Convergence criteria and computational
efficiency of a dynamic traffic assignment model
Fabien Leurent (1), Vincent Aguiléra, Hai-Dang Mai
Laboratoire Ville, Mobilité, Transport
INRETS-ENPC-UMLV
19, rue Alfred Nobel, Cité Descartes, Champs-sur-Marne
F-77 455 MARNE-LA-VALLEE CEDEX 2
Abstract
In this paper, a high-level formulation of the dynamic traffic assignment problem is
used to analyse its specific complexities, most notably the chronological relativity of the
volume loading. This applies in turn to the time-flow relationship at the path level. This
induces significant discrepancies to static assignment, in particular as concerns the
convergence criterion to be used in an equilibration algorithm. Convex combination
algorithms, either arc-based or path-based, are analyzed in order to complement them
with rigorous convergence criteria. A novel “hybrid” algorithm is introduced. Lastly,
Leurent’s LADTA model is applied to a large size problem, demonstrating its
computational efficiency.
Keywords
Dynamic Traffic Assignment, equilibrium formulation, hybrid algorithm
1
Corresponding author. Email : fabien.leurent@enpc.fr. Tel : +33 164 152 111. Fax : +33 164 152 140
1. INTRODUCTION
This paper considers the problem of dynamic traffic assignment under the User
Equilibrium principle. This topic has been the subject of a considerable amount of
research during the last two decades, both in the transportation and operations research
communities. However, it is acknowledged by many researchers that the theory of
Dynamic Traffic Assignment (DTA) is still relatively under-developped, while
practitioners manifest a heightened interest for large scale applications, with both real
time and planning applications in mind. As a consequence, dynamic traffic assignment
tools have been given few applications, as compared to static traffic assignment models.
The work presented here — based on Leurent’s Lumpded Analytical Dynamic Traffic
Assignement model (LADTA) — is an attempt to bridge the gap between, on one side,
a sound analytical problem formulation and, on the other side, computational
tractabiblity and efficiency. The sequel is structured as follows.
Section 2 introduces some notations and presents the equilibration algorithm in
LADTA. Section 3 concentrates on some specific complexities of dynamic assignment,
most notably the chronological relativity of the volume loading function, which affects
in turn the time-flow relationship at the path level. This induces important discrepancies
to static assignment, in particular as concerns the convergence criterion: this issue is
discussed in Section 4. The design of a novel equilibration algorithm, called the hybrid
algorithm, is the topic of Section 5. In Section 6, results from computational
experiments are presented. Lastly, Section 7 concludes and indicates directions for
research development.
2. ON THE FORMULATION OF DYNAMIC ASSIGNMENT
Leurent (2003, 2004) provided a high-level formulation of dynamic traffic assignment
which we shall recall hereafter in order to prepare the analysis in the following sections.
The formulation is high-level as it makes explicit only the main traffic variables and
their mutual dependencies, which are indicated in an abstract way that leaves place for
flexible forms. It is generic in that it may describe most dynamic traffic assignment
models: Leurent also provided a particular low-level formulation called LADTA for
Lumped Analytical Dynamic Traffic Assignment.
We shall first give the notation (section 2.1). Then the high-level formulation is
provided (section 2.2) and it is applied to static assignment (section 2.3). Lastly, the
equilibration algorithm in LADTA is described (section 2.4).
2.1
Notation
Let us recall the elements of Leurent’s model (Leurent, 2003) in a simplified version in
which the following assumptions are made:
-
On the demand side, there is a single user class with path choice behaviour that
is homogeneous and deterministic. Each user is assumed to select the path with
minimum generalized cost.
-
On the supply side, loop-free paths are considered, and congestion is modelled
at the arc level on the basis of a vertical queue.
List of notation:
G  [ N , A]
the transport network with N the set of nodes indexed by n and A the set
of arcs indexed by a  (n, m) .
An , An
the subsets of the arcs that respectively come out of, or into, node n.
h
instant in a period H.
X a (h)
the cumulated flow that entered arc a until h.
X a ( )
the cumulated flow that exited arc a until  .
ta (h)
the actual travel time of arc a having entered it at h.
H a (h)  h  ta (h)
the time of exiting a knowing entry at instant h: this is a function
for “chronological transfer”.
H a ( )
the inverse function of H a : it maps an exit instant  into the
corresponding entry instant h.
S
the set of destination nodes indexed by s.
Os
the set of origin nodes o connected to destination s.
I   sS Os the set of origin-destination pairs i  (o, s ) .
Ri
the set of loop-free paths r on the O-D pair i.
Qi (h) the O-D volume of pair i cumulated until departure instant h.
X r (h)
the cumulated flow that entered path r until h.
X r ()
the cumulated flow that exited path r until  .
tr (h) the actual travel time of route r having entered it at h.
Hr (h)  h  tr (h)
the time of exiting r knowing entry at instant h.
H r () the inverse function of H r : it maps an exit instant  into the corresponding
entry instant h.
r  a for a path r and an arc a which belongs to r, designates the part of r that lies
strictly upstream of a.
r  a for a path r and an arc a which belongs to r, designates the part of r that lies
strictly downstream of a.
The model endogenous variables are the entry and exit volumes of the paths, X R and
X R , and of the arcs, X A and X A : these are called the primal variables; and also the
travel times, t R and t A , which we call dual variables since these times induce the user
costs.
2.2
A high-level formulation
Leurent (2003) decomposed dynamic assignment into four parts as follows:
Service Formation
tR  FS (t A )
(1a)
User Choice
X R  FU (t R )
(1b)
X A  FV ( X R , t R )
(1c)

