Super

n
etwork
Based
Equilibrium Model
and
Algorithm for M
ulti

mode
Urban
Transport System
Bingfeng Si
1
, Ming Zhong
2
, and Ziyou Gao
1
1 S
chool
of
Traffic and Transportation,
Beijing Jiaotong University, Beijing,
P.O.
Box
100044
;
PH (
8610
)
51687143
; FAX (
86
10
) 51688421911; email:
bfsi
@
bjtu
.edu
.cn
2 Department of Civil Engineering, University of New Brunswick, Fredericton, N.B.,
Canada E3B 5A3
;
Phone:
(
1506
)
4526324
;
F
AX
(
1506
)
4533568
;
Email:
ming@unb.ca
ABSTRACT
In this pap
er, the structure of m
ulti

mode
urban transport system is fully analyzed
and then a
super

network model is proposed to describe such a system. Based on the
analysis of travelers’ combined choices, the generalized travel cost function
s
and the
link impedanc
e function
s
are formulated, where the interferences between different
modes
on the same road
segments are taken into account. O
n the basis of these, a
variational inequality model is proposed
to
describe
equilibrium assignment
for
m
ulti

mode
urban transpor
t system. The corresponding solution algorithm is
also
presented. Finally, a numerical example is provided to illustrate the proposed model
and algorithm.
INTRODUCTION
With rapid urbanization and motorization, travel demand and travel distance increase
co
nsiderably
in most cities of China. The transportation planning theories and
methods, which were primarily developed for pure motorized

traffic transport
system and used extensively in developed countries, are deemed inadequate for
mixed

traffic transport
system found in
China
. Generally, mixed

traffic urban
transport system consist
s
of
several
modes such as
automobile
,
transit
and bicycle
and it
is much more complicated than purely motorized
system
in developed
countries.
Therefore, assignment model for
si
mulat
ing/predicting
travelers’
combined choice
behaviors in
cluding
mode and route choices within a mixed

traffic
network
is
pressingly needed and deserves a special attention.
In
the
past decades, various assignment models and algorithms
(
Beckmann
et al
,
1
9
5
6;
Smith
,1979;
Dafermos
,1980;
Dial
, 1996; Yang
and Huang
, 2004
)
have been
proposed
for purely motorized

traffic
urban transport
system
based on
Wordrap
principle
(
Wordrap
, 1952
)
, which implies that they paid primary attention to
automobile
driver
’
s route
choice
,
while the
travel
er
’
s mode choice was
largely
neglected. The reason for such an outcome is the insignificance of non

motorized
traffic in developed countries. A
ssignment models for
mixed

traffic urban
transport
system
(
Florian and Nguyen
,
19
7
8;
Fis
k and Boyce
,
1
983;
Lam and Huang
, 1992;
Abrahamsson and Lundqvist
, 1999;
Nagurney and Dong
, 2002
)
were proposed,
however,
the
topology
structur
e
of such
a system and
the interferences between
different transport modes were not
considered
in the
se previous
research
work
s
. Si
et
al
.
(
2
008
)
moved a step forward by
developing a combined model for mixed

traffic
transport
system, in which the interference between motorized and non

motorized
traffic was
considered
.
However, the structure of such a system as well a
s the
generalized travel cost functions of different modes
was
not
well addressed.
In this paper, the structural
characteristic
of mixed

traffic urban transport
system
is
fully analyzed
,
and the system is then decomposed into a set of sub

networks
. Based
o
n
super

graph theory, a
super

network model is proposed to describe such a system.
The generalized travel cost functions and the link impedance functions for different
transport modes within such a system are formulated and the interferences between
diffe
rent modes are considered. The traveler’s
combined choice
behaviors including
mode and route choice are then analyzed in such a
super

network. On the basis of
these, a
n
equilibrium assignment model based on
variational inequality (VI) is
proposed. The solu
tion algorithm is also presented. Finally, a numerical example is
provided to illustrate the proposed model and algorithm.
SUPER

