Feature Article: The Definition of Optimal Solution and an Extended Kunhn-Tucker Approach…
1
The Definition of Optimal Solution and an
Extended Kuhn-Tucker Approach for Fuzzy
Linear Bilevel Programming
Guangquan Zhang and Jie Lu
Abstract— Bilevel decision techniques are mainly developed for
solving decentralized management problems with decision makers in
a hierarchical organization. Organizational bilevel decision-making,
such as planning of land-use, transportation and water resource, all
may involve uncertain factors. The parameters shown in a bileved
programming model, either in the objective functions or constraints,
are thus often imprecise, which is called fuzzy parameter bilevel
programming (FPBLP) problem. Following our previous work [1, 2],
this study first proposes a model of FPBLP. It then gives the
definition of optimal solution for an FPBLP problem. Based on the
definition and related theorems, this study develops a fuzzy number
based Kuhn-Tucher approach to solve the proposed FPBLP problem.
Finally, an example further illustrates the power of the fuzzy number
based Kuhn-Tucher approach.
Index Terms— Linear bilevel programming, Kuhn-Tucker
approach, Fuzzy set, Optimization.
I. INTRODUCTION
T
HE execution of many decisions in businesses is sequential,
from a higher level (leader) to a lower level (follower); each
unit independently optimizes its own objective, but is affected by
other unit’s actions through externalities. This is called bilevel
programming (BLP) problem (also called bilevel decision or
bilevel optimization problems). BLP was first introduced by Von
Stackelberg [3] in the context of unbalanced economic markets [4,
5]. In a BLP problem, each decision maker (leader or follower)
tries to optimize his/her own objective function with partially or
without considering the objective of the other level, but the
decision of each level affects the objective optimization of the
other level [6].
There have been nearly two dozen algorithms [5, 7-10]
proposed for solving BLP problems since the field caught the
attention of researchers in the mid-1970s [11-19]. Although BLP
theory and technology have been applied with remarkable success
in different domains [20-22], existing approaches mainly support
the decision situation in which the objective functions and
constraints are characterized with precise parameters. Therefore,
the parameters are required to be fixed at some values in an
experimental and/or subjective manner through the experts’
understanding of the nature of the parameters in the
problem-formulation process. It has been observed that, in most
Authors are with Faculty of Information Technology, University of Technology,
Sydney, PO Box 123, Broadway, NSW 2007, Australia (e-mail: {jielu,
zhangg}@it.uts.edu.au).
IEEE Intelligent Informatics Bulletin
real-world situations, particularly in critical resource planning,
such as planning of land-use, transportation and water resource,
the possible values of these parameters are often only imprecisely
or ambiguously known to these experts. It results in a difficulty to
fix parameters in the objective functions or constraints of a bilevel
programming model. With this observation, it would be certainly
more appropriate to interpret the experts’ understanding of
parameters as fuzzy numerical data which can be represented by
means of fuzzy sets theory [23]. A bilevel programming problem
in which the parameters either in objective function or in
constrains are described by fuzzy values is called a fuzzy
parameter bilevel programming (FPBLP) problem.
The FPBLP problem was first explored by Sakawa et al. in 2000
[24]. Sakawa et al. formulates bilevel programming problems with
fuzzy parameters from the perspective of experts’ imprecision and
proposes a fuzzy programming method for fuzzy bilevel
programming problems. However, Sakawa’s work is mainly
based on the definition of solution for bilevel programming
proposed by Bard [5, 15]. One deficiency of Bard’s linear BLP
theory is that it could not well solve a linear bilevel programming
problem when the upper-level constraint functions are of arbitrary
linear form. Our recent research work has extended Bard’s theory
of bilevel programming by proposing a new definition of optimal
solution for linear bilevel programming which can overcome the
arbitrary linear form problem indicated above [1]. We have then
proposed an extended Kuhn-Tucher approach, based on our
definition of optimal solution, for solving linear bilevel problems
[2].
Following our previous research results shown in [1, 2], this
study aims at solving a FPBLP problem by transferring it into a
non-fuzzy bilevel programming problem. This paper first
proposes a model of FPBLP problem, then gives a definition of the
optimal solution for the FPBLP problem. Based on the definition
and related theorems, this paper develops a fuzzy number based
Kuhn-Tucher approach to solve the proposed FPBLP problem. As
this paper only deals with linear bilevel problem, so bilevel
programming means linear bilevel programming in this paper.
Following the introduction, Section 2 reviews related
definitions, theorems and properties of fuzzy number, BLP
solution and Kuhn-Tucher approach for solving an BLP problem.
A definition of optimal solution and a fuzzy number based
Kuhn-Tucher approach for solving FPBLP problems are
presented in Section 3. A numeral example is shown in Section 4
for illustrating the proposed fuzzy number based Kuhn-Tucher
approach. Conclusion and further study are discussed in Section 5.
November 2005 Vol.6 No.2
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Feature Article: Guangquan Zhang and Jie Lu
~
a~ + b =
II. PRELIMINARIES
U λ[aλ + bλ , aλ + bλ ],
L
L
R
R
λ ∈[ 0 ,1]
A. Fuzzy Numbers
In this section, we present some basic concepts, definitions and
theorems that are to be used in the subsequent sections. The work
presented in this section can also be found from our recent paper
in [25].