t A  FF ( X A
)
(1d)
Volume loading
Flowing
where  is used in place of = when the mapping is multi-valued.

A state ( X R , X A
, t A , t R ) which satisfies the system of equations (1) is a dynamic traffic
equilibrium. This abstract formulation is generic for many analytical dynamic traffic
assignment models, including those of Akamatsu and Kuwahara (1998), Tong and
Wong (2000); Friesz et al (2005) etc.
2.3
Application to static assignment
The generic formulation also applies to static assignment, provided that we omit the
temporal variations and that we replace cumulated flows X by flow rates x: then the
service formation problem amounts to building paths on the basis of arcs, the user
choice problem amounts to assigning O-D flows on optimal paths only, the volume
loading problem amounts to loading path flows on the network arcs yielding arc flows,
and the flowing problem amounts to evaluating the arc travel time functions with
respect to the arc flows.
2.4
The LADTA equilibration algorithm
The original approach to equilibrium in the LADTA model, proposed independently by
Leurent (2003) and Gentile et al (2003), is to cast the equilibrium conditions into a
fixed-point problem with respect to the arc cumulated flows X A , which are the basic
endogenous variables. The fixed-point problem is obtained by linking the other
endogenous variables to the basic endogenous variables, X A :

t A  FF ( X A
)
(2a)
t R  FS  FF ( X A )
(2b)
X R  FU  FS  FF ( X A )
(2c)
X A  FV (FU (),)  FS  FF ( X A )
(2d)
The last condition is a fixed-point condition for the following mapping:
FX  FV (FU (),)  FS  FF
(3)
Then the equilibrium is searched for using a method of convex combination, which is
very parsimonious as it only requires storing arc variables.
The convergence criterion is an inter-iteration gap pertaining to the arc volumes:
X A ( k 1)  X A ( k )
2