NETWORK MODEL
F
OR
MIXED

TRAFFIC
TRANSPORT
SYSTERM
In this paper, the
mixed

traffic
transport
system
is expressed as
G
= (
N
,
A
,
K
), where
N
is the set of nodes
,
A
is the set of road segments
and
K
is the set of transport
modes.
T
here are
K
sub

net
work
s
in
mixed

traffic
transport
system
and each
sub

net
work
, represented by
G
k
= (
N
k
,
A
k
,
k
), where
k
K
, is correspond
ing
to the
sub

netwo
rk of transport mode
k
.
The following example is used to
illustrate the
structural
characteristic
of
mixed

traffic
transport
system
.
The urban road traffic network is shown in Figure 1,
which consists of one O

D pair (
i
,
j
), 9 nodes, 12 road sections, two
motor modes
(
automobile
and
transit
) and one non

motor mode (bi
cycle
).
Figure 2 show
s
the
structures of sub

networks of
different transport modes
by assuming that automobile
and
transit
can only access a part of all 12 links.
F
igure
1
.
An example of
mix
ed

traffic transport
network
F
igure
2
.
The
sub

networks of different transport modes
In such
mixed

traffic
transport system,
traveler
s
from original
i
to destination
j
should make two successive decision
s
. The first
one
is the mode choice and the
second
is the route choice in the corresponding sub

net
work
once the transport mode
is selected.
At th
e first
stage, the
mixed

traffic
transport
system
can be
described
with
simplifi
cation
as
the
follow
ing three simplified mode choices, one for each
available mod
e
s
between the O

D pair (
i
,
j
)
.
F
igure
3
.
Level 1: Mode choices of the
mixed

traffic
network
I
t
can be
found here
that
possible routes between
the O

D
pair
(
i
,
j
) of different
modes in Figure 3 can be
described by
the
sub

net
work
s
of the corresponding m
odes
shown in Figure 2
.
According to the structural
characteristic
of
mixed

traffic
transport
system
, a
super

network model is proposed to describe
such
a
system in this paper. In
the
proposed
super

network
model
, each node is described by two variables (
n
,
k
), where
N
,
denotes the physical
nodes of
the road network and
k
K
, denotes the
transport mode
that
can
run through node
n
.
Similarly, the link in such
super

network
is described by two variables
(
a
,
k
), where
a
A
,
denotes th
e
physical road
links
and
k
K
,
denotes the
transport mode
that
can access
road
link
a
.
T
he set of links
connecting the different nodes can be divided into two categories. One category
includes loading
/unloading
link
s
and
one end of
such a link
is original
or destination
.
T
he other category
consists of
in

vehicle link
s
that
are
the accessible links
for each
mode
in
each
sub

net
work
.
By constructing such
a
super

network,
the
mixed

traffic
transport system can
be simplified as
a simple
“
road
”
network
. It then
can be
used
directly
as a generalized network for traffic assignment or network analysis
purposes
.
Based on the discussions above, the
mixed

traffic
transport system
in Figure 1
can
be
represented
as the following
super

network
as shown
in Figure 4.
F
igu
re
4
.
The
proposed
super

network
for the mixed transport system
CONSERVATION CONDITIONS IN
MIXED

TRAFFIC SYSTEM
First of all,
without losing any generality,
it is assumed that the O

D demands in
a
mixed

traffic
transport
system
are given and fixed.
F
or
a
given
O

D pair, the sum of
travel demands of different modes equals to the total travel demand, that is
,
(1)
where
is the total demand between O

D
pair
w
;
is the demand that select
s
the transport mode
k
between O

D pair
w
.
Secondly, for
a given
O

D pair and mode, the sum of travel demands on different
routes in the corresponding sub

net
work
equals to the travel demand of the
c
orresponding mode, that is
,
(2)
where
is the travel demand on the route
r
in sub

net
work
k
(for mode
k
)
between O

D pair
w
.
In addition, in the sub

net
work
k
between O

D pair
w
, the travel demand on road
link
a
can be represented by the travel demand on the routes, that is
,
(3)
where
is the travel dema
nd of mode
k
on
the
road
link
a
;
is route and
road incidence variable in the sub

net
work
k
between O

D pair
w
, if
the mode
k
can
access the
road
link
a
and it
is on the route
r
that connects the O

