A New Algorithm for Solving Fully Fuzzy Bi-Level Quadratic Programming Problems

Operations Research and Applications : An International Journal (ORAJ), Vol.5, No.2, May 2018
DOI: 10.5121/oraj.2018.5201 1
A NEW ALGORITHM FOR SOLVING FULLY
FUZZY BI-LEVEL QUADRATIC PROGRAMMING
PROBLEMS
Azza H. Amer
Department of Mathematics, Faculty of Science,
Helwan University, Cairo, Egypt
ABSTRACT
This paper is concerned with new method to find the fuzzy optimal solution of fully fuzzy bi-level non-linear
(quadratic) programming (FFBLQP) problems where all the coefficients and decision variables of both
objective functions and the constraints are triangular fuzzy numbers (TFNs). A new method is based on
decomposed the given problem into bi-level problem with three crisp quadratic objective functions and
bounded variables constraints. In order to often a fuzzy optimal solution of the FFBLQP problems, the
concept of tolerance membership function is used to develop a fuzzy max-min decision model for generating
satisfactory fuzzy solution for FFBLQP problems in which the upper-level decision maker (ULDM)
specifies his/her objective functions and decisions with possible tolerances which are described by
membership functions of fuzzy set theory. Then, the lower-level decision maker (LLDM) uses this
preference information for ULDM and solves his/her problem subject to the ULDMs restrictions. Finally,
the decomposed method is illustrated by numerical example.
KEYWORDS
Fully Fuzzy Quadratic Non-Linear Programming Problems. Triangular Fuzzy Numbers, Decomposed
Method, Bi-Level Programming.
1. INTRODUCTION
The bi-level programming (BLP) problem is considered a useful optimization problems in which
there are independent decision makers (DMs) and the feasible region of the upper-level (UL)
problem is determined implicitly by the solution set of the lower-level (LL) problem. In the past
few decades, the BLP problem has been covered the theoretical and computational points [1-11]
and has been applied indifferent fields such as finance budget, transport network design [12],
supply chain management [13], principal-agent problem [14] engineering design [15], price
control and electricity markets.
In recent decades, the bi-level decision making problems became very hard to find the values of
the coefficients because of imprecise information when finding these models. So, fuzzy set theory
has been applied to handle imprecise data [16] where the coefficients in both objective functions
and the constraints are described by fuzzy numbers.

2
Tanaka et al. [17] was first proposed the concept of fuzzy mathematical programming (FMP) on
general level in the framework of the fuzzy decision of Bellman and Zadeh [18]. Zimmermann
[19] was first proposed the formulation of fuzzy linear programming. Also, Sakawa et al. [20-23]
first formulated the fuzzy bi-level programming problem and developed a fuzzy programming
method to solve it. Many researchers adopted this concept for solving FLP problems. It is notice
that the decision variables of all the above works are not fuzzy and the crisp solutions are
ineligible to describe the fuzzy advantage of the decision making process in an uncertain
environment.
In recent years, the fully fuzzy linear programming (FFLP) problems in which the coefficients as
well as decision variables are described by fuzzy numbers has been an attractive topic for the
researchers. Liou and Wang [24] proposed the concepts of ranking fuzzy numbers which is
playing a very important role in decision making. Also, there are a number of researchers who
have developed and presented new methods in this field of FFLP such as [25-32]. For the fully
fuzzy non-linear programming problems, Walaa Ibrahim Gabr [33] presents a comprehensive
methodology for solving and analyzing Quadratic and non-linear programming problems in fully
fuzzy environment. It should be noted that all these works are considered in the case of one-
single-level FFLP.
To our knowledge, until now there are few researcher studies the type of fully fuzzy bi-level
linear programming (FFBLLP) problem in which all coefficients and variable of both objective
functions and the constraints are expressed as fuzzy number such as [34-37].
The aim of this paper is to develop a new method to deal with the fully fuzzy bi-level quadratic
programming problems by applying the concept of tolerance membership function to show that,
the satisfactory solution obtained by fuzzy max-min decision model are always fuzzy optimal
solution [38, 39]. A numerical example is given to illustrate the proposed method.
2. PRELIMINARIES
In this section, some basic definitions of the fuzzy number and fuzzy arithmetic operations
depending on fuzzy numbers are reviewed.
Definition 1[38]
The characteristic function A
µ of a crisp set X
A ⊆ assigns a value either 0 or 1 to each member
in X. This function can be generalized to a function A
~
µ such that the value assigned to the
element of the universal set X fall within a specified range ].
1
,
0
[
:
.
. ~ →
X
e
i A
µ The assigned value
indicate the membership grade of the element in the set A. The function A
~
µ is called the
membership function and the set }
:
))
(
,
{( ~ X
x
x
x
A A
∈
= µ defined by A
~
µ (x) for each X
x∈ is
called a fuzzy set.
Definition 2[38]
A fuzzy number )
,
,
(
~
c
b
a
A = is said to be a triangular fuzzy number if its membership function is
given by