2
1
  X a(k 1) (h)  X a( k ) (h) dh
A aA
We shall establish more rigorous convergence criteria in Section 5.
3. THE SPECIFIC COMPLEXITIES OF DYNAMIC ASSIGNMENT
Let us now use the high-level formulation so as to detect and characterize the specific
complexities of dynamic assignment: not only does the temporal dimension drive us to
consider temporal profiles as model variables, but more importantly it makes the
loading of path flows onto the arcs depend on a time field, in other words a chronology.
We call this effect “the chronological relativity” of the volume loading (section 3.1).
Then the time-flow relationship is relative to the chronology, too (section 3.2): path
times stem from both path flows and the time field, and the circumstance under which
the path times correspond to the time field is a special one, which we call the “time-flow
consistency” (section 3.3). We also study the dual relationship from time to flow to
obtain weaker conditions of “dual consistency” (section 3.4). Lastly, we define a
consistency property that pertains to “flow energies” (total cost), for which we provide
some sufficient conditions (section 3.5).
3.1
The chronological relativity of volume loading
In static assignment, arc flows are derived from path flows in a straightforward way:
xa  ra xr
(4)
In dynamic assignment, the arc volumes are derived from the path flows through the
volume loading function, X A  FV ( X R , t R ) .
More precisely, Leurent (2003) stated the volume loading equation as follows:
X a
h

  X r  H r a
ra
h

It means that the volume that enters a from instant  to instant h is the sum of the
volumes of the paths r which traverse a: and that the volume of path r that enters a
during [, h] correspond to the departure interval [h, h ] such that h  tr a (h)   and
h  tr  a (h )  h : hence h  H r a () and h  H r  a (h) .
Thus the functions t R are involved in the volume loading problem through the
associated time transfer functions, H R , and their inverse functions, H R .
We call this effect the “chronological relativity of volume loading”, as the time transfer
functions constitute a chronological field.
3.2
The chronological relativity of the time-flow relationship
In static assignment the time-flow relationship at the arc level has a simple form:
denoting by t a the travel time function,
ta  t a ( xa )
At the path level, the relationship of time to flow is still quite simple:
tr  ar ta where ta  t a ( xa ) and xa   ra xr
In dynamic assignment the relationship of time to flow may also be considered as a
simple one, by making use of a local flowing model: the arc travel times [ta (h) : h  H ]
(5)
stem from the arc flows [ X a (h) : h  H ] which are constrained by the local flowing
conditions (including the minimal travel time ta 0 (h) and the capacity flow rate a (h)
in the LADTA model). Concerning the paths however, the relationship of time to flow
takes on the following form:
t R  FS (t A )
 FS  FF ( X A )
(6)
 FS  FF  FV ( X R , ~
tR )
which
shows
that
the
relationship
is
conditional
on
the
chronology
~
~
H R  [h  tr (h) : h  H ]rR that influences the volume loading.
3.3
On the time-flow consistency
Let us define the “time-flow consistency” as the condition of identity between t R and
~
tR in Eq. (6):
t R  FS  FF  FV ( X R , t R )
(7)
This is a fixed-point condition on t R (which is parameterized by X R ).
The consistency is a necessary condition for dynamic traffic equilibrium. It is not a
sufficient condition, since it indicates nothing about the user choice.
Under time-flow consistency, the convergence of a traffic state to equilibrium can be
evaluated on the basis of the distance from X R to FU (t R ) , which can be measured
using the following criterion of complementary slackness:
 hH [tr (h)  ui (r ) (h)] d X r (h)
rR
where ui (h)  min i t (h) .
(8)

A consistent pair ( X R , t R ) and the related pair ( X A
, t A ) satisfy the consistency of
energies, namely that:
 hH tr (h) d X r (h)   hH ta (h) d X a (h)
rR
aA
This is because X A stems from X R according to the chronology associated to t R ,
which is derived from t A : then the sufficient conditions of §.5 are satisfied.
To impose the time-flow consistency, the “natural” approach is to assign the path flows
onto the network in a step-by-step, instantaneous way as in performing a time
integration and solving for a differential equation; this yields also X A and t A . Tong
and Wong (2000) implemented that treatment, which amounts to integrate the volume
loading and the flowing problems, by using a mesoscopic simulation.
3.4
Dual time-flow consistency
Using the arc times t A as the basic endogenous variables, the model cycle is stated as:
tR  FS (t A )
(9a)
YR  FU  FS (t A )
(9b)
YA  FV (YR , t R )
(9c)
~
t A  FF (YA )
(9d)
As t R stems from t A , and YA is derived from YR under the chronology associated to
t R , the consistency of energies holds:
 hH tr (h) dYr (h)   hH ta (h) dYa (h)
rR
aA
(10)
When the YR are obtained by an assignment to paths with minimal time, (10) may be
used to evaluate the overall minimal cost: by definition of YR , the O-D minimal cost
ui (h)  min ri tr (h) satisfies that ui (h) dQi (h)  ri tr (h) dYr (h) , hence
 hH ui (h) dQi (h)   hH tr (h) dYr (h)
iI
rR
  hH ta (h) dYa (h)
(11)
aA
3.5
The consistency of energies
In static assignment, the “energetic power” of total cost takes on the same value at the
arc and route levels, provided that tr  ar ta and xa   ra xr :
 tr . xr   ta . xa
rR
aA
A similar result holds in the dynamic case under the two following conditions: first, that
t R is derived from t A (through the service formation) and second, that X A stems from
X R under the chronology induced by t R . Then tr (h)   ar ta ( H r  a (h)) and
h


