A
Genetic Alg
orithm
for
Solvin
g a
Clas
s of
M
u
lti

objective
B
i
level
P
rogramming
P
r
oblems
Shanfeng Zhang
,
Keqiang
Li
Department of
Computer
,
******
University,
Guangzhou,
810008
, China
Abstract
At present, most of the research
es
on
bilevel programming prob
lems
are
focused
on
s
ingle object
ive cases. This paper discusses a bilevel
programming
problem
with upper level multi

objective optimization. In order to solve the problem
efficiently, we present a genetic algorithm using interpolation. This method does
no
t require
s
olving
lower optimization problem
frequently
. In the proposed
algorithm, firstly, the interpolation functions are adopted to approximate lower
level solution functions. As a result, the original problem can be approximated
by a single

level mult
i

objective programming. In addition,
the corresponding
interpolation functions
are
updated
such that these functions can approximate the
lower level solution function very well
.
F
inally, the mult i

objective
programming is solved for obtaining an optimal s
olution set of the original
problem. The
simulation on two
examples
indicates the proposed algorithm is
e
ffective and feasible
.
Keywords
:
m
ulti

objective bilevel programming
,
g
enetic algorithm
,
interpolation
.
1.
Introduction
Bi
level programming problem (BLPP)
is a
hierarchical optimization problem
,
which
include
s an
upper levels
deci
sion maker and
a
lower levels
deci
sion
maker. Both of them
have
the
objective function
s and
constraint condition
s.
This
problem
was
proposed by Stackelberg
[1], and known as
Stackelb
erg
problem.
The
mathematical model
of this kind of problems can be
written
as follows:
(1)
W
here
are
decision variable
s
,
are
objective function
s and
are
constraint
functions;
are the others
constraint
, such as upper
and lower
bound
s or
integer constraints
. Bilevel programming problems occur in
various applications, such as
e
conomics and
m
anagement
[2],
engineering[3],
t
ransportation
and others.
Most of these
model
s are bilevel
single
objective
problems. In fact,
t
here are
lots of real

world problem
s
that can be modeled as m
ult i

object ive
b
ilevel
p
rogramming
problems [4]
. In a mult i

object ive bilevel pro
blem at least one
deci
sion maker has more than one
objective function
, and these objectives
conflict with each other.
In recent years, mult i

objective bilevel programming
problems
ha
ve
been discussed
,
a
nd
some
efficient
methods have been
reported.
Ary
anezhad and Roghanian proposed a method for bilevel linear mult i

objective
decision making model with interval
coefficients
[4]. In this method the
tolerance
membership function is applied
.
Feng and Wen use
d a
fuzzy approach to solve
bilevel multi

objective
model[5].
In this paper,
b
ased on
the
existing
approach and
interpolation
techniques[6],
we propose a genetic algorithm
t
o solve
bilevel programming problem
s
with
upper level mult i

objective optimization
.
First,
some
sample points of the
interpolation fu
nctions are gotten by solving the lower level problems for some
given upper level values, and the interpolation functions are
calculated
by using
these interpolation points. Then, these interpolation functions
are
substituted into
the u
pper
level
problem
.
As a result, the original
mult i

objective bilevel
programming can be approximated by a single

level program. Finally, we
s
olv
e
this single

l
evel
problem by using
NSGA

II
[7] to obtain the optimal solution set
of the original problem.
The paper is organized
as follows.
In the
next section, we propose the model
that we
r
esearch
in this paper, a
nd some definitions and assumptions
are given.
In Section 3, the lower level solution based on interpolation functions is
presented. Section 4 presents the genetic algo
rithm steps
based on NSGA II
.
I
n
Section 5, the computational simulation is made to illustrate
the efficiency of the
algorithm
. Finally, we conclude our paper in section 6.
2.
Discussed problems
T
he discussed model is given as follow:
(2)
Some basic concepts associated with the problem are presented as follows
Search space
：
;
Constraint
region
：
;
Projection of
onto the leader’s decision space:
;
Feasible set for the follower for each fixed
:
;
Inducible region:
;
Definition 1
For any two
points
,
if
the following conditions are
satisfied
:
i
)
For any
, we have
;
ii
)There exists a
such that
;
T
hen, we called
dominates
.
Definition 2
For point
, if any solution in
does not dominate
s
,
then
is
a P
areto

optimal solu
tion
.
Two
assumptions
are given as follows:
i
)
is
Non

empty and compact
.
ii
)
For each variable
value
,
is
unique.
3.
The
l
ower
s
olution
b
ased
on
i
nterpolatio
n
3.1
Interpolation function
F
or
each
feasible point
,
it is necessary to
solve
lower
one
level
problem
.
F
or large

scale problems
the
amount of
calculation increased significantly
.
F
or each
,
the lower
level
optima
l solution
is
unique.
T
he problem can
be transformed into
a
multi

objective single

level
optimization problem
:
(3)
W
here,
,
can be seen as
a
function
of
upper variables
.
H
owever
, it is
difficult to obtain
.W
e use
interpolation
f
unction
to approximate
the optimal solution function
.
Hence, (3) can be transformed into (4)
(
4)
Given
points
,
,
t
hese points are
fixed in
the
lower
level
function
s,
and the lower level problems are solved to
obtain
the
lower
level
optimal solution
via a
genetic algorithm
. As a re
sult,
interpolat ion
node
s are
obtain
ed
as follows
,
Then, we
u
s
e
a cubic spline interpolation function
in the MATLAB toolbox to
get an
a
pproximate function
that is
.
In the
proposed
algorithm
, w
e
update the interpolation nodes and interpolation functions
.
This process makes
the interpolation function
a
pproximation
get better and better.
3.2
Fast non

