Network User Equilibrium with Elastic Demand:
Formulation, Qualitative Analysis and
Computation∗
Ke Han
(Corresponding Author, Email: [email protected])
Department of Mathematics
Pennsylvania State University, University Park, USA
Terry L. Friesz
Department of Industrial and Manufacturing Engineering,
Pennsylvania State University, University Park, USA
Tao Yao
Department of Industrial and Manufacturing Engineering,
Pennsylvania State University, University Park, USA
1
Introductory Remarks
This paper presents an elastic demand extension of the fixed-demand simultaneous routeand-departure choice dynamic user equilibrium model originally presented in Friesz et al.
(1993). Although Friesz et al. (2011) show that analysis and computation of dynamic user
equilibrium with constant travel demand is tremendously simplified by stating it as a a
differential variational inequality (DVI), they do not discuss how elastic demand may be
accommodated within a DVI framework. The DVI formulation of elastic demand DUE
(E-DUE) is not straightforward.
In this paper, we advance knowledge of E-DUE in its mathematical formulation, existence, and computation. We first present an infinite-dimensional variational inequality
formulation in a Hilbert space using measure-theoretic argument. Existence result for
such variational inequality is show by invoking a version of Brouwer’s fixed point theorem
(Browder, 1968). Finally, we employ the mathematical paradigm of differential variational
inequality to formally establish a fixed point formulation of E-DUE in an extended Hilbert
∗
The complete work is reported in two papers available at http://arxiv.org/abs/1304.5286
http://arxiv.org/abs/1305.1276
1
and
space. An iterative fixed point algorithm based on such formulation is proposed with convergence result shown for a limited class of delay operators. The algorithm is tested on
the Sioux Falls network with confirmed numerical convergence.
2
Notation and Essential Background
We assume a single commuting period or “day” expressed as [t0 , tf ] ⊂ <1 . Here, as in
all DUE modeling, the single most crucial ingredient is the path delay operator, which
provides the delay on any path p per unit of flow departing from the origin of that path;
it is denoted by Dp (t, h) for all p ∈ P; where P is the set of all paths employed by
travelers; t denotes departure time; and h is a vector of departure rates. From these,
we construct effective unit path delay operators Ψp (t, h) by adding the schedule delay
f [t + Dp (t, h) − TA ]; that is Ψp (t, h) = Dp (t, h) + f [t + Dp (t, h) − TA ] for all p ∈ P where
TA is the desired arrival time. We stipulate that each Ψp (·, h) : [t0 , tf ] −→ <1++ for all
p ∈ P is measurable and strictly positive. We employ the obvious notation Ψ(·, h) =
(Ψp (·, h) : p ∈ P) ∈ <|P| to express the complete vector of effective delay operators.
Transportation demand is assumed to be expressed as the following invertible function
Qij (tf ) = Fij [v] for each origin-destination pair (i, j) ∈ W, where W is the set of all origindestination pairs and v is a concatenation of origin-destination minimum travel costs vij
associated with (i, j) ∈ W. That is, we have that vij ∈ <1+ so that v = (vij : (i, j) ∈ W) ∈
<|W| . Note that to say vij is a minimum travel cost means it is the minimum cost for all
departure time choices and all route choices pertinent to origin-destination pair (i, j) ∈ W.
Further note that Qij (tf ) is the unknown cummulative travel demand between (i, j) ∈ W
that must ultimately arrive by time tf .
It is convenient to form the complete vector of travel demands by concatenating
|W|
the origin-specific travel demands to obtain Q = (Qij : (i, j) ∈ W) ∈ <+ . The inverse demand function for every (i, j) ∈ W is vij = Θij [Q], and we naturally define
|W|
Θ = (Θij : (i, j) ∈ W) ∈ <++ . As a consequence, we employ the following set of feasible
departure flows when the travel demand between each origin-destination pair is unknown.




X Z tf
|P|
|W|
e = (h, Q) : h ≥ 0,
Λ
hp (t) dt = Qij ∀(i, j) ∈ W ⊂ L2 [t0 , tf ]
× <+