3







≤
≤
−
−
=
≤
≤
−
−
=
.
,
,
,
1
,
,
)
(
~
c
x
b
c
b
c
x
b
x
b
x
a
a
b
a
x
x
A
µ
Definition 3[38]
A triangular fuzzy number (a, b, c) is said to be non-negative fuzzy number iff .
0
≥
a
Definition 4[38] (Arithmetic Operations)
Let )
,
,
(
~
and
)
,
,
(
~
g
f
e
B
c
b
a
A =
= be two triangular fuzzy numbers. The algebraic operations
between any two triangular fuzzy numbers A
~
and B
~
can be defined by:
1) )
,
,
(
~
~
g
c
f
b
e
a
B
A +
+
+
=
⊕
2) )
,
,
(
)
,
,
(
~
a
b
c
c
b
a
A −
−
−
=
−
=
−
3) Θ
A
~
)
,
,
(
~
e
c
f
b
g
a
B −
−
−
=
4)



<
≥
=
0
,
)
,
,
(
,
0
,
)
,
,
(
~
k
ka
kb
kc
k
kc
kb
ka
A
k
5) Let )
,
,
(
~
c
b
a
A = be an arbitrary triangular fuzzy number and let )
,
,
(
~
g
f
e
B = be a non-
negative triangular fuzzy number, then
,
.
0
,
)
,
,
(
0
,
0
,
)
,
,
(
,
0
,
)
,
,
(
~
~





<
≥
<
≥
=
⊗
c
ce
bf
ag
c
a
cg
bf
ag
a
cg
bf
ae
B
A
3. FORMULATION OF THE FULLY FUZZY BI-LEVEL QUADRATIC
PROGRAMMING PROBLEM
Consider the following FFBLQP problem in which all the coefficients and the decision variables
are fuzzy numbers:
FFULDM:
( )
2
2
12
2
1
11
2
1
1
~
~
~
~
~
max
)
~
,
~
(
~
max
1
~
1
x
c
x
c
x
x
F
x
x
⊗
⊕
⊗
= (1)
where 2
~
x solves
FFLLDM:
( )
2
2
22
2
1
21
2
1
2
~
~
~
~
~
max
)
~
,
~
(
~
max
2
~
2
x
c
x
c
x
x
F
x
x
⊗
⊕
⊗
= (2)
s.t.

4
{
}.
0
~
~
,
0
~
~
,
,...,
2
,
1
,
~
~
~
~
~
)
~
,
~
(
~
2
1
2
2
1
1
2
1
≥
≥
=
≤
⊗
⊕
⊗
=
x
x
m
i
b
x
a
x
a
x
x
G i
i
i
(3)
where ( ) 1
1
1
12
11
1
~
,...,
~
,
~
~ n
n R
x
x
x
x ∈
= is an 1
n -dimensional fuzzy decision vector of the upper-level,
and ( ) 2
2
2
22
21
2
~
,...,
~
,
~
~ n
n R
x
x
x
x ∈
= is an 2
n -dimensional fuzzy decision vector of the lower-level.
Let 1
2
1
:
~
1
N
n
n
R
R
R
F →
× be the upper-level objective functions and 2
2
1
:
~
2
N
n
n
R
R
R
F →
× be the
lower-level objective functions, ,
2
, 2
1 ≥
N
N elements j
j
jk n
k
j
x j
,...,
2
,
1
,
2
,
1
,
~ =
= of decision
vectors j
x
~ are non-negative fuzzy triangular fuzzy numbers; ( ),
~
,...,
~
,
~
~
2
1 j
n
Lj
Lj
Lj
Lj c
c
c
c =
,
,...,
2
,
1
),
~
,...,
~
,
~
(
~
and
2
,
1
,
2
,
1 2
1
m
i
a
a
a
a
j
L j
ijn
ij
ij
ij =
=
=
= are j
n -dimensional fuzzy vectors;
elements i
j
ij
Lj b
n
t
s
a
c t
s
~
and
,...,
2
,
1
,
,
~
,
~ = are fuzzy numbers; G
~
is the constraint region of problem
(1)-(3).
3.1. Decomposition Method of FFBLQ P problem.
In this section, a new method is proposed to find the fuzzy optimal solution of FFBLQP problem.
In this way, one first gets the fuzzy satisfactory solution is acceptable to FFULDM, then
FFULDM gives fuzzy decision variables and fuzzy goals with some information to the FFLLDM
for him/her to seek the fuzzy satisfactory solution and to arrive at the fuzzy solution which is
closest to the fuzzy satisfactory solution of the FFULDM. This due to, the FFLLDM should not
only optimize his/her objective functions but also try to satisfy the FFULDM’s goals and
preference as much as possible [1].
Let the fuzzy parameters m
i
L
j
b
a
c
x
F i
ij
Lj
j
j ,...,
2
,
1
,
2
,
1
,
2
,
1
,
and
,
,
~
,
~
=
=
= be the triangular fuzzy
numbers which is represented by )
0
,
0
,
0
( u
m
L
, then problem (1) – (3) can be written as:
FFULDM:




⊗
= ∑
=
)
,
,
(
)
,
,
(
max
)
,
,
(
max 2
1
1
2
1
2
1
11
11
11
1
)
,
,
(
1
1
1
)
,
,
( 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
u
k
m
k
L
k
u
k
m
k
L
k
n
k
x
x
u
m
L
x
x
x
x
x
x
c
c
c
F
F
F u
k
m
k
L
k
u
k
m
k
L
k
)



⊗
⊕ ∑
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
x
x
x
c
c
c 2
2
2
2
2
2
12
12
12
1
2
2
2
2
2
2
2
2
,
,
(
)
,
,
( , (4)
where )
,
,
( 2
2
2
u
m
L
x
x
x solves
FFLLDM:




⊗
= ∑
=
)
,
,
(
)
,
,
(
max
)
,
,
(
max 2
1
2
1
2
1
21
21
21
1
)
,
,
(
2
2
2
)
,
,
( 1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
u
k
m
k
L
k
u
k
m
k
L
k
n
k
x
x
x
u
m
L
x
x
x
x
x
x
c
c
c
F
F
F u
k
m
k
L
k
u
k
m
k
L
k




⊗
⊕ ∑
=
)
,
,
(
)
,
,
( 2
2
2
2
2
2
22
22
22
1
2
2
2
2
2
2
2
2
u
k
m
k
L
k
u
k
m
k
L
k
n
k
x
x
x
c
c
c , (5)

5
s.t.
( ) ( )





⊗
⊕
⊗
= ∑
∑ =
=
)
,
,
(
,
,
)
,
,
(
,
, 2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
1
1
1
1
u
k
m
k
L
k
u
i
m
i
L
i
n
k
u
k
m
k
L
k
u
i
m
i
L
i
n
k
x
x
x
a
a
a
x
x
x
a
a
a
G k
k
k
k
k
k
,
,...,
2
,
1
),
,
,
( m
i
b
b
b u
i
m
i
L
i =
≤



≥
≥ )
0
,
0
,
0
(
)
,
,
(
,
)
0
,
0
,
0
(
)
,
,
( 2
2
2
1
1
1
u
m
L
u
m
L
x
x
x
x
x
x (6)
Now, using the arithmetic operations (see def. 4), then problem (4)-(6) can be decomposed into
three crisp bi-level quadratic programming (BLQP) problem as:
FULDM: (7)
( ) ( )
( ) ( ) ,
,
,
,
,
,
,
,
,
of
value
lower
max
2
2
2
2
2
2
12
12
12
1
2
1
2
1
2
1
11
11
11
1
1
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1




⊗
⊕




⊗
=
∑
∑
=
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
k
m
k
L
k
u
k
m
k
L
k
n
k
L
x
x
x
x
c
c
c
x
x
x
c
c
c
F
k
( ) ( )
( ) ( ) ,
,
,
,
,
,
,
,
,
of
value
middle
max
2
2
2
2
2
2
12
12
12
1
2
1
2
1
2
1
11
11
11
1
1
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1




⊗
⊕




⊗
=
∑
∑
=
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
k
m
k
L
k
u
k
m
k
L
k
n
k
m
x
x
x
x
c
c
c
x
x
x
c
c
c
F
k
( ) ( )
( ) ( ) ,
,
,
,
,
,
,
,
,
of
value
upper
max
2
2
2
2
2
2
12
12
12
1
2
1
2
1
2
1
11
11
11
1
1
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1




