Multiclass Dynamic Transportation Model for User Equilibria

A Link-Node Nonlinear Complementarity Model
for a Multiclass Simultaneous Transportation
Dynamic User Equilibria

Mohamad K. Hasan
Associate Professor
Department of Quantitative Methods and Information Systems
College of Business Administration, Kuwait University
P.O. Box 5486, Safat, 13055, Kuwait
Fax: (965) 2483-9406
Tel.: Work: + (965)2498-8453, Mobile: + (965)9782-2073
Email: mkamal@cba.edu.kw

And

Xuegang (Jeff) Ban
Assistant Professor
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
Jonsson Engineering Center, room: 4034
110 8th Street, Troy, New York 12180
Fax: 518-276-4833, Tel.: 518-276-8043
Email: banx@rpi.edu

Accepted for Publication in International Journal of Operations Research and Information
Systems (IJORIS), IGI Global, USA, May 2011

Presentation Outlines
1. Introduction

2. The Nonlinear Complementarity Formulation for DUE

3. Multiclass Simultaneous Transportation Equilibrium Model
(MSTEM)

4. Discrete-Time Domain Finite-Dimensional Nonlinear
Complementarity DUE Multiclass Simultaneous
Transportation Equilibrium Model (DMSTEM)

5. Solution Algorithm

6. Conclusions and Future Research

1. Introduction

Transportation Network Equilibrium Models

Traffic (Passengers ) Models
Freight Models (Hasan UN-ESCWA)

Dynamic Models Stochastic Models Deterministic Models

Traffic Assignment LU-MUE
DSTEM (Hasan and Departure Traffic MUE LU-MUE MSTEM
Time (Jeff and et. Assignment (Hasan)Work in
and Jeff ) Simultaneous
al. 2008) (Jeff and et. al. progress Sequential Models
2008) Models

1- Trip Generation
2- Trip Distribution
3- Modal Split
Multiclass Single Class 4- Traffic assignment
Variational Inequality

Variational Equivalent
Generation, Distribution, Inequality Convex program
Distribution , Modal Split
Modal split, Assignment, and and Assignment (De Cea PGSTEM, GSTEM
Departure Time (Hasan and et. al 2003) (Hasan 1991)
Dashti 2007) ESTRAUS
MSTEM Generation, Distribution, Distribution , Modal Distribution
Modal split, and Assignment Split and Assignment and
(Safwat and Magnanti 1988) ( Florian and Nguyen Assignment
STEM 1978) (Evans 1976)

Dynamic Traffic Assignment (DTA) models, which aim to predict
future dynamic traffic states in a short-term fashion, have been
extensively studied for decades. DTA studies accelerated in the last
fifteen years with the advent of intelligent transportation systems
(ITS).

Within DTA studies, the dynamic user equilibrium (DUE)
problem, which tries to determine the distribution of time-dependent
traffic flow of a traffic network assuming travelers
follow rational route choices (such as to minimize travel time or other
forms of cost), is one of the most challenging.

Two distinct approaches have dominated the methodologies
applied to DUE research:

The simulation-based (microscopic/mesoscopic) approach.

The analytical (macroscopic) approach. Variational Inequality (VI)
and Nonlinear Complementarity (NC) have been shown to be the most
effective approaches.

The simulation-based (microscopic/mesoscopic) approach.

Multiclass DUE models have also been studied with these
analytical approaches (Ran and Boyce, 1996; Belimer et
al., 2003) or simulation-based approaches
(Ben-Akiva et al, 2001; Mahmassani et al., 2001).

The early studies, however, focused on different vehicle
classes and a few social-economic factors of drivers, such
as value of times (VOD) preferences (Lu et al., 2008).
With this goal, Ramadurai and Ukkusuri (2010)
integrate activity-based models for demand analysis into
DUE. The model is solved using a super-network
representation of activities and traffic networks.

By investigating further how socio-economic factors of
drivers may affect trip generation, trip distribution, mode
choice, and departure-times in a dynamic (i.e. time-
dependent) fashion, and integrating the findings into DUE
modeling, we expect to improve the practicality and
accuracy of DUE models.