Let R be the set of all real numbers, Rn be n-dimensional
Euclidean space, and x = (x1, x2, …, xn)T, y = (y1, y2, …, yn)T ∈ Rn
be any two vectors, where xi, yi ∈ R, i = 1,2, …, n and T denotes
the transpose of the vector. Then we denote the inner product of x
and y by x, y . For any two vectors x, y ∈ Rn, we write x > y iff
xi > y i , ∀ i = 1,2, L , n;
x > y iff
x > y and x ≠ y ;
x > y iff xi > yi, ∀ i =
1,2, …, n.
Definition 2.1 A fuzzy number a~ is defined as a fuzzy set on R,
whose membership function µ a~ satisfies the following conditions:
1.
µ a~ is a mapping from R to the closed interval [0, 1];
2.
it is normal, i.e., there exists x ∈ R such that µa~ (x ) = 1 ;
3.
for any λ ∈ (0, 1], aλ = {x; µ a~ (x) ≥ λ} is a closed interval,
denoted by [aλL , aλR ] .
Let F(R) be the set of all fuzzy numbers. By the decomposition
theorem of fuzzy set, we have
(2.1)
a~ = U λ[a L , a R ],
λ ∈[ 0 ,1]
λ
λ
for every a~ ∈ F(R).
Let F * ( R ) be the set of all finite fuzzy numbers on R.
Theorem 2.1 Let a~ be a fuzzy set on R, then a~ ∈ F(R) if and
only if µ a~ satisfies
x ∈ [m, n]
⎧ 1
⎪
µ a~ ( x) = ⎨ L( x)
⎪ R ( x)
⎩
x<m ,
x>n
λ ∈[ 0 ,1]
αa~ =
L
λ
− bλR , aλR − bλL ],
U λ[αaλ , αaλ ].
L
R
λ ∈[ 0 ,1]
Definition
2.3
a~i ∈ F ( R), i = 1,2,L, n.
Let
a~ = (a~1 , a~2 , L, a~n )
We
define
µ a~ : R n → [0,1]
n
x a ∧ µ a~i ( xi ),
i =1
where x = (x1, x2, …, xn)T ∈ Rn, and a~ is called an n-dimensional
fuzzy number on Rn. If a~i ∈ F * ( R), i = 1,2,L, n, a~ is called an
n-dimensional finite fuzzy number on Rn.
Let F ( R n ) and F*(Rn) be the set of all n-dimensional fuzzy
numbers and the set of all n-dimensional finite fuzzy numbers on
Rn respectively.
Proposition 2.1 For every a~ ∈ F ( R n ), a~ is normal.
Proposition 2.2 For every a~ ∈ F ( R n ), the λ-section of
n-dimensional closed rectangular region for any λ ∈ [0,1] .
a~ is
an
Proposition 2.3 For every a~ ∈ F ( R n ) and λ1, λ2 ∈[0,1], if
λ1 < λ2, then aλ ⊂ aλ .
2
1
Definition 2.4 For any n-dimensional fuzzy numbers
~
~
a , b ∈ F ( R n ), we define
1. a~ f b~ iff ai λL > bi λL and ai λR > bi λR , i = 1,2, L , n, λ ∈ (0,1];
2.
3.
~
a~ f b iff ai λL > bi λL and ai λR > bi λR , i = 1,2, L , n, λ ∈ (0,1];
~
a~ f b iff ai λL > bi λL and ai λR > bi λR , i = 1,2,L , n, λ ∈ (0,1].
We call the binary relations f , f and f a fuzzy max order, a
strict fuzzy max order and a strong fuzzy max order, respectively.
where L(x) is the right-continuous monotone increasing
function, 0 < L(x) < 1 and lim x → −∞ L( x) = 0 , R(x) is the
left-continuous monotone decreasing function, 0 < R(x) < 1 and
lim x → ∞ R( x) = 0 .
B. The Extended Kuhn-Tucker Approach for Linear Bilevel
Programming
Let write a linear programming (LP) as follows.
min f ( x ) = cx
subject to Ax < b
Corollary 2.1 For every
a~ ∈ F ( R ) and
λ1, λ2 ∈ [0, 1], if λ1 < λ2,
then aλ ⊂ aλ .
2
( ) U λ[ a
~
~
a~ − b = a~ + − b =
1
~
a~, b ∈ F ( R)
and 0 < λ ∈ R, the sum of
~
a~ and b and the scalar product of λ and a~ are defined by the
Definition 2.2 For any
membership functions
µ a~ + b~ (t ) = sup min{µ a~ (u ), µb~ (v)},
(2.2)
µ a~ −b~ (t ) = sup min
{µ a~ (u ), µ b~ (v)},
t =u −v
(2.2)
µλa~ (t ) = sup µ a~ (u ).
(2.3)
t =u + v
t = λu
Theorem 2.2 For any a~, b~ ∈ F ( R ) and
0 < α ∈ R,
x > 0,
where c is an n-dimensional row vector, b an m-dimensional
column vector, A an m × n matrix with m ≤ n , and x ∈ R n .
