Generalization of Beckmann’s Transformation for Traffic Assignment Models with Asymmetric Cost Functions

Abstract

An optimization model is developed to solve the deterministic traffic assignment problem under congested transport networks with cost functions that have an asymmetric Jacobian. The proposed formulation is a generalization of Beckmann’s transformation that can incorporate network links with multivariate vector cost functions to capture the asymmetric interactions between the flows and costs of the different links. The objective function is built around a line integral that generalizes the simple definite integral in Beckmann’s transformation and is parameterised to ensure the solution of the new problem satisfies Wardrop’s first principle of network equilibrium. It is shown that this method is equivalent to the variational inequality approach. Our new approach could be extended to supply-demand equilibria models in other markets than transportation, with complementary or substitute goods/services in which there are asymmetric interactions between prices.

Full text

Research Article
Generalization of Beckmann’s Transformation for Traffic
Assignment Models with Asymmetric Cost Functions
Matthieu Marechal
1
and Louis de Grange
2
1
Institute of Basic Sciences, Faculty of Engineering, Diego Portales University, Santiago, Chile
2
School of Industrial Engineering, Diego Portales University, Santiago, Chile
Correspondence should be addressed to Matthieu Marechal; matthieu.marechal@udp.cl
Received 29 September 2023; Revised 31 May 2024; Accepted 24 June 2024
Academic Editor: Tomio Miwa
Copyright ©2024 Matthieu Marechal and Louis de Grange. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
An optimization model is developed to solve the deterministic trac assignment problem under congested transport networks
with cost functions that have an asymmetric Jacobian. e proposed formulation is a generalization of Beckmann’s trans-
formation that can incorporate network links with multivariate vector cost functions to capture the asymmetric interactions
between the ows and costs of the dierent links. e objective function is built around a line integral that generalizes the simple
denite integral in Beckmann’s transformation and is parameterised to ensure the solution of the new problem satises Wardrop’s
rst principle of network equilibrium. It is shown that this method is equivalent to the variational inequality approach. Our new
approach could be extended to supply-demand equilibria models in other markets than transportation, with complementary or
substitute goods/services in which there are asymmetric interactions between prices.
1. Introduction
is article develops a nonlinear optimization model for
solving the deterministic trac assignment problem under
congested transport networks that incorporates the general
case of network links with asymmetric multivariate cost
functions (i.e., the cost vector has an asymmetric Jacobian
matrix). e formulation we propose is thus a generalization
of the classic Beckmann transformation [1] in which the
objective function is based on a line integral instead of
a simple denite integral. e constraints, however, remain
exactly the same.
To solve our proposed optimization problem, we devise
a parameterisation that ensures the solution satises War-
drop’s rst principle of trac network equilibrium [2]. is
is proved by a new theorem. We also show that with this
parameterisation, the problem’s optimality conditions are
equivalent to the typical variational inequalities formulated
for solving trac assignment problems with asymmetric
interactions. Traditionally, these problems have been
addressed either by solving the variational inequality or
applying some alternative approach such as xed-point
methods, diagonalization, decomposition (partitionable,
transfer, simplicial, cobweb, etc.), relaxation methods, or
projection methods.
Our article corresponds to a theoretical and methodo-
logical development. We generalize the formulation of the
trac equilibrium problem to the case with asymmetric
interactions, which can eventually be extended or adapted to
other market equilibria other than transportation; for ex-
ample, supply-demand balances with complementary or
substitute goods/services in which there are asymmetric
interactions between prices. at is, when the demand for
a certain good or service depends not only on its own price
but may also depend on the price of substitute and com-
plementary goods. is last aspect is relevant both to esti-
mate market equilibria (e.g., Nash equilibria) and to obtain
changes in consumer surplus (e.g., social evaluation of
projects).
e main contribution of this article is of a theoretical
nature since it allows us to consolidate, under the same
theoretical framework, the approach based on the equivalent
Wiley
Journal of Advanced Transportation
Volume 2024, Article ID 2921485, 13 pages
https://doi.org/10.1155/2024/2921485