With this objective, we combine the NCP-based DUE model
in Ban et al. (2008) and the Multiclass Simultaneous
Transportation Equilibrium Model (MSTEM) in Hasan and
Dashti (2007) to develop a multiclass combined Trip
Generation (TG)-Trip Distribution (TD)-Nested Modal Split
(NMS)-Modal Split (MS)-Dynamic Trip Assignment (DTA)-
Departure Time (DT) Model (NCPDUE-TG-TD-NMS-MS-
DTA-DT).

2. The Nonlinear Complementarity Formulation for DUE
Ban et al. (2008) evenly divided the entire study period into K ' time intervals by introducing
th
the length of each time interval such that K ' T ' . We use index k to denote the k- time
interval [(k 1) , k ) for 1 k K ' . The notation for the discrete-time model is first listed as
follows (Ban et al., 2008):
k
u as u as ((k 1) ) : t he inflow rat e t o link a t owardsdest inat io s at t hebeginningof t imeint erval ,
n k
k
wh is also assumed t o be const antduring t heent iret imeint erval , u
ich k (u as ) a , s ,k
k
v as vas ((k 1) ) : t heexit flow rat efrom link a t owardsdest inat io s at t hebeginningof t imeint erval ,
n k
k
whi is also assumed t o be const antduring t heent iret imeint erval , v
ch k (vas ) a ,s ,k
k
u as : t heinflows t o link a t owardsdest inat io s during t imeint erval
n k
k
vas : t heexit flows from link a t owardsdest inat io s during t imeint erval
n k
k
xas xas ((k 1) ) : t heflows of link a t owardsdest inat io s at t hebeginningof t imeint erval ,
n k
k
which is assumed t o be const antduring t heent ireint erval , x
k ( xas ) a ,s ,k
k k
u s u : t heaggregat edinflow rat e t o link a at t hebeginningof t imeint erval ,
as k
s S

which is assumed t o be const antduring t heent ire int erval , u A
k k
(u a ) a ,k
k k
v a v : t heaggregat edexit flow rat efrom link a at t hebeginningof t imeint erval ,
as k
s S

wh is assumed t o be const antduring t heent ire int erval , v A
ich k k
( va ) a ,k
k k
xa xas : t heaggregat edflows of link a t owardsdest inat io s at t hebeginningof t imeint erval ,
n k
s S

wh is assumed t o be const antduring t heent ire int erval , x A
ich k k
( xa ) a ,k
k
d is d is ((k 1) ) : t hedemand rat egenerat edfrom node i t o dest inat io s at t hebeginningof t imeint erval ,
n k
k
wh is assumed t o be const antduring t heent ire int erval , d
ich k ( d is ) i , s , k ,i s
k
d is : t hedemands generat ed from node i t o dest inat io s during t imeint erval
n k

C k (u ) C (k ) : t he t ravelt imeof link a at t he end of t imeint ervalk , a funct ionof u, C (C k )
a a a a, k
k (k ) : t he minimum t ravelt imefrom node i t o dest inat io s at t heend of t imeint ervalk ,
n ( k)
is is is i, s, k , i s
e k (u ) (k 1) C k 1(u ) : t heexit t ime
for vehicl ent eringa at t he beginningof t imeint ervalk ,
es
a a
a funct ionof u, e (e k )
a a, k

Ban et al. (2008) showed that the above discretization scheme leads to the following NCP formulation for DUE
with fixed demand: find (u, ) such that

0 u u (u, ) 0
0 (u, ) 0

where the two functions u (u, ) and (u, ) are defined as:

(u, ) C k (u ) 3, k , l (u ) l [1 3, k , l (u )] l 1 k ,
u a a h s a h s l s
l 1 ek 1(u ) / l a a a
a a, s , k (1)

(u, ) uk dk [ 1, k ' (u ) 2, k ' , k (u ) u k ' 1, k ' 1(u ) (1 2, k ' , k (u )) u k ' 1]
as is a a as a a as
a A(i ) a B(i ) k ': ek ' (u ) (k 1) ek ' 1(u )
a a i, s, i s, k (2)

�� (�� −1)
��ℎ�� = �� − 1 ∆ + �� is the exit time of vehicles entering a at the
(�� −1)
beginning of the k-th interval, which is a function of the inflow vector �� since �� is
so.