⊗
⊕




⊗
=
∑
∑
=
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
x
x
x
x
c
c
c
x
x
x
c
c
c
F
k
where )
,
,
( 2
2
2
u
m
L
x
x
x solves
FLLDM: (8)
( ) ( )
( ) ( ) ,
,
,
,
,
,
,
,
,
of
value
lower
max
2
2
2
2
2
2
22
22
22
1
2
1
2
1
2
1
21
21
21
1
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2




⊗
⊕




⊗
=
∑
∑
=
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
k
m
k
L
k
u
k
m
k
L
k
n
k
L
x
x
x
x
c
c
c
x
x
x
c
c
c
F
k
( ) ( )
( ) ( ) ,
,
,
,
,
,
,
,
,
of
value
middle
max
2
2
2
2
2
2
22
22
22
1
2
1
2
1
2
1
21
21
21
1
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2




⊗
⊕




⊗
=
∑
∑
=
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
k
m
k
L
k
u
k
m
k
L
k
n
k
m
x
x
x
x
c
c
c
x
x
x
c
c
c
F
k

6
( ) ( )
( ) ( ) ,
,
,
,
,
,
,
,
,
of
value
upper
max
2
2
2
2
2
2
22
22
22
1
2
1
2
1
2
1
21
21
21
1
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2




⊗
⊕




⊗
=
∑
∑
=
=
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
k
m
k
L
k
u
k
m
k
L
k
n
k
u
x
x
x
x
c
c
c
x
x
x
c
c
c
F
k
s.t.
( ) ( )
( ) ( ) ,
,...,
2
,
1
,
,
,
,
,
,
,
,
,
of
value
lower
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
1
1
1
1
m
i
b
x
x
x
a
a
a
x
x
x
a
a
a
G
L
i
u
k
m
k
L
k
u
i
m
i
L
i
n
k
u
k
m
k
L
k
u
i
m
i
L
i
n
k
k
k
k
k
k
k
=
≤




⊗
⊕









⊗
=
∑
∑
=
=
(9)
middle value of ( ) ( )




⊗
∑
=
u
k
m
k
L
k
u
i
m
i
L
i
n
k
x
x
x
a
a
a k
k
k 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
,
,
,
,
( ) ( ) ,
,...,
2
,
1
,
,
,
,
, 2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
m
i
b
x
x
x
a
a
a m
i
u
k
m
k
L
k
u
i
m
i
L
i
n
k
k
k
k
=
≤




⊗
⊕∑
=
upper value of ( ) ( )




⊗
∑
=
u
k
m
k
L
k
u
i
m
i
L
i
n
k
x
x
x
a
a
a k
k
k 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
,
,
,
,
( ) ( ) ,
,...,
2
,
1
,
,
,
,
, 2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
m
i
b
x
x
x
a
a
a u
i
u
k
m
k
L
k
u
i
m
i
L
i
n
k
k
k
k
=
≤




⊗
⊕∑
=



=
=
≥
−
≥
− m
k
j
x
x
x
x m
jk
u
jk
L
jk
m
jk j
j
j
j
,...,
2
,
1
,
2
,
1
,
0
,
0
Definition 5 [2]
For any { }
( )
G
x
x
G
x
x ∈
=
∈ )
~
,
~
(
~
~
~
2
1
0
1
1 given by FULDM, if the fuzzy decision making variable
{ }
( )
G
x
x
G
x
x ∈
=
∈ )
~
,
~
(
~
~
~
2
1
1
2
2 at the lower- level (LL) is the fuzzy optimal solution of FLLDM, then
)
~
,
~
( 2
1 x
x is a fuzzy feasible solution of problem (1) – (3).
Definition 6 [2]
)
~
,
~
( *
2
*
1 x
x is a fuzzy feasible solution of FFBLQP problem (1)-(3); no other fuzzy feasible solution
G
x
x
~
)
~
,
~
( 2
1 ∈ exists, such that )
~
,
~
(
~ *
2
*
1
1 x
x
f s s
f1
~
≤ );
~
,
~
( 2
1 x
x at least one s (s = 1, 2, …,N1) is strict
inequality, then )
~
,
~
( *
2
*
1 x
x is the fuzzy optimal solution of problem (1)-(3).
3.2 Fuzzy Decision Models for BLQP Problem
In this section, the solution method simplifies a BLQP problem by transforming it into separate
quadratic decision making (QDM) problems at upper and lower- levels as:

7
3.2.1 FULDM Problem
The FULDM solves the following fuzzy multi-objective decision making (FMODM) problem as:
( )
)
~
(
),
~
(
),
~
(
max
)
~
(
~
max 1
1
1
~
1
~
x
F
x
F
x
F
x
F u
m
L
x
x
=
s.t.
G
x ∈
~ (10)
where 2
1
~
),
~
,
~
(
~
2
1
n
n
R
x
x
x
x +
∈
=
We should first find individual best fuzzy solutions
+
1
~
F and individual worst fuzzy
solutions −
1
~
F for each objective of (10), where [4]:
)
~
(
min
~
and
)
~
(
max
~
1
~
1
1
~
1 x
F
F
x
F
F
G
x
G
x ∈
−
∈
=
=
+
(11)
Goals and tolerances can then be reasonably set for individual best fuzzy solutions and the
differences of the best and worst fuzzy solutions, respectively. This data can then be formulated
as the following membership functions of fuzzy set theory [18, 19] as:
[ ]







≥
≤
≤
−
−
>
=
−
+
−
−
+
−
+
.
)
~
(
~
~
if
,
0
,
~
)
~
(
~
~
if
,
~
~
~
)
~
(
~
,
~
)
~
(
~
if
,
1
)
~
(
~
1
1
1
1
1
1
1
1
1
1
1
1
x
F
F
F
x
F
F
F
F
F
x
F
F
x
F
x
F
µ (12)
Also, we can find the fuzzy solution of the FULDM problem by solving the following
Tchebycheff problem [18,19] as:
[ ] ].
1
,
0
[
~
,
~
)
~
(
~
,
~
.
.
~
max
1
1
1
1
∈
≥
∈
λ
λ
µ
λ
x
F
G
x
t
s
(13)
where ( )
u
m
L
1
1
1
1
~
,
~
,
~
~
λ
λ
λ
λ = is satisfactory level and the fuzzy solution is assumed to be
( )
u
u
u
u
and
F
x
x 1
1
2
1
~
~
,
~
,
~ λ for the upper-level.
3.2.2. FLLDM Problem
In the same way, the FLLDM independently solves:
( )
,
~
.
.
)
~
(
),
~
(
),
~
(
max
)
~
(
~
max 2
2
2
~
2
~
G
x
t
s
x
F
x
F
x
F
x
F u
m
L
x
x
∈
=
(14)
where 2
1
~
),
~
,
~
(
~
2
1
n
n
R
x
x
x
x +
∈
=
The individual best fuzzy solutions
+
2
~
F and individual worst fuzzy solutions
−
2
~
F for each
objective of (14) are:

8
)
~
(
min
~
)
~
(
max
~
2
~
2
2
~
2 x
F
F
and
x
F
F
G
x
G
x ∈
−
∈
+
=
= (15)
From this information, the membership functions can be formulated by using fuzzy theorem as:
[ ]







≥
≤
≤
−
−
>
=
−
+
−
−
+
−
+
.
)
~
(
~
~
if
,
0
,
~
)
(
~
~
if
,
~
~
~
)
(
~
,
~
)
~
(
~
if
,
1
)
~
(
~
2
2
2
2
2
2
2
2
2
2
2
2
x
F
F
F
x
F
F
F
F
F
x
F
F
x
F
x
F
µ (16)
Now, we can obtain the fuzzy solution of the FLLDM problem by solving the following
Tchebycheff problem as [18, 19]:
[ ] ].
1
,
0
[
~
,
~
)
~
(
~
,
~
.
.
~
max
2
2
2
2
∈
≥
∈
λ
λ
µ
λ
x
F
G
x
t
s
(17)
Whose fuzzy solution is assumed to be )
~
and
~
,
~
,
~
( 2
2
2
1
L
L
L
L
F
x
x λ for the lower-level and L
2
~
λ is
satisfactory level.
3.2.3 FFBLQDM Problem
Finally, in order to generate the fuzzy satisfactory solution which is also a fuzzy optimal solution
with an overall satisfaction for both DMs, we can solve the following Tchebycheff problem [4,
18, 19] as:
,
~
.
.
~
max
G
x
t
s
∈
λ
(18)
[ ] [ ]
[ ] [ ] [ ]
].
1
,
0
[
~
and
0
~
~
,
0
~
~
,
~
~
~
/
~
)
~
(
~
)
~
(
~
,
~
~
~
/
~
)
~
(
~
)]
~
(
~
[
,
~
~
/
)]
~
~
(
~
[
,
~
~
/
~
)]
~
~
[(
2
1
2
2
2
2
2
~
1
1
1
1
1
~
1
1
1
1
1
1
1
2
1
∈
≥
≥
≥
′
−
′
−
=
′
≥
′
−
′
−
=
′
≥
−
−
≥
−
+
λ
λ
µ
λ
µ
λ
λ
x
x
F
F
F
x
F
x
F
F
F
F
x
F
x
F
I
t
t
x
x
I
t
x
t
x
L
F
u
F
u
u
where λ
~
is the overall satisfaction, I is the column vector with all elements equal to 1s and the
same dimension as
u
x
x 1
1
~
;
~ is the fuzzy optimal decision variable of the FULDM, 1
~
t is
maximum tolerances of the range of the decision variable 1
~
x around
u
x1
~ and the membership
function of 1
~
x can be stated as follows:

9









+
≤
≤
−
+
≤
≤
−
−
−
=
′
.
if
,
0
,
~
~
~
~
if
,
~
~
)
~
~
(
,
~
~
~
~
if
,
~
)
~
~
(
~
)
~
( 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
otherwise
t
x
x
x
t
x
t
x
x
x
t
x
t
t
x
x
x u
u
u
u
u
u
µ (19)
The following membership functions of the FULDM can be stated as:
[ ]







′
≤
≤
≤
′
′
−
′
−
>
=
′
.
~
)
~
(
~
if
,
0
,
~
)
(
~
~
if
,
~
~
~
)
~
(
~
,
~
)
~
(
~
if
,
1
)
~
(
~
1
1
1
1
1
1
1
1
1
1
1
1
~
1
F
x
F
F
x
F
F
F
F
F
x
F
F
x
F
x
F u
u
u
F
µ (20)
where ).
,
(
~
~
2
1
1
1
L
L
x
x
F
F =
′
Also, the FLLDM has the following membership functions for his/her goals as:
[ ]







′
≤
≤
≤
′
′
−
′
−
>
=
′
.
~
)
~
(
~
if
,
0
,
~
)
~
(
~
~
if
,
~
~
~
)
~
(
~
,
~
)
~
(
~
if
,
1
)
~
(
~
2
2
2
2
2
2
2
2
2
2
2
2
~
2
F
x
F
F
x
F
F
F
F
F
x
F
F
x
F
x
F L
u
L
F
µ (21)
where ).
,
(
~
~
2
1
2
2
u
u
x
x
F
F =
′
By solving problem (18) for three decomposition problems, if the FULDM is satisfied with this
solution, then a fuzzy satisfactory solution is reached. Otherwise, he/she should provide new
membership functions for the fuzzy control variable and objectives to FLLDM until a fuzzy
satisfactory solution is reached.
4. A NEW ALGORITHM FOR SOLVING FFBLQP PROBLEM
The steps for the computation of FFBLQP problem can be summarized as:
Step 1: Consider the variables and parameters of FFBLQP problem as triangular fuzzy number.
Step 2: Formulate the FFBLQP problem (1)-(3).
Step 3: Transform problem (1)-(3) to problem (4)-(6).
Step 4: Use the decomposition method to convert problem (4)-(6) into three crisp BLQP problems
(7)-(9) by using the arithmetic operations on fully fuzzy.
Step 5: Calculate the individual fuzzy best and worst fuzzy solutions for each objective of
problem (10) and problem (14) as in (11) and (15).
Step 6: Formulate the membership function of the fuzzy ULDM and fuzzy LLDM problems as in
(12) and (16).
Step 7: Find the fuzzy solution of the fuzzy BLQP problem by solving the Tchebycheff problem
as in (13) and (17).