Let λ ∈ R m and µ ∈ R n be the dual variables associated with
constraints Ax > b and x ≥ 0 , respectively. Bard [5] gave the
following proposition.
Proposition 2.4 [5] A necessary and sufficient condition that
*
( x * ) solves above LP is that there exist (row) vectors λ , µ * such
that ( x * , λ* , µ * ) solves:
λA − µ = −c
Ax − b > 0
λ ( Ax − b) = 0
µx = 0
x > 0, λ > 0, µ > 0
November 2005 Vol.6 No.2
IEEE Intelligent Informatics Bulletin
Feature Article: The Definition of Optimal Solution and an Extended Kunhn-Tucker Approach…
, y ∈ Y ⊂ R m , F : X × Y → R 1 , and
f : X × Y → R1 , a linear BLP problem is given by Bard [4]:
(2.5a)
min F ( x, y ) = c1 x + d1 y
For
x ∈ X ⊂ Rn
x∈ X
subject to A1 x + B1 y < b1
(2.5b)
min f ( x, y ) = c 2 x + d 2 y
(2.5c)
subject to A2 x + B2 y < b2
(2.5d)
y∈Y
where c1 , c2 ∈ R n , d1 , d 2 ∈ R m , b1 ∈ R p , b2 ∈ R q , A1 ∈ R p×n ,
B1 ∈ R p×m , A2 ∈ R q×n , B2 ∈ R q×m .
Definition 2.5 [1]
(a)Constraint region of the linear BLP problem:
S = {( x, y ) : x ∈ X , y ∈ Y , A1 x + B1 y < b1 , A2 x + B2 y < b2 }
(b) Feasible set for the follower for each fixed x ∈ X :
S ( X ) = {x ∈ X : ∃y ∈ Y , A1 x + B1 y < b1 , A2 x + B2 y < b2 }
(c)Projection of S onto the leader’s decision space:
S ( X ) = {x ∈ X : ∃y ∈ Y , A1 x + B1 y < b1 , A2 x + B2 y < b2 }
3
min{F ( x, y ) : ( x, y ) ∈ IR}
(2.6)
We also present the following theorem to characterize the
condition under which there is a Pareto optimal solution for a
linear BLP problem.
Theorem 2.3 [1] If S is nonempty and compact, there exists a
Pareto optimal solution for a linear BLP problem
Theorem 2.4 [2] [Extended Kuhn-Tucher Theorem] A
necessary and sufficient condition that ( x * , y * ) solves the linear
*
*
BLP problem (2.5) is that there exist (row) vectors u , v and
w* such that ( x * , y * , u * , v * , w* ) solves:
min F ( x, y ) = c1 x + d1 y
(2.7a)
subject to A1 x + B1 y ≤ b1
(2.7b)
(2.7c)
A2 x + B2 y ≤ b2
(2.7d)
uB1 + vB2 − w = −d 2
u (b1 − A1 x − B1 y ) + v(b2 − A2 x − B2 y) + wy = 0 (2.7e)
(2.7f)
x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0, w ≥ 0
(d) Follower’s rational reaction set for x ∈ S ( X ) :
P( x) = { y ∈ Y : y ∈ arg min[ f ( x, yˆ ) : yˆ ∈ S ( x)]}
III. FUZZY PARAMETER LINEAR BILEVEL PROGRAMMING
PROBLEM
where
arg min[ f ( x, yˆ ) : yˆ ∈ S ( x)] = { y ∈ S ( x) : f ( x, y ) < f ( x, yˆ ), yˆ ∈ S ( x)}
Definition 2.6 [1] ( x * , y * ) is said to be a complete optimal
solution, if and only if there exists ( x * , y * ) ∈ S such that
F ( x * , y * ) < F ( x, y ) and f ( x * , y * ) < f ( x, y ) for all ( x, y ) ∈ S .
Consider the following fuzzy parameter linear bilevel
programming (FPBLP) problem:
For x ∈ X ⊂ R n , y ∈ Y ⊂ R m , F : X × Y → F * ( R) , and
f : X × Y → F * ( R) ,
~
(3.1a)
min F ( x, y ) = c~1 x + d1 y
However, in general, such a complete optimal solution that
simultaneously minimizes both the leader’ and follower’s
objective functions does not always exist. Instead of a complete
optimal solution, a new solution concept, called Pareto optimality,
is introduced in linear BLP.
Definition 2.7 [1] ( x * , y * ) is said to be a Pareto optimal
solution, if and only if there does not exist ( x, y ) ∈ S such that
*
*
F ( x, y ) < F ( x * , y * ) , f ( x, y) < f ( x * , y * ) and F ( x, y ) ≠ F ( x , y ) or
~
(3.1c)
min f ( x, y ) = c~2 x + d 2 y
y∈Y
~
~
~
subject to A
(3.5d)
x + B 2 y p b2
2
~
~
~
*
m
*
n
where c~1 , c~2 ∈ F ( R ) , d 1 , d 2 ∈ F ( R ) , b1 ∈ F * ( R p ) ,
~
~
~
~
~
b2 ∈ F * ( R q ) , A1 = (a~ij ) , a~ij ∈ F * ( R ), B1 = bij p×m , bij ∈ F * ( R ),
p× n
~
~
A2 = (e~ij )q×n , e~ij ∈ F * ( R ), B2 = (~
sij )q×m , ~
sij ∈ F * ( R ) .
