Fuzzy system of linear equations - Dr. Karam Ouharou Department of nuclear physics, department of applied mathematics

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FUZZY SYSTEM OF LINEAR
EQUATIONS
Karam OUHAROU
Department of Applied Mathematics, Department of Nuclear physics
LHC, Large Hadron collider for beauty experiment at CERN
CMS experiment at CERN
Weizmann Institute of Science
Abstract
Real life applications arising in various fields of Engineering and Sciences like
Electrical, Civil, Economics, Dietary etc. can be modeled using system of linear
equations. In such models it may happen that the values of the parameters are
not known or they cannot be stated precisely only their estimation due to
experimental data or experts knowledge is available. In such situation it is
convenient to represent such parameters by fuzzy numbers (refer [22]). Klir,
[15] gave the results for the existence of solution of linear algebraic equation
involving fuzzy numbers. The method to obtain solution of system of equation
involving fuzzy numbers. The method to obtain solution of system of linear
equations with all the involved parameters being fuzzy is proposed here. fully
fuzzy systems of linear equations (FFSL). In this paper, the conditions for the
existence and uniqueness of the fuzzy solution are proved.
Introduction
The primitive applications in Science and Engineering give rise to system of linear
equations. Many times the practical realization of such problems involve the imprecise
and non-probabilistic uncertainties in the parameters, where in their estimates are
known due to the experimental data or experts knowledge. To obtain the realistic model
of such systems it is convenient to treat the coefficients and the resources, right hand
side (RHS) in the equations as fuzzy numbers (e.g. Triangular, Trapezoidal, Gaussian
etc.).
System of linear equations with fuzzy parameters have become more pervasive in various
fields instead of their crisp counter parts, as in [20] wherein circuit analysis is done
using fuzzy complex system, [17] demonstrates the use of fuzzy system of linear

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equations in Economics but such systems involve negative fuzzy numbers. Also in
obtaining the solution of fuzzy differential equation, intermediately fuzzy system of
linear equation is to be solved, [21], [8].
Freidman et al. [14] were first to propose the model for solving the fuzzy systems with
the crisp coefficient matrix and fuzzy right-hand side column. Fully fuzzy system of
linear equations is another important class of systems in which all the parameters
involved are fuzzy. Solution of such systems using QR decomposition [16], LU
decomposition [2], Gram-Schmidt approach to solve such systems [18], and other
approaches [1], [4], [5], [6], [3], [10], [9], [11], [12] etc. were proposed. Recently
2010 Mathematics Subject Classification. Primary: 03E72.
Key words and phrases. Fuzzy numbers, fully fuzzy linear equations, 𝛼-cut, level cut,
weak solutions, parallel computing.
The author is supported by AICTE, MSU-UGC un-assigned grant. an article by
Allahviranloo [7] have made a note that technique suggested by Friedman [14] gives only
the weak solutions for the fuzzy systems, which may be fuzzy in some cases.
In this paper, fully fuzzy linear system of the form 𝐴
˜𝐱
˜ = 𝐛
˜ , wherein the coefficients and
RHS both, are represented by fuzzy numbers are considered. A method to solve these
systems is proposed and the result for the existence and uniqueness of the fuzzy
solutions for such systems is proved. The proposed technique converts the system into
the form that can be partitioned into two subsystems. The solution of these subsystems
are used to obtain the fuzzy solution. The benefit for the partitioned form of the crisp
system resulting due to 𝛼-cut is that it may be useful to obtain the fuzzy solutions of
large systems using parallel computing algorithms [19].
The paper is organized in the following manner, initially the preliminaries are listed,
then the reduction of fuzzy system into the crisp using 𝛼-cut is computed. The next
section holds the main result. The result is substantiated with two illustrations one
which satisfies the condition and hence has fuzzy solution, and the other which does not.
The result presented here can be generalized to 𝑛 dimensional system without loss of
generality.
1. Preliminaries.
1.1. Definition.
2.1.1. Fuzzy number. Let, 𝑅𝐹 be the class of fuzzy subsets on the real axis (i.e. 𝑢: 𝑅 →
[0,1]) satisfying the following properties:
I. 𝑢 ∈ 𝑅𝐹, 𝑢 is normal, i.e. ∃𝑥0 ∈ 𝑅 with 𝑢(𝑥0) = 1;

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II. 𝑢 ∈ 𝑅𝐹, 𝑢 is convex fuzzy set, That is, 𝑢(𝑡𝑥 + (1 − 𝑡)𝑦) ≥ min{𝑢(𝑥), 𝑢(𝑦)}, ∀𝑡 ∈
[0,1], 𝑥, 𝑦 ∈ 𝑅
III. 𝑢 ∈ 𝑅𝐹, 𝑢 is upper semicontinuous on 𝑅;
IV. {𝑥 ∈ 𝑅; 𝑢(𝑥) > 0
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅} is compact, where 𝐴
‾ denotes the closure of 𝐴.
Then 𝑅𝐹 is called the space of fuzzy numbers 𝑢
˜, (refer [13]). Obviously 𝑅 ⊂ 𝑅𝐹. Here, 𝑅 ⊂
𝑅𝐹 is obvious as 𝑅 can be regarded as {𝜒𝑥: 𝑥 is any usual real number }.
2.1.2. Non negative fuzzy number. A fuzzy number 𝑢
˜ is said to be non-negative fuzzy
number if 𝑢
˜(𝑥) = 0, ∀𝑥 < 0 and 𝑢
˜ is said to be positive if 𝑢
˜(𝑥) = 0, ∀𝑥 ≤ 0.
2.1.3. Non positive fuzzy number. A fuzzy number 𝑢
˜ is said to be non-positive fuzzy
number if 𝑢
˜(𝑥) = 0, ∀𝑥 > 0 and 𝑢
˜ is said to be negative if 𝑢
˜(𝑥) = 0, ∀𝑥 ≥ 0
2.1.4. Fuzzy matrix. A matrix 𝐴
˜ = (𝑎
˜𝑖𝑗) is called a fuzzy matrix, if each element of 𝐴
˜ is a
fuzzy number. 𝐴
˜ will be positive (negative) and denoted by 𝐴
˜ > 0(𝐴
˜ < 0) if each element
of 