optimization problem for trac assignment models without
ow interactions in their costs (e.g., Beckmann transform),
with the general case in which the trac assignment model
(with asymmetric interactions) is formulated through
a variational inequality. at is, our contribution is to
generalize an equivalent deterministic assignment optimi-
zation problem to obtain trac equilibrium including
multivariate cost functions with asymmetric interactions.
Our new approach can eventually be extended to model
supply-demand equilibria in markets other than trans-
portation, with complementary or substitute goods/services
in which there are asymmetric interactions between prices.
In urban areas, it has been estimated that about 40% of
travel delays are attributed to intersections [3]. Some cases
when the cost on a link depends, besides on its own ow, on
the ow on other links (normally in an asymmetric way), are
the following [4]:
(i) Highway on-ramps (Figure 1): as the ow on the
highway increases, the ow that enters through the
on-ramp takes a longer time to enter the highway.
Note that in this case, the ow on the highway
aects the ow on the on-ramp but not the other
way around, and thus, the relationship is
asymmetric.
(ii) Priority intersections (Figure 2): at intersections in
which there are yield or stop signs, there is also
a cost asymmetry. Note that in this case, the ow on
the priority access aects the ow on the secondary
access; thus, the relationship is asymmetric.
(iii) Two-way streets (Figure 3): in two-way streets, and
particularly in cases with one lane in each direction,
the overpassing maneuver is more dicult. In this
case, it is also feasible that the interactions be
asymmetric.
Considering the limitation of the trac assignment
problem raised by Beckmann et al. [1], which does not allow
considering vehicular ow interactions, some developments
emerged that sought to include some types of interactions. A
particular case is observed in She [5] when the Jacobian of
the cost vector is symmetric (i.e., the vector eld dened by
the cost vector in the links of the network is conservative),
and it can be solved directly since it has a problem equivalent
optimization.
e remainder of this article is divided into three sec-
tions. In Section 2, we present a review of the literature
focused on equilibrium models for transport networks with
asymmetric interactions. Section 3 introduces the formu-
lation of a generalized trac assignment model with mul-
tivariate cost functions having an asymmetric Jacobian as an
optimization problem with an objective function containing
a line integral. Also explained in this section is the
parameterisation (or integration path) for transforming the
line integral into a summation of denite integrals from
which a direct solution of the problem is obtained that is
consistent with Wardrop’s rst principle. Section 4 presents
a numerical example of the proposed model, and Section 5
wraps up with a brief summary of our main conclusions.
2. Literature Review
Perhaps the rst article that formalizes the need to model
asymmetric intersections in transportation network models
was that of Prager [6], in which a formulation of the network
equilibrium problem, which includes the travel time integral,
is conceptually described as a linear function of opposing
ows on a two-way highway. Motivated by the behavioral
criteria presented by Wardrop [2], Prager [6] describes the
equality of travel times of routes as a function of potential
dierences at the nodes of the networks. In this work, the
need to model unbalanced interactions between ows of
opposite directions on a two-way street is formalized for the
rst time. Subsequently, several studies investigated asym-
metric modeling under the trac allocation framework.
Before it was announced that the formulation of a var-
iational inequality (VI) was suitable for modeling costs in
Figure 1: Highway on-ramps.
Figure 2: Priority intersections.
Figure 3: Two-way streets.
2Journal of Advanced Transportation

links with asymmetric interactions within transportation
network models [7–9], Netter and Sender [10] identied the
existence of the equilibrium solution in the transport net-
work problem with multiple classes using the xed
point model.
e general formulation of VI for modeling asymmetric
intersections in transportation network models motivated
dierent researchers to examine the application of existing
solution techniques designed for conventional trac as-
signment problems and investigate the convergence con-
ditions for the asymmetric case.
In practice, the most general solution approach to address
the computational implementation of VI-based trac as-
signment models with asymmetric interactions is to sym-
metrize the asymmetric interactions and solve the resulting
separable subproblems with conventional solution algorithms
used for general trac problems, trac allocation [11].
Among general solution algorithms, the Jacobi method
(also known as diagonalization method) is perhaps the most
popular. An investigation on convergence using the di-
agonalization method for the trac assignment problem
with asymmetric interactions based on VI can be found in
Beckmann et al. [12]. Gabriel and Bernstein [13] solved the
asymmetric intersections model for transportation networks
associated with the nonadditive route cost, using the column
generation method. Lo and Chen [14] also applied column
generation, in which the route-specic toll is modeled.
Other classic works that analyze the formulation of the
equilibrium problem, the uniqueness of the solution and the
solution algorithm are Dafermos [9, 15], Florian and Spiess
[16], Fisk and Nguyen [17], Fisk and Boyce [18], Nagurney
[19], Hammond [20], Nguyen and Dupuis [21], Marcotte
and Guelatt [22], and Auchmuty [23]. More recent analyses,
focusing for the most part on the design, implementation,
and comparison of solution algorithms for the trac as-
signment problem, including cases with asymmetric in-
teractions, are found in Chen et al. [24], Panicucci et al. [25],
and Sancho et al. [26].
A recent article that reviews the main advances in the
eld of trac assignment models corresponds to that of
Elimadi et al. [27]. e authors present the classic ap-
proaches with user balance and stochastic balance and also
review dynamic assignment approaches. However, they do
not include in their analysis the problems with asymmetric
interactions, which represent a more realistic (and more
complex) approach to trac assignment models. In the
article presented by Ashfaq et al. [28], they compare results
from simulation-based dynamic assignment models based
on route choice, which however represent a dierent ap-
proach than the one we address in this work. ese both
articles conrm that the study of mathematical models of
trac assignment with asymmetric interactions has not been
the priority among transportation researchers.
Deterministic approaches with asymmetric interactions
that include both the trac assignment problem and the
choice of combined modes are reviewed in the work of Fan
et al. [29]. e authors develop a general xed-point model
that combines mode choice under a cross-nested logit
framework and trac assignment based on variational
inequality (used for asymmetric interactions) to formulate
a more realistic problem. e numerical results show the
eectiveness of the proposed model and algorithm. Fan et al.
[29] highlight that few works consider the problem of
asymmetric trac assignment and that it is an aspect that
cannot be ignored in practice, especially when considering
interactions between road links or between dierent travel
modes. e usual approach developed in the literature
corresponds to VI.
In De Grange and Muñoz [4], the authors proposed the
idea of parameterising a line integral in the objective
function for a problem similar to Beckmann’s trans-
formation in order to tackle the asymmetric multivariate
cost function case. In that paper, however, authors did not
generalize the result, limiting to particular examples. What
was only a research proposal in that work will now be
demonstrated here, in the following section.
3. The Model, Its Parametrization, and
Optimality Conditions
Consider the following general formulation of the de-
terministic trac assignment problem for a transport net-
work (we summarize the notation used in Table 1):
min
F{ } Z�∳
P
C(x) · dx�􏽘
a∈L
∳
P
cax1, x2,. . . , xL
 􏼁dxa,(1)
s.t.:F∈ Ω,(2)
where Lis the set of network links and L� |L|is the car-
dinality of the set of links, Pis an integration path from ow
vector FI≔(fI,a)a∈lto ow vector F≔(fa)a∈l,Ωis the
typical set of ow conservation and nonnegativity equations
for the links and the corresponding trip matrix, cais the cost
function of link adependent on its own ow faand that of
all the other links in the network (ca�ca(f1, f2,. . . , fL)),
and C(x) � (c1(x), c2(x),..., cL(x)). is notation is
consistent with classical spatial aggregation approaches used
in equilibrium modeling in congested transport
networks [30].
e above expression is clearly a generalization of the
objective function in the classic Beckmann transformation,
which is dened simply as 􏽐a∈L􏽒fa
0ca(xa)dxa, a particular
case of (1).
To dene the set Ω, we rst assume that our network can
be represented by a graph (N, L), where N�1,2,..., N
{ } is
the set of nodes and Lis the set of links, with L≥N.
We dene a set O-D of Rorigin-destination pairs, each
pair r≔ij of which has a vector Ersuch that Er
k�0 if
k∉i, j
􏼈 􏼉, and Er
i� − Dr, Er
j�Dr, where Dris the ow be-
tween pair r. For each pair r≔ij, the ow in link agenerated
by trips from origin ito destination jis fr
a. We thus have the
equality fa�􏽐r∈O−Dfr
a. If we now dene Fr� (fr
a)a∈L, we
get F�􏽐r∈O−DFr. Moreover, we set F� (Fr)r∈O−D.
Finally, we dene an incidence matrix Δ ≔ (δna)n∈N,a∈L
in the following manner:
Journal of Advanced Transportation 3