(e)Inducible region:
IR = {( x, y ) : ( x, y ) ∈ S , y ∈ P ( x)}
f ( x, y ) ≠ f ( x * , y * ) .
Definition 2.8 A topological space is compact if every open
cover of the entire space has a finite subcover. For example, [ a, b]
is compact in R (the Heine-Borel theorem) [26].
To ensure that (2.5) has a Pareto optimal solution, Bard gave the
following assumption.
Assumption 2.1
(a) S is nonempty and compact.
(b) For decisions taken by the leader, the follower has some
rooms to respond; i.e, P(x) ≠ φ .
(c) P (x) is a point-to-point map.
To ensure that (2.5) is well posed we assume that S is
nonempty and compact, and that P (x) is a point-to-point map.
The rational reaction set P (x ) defines the response while the
inducible region IR represents the set over which the leader may
optimize his objective. Thus in terms of the above notations, the
linear BLP problem can be written as
IEEE Intelligent Informatics Bulletin
x∈ X
~
~
~
subject to A
x + B1 y p b1
1
(3.1b)
( )
Associated with the FPBLP problem, we now consider the
following linear multi-objective multi-follower bilevel
programming (LMMBLP) problem:
For x ∈ X ⊂ R n , y ∈ Y ⊂ R m , F : X × Y → F * ( R) , and
f : X × Y → F * ( R) ,
min (F ( x, y ) )λ = c1 λ x + d 1 λ y, λ ∈ [0, 1]
L
L
L
(3.2a)
x∈ X
min (F ( x, y ) )λ
R
x∈ X
= c1 λ x + d 1 λ y, λ ∈ [0, 1]
R
R
subject to A1λ x + B1λL y < b1λL , A1λR x + B1λR y < b1 λR , λ ∈ [0, 1]
(3.2b)
L
min ( f ( x, y ) )λ = c 2 λ x + d 2 λ y, λ ∈ [0, 1]
L
L
L
(3.2c)
y∈Y
( f ( x, y) )λ = c2 λ x + d 2 λ y, λ ∈ [0, 1]
min
y∈Y
R
R
R
subject to A2 λL x + B2 λL y < b2λL , A2λR x + B2 λR y < b2 λR , λ ∈ [0, 1] (3.5d)
where
L
b1 λ , b1 λ ∈ R p
L
R
c1 λ , c1 λ
R
( )
,
L
R
c2 λ , c2 λ ∈ R n ,
,
b2 λ , b2 λ ∈ R q
L
R
( )
L
R
d1 λ , d1 λ ,
L
R
d2λ , d2λ ∈ Rm ,
( )
( )
= (s ), B = (s )∈ R
A1 λ = aij λ , A1 λ = aij λ ∈ R p×n ,
,
L
L
L
R
R
L
q× n
B1 λ = bij λ , B1 λ = bij λ ∈ R p×m , R , B2 λ
L
R
R
L
R
R
ij λ
2λ
ij λ
q× m
.
Theorem 3.1 Let ( x , y ) be the solution of the LMMBLP
*
*
November 2005 Vol.6 No.2
4
Feature Article: Guangquan Zhang and Jie Lu
problem (3.2). Then it is also a solution of the FPBLP problem
defined by (3.1).
Proof. The proof is obvious from Definition 2.4.
Lemma 3.1 If there is ( x * , y * ) such that cx + dy > cx * + dy * ,
c x+d y >c x +d y
L
0
L
0
L
0
*
L
0
then it satisfies (3.2a) and (3.2c). Then we need only prove,
if ( x * , y * ) satisfies (3.4b) and (3.4d), then it satisfies (3.2b) and
(3.2d). In fact, for any λ ∈ (0, 1),
aij λ = aij λ + aij 0 (1 − λ ),
L
and c x + d y > c x + d y , for any
*
R
0
R
0
R
0
*
R
0
*
~
( x, y ) and isosceles triangle fuzzy numbers c~ and d , then
c λL x + d λL y > c λL x * + d λL y * ,
c λR x + d λR y > c λR x * + d λR y * ,
bij λ = bij λ + bij 0 (1 − λ ) and
L
L
b1λ = b1λ + b10 (1 − λ ),
L
L
we have
A1 λ x * + B1 λ y * = ( a ij λ ) x * + (bij λ ) y *
L
~
for any λ ∈ (0, 1) , where c and d are the centre of c~ and d
L
L
L
(
)
L
( )
= ((a )x + (b )y )λ + ((a )x + (b )y )(1 − λ )
= ( A x + B y )λ + (A x + B y )(1 − λ )
*
L
ij
*
L
*
ij 0
L
*
1
L
*
*
ij
1
*
ij 0
L
*
10
*
10
< b1 λ + b1 0 (1 − λ ) = b1 λ ,
L
L
0
L
A1 λ x * + B1 λ y * < b1 λ ,
R
L
0
L
*
L
prove c λR x + d λR y > c λR x * + d λR y * from similar reason.