He also define three types of indicator functions on ��: ��1,�� , ��2,�� ,�� and ��3,�� ,�� .
��
1,��
First, flow propagation constraints can be represented by �� , which is defined as:

∆
��1,��
�� = �� −1
, ∀��, �� (3)
��
�� −�� +∆

The relation between link inflow and exit flow rates can be represented by both ��1,��
��
and ��2,�� ,��
�� with the latter defined as:

��
�� + �� +1−�� ∆ �� +1
��2,��
�� = �� −1
��
∀��, ��, �� , �� ≤ �� − 1 Δ < �� (4)
��
�� −�� +∆

3,��
Similarly, �� is used to discretize (and interpolate) the minimum travel time
��ℎ �� + �� in equation (1). It can be defined as:

�� +1
��
��3,��
�� = �� − �� − , ∀�� − 1 ≤ < �� (5)
∆ ∆

The three indicator functions satisfy the following:

��1,��
�� > 0, ∀��, ��

�� +1
0 < ��2,�� ,��
�� ≤ 1, ��
∀��, ��, �� , �� ≤ �� − 1 Δ < ��
�� +1
3,�� ,��
0 < �� ≤ 1, ∀ �� − 1 ≤ < ��
∆

Φ�� is a vector function for any combination of link a, destination s, and time interval k.
Each of its component represents, for the given a; s; k, the difference of the minimum
travel times via two sets of paths from the starting node (or tail node) of link a to
destination s at k. The first set of paths must traverse link a, which is a subset of the
second set that includes all paths from the starting node of a to destination s. This
��
difference should be zero if link a is selected (i.e. the inflow to link a at time k,�� , is
nonzero). Φ�� , on the other hand, simply represents the flow conservation constraint at
any node i, destination �� ≠ �� and time k. Detailed discussion on how Φ�� and Φ�� are
derived can be found in Ban et al. (2008), which also showed (Theorem 3) that NCPDUE
has a nonempty and compact solution set under certain conditions.

3. Multiclass Simultaneous Transportation Equilibrium Model (MSTEM)
A brief description of MSTEM model that developed by Hasan and Dashti (2007).
The notation for MSTEM model is first listed as follows:

( N , A) = A multimodal traffic network consisting of a set of N nodes and a set of A links
l = User class (e.g., income level, car availability, etc.)
L = Set of all user classes
o = Trip purpose (e.g., home-based-work, home-based-shopping, etc.)
O = Set of all trip purpose
I lo = Set of origin nodes for user class l and trip purpose o
i = An origin node in the set I lo for user class l with trip purpose o
Dilo = Set of destination nodes that are accessible from a given origin i for user
class l with trip purpose o
s = A destination node in the set Dilo for user class l with trip purpose o
R lo = Set of origin-destination pairs is for user class l with trip purpose o , i.e., the
set of all origins i I lo and destinations s Dilo
m = Any transportation mode in the urban area
n = Nest of transportation modes m that has a specific characteristics (e.g., pure modes
including private and public or combined modes) that are available for user class l
with trip purpose o travel between origin-destination pairs is
lo
is = Set of all nests of modes n that are available for user class l with trip purpose o
travel between origin-destination pairs is

lo
M n = Set of all transportation modes m in the nest n for user class l with trip purpose o
travel between origin-destination pairs is
k = Departure time period for user class l with trip purpose o using mode m in the nest
n to travel between origin-destination pairs is

lo
Km = Time horizon of the departure time periods k for users of class l with trip purpose
o using mode m between origin-destination pairs is
a = A link in the set A in the multimodal network ( N , A)

�� = the travel time of vehicles entering link a at the end of the ��th time interval
��
(=Departure time period k ) for user class l with trip purpose o using mode m in
the nest n to travel between origin-destination pairs is
Awslo
= the value of the w th socio-economic variable that influences trip attraction
at destination s for users of class l with trip purpose o

W
lo lo lo lo
A s iw g w ( Aws ) =a composite measure of the effect that socio-economic variables, which,
w 1
are exogenous to the transport system, have on trip attraction at destination
s for users of class l with trip purpose o .
lo
gw ( Aws ) = a given function specifying how the w th socio-economic variable Awj
lo lo

influences trip attraction at destination s for users of class l with trip purpose
o , and

The quantities ilo and iw for w 1, 2, ...,W are coefficients to be estimated, where ilo 0 .
lo

During the time period required to achieve the short-run equilibrium predicted in the model,
socio-economic activities in the system will remain essentially unchanged; the composite effect