δna �
1,if link aenters node n,
−1,if link aexits node n,
0,otherwise.
⎧
⎪
⎪
⎨
⎪
⎪
⎩(3)
We can now dene set Ωas
Ω�F∈RL|F�􏽘
r∈O−D
Fr,ΔFr�Er, Fr≥0
⎧
⎨
⎩⎫
⎬
⎭.(4)
Equality ΔFr�Erimplies that 􏽐aδna ·fr
a�Er
n
Note that the constraints of the classic Beckmann
problem consider ows in paths (although in simple net-
works of small size they can coincide with ows in links).
However, the set of constraints Ωcan also be expressed as
a function of the ows in links and the topology of the
network (i.e., nodes and xed trip matrix).
Assume that Ω≠Ø. If the Jacobian matrix of Cis
symmetric, the integral in (1) can be written as
∳PC(x) · dx�V(F) − V(FI), where Vis a potential func-
tion of C, that is, ∇V(F) � C(F). en, the optimality
conditions of problem equations (1) and (2) can be written
directly as follows:
C(F) + tΔλr−r�0,
ΔFr�Er,
r⊗Fr�0,
Fr≥0,
r≥0,
(5)
where λris the Lagrange multiplier associated with the
constraint ΔFr�Erand ris the Lagrange multiplier
associated with the constraint Fr≥0. e notation r⊗Fris
the Hadamard product between the vectors rand Fr, which
means that r⊗Fr� (r
a·fr
a)a.
If, however, the Jacobian of Cis not symmetric, the
integral in (1) will depend on the integration path P. In
what follows, we propose a way of interpreting the integral
and then build an equilibrium model whose every solution
satises the conditions in (5). It is possible to include
additional restrictions to the problem, associated for
example with restrictions on environmental emissions
[31], but this is outside the scope and objective of
this work.
To dene the parameterisation of the problem, in eect,
the line integral in objective function (1), consider a curve
P0≔x(t)|t∈R
{ } ⊂RL(RLis the set of vectors with Lreal
components) that satises the following properties:
(i) xa(0) � 0, xa
′(t)>0, for all a∈L,t∈R
(ii) limt⟶+∞xa(t) � +∞and limt⟶−∞xa(t) � − ∞
Consider also two ow vectors FIand F. For every link a,
there exists a unique number pa(fI,a, fa)such that
fI,a +xa(pa(fI,a, fa)) � fa. is is so because xais a bi-
jective function from Rto R. erefore, Fbelongs to curve
FI+P0if and only if p1(fI,1, f1) � · · · � pL(fI,L , fL). If this
holds, its value is denoted p(FI, F). Since the derivative of xa
is dierent from 0 over all R, by the global inverse function
theorem, the function pais dierentiable on R2.
It follows that if Pis an integration path between F
I
and F
along the curve FI+P0, by the denition of the line integral
in a vector eld [32], Section 16.2 and given that
dx�x′(t)dt, the integral in (1) becomes
Table 1: Table of notations.
NNumber of nodes
LNumber of links
RNumber of pairs origin-destination
N�1,2,..., N
{ } Set of nodes
LSet of links
DrFlow between pair origin-destination r
Er� (Er
n)n∈If r�ij, for all n∉i, j
􏼈 􏼉,Er
n�0, Er
i� − Dr, Er
j�Dr
fr
aFor r≔ij, it is the ow in link agenerated by trips from origin ito destination j
Fr� (fr
a)a
For r≔ij, it is the vector of ows in links generated by the trip from origin ito
destination j
F� (Fr)rMatrix of the ows in links generated by the trips of each pair origin-destination
fa�􏽐rfr
aFlow in link a
F� (fa)aVector of ows in links
Δ ≔ (δna)n∈N,a∈LIncidence matrix nodes-links
ca�ca(f1, f2,. . . , fL) � ca(F)Cost function in link a
C(F) � (c1(F), c2(F),..., cL(F)) Vector of cost in links
λrVector of Lagrange multipliers associated with the constraints ΔFr�Er
rVector of Lagrange multipliers associated with the constraints Fr≥0
4Journal of Advanced Transportation