D pair
w
, then
=1, otherwise,
=0.
In general
,
traffic
flow is defined as the number of all vehicles
passing by a
road
section during
a
time unit
(minute or hour)
for the purely
motorized
system
.
I
n the
context of mixed

traffic system,
the
flow
of e
ach mode
can be
converted from
the
corresponding
travel demand, that is
(
Si
et al
,
2
008
)
=
,
(4)
where
is the flow of mode
k
on r
oad
link
a
;
is
passenger car unit (
pcu
)
conversion coefficient of mode
k
;
is occupancy rate of mode
k
,
which
indicates
the average number of
travelers
within each vehicle of mode
k
.
TRAVEL

DEMAND BASED GENERALI
ZED TRAVEL COST FUNCTION
In this paper, the
level
of
congestion in
each
sub

net
work
, travel time and
fare
of
transport mode are
all assumed
included in the generalized travel cost. The
generalized travel cost function can
then
be written as
the
follow
ing
:
,
(5)
where
is the generalized travel cost
for the
mode
k
between O

D pair
w
;
is
the vector of travel demands between O

D pair
w
, that is,
=
;
represents the
level
of
congestion
in sub

net
work
k
between O

D pair
w
,
which is
a function of
the travel demand between O

D pair
w
;
denotes the
fare
of transport mod
e
k
between O

D pair
w
;
denotes the equilibrium travel time of
mode
k
between O

D pair
w
;
and
are parameters
concerned with
mode
k
.
Let
represent the travel cost ex
cluding travel time of mode
k
between
O

D pair
w
. That is
,
(6)
Then, the function
(5)
can be rewritten as
,
(7)
T
he travel time on route
r
in sub

net
work
k
between the O

D pair
w
, denoted by
, can be expressed as
,
(8)
where
denotes the
travel time of transport mode
k
on road
link
a
, which can be
computed by the link impedance function.
In
mixed

traffic
transport system,
the interferences among different transport
modes
will
present
if there
is
no physical
barrier
between
different modes
. Therefore,
the link impedance function
of the
mixed

traffic
transport
system
should be
different
from
that for
single

mode transport
system
, which
can be formulated as:
,
(9)
where
is
the free

flow travel time of mode
k
on road
link
a
;
is the
practical capacity
of
road
link
a
;
is the vector of
the
flow
of different modes
on
road
link
a
, that is
.
In general,
and
can be
assumed as
the
constants
and
can be treated as a function of
. According to
the Equation
(4), the flows of different modes on road
l
ink
a
can be expressed by the travel
demand of corresponding modes. Consequently, the
Function
(9) can be rewritten as
follows:
,
(10)
where
is the
vector of the travel demand on road
link
a
, that is
.
SUPER

NETWORK BASED
EQUILIBRIUM
ASSIGNMENT MODEL
In this paper, the assignment
model
for
mixed

traffic
tr
ansport system is formulated
based on
user equilibrium (UE) principle.
In o
rder to
be
consistent with choice
behavior
theories
, the
user equilibrium
defined
in
mixed

traffic
transport system
is
divided into
the following
two categories. One category equilibrium exits
among
different transport modes, namely the generalized travel
costs of
the
selected
transport modes is the same and the minimum while the generalized travel costs of
unselected transport modes must be not less than the minimum travel cost between
a
given
O

D pair. The other category is the traditional equilibrium amo
ng different
routes in each sub

net
work
between
the
O

D pair.
Therefore, t
he user equilibrium in
mixed

traffic
transport system can be described as:
, if
,
(11)
, if
,
(12)
where
is the generalized travel cost between O

D pair
w
at equilibrium.
Note that the generalized travel cost functions
and the link
imped
ance functions
proposed in this paper
are all
asymmetric
,
which
means
that
the
generalized
travel
cost
s
or the link travel time
s
of each mode
are influenced by
not only
its own flow
but also
those from the other competing modes
.
In this
paper
,
the following VI
model is proposed to describe the
super

network based
assignment problem
for
mixed

traffic transport system
: to find
such that:
+
(13
a
)
where
(
13b
)
SOLUTION
ALGORITHM
“Diagonalization” is the o
ne of the
most commonly used
approaches
for
solv
ing
VI
model because of its easy implementation. The “diagonalization” algorithm that
provides the solution for
the assignmen
t problem (1
3
) is based on solving a series of
mathematical programs. At each iteration, the vector function
and
are “diagonalized” at the current solution, yielding a symmetric assignment
problem.
Based on the
same approach
, the following mathematical program
is
formulated:
min
=
+
(1
4
a)
s.t
(1
4
b)
In this paper, the method of successiv
e average (MSA) is
used
to solve the
“diagonalized”
minimization problem (1
4
a)