Alo for users of class l with trip purpose o of these activities is assumed to be a fixed constant.
j

E loi = the value of the th socio-economic variable that influences the number of
trips generated from origin i for users of class l with trip purpose o ,
q Eloi = a given function specifying how the th socio-economic variable, E loi ,
influences the number of trips generated from origin i for users of class l with
trip purpose o , and
Eilo lo
q ( E loi ) = a composite measure of the effect the socio-economic variables,
1
which are exogenous to the transport system, have on the number of
trips generated from origin i for users of class l with trip purpose o

The quantities lo
and lo
for 1, 2, ..., are coefficients to be estimate l L, o O .

lo
Similar to As , Eilo is assumed to be a fixed constant during the time period required to
achieve short-run equilibrium.

��
�� = the inflow rate to link a towards destination s at the beginning of the ��-th time

interval for user class l with trip purpose o using mode m in the nest n ,which
is assume to be constant during the entire �� th time interval
��
�� = the minimum travel time from node i to s destination at the end of time

interval k for user class l with trip purpose o using mode m in the nest n

S ilo = the accessibility of origin i I lo as perceived from user of class l with trip purpose
o traveling from that origin.

Gilo = the number of trips generated from origin i for users of class l with trip purpose o .

d is k = the number of trips of users of class l with trip purpose o traveling from the origin
lonm

node i I lo to the destination node s Dilo and whose already chose the mode of
lo lo
transport m M n from the nest of modes n is and start their trip at the time
lo
interval k Km .

lonm
d is = the number of trips of users of class l with trip purpose o traveling from the origin
node i I lo to the destination node j Dilo and whose already chose the mode of
lo lo
transport m M n from the nest of modes n is

lon
node i I lo to the destination node s Dilo and whose already chose the nest of
lo
modes n is .

lo
node i I lo to the destination node s Dilo .

MSTEM:

S ilo max 0, ln exp ( θ ilo lonmk
is
lo
As ) i I lo , l L, o O
s Dilo n lo
is
lo
m Mn k lo
Km

Gilo lo
Silo Eilo i I lo , l L, o O

exp( θ ilo lonmk
is
lo
As )
lo lo lo
lo n m M n k Km
d is Gilo is
is R lo , l L, o O
exp( θ i
lo lonmk
is A lo
s )
s Dilo n lo
is
lo lo
m M n k Km

exp( θ ilo lonmk
is
lo
As )
lo lo
lon lo m M n k Km lo
d is d is , n is , is R lo , l L, o O
exp( θ i
lo lonmk
is A lo
s )
lo lo lo
n is m M n k Km

exp( θ ilo lonmk
is
lo
As )
lo
lonm lon k Km lo lo
d is d is , m Mn , n is , is R lo , l L, o O
exp( θ i
lo lonmk
is A lo
s )
lo lo
m M n k Km

lo lonmk lo
lonmk lonm exp( As ) lo lo lo
d is d is i is
, k Km , m Mn , n is , is R lo , l L, o O
exp ( θ i
lo lonmk
is
lo
As )
lo
k Km
lonmk
d is can also be given by the following combined trip generation, trip distribution, modal split,
and departure time demand model (function):
lo lonmk lo
lonmk lo lo lo exp( i is As )
d is ( S i E )
i
exp( θilo lonmk
is
lo
As )
j Dilo n lo
is
lo lo
m M n t Km
lo lo lo
k Km , m Mn , n is , is R lo , l L, o O

4. Discrete-Time Domain Finite-Dimensional Nonlinear
Complementarity DUE Multiclass Simultaneous
Transportation Equilibrium Model (DMSTEM)
Following the same thought of Ban et al. (2008), the dynamic user equilibrium traffic assignment (NCPDUE-DTA) can be represented by the
following two NCPs:

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��, ��, �� (6)
��

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��, ��, �� (7)
��

To extend the DTA model above to model simultaneous route and departure time, we assume the early/late arrival penalty function for each
��
user class l with trip purpose o using mode m in the nest n to travel between origin-destination pairs is departing at time �� (denoted as �� )i

