DOI 10.1007/s10598-015-9283-0 Computational Mathematics and Modeling, Vol. 26, No. 3, July, 2015
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM 1,2 1 1 G. Malkawi , N. Ahmad , and H. Ibrahim
This paper proposes a new computational method to obtain a positive solution for arbitrary fully fuzzy linear system (FFLS). The new method transforms the coefficients in FFLS to a one-block matrix. As a result, none of the fuzzy operations are needed. This method can provide a solution regardless of the size of a system. Some necessary theorems are proved and new numerical examples are presented to illustrate the proposed method. Keywords: fuzzy linear system; fully fuzzy linear system; fuzzy number; triangular fuzzy number; block matrix.
1. Introduction Linear systems of equations are considered the simplest model in solving mathematical problems. However, the coefficients of these systems are not completely obtainable. Therefore, linear systems are replaced by fuzzy systems, replacing crisp numbers by fuzzy numbers. Linear systems of equations are called fuzzy linear systems (FLS) if the elements of the matrix on the left-hand side are crisp numbers and the elements for vector on the right-hand side are represented by fuzzy numbers. In addition, a linear system of equations is called a fully fuzzy linear system (FFLS) when all of the elements in both sides are fuzzy numbers. The first feasible approach to solving FLS was presented by Friedman et al. [31], where they proposed a generic model for solving an n × n FLS by transferring the FLS to 2n × 2n linear system. Dehghan and his colleagues [10–13] obtained the solution for FFLS where the coefficient and parameters are positive. Furthermore, for the same scenario, several scholars [1, 17, 34–38] suggested new methods for solving FFLS in a similar way to Dehghan. Malkawi and his colleagues [26–28] recommended new matrix methods for solving a positive FFLS. Their methods and results were also capable of solving left–right fuzzy linear system (LR-FLS) and FLS. The necessary and sufficient condition to obtain a positive solution was discussed. In addition, methods to obtain positive solution for FFLS and fully fuzzy matrix equations (FFME) have been proposed [5, 20, 21, 39], where the coefficients are arbitrary. All of the examples in their methods are limited to n = 3 or 3 indicating difficulties in real applications. In this paper, we propose a method that can provide a positive solution for arbitrary coefficients in FFLS using a one-block matrix. This method can deal with any system regardless of its size. The verification of solution is provided for n = 5 and 10. This paper is organized into five sections. Section 2 reviews the basic definitions of fuzzy set theory, and Section 3 introduces the proposed model. Section 4 contains some numerical examples to illustrate the work, and Section 5 provides a summary of the method. 1 2
College of Arts & Sciences, University Utara Malaysia, 06010 Sintok, Kedah, Malaysia. Mathematics Department, college of Arts & Sciences, Northern Borders University, Rafha, Kingdom of Saudi Arabia; e-mail:
[email protected].
436
1046–283X/15/2603–0436
© 2015
Springer Science+Business Media NewYork
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
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2. Preliminaries In this section, basic definitions and notions of fuzzy set theory are reviewed [9, 18]. Definition 2.1. Let X be a universal set, the fuzzy subset A of X , with its membership function µ A (x) : R → [ 0;1 ] , which assigns to each element x ∈ X of real numbers µ A (x) in the interval [ 0,1 ] , where the value µ (x) represents the grade of membership of x in A . A
A fuzzy set A is written as A = { ( x, µ A (x) ) , x ∈ X, µ A (x) ∈[ 0,1 ] } . Definition 2.2. A fuzzy set A in X = R n is a convex fuzzy set if:
∀x1, x 2 ∈ X,
∀λ ∈[ 0,1 ] ,
µ A ( λx1 + ( 1 − λ ) x 2 ) ≥ min ( µ A (x1 ), µ A (x 2 ) ) . Definition 2.3. Let A be a fuzzy set defined on the set of real numbers R . A is called a normal fuzzy set if there exist x ∈R such that µ A (x) = 1 . Definition 2.4. A fuzzy number is a normal and convex fuzzy set when its membership function µ A (x) is
defined in real line R and piecewise continuous.
Definition 2.5 (LR fuzzy number). A fuzzy number m is called an LR fuzzy number when its membership function satisfies
⎧ ⎛ m−x⎞ ⎪ L ⎜⎝ α ⎟⎠ , ⎪ µ m (x) = ⎨ ⎪ ⎛ x−m⎞ ⎪ R ⎜⎝ β ⎟⎠ , ⎩
for
x ≤ m,
α > 0, (2.1)
for
m ≤ x,
β > 0,
where m, α, β ∈R , and the function L(⋅) is called a left shape function if the following conditions are satis-
fied:
(1)
L(x) = L(−x) ;
(2)
L(0) = 1 , L(1) = 0 ,
(3)
L is nonincreasing on
[ 0, ∞ ] .
Similar to the right shape function R(⋅) , an LR fuzzy number is symbolically written m = ( m, α, β ) LR , where m represents the mean value, whereas α and β are the left and right spreads, respectively. We denote the set
G. MALKAWI, N. AHMAD,
438
AND
H. IBRAHIM
of LR fuzzy numbers as F(R) . The sign of m = ( m,α,β) LR is classified as follows: –
m! is called positive (negative) iff m − α ≥ 0
–
m! is called zero iff (m = 0 , α, β = 0) .
–
m! is called near zero iff m − α < 0 < β + m.
(β+ m ≤ 0 ) .
Definition 2.6. Two fuzzy numbers n = ( n, γ, δ ) LR and m = ( m, α, β ) LR are called equal if n = m ,
γ = α, δ = β.
Definition 2.7 (Arithmetic operations on LR fuzzy numbers). We will represent arithmetic operations for two LR Fuzzy numbers m = ( m, α, β ) LR and n = ( n, γ, δ ) LR as follows: Addition:
( m, α, β )LR ⊕ ( n, γ, δ ) LR = ( m + n, α + γ, β + δ ) LR .
(2.2)
Opposite:
− ( m, α, β ) LR = − ( m, α, β ) LR =
( −m, β, α )RL .
(2.3)
Subtraction:
( m, α, β )LR ( n, γ, δ ) RL = ( m − n, α + δ, β + γ ) LR .
(2.4)
Definition 2.8. An LR fuzzy number is a triangular fuzzy number (TFN), where L = R = max(0,1 − x) .
Consequently, using (2.1), we write its membership function as
m−x ⎧ ⎪1 − α , ⎪ ⎪ x−m µ A (x) = ⎨ 1− , ⎪ β ⎪ ⎪ 0, ⎩
m − α ≤ x m, α 0, m ≤ x m + β, β 0,
(2.5)
otherwise,
= ( m, α, β ) . which is symbolically written as a triangular fuzzy number m Definition 2.9 (Kaufmann’s approximation for multiplication of TFN).
(n, γ,δ) , be two arbitrary triangular fuzzy numbers,
m ⊗ n =
( mn, f1, f2 ) ,
Let m = ( m, α, β ) and n =
(2.6)
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
439
where
f1 = mn − min { ( m − α ) ( n − γ ) , ( m + β ) ( n − γ )
( m + β )( n + δ )( m − α )( n + δ )} , and
f2 = max { (m − α)(n − γ ), (m + β)(n − γ ), (m + β)(n + δ), (m − α)(n + δ) } − mn . Definition 2.10. A vector X = ( x1, x 2 ,…, x n )T is called a fuzzy vector if x i ∈ F(R) , ∀i = 1,…, n .
Definition 2.11. Let A = (a ij ) and B = (bij ) be two m × n and n × p linear systems, respectively. We define A ⊗ B = C = (cij ) , which is the m × p matrix, where
cij =
⊕
∑
a ik ⊗ bkj .
k=1, …, n
Definition 2.12 (FFLS). Consider the n × n linear system,
⎧ a11x1 + a12 x 2 + … + a1n x n = b1, ⎪ ⎪⎪ a 21x1 + a 22 x 2 + … + a 2n x n = b2 , ⎨ ⎪ ⎪ ⎪⎩ a n1x1 + a n2 x 2 + … + a nn x n = bn ,
(2.8)
where ∀a ij , b j ∈ F (R ) . This system is an FFLS. If the coefficients are arbitrary, then it is a general FFLS. n n The matrix A = (a ij )i, j=1 and the vector B = (b j ) may be represented as j=1
A ⊗ X = B .
(2.9)
n The vector X = ( x j ) j=1 is called an exact fuzzy solution if ∀x j ∈ F(ℜ) , j = 1, 2,…, n , otherwise it is called
a nonfuzzy solution.
3. The Proposed Method In this section a novel method will be constructed to obtain a positive solution for unrestricted FFLS. Consider the following unrestricted FFLS,
A ⊗ X = B ,
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440
( )n×1 = (b,h, g) ,
where A = (a ij )n×n = ( A, M , N ) , B = bi
AND
H. IBRAHIM
and X = ( x j )n×1 = ( x, y, z) ≥ 0 ,
( A, M , N ) ⊗ ( m x , α x , β x ) = ( m b , α b , β b ) .
(3.1)
Let
( mi,a j , α i,a j , βi,a j ) ,
a ij =
x j =
( m xj , α xj , β xj ) ,
and
bi =
( mib , α bi , βbi ) .
Then, the n × n FFLS may be written as ⊕
∑aij ⊗ x j j=1
= bi ,
⊕
∑ ( mi,a j , α i,a j , βi,a j ) ⊗ ( m xj , α xj , β xj ) j=1
∀i = 1, 2,…, n ,
=
( mib , α bi , βbi ) ,
∀i = 1, 2,…, n .
(3.2)
The algorithm of the proposed method is given in the following five steps. Step 1: Apply the arithmetic operators of LR fuzzy numbers in Definition 2.7 and 2.9 on Eq. (3.2)
mi,b j = mi,a j m xj ,
(
(3.3a)
)(
)(
)(
)(
)(
)(
)
α bi, j = mi,b j − min ⎡⎣ mi,a j − α i,a j m xj − α xj mi,a j − α i,a j m xj + β xj ⎤⎦ ,
(
)
β bi, j = −mi,b j + max ⎡⎣ mi,a j + β i,a j m xj − α xj , mi,a j + β i,a j m xj + β xj ⎤⎦ .
(3.3b) (3.3c)
Step 2: Find α i,b j .
Let
fi,αj = min $% mi,a j − α i,a j
(
) ( m xj − α xj ) , ( mi,a j − αi,a j ) ( m xj + β xj ) &' ,
hence, (3.3b) may be written as
α i,b j = mi,a j mix − fijα , ci,a j is defined as follows: ci,a j = mi,a j − α i,a j ,
(3.4)
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
441
where
ci,a j
# c+a , % i, j = $ % c −a , & i, j
ci,a j ≥ 0, ci,a j
(3.5)
< 0.
Then, either ci,+aj = 0 or ci,−aj = 0 (or both),
( ci,+aj ) ( ci,−aj ) Since
( m xj , α xj , β xj )
and
ci,a j = ci,+aj + ci,−aj .
(3.6)
is positive, then
0 ≤ According to the sign of
Case 1: If
= 0
( m xj − α xj ) ≤ ( m xj + β xj ) .