(1
4
b). The
procedures
of MSA are
described as
the
follow
ing
:
Step 0:
initialization. Set
=0 and
=0 for any
w
,
k
and
a
. Compute
and
. Find the shortest route in the sub

net
work
k
between O

D pair
w
and get the
corresponding minimum travel time
. Then calculate the
generalized travel cost
of mode
k
between O

D pair
w
. Execu
te the travel demand assignment
between O

D
pair
w
according to the following rules:
=
(
1
5
)
Subsequently, perform all

or

nothing assignment to load
in sub

net
w
ork
k
and
obtain the travel demand of mode
k
on road
link
a
,
. Set iteration
n
=1.
Step 1: compute
and
based on
and
respectively.
Step 2: find
the shortest route in the sub

net
work
k
between O

D pair
w
and get the
corresponding minimum travel time
. Then calculate the
generalized travel cost
of mode
k
between O

D pair
w
and execute the travel demand assignment
between
O

D pai
r
w
according to the following rules:
=
(
16
)
Perform all

or

nothing assignment to load
in sub

net
work
k
and obtain
.
Step 3: compute
,
(
1
7
a
)
,
(
1
7
b
)
Step 4: Convergence test. If a convergence criterion is met, stop
.
T
he current
solutions, {
} and {
}, are the sets of equilibrium travel demands for
the
transport modes and
the sub

networks considered
respectively; otherwise, set
n
=
n
+1
and go
to step 1.
NUMERICAL EXAMPLE
A simple numerical example is used to illustrate th
e effectiveness of the model and
the solution
algorithm proposed in this paper. The
numerical example is based on a
simple
m
ixed

traffic
transport system
a
s shown in
the
Figure 1. The corresponding
super

network
is given in
the F
igure 4.
In this paper, t
he
generalized travel cost functions use the logarithmic form
as
the
follow
ing
:
,
(
18
)
T
he following link imped
ance function
is
proposed for
the
mixed

traffic
transport
system
(
Si
et al
,
2008
)
:
,
(
19
)
Here
we
assume that
= 0.15,
= 4,
= 0.5 and
= 0.2
respectively.
The
first two param
eters are borrowed from the original BPR function and the last two
are determined based on typical value of time
(VOT)
in China.
These parameters are
calibrated so that the calculated VOTs are close to
the
observed values.
For brevity,
the cases of
=10
,
000
passengers per hour
(
P.h

1
)
and
=20
,
000
(
P
.h

1
)
are
used
to represent the
mixed

traffic
transport system
without congestion and with
congestion respectively.
The relevant
parameters
of
12
road
link
s
in the Figur
e 1
are given in Table 1,
while the pcu conversion coefficient, the average occupancy rate, potential fee,
which are pertinent to different modes, are
listed
in Table 2.
T
able
1
.
The Relevant Data of Different Roads
R
oad
/(h)
/(h)
/(h)
/(P.h

1
)
/(P.h

1
)
/(P.h

1
)
(1,2)
0.111
0.178
0.261
1000
1000
600
(2,3)
0.128
—
0.278
700
—
400
(1,4)
0.100
0.167
0.250
1500
1500
800
(2,5)
0
.106
0.172
0.256
700
700
400
(3,6)
0.089
—
0.239
700
—
400
(4,5)
—
0.144
0.228
—
1000
600
(5,6)
—
—
0.244
—
—
600
(4,7)
0.133
0.200
0.283
900
900
500
(5,8)
0.111
0.178
0.261
700
700
400
(6,9)
0.144
—
0.294
700
—
400
(7,8)
0.094
0.161
0.244
900
900
5
00
(8,9)
0.100
0.167
0.250
900
900
500
T
able
1
.
The Relevant Data of Different Modes
Mode
U
k
A
k
car
1
4
7
bus
1.5
20
3
bike
0.25
1
0
The convergence of MSA algorithm for the “diagonalization” model (1
4
a)

(1
4
b) is
firstly analyzed by studying the variations of
automobile travel
demand from
i
to
j
against iterations. Figures 5 shows the two variations against the iteration number for
different O