�� 2
�� = 0.5 ��∇ + �� − �� , ∀��, ��, ��, �� ≠ �� ∀ ��, ��, ��, ��
��
where �� , ∀��, ��, ��, �� ≠ �� is the desired arrival time for user class l with trip purpose o using mode m in the nest n to travel between
��
origin-destination pairs is departing at time ��. Therefore, ��∇ + �� − �� represents early or late arrival for travelers of class l
with trip purpose o using mode m in the nest n departing �� to �� at time ��. In other words, the cost (disutility) for such a traveler at time �� is
�� 2
Ψ�� = �� + �� = �� 0.5(��Δ + �� − �� ) , ∀��, ��, ��, �� ≠ ��, ∀��, ��, ��, ��

Further, we denote the minimum disutility between �� and �� for user class l with trip purpose o using mode m in the nest n
(for all time intervals) as

��
�� = min Ψ�� , ∀��, ��, �� ≠ ��, ∀��, ��, ��, ��
��
��

Therefore, the departure time choice condition can then be expressed as:

��
�� ≥ 0, �� Ψ�� = �� , ∀��, ��, ��, �� ≠ ��, ∀��, ��, ��, ��

��
�� = 0, �� Ψ�� > ��
��
��
, ∀��, ��, ��, �� ≠ ��, ∀��, ��, ��, ��

This condition simply states that travelers are trying to minimize their disutility when choosing departure times.

Combining equations (6) and (7) with the following two NCPs we can combined the dynamic traffic assignment with
departure time ((NCPDUE-DTA-DT) :

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��, ��, �� (8)
��

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��, ��, �� (9)
��

Similarly, combining equations (6)-(9) with the following two NCPs we can combined the Dynamic Traffic
Assignment, Departure Time and Modal Split ((NCPDUE-MS-DTA-DT) :

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��, ��
��
(10)

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��, ��
��
(11)

Combining equations (6)-(11) with the following two NCPs we can combined the Dynamic Traffic
Assignment, Departure Time, Modal Split, and Nested Modal Split (NCPDUE-NMS-MS-DTA-DT):

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��
��
(12)

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��
��
(13)

Combining equations (6)-(13) with the following two NCPs we can combined the Dynamic Traffic
Assignment, Departure Time, Modal Split, Nested Modal Split, and Trip Distribution
(NCPDUE-TD-NMS-MS-DTA-DT):

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��
��
(14)
��

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��
��
(15)
��

Combining equations (6)-(15) with the following two NCPs we can combined the Dynamic Traffic Assignment, Departure Time,
Modal Split, Nested Modal Split, Trip Distribution, and Trip Generation (NCPDUE-TG-TD-NMS-MS-DTA-DT):

0 ≤ �� ⊥ Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, �� (16)
��
��

0 ≤ �� ⊥ Φ��(�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) ≥ 0, ∀��, ��
��
(17)

where the decision variables are defined as follows:
��
�� = �� , ∀��, ��, �� , ∀��, ��, ��, �� , �� = �� , ∀��, ��, ��, �� ≠ �� , ∀��, ��, ��, ��
��

��
�� = �� , ∀��, ��, ��, �� ≠ �� , ∀��, ��, ��, �� , �� = �� , ∀��, ��, �� ≠ �� , ∀��, ��, ��, ��
��
�� = �� , ∀��, ��, ��, �� ≠ �� , ∀��, ��, ��, �� = �� , ∀��, ��, �� ≠ �� , ∀��, ��, ��
��
�� = �� , ∀��, ��, ��, �� ≠ �� , ∀��, �� , �� = �� , ∀��, ��, �� ≠ �� , ∀��, ��

�� = (�� , ∀��, ��, �� ≠ ��), ∀��, ��, �� = (�� , ∀��), ∀��, ��
��
��
��

�� = (�� , ∀��), ∀��, ��, �� = (�� , ∀��, ��)
��

��
�� = min�� μ�� , ∀��, ��, �� ≠ ��, ∀��, ��, ��, �� = min�� μ�� , ∀��, ��, �� ≠ ��, ∀��, ��
��

�� = min��∈�� μ�� , ∀��, �� ≠ ��, ∀��, ��, �� = min��∈�� μ�� , ∀��, �� .
��
��

Similarly to the formulation of Ban et al. (2008), all Φs functions can be defined as follows:

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

�� 3,�� ,��
(�� + {�� ℎ ��
�� +1
��
�� −1≤ �� <��
∆

+[1 − ��3,��
��
,��
��
]��ℎ �� ,�� +1 ��
} − ��ℎ �� ), ∀��, ��, ��), ∀��, ��, ��, �� (18)