( m aj , α aj , βaj ) , we have two possibilities for
( mi,a j − αi,a j )
(3.7)
fijα :
is non-negative, using (3.7), we have
0 ≤
( m xj − α xj ) ( mi,a j − αi,a j ) ≤ ( m xj + β xj ) ( mi,a j − αi,a j ) ;
then
fi,αj = Case 2: If
( mi,a j − αi,a j )
( m xj − α xj ) ( mi,a j − αi,a j ) .
(3.8a)
is negative,
0 >
( m xj − α xj ) ( mi,a j − αi,a j ) ≥ ( m xj + β xj ) ( mi,a j − αi,a j ) ;
then
fi,αj =
( m xj + β xj ) ( mi,a j − αi,a j ) .
Hence, Eqs. (3.8a) and (3.8b) may be written as piecewise functions
fi,αj
% ma − αa i, j ' i, j = & ' ma − αa i, j ( i, j
(
) ( m xj − α xj ) ,
( mi,a j − αi,a j ) ≥ 0,
(
) ( m xj + β xj ) ,
( mi,a j − αi,a j ) < 0,
(3.8b)
G. MALKAWI, N. AHMAD,
442
% c+a m x − α x , j j ' i, j = & ' c −a m x + β x , j j ( i, j
(
)
ci,a j ≥ 0,
(
)
ci,a j < 0.
AND
H. IBRAHIM
Using Eq. (3.6), we have
fi,αj = ci,+aj m xj − α xj + ci,−aj m xj + β xj
(
)
(
)
= ci,+aj m xj − ci,+aj α xj + ci,−aj m xj + ci,−aj β xj = ci,+aj + ci,−aj m xj + ci,−aj β xj − ci,+aj α xj
(
)
= ci,a j + m xj + ci,−aj β xj − ci,+aj α xj = mi,a j − α i,a j m xj + ci,−aj β xj − ci,+aj α xj .
(
)
Then, Eq. (3.3b) may be written as
α i,b j = mi,b j −
{ ( mi,a j − αi,a j ) m xj + ci,−aj β xj − ci,+aj α xj }
= mi,a j m xj − mi,a j m xj − α i,a j m xj + ci,−aj β xj − ci,+aj α xj
{
}
= mi,a j m xj − mi,a j m xj + α i,a j m xj − ci,−aj β xj + ci,+aj α xj
(
)
= α i,a j m xj − ci,−aj β xj + ci,+aj α xj . Hence,
α i,b j = α i,a j m xj − ci,−aj β xj + ci,+aj α xj ;
(3.9)
using Eq. (3.6), we have either ci,−aj β xj = 0 or ci,+aj α xj = 0 (or both). Then
( ci,−aj β xj ) ( ci,+aj α xj )
= 0.
Step 3: Find β i,b j .
Let
fi,βj = max $% mi,a j + βi,a j
(
) ( m xj − α xj ) , ( mi,a j + βi,a j ) ( m xj + β xj ) &' ;
(3.10)
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
443
hence, Eq. (3.3c) may be written as
β i,b j = −mi,a j m xj + fi,βj ;
(3.11)
di,a j is defined as follows: di,a j = mi,a j + β i,a j ,
di,a j
mi,a j + β i,a j ≥ 0,
⎧ di,+aj , ⎪ = ⎨ ⎪ di,−aj , ⎩
mi,a j − β i,a j ≤ 0;
hence, either di,+aj = 0 or di,−aj = 0 (or both). Using Eq. (3.7), we have two possibilities for fi,βj :
( di,+aj ) ( di,−aj ) Case 1: If
( mi,a j + βi,a j )
= 0
and
di,a j = di,+aj + di,−aj .
(3.12)
is non-negative, using Eq. (3.7) we have
0 ≤
( mi,a j + βi,a j ) ( m xj − α xj ) ≤ ( mi,a j + βi,a j ) ( m xj + β xj ) ;
then
fi,βj = Case 2: If
( mi,a j + βi,a j )
( mi,a j + βi,a j ) ( m xj + β xj ) .
(3.13a)
is negative,
0 >
( mi,a j + βi,a j ) ( m xj − α xj ) ≥ ( mi,a j + βi,a j ) ( m xj + β xj ) ;
then
fi,βj =
( mi,a j + βi,a j ) ( m xj − α xj ) .
Hence, Eqs. (3.13a) and (3.13b) may be written as piecewise functions
fi,βj
(
) ( m xj + β xj ) ,
mi,a j + β i,a j ≥ 0,
(
) ( m xj − α xj ) ,
mi,a j + β i,a j ≤ 0,
⎧ mi,a j + β i,a j ⎪ = ⎨ ⎪ mi,a j + β i,a j ⎩
(3.13b)
G. MALKAWI, N. AHMAD,
444
% d +a m x + β x , j j ' i, j = & ' d −a m x − α x , j j ( i, j
(
)
di,a j ≥ 0,
(
)
di,a j < 0,
AND
H. IBRAHIM
β Using Eq. (3.12), we represent fi, j as
= di,+aj m xj + β xj + di,−aj m xj − α xj
(
)
(
)
= di,+aj m xj + di,+aj β xj + di,−aj m xj − di,−aj α xj = di,+aj m xj + di,−aj m xj + di,+aj β xj − di,−aj α xj
(
) (
= di,+aj + di,−aj m xj + di,+aj β xj − di,−aj α xj
(
)
(
= di,a j m xj + di,+aj β xj − di,−aj α xj
(
) (
)
)
)
= mi,a j + βi,a j m xj + di,+aj β xj − di,−aj α xj .
(
)
(
)
Then, Eq. (3.3c) may be written as
βi,b j = −mi,b j + fi,βj , βi,b j = −mi,a j m xj +
{ ( mi,a j + βi,a j ) m xj + ( di,+aj β xj − di,−aj α xj ) }
= −mi,a j m xj + mi,a j m xj + βi,a j m xj + di,+aj β xj − di,−aj α xj
(
)
(
)
= 0 + βi,a j m xj + di,+aj β xj − di,−aj α xj ,
(
)
hence,
βi,b j = βi,a j m xj + di,+aj β xj − di,−aj α xj .
(3.14)
Using Eq. (3.12), we obtain
( ci,−aj β xj ) ( ci,+aj α xj )
= 0.
(3.15)
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
Step 4: Compute the LR fuzzy numbers
445
( mi,b j , αi,b j , βi,b j ) .
Distribute Eq. (3.2) using Eqs. (3.3a), (3.9), and (3.14) to obtain ⊕
∑ ( mi,a j , αi,a j , βi,a j ) ⊗ ( m xj , α xj , β xj ) j=1
⊕
=
∑ ( mi,b j , αi,b j , βi,b j )
=
∑ ( mi,a j m xj , αi,a j m xj − ci,−aj β xj + ci,+aj α xj , βi,a j m xj + di,+aj β xj − di,−aj α xj ) ,
j=1 ⊕
j=1
∀i = 1, 2,…, n ,
(3.16)
which is equivalent to the following crisp linear system:
% n a x b ' ∑mi, j m j = mi , ' j=1 ' '' n a x −a x +a x b & ∑α i, j m j − ci, j β j + ci, j α j = α i , ' j=1 ' ' n ' ∑βi,a j m xj + di,+aj β xj − di,−aj α xj = βbi , '( j=1
(3.17)
where i = 1,…, n .
Step 5: Convert the FFLS for two separated linear systems. Let
Ca =
( ci,a j )n×n ,
C +a =
( ci,+aj )n×n ,
C −a =
( ci,−aj )n×n ;
the other entries do not determine zeros; then
C a = C +a + C −a = A − M because n
∑ci,a j j=1
=
n
∑ci,+aj + ci,−aj j=1
=
n
∑mi,a j − αi,a j , j=1
∀i = 1, 2,…, n.
G. MALKAWI, N. AHMAD,
446
AND
H. IBRAHIM
Similarly,
Da =
( di,a j )n×n ,
D +a =
( di,+aj )n×n ,
d −a =
( di,−aj )n×n ;
the other entries do not determine zeros; then
D a = D +a + D −a = A + M , because n
∑di,a j j=1
n
∑di,+aj + di,−aj
=
j=1
=
n
∑mi,a j + βi,a j , j=1
∀i = 1, 2,…, n .
Hence, the parameters m xj , α xj , β xj for Eq. (3.2) may be obtained as follows:
– The mean values m xj are obtained separately using the following n × n linear system: n
∑mi,a j m xj j=1
= mib ,
∀i = 1,…, n .
(3.18a)
– The spreads α xj , β xj are obtained jointly using the following 2n × 2n linear system, where the values m xj are obtained from the linear system (3.18a),
& n a x −a x +a x b ( ∑α i, j m j − ci, j β j + ci, j α j = α i , (( j=1 ' ( n ( ∑βi,a j m xj + di,+aj β xj − di,−aj α xj = βbj , () j=1
∀i = 1,…, n, (3.18b)
∀i = 1,…, n.
Solving the linear systems (3.18a) and (3.18b), we get the solutions of vectors m xj , α xj , β xj .
Definition 3.1 (The general associated linear system). Let A = ( A, M , N ) be a fuzzy matrix, and B =
(m b , α b , βb )
(
and X = m x , α x , β x
)
be fuzzy vectors.
Let
( )n×n ,
C +a = ci,+aj
( )n×n ,
C −a = ci,−aj
The 3n × 3n linear systems are defined as
(
D +a = di,+aj
)n×n ,
and
(
D −a = di,−aj
)n×n .
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
GX = B ,
447
(3.19)
where
⎛A ⎜ G = ⎜M ⎜ ⎜ ⎝N
0
0
⎞ ⎟ −C −a ⎟ , ⎟ ⎟ D +a ⎠
C +a −D −a
(
X = Vec m x , α x , β x
(
B = Vec m b , α b , β b
)
⎛ mx ⎞ ⎟ ⎜ ⎜ x⎟ = ⎜α ⎟ , ⎟ ⎜ ⎜⎝ β x ⎟⎠
)
⎛ mb ⎞ ⎟ ⎜ ⎜ b⎟ = ⎜α ⎟. ⎟ ⎜ ⎜⎝ β b ⎟⎠
Then, GX = B may represented as follows:
⎛A ⎜ ⎜M ⎜ ⎜ ⎝N
0 C +a −D −a
0
x
⎞⎛m ⎟⎜ −a ⎟ ⎜ x −C α ⎟⎜ ⎜ ⎟ D +a ⎠ ⎜⎝ β x
⎞ ⎛ mb ⎞ ⎟ ⎟ ⎜ ⎟ ⎜ b⎟ ⎟ = ⎜α ⎟. ⎟ ⎟ ⎜ ⎟⎠ ⎜⎝ β b ⎟⎠
(3.20)
In this paper, the linear system GX = B is called a general associated linear system. The following theorem shows the relation between the components of the 3n dimensional crisp solution vector X in (3.19) and the solution X of FFLS in Eq. (2.9).
Theorem 3.1. The unique solution of the crisp system GX = B and the positive solution for an arbitrary FFLS A ⊗ B = X are equivalent. Proof.