h
h tr (h) d X r (h)  ar h ta ( H r a (h)) d X r (h)
 ar H r  a( h) ta (h) d X r ( H r a (h))
H
(h )
r a
Summing over the network paths, we obtain that
 hH tr (h) d X r (h)   hH ta (h) [  d X r ( H r a (h))]
rR
aA
 
aA
since X A  FV ( X R , t R ) .
ra

hH ta (h) d X a (h)
(12)
4. CONVERGENCE CRITERIA FOR PATH-BASED MODELS
Out of the numerous papers dealing with dynamic assignment and equilibrium, none has
addressed explicitly the design of a convergence criterion that would rigorously indicate
whether a traffic state is in equilibrium or not. In most papers, the convergence criterion
is either the variation in arc flows from one iteration to the next (in an intuitive approach
that may conduct to miss the equilibrium or to underestimate the gap to it), or the
complementary slackness, which has more theoretical ground. However the
complementary slackness alone is not a rigorous convergence criterion in the context of
dynamic assignment, unless the time-flow consistency is achieved.
We shall first state a rigorous overall convergence criterion in an abstract way (section
4.1). Then we show how it may be reduced to complementary slackness if the solution
algorithm yields time-flow consistency, as in the model of Tong and Wong (2001)
(section 4.2). In the absence of consistency, an additional term must be included
(section 4.3). Lastly, to avoid the enumeration of paths in the evaluation of the
complementary slackness, we design an auxiliary complementary slackness that may be
evaluated economically (section 4.4).
4.1
An overall convergence criterion
As an equilibrium state is characterized as the solution to the system expressed in Eq.
(1) which is composed of four conditions, the distance to equilibrium of a given traffic
state may be evaluated as the sum of four terms that capture the violation of each
condition in Eq. (1):

d S [t R , FS (t A )]  dU [ X R , FU (t R )]  dV [ X A , FV ( X R , t R )]  d F [t A , FF ( X A
)]
(13)
where the notation “small d” means some measure of distance, and the index taken in
set {S, U, V, F} indicates the equilibrium condition which is addressed.
This makes a rigorous convergence criterion. In an algorithmic scheme, some
components would fall down because of definitional linkages between such or such
endogenous variable. For instance, if t R  FS (t A ) then d S [t R , FS (t A )]  0 .
4.2
Complementary slackness under time-flow consistency
Time-flow consistency implies that