Theorem 3.2 For x ∈ X ⊂ R n , y ∈ Y ⊂ R m , If all the fuzzy
coefficients a~ij , b~ij , e~ij , ~sij , c~i and d~i have triangle membership
functions of the FPBLP problem (3.1).
t<z
*
vectors u , v * and w* such that ( x * , y * , u * , v * , w* ) solves:
(
(3.3)
L
L
min (F ( x, y ) )0 = c1 0 x + d 1 0 y,
R
L
min (F ( x, y ) )0 = c1 0 x + d 1 0 y,
L
L
L
(
R
(3.4b)
L
A1 0 x + B1 0 y < b1 0 ,
R
( f ( x, y ) )c = c 2 x + d 2 y,
min
y∈Y
( f ( x, y) )0 = c2 0 x + d 2 0 y,
min
y∈Y
L
( f ( x, y) )λ = c2 0 x + d 2 0 y,
min
y∈Y
(3.4c)
R
(
v (b
L
)
y ) + v (b
L
R
10
)
(3.5e)
− A2 0 x − B2 0
L
L
3
R
20
)
− A2 0 x − B2 0 y + wy = 0
R
R
(3.5f)
Proof: (1) From Theorem 3.2, we know that we need only to
solve the problem (3.4). In fact, to solve the problem (3.4), we can
use the method of weighting [27] to this problem, such that it is the
following problem:
L
L
R
R
(3.6a)
min (F ( x, y ) ) = (c1 x + d1 y ) + (c1 x + d1 y ) + (c1 x + d1 y )
0
x∈ X
(3.4d)
Proof. From Lemma 3.1, if ( x , y ) satisfies (3.4a) and (3.4c),
November 2005 Vol.6 No.2
L
0
0
0
A1 0 x + B1 0 y < b1 0 ,
R
*
(
(3.5d)
subject to A1 x + B1 y < b1 ,
A2 0 x + B2 0 y < b2 0 .
*
L
20
R
10
L
A2 0 x + B2 0 y < b2 0 ,
)
x ≥ 0, y ≥ 0, u ≥ 0, v ≥ 0, w ≥ 0
subject to A2 x + B2 y < b2 ,
L
R
R
u3 b − A x − B y + v1 (b2 − − A2 x − B2 y ) +
R
10
2
A1 0 x + B1 0 y < b1 0 ,
L
L
u1 (b1 − A1 x − B1 y ) + u2 b10 − A10 x − B10 y +
x∈ X
subject to A1 x + B1 y < b1 ,
(3.5c)
L
u1B1 + u2 B10 + u3 B10 + v1B2 + v2 B2 0 + v3 B2 0 − w
R
R
(3.5b)
A2 0 x + B2 0 y < b2 0 ,
(3.4a)
x∈ X
R
(3.5a)
R
L
L
L
L
)
A2 0 x + B2 0 y < b2 0 ,
(F ( x, y) )c = c1 x + d1 y,
min
x∈ X
L
R
L
= − d 2 + d 20 + d 20
R
R
A2 x + B2 y < b2 ,
~z respectively. Then, it is the solution of the problem (3.1) that
n
m
( x * , y * ) ∈ R × R satisfying
R
L
A1 0 x + B1 0 y < b1 0 ,
R
where ~z denotes a~ij , b~ij , e~ij , ~sij , c~i and d~i and z are the centre of
L
L
A1 0 x + B1 0 y < b1 0 ,
z 0R < t
L
) (
min (F ( x, y ) ) = (c1 x + d1 y ) + c10 x + d10 y + c10 x + d10 y
z < t < z 0R
R
R
for any λ ∈ (0, 1) from (3.4b) and (3.4d). The proof is complete.
Theorem 3.3 [Extended Kuhn-Tucher Theorem] A necessary
and sufficient condition that ( x * , y * ) solves the FPBLP problem
(3.1) with triangle fuzzy numbers is that there exist (row)
,
L
R
subject to A1 x + B1 y < b1 ,
z <t < z
R
L
x∈ X
L
0
L
0
L
L
R
*
from cx + dy > cx * + dy * and c 0L x + d 0L y > c 0L x * + d 0L y * . , we can
⎧0
⎪ t − zL
0
⎪
⎪ z − z 0L
µ ~z (t ) = ⎨
R
⎪ − t + z0
⎪ z 0R − z
⎪0
⎩
R
A2 λ x * + B2 λ y * < b2 λ ,
= cλ x + d λ y ,
L
R
A2 λ x * + B2 λ y * < b2 λ ,
> (c0L x * + d 0L y * )(1 − λ ) + (cx * + dy * )λ
R
( )
*
= (c0L x + d 0L y )(1 − λ ) + (cx + dy )λ
L
(
= (a ij )x λ + a ij 0 x (1 − λ ) + (bij )y λ + bij 0 y * (1 − λ )
L
*
c λ x + d λ y = c (1 − λ ) x + cλx + d (1 − λ ) y + dλy
L
R
L
from (3.4b). Similarly, we can prove
Therefore, we have
L
)
= aij λ + aij 0 (1 − λ ) x * + bij λ + bij 0 (1 − λ ) y *
respectively.
Proof. As λ-section of isosceles triangle fuzzy numbers c~ and
~
d are
cλL = c0L (1 − λ ) + cλ and cλR = c0R (1 − λ ) + cλ
d λL = d 0L (1 − λ ) + dλ and d λR = d 0R (1 − λ ) + dλ .