∳PC(x) · dx�􏽚p FI,F
( )
0
C FI+x(t)
 􏼁·x′(t)dt�􏽘
a∈L􏽚p FI,F
( )
0
caFI+x(t)
 􏼁xa
′(t)dt,
∳PC(x) · dx�􏽘
a∈L􏽚pafI,a,fa
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt.
(6)
en, the optimization problem equations (1) and (2)
can then be written as
min 􏽘
a∈L􏽚pafI,a,fa
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt
s.t. :ΔFr�Er,∀r∈O−Dλr
 􏼁,
Fr≥0,∀r∈O−Dr
 􏼁,
F�􏽘
r∈O−D
Fr,
(7)
Note that this formulation does not include a Lagrange
multiplier for the constraint F�􏽐r∈O−DFrgiven that this
equality can be included directly in the objective function.
e existence and uniqueness of problem (7) are reported in
Appendix A.
e optimality conditions for (7) are
zpafI,a, fa
􏼐 􏼑
zfa
·caFI+x pafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑·xa
′pafI,a, fa
􏼐 􏼑􏼐 􏼑+tΔλr
􏼐 􏼑a−r
a�0,∀(a∈L, r ∈O−D),
ΔFr�Er,∀r∈O−D,
r⊗Fr�0,∀r∈O−D,
Fr≥0,∀r∈O−D,
r≥0,∀r∈O−D.
(8)
where (tΔλr)ais component aof vector tΔλr.
If we calculate (zpa(fI,a, fa)/zfa), we obtain fI,a +
xa(pa(fI,a, fa)) � fa, implying that dierentiating with
respect to fagives xa
′(pa(fI,a, fa))(zpa(fI,a , fa)/zfa) � 1.
e optimality conditions above then become
caFI+x pafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑+tΔλr
􏼐 􏼑a−r
a�0,∀a∈L, ∀r∈O−D,
ΔFr�Er,∀r∈O−D,
r⊗Fr�0,∀r∈O−D,
Fr≥0,∀r∈O−D,
r≥0,∀r∈O−D.
(9)
It can be seen that the set of solutions for (9) depends on
the ow vector FIand that in general, they do not belong to
the curve FI+P0. If there exists a solution Fthat does belong
to the curve, then for all links a, it is the case that
FI+x(pa(fI,a, fa)) � FI+x(p(FI, F)) � F. Also observ-
able is that Fbelongs to FI+P0if and only if
FI+x(p1(fI,1, f1)) � F.
us, (9) can be written as the following system of
equations with unknowns F and F0:
C(F) + tΔλr−r�0,∀r∈O−D,
ΔFr�Er,∀r∈O−D,
r⊗Fr�0,∀r∈O−D,
Fr≥0,∀r∈O−D,
r≥0,∀r∈O−D,
FI+x p1fI,1, f1
􏼐 􏼑􏼐 􏼑�F.
(10)
Journal of Advanced Transportation 5