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

��
( �� – ��
��

− { [��1,��
��
��
�� 2,��
��
,��
��
��

�� +1
�� :�� ≤ �� −1 Δ<��
1,�� +1 �� +1
+ �� 1 − ��2,��
��
,��
�� ]}, ∀��, ��, ��), ∀��, ��, ��, �� (19)

where

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

��
Ψ�� − �� , ∀��, ��, ��, �� ≠ �� ∀��, ��, ��, �� (20)

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

lo lonmk lo
�� exp( As )
�� − (
lo
S ilo E ilo ) i is
0 , ∀��, ��, �� ≠ �� ∀��, ��, ��, ��, ��
exp ( θ ilo lonmk
is
lo
As )
s Dilo n lo
is m lo
Mn k lo
Km

(21)
��
Equations (21) ensure that if the demand �� and a period time ( departure time ) �� is positive it should
be given by the MSTEM combined trip generation, trip distribution, modal split, and the departure time deman
model (function) mentioned before.

Following the same concept we can define all others Φ�� ‘s and Φ�� ‘s functions where all Φ�� ‘s given by
MSTEM model equations.

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

��
μ�� − ��
�� , ∀��, ��, ��, �� ≠ �� ∀��, ��, �� (22)

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

exp( θ ilo lonmk
is
lo
As )
lo
k Km
lonm
d is lon
d is 0 ∀��, ��, ��, �� , ∀��, ��, �� ≠ �� (23)
exp( θ ilo lonmk
is
lo
As )
lo lo
m M n k Km

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��

��
μ�� − �� , ∀��, ��, ��, �� ≠ ��
�� ∀��, �� (24)

Φ�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��
��
=
exp( θ ilo lonmk
is
lo
As )
lo lo
lon lo m M n k Km lo
d is d is 0 , n is , is R lo , l L, o O
exp( θ i
lo lonmk
is A lo
s )
lo lo lo
n is m M n k Km

, ∀��, ��, �� ≠ �� (25)

Φ�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��
��
=
��

��
μ�� − ��
�� , ∀��, ��, �� ≠ �� ∀��, �� (26)

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��
��

exp( θ ilo lonmk
is
lo
As )
lo lo lo
lo lo n m Mn k Km
d is ( S ilo E ilo ) is
0 is R lo , l L, o O
exp( θ i
lo lonmk
is A lo
s )
s Dilo n lo
is
lo
m Mn k lo
Km

, ∀��, �� ≠ �� (27)

Φ�� (�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��) =
��
��

��
�� − �� , ∀�� ∀��, �� (28)

Φ�� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , ��
��
=

([ S ilo max 0, ln exp ( θ ilo lonmk
is
lo
As ) 0] i I lo , l L, o O (29)
s Dilo n lo
is
lo
m Mn k lo
Km

Theorem: The NCPDUE-TG-TD-NMS-MS-DTA-DT model (6) - (17) has a nonempty and compact solution set if the following four
conditions are satisfied.

(a) The link travel time is positive and finite for any finite ��;

(b) ��1 is bounded from above;

��
(c) �� ∀��, ��, ��, �� ≠ �� ∀ ��, ��, ��, �� is positive and bounded from above; and

(d) �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� are
��

continuous with respect to ��
, ��
, ��
, ��
, ��
, ��
, �� , �� , �� , �� , �� , �� .
��
��

5. Solution Algorithm
we presented a heuristic approach to solve the link-node based simultaneous Trip
Generation (TG) - Trip Distribution (TD) - Nested Modal Split (NMS) - Modal Split
(MS) - Dynamic Trip Assignment (DTA) - and Departure Time (DT) Model
(NCPDUE-TG-TD-NMS-MS-DTA-DT). The solution convergence condition of the
algorithm cannot be established mainly due to the fact, as shown in the definitions of
functions �� , �� , �� , �� , �� , �� , �� , �� , �� , �� , �� ,
��
�� in Equations (18)-(19), that the model is not close-formed. However, for a
given demands vectors �� , �� , �� , �� , �� (called base demand) and its
��
resulting inflow vector �� (called base inflow), all the indicator functions and the
exit times (e) can be calculated and fixed via a so-called dynamic network loading
procedure (see Ban et al. 2008). This leads to the so-called relaxed sub-problems that
are explained in details in the appendix.