GX = B , ⎛A ⎜ ⎜M ⎜ ⎜ ⎝N
0 C +a −D −a
0
x
⎞⎛m ⎟⎜ −a ⎟ ⎜ x −C α ⎟⎜ ⎟⎜ D +a ⎠ ⎜⎝ β x
⎞ ⎛ mb ⎞ ⎟ ⎟ ⎜ ⎟ ⎜ b⎟ ⎟ = ⎜α ⎟, ⎟ ⎟ ⎜ ⎟⎠ ⎜⎝ β b ⎟⎠
G. MALKAWI, N. AHMAD,
448
⎛ a ⎛ ⎜ 11 ⎜ ⎜ ⎜⎜ ⎜⎜ ⎜ ⎝ an1 ⎜ ⎜ ⎛ m11 ⎜⎜ ⎜ ⎜ ⎜⎜ ⎜ ⎝ mn1 ⎜ ⎜ ⎜ ⎛ n11 ⎜⎜ ⎜ ⎜ ⎜⎜ ⎜ ⎝ nn1 ⎝
… a1n ⎞ ⎟ ⎟ ⎟ … ann ⎠
⎛0 ⎜ ⎜ ⎜ ⎝0
0⎞ ⎟ ⎟ ⎟ 0⎠
… m1n ⎞ ⎟ ⎟ ⎟ … mnn ⎠
+ ⎛ c11 ⎜ ⎜ ⎜ ⎜⎝ c + n1
+ ⎞ … c1n ⎟ ⎟ ⎟ + ⎟ … cnn ⎠
… n1n ⎞ ⎟ ⎟ ⎟ … nnn ⎠
− − ⎞ ⎛ −d11 … −d1n ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ −d − … −d − ⎟⎠ n1 nn
…
…
⎛0 ⎜ ⎜ ⎜ ⎝0
… …
− ⎛ −c11 … ⎜ ⎜ ⎜ ⎜⎝ −c − … n1 + ⎛ d11 ⎜ ⎜ ⎜ ⎜⎝ d + n1
… …
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ − ⎞ ⎟ −c1n ⎟⎟ ⎟⎟ ⎟⎟ − ⎟⎟ −cnn ⎠ ⎟ ⎟ + ⎞ ⎟ d1n ⎟⎟ ⎟⎟ ⎟⎟ + ⎟ ⎟ d nn ⎠ ⎠ 0⎞ ⎟ ⎟ ⎟ 0⎠
AND
⎛ ⎛ m1x ⎞ ⎞ ⎛ ⎛ m1b ⎞ ⎞ ⎜⎜ ⎜⎜ ⎟⎟ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎜⎜ ⎟⎟ ⎟⎟ ⎜ ⎜⎝ mnx ⎟⎠ ⎟ ⎜ ⎜⎝ mnb ⎟⎠ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎛ αx ⎞ ⎟ ⎜ ⎛ αb ⎞ ⎟ ⎜⎜ 1⎟ ⎟ ⎜⎜ 1⎟ ⎟ ⎜⎜ ⎟ ⎟ = ⎜⎜ ⎟ ⎟. ⎜⎜ ⎜⎜ ⎟⎟ ⎟⎟ ⎜ ⎜⎝ α x ⎟⎠ ⎟ ⎜ ⎜⎝ α x ⎟⎠ ⎟ 1 ⎟ 1 ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ x b ⎜ ⎛ β1 ⎞ ⎟ ⎜ ⎛ β1 ⎞ ⎟ ⎜⎜ ⎟ ⎟ ⎜⎜ ⎟ ⎟ ⎜⎜ ⎟ ⎟ ⎜⎜ ⎟ ⎟ ⎜ ⎟ ⎜⎜ x⎟ ⎟ ⎜ ⎜⎜ b ⎟⎟ ⎟ ⎝ ⎝ β1 ⎠ ⎠ ⎝ ⎝ β1 ⎠ ⎠
GX = B is split into the following three linear systems: – First linear system: for i = 1,…, n ,
⎛ a11 ⎜ ⎜ ⎜ ⎜⎝ a n1
⎛ m1b ⎞ … a1n ⎞ ⎛ m1x ⎞ ⎜ ⎟ ⎟ ⎟⎜ ⎟⎜ ⎟ = ⎜ ⎟; ⎜ ⎟ ⎟ ⎟⎜ ⎜ x⎟ ⎜ b⎟ ⎟ … ann ⎠ ⎝ mn ⎠ ⎝ mn ⎠
then n
∑ai, j m xj j=1
= mib ,
∀i = 1,…, n .
If ai, j = mi,a j , we get (3.18a). – Second linear system: for i = n + 1,…, 2n , x + ⎛ m11 … m1n ⎞ ⎛ m1 ⎞ ⎛ c11 ⎟ ⎜ ⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟ +⎜ ⎟ ⎜ ⎜ ⎟⎜ ⎜ x⎟ ⎜ + ⎜⎝ m ⎟ n1 … mnn ⎠ ⎝ mn ⎠ ⎝ cn1
+ ⎞⎛ x⎞ ⎛ − − ⎞⎛ x⎞ ⎛ α1b ⎞ … c1n … −c1n α1 −c11 β1 ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎟ ⎟ ⎜ ⎟ +⎜ ⎟⎜ ⎟ = ⎜ ⎟, ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎟ + − − … cnn … −cnn ⎝ α bn ⎠ ⎠ ⎝ α1x ⎠ ⎝ −cn1 ⎠ ⎝ β1x ⎠
then n
∑mi,a j m xj − ci,−aj β xj + ci,+aj α xj j=1
= α bi ,
∀i = 1,…, n ,
H. IBRAHIM
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM n
∑mi,a j m xj + ci,+aj α xj − ci,−aj β xj j=1
= α bi ,
∀i = 1,…, n .
449
(3.21)
– Third linear system: for i = 2n + 1,…, 3n ,
⎛ n11 ⎜ ⎜ ⎜ ⎜⎝ n n1
− − ⎞⎛ x⎞ ⎛ + … n1n ⎞ ⎛ m1x ⎞ ⎛ −d11 … −d1n α1 d11 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ +⎜ ⎟ ⎜ ⎟ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ + − − … nnn ⎟⎠ ⎝ mnx ⎠ ⎝ −d n1 … −d nn ⎠ ⎝ α1x ⎠ ⎝ d n1
+ ⎞⎛ x⎞ ⎛ β1b ⎞ … d1n β1 ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ = ⎜ ⎟, ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎜ b⎟ ⎟ ⎟ + … d nn ⎝ βn ⎠ ⎠ ⎝ β1x ⎠
then, n
∑ni,a j m xj + di,+aj β xj − di,−aj α xj j=1 n
∑ni,a j m xj − di,−aj α xj + di,+aj β xj j=1
= β bj ,
∀i = 1,…, n ,
= β bj ,
∀i = 1,…, n .
(3.22)
If mi,a j = α i,a j and ni,a j = β i,a j , then (3.21) and (3.22) are equivalent to (3.18b). The proof is completed. 4. Numerical Examples In this section, numerical examples are illustrated to show the ease and efficiency of the proposed method in obtaining a positive solution for an arbitrary FFLS. Examples 4.1 and 4.2 are obtained from previous studies for comparison. Example 4.1. Consider the following FFLS [20, 5]:
(
)
(
)
(
)
⎧ ( 4, 6,1 ) ⊗ m1x , α1x , β1x ⊕ ( 4, 2, 4 ) ⊗ m2x , α 2x , β 2x = ( 24, 26, 31 ) , ⎪ ⎨ ⎪ ( 3, 2,1 ) ⊗ m1x , α1x , β1x ⊕ ( 2, 3,1 ) ⊗ m2x , α 2x , β 2x = ( 14,18,13 ) , ⎩
(
(
)
)
where x i = mi x , α i x , β i x ≥ 0 , i = 1, 2 .
The system may be written in matrix form:
⎛ (4, 6,1) ⎜ ⎜⎝ (3, 2,1)
(4, 2, 4) ⎞ ⎛ x1 ⎞ ⎛ ( 24, 26, 31 ) ⎞ = ⊗ ⎟, ⎜ ⎟ ⎜ ⎟ ⎜⎝ ( 14,18,13 ) ⎟⎠ (2, 3,1) ⎟⎠ ⎜⎝ x 2 ⎟⎠
G. MALKAWI, N. AHMAD,
450
AND
where
(
⎛ m1x , α1x , β1x ⎛ x1 ⎞ X = ⎜ ⎟ = ⎜⎜ ⎜⎝ x ⎟⎠ ⎜⎝ m2x , α 2x , β 2x 2
(
) ⎞⎟ )
⎟; ⎟⎠
then the subvectors for the vector X are
m
x
⎛ m1x ⎞ ⎟, = ⎜ ⎜ x⎟ ⎝ m2 ⎠
α
x
⎛ α1x ⎞ ⎟, = ⎜ ⎜ x⎟ ⎝ α2 ⎠
β
x
⎛ β1x ⎞ = ⎜ ⎟. ⎜ x⎟ ⎝ β2 ⎠
The submatrices for block matrix G are
⎛4 A = ⎜ ⎜⎝ 3
4⎞ ⎟, 2 ⎟⎠
⎛6 M = ⎜ ⎜⎝ 2
⎛ −2 A−M = ⎜ ⎜⎝ 1
2⎞ ⎟, −1 ⎟⎠
⎛0 C +a = ⎜ ⎜⎝ 1
2⎞ ⎟, 0 ⎟⎠
⎛ −2 C −a = ⎜ ⎜⎝ 0
0⎞ ⎟, −1 ⎟⎠
⎛5 A−N = ⎜ ⎜⎝ 4
8⎞ ⎟, 3 ⎟⎠
⎛0 D −a = ⎜ ⎜⎝ 0
0⎞ ⎟, 0 ⎟⎠
⎛5 D +a = ⎜ ⎜⎝ 4
8⎞ ⎟, 3 ⎟⎠
(
α1b ,
(
α b2 ,
⎛ m1b , B = ⎜⎜ ⎜⎝ m2b ,
2⎞ ⎟, 3 ⎟⎠
⎛1 N = ⎜ ⎜⎝ 1
)
26,
)
18,
β1b ⎞ ⎛ ( 24, ⎟ = ⎜ ⎟ ⎜⎝ ( 14, β b2 ⎟⎠
4⎞ ⎟, 1 ⎟⎠
31 ) ⎞ ⎟. 13 ) ⎟⎠
The subvectors for the vector B are
⎛ 24 ⎞ mb = ⎜ ⎟ , ⎜⎝ 14 ⎟⎠
⎛ 26 ⎞ αb = ⎜ ⎟ , ⎜⎝ 18 ⎟⎠
⎛ 31 ⎞ βb = ⎜ ⎟ . ⎜⎝ 13 ⎟⎠
The given FFLS can be converted into a crisp linear system using the following matrix form in (3.19):
GX = B ,
H. IBRAHIM
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
⎛A ⎜ ⎜M ⎜ ⎜ ⎝N ⎛⎛4 ⎜⎜ ⎜⎝3 ⎜ ⎜⎛6 ⎜⎜ ⎜⎝2 ⎜ ⎜ ⎛1 ⎜ ⎜⎝ ⎜⎝ 1
0
x
0
C +a −D −a
⎞⎛m ⎟⎜ −a ⎟ ⎜ x −C α ⎟⎜ ⎟⎜ D +a ⎠ ⎜⎝ β x
4⎞ ⎟ 2⎠
⎛0 ⎜ ⎝0
0⎞ ⎟ 0⎠
⎛0 ⎜ ⎝0
2⎞ ⎟ 3⎠
⎛0 ⎜ ⎝1
2⎞ ⎟ 0⎠
⎛2 ⎜ ⎝0
4⎞ ⎟ 1⎠
⎛0 ⎜ ⎝0
0⎞ ⎟ 0⎠
⎛5 ⎜ ⎝4
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
4
4
0
0
0
3
2
0
0
0
6
2
0
2
2
2
3
1
0
0
1
4
0
0
5
1
1
0
0
4