d S [t R , FS (t A )]  dV [ X A
, FV ( X R , t R )]  d F [t A , FF ( X A
)]  0
since each of the three terms is null.
Then the overall criterion is reduced to dU [ X R , FU (t R )] , which may be evaluated by
the complementary slackness criterion.
4.3
In the absence of time-flow consistency
When time-flow consistency does not hold, the terms in the overall criterion are likely
to intervene in the following way. Firstly, it seems easy to zero d S [t R , FS (t A )] because
the service formation is a relatively simple problem. However some caution is in order
because, as convergence is measured at a given stage in the algorithm, the value of t A
may have changed since t R was evaluated, yielding a discrepancy between t R and
FS (t A ) .
Secondly, the term dU [ X R , FU (t R )] is difficult to zero. In a convex combination
method the auxiliary path flow vector YR satisfies that dU [YR , FU (t R )]  0 but this
does not apply to the current path flow vector unless equilibrium is reached. The term is
costly to evaluate because it involves path variables. It may be stated as the
complementary slackness criterion, which yields no significant computational saving
unless the consistency of energies is satisfied, in which case the criterion can be
evaluated using only arc variables.
Thirdly, the term dV [ X A , FV ( X R , t R )] has to be evaluated if the path flow vector X A
does not come from the volume loading function FV ( X R , t R ) . The volume loading is
costly to perform, not the evaluation of dV between two arc flow vectors.
Lastly, the term d F [t A , FF ( X A )] is relatively easy to evaluate since it amounts to
compare two arc-based vectors.
4.4
An auxiliary complementary slackness criterion
In an equilibration algorithm that takes the arc flows X A as the basic endogenous
variables, an issue arises of the existence of path flows X R and path times t R that
generate the arc flows, i.e.
X A  FV ( X R , t R )
If this condition could not be met, there would be a “time discrepancy” of the arc flows
that would preclude not only equilibrium but also the primal feasibility!
Let us show that, in equilibration algorithms that handle arc flows by convex
combination, there is no risk of time discrepancy. We may assume that there are flows
by arcs and destinations, X AS  , underlying X A since in the total arc flow vector the
destination/arc flow vectors are summed up. This decomposition is left unchanged by
convex combination. For each destination/arc flow vector, the primal feasibility
condition is that, denoting by dX the elemental variation of X during [h, h  dh[ ,
d X as   0 , s  S , a  A, h  H
(14a)
d X as  (h)  d X as  (h   as (h)) , s  S , a  A, h  H
(14b)
 a An d X as  (h)  dQns (h)   a An d X as  (h) , s  S , n  N , h  H
(14c)
In Eq. (14) an arc time profile  as is associated to each arc a, by destination node s: let
us call it a chronological gap between the entry and exit times for a given flow unit.
Then, if X as  and X as  are known, a chronological gap  as may be obtained by
application of the following procedure:
1. Let X as [ 1] ( z )  inf { h : X as  (h)  z } .
2. Let H as (h)  X as [1]  X as (h) .
3. Let  as (h)  H as (h)  h .
Let us call this procedure the “method of chronological inference”.
We may use the set of auxiliary time mappings ( sA ) sS in order to evaluate an
auxiliary complementary slackness criterion: rather than measuring dU [ X Rs  , FUs (t R )]
which requires to store X Rs  , we can evaluate
dU [ X Rs  , FUs ( sR )]  d[t A ,  sA ]
(15)
because if d[t A ,  sA ]  0 then  sA  t A and dU [ X Rs  , FUs ( sR )]  dU [ X Rs  , FUs (t R )] .
Let us now consider  sR  FS ( sA ) , Z Rs   FU ( sR ) and Z As   FV (Z Rs  ,  sR ) . As
X As   FV ( X Rs  ,  sR ) , the term dU [ X Rs  , FUs ( sR )] can be evaluated as