L
L
L
(3.6b)
L
A1 0 x + B1 0 y < b1 0 ,
R
R
R
min ( f ( x, y ) ) = c 2 x + d 2 y + c 2 0 x + d 2 0 y + c 2 0 x + d 2 0 y (3.6c)
L
L
R
R
y∈Y
IEEE Intelligent Informatics Bulletin
Feature Article: The Definition of Optimal Solution and an Extended Kunhn-Tucker Approach…
subject to A2 x + B2 y < b2 ,
A2 0 x + B2 0 y < b2 0 ,
L
L
(3.6d)
L
A2 0 x + B2 0 y < b2 0 .
R
R
R
Therefore, the linear BLP problem can be written as
min{F ( x, y ) : ( x, y ) ∈ IR}
(3.7)
Let us get an explicit expression of (3.7) and rewrite (3.7) as
follows:
(3) Sufficiency. If ( x * , y * ) is the optimal solution of (3.6), we
need to show that there exist (row) vectors u1* , u 2* , u3* , v1* , v2* , v3* and
w* such that ( x * , y * , u1* , u 2* , u 3* , v1* , v 2* , v3* , w* ) to solve (3.5). Going
one step farther, we only need to prove that there exist (row)
,
and
such
that
vectors
w*
u1* , u 2* , u3*
v1* , v 2* , v3*
*
*
*
*
*
*
*
*
*
( x , y , u1 , u 2 , u 3 , v1 , v 2 , v3 , w ) satisfy the follows
u1B1 + u2 B10 + u3 B10 + v1B2 + v2 B2 0 + v3 B2 0 − w
L
min F ( x, y )
R
subject to ( x, y ) ∈ IR .
R
u 2 (b1 0 − A1 0 x − B1 0 y ) = 0
(3.9b)
(3.9c)
u 3 (b1 0 − A1 0 x − B1 0 y ) = 0
(3.9d)
v1 (b2 − A2 x − B2 y ) = 0
v 2 (b2 0 − A2 0 x − B2 0 y ) = 0
(3.9e)
(3.9f)
v3 (b2 0 − A2 0 x − B2 0 y ) = 0
(3.9g)
L
min F ( x, y )
L
R
subject to ( x, y ) ∈ S
y ∈ P (x)
min F ( x, y )
R
L
R
subject to ( x, y ) ∈ S
L
R
L
by Definition 2.5(e). Then, we have
L
R
R
wy = 0 ,
y ∈ arg min[ f ( x, yˆ ) : yˆ ∈ S ( x)]
where u1 , u 2 , u 3 ∈ R , v1 , v 2 , v3 ∈ R , w ∈ R
p
by Definition 2.5(d). We rewrite it as:
subject to ( x, y ) ∈ S
min f ( x, y )
subject to y ∈ S (x) .
We have
min F ( x, y )
subject to ( x, y ) ∈ S
min f ( x, y )
y∈Y
subject to ( x, y ) ∈ S ,
by Definition 2.5(c). Consequently, we can have
) (
L
R
R
)
(3.8a)
subject to A1 x + B1 y < b1 ,
L
R
(3.8b)
R
L
A2 0 x + B2 0 y < b2 0 .
R
R
R
min ( f ( x, y ) ) = c2 x + d 2 y + c2 0 x + (3.8c)
y∈Y
L
d 2 0 y + c2 0 x + d 2 0 y
R
R
L
L
A1 0 x + B1 0 y < b1 0 ,
R
R
(3.8d)
A2 x + B2 y < b2 ,
A2 0 x + B2 0 y < b2 0 ,
L
L
L
A2 0 x + B2 0 y < b2 0 .
R
(3.11b)
R
R
by Definition 2.5(a).
This simple transformation has shown that solving the fuzzy
linear BLP (3.1) is equivalent to solving (3.8).
IEEE Intelligent Informatics Bulletin
R
A1 0 x + B1 0 y < b1 0 ,
(3.11c)
A1 0 x + B1 0 y < b1 0 ,
(3.11d)
A2 x + B2 y < b2 ,
(3.11e)
L
L
R
R
A2 0 x + B2 0 y < b2 0 ,
(3.11f)
A2 0 x + B2 0 y < b2 0 .
(3.11g)
L
L
R
L
R
R
(3.11h)
(3.11i)
To simplify (3.11), we can have
min g ( y ) = (d 2 + d 2 0 + d 2 0 ) y
(3.12a)
subject to − B1 y > − (b1 − A1 x * ),
(3.12b)
L
R
− B1 0 y > − (b1 0 − A1 0 x * ),
(3.11c)
− B1 0 y > − (b1 0 − A1 0 x * ),
(3.12d)
− B2 y > − (b2 − A2 x * ),
(3.12e)
L
R
L
R
L
R
− B2 0 y > − (b2 0 − A2 0 x ),
(3.12f)
− B2 0 y > − (b2 0 − A2 0 x ),
(3.12g)
y > 0.
(3.12h)
L
R
(2) Necessity is obvious from (3.8).
