A Genetic Algorithm for Solving a Class of Multi-objective Bilevel Programming Problems

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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.

Corresponding

Author

You are
strongly

suggested to provide the informat ion about corresponding
author including: Name, Email, Mobile phone

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