0⎞ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ 8⎟ ⎟ ⎟ 3⎠
⎞ ⎛ mb ⎞ ⎟ ⎟ ⎜ ⎟ ⎜ b⎟ ⎟ = ⎜α ⎟, ⎟ ⎟ ⎜ ⎟⎠ ⎜⎝ β b ⎟⎠ 0⎞ ⎞ ⎟⎟ 0⎠ ⎟ ⎟ 0⎞ ⎟ ⎟⎟ 1⎠ ⎟ ⎟ 8⎞ ⎟ ⎟ ⎟⎟ 3⎠ ⎠
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
451
⎛ ⎛ m1x ⎞ ⎞ ⎛ ⎛ 24 ⎞ ⎞ ⎜⎜ ⎟⎟ x ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎝ m2 ⎠ ⎟ ⎜ ⎝ 14 ⎠ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎛ αx ⎞ ⎟ ⎜ ⎜ 1 ⎟ ⎟ = ⎜ ⎛ 26 ⎞ ⎟ , ⎜⎜ ⎟⎟ ⎜⎜ x⎟ ⎟ ⎜ ⎝ 18 ⎠ ⎟ ⎜ ⎝ α2 ⎠ ⎟ ⎜ ⎟ ⎜ ⎟ x ⎜ ⎛ 31 ⎞ ⎟ ⎜ ⎛ β1 ⎞ ⎟ ⎜⎝ ⎜⎝ 13 ⎟⎠ ⎟⎠ ⎜⎜ ⎟ ⎟ ⎜⎝ ⎜⎝ β x ⎟⎠ ⎟⎠ 2
m1x ⎞ ⎟ ⎛ 24 ⎞ x ⎟ ⎜ ⎟ m2 ⎟ ⎜ 14 ⎟ ⎟ ⎜ ⎟ α1x ⎟ ⎜ 26 ⎟ ⎟ = ⎜ ⎟. x ⎟ ⎜ ⎟ 18 α2 ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ 31 β1x ⎟ ⎜ ⎟ ⎟ ⎜⎝ ⎟⎠ 13 ⎟ x ⎟ β2 ⎠
Using the inversion matrix method, we have
X = G −1B,
⎛ 1 − ⎜ 2 ⎜ ⎛ m1x ⎞ ⎜ 3 ⎜ ⎟ ⎜ ⎜ x⎟ ⎜4 ⎜ m2 ⎟ ⎜ ⎜ ⎟ ⎜ 25 x ⎜ α1 ⎟ ⎜ 34 ⎜ ⎟ = ⎜ ⎜ x⎟ ⎜ 29 ⎜ α2 ⎟ ⎜ ⎜ ⎟ ⎜ 68 ⎜ βx ⎟ ⎜ ⎜ 1 ⎟ ⎜ 11 ⎜ x ⎟ ⎜ 34 ⎝ β2 ⎠ ⎜ ⎜ 35 ⎜⎝ − 68
1
0
0
0
−1
0
0
0
5 17
0
1
−
25 17
1 2
0
−
9 17
0
0
0
0
12 17
−
4 17
3 17 −
3 17
4 17
⎞ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 5 ⎟ 17 ⎟ ⎟ 8 ⎟ − ⎟ 17 ⎟ ⎟ 8 ⎟ 17 ⎟ ⎟ 5 ⎟ − ⎟ 17 ⎠ 0
⎛ 24 ⎞ ⎜ ⎟ ⎜ 14 ⎟ ⎜ ⎟ ⎜ 26 ⎟ ⎜ ⎟. ⎜ ⎟ ⎜ 18 ⎟ ⎜ ⎟ ⎜ 31 ⎟ ⎜ ⎟ ⎜⎝ 13 ⎟⎠
G. MALKAWI, N. AHMAD,
452
Then, the crisp solution is
⎛ m1x ⎞ ⎛ 2⎞ ⎜ ⎟ ⎜ ⎟ ⎜ x⎟ m 2 ⎜ 4⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 ⎟ ⎜ α1x ⎟ ⎜ ⎟ X = = ⎜⎜ ⎟⎟ ⎜ x⎟ ⎜ 2⎟ ⎜ α2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 ⎟ ⎜ βx ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜⎝ 1 ⎟⎠ ⎜ x ⎟ ⎝ β2 ⎠
⎛ ⎛ m1x ⎞ ⎞ ⎛⎛ 2⎞⎞ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ x⎟⎟ ⎜ ⎜⎝ 4 ⎟⎠ ⎟ ⎜ ⎝ m2 ⎠ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎛ α1x ⎞ ⎟ ⎛ ⎞ ⎟ ⎟ = ⎜⎜ ⎜ ⎟ ⎟⎟ , X = ⎜⎜ ⎜ ⎜ ⎟ ⎜ x⎟ ⎟ ⎜ ⎝ 2⎠ ⎟ ⎜ ⎝ α2 ⎠ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎛ 1⎞ ⎟ ⎜⎛ x⎞ ⎟ ⎜⎜ ⎟ ⎟ ⎜ β1 ⎟ ⎜ ⎜⎝ 1 ⎟⎠ ⎟ ⎜⎜ ⎟ ⎟ ⎝ ⎠ ⎜⎝ ⎜⎝ β x ⎟⎠ ⎟⎠ 2
or
which is equivalent to a fuzzy solution in [20],
(
⎛ m1x , α1x , β1x ⎛ x1 ⎞ X = ⎜ ⎟ = ⎜⎜ ⎜⎝ x ⎟⎠ ⎜⎝ m2x , α 2x , β 2x 2
(
) ⎞⎟ )
⎛ ( 2, 1, 1 ) ⎞ ⎜ ⎟. = ⎟ ⎜ 4, 2, 1 ) ⎟⎠ ⎝( ⎟⎠
Example 4.2. Consider the following FFLS [5]:
(
)
(
)
=
( 5,16,17 ) ,
(
)
=
( −4,12, 22 ) ,
⎧ ( 3, 2, 3 ) ⊗ m1x , α1x , β1x ⊕ ( −2,1,1 ) ⊗ m2x , α 2x , β 2x ⎪ ⎨ ⎪ ( −4,1, 2 ) ⊗ m1x , α1x , β1x ⊕ ( 4, 2,1 ) ⊗ m2x , α 2x , β 2x ⎩
(
(
)
)
where x i = mi x , α i x , β i x ≥ 0 , i = 1, 2 .
The system may be written in matrix form:
⎛ ( 3, 2, 3 ) ⎜ ⎜⎝ ( −4, 1, 2 )
( −2, 1, 1 ) ⎞
⎛ ( 5, 16, 17 ) ⎞ ⎛ x1 ⎞ ⎟ ⊗⎜ ⎟ = ⎜ ⎟, ( 4, 2, 1 ) ⎟⎠ ⎜⎝ x 2 ⎟⎠ ⎜⎝ ( −4, 12, 22 ) ⎟⎠
where
(
⎛ m1x , α1x , β1x ⎛ x1 ⎞ X = ⎜ ⎟ = ⎜⎜ ⎜⎝ x ⎟⎠ ⎜⎝ m2x , α 2x , β 2x 2
(
) ⎞⎟ )
⎟. ⎟⎠
Then
m
x
⎛ m1x ⎞ ⎟, = ⎜ ⎜ x⎟ ⎝ m2 ⎠
α
x
⎛ α1x ⎞ ⎟, = ⎜ ⎜ x⎟ ⎝ α2 ⎠
β
x
⎛ β1x ⎞ = ⎜ ⎟, ⎜ x⎟ ⎝ β2 ⎠
AND
H. IBRAHIM
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
⎛3 A = ⎜ ⎜⎝ −4
−2 ⎞ ⎟, 4 ⎟⎠
⎛1 A−M = ⎜ ⎜⎝ −5
−3 ⎞ ⎟, 2 ⎟⎠
C
⎛6 A−N = ⎜ ⎜⎝ −2
−1 ⎞ ⎟, 5 ⎟⎠
⎛0 D −a = ⎜ ⎜⎝ −2
⎛2 M = ⎜ ⎜⎝ 1 +a
(
⎛1 = ⎜ ⎜⎝ 0
⎛ m1b , α1b , β1b B = ⎜⎜ ⎜⎝ m2b , α b2 , β b2
(
⎛5 ⎞ mb = ⎜ ⎟ , ⎜⎝ −4 ⎟⎠
1⎞ ⎟, 2 ⎟⎠
453
⎛3 N = ⎜ ⎜⎝ 2
0⎞ ⎟, 2 ⎟⎠
C
−a
⎛0 = ⎜ ⎜⎝ −5
⎛6 D +a = ⎜ ⎜⎝ 0
−1 ⎞ ⎟, 0 ⎟⎠
) ⎞⎟ )
⎛ ( 5, 16, 17 ) ⎞ ⎜ ⎟, = ⎟ ⎜⎝ ( −4, 12, 22 ) ⎟⎠ ⎟⎠
⎛ 16 ⎞ αb = ⎜ ⎟ , ⎜⎝ 12 ⎟⎠
⎛ 17 ⎞ βb = ⎜ ⎟ . ⎜⎝ 22 ⎟⎠
The given FFLS can be converted into a crisp linear system using (3.20):
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
3
−2
0
0
0
−4
4
0
0
0
2
1
1
0
0
1
2
0
2
5
3
1
0
1
6
2
1
2
0
0
0⎞ ⎟ 0⎟ ⎟ 3⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 5⎠
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝
m1x ⎞ ⎟ ⎛ 5 ⎞ x ⎟ ⎜ ⎟ m2 ⎟ ⎜ −4 ⎟ ⎟ ⎜ ⎟ α1x ⎟ ⎜ 16 ⎟ ⎟ = ⎜ ⎟, ⎟ x ⎜ ⎟ 12 α2 ⎟ ⎜ ⎟ ⎟ ⎜ 17 ⎟ β1x ⎟ ⎜ ⎟ ⎟ ⎜⎝ ⎟⎠ 22 ⎟ x ⎟ β2 ⎠
X = G −1B ; then ⎛ m1x ⎞ ⎛ 3⎞ ⎜ ⎟ ⎜ ⎟ ⎜ mx ⎟ ⎜ 2⎟ ⎜ 2⎟ ⎜ ⎟ ⎜ x⎟ ⎜ 2⎟ ⎜ α1 ⎟ X = ⎜ ⎟ = ⎜ ⎟ ⎜ 0⎟ ⎜ α 2x ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1⎟ ⎜ βx ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜⎝ 2 ⎟⎠ ⎜ x ⎟ ⎝ β2 ⎠
1⎞ ⎟, 1 ⎟⎠ −3 ⎞ ⎟, 0 ⎟⎠ 0⎞ ⎟, 5 ⎟⎠
G. MALKAWI, N. AHMAD,
454
AND
H. IBRAHIM
or
⎛ ⎛ m1x ⎞ ⎞ ⎛ ⎛ 3⎞ ⎞ ⎜⎜ ⎟⎟ x ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎝ m2 ⎠ ⎟ ⎜ ⎝ 2⎠ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎛ αx ⎞ ⎟ 2 ⎜⎜ 1⎟ ⎟ = ⎜⎛ ⎞⎟, ⎜⎜ ⎟⎟ ⎜⎜ x⎟ ⎟ ⎜ ⎝ 0⎠ ⎟ ⎜ ⎝ α2 ⎠ ⎟ ⎜ ⎟ ⎜ ⎟ x ⎜ ⎛1⎞ ⎟ ⎜ ⎛ β1 ⎞ ⎟ ⎜⎝ ⎜⎝ 2 ⎟⎠ ⎟⎠ ⎜⎜ ⎟ ⎟ ⎜⎝ ⎜⎝ β x ⎟⎠ ⎟⎠ 2 which is equivalent to a fuzzy solution of FFLS in [5],
! m x , α x , βx ! x!1 $ # 1 1 1 X! = # & = # # & # m2x , α 2x , β 2x " x! 2 % "
( (
) $& & ) &%
! ( 3, 2, 1 ) $ &. = # # & 2, 0, 2 " %
In Examples 4.3 and 4.4, we show the efficiency of the proposed method in obtaining a solution for large systems, where all of the examples in [5] and [20] do not exceed n = 3 . The verification of the solution is provided in the Appendix. Example 4.3. Consider the following 5 × 5 FFLS:
& (−9, 8,1) ⊕ m x , α x , β x ⊕ ( −12, 2, 3 ) ⊗ m x , α x , β x ⊕ ( −12, 3, 7 ) ⊗ m x , α x , β x 1 1 1 2 2 2 3 3 3 ( ( ( ⊕ ( 1, 5, 6 ) ⊕ m 4x , α 4x , β 4x ⊕ ( 9, 3,1 ) ⊕ m5x , α 5x , β 5x = ( −186, 291, 289 ) , ( ( x x x x x x x x x ( (−1, 3, 7) ⊕ m1 , α1 , β1 ⊕ ( 7, 8, 9 ) ⊗ m2 , α 2 , β 2 ⊕ ( −12, 7, 3 ) ⊗ m 3 , α 3 , β 3 ( ( ⊕ ( 3, 3, 7 ) ⊕ m 4x , α 4x , β 4x ⊕ ( −3, 4, 5 ) ⊕ m5x , α 5x , β 5x = ( −29, 235, 386 ) , ( ( ( (7, 3, 5) ⊕ m1x , α1x , β1x ⊕ ( 2,1, 5 ) ⊗ m2x , α 2x , β 2x ⊕ ( −6, 2,1 ) ⊗ m 3x , α 3x , β 3x ( ' ( ⊕ ( 7, 4, 8 ) ⊕ m 4x , α 4x , β 4x ⊕ ( 2, 9, 7 ) ⊕ m5x , α 5x , β 5x = ( 93, 203, 309 ) , ( ( ( (2, 5, 0) ⊕ m1x , α1x , β1x ⊕ ( 1, 6, 9 ) ⊗ m2x , α 2x , β 2x ⊕ ( −1, 5, 2 ) ⊗ m 3x , α 3x , β 3x ( ( ⊕ ( −9, 8, 4 ) ⊕ m 4x , α 4x , β 4x ⊕ ( 3,1, 3 ) ⊕ m5x , α 5x , β 5x = ( −20, 218, 228 ) , ( ( ( (1, 4, 7) ⊕ m x , α x , β x ⊕ ( 1, 9,1 ) ⊗ m x , α x , β x ⊕ ( −2,1, 6 ) ⊗ m x , α x , β x 1 1 1 2 2 2 3 3 3 ( ( () ⊕ ( −2,1, 7 ) ⊕ m 4x , α 4x , β 4x ⊕ ( 5, 8, 6 ) ⊕ m5x , α 5x , β 5x = ( 17, 231, 270 ) ,
(
)
(
(
( ( (
)
(
)
)
) ) )
(
(
)
(
)
(
)
where x i = mi x , α i x , β i x , for i = 1,…, 5 .
)
)
( ( (
) ) )
(
)
)
(
)
)
(
)
(
)
(
)
(
)
(
(
(
)
(
(
)
(
)
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
455
The system may be written in matrix form:
" ( −9, 8,1 ) $ $ $ ( −1, 3, 7 ) $ $ ( 7, 3, 5 ) $ $ $ ( 2, 5, 0 ) $ $ # ( 1, 4, 7 )
( −12, 2, 3 )
( −12, 3, 7 )
( 1, 5, 6 )
( 7, 8, 9 )
( −12, 7, 3 )
( 3, 3, 7 )
( 2,1, 5 )
( −6, 2,1 )
( 7, 4, 8 )
( 1, 6, 9 )
( −1, 5, 2 )
( −9, 8, 4 )
( 1, 9,1 )
( −2,1, 6 )
( −2,1, 7 )
( 9, 3,1 ) % "$ x!1 %' '
" ( −186, 291, 289 ) % ' $ ' ' $ ' $ ( −3, 4, 5 ) ' $ x! 2 ' $ ( −29, 235, 386 ) ' ' ' $ ' $ ( 2, 9, 7 ) ' ⊗ $$ x! 3 '' = $ ( 93, 203, 309 ) ' , ' $ ' ' ' $ ' $ ( 3,1, 3 ) ' $ x! 4 ' $ ( −20, 218, 228 ) ' ' ' $ ' $ ' ' $ ' $ ( 5, 8, 6 ) & # x! 5 & # ( 17, 231, 270 ) &
where
! mx, # 1 ! x!1 $ # # & # m2x , # x! & 2 # # & # # & X! = # x! 3 & = # m 3x , # # & # x # x! 4 & # m4 , # & # # & " x! 5 % ## x " m5 ,
" −9 $ $ −1 $ A = $$ 7 $ $ 2 $$ # 1
mx
! mx $ # 1& & # # m2x & & # # x& = # m3 & , & # # m 4x & & # # x& # m5 & % "
−12
−12
1
7
−12
3
2
−6
7
1
−1
−9
1
−2
−2
αx
α1x , β1x $ & & α 2x , β 2x & & & α 3x , β 3x & , & & α 4x , β 4x & & x x & α 5 , β 5 &%
" αx % $ 1' ' $ $ α 2x ' ' $ $ x' = $ α3 ' , ' $ $ α 4x ' ' $ $ x' $ α5 ' & # 9 % ' −3 '' ' 2 ', ' 3 ' '' 5 &
βx
" βx % $ 1' $ ' $ β 2x ' $ ' $ ' = $ β 3x ' , $ ' $ β 4x ' $ ' $ x' $ β5 ' # &
!8 # #3 # M = ## 3 # #5 ## "4
2
3
5
8
7
3
1
2
4
6
5
8
9
1
1
3$ & 4 && & 9&, & 1& && 8%
G. MALKAWI, N. AHMAD,
456
⎛ ⎜ ⎜ ⎜ N = ⎜ ⎜ ⎜ ⎜ ⎜⎝
C +a
1
3
7
6
7
9
3
7
5
5
1
8
0
9
2
4
7
1
6
7
⎛ ⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎜ ⎜⎝
1⎞ ⎟ 5⎟ ⎟ 7⎟, ⎟ 3⎟ ⎟ 6 ⎟⎠
0
0
0
0
0
0
0
0
4
1
0
3
0
0
0
0
0
0
0
0
6⎞ ⎟ 0⎟ ⎟ 0⎟ , ⎟ 2⎟ ⎟ 0 ⎟⎠
⎛ ⎜ ⎜ ⎜ A−N = ⎜ ⎜ ⎜ ⎜ ⎜⎝
D −a
# mb , % 1 % b % m2 , % B! = % m 3b , % % b % m4 , % % m5b , $
⎛ ⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎜ ⎜⎝
" −17 $ $ −4 $ A−M = $ 4 $ $ −3 $ $ # −3
−8
−9
−5
0
0
0
−9
0
0
0
−5
0
0
0
0
−5
0
0
0
0
C −a
⎛ ⎜ ⎜ ⎜ = ⎜ ⎜ ⎜ ⎜ ⎜⎝
−14
−15
−4
−1
−19
0
1
−8
3
−5
−6
−17
−8
−3
−3
−17
−14
−15
−4
−4
−1
−19
0
0
0
−8
0
−3
−5
−6
−17
−3
−8
−3
−3
−8
−9
−5
7
6
16
−9
10
12
7
−5
15
2
10
1
−5
8
2
4
5
α1b , β1b & # ( −186, 291, 289 ) & ( ( % ( % ( −29, 235, 386 ) ( α b2 , βb2 ( ( % ( α b3 , βb3 ( = % ( 93, 203, 309 ) ( , ( % ( ( % ( α b4 , βb4 ( % ( −20, 218, 228 ) ( ( % ( $ ( 17, 231, 270 ) ' α b5 , βb5 ('
0⎞ ⎟ 0⎟ ⎟ 0⎟ , ⎟ 0⎟ ⎟ 0 ⎟⎠
mb
D +a
⎛ 0 ⎜ ⎜ 6 ⎜ = ⎜ 12 ⎜ ⎜ 2 ⎜ ⎜⎝ 8
⎛ −186 ⎞ ⎜ ⎟ ⎜ −29 ⎟ ⎜ ⎟ = ⎜ 93 ⎟ , ⎜ ⎟ ⎜ −20 ⎟ ⎜ ⎟ ⎜⎝ 17 ⎟⎠
AND
H. IBRAHIM
6 % ' −7 ' ' −7 ' , ' 2 ' ' ' −3 & 0 ⎞ ⎟ −7 ⎟ ⎟ −7 ⎟ , ⎟ 0 ⎟ ⎟ −3 ⎟⎠
10 ⎞ ⎟ 2 ⎟ ⎟ 9 ⎟, ⎟ 6 ⎟ ⎟ 11 ⎟⎠ 0
0
7
16
0
10
7
0
15
10
1
0
2
4
5
αb
⎛ 291 ⎞ ⎜ ⎟ ⎜ 235 ⎟ ⎜ ⎟ = ⎜ 203 ⎟ , ⎜ ⎟ ⎜ 218 ⎟ ⎜ ⎟ ⎜⎝ 231 ⎟⎠
10 ⎞ ⎟ 2 ⎟ ⎟ 9 ⎟, ⎟ 6 ⎟ ⎟ 11 ⎟⎠
βb
⎛ 289 ⎞ ⎜ ⎟ ⎜ 386 ⎟ ⎜ ⎟ = ⎜ 309 ⎟ . ⎜ ⎟ ⎜ 228 ⎟ ⎜ ⎟ ⎜⎝ 270 ⎟⎠
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
457
The given FFLS can be converted into a crisp linear system using (3.20),
" −9 $ $ −1 $ $ 7 $ $2 $ $1 $8 $ $ 3 $ $ 3 $ $5 $ $4 $ $1 $7 $ $5 $ $0 $ #7
−12
−12
1
9
0
0
0
0
0
0
0
0
0
7
−12
3
−3
0
0
0
0
0
0
0
0
0
2
−6
7
2
0
0
0
0
0
0
0
0
0
1
−1
−9
3
0
0
0
0
0
0
0
0
0
1
−2
−2
5
0
0
0
0
0
0
0
0
0
2
3
5
3
0
0
0
0
6
17
14
15
4
8
7
3
4
0
0
0
0
0
0
0
0
0
1
2
4
9
4
1
0
3
0
0
0
0
0
6
5
8
1
0
0
0
0
2
3
5
6
17
9
1
1
8
0
0
0
0
0
3
8
3
3
3
7
6
1
8
9
5
0
0
0
0
0
7
9
3
7
5
0
0
9
0
0
6
16
0
10
5
1
8
7
0
0
5
0
0
12
7
0
15
9
2
4
3
0
0
0
5
0
2
10
1
0
1
6
7
6
0
0
0
0
0
8
2
4
5
0% ' 0' ' 0' ' 0' ' 0' 0' ' 7' ' 7' ' 0' ' 3' ' 10 ' ' 2' 9' ' 6' ' 11 &
Then the crisp solution is
# mx & % 1( #9& % ( % m2x ( %7( % x( %6( % m3 ( % ( % x( %6( % m4 ( %5( % mx ( % ( % 5( %8( % α1x ( % 3( % x ( % ( % α2 ( %5( % x ( X = % α3 ( = % ( 1 % ( % αx ( 4 ( %2( % %1( % α 5x ( % ( % x ( %7( % β1 ( % ( % x ( %1( % β2 ( %1( % βx ( % ( % 3 ( %0( % β 4x ( % ( % x ( $6' %β ( $ 5 '
or