R
y>0
A1 0 x + B1 0 y < b1 0 ,
R
subject to A1 x + B1 y < b1 ,
L
x = x*
subject to A1 x + B1 y < b1 ,
L
(3.11a)
R
A2 0 x + B2 0 y < b2 0 ,
L
min ( f ( x, y ) ) = c 2 x + d 2 y + c 2 0 x + d 2 0 y + c 2 0 x + d 2 0 y
L
A2 x + B2 y < b2 ,
L
subject to y ∈ S (x)
x = x* .
From Definition 3.2(b), we have
L
L
A1 0 x + B1 0 y < b1 0 ,
R
(3.9h)
and they are not
y∈Y
A1 0 x + B1 0 y < b1 0 ,
L
m
min f ( x, y )
min (F ( x, y ) ) = (c1 x + d1 y ) + c10 x + d10 y + c10 x + d10 y
L
q
negative variables.
Because ( x * , y * ) is the optimal solution of (3.6), we have
( x * , y * ) ∈ IR ,
by (3.7). Thus we have
y * ∈ P( x * ) ,
by Definition 2.5(e). y * is the optimal solution to the following
problem
(3.10)
min( f ( x * , y ) : y ∈ S ( x * )) ,
by Definition 2.5(d). Rewrite (10) as follows
min F ( x, y )
(
(3.9a)
R
u1 (b1 − A1 x − B1 y ) = 0
We have
L
L
= −( d 2 + d 2 0 + d 2 0 )
L
x∈ X
5
L
R
L
R
*
*
November 2005 Vol.6 No.2
6
Feature Article: Guangquan Zhang and Jie Lu
Let we note
⎛ b1 ⎞
⎛ A1 ⎞
⎜ L⎟
⎜ L⎟
⎜ b1 0 ⎟
⎜ A1 0 ⎟
⎜b R ⎟
⎜AR⎟
1
1
A = ⎜ 0 ⎟, and b = ⎜ 0 ⎟.
⎜ b2 ⎟
⎜ A2 ⎟
⎜b L ⎟
⎜A L⎟
⎜ 2 0R ⎟
⎜ 2 0R ⎟
⎜b ⎟
⎜A ⎟
⎝ 20 ⎠
⎝ 20 ⎠
⎛ B1 ⎞
⎜ L⎟
⎜ B1 0 ⎟
⎜B R ⎟
1
B = ⎜ 0 ⎟,
⎜ B2 ⎟
⎜B L ⎟
⎜ 2 0R ⎟
⎜B ⎟
⎝ 20 ⎠
(3.13)
We rewrite (3.12) by using (3.13) and we get
min g ( y ) = (d 2 + d 2 0 + d 2 0 ) y
(3.14a)
subject to − By > − (b − Ax )
y > 0.
(3.14b)
L
R
*
(3.14c)
*
Now we see that y is the optimal solution of (3.14) which is a
LP problem. By Proposition 2, there exists vector λ* , µ * , such that
( y * , λ* , µ * ) satisfy a system below
L
R
(3.15a)
λB − µ = −(d 2 + d 2 0 + d 2 0 )
− By + (b − Ax )> 0
*
(3.15b)
(3.15c)
λ (− By + (b − Ax )) = 0
,
(3.15d)
µy = 0
where λ ∈ R 3 p +3q and µ ∈ R m .
Let u1 , u2 , u3 ∈ R p , v1 , v 2 , v3 ∈ R q and w ∈ R m and define
λ = (u1 , u 2 , u 3 , v1 , v 2 , v3 )
w=µ.
Thus we have ( x * , y * , u1* , u 2* , u 3* , v1* , v 2* , v3* , w* ) that satisfy (3.9).
*
t<2
⎧0
⎪t − 2
⎪
µ ~3 (t ) = ⎨
⎪4−t
⎪⎩ 0
2<t <3
,
3<t <4
4<t
⎧0
⎪t − 3
⎪
µ ~4 (t ) = ⎨
⎪5 − t
⎪⎩0
t <3
3<t <4
,
4<t <5
5<t
⎧0
⎪t + 1
⎪
µ ~0 (t ) = ⎨
⎪1 − t
⎪⎩0
t < −1
−1 < t < 0
0 < t <1
1< t
.
Step 1 The problem is transferred to the following LMMBLP
problem by using Theorem 3.2
(F ( x, y) )c = 1x − 2 y
min
x∈ X
min(F ( x, y ) )0 = 0 x − 3 y
L
x∈ X
(F ( x, y) )0 = 2 x − 1y
min
x∈ X
R
subject to − 1x + 3 y ≤ 4
− 2x + 2 y ≤ 3
0x + 4 y ≤ 5
( f ( x, y ) )c = 1x + 1 y
min
y∈Y
( f ( x, y ) )0 = 0 x + 0 y
min
y∈Y
L
min
( f ( x, y))0 = 2 x + 2 y
y∈Y
R
subject to 1x − 1 y ≤ 0
Our proof is completed.