erefore, the optimization problem stated in (7) is
equivalent to the variational inequality described in (9) if we
require that the integration path in (7) is dened such that
FI+x(p1(fI,1, f1)) � Fholds. e existence of a ow
vector F
I
such that a solution of (7) belongs to the curve
FI+P0is proved in Appendix B under the assumption that
(zca/zfa′)≥0,∀a, a′∈L.
e following proposition allows us to eliminate the
unknown FI:
Proposition 1. A ow vector F is a solution of the system
C(F) + tΔλr−r�0,
ΔFr�Er,
r⊗Fr�0,
Fr≥0,
r≥0,
(11)
if and only if there exists a ow vector FIsuch that (F, FI)is
a solution of system (10) and we can chose FI�F.
Proof. If (F, FI)is a solution of system (10), then F is
a solution of system (11). If Fis a solution of system (11),
then (F, F)is a solution of system (10).
e following proposition relates system (10) to a vari-
ational inequality. □
Proposition 2. A ow vector F is a solution of system (11) if
and only if F∈ and for all F′∈, C(F) · (F′−F)≥0.
Proof. F is a solution of system (11) if and only if Fis
a solution of the following optimization problem:
min
F′∈Ω C(F) · F′.(12)
is is equivalent to stating that Fis the solution of the
following variational inequality:
∀F′∈ Ω, C(F) · F′−F
􏼐 􏼑≥0,(13)
Given the equivalence of (11) with the IV of the previous
proposition, we can observe that when C is strongly
monotone (that is, there is a constant >0 such that if Fand
F′are two ow vectors, we will have (C(F′) −
C(F)) · (F′−F)≥‖F′−F‖2), and then system (11) has
a unique solution in arc ows (see, e.g., [33].
A sucient condition for the vector function C to be
strongly monotone is that there exists a constant α>0 such
that the eigenvalues of the symmetric matrix 1/2(∇C(F) +
t∇C(F)) are greater than or equal to αfor every ow vector
F.
Finally, the theorem below proves that the system of
optimality conditions (11) satises Wardrop’s rst principle.
Before developing the proof, we set out the necessary no-
tation. For each O–D pair r≔ij, a route pbetween iand j
consists of a nite sequence of links (a0, a1,. . . , aq)such that
iis the origin of link a0,jis the destination of link aq, and for
all k∈0,1, ..., q
􏼈 􏼉, the destination of link akis the origin of
link ak+1. e quantity hr
prepresents the ow on route p
between origin iand destination jand is dened as
hr
p�mink�0,...,q fr
ak. e cost of route pbetween O-D pair ris
given by gr
p(F) � 􏽐q
k�0cak(F).□
Theorem 3. Let F be a solution of (11) and r≔ij an origin-
destination pair. Also, let p and p′be two routes between
origin i and destination j. If hr
p≠0and hr
p′≠0, then
gr
p(F) � gr
p′(F).
Proof. Let λr:� (λr
n)n∈Nbe a vector of Lagrange multipliers
associated with the subset of constraints ΔFr�Er. Recall
that route pcan be written as p� (a0, a1, ..., aq). For all
k∈0, ..., q
􏼈 􏼉, we have
cak(F) + tΔλr
􏼐 􏼑ak
−r
ak�0.(14)
If hr
p≠0, then fr
ak>0 for all k∈0,1, ..., q
􏼈 􏼉, implying in
turn that r
ak�0. Also, (tΔλr)ak�􏽐n∈Nδnakλr
n.
Now, let there be nodes (n0, n1,. . . , nq+1)such that for
each k∈0,1,. . . , q
􏼈 􏼉, link akexits node nkand enters node
nk+1. en, for all k∈0,1, ..., q
􏼈 􏼉, we have
􏽘
n∈N
δnakλr
n�λr
nk+1−λr
nk,(15)
cak(F) � λr
nk−λr
nk+1,(16)
we then obtain
gr
p(F) � 􏽘
q
k�0
cak(F) � λr
n0−λr
nq+1�λr
i−λr
j.(17)
Since the Lagrange multiplier vector λ
r
does not depend
on the route between origin iand destination j, we deduce
that sum (17) above is also independent of that route.
Observe also that the Lagrange multiplier set is not
unique, but the quantities λr
i−λr
jdo not depend on the
choice of Lagrange multipliers. ese quantities are inter-
preted as the trip cost between origin iand destination j.□
3.1. Small Example. Assume there is a network with two
nodes A and B and two parallel links 1 and 2 that join them
(see Figure 4). e links’ respective asymmetric cost func-
tions are c1(f1, f2) � 20 +f1+f2,c2(f1, f2) � 2+
2f1+3f2. Assume also that total demand between the two
nodes is 10 and is distributed between the two links.
Given the cost functions as just dened, the equilibrium
ows f1and f2that satisfy them, or in other words, that
satisfy Wardrop’s rst principle, are (f1, f2) � (2,8).
is equilibrium is determined using our construction
with the line integral in the following manner. Let x(t) �
(at, bt)be a vector function where a>0 and b>0 are real
numbers. If FI� (fI,1, fI,2)is a ow vector, problem (7) can
then be written as
6Journal of Advanced Transportation

min
f1,f2≥0
f1+f2�10
a􏽚f1−fI,1
( )/a
0
20 +fI,1+fI,2+(a+b)t
􏼐 􏼑dt+b􏽚f2−fI,2
( )/b
0
2+2fI,1+3fI,2+(2a+3b)t
􏼐 􏼑dt. (18)
As we saw in (10), solving the trac assignment problem
is the equivalent of nding a ow vector FIsuch that F�FI,
where Fis a solution of (18). Assume then that FI∈ Ω, which
implies that fI,1+fI,2�10. Eliminate variable f2by
replacing it with f2�10 −f1and simplify using X�
f1−fI,1. Problem (18) then becomes
min
−fI,1≤X≤10−fI,1
a􏽚X/a
0
(30 +(a+b)t)dt+b􏽚− (X/b)
0
32 −fI,1+(2a+3b)t
􏼐 􏼑dt. (19)
e upper bound of the second integral in (19) is
a consequence of the following equalities:
f2−fI,2�10 −f1−10 −fI,1
􏼐 􏼑�fI,1−f1� − X(20)
e objective function of (19) is a convex quadratic
function and its unique solution is given by
X�min ab
2a2+4ab +b22−fI,1
􏼐 􏼑,10 −fI,1
􏼠 􏼡 (21)
is implies that
f1�min ab
2a2+4ab +b22−fI,1
􏼐 􏼑+fI,1,10
􏼠 􏼡 (22)
e function fI,1⟶f1wheref1is given by the
equality immediately above is a Lipschitz function with
a Lipschitz constant equal to (2a2+3ab +b2)/(2a2
+4ab +b2). Since the latter is less than 1, the function is
a contraction and therefore has a unique xed point. Every
sequence dened by
fn
1�min ab
2a2+4ab +b22−fn−1
1
􏼐 􏼑+fn−1
1,10
􏼠 􏼡 (23)
converges to this point, which is equal to f1�2. We then
obtain the value for f2�8 and f1�2.
is little example gives a simple demonstration of the
operation of our generalization of Beckmann’s trans-
formation based on the line integral dened in (1), which
allows us to obtain a xed-point iteration that converges.
4. Analysis of a Numerical Example
In this section, we dene an example for a network and an
exogenous trip matrix, to obtain a balance considering the
optimization problem exposed in equation (7).
e network used for our numerical example was in-
spired by the classic version due to Nguyen and Dupuis [21]
in Figure 5. is network has been widely used in the lit-
erature associated with the modeling of transport networks
(some recent works are [34–36]).
We assume that there are four O-D pairs (r), each with
its respective xed demand level (D), as follows:
r1�1−13, Dr1�40; r2�1−11, Dr2�70; r3�3−13,
Dr3�30; r4�3−11, Dr4�40. A visual representation of
the network with its various links and nodes is given in
Figure 5.
e (asymmetric) cost functions for the links that we
consider in this formulation are set out in Table 2 and have
the form ca(F) � ea+􏽐a′ba,a′fa′(notice that the cost
functions of the links in these deterministic equilibrium
models must always be monotonically increasing).
A
c1 ( f1,f2) = 20 +f1 + f2
c2 ( f1,f2) = 2 +2f1 +3f2
B
10
10
Figure 4: Example network with asymmetric cost functions.
Journal of Advanced Transportation 7