!! mx $ $ ## 1 && !! 9 $$ # # m2x & & ## && ## x && ## 7 && m ## 3 && ## 6 && ## x && ## && # # m4 & & ## 6 && x & # # m5 & % " # #" 5 &% & & # & # # ! αx $ & 1 $ ! 8 & # ## && ## 3&& # # α 2x & & ## && ## x & & ##5 &&, = α ## 3 & & ## && ## x & & α 4 ##1 && ## && # #" 2 &% & # # α 5x & & % " & # & # #! 7 $& # ! βx $ & ## 1 && ## 1 & & ## && # # β 2x & & ## 1 && ## x & & ## 0 && β ## 3 & & ## ## && && ## x & & β 4 "" 6 %% ## & & x # # β5 & & "" % %
# mx & " −186 % % 1( ' $ x % m2 ( $ −29 ' ( % ' $ % m 3x ( 93 ' $ ( % x ' $ ( % m4 $ −20 ' ( % $ 17 ' % m5x ( ' $ % x ( $ 291 ' % α1 ( ' $ % x ( ' $ α 235 % 2 ( ' $ % x ( % α 3 ( = $ 203 ' . ' $ % x ( $ 218 ' % α4 ( ' $ % x ( 231 ' α $ 5 ( % ' $ % βx ( $ 289 ' % 1 ( $ 386 ' % βx ( ' $ % 2 ( $ 309 ' % β 3x ( ' $ ( % $ 228 ' % β 4x ( '' $$ ( % # 270 & % βx ( $ 5 '
G. MALKAWI, N. AHMAD,
458
AND
H. IBRAHIM
which is equivalent to the fuzzy solution
! mx, ! x!1 $ # 1 # & # mx, # x! 2 & # 2 # & # X! = # x! 3 & = # m 3x , # & # x # x! 4 & # m4 , # & # x # & # m5 , " x! 5 % "
! ( 9, 8, 1 ) $ α1x , β1x $ & & # x x & # α 2 , β2 ( 7, 3, 7 ) && & # & α 3x , β 3x & = # ( 6, 5, 1 ) & ; & # & # x x & α 4 , β4 & # ( 6, 1, 0 ) & & & # α 5x , β 5x &% " ( 5, 2, 6 ) %
the verification of the solution is shown in the Appendix. Example 4.4. Consider the following 10 × 10 FFLS:
" (−3,2,13) $ $ (−7, 3,10) $ $ (−3,2, 7) $ $ (−6,1, 4) $ (−8,1,2) $ $ (−9,2,1) $ $(−10, 3,0) $ $ (−11, 4,1) $ $(−12, 3,2) $ # (4,2, 3)
(−2,2,11) (−1,0,9)
(0,2, 7)
(1, 4,5)
(−2, 4, 3)
(3, 4,1)
(4, 4,1)
(5, 4, 3)
(0,1,6)
(−8, 3, 4) (−2,5,2)
(3, 7,0)
(−4, 7,2)
(1,2, 3)
(2, 4,1)
(3,6,1)
(1,1,12)
(0,1,10)
(−1,1,8)
(0,2,9)
(3,0,11)
(2,0,9)
(3, 3,6)
(2,1,8) (−5,1,10)
(4,1,8)
(3,1,6)
(−2, 3, 4)
(1,5,2)
(2,5,0)
(−6,2, 3)
(3,0,5)
(−4,0, 7)
(7,2,9)
(6,2, 7)
(5,2,5)
(4, 4, 3)
(−3,6,1)
(8,2,1)
(6,1,2)
(3, 3, 4)
(6,1,6)
(9, 3,8)
(8, 3,6)
(7, 3, 4)
(6,5,2)
(9, 3,0)
(8,1,0)
(6,2,1)
(5, 4, 3) (−8,2,5) (−11, 4, 7) (10, 4,5)
(9, 4, 3)
(10,2,1)
(−9,0,1)
(6, 4,1)
(−6,5,2) (7,5,2)
(10, 3, 4)
(4,5, 4)
(−6,1,2)
(8,1,2)
(−7,5,2)
(8,5,2)
(10, 4, 3)
(11, 4, 3)
(−12, 4, 3) (5, 4,5)
(3,0, 3)
(−1,2, 3) (−7,6,1)
(−3,6,1)
(2, 4, 3)
(14, 3, 4)
(−8, 3, 4) (−9,5, 4)
(−1,0, 7) (0,2,5)
" ( 86, 211, 371 ) % " x!1 % ' $ ' $ ' $ −99, 373, 305 ( ) $ x! 2 ' ' $ ' $ $ ( 50, 240, 377 ) ' ! x $ 3' ' $ ' $ ' 31, 269, 329 $ ( ) ! x $ 4 ' ' $ $ x! ' 77, 354, 324 ' $ ( ) 5 ' = $ ⊗$ '. $ x! ' 186, 288, 418 ( ) ' $ $ 6 ' ' $ $ x! 7 ' $ ( 11, 302, 429 ) ' ' $ ' $ $ x! 8 ' $ ( 226, 403, 410 ) ' ' $ ' $ $ x! 9 ' $ ( 52, 471, 371 ) ' ' $ ' $ # x!10 & # ( −4, 474, 292 ) & The given FFLS can be converted into a crisp linear system using (3.20).
(13,5,6)
(6,6,5) % ' (−5, 7, 4)' ' (4,8, 3) ' ' (3,5,2) ' (2,8,1) ' ' (5, 7,0) ' ' (8,6,1) ' ' (11,5,2) ' ' (−8,6, 3)' ' (−6, 7, 4)&
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
459
Then the crisp solution can be obtained as
X = G −1B . The fuzzy solution is
! m x , α x , β x 1 1 1 # ! x!1 $ # & # x x x # m2 , α 2 , β 2 # x! & # # 2 & # m x , α x , β x & # 3 3 3 ! x # 3 & # # & # x x x # m 4 , α 4 , β 4 # x! 4 & # & # # m5x , α 5x , β 5x # x! 5 & & = # X! = # # # x! & x x x # m6 , α 6 , β 6 # 6 & # & # ! x # m 7x , α 7x , β 7x 7 & # # & # # x x x # x! 8 & # m8 , α 8 , β 8 & # # # x! 9 & # m9x , α 9x , β 9x & # # & # " x!10 % # x x x " m10 , α10 , β10
( ( ( ( ( ( ( ( (
(
) ) ) ) ) ) ) ) ) )
$ ! ( 1, 2, 3 ) $ & & # & # 0, 2, 5 & & )& #( & & # & # ( 0, 1, 4 ) & & & # & # ( 0, 2, 8 ) & & # & & # & & ( 1, 4, 5 ) & & = # # &, & 5, 1, 1 # & ( ) & # & & # ( 7, 4, 1 ) & & # & & # & & # ( 4, 4, 1 ) & & # & & # & 5, 4, 3 ( ) & # & & # & & " ( 6, 6, 5 ) % %
the verification of the solution is shown in the Appendix.
Conclusion FFLSs are transformed to equivalent linear systems without any fuzzy operation. As a result, many steps from fuzzy operation are reduced. This flexibility in transforming provides many advantages such as solving any unrestricted FFLS regardless of the system size. REFERENCES 1. S. Abbasbandy and M. S. Hashemi, “Solving fully fuzzy linear systems by using implicit Gauss–Cholesky algorithm,” Comput. Math. Model., 23, No. 1, 107–123 (2012). 2. T. Allahviranloo and N. Mikaeilvand, “Fully Fuzzy Linear Systems Solving Using MOLP,” World Appl. Sci. J., 12, No. 12, 2268– 2273 (2011). 3. T. Allahviranloo and N. Mikaeilvand, “Nonzero solutions of the fully fuzzy linear systems,” Appl. Comput. Math., 10, No. 2, 271– 282 (2011). 4. T. Allahviranloo, E. Haghi, and M. Ghanbari, “The nearest symmetric fuzzy solution for a symmetric fuzzy linear system,” An. St. Univ. Ovidius Constanta, 20, 151–172 (2012). 5. N. Babbar, A. Kumar, and A. Bansal, “Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers (m, α, β) ,” Soft Comput, 17, No. 4, 691–702 (2013). 6. J. Buckley, “Solving fuzzy equations in economics and finance,” Fuzzy Sets Syst., 48, 289–296 (1992).
460 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.
