Super-network Based Equilibrium Model and Algorithm for Multi-mode Urban Transport System

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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