We consider, for each link aand for each initial ow fI,a,
the function xa(t) � fI,a +cat, with ca>0. We can observe
that for each ow fa,xa(t) � faif and only if t�
(fa−fI,a)/ca. We then have pa(fI,a , fa) � (fa−fI,a)/ca.
Taking into account the general form of cost functions ca(F),
formulation (7) can be written as follows:
min 􏽘
a∈L
ca􏽚fa−fI,a
 􏼁/ca
0
ea+􏽘
a′
ba,a′fI,a′+ca′t
􏼐 􏼑
⎛
⎝⎞
⎠dt,
s.t.:ΔFr�Er,∀r∈O−D,
Fr≥0,∀r∈O−D,
F�􏽘
r∈O−D
Fr,
(24)
which leads to
min 􏽘
a∈L
caea+􏽘
a′
ba,a′fI,a′
⎛
⎝⎞
⎠·fa−fI,a
ca
+􏽘
a′
ca′
2
fa−fI,a
ca
􏼠 􏼡2
⎡
⎢
⎢
⎣⎤
⎥
⎥
⎦
s.t.:ΔFr�Er,∀r∈O−D,
Fr≥0,∀r∈O−D,
F�􏽘
r∈O−D
Fr.
(25)
1 2
7
4
35 6
89 10 11
12 13
1
2
34
56
7
8910 11 12
13 14 15
16
17
18
19
Figure 5: Network for the numerical example: 13 nodes and 19 links.
Table 2: Cost functions for the example network.
c1�1 + 2 f1c11 �1 + f11 + 3f14
c2�1 + 4f3+ 4f2c12 �1 + f12
c3�1 + f3c13 �1 + f13
c4�1 + f4c14 �1 + f14
c5�1 + f5c15 �1 + f15
c6�1 + f6+ 4f7c16 �2 + f16
c7�8 + 2f7c17 �1 + f17 + 8f18
c8�10 + f8c18 �1 + 0.5f18
c9�1 + 3f8+f9c19 �1 + f19
c10 �1 + f10 + 4f13
8Journal of Advanced Transportation

As we have seen in (11), solving the trac assignment
problem is the equivalent of nding a ow vector FIsuch
that F�FI, where Fis a solution of (25). e algorithm
consists of an iteration of xed point.
Step 1: Choose an initial ow vector FIand ε>0.
Step 2: Take F as a solution of (25).
Step 3: If ‖F−FI‖<ε, then STOP, else take FI�Fand
come back to Step 2.
In our simulation, we chose ε�10−3. e algorithm
converges in 11 iterations. For solving the optimization
problem, we use fmincon in MATLAB. e equilibrium
ows and costs for the 19 links in our example are set out in
Table 3: Equilibrium ows and costs for the example network links.
Link fr1
afr2
afr3
afr4
afa�􏽐4
i�1fri
aca(F)
1 0 119.9989 0 0 119.9989 241
2 120 0.0011 0 0 120.0011 747,4
3 0 0.0000 0.0011 66.6 66.6011 67,6
4 89.7055 0.0007 0.0009 66.5998 156.3068 157,3
5 89.7050 0.0005 0.0007 66.5996 156.3059 157,3
6 0 0.0003 0 66.5994 66.5997 187,6
7 0 119.9989 0 0 119.9989 248
8 0 0.0000 129.9989 73.4000 203.3989 213,4
9 30.2945 0.0004 0.0002 0.0002 30.2954 641,5
10 0.0005 0.0002 0.0002 0.0001 0.0009 935,8
11 89.7050 0.0002 0.0007 0.0002 89.7061 791,8
12 0 119.9992 0 66.5994 186.5987 187,6
13 30.2944 0.0003 129.9990 73.4001 233.6937 234,7
14 30.2948 0.0005 129.9992 73.4002 233.6947 234,7
15 0 0.0003 0 73.3967 73.3970 74,4
16 0 0.0001 0 0.0001 0.0005 2
17 0 0.0001 0 0.0001 0.0005 2001
18 119.9999 0.0004 129.9999 0.0037 250.0039 126
19 0 0.0005 0 0.0039 0.0044 1
Table 4: Equilibrium costs for the routes used between each network example O-D pair.
Origin-destination Route Route cost
1–13 2-4-5-11-18∗1979.8
1–13 2-9-13-14-18∗1984.3
1–13 2-9-16-17 3391.9
1–13 2-4-10-14-18 2201.2
1–11 1-7-12∗676.6
1–11 2-4-5-6-12 1437.2
1–11 2-4-5-11-15 1928.2
1–11 2-4-5-11-18-19 1928.2
1–11 2-4-10-14-15 2149.6
1–11 2-4-10-14-18-19 2202.2
1–11 2-9-13-14-15 1932.7
1–11 2-9-13-14-18-19 1985.3
1–11 2-9-16-17-19 3392.9
3–13 3-4-5-11-18 1300
3–13 3-9-13-14-18 1304.5
3–13 8-16-17 2216.4
3–13 3-4-10-14-18 1521.4
3–13 8-13-14–18∗808.8
3–11 8-13-14-15∗757.2
3–11 3-4-5-6-12∗757.4
3–11 3-4-5-11-15 1248.4
3–11 3-4-5-11-18-19 1301
3–11 3-4-10-14-15 1469.8
3–11 3-4-10-14-18-19 1522.4
3–11 3-9-13-14-15 1252.9
3–11 3-9-13-14-18-19 1305.5
3–11 8-16-17–19 2217.4
e symbol (∗) indicates the routes used.
Journal of Advanced Transportation 9

