real-time
operations, is still elusive. Consequently, current research still focuses on fundamental
problem characteristics while venturing into deployment issues that are motivated by
current and future operational needs.
Researchers in the next few years will debate and research the issue of applicability of
DTA for planning applications. Early attempts were consumed with comparatively
evaluating static and dynamic models applied to the same network and total demand;
this is a futile effort, since the ground truth will never be known to actually compare the
outcome of the models with. Debating on whether dynamic models are better than static
ones is hardly the issue; dynamic models are obviously superior, since they relax more
assumptions and capture more realities than the static approaches. Dynamic models are
simply the natural evolution in the transportation ®eld that like any other new effort suffers
from early development shortcomings; it is only a matter of time before they are improved
and ultimately adopted by the industry. Academics and researchers should look ahead and
be concerned with some of the fundamental questions posed above, especially those
related to the realism of assumptions and mathematical tractability.
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