t0
p∈Pij
where
L2 [t
(1)
|P|
|W|
× <+ is the direct product of the |P|-fold product of Hilbert spaces
0 , tf ]
consisting of square-integrable path flows, and the |W|-dimensional Euclidean space consisting of vectors of elastic travel demands.
e where h∗ is a vector of departure rates
Definition 2.1 (E-DUE) A pair (h∗ , Q∗ ) ∈ Λ,
(path flows) and Q∗ is the associated vector of travel demands, is said to be a dynamic
user equilibrium with elastic demand if for all (i, j) ∈ W,
3
h∗p (t) > 0, p ∈ Pij =⇒ Ψp (t, h∗ ) = Θij [Q∗ ]
∀ν (t) ∈ [t0 , tf ]
Ψp (t, h∗ ) ≥ Θij [Q∗ ]
∀ν (t) ∈ [t0 , tf ],
(2)
∀p ∈ Pij
(3)
The Variational Inequality Formulation and Existence Result for E-DUE
The DUE problem with elastic demand can be expressed as a variational inequality, as
shown in the theorem below.
Theorem 3.1 (E-DUE equivalent to a variational inequality) Assume Ψp (·, h) :
[t0 , tf ] → <++ is measurable and strictly positive for all p ∈ P and all h such that
e Also assume that the elastic travel demand function is invertible with inverse
(h, Q) ∈ Λ.
e is a DUE with elastic demand as in
Θij [·] for all (i, j) ∈ W. Then a pair, (h∗ , Q∗ ) ∈ Λ,
Definition 2.1 if and only it solves the following variational inequality:

e such that

find (h∗ , Q∗ ) ∈ Λ



X Z tf
X

∗
∗
∗
∗
Ψp (t, h )(hp − hp )dt −
Θij [Q ] Qij − Qij ≥ 0
V I(Ψ, Θ, t0 , tf ) (4)


p∈P t0
(i, j)∈W



e
∀(h, Q) ∈ Λ
Proof
See Han et al. (2013a).
Existence result for V I(Ψ, Θ, t0 , tf ) is formally established in this paper. Our proposed approach is meant to incorporate the most general dynamic network loading submodel with minimum regularity requirements, and to yield existence of E-DUE without
invoking the a priori upper bound on path flows. To this end, we invoke a version of
Brouwer’s fixed point theorem provided by Browder (1968) and make the following two
minor assumptions:
(A1). The schedule-delay function f (·) is continuous on [t0 , tf ] and satisfies
f (t2 ) − f (t1 ) ≥ ∆(t2 − t1 )
∀t0 ≤ t1 < t2 ≤ tf
(5)
for some ∆ > −1
(A2). The first-in-first-out (FIFO) rule is obeyed on a path level. In addition, each link
a ∈ A in the network has a finite exit flow capacity Ma < ∞.
Theorem 3.2 (Existence of E-DUE) Assume that the effective delay operator is continuous. In addition, let assumptions (A1) and (A2) hold. If the inverse demand function
|W|
|W|
Θ : <+ → <++ is continuous, then the variational inequality V I Ψ, Θ, t0 , tf has a solution.
Proof
See Han et al. (2013a).
4
Computation of E-DUE
Computation of E-DUE is most facilitated by the mathematical paradigm of differential
.
variational inequality and optimal control theory. We define the product space E =
|P|
L2 [t0 , tf ]
× <|W| with a naturally induced norm h·, ·iE . In addition, we define the
mapping
e −→ E,
F :Λ
(h, Q) 7→ (Ψ(·, h), −Θ(Q))
(6)
|W|
|W|
e Ψ(·, h) ∈ L2+ [t0 , tf ]
where (h, Q) ∈ Λ,
, and Θ(Q) ∈ <++ . Such a mapping is clearly
well-defined.
Theorem 4.1 (E-DUE equivalent to a fixed-point problem) The fixed point problem
X ∗ = PΛe X ∗ − αF(X ∗ )
(7)
|W|
is equivalent to E-DUE, where α ∈ <++ , X ∗ = (h∗ , Q∗ ) ∈ (L2+ [t0 , tf ])|P| ×<+ , and PΛe [·]
e
is the minimum norm projection onto Λ.
Proof
See Han et al. (2013b).
The fixed point formulation suggests an iterative algorithm where each iteration amounts
to a linear-quadratic optimal control problem, for which we able to obtain closed-form solution using necessary conditions for optimal control problems. The complete fixed-point
algorithm, together with its convergence theorem, are provided in Han et al. (2013b). Such
algorithm is tested on the Sioux Falls network with demonstrated convergence result.
References
F.E. Browder. The fixed point theory of multi-valued mappings in topological vector
spaces. Mathematische Annalen, 177, 283 – 301, 1968.
T.L. Friesz, D. Bernstein, T. Smith, R. Tobin, B. Wie. A variational inequality formulation
of the dynamic network user equilibrium problem. Operations Research, 41 (1), 80 – 91,
1993.
T.L. Friesz, T. Kim, C. Kwon, M.A. Rigdon. Approximate network loading and dualtime-scale dynamic user equilibrium. Transportation Research Part B, 45 (1), 176 –
207, 2011.
K. Han, T.L. Friesz, T. Yao. An infinite-dimensional variational inequality formulation and
existence result for dynamic user equilibrium with elastic demands arXiv:1305.1276,
2013a.
K. Han, T.L. Friesz, T. Yao. Dynamic User Equilibrium with Elastic Demand and Bounded
Rationality: Formulation, Qualitative Analysis and Computation arXiv:1304.5286,
2013b.