Table 3, while the equilibrium costs for the routes used
between each O-D pair are given in Table 4. e routes
indicated with a star correspond to the used routes. e
assumptions of the uniqueness result (explained in Ap-
pendix A) are satised in the case of this simulation, so the
convergence of the algorithm depends only on the initial
ow FIof Step 1 and not on the choice of the solution in Step
2. e main advantage of this method is the simplicity of the
auxiliary problem (25) and, in addition, the exibility that
the method oers by being able to choose the parameter-
isation for the line integral.
Notice that some analytical properties, such as the
uniqueness and stability of a deterministic trac assignment
problem, are actually not a general property of trans-
portation problems [37]. Furthermore, the existence of
multiple equilibria in the case of a model with asymmetric
interactions between road intersections is well known [38].
5. Conclusions
is article presented a generalization of the classic Beck-
mann transformation for specifying and solving the trac
assignment problem. e proposed optimization problem is
similar to Beckmann’s except that the objective function is
built around a line integral that incorporates a cost vector
function with multivariate and asymmetric costs. e
constraints in the two versions are identical.
e main contribution of this article is theoretical/
methodological since it allows consolidating, under the same
theoretical framework, the approach based on the equivalent
optimization problem for trac assignment models without
ow interactions in their costs (e.g., transformation of
Beckmann), with the general case in which the trac as-
signment model (with asymmetric interactions) is formu-
lated through a variational inequality. at is, our
contribution is to generalize an equivalent deterministic
assignment optimization problem to obtain trac equilib-
rium including multivariate cost functions with asymmetric
interactions. Our new approach can eventually be extended
to model supply-demand equilibria in markets other than
transportation, with complementary or substitute goods/
services in which there are asymmetric interactions between
prices.
To solve the optimization problem the line integral must
be parameterised, which is done by dening an integration
path based on a ow vector. is converts the new for-
mulation into a xed-point equilibrium problem. Its solu-
tions are also the solutions of a classic variational inequality
and satisfy Wardrop’s rst principle.
e proposed approach contributes to a better un-
derstanding of equilibrium in transport networks and
provides a common trac assignment framework for
analyzing both the simple case captured by Beckmann’s
transformation and the more general case represented by
a variational inequality.
e theoretical development of the new optimization
problem is complemented by a numerical example based on
the classic trac network due to Nguyen and Dupuis but
with multivariate cost functions for the network links and
asymmetric interactions. It was shown that the solution of
the problem via a parameterised line integral generates
a xed-point equilibrium condition equivalent to a varia-
tional inequality whose solution satises Wardrop’s rst
principle.
Future research could include the analysis of the eect
that dierent integration paths, considered when parame-
terising the line integral, could have on the uniqueness of the
equilibrium solution or also study sensitivity analysis of the
equilibrium.
Appendix
A. Uniqueness of the Solution to the
Auxiliary Problem
Theorem A.1. If there exists a positive real number α such
that, for every link a, we have (zca/zfa)≥α, and if also for all
links a and a′, we have (zca/zfa′)≥0, then given a vector of
ows in initial links FI, optimization problem (7) has
a unique solution.
Proof. Let FIbe a vector of ows in links. We dene, in each
link a, the function
Va:F⟶􏽚pafI,a,fa
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt. (A.1)
Optimization problem (7) is rewritten:
min 􏽘
a∈L
Va(F),
s.t.:ΔFr�Er,∀r∈O−D,
Fr≥0,∀r∈O−D,
F�􏽘
r∈O−D
Fr.
(A.2)
In the next Appendix B, we have calculated the second
derivatives of Va: for all links a, a′, a″, if a′≠aor a″≠a, then
z2Va
zfa′fa″
�0,(A.3)
and also the following equality is fullled:
10 Journal of Advanced Transportation