 sR .( X Rs  Z Rs )   sA .( X As  Z As )
which shows that the auxiliary complementary slackness is easy to evaluate.
Then an overall criterion is as follows:
  sA .( X As  Z As )  d[ sA , FF ( X A )]
(16)
sS
5. EQUILIBRATION ALGORITHMS
Here we shall consider primal algorithms that handle flow vectors as basic endogenous
variables. More specifically, as in most models of dynamic assignment, we shall
consider convex combination methods: firstly at the arc level as in the LADTA model
(section 5.1), secondly at the path level with separation of volume loading and flowing
(section 5.2), thirdly at the path level with integration of volume loading and flowing as
in the model by Tong and Wong (section 5.3). Lastly we shall introduce a novel
algorithm called the “hybrid algorithm” that combines the chronological gap vectors
 sA to the travel time vector t A  FF ( X A ) that results from the time-flow relationship
into the current time vector  A (section 5.4).
5.1
Arc-based convex combination method
The algorithm is stated shortly as:
(k 1)
( k )
( k )
XA
 (1  k ) X A
 k FV (FU (),)  FS  FF ( X A
)
(17)
This algorithm is simple to implement and it requires limited memory space since it
does not require storing path information.
To the best of our knowledge, criterion expressed by Eq. (16) is the simplest rigorous
convergence criterion for that algorithm. Its implementation requires the additional
computational effort of inferring the chronological gap, computing Z R and assigning it
onto the network, yielding arc flows Z A . This doubles the base computation cost of the
algorithm; a strategy to limit the increase is to measure convergence only at one out of
N iterations.
5.2
Path-based convex combination under time-flow consistency
This algorithm is stated shortly as:
t R( k )  FS  FF  FV ( X R ( k ) , t R( k 1) )
X R ( k 1)  (1   k ) X R ( k )   k FU (t R( k ) )
(18)
In general the arc flows obtained using this method, denoted as X A / R , differ from
those obtained by the arc-based convex combination, X A / A , even if the coefficients
 k are identical. This is in contrast to static assignment.
The overall criterion in Eq. (13) simplifies into
( k )
dU [ X R(k ) , FU (t R(k ) )]  dV [ X A
, FV ( X R(k ) , t R(k ) )]
( k )
As X A
 FV ( X R(k ) , t R(k 1) ) , the second term can be replaced by a distance between
)
t R(k 1) and t R(k ) , hence also by a distance between t (Ak 1) and t (k
A , yielding a simpler
overall criterion:
dU [ X R(k ) , FU (t R(k ) )]  d[t (Ak 1) , t (Ak ) ]
(19)
As shown in Mai (2006), the first term may be evaluated as follows, using auxiliary
(k 1)
( k )
flow YA(k )  FV ( X R(k ) , t R( k 1) ) and next state X A
:
  k YA(k )  (1   k ) X A
Z
1
 hH t a(k ) (h)[d X a (k 1) (h)  dYa (k ) (h)]
1   k a A
(20)
5.3
Path-based convex combination relaxing time-flow consistency
Denoting by FFV : X R  t A the function that operates volume loading and flowing in
an integrated way, this algorithm is stated shortly as:
X R(k 1)  (1   k ) X R(k )   k FU  FS  FFV ( X R(k ) )
(21)
This method, which was proposed by Tong and Wong (2000), has two attractive
features: first, its theoretical convergence towards equilibrium ought to be robust;
second, the complementary slackness criterion is rigorous on its own right.
But the computational cost is very heavy because of the mesoscopic simulation to
perform FFV ; moreover, the accuracy of FFV is no trivial matter, which could imped
convergence.
5.4
A “hybrid” algorithm
Using the same notation k as iteration counter and  k as coefficient in the convex
combination, our hybrid algorithm is specified as follows:
Initialisation. Let
k  0 , t A  [t a 0 (h)]hH , a A ,  A  t A ,  A  t A ,
X A  0 ,
X A  0 .
Service formation. Let
 R  FS ( A ) .