10
Step 8: Formulate the membership function of the fuzzy BLQP problem as in (19), (20) and (21)
after defined the value of fuzzy control decision variables )
~
( 1
u
x and the maximum
tolerance )
~
( 1
t .
Step 9: Formulate a Tchebycheff problem for BLQDM problem as in (18) to reach the fuzzy
satisfactory solution.
Step 10: If λ
~
>0.5, stop and the fuzzy compromise solution is obtained, λ
~
is the overall
satisfaction for all DMs, otherwise go to step8 after changing of the values 1
~
t .
5. NUMERICAL EXAMPLE
To demonstrate the solution method for FFBLQP problem, let us consider the following example:
,
~
)
10
,
4
,
2
(
~
)
12
,
5
,
3
(
~
max 2
2
2
1
1
~
1
x
x
F
x
⊗
⊕
⊗
=
where 2
~
x solves
0
~
)
~
,
~
(
~
),
8
,
5
,
2
(
~
)
5
,
2
,
1
(
~
)
4
,
3
,
2
(
),
20
,
10
,
4
(
~
)
4
,
7
,
2
(
~
)
6
,
5
,
4
(
.
.
,
~
)
8
,
7
,
1
(
~
)
10
,
5
,
3
(
~
max
2
1
2
1
2
1
2
2
2
1
2
~
2
≥
=
≤
⊗
+
⊗
≤
⊗
+
⊗
⊗
⊕
⊗
=
x
x
x
x
x
x
x
t
s
x
x
F
x
where 2
1
~
,
~ x
x are triangular fuzzy numbers,
Assume that )
,
,
(
~
),
,
,
(
~
2
2
2
2
1
1
1
1 z
y
x
x
z
y
x
x =
= and ),
,
,
(
~ u
j
m
j
L
j
j F
F
F
F =
.
2
,
1
=
j
According to arithmetic operations of triangular fuzzy numbers, the FFBLQP problem (7)-(9)
can be rewritten as:
FULDM: ,
2
3
max 2
2
2
1
1
1
x
x
F L
x
+
=
,
10
12
max
,
4
5
max
2
2
2
1
1
2
2
2
1
1
1
1
z
z
F
y
y
F
u
z
m
y
+
=
+
=
where )
,
,
(
~
2
2
2
2 z
y
x
x = solves
FLLDM: ,
3
max 2
2
2
1
2
2
x
x
F L
x
+
=
,
8
10
max
,
7
5
max
2
2
2
1
2
2
2
2
1
2
2
2
z
z
F
y
y
F
u
z
m
y
+
=
+
=
s.t.

11
{ ,
2
2
,
4
2
4 2
1
2
1 ≤
+
≤
+
= x
x
x
x
G
}
0
,
0
0
,
0
0
,
0
,
8
5
4
,
20
4
6
,
5
2
3
,
10
7
5
2
2
1
1
2
2
1
1
2
1
2
1
2
1
2
1
2
1
≥
−
≥
−
≥
−
≥
−
≥
≥
≤
+
≤
+
≤
+
≤
+
y
z
y
z
x
y
x
y
x
x
z
z
z
z
y
y
y
y
where )
,
,
(
and
)
,
,
( 2
2
2
1
1
1 z
y
x
z
y
x are triangular fuzzy numbers.
First: The FULDM solves problem (10) as follows:
.
~
.
.
),
,
,
(
max
~
max
1
1
1
1
~
1
~
1
1
G
x
t
s
F
F
F
F u
m
L
x
x
∈
=
1- Find individual best fuzzy solution +
1
~
F and individual worst fuzzy solution −
1
~
F by
solving (11), we get
Upper level )
,
,
(
~
1
1
1
1 z
y
x
x = )
,
,
(
~
2
2
2
2 z
y
x
x = )
,
,
(
max
~
1
1
1
1
1
u
m
L
G
x
F
F
F
F
∈
+
= )
,
,
(
max
~
1
1
1
1
1
u
m
L
G
x
F
F
F
F
∈
−
=
Best fuzzy
solution
(1,1.667,2) (0,0,0) )
48
,
889
.
13
,
3
(
~
1 =
+
F
Worst fuzzy
solution
(0,0,0) (0,0,0) −
1
~
F (0,0,0)
2- Use (12) to build the membership functions )]
~
(
~
[ 1
~
1
x
F
F
µ and solve (13) as follows:
.
1
0
,
2
2
,
4
2
4
,
3
2
3
.
.
,
max
1
2
1
2
1
1
2
2
2
1
1
≤
≤
≤
+
≤
+
≥
+
L
L
L
x
x
x
x
x
x
t
s
λ
λ
λ
.
1
0
,
5
2
3
,
10
7
5
,
889
.
13
4
5
.
.
,
max
1
2
1
2
1
1
2
2
2
1
1
≤
≤
≤
+
≤
+
≥
+
m
m
m
y
y
y
y
y
y
t
s
λ
λ
λ
.
1
0
,
8
5
4
,
20
4
6
,
48
10
12
.
.
,
max
1
2
1
2
1
1
2
2
2
1
1
≤
≤
≤
+
≤
+
≥
+
u
u
u
z
z
z
z
z
z
t
s
λ
λ
λ
whose solution is:
)
499
.
26
,
168
.
8
,
501
.
4
(
~
)
1
,
1
,
1
(
~
)
48
,
894
.
13
,
329
.
3
(
~
),
0
,
0
,
253
.
1
(
~
),
2
,
667
.
1
,
251
.
0
(
~
1
1
1
2
1
=
=
=
=
=
L
u
u
u
F
and
F
x
x
λ
Second: In the same way, the FLLDM solves problem (14) as follows:
1- Find individual best fuzzy solutions +
2
~
F and individual worst fuzzy solutions −
2
~
F by
solving (15), we get

12
Lower level )
,
,
(
~
1
1
1
1 z
y
x
x = )
,
,
(
~
2
2
2
2 z
y
x
x = )
,
,
(
max
~
2
2
2
2
2
u
m
L
G
x
F
F
F
F
∈
+
= )
,
,
(
max
~
21
2
2
2
2
u
m
L
G
x
F
F
F
F
∈
−
=
Best fuzzy
solution
(0,0,0) (1.429,1.429,1.6) (2.042,14.286,20)
Worst fuzzy
solution
(0,0,0) (0,0,0) (0,0,0)
2- By using (16), build the membership functions )]
~
(
~
[ 2
~
2
x
F
F
µ and solve (17) we get:
)
40
,
894
.
13
,
759
.
1
(
~
,
)
029
.
22
,
294
.
14
,
272
.
2
(
~
),
1
,
1
,
1
(
~
),
402
.
0
,
429
.
1
,
493
.
1
(
~
,
)
44
.
1
,
0
,
119
.
0
(
~
2
2
2
2
1
=
=
=
=
=
u
L
L
L
F
F
x
x λ
Third: In order to generate the fuzzy satisfactory solution,
1- Assume the FULDM's fuzzy control decision u
x1
~ is around (0.251,1.667,2) with the
tolerance = 0.2.
2- By (19), (20) and (21) and from membership functions, )]
~
(
~
[
)],
~
(
~
[
),
~
( 2
~
1
~
1 2
1
x
F
x
F
x F
F
µ
µ
µ ′
′
′ ,
the FLLDM solves the following Tchebycheff problem (18) as:
.
1
0
,
759
.
1
513
.
0
3
,
501
.
4
172
.
1
2
3
,
051
.
0
2
.
0
,
451
.
0
2
.
0
.
.
,
max
2
2
2
1
2
2
2
1
1
1
1
≤
≤
≥
−
+
≥
+
+
≥
−
≤
+
L
L
L
L
L
L
x
x
x
x
x
x
t
s
λ
λ
λ
λ
λ
λ
.
1
0
,
894
.
13
4
.
0
7
5
,
168
.
8
726
.
5
4
5
,
467
.
1
2
.
0
,
867
.
1
2
.
0
.
.
,
max
2
2
2
1
2
2
2
1
1
1
≤
≤
≥
−
+
≥
−
+
≥
−
≤
+
m
m
m
m
m
m
y
y
y
y
y
y
t
s
λ
λ
λ
λ
λ
λ
.
1
0
,
40
971
.
17
8
10
,
499
.
26
501
.
21
10
12
,
8
.
1
2
.
0
,
2
.
2
2
.
0
.
.
,
max
2
2
2
1
2
2
2
1
1
1
≤
≤
≥
+
+
≥
−
+
≥
−
≤
+
u
u
u
u
u
u
z
z
z
z
z
z
t
s
λ
λ
λ
λ
λ
λ
Therefore, the compromise fuzzy solutions are
).
DM
both
for
on
satisfacti
overall
(
)
1
,
1
,
1
(
)
,
,
(
~
and
,
)
034
.
463
,
635
.
56
,
779
.
1
(
~
),
792
.
126
,
318
.
38
,
369
.
3
(
~
,
)
807
.
2
,
471
.
2
,
261
.
1
(
~
,
)
2
,
667
.
1
,
251
.
0
(
~
s
2
1
2
1
=
=
=
=
=
=
u
m
L
F
F
x
x
λ
λ
λ
λ
6. CONCLUSION
In this paper, a fully fuzzy bi-level quadratic programming problem in which all the coefficients
and decision variables are fuzzy numbers is introduced. In order to obtain a fuzzy optimal
solution of the FFBLQP problem, the concepts of tolerance membership functions at each level to
develop a fuzzy max-min decision model for generating fuzzy satisfactory solution for FFBLQP
problem. Then the fully fuzzy bi-level quadratic programming can be converted into a
deterministic bi-level programming problem by using the bound and decomposition method.
Also, a new algorithm is based on the fuzzy decision approach, bound and decomposition method

13
are introduced to obtain the fuzzy satisfactory solution for FFBLQP problem. Finally, an
illustrative numerical example has been given to clarify the proposed solution method.
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