Theorem 3.3 means that the most direct approach to solving
(3.1) is to solve the equivalent mathematical program given in
(3.5). One advantage that it offers is that it allows for a more
robust model to be solved without introducing any new
computational difficulties
0 x − 2 y ≤ −1
2x − 0 y ≤ 1
− 1x − 1 y ≤ 0
0x − 0 y ≤ 0
− 2 x − 2 y ≤ −1
Step 2. The problem is transferred to the following linear BLP
problem by using method of weighting [27].
min F ( x, y ) = 3 x − 6 y
IV. AN ILLUSTRATIVE EXAMPLE
x∈ X
Example 1 Consider the following FPBLP problem with
x ∈ R 1 , y ∈ R1 , and X = { x ≥ 0} , Y = { y ≥ 0} ,
~
~
(4.1a)
min F ( x, y ) = 1 x − 2 y
x∈ X
subject to
~ ~ ~
−1x+3yp4
~
~
min f 1 ( x, y ) = 1 x + 1 y
y∈Y
subject to ~1 x − ~1 y p ~0
~
~ ~
− 1 x − 1 yp0
where
⎧0
⎪t
⎪
µ ~1 (t ) = ⎨
⎪2 − t
⎪⎩ 0
⎧0
⎪t − 1
⎪
µ ~2 (t ) = ⎨
⎪3 − t
⎪⎩0
t<0
0 < t <1
,
1< t < 2
2<t
t <1
1< t < 2
,
2<t <3
3<t
November 2005 Vol.6 No.2
subject to − 1x + 3 y ≤ 4
− 2x + 2 y ≤ 3
0x + 4 y ≤ 5
min
f ( x, y ) = 3 x + 3 y
y∈Y
subject to 1x − 1 y ≤ 0
(4.1b)
0 x − 2 y ≤ −1
2x − 0 y ≤ 1
− 1x − 1 y ≤ 0
(4.1c)
(4.1d)
− 2 x − 2 y ≤ −1 .
0x − 0 y ≤ 1
(4.1e)
Step 3 Solve this linear BLP problem
min F ( x, y ) = 3 x − 6 y
x∈ X
subject to − 1x + 3 y ≤ 4
− 2x + 2 y ≤ 3
0x + 4 y ≤ 5
1x − 1 y ≤ 0
0 x − 2 y ≤ −1
2x − 0 y ≤ 1
− 1x − 1 y ≤ 0
IEEE Intelligent Informatics Bulletin
Feature Article: The Definition of Optimal Solution and an Extended Kunhn-Tucker Approach…
− 2 x − 2 y ≤ −1 .
0x − 0 y ≤ 1
3u1 + 2u 2 + 4u 3 − u 4 − 2u 5 − 0u 6 − u 7 − 2u 8 − 0u 9 − u10 = −3
u1 (4 + 1x − 3 y ) + u2 (3 + 2 x − 2 y ) + u3 (5 − 4 y ) +
u4 (− x + y ) + u5 (−1 + 2 y ) + u6 (1 − 2 x) +
[2]
[3]
[4]
u7 ( x + y ) + u8 (−1 + 2 x + 2 y ) + u9 + u10 y = 0
[5]
x ≥ 0, y ≥ 0, u1 ≥ 0, K , u10 ≥ 0 .
[6]
Step 4 The result is
min(F ( x, y ) )c = 1x − 2 y = −1
[7]
min(F ( x, y ))0L = 0 x − 3 y = −1.5
[8]
x∈ X
x∈ X
min(F ( x, y ) )0 = 2 x − 1y = −0.5
R
x∈ X
and
[9]
( f ( x, y ) )c = 0.5
min
y∈Y
[10]
min ( f ( x, y ) )0 = 0
L
y∈Y
[11]
min( f ( x, y ) )0 = 1
R
y∈Y
[12]
x = 0, y = 0.5
Consequently, we have the solution of the problem (4.1)
~ ~
min
F ( x, y ) = 1 x − 2 y = c~
x∈ X
~
~
~
min f 1 ( x, y ) = 1 x + 1 y = d
[13]
x = 0, y = 0.5 ,
[15]
[14]
y∈Y
and
where
⎧ 0
⎪ t + 1.5
⎪⎪
µ ~c (t ) = ⎨ 0.5
− 0.5 − t
⎪
⎪ 0.5
⎩⎪ 0
[16]
⎧0
⎪ t
− 1.5 ≤ t < −1
⎪⎪
, µ d~ (t ) = ⎨10−.5t
⎪
− 1 ≤ t < −0.5
⎪ 0.5
− 0.5 ≤ t
⎩⎪ 0
t < −1.5
t<0
0 ≤ t < 0.5
[17]
.
0.5 ≤ t < 1
[18]
1≤ t
[19]
V. CONCLUSION
Many organizational decision problems can be formulated by
bilevel programming models. Following our previous research [1,
2], this paper proposes the definition of optimal solution and
related theorems for fuzzy parameter based linear bilevel
programming. By using the proposed definition and theorems, this
study develops a fuzzy number based Kuhn-Tucher approach to
solve proposed FPBLP problem. A numeral example illustrates
the power and details of the proposed approach. Further study
includes the development of the model and related solving
approaches for fuzzy parameter based multi-follower bilevel
programming problems.
[20]
[21]
[22]
[23]
[24]
[25]
[26]
ACKNOWLEDGMENTS
The work presented in this paper was supported by Australian
Research Council (ARC) under discovery grants DP0557154 and
DP0559213.
[27]
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IEEE Intelligent Informatics Bulletin
November 2005 Vol.6 No.2