User choice. Let YR  FU ( R ) .


Volume loading. Let Y A  FV (YR ,  R ) and
YA  YA  ( Id A   A )[1] .






Convex combination. Let W A  (1   k ) X A   k Y A and W A  (1   k ) X A   k Y A .
Complementary
slackness.
For
each
destination
s
in
S,
let
 sR  FS ( sA ) ,
Z Rs   FUs ( sR ) , Z As   FV ( Z Rs  ,  sR ) and CSs   sA .( X As   Z As  ) . Then, let
CS  sS CSs .
d[ A , t A ]  sS CSs  d[ A ,  sA ]   then terminate, otherwise
Convergence test. If
replace
X As  WAs , X As  WAs for each destination s in S.
Traffic flowing. Let t A
 FF ( X A ) .
Chronological inference. Derive
Time update. Set
 sA from X As  and X As  for each destination s in S.
 A on the basis of t A , ( sA ) sS and eventually some previous values.
Progression. Increment
k  k  1 and go to Service formation.
The time update may be a convex combination (not to be confused with that on flows):
 A  (1  )t A  .
1
.   sA
S sS
6. A LARGE-SIZE EXPERIMENT
The main objective of this experiment is to asses if our implementation of the LADTA
model can be of practical use for real world networks. To this end, we first selected a
large interurban road network on which congestion frequently occurs. More detailed
motivations concerning the choice of the network are given in section 6.1. Second, we
collected the data (section 6.2) and processed them to provide appropriate inputs to the
simulator (section 6.3). Finally, we present some results related to the evolution of the
computation time per iteration (section 6.4).
6.1
Motivations
The Vallée du Rhône (VDR) area, located between Lyon and Marseille, is of main
concern for the French DOT, not only because some wine is produced there, but also
because a significant part of the trans-european road traffic Europe concentrates on it.
This is particularly noticeable in summer time, when tourists coming from northern
Europe, including Belgium, Germany, Netherlands and the U.K. travel accross France
to reach southern countries like Greece, Italy and Spain. When concerned with their
choice of itinerary, travellers certainly do not only concentrate on wine tasting
opportunities on the way. A quick glance at the structure of the demand (see Fig. 1)
explains why a large part of the traffic concentrates on the A7 highway, in the VDR
area between Lyon and Orange. Also, the VDR road network supports a very high load
of traffic during the whole year, due to goods transportation by trucks. Moreover, traffic
forecasts do show an increase of demand for the coming 20 years, for both passenger
cars and trucks.
Besides high cost measures (i.e. investments in infrastucture), road authorities and
highway operators are highly interested by fine-grain (i.e. within an hour) and dynamic
(i.e. traffic responsive) time varying exploitation measures to better operate the network.
A dynamic traffic assignment tool can be of great help in the design and implementation
of dynamic traffic control measures, both at the conception stage (i.e. cost-benefit
assesment, design of operating rules, ...) and at the operation stage, as a decision-aid
tool. This last use is certainly the more demanding. It implies to be able to compute and
keep up to date several scenarios, within a few minutes, in order to provide the roadoperator with accurate traffic forecasts for the coming hours.
6.2
Data acquisition
Two services of the French DOT, SETRA and CETE Méditerrannée, provided us with
the following data :
-
a geocoded model of the road network, comprising 732 (undirected) arcs and
595 nodes, including connectors;
-
for each arc a , its capacity qa , its length la , together with its measured average
free flow speed, va ; among the 732 arcs, 471 have a capacity of 1800 veh/h, 26
a capacity of 2600 veh/h, 145 a capacity of 3600 veh/h (corresponding to 2-lanes
highway sections), 81 a capacity of 5400 veh/h (corresponding to 3-lanes
highway sections). 7 connectors (to Belgium, Germany, the U.K, Spain, Italy,
etc) have an infinite capacity.
-
a static O-D matrix, representing the average annual daily demand for 1137 O-D
pairs, in year 2000, for both passenger cars and trucks, including demand from
connectors.
6.3
Data generation
For the purpose of our experiment, those data have been used to generate the
appropriate inputs to the LADTA implementation, following the three steps below :
Step 1: generation of directed arcs. For each undirected arc a linking the set of nodes
m, n , two directed arcs
a1  (m, n) and a2  (n, m) have been created.
Step 2: generation of arc data. For each undirected arc a , the free flow travel time and
capacity profiles of the associated directed arcs a1 and a2 have been set up as follows:
ta1 ,0 (h)  ta2 ,0 (h) 
ta
va
K a1 (h)  K a2 (h)  qa  h .
Step 3: generation of demand. For each O-D pair i , a cumulated volume profile X i (h)
has been generated. The annual average daily demand Qi has been distributed on 24
consecutive periods of one hour each, according to the following equations
dX i
(h)  wk ( h )Qi , with k (h)  k for k 1  h  k and
dt
24
w
k 1
k
 1,8
The sum of the wk coefficients is set to 1,8 in order to simulate a summer saturday day
in year 2020. The distribution of the wk coefficients during the day has been choosen to
simulate a morning peak and an evening peak within the simulated day. The shape of
the corresponding flow functions is illustrated by Fig. 2. Note that wk coefficients do
not depend on the O-D pair i , i.e. we assume that the distribution of the demand within
the day is the same for each O-D. This is certainly a very bad approximation of the
reality, but does not conflict with our main objective.
6.4
Results
The Convex combination operation at the core of the hybrid algorithm presented in
section 5.4 is a classical Method of Successive Averages. When applied on scalar
values, as it is the case for static assignment, there is nothing special to mention. But in
the case of dynamic assignment, the values we are dealing with are (piece-wise linear)
functions of time. When computing W A  (1   k ) X A   k Y A , the number of pieces
in WA , denoted as #(WA ) , is in the range  max{#( X A ), #(YA )}; #( X A )  #(YA )  . At the
end of the main loop, X A is assigned the value of WA . In other words, #( X A ) increases
with the number of iterations, in a way which is hard to predict and control. This remark
has two bad consequences in practice :