DSTEM Algorithm:

Step 1: Initialization. Assign a base demands vectors

�� (��) �� (��) �� (��) �� (��) �� (��) (��)
�� , �� , �� , �� , �� and a base inflow vector ��

Step 2: Major Iteration. Set �� = ��.
Step 2.1: Construct and solve RNCPDUE-DTA (A1) - (A2). Repeat this
(��)
�� times. Each time using �� as the fixed demand. For the first
(��)
time, use �� as the base inflow. For other times, use inflow from
the previous solve as the base inflow. Denote the final inflow (after
(��)
�� the solves) as �� .

Step 2.2: Construct and solve RNCPDUE-DTA-DT (A5) - (A8). Repeat this
�� (��)
�� times. For the first time, use �� as the base inflow and use
(��)
�� as the initial demand. For other times, use inflow and demand
obtained from the previous solve as the base inflow and initial demand.
��
Denote the final inflow and demand (after the �� solves) as ��
��
and �� respectively.

Step 2.3: Construct and solve RNCPDUE-DTA-DT-MS (A10) - (A15). Repeat
(��) (��)
this �� times. For the first time, use �� and �� as the
(��)
base inflow and base demand respectively and use �� as the initial
demand. For other times, use inflow and demands obtained from the
previous solve as the base inflow and initial demand. Denote the final
(��) (��)
inflow and demands (after the �� solves) as �� , �� , and
(��)
�� respectively.

Step 2.4: Construct and solve RNCPDUE-DTA-DT-MS-NMS (A17) - (A24).
(��) (��)
Repeat this �� times. For the first time, use �� , �� , and
(��) (��)
�� as the base inflow and base demands respectively and use ��
as the initial demand. For other times, use inflow and demand obtained
from the previous solve as the base inflow and initial demand. Denote
(��)
the final inflow and demands (after the �� solves) as �� ,
(��) (��) (��)
�� , �� , and �� respectively.

Step 2.5: Construct and solve RNCPDUE-DTA-DT-MS-NMS-TD (A26) -
(��)
(A35). Repeat this �� times. For the first time, use �� ,
(��) (��) (��)
�� , �� , and �� as the base inflow and base demands
(��)
respectively and use �� as the initial demand. For other times, use
��
inflow and demand obtained from the previous solve as the base inflow
and initial demand. Denote the final inflow and demands (after the ��
(��) (��) (��) (��) (��)
solves) as �� , �� , �� , �� , and �� respectively .
��

Step 2.6: Construct and solve RNCPDUE-DTA-DT-MS-NMS-TD-TG (A37) - (A48).
(��) (��) (��) (��)
Repeat this for �� times. For the first time, use �� , �� , �� ,�� ,
(��) (��)
and ��
��
as the base inflow and base demands respectively and use �� as the ��
initial demand. For other times, use inflow and demand obtained from the
previous solve as the base inflow and initial demand. Denote the final inflow and
(��) (��) (��) (��) (��)
demands (after the �� solves) as �� ,�� , �� , �� , �� ,
��
(��)
and ��
��
respectively . Call them the candidate solution

Step 2.7: Convergence Test. If certain convergence criterion is satisfied at the
candidate solution, go to Step 3; otherwise, go to Step 2.8.

Step 2.8: Update and Move. Set
(��+��) (��) ��
�� = �� + ��(�� − �� )
(��+��) (��) ��
�� = �� + ��(�� − �� )
(��+��) (��) ��
�� = �� + ��(�� − �� )
(��+��) (��) ��
�� = �� + ��(�� − �� )
(��+��) (��) (��) (��)
��
��
= ��
��
+ ��(��
��
− �� )
��

(��+��) (��) (��) (��)
��
��
= ��
��
+ ��(��
��
− ��
��
)

Set n = n + 1 and go to Step 2.1.