G. MALKAWI, N. AHMAD,
AND
H. IBRAHIM
J. Buckley and Y. Qu, “Solving systems of linear fuzzy equations,” Fuzzy Sets Syst., 43, 33–43 (1991). D. Dubois and H. Prade, “Systems of linear fuzzy constraints,” Fuzzy Sets Syst., 3, 37–48 (1980). D. Dubois and H. Prade, “Operations on fuzzy numbers,” Int. J. Sys. Sci., 9, No. 6, 613–626 (1978). M. Dehghan and B. Hashemi, “Iterative solution of fuzzy linear systems,” Appl. Math. Comput., 175, 645–674 (2006). M. Dehghan and B. Hashemi, “Solution of the fully fuzzy linear systems using the decomposition procedure,” Appl. Math. Comput. 182, 1568–1580 (2006). M. Dehghan, B. Hashemi, and M. Ghatee, “Computational methods for solving fully fuzzy linear systems,” Appl. Math. Comput., 179, 328–343 (2006). M. Dehghan, B. Hashemi, and M. Ghatee, “Solution of the fully fuzzy linear systems using iterative techniques,” Chaos, Solitons Fractals, 34, 316–336 (2007). R. Ezzati, “Solving fuzzy linear systems,” Soft Comput., 15, 193–197 (2011). R. Ezzati, S. Khezerloo, and A. Yousefzadeh, “Solving fully fuzzy linear system of equations in general form,” J. Fuzzy Set Valued Anal., 2012, 1–11 (2012). M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets Syst., 96, 201–209 (1998). J. Gao and Q. Zhang, “A unified iterative scheme for solving fully fuzzy linear system,” Global Congress on Intelligent Systems, 431–435 (2009). A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold, New York (1991). A. Kumar and N. Babbar, “A new computational method to solve fully fuzzy linear systems for negative coefficient matrix,” Int. J. Manuf. Technol. Manag., 25, 19–23 (2012). A. Kumar, A. Bansal, and Neetu, “Solution of fully fuzzy linear system with arbitrary coefficients,” Int. J. Appl. Math. Comput., 3, No.3, 232–237 (2011). A. Kumar, Neetu, and A. Bansal, “A new computational method for solving fully fuzzy linear systems of triangular fuzzy numbers,” Fuzzy Inform. Eng., 4, No. 1, 63–73 (2012). A. Kumar, Neetu, and A. Bansal, “A new approach for solving fully fuzzy linear systems,” Adv. Fuzzy Syst., 1–8 (2011). A. Kumar, Neetu, and A. Bansal, “A new method to solve fully fuzzy linear system with trapezoidal fuzzy numbers,” Can. J. Sci. Eng. Math., 1, No. 3, 45–56 (2010). H.-K Liu, “On the solution of fully fuzzy linear systems,” Int. J. Comput. Math. Sci., 4, No. 1, 29–33 (2010). G. Malkawi, N. Ahmad, and H. Ibrahim, “Revisiting fuzzy approach for solving system of linear equations,” ICDeM 2012, 13–16 March, Kedah, Malaysia, 157–164 (2012). G. Malkawi, N. Ahmad, and H. Ibrahim, “Solving fully fuzzy linear system with the necessary and sufficient condition to have a positive solution,” Appl. Math., 8, No. 3, 1003–1019 (2014). G. Malkawi, N. Ahmad, and H. Ibrahim, “A note on solving fully fuzzy linear systems by using implicit Gauss–Cholesky algorithm,” Comput. Math. Model., Accepted (2014). G. Malkawi, N. Ahmad, and H. Ibrahim, “Row reduced echelon form for solving fully fuzzy system with unknown coefficients,” J. Fuzzy Set Valued Anal., 2014, 1Р18 (2014). G. Malkawi, N. Ahmad, and H. Ibrahim, “On the weakness of linear programming to interpret the nature of solution of fully fuzzy linear system,” J. Uncertainty Anal. Appl., 2, No. 1, 1Р23 (2014). G. Malkawi, N. Ahmad, and H. Ibrahim, “A note on the nearest symmetric fuzzy solution for a symmetric fuzzy linear system,” An. St. Univ. Ovidius Constanta, Accepted (2013b). M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets Syst., 96, 201–209 (1998). M. Mosleh, M. Otadi, and A. Khanmirzaie, “Decomposition method for solving fully fuzzy linear systems,” Iran. J. Optim., 1, 188– 198 (2009). S. Muzziolia and H. Reynaertsb, “Fuzzy linear systems of the form A1x + b1 = A2 x + b2 ,” Fuzzy Sets Syst., 157, 939–951 (2006). S. H. Nasseri and F. Zahmatkesh, “Huang method for solving fully fuzzy linear system of equations,” J. Math. Comput. Sci., 1, No. 1, 1–5 (2010). S. H. Nasseri, M. Sohrabi, and E. Ardil, “Solving fully fuzzy linear systems by use of a certain decomposition of the coefficient matrix,” Int. J. Comput. Math. Sci., 3, 140–142 (2008). S. Nasseri and M. Sohrabi, “Cholesky decomposition for solving the fully fuzzy linear system of equations,” Int. J. Appl. Math., 22, No. 5, 689–696 (2009). S. Nasseri, M. Matinfar, and Z. Kheiri, “Greville’s method for the fully fuzzy linear system of equations,” Adv Fuzzy Sets Syst., 4, 301–3011 (2009). S. Nasseri, F. Taleshian, E. Behmanesh, and M. Sohrabi, “A Qr–decomposition of the mean value matrix of the coefficient matrix for solving the fully fuzzy linear system,” Int. J. Appl. Math., 25, 473–480 (2012). M. Otadi and M. Mosleh, “Solving fully fuzzy matrix equations,” Appl. Math. Model., 36, No. 12, 6114–6121 (2012). M. Otadi, M. Mosleh, and S. Abbasbandy, “Numerical solution of fully fuzzy linear systems by fuzzy neural network,” Soft Comput., 15, 1513–1522 (2011). L. A. Zadeh, “Fuzzy sets,” Inform. Control, 8, 338–353 (1965). L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning–II,” Inform. Sci., 8, 301–357 (1975).
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
Appendix Verification of solution — Example 4.3
( −9, 8, 1 ) ⊗ ( 9, 8, 1 ) ⊕ ( −12, 2, 3 ) ⊗ ( 7, 3, 7 ) ⊕ ( −12, 3, 7 ) ⊗ ( 6, 5, 1 ) ⊕ ( 1, 5, 6 ) ⊗ ( 6, 1, 0 ) ⊕ ( 9, 3, 1 ) ⊗ ( 5, 2, 6 ) = ( −81, 89, 73 ) ⊕ ( −84, 112, 48 ) ⊕ ( −72, 33, 67 ) ⊕ ( 6, 30, 36 ) ⊕ ( 45, 27, 65 ) = ( −186, 291, 289 ) ,
( −1, 3, 7 ) ⊗ ( 9, 8, 1 ) ⊕ ( 7, 8, 9 ) ⊗ ( 7, 3, 7 ) ⊕ ( −12, 7, 3 ) ⊗ ( 6, 5, 1 ) ⊕ ( 3, 3, 7 ) ⊗ ( 6, 1, 0 ) ⊕ ( −3, 4, 5 ) ⊗ ( 5, 2, 6 ) = ( −9, 31, 69 ) ⊕ ( 49, 63, 175 ) ⊕ ( −72, 61, 63 ) ⊕ ( 18, 18, 42 ) ⊕ ( −15, 62, 37 ) = ( −29, 235, 386 ) ,
( 7, 3, 5 ) ⊗ ( 9, 8, 1 ) ⊕ ( 2, 1, 5 ) ⊗ ( 7, 3, 7 ) ⊕ ( −6, 2, 1 ) ⊗ ( 6, 5, 1 ) ⊕ ( 7, 4, 8 ) ⊗ ( 6, 1, 0 ) ⊕ ( 2, 9, 7 ) ⊗ ( 5, 2, 6 ) = ( 63, 59, 57 ) ⊕ ( 14, 10, 84 ) ⊕ ( −36, 20, 31 ) ⊕ ( 42, 27, 48 ) ⊕ ( 10, 87, 89 ) = ( 93, 203, 309 ) ,
( 2, 5, 0 ) ⊗ ( 9, 8, 1 ) ⊕ ( 1, 6, 9 ) ⊗ ( 7, 3, 7 ) ⊕ ( −1, 5, 2 ) ⊗ ( 6, 5, 1 ) ⊕ ( −9, 8, 4 ) ⊗ ( 6, 1, 0 ) ⊕ ( 3, 1, 3 ) ⊗ ( 5, 2, 6 ) = ( 18, 48, 2 ) ⊕ ( 7, 77, 133 ) ⊕ ( −6, 36, 13 ) ⊕ ( −54, 48, 29 ) ⊕ ( 15, 9, 51 ) = ( −20, 218, 228 ) ,
( 1, 4, 7 ) ⊗ ( 9, 8, 1 ) ⊕ ( 1, 9, 1 ) ⊗ ( 7, 3, 7 ) ⊕ ( −2, 1, 6 ) ⊗ ( 6, 5, 1 ) ⊕ ( −2, 1, 7 ) ⊗ ( 6, 1, 0 ) ⊕ ( 5, 8, 6 ) ⊗ ( 5, 2, 6 ) = ( 9, 39, 71 ) ⊕ ( 7, 119, 21 ) ⊕ ( −12, 9, 40 ) ⊕ ( −12, 6, 42 ) ⊕ ( 25, 58, 96 ) = ( 17, 231, 270 ) .