dominated sorting
.
These
individuals in the population are sorted based on
non

domination, the fast
non

dominated sorting algorithm is
adopted
as follows [7]:
i
)
For each individual
,
there are t wo parameters
(
and
)
have
been
defined
.
r
epresent
s the number of individuals that d
ominate
i
n the population
,
whereas
r
epresent
s the set of individuals that are
dominated
by
.
ii
) We find out all indiv
iduals which
,and add them to the set
which
stores the individual rank is one, i.e.
.
iii
) We consider each individual
in
,
check
the set
which stores the
individuals
dominated
by
individual
.
For each individual
in
,
if
,
they will be stored
in
another
collection
.Set rank of individual
to second
iv
)
This process is repeated
until a
ll individuals
get their rank
values
.
3.3
D
efine the crowding distance
.
I
n the population,
t
he density of the surrounding individual
s
of
, is
expressed
by
,which is
the smallest range
that
contains
but doesn’t contain other points
around the individual
.
The specific calculat ion process is
presented
as fol
low
[7]:
i
) For each front
,
is the number of
individual
i.e.
.
ii
) For every
individual
,
set the initial
crowding
distance
.
iii
) Set
. For each individual
,
denotes the
value of the
o
bjective function
.
iv
) Let
cycle f
rom
2 to
, and
calculat
ing the follow
expression
to define
the crowding distance for each individual
.
4.
Genetic
a
lgorithm
Genetic algorithm is widely used in the
multi

objective optimizat ion problems
with
g
lobal
search capability
and robustness
[7].
In order to solve the problem
(3),
we
encod
e
the upper variable
values using real coding scheme,
a
nd
give a
fitness
function
based on n
on

dominated solutions
s
ort
ing method
and
crowding
distance
,
i
t can distinguish diff
erent individuals
e
ffectively
. Our algorithm is
developed:
Step1:
(
I
nterpol ation
function
)
Get
individual
s
in
r
and
o
mly
. By
u
s
ing
genetic algorithms
, we
o
btained
the
corresponding lower
level
optimal
solution
and
get
the i
nterpolation
function
,
which
are
nodes
.
Step2:
(I
ni tial populati on
)
Take
points randomly,
and
s
ubstitut
e
these
points
to i
nterpolation
function
to
obtain
the initial population
with a
p
opulation size
Step3:
A
rithmetic crossover operator
.
[8]
Step4:
Non

uniform mutation
operator
.
[8]
Step5:
(
Select
)
S
et
,
execute the fast non

dominated
sorting, determine their
ordinal values
, and calculate the crowding distance of the
individuals with the same
ordinal value
. Define the relationship
:For two
different individuals
and
,
if
or
,
we
call
.
Choose
individuals from
as
.
Step6
:
(
U
pdate
interpolation function
)
The
i
ndividual
s with
are
selected from
,
and we
s
uppose there are
points. For these
points, the
lower level problems are solved, and other
nodes are
gotten.
T
hese points are
used to update the
interpolation function.
In order to reduce the amount of
computation of obtaining the lower level solutions, we design a mult i

criteria
evolutionary scheme. Firstly, for each point
, generat
e
points according to
Gaussian distribution. Hence, w
e g
e
t a population
with population
size
.
fitness function
s are obtained by
upper level variable values in
the eva
luation and
selection process
, which makes
runs of genetic algorithm
are finished in one execution.
Step7:
(T
erminati on conditi on
)
If the algorithm reach the maximum
generation
, then
stops
,
and o
utput
the
best
non

dominated individuals
;
o
therwise
let
,
t
urn
to Step3
.
5.
Computational
e
xamples and
a
nalysis
I
n order t
o
illustrate
the
f
easibility and effectiveness
of the a
lgorithm
, we
construct
two
examples
according to examples in
literature
[
9
,
10
]. We solve
them by two different approaches. The first is the approach proposed in this
paper,
and the
second approach is the same as the first one except using the
MATLAB toolbox function to solve the lower level problem.
The
p
arameter
s
are
set as fo
llows
: the p
opulation size
is 100, the m
aximum generation
is 50, and the
crossover
and mutation probability
is 0.8 and 0.1.
E
xample 1
：
E
xample 2
：
The
pareto frontier
of t
he t
wo
examples
are
shown
as follows:
Fig.1
:
P
areto front of example 1
Fig.2
:
P
areto
front of example 2
Table 1
:
C
pu
time
of the two
approaches for each problem
No.
Approach1
for problem1
Approach2
for problem1
Approach1
for problem2
Approach2
for problem2
1
10.3594
64.4531
11.0781
64.0937
2
10.1719
65.6406
11.5625
59.8445
3
10.3438
6
3.3750
11.7188
61.9219
4
11.2188
66.0469
11.3281
58.6250
5
103594
65.2656
11.3125
59.4063
As can be seen from Fig.
1
,Fig.2
and Table 1
,
The
optimal
r
esults are very
close
. But the
C
pu
time
of Approach 1 is
f
ar less than
that of Approach 2
.
It can
be see
n that the proposed algorithm is feasible
and effient
.
6.
Conclusions
In t
he proposed
algorithm, the m
ulti

objective
b
ilevel
p
rogramming
problem is
t
ransformed into
a s
ingle

l
evel problem by using
interpolation
functions of the
lower level solutions. The proc
ess avoids solving the lower level problems
frequently, and reduces the computational cost.
The
major
advantage of this
algorithm is
that it can solve some c
omplex issues
, in which the lower level
problems are non

convex and
non

differentiable
.
Hence, it c
an be
used to deal
with hard
multi

objective
bilevel programming problems
.
Acknowledgements
The research work was supported by National Natural Science
Foundation of
China under Grant No. 61065009 and Natural Science
Foundation of Qinghai
Provincial under
Grant No. 2011

z

756.
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