(��) (��) (��) (��) (��) (��)
Step 3 Find an optimal solution �� ,�� , �� , �� , �� , and ��
��

In Step 2.1 and 2.2 of DSTEM Algorithm, the calculation of the indicator functions
and exit times are done via dynamic network loading, as detailed in Nie and Zhang
(2005) and Ban et al. (2008). The relaxed sub-problems RNCPDUE-DTA and
RNCPDUE-DTA-DT, RNCPDUE-DTA-DT-MS, RNCPDUE-DTA-DT-MS-NMS,
RNCPDUE-DTA-DT-MS-NMS-TD, and RNCPDUE-DTA-DT-MS-NMS-TD-TG are
solved directly by the PATH solver in GAMS (Ferris and Munson 1998). For detailed
discussions, one can refer to Ban et al. (2008)

In Step 2.7, there are several commonly used gap functions that can be used in
the convergence test. The first one is based on the difference of the candidate solution
and the base solution. In particular, we can check whether the following condition
holds:
��
�� = �� − �� + �� − �� + �� − ��
��
��
��
�� − �� + ��
�� − ��
�� + ��
�� − ��
�� < �� ∀��, �� (30)
��

��
Especially if �� = �� , �� = �� , �� − �� ,
��
�� = �� , = , and ��
= ��
we obtain a fix point solution
��
��
��
��
��
to the original problem NCPDUE-DTA-DT-MS-NMS-TD-TG.

We can also check directly the conditions of route choice, departure time choice,
nest mode choice, mode choice from the nest, trip distribution choice, and trip
generation choice by defining the second gap function:

��
�� = �� + �� + �� + ��
��
+ ��
��
�� + ��
��
�� < �� ∀��, �� (��)
��

Similarly, if �� = ��, both conditions are satisfied, the obtained candidate
solution is indeed an optimal solution. In (30) and (31), �� and �� are both small
scalars representing user-defined convergence accuracy.

6. Conclusions and Future Research
In this paper we presented a link-node based discrete-time NCP model for
DUE that combined multiclass trip generation, trip distribution, nested
modal split, specific modal split, dynamic trip assignment and departure
time (DMSTEM) in a unified formulation.

This model combined the dynamic link-node based discrete-time NCP
model that developed by Ban et al. (2008) and the static Multiclass
Simultaneous Transportation Equilibrium Model (MSTEM) that developed b
Hasan and Dashti (2007).

We also developed an iterative solution algorithm for the proposed model
by solving several relaxed NCP in each iteration.

Most of the recent DUE models deal only with the traffic assignment and fixed
demand flows between each O-D pair at each time period of the time horizon or an
elastic dynamic demand (input to DTA models) that is usually modeled as departure-
time choice for single user (traveler) class.

To apply DTA with departure-time choice (DTA-DT) in practice, we require fixed
demand flows between each O-D pair during the whole time horizon and the DTA-DT to
be embedded in more comprehensive transportation planning framework that results in
an inconsistence prediction process.

The DMSTEM model will generate a better solution than this inconsistent prediction
process within a comprehensive planning and operations framework.

We would apply the DMSTEM and it solution algorithm to a prototype
example to get more insight and implication of its predictions and
validity.

This prototype solution will encourage applying the model to real
world transportation network. A comparative study between
DMSTEM as a dynamic model and MSTEM as static model in term of
their prediction accuracy and solution complexity is very desirable in
the transportation planning in general and in particular, the
Intelligence Transportation System (ITS).

Transportation network analysis tools can be broadly categorized as
static analysis tools and dynamic analysis tools. The former focuses on
long-term, steady traffic states, often referred as the four-step demand
analysis models (Sheffi, 1985), while the latter focuses on short-term,
dynamic traffic states, often called dynamic network analysis or
dynamic traffic assignment (DTA, see Peeta and Ziliaskopoulos (2001)).

Traditionally, these two types of models have been applied in quite
distinct scenarios: static tools for transportation planning purposes and
dynamic tools for traffic operation purposes.

It would be ideal to combine the planning and operation tools to address
transportation problems in a more comprehensive manner. In practice,
such combination has been recently experimented and tested. Examples
include the corridor management plan demonstration project (CCIT,
2006) and the state-wide corridor system management plan that is
currently underway in California, which aims to integrate long-term
planning tools (such as the static four-step demand analysis models) with
operational tools such as microscopic traffic simulation and dynamic
traffic assignment. The ICM program by FHWA also focuses on
combining different transportation analysis tools for integrated corridor
planning and operations.

Combining planning and DTA models in theory is not trivial because the
two types of models are based on quite distinct assumptions. In this
paper we present a combined model to simultaneously consider the
origin, destination, mode, departure, and route choices of multiclass
travelers.

Multiclass Dynamic Transportation Model for User Equilibria

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