the computation time par iteration increases with the number or iterations;

the memory needed to store the variables increases with the number of
iterations.
This contrasts heavily with the static assignment problem, and typically results from the
time dimension of the dynamic assignment problem.
To better figure out the computational behavior of our implementation, we recorded the
time per iteration and the memory usage during a simulation of the VDR network. The
simulation was run an a laptop computer equipped with a 1.6 GHz Intel Pentium M
processor, 512MB memory, and running under Windows XP. The code, written in C++,
was compiled with g++, with all stantard optimisations turned on. Results are presented
in Fig. 3 and Fig. 4.
Fig. 3 shows the computation time taken from iteration 1 to iteration 4. As expected, the
time per iteration increases with the number of iterations, but in a reasonable way,
growing slowly from 6 seconds to 8 seconds.
In Fig. 4 is plotted the memory occupancy during the simulation. It grows significantly,
starting from 50 Mb up to nearly 350 Mb at iteration 40.
7. CONCLUSION
Using a high-level formulation of dynamic traffic assignment, we analyzed its specific
complexities, most notably:
-
The chronological relativity of the volume loading.
-
The chronological relativity of the time-flow relationship.
-
That the time-flow relationship at the path level does not imply a time-flow
consistency.
-
That the time-flow consistency i.e. t R  FS  FF  FV ( X R , t R ) is a necessary
condition for equilibrium.
We established some related properties of consistency.
As concerns the measurement of convergence, we showed that:
-
the chronological relativity impedes the computation of the complementary
slackness criterion.
-
An auxiliary complementary criterion is available, which is much easier to
evaluate when the volume loading and traffic flowing problems are addressed
separately.
-
There is a sound approach to design a rigorous convergence criterion, which will
be composed of several components except when the volume loading and traffic
flowing problems are integrated.
The algorithmic approach of Tong and Wong (2000) deserves emphasis as it integrates
those two problems. We analysed it as well as other convex combination algorithms,
either arc-based or path-based, in order to complement them with rigorous convergence
criteria. We introduced a novel “hybrid” algorithm. Lastly, we reported on numerical
application of the LADTA model to a large size problem, demonstrating its
computational efficiency. Further developments are reported in our technical report
(Leurent et al, 2006), in which path-based as well as arc-based formulations are
introduced, with identification of more time variables, and introduction of various
algorithmic frameworks that may be used to design novel algorithms for dynamic
assignment.
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Figure 1: Structure of the traffic demand on the VDR area during summer time.
The VDR area
Figure 2: Values of the wk coefficients used to generate the demand profiles.
0.12
0.1
wk
0.08
0.06
0.04
0.02
0
1
3
5
7
9
11
13
h
15
17
19
21
23
Figure 3 : Evolution of the time per iteration during the simulation.
9
8
7
Temps (s)
6
5
4
3
2
1
0
0
10
20
30
Itération
40
50
Figure 4: Evolution of the memory occupancy during the simulation.
350
Mémoire (Mo)
300
250
200
150
100
50
0
0
10
20
30
Itération
40
50