D demands. It is obvious that the
automobile travel
demand rea
ches the
convergence after 8 iterations
as the solutions tend to
be
very stable after that
.
I
n
addition, i
t can be
seen
that the solution algorithm has a
good
convergence especially
for the
scenario without congestion (the
low
er
demand
case)
.
F
igure
5
.
A
utomobile
O

D demand variations against iterations
Table 3 show
s
equilibrium demands and the corresponding travel times of
different modes
on each
of the 12
road
links
when
=10000/p.h

1
.
T
able
3
.
The Equilibrium Demands and Travel Tim
es
on Each Road
Link
When
qw=10000/P.h

1
roads
/(P.h

1
)
/(P.h

1
)
(/P.h

1
)
/(h)
/(h)
/(h)
(1,2)
392.18
1.40
1338.37
0.1122
0
.1795
0.2649
(2,3)
0.38
—
253.85
0.1278
—
0.2778
(1,4)
991.8
2
4211.17
3065.06
0.1087
0.1812
0.2821
(2,5)
391.80
1.40
1084.52
0.1078
0.1759
0.2637
(3,6)
0.38
—
253.85
0.0889
—
0.2389
(4,5)
—
4203.80
2056.99
—
0.1525
0.2465
(5,6)
—
—
1647.20
—
—
0.2526
(4,7)
991.8
2
7.37
1008.06
0.1343
0.2015
0.2862
(5,8)
391.80
4205.20
1494.32
0.1217
0.1948
0.2944
(6,9)
0.38
—
1901.05
0.1732
—
0.3825
(7,8)
991.8
2
7.37
1008.06
0.0951
0.1623
0.2469
(8,9)
138
3
.
62
4212.57
2502.38
0.1299
0.2166
0.3519
According to the
results in Table 3, the feasible flows and corresponding travel
costs on all feasible routes
in each sub

network
are
obtained and
shown in Table 4.
T
able
4
.
The Equilibrium Flows and Travel Costs on Feasible Routes in
Different Sub

network
modes
routes
(no
de serial)
F
lows
/P.h

1
travel
times/h
mode split
/P.h

1
general travel
cost/h
automobile
1

2

3

6

9
0.38
0.
47
21
1384.00
5.38
2
9
1

2

5

8

9
391.80
0.47
0
6
1

4

7

8

9
991.82
0.46
95
transit
1

2

5

8

9
1.40
0.7518
4212.57
5.3824
1

4

5

8

9
4203.80
0.7495
1

4

7

8

9
7.37
0.7506
bicycle
1

2

3

6

9
253.85
1.1641
4403.43
5.3818
1

2

5

6

9
1000.00
1.1637
1

2

5

8

9
1000.00
1.1649
1

4

5

6

9
647.20
1.1637
1

4

5

8

9
494.32
1.1649
1

4

7

8

9
1008.06
1.1671
It can be seen that the travel
time
s o
n
such routes selected by
travel
ers are basically
the same when the
travel demand
in
each
sub

n
etwork reach
es
equilibrium. In
addition, the
general travel costs of different modes are also the same at equilibrium
.
These results above are consistent with
the
conditions (11) and (12)
, which indicate
that the algorithm proposed in this paper is completely effective.
CONCLUSIONS
At present, lots of
traffic
assignment models and algorithms
have been proposed
for
the transport systems dominated by
motorized

tr
affic
based on
Wordrap principle
while
those
for
mixed

traffic
transport system
found in most developing countries
are
rare
.
In this paper, based on
a proposed
super

network, the generalized travel
cost functions and the link impedance functions for differ
ent transport modes are
formulated while the interferences between different modes are considered.
A
ssignment problem for the mixed

traffic urban system is approached by solving
traveler’s
combined
choice
s of
mode and route. On the basis of these, a VI mod
el is
proposed to
solve
the
equilibrium assignment
problem of
mixed

traffic
transport
system. The solution algorithm is presented.
T
he results of the numerical example
indicate that the algorithm proposed in this paper is completely effective.
ACKNOWLEDGEM
ENTS
The
work described in this paper i
s mainly supported by the grants from
the
National
Basic
Research
Program
of
China (Project Nos. 2006CB705500)
and
the
National Natural Science Foundation of China (Project No. 70631001). It is also
partially funded b
y a Discovery Grant (Application No. 342485

07) from the Natural
Science and Engineering Research Council (NSERC), Canada.
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