z2Va
zf2
a
�􏽘
a′∈L,a≠a
zcaFI+xpafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑
zxa′
􏽼√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√􏽽
≥0
x′
a′pafI,a, fa
􏼐 􏼑􏼐 􏼑
x′apafI,a, fa
􏼐 􏼑􏼐 􏼑+
􏽼√√√√√√√√􏽻􏽺√√√√√√√√􏽽
>0
zcaFI+xpafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑
zxa
􏽼√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√􏽽
≥α
≥α(A.4)
Finally, the eigenvalues of the Hessian matrix of
F⟶􏽐a∈LVa(F)are greater than or equal to α, and, in this
way, the objective function of problem (7) is strongly
convex; therefore, (7) has a unique solution since the re-
striction set is convex. □
B. Demonstration of Existence
Recall that the optimization problem we are considering is
the following:
min F􏽘
a∈L􏽚pafI,a,fa
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt,
s.t.:ΔFr�Er,∀r∈O−Dλr
 􏼁
Fr≥0,∀r∈O−Dr
 􏼁,
F�􏽘
r∈O−D
Fr.
(B.1)
e problem depends on the ow vector F
I
and has a set
of solutions denoted K(FI). We will prove that there exists
an FIsuch that FI∈K(FI), which in turn proves that there
exists an FIsuch that a solution to (B.1) belongs to the curve
FI+P0. First, however, we state the Kakutani xed-point
theorem.
Theorem B.1. Let T: S ⇉S be a multifunction where S is
a nonempty compact convex set in a Euclidean space. If the
graph of S is closed and its values are convex and not empty,
then there exists an x ∈S such that x∈T(x).
e multifunction Tis said to have a closed graph if, for
every sequence xn⟶x, with xn∈Sand for every sequence
yn⟶ywith yn∈T(xn), we have y∈T(x). In what fol-
lows, we show that the multifunction Kveries the Kakutani
theorem hypothesis.
Theorem A.3. Suppose that (zca/zfa′)≥0,∀a, a′∈L. en,
there exists a ow vector FIsuch that FI∈K(FI).
Proof. Recall that the set of feasible ows of the problem is
denoted Ω, that is,
Ω�F∈RL|F�􏽘
r∈O−D
Fr, Fr≥0,ΔFr�Er
⎧
⎨
⎩⎫
⎬
⎭,(B.2)
We therefore observe that K:Ω⇉Ω. Since Ωis dened
by linear relationships, it is a convex set. Furthermore, since
the ow on a network link is less than or equal to the sum of
the ows entering the network, then for every link a, we have
0≤fa≤􏽘
r∈O−D
Dr.(B.3)
Since Ωis a closed set, the above inequality implies that it
is compact. Moreover, Ωis nonempty by hypothesis.
We now show that the graph of Kis closed. Let (Fn
I)be
a sequence such that Fn
I∈ Ω and Fn
I⟶FI, and let Fnbe
a sequence such that Fn∈K(Fn
I)and Fn⟶F. We will
show that F∈K(FI). For all F′∈ Ω and all integers n, we
have
􏽘
a∈L􏽚pafn
I,a,fn
a
 􏼁
0
caFn
I+x(t)
 􏼁xa
′(t)dt≤􏽘
a∈L􏽚pafn
I,a,fa
′
 􏼁
0
caFn
I+x(t)
 􏼁xa
′(t)dt. (B.4)
By continuity of the function pa, if we let ntend to +∞,
then
􏽘
a∈L􏽚pafI,a,fa
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt ≤􏽘
a∈L􏽚pafI,a,fa
′
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt. (B.5)
Since this inequality holds for all F′∈ Ω, we have
F∈K(FI), which proves that the graph of Kis closed. We
now show that K(FI)is a convex set by proving that the
objective function of the optimization problem is convex.
Since the sum of convex functions is itself a convex function,
it is sucient to show that for each link a, the function
Va:F⟶􏽚pafI,a,fa
 􏼁
0
caFI+x(t)
 􏼁xa
′(t)dt, (B.6)
is convex. Observe that since Vadoes not depend on fa′
when a′≠a, then
Journal of Advanced Transportation 11

zVa
zfa′
�0,∀a′≠a. (B.7)
From this, we deduce that for all links a′, a″, if a′≠aor
a″≠a, then
z2Va
zfa′fa″
�0,(B.8)
is implies that the Hessian matrix of Vahas one el-
ement not equal to zero, which is the element (z2Va/zf2
a).
We have
zVa
zfa
�zpafI,a, fa
􏼐 􏼑
zfa
·caFI+x pafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑·xa
′pafI,a, fa
􏼐 􏼑􏼐 􏼑�caFI+x pafI,a , fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑,(B.9)
since, as we saw in Section 3, xa
′(pa(fI,a, fa))(zpa
(fI,a, fa)/zfa) � 1.
By the chain rule, we obtain
z2Va
zf2
a
�􏽘
a′∈L
zcaFI+xpafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑
zxa′xa′
′pafI,a, fa
􏼐 􏼑􏼐 􏼑zpafI,a , fa
􏼐 􏼑
zfa
,
z2Va
zf2
a
�􏽘
a′∈L
zcaFI+xpafI,a, fa
􏼐 􏼑􏼐 􏼑􏼐 􏼑
zxa′
􏽼√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√􏽽
≥0
xa′
′pafI,a, fa
􏼐 􏼑􏼐 􏼑
xa
′pafI,a, fa
􏼐 􏼑􏼐 􏼑
􏽼√√√√√√√􏽻􏽺√√√√√√√􏽽
>0
.
(B.10)
Since by hypothesis (zca/zxa′)≥0, then (z2Va/zf2
a)≥0
implying Vais a convex function, thus proving that K(FI)is
a convex set. e set K(FI)is not empty because the ob-
jective function is continuous and the set Ωis compact.
Finally, by the Kakutani xed-point theorem, there exists
a ow vector FIsuch that FI∈K(FI).□
Data Availability
e data used to support the numerical study are included
within the article.
Disclosure
A partial preprint has previously been published, Marechal
and De Grange [39, 40]. is work has been presented
during the 18th Chilean Congress of Transportation Engi-
neering in La Serena, Chile, 2017.
Conflicts of Interest
e authors declare that they have no conicts of interest.
Acknowledgments
is study has been funded by Diego Portales University.
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