461
G. MALKAWI, N. AHMAD,
462
AND
H. IBRAHIM
Verification of solution — Example 4.4
( −3, 2, 13 ) ⊗ ( 1, 2, 3 ) ⊕ ( −2, 2, 11 ) ⊗ ( 0, 2, 5 ) ⊕ ( −1, 0, 9 ) ⊗ ( 0, 1, 4 ) ⊕ ( 0, 2, 7 ) ⊗ ( 0, 2, 8 ) ⊕ ( 1, 4, 5 ) ⊗ ( 1, 4, 5 ) ⊕ ( −2, 4, 3 ) ⊗ ( 5, 1, 1 ) ⊕ ( 3, 4, 1 ) ⊗ ( 7, 4, 1 ) ⊕ ( 4, 4, 1 ) ⊗ ( 4, 4, 1 ) ⊕ ( 5, 4, 3 ) ⊗ ( 5, 4, 3 ) ⊕ ( 6, 6, 5 ) ⊗ ( 6, 6, 5 ) = ( −3, 17, 43 ) ⊕ ( 0, 20, 45 ) ⊕ ( 0, 8, 32 ) ⊕ ( 0, 16, 56 ) ⊕ ( 1, 19, 35 )
⊕ ( −10, 26, 16 ) ⊕ ( 21, 29, 11 ) ⊕ ( 16, 16, 9 ) ⊕ ( 25, 24, 39 ) ⊕ ( 36, 36, 85 ) = ( 86, 211, 371 ) ,
( −7, 3, 10 ) ⊗ ( 1, 2, 3 ) ⊕ ( 1, 1, 12 ) ⊗ ( 0, 2, 5 ) ⊕ ( 0, 1, 10 ) ⊗ ( 0, 1, 4 ) ⊕ ( −1, 1, 8 ) ⊗ ( 0, 2, 8 ) ⊕ ( 0, 1, 6 ) ⊗ ( 1, 4, 5 ) ⊕ ( −8, 3, 4 ) ⊗ ( 5, 1, 1 ) ⊕ ( −2, 5, 2 ) ⊗ ( 7, 4, 1 ) ⊕ ( 3, 7, 0 ) ⊗ ( 4, 4, 1 ) ⊕ ( −4, 7, 2 ) ⊗ ( 5, 4, 3 ) ⊕ ( −5, 7, 4 ) ⊗ ( 6, 6, 5 ) = ( −7, 33, 19 ) ⊕ ( 0, 26, 65 ) ⊕ ( 0, 10, 40 ) ⊕ ( 0, 16, 56 ) ⊕ ( 0, 18, 36 )
⊕ ( −40, 26, 24 ) ⊕ ( −14, 42, 14 ) ⊕ ( 12, 32, 3 ) ⊕ ( −20, 68, 18 ) ⊕ ( −30, 102, 30 ) = ( −99, 373, 305 ) ,
( −3, 2, 7 ) ⊗ ( 1, 2, 3 ) ⊕ ( 0, 2, 9 ) ⊗ ( 0, 2, 5 ) ⊕ ( 3, 0, 11 ) ⊗ ( 0, 1, 4 ) ⊕ ( 2, 0, 9 ) ⊗ ( 0, 2, 8 ) ⊕ ( −1, 0, 7 ) ⊗ ( 1, 4, 5 ) ⊕ ( 0, 2, 5 ) ⊗ ( 5, 1, 1 ) ⊕ ( 1, 2, 3 ) ⊗ ( 7, 4, 1 ) ⊕ ( 2, 4, 1 ) ⊗ ( 4, 4, 1 ) ⊕ ( 3, 6, 1 ) ⊗ ( 5, 4, 3 ) ⊕ ( 4, 8, 3 ) ⊗ ( 6, 6, 5 ) = ( −3, 17, 19 ) ⊕ ( 0, 18, 45 ) ⊕ ( 0, 14, 56 ) ⊕ ( 0, 22, 88 ) ⊕ ( −1, 17, 37 )
⊕ ( 0, 12, 30 ) ⊕ ( 7, 15, 25 ) ⊕ ( 8, 18, 7 ) ⊕ ( 15, 39, 17 ) ⊕ ( 24, 68, 53 ) = ( 50, 240, 377 ) ,
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
( −6, 1, 4 ) ⊗ ( 1, 2, 3 ) ⊕ ( 3, 3, 6 ) ⊗ ( 0, 2, 5 ) ⊕ ( 2, 1, 8 ) ⊗ ( 0, 1, 4 ) ⊕ ( −5, 1, 10 ) ⊗ ( 0, 2, 8 ) ⊕ ( 4, 1, 8 ) ⊗ ( 1, 4, 5 ) ⊕ ( 3, 1, 6 ) ⊗ ( 5, 1, 1 ) ⊕ ( −2, 3, 4 ) ⊗ ( 7, 4, 1 ) ⊕ ( 1, 5, 2 ) ⊗ ( 4, 4, 1 ) ⊕ ( 2, 5, 0 ) ⊗ ( 5, 4, 3 ) ⊕ ( 3, 5, 2 ) ⊗ ( 6, 6, 5 ) = ( −6, 22, 13 ) ⊕ ( 0, 18, 45 ) ⊕ ( 0, 10, 40 ) ⊕ ( 0, 48, 40 ) ⊕ ( 4, 40, 68 )
⊕ ( 15, 7, 39 ) ⊕ ( −14, 26, 30 ) ⊕ ( 4, 24, 11 ) ⊕ ( 10, 34, 6 ) ⊕ ( 18, 40, 37 ) = ( 31, 269, 329 ) ,
( −8, 1, 2 ) ⊗ ( 1, 2, 3 ) ⊕ ( −6, 2, 3 ) ⊗ ( 0, 2, 5 ) ⊕ ( 3, 0, 5 ) ⊗ ( 0, 1, 4 ) ⊕ ( −4, 0, 7 ) ⊗ ( 0, 2, 8 ) ⊕ ( 7, 2, 9 ) ⊗ ( 1, 4, 5 ) ⊕ ( 6, 2, 7 ) ⊗ ( 5, 1, 1 ) ⊕ ( 5, 2, 5 ) ⊗ ( 7, 4, 1 ) ⊕ ( 4, 4, 3 ) ⊗ ( 4, 4, 1 ) ⊕ ( −3, 6, 1 ) ⊗ ( 5, 4, 3 ) ⊕ ( 2, 8, 1 ) ⊗ ( 6, 6, 5 ) = ( −8, 28, 17 ) ⊕ ( 0, 40, 16 ) ⊕ ( 0, 8, 32 ) ⊕ ( 0, 32, 24 ) ⊕ ( 7, 55, 89 )
⊕ ( 30, 14, 48 ) ⊕ ( 35, 26, 45 ) ⊕ ( 16, 16, 19 ) ⊕ ( −15, 57, 13 ) ⊕ ( 12, 78, 21 ) = ( 77, 354, 324 ) ,
( −9, 2, 1 ) ⊗ ( 1, 2, 3 ) ⊕ ( 0, 2, 5 ) ⊗ ( 0, 2, 5 ) ⊕ ( 6, 1, 2 ) ⊗ ( 0, 1, 4 ) ⊕ ( 3, 3, 4 ) ⊗ ( 0, 2, 8 ) ⊕ ( 6, 1, 6 ) ⊗ ( 1, 4, 5 ) ⊕ ( 9, 3, 8 ) ⊗ ( 5, 1, 1 ) ⊕ ( 8, 3, 6 ) ⊗ ( 7, 4, 1 ) ⊕ ( 7, 3, 4 ) ⊗ ( 4, 4, 1 ) ⊕ ( 6, 5, 2 ) ⊗ ( 5, 4, 3 ) ⊕ ( 5, 7, 0 ) ⊗ ( 6, 6, 5 ) = ( −9, 35, 20 ) ⊕ ( 0, 18, 45 ) ⊕ ( 0, 8, 32 ) ⊕ ( 0, 14, 56 ) ⊕ ( 6, 42, 66 )
⊕ ( 45, 21, 57 ) ⊕ ( 56, 41, 56 ) ⊕ ( 28, 28, 27 ) ⊕ ( 30, 29, 34 ) ⊕ ( 30, 52, 25 ) = ( 186, 288, 418 ) ,
463
G. MALKAWI, N. AHMAD,
464
AND
H. IBRAHIM
( −10, 3, 0 ) ⊗ ( 1, 2, 3 ) ⊕ ( 9, 3, 0 ) ⊗ ( 0, 2, 5 ) ⊕ ( 8, 1, 0 ) ⊗ ( 0, 1, 4 ) ⊕ ( 6, 2, 1 ) ⊗ ( 0, 2, 8 ) ⊕ ( 5, 4, 3 ) ⊗ ( 1, 4, 5 ) ⊕ ( −8, 2, 5 ) ⊗ ( 5, 1, 1 ) ⊕ ( −11, 4, 7 ) ⊗ ( 7, 4, 1 ) ⊕ ( 10, 4, 5 ) ⊗ ( 4, 4, 1 ) ⊕ ( 9, 4, 3 ) ⊗ ( 5, 4, 3 ) ⊕ ( 8, 6, 1 ) ⊗ ( 6, 6, 5 ) = ( −10, 42, 23 ) ⊕ ( 0, 18, 45 ) ⊕ ( 0, 8, 32 ) ⊕ ( 0, 14, 56 ) ⊕ ( 5, 29, 43 )
⊕ ( −40, 20, 28 ) ⊕ ( −77, 43, 65 ) ⊕ ( 40, 40, 35 ) ⊕ ( 45, 40, 51 ) ⊕ ( 48, 48, 51 ) = ( 11, 302, 429 ) ,
( −11, 4, 1 ) ⊗ ( 1, 2, 3 ) ⊕ ( 10, 2, 1 ) ⊗ ( 0, 2, 5 ) ⊕ ( −9, 0, 1 ) ⊗ ( 0, 1, 4 ) ⊕ ( 6, 4, 1 ) ⊗ ( 0, 2, 8 ) ⊕ ( −6, 5, 2 ) ⊗ ( 1, 4, 5 ) ⊕ ( 7, 5, 2 ) ⊗ ( 5, 1, 1 ) ⊕ ( 10, 3, 4 ) ⊗ ( 7, 4, 1 ) ⊕ ( 13, 5, 6 ) ⊗ ( 4, 4, 1 ) ⊕ ( 4, 5, 4 ) ⊗ ( 5, 4, 3 ) ⊕ ( 11, 5, 2 ) ⊗ ( 6, 6, 5 ) =
( −11, 49, 26 ) ⊕ ( 0, 22, 55 ) ⊕ ( 0, 36, 9 ) ⊕ ( 0, 14, 56 ) ⊕ ( −6, 60, 39 ) ⊕ ( 35, 27, 19 ) ⊕ ( 70, 49, 42 ) ⊕ ( 52, 52, 43 ) ⊕ ( 20, 28, 44 ) ⊕ ( 66, 66, 77 )
= ( 226, 403, 410 ) ,
( −12, 3, 2 ) ⊗ ( 1, 2, 3 ) ⊕ ( −6, 1, 2 ) ⊗ ( 0, 2, 5 ) ⊕ ( 8, 1, 2 ) ⊗ ( 0, 1, 4 ) ⊕ ( −7, 5, 0 ) ⊗ ( 0, 2, 8 ) ⊕ ( 8, 5, 2 ) ⊗ ( 1, 4, 5 ) ⊕ ( 10, 4, 3 ) ⊗ ( 5, 1, 1 ) ⊕ ( 11, 4, 3 ) ⊗ ( 7, 4, 1 ) ⊕ ( −12, 4, 3 ) ⊗ ( 4, 4, 1 ) ⊕ ( 5, 4, 5 ) ⊗ ( 5, 4, 3 ) ⊕ ( −8, 6, 3 ) ⊗ ( 6, 6, 5 ) = ( −12, 48, 27 ) ⊕ ( 0, 35, 14 ) ⊕ ( 0, 10, 40 ) ⊕ ( 0, 96, 24 ) ⊕ ( 8, 38, 52 )
⊕ ( 50, 26, 28 ) ⊕ ( 77, 56, 35 ) ⊕ ( −48, 32, 48 ) ⊕ ( 25, 24, 55 ) ⊕ ( −48, 106, 48 ) = ( 52, 471, 371 ) ,
AN ALGORITHM FOR A POSITIVE SOLUTION OF ARBITRARY FULLY FUZZY LINEAR SYSTEM
465
( 4, 2, 3 ) ⊗ ( 1, 2, 3 ) ⊕ ( 3, 0, 3 ) ⊗ ( 0, 2, 5 ) ⊕ ( −1, 2, 3 ) ⊗ ( 0, 1, 4 ) ⊕ ( −7, 6, 1 ) ⊗ ( 0, 2, 8 ) ⊕ ( −3, 6, 1 ) ⊗ ( 1, 4, 5 ) ⊕ ( 2, 4, 3 ) ⊗ ( 5, 1, 1 ) ⊕ ( 14, 3, 4 ) ⊗ ( 7, 4, 1 ) ⊕ ( −8, 3, 4 ) ⊗ ( 4, 4, 1 ) ⊕ ( −9, 5, 4 ) ⊗ ( 5, 4, 3 ) ⊕ ( −6, 7, 4 ) ⊗ ( 6, 6, 5 ) = ( 4, 11, 24 ) ⊕ ( 0, 12, 30 ) ⊕ ( 0, 12, 8 ) ⊕ ( 0, 104, 26 ) ⊕ ( −3, 51, 30 )
⊕ ( 10, 22, 20 ) ⊕ ( 98, 65, 46 ) ⊕ ( −32, 23, 32 ) ⊕ ( −45, 67, 40 ) ⊕ ( −36, 107, 36 ) = ( −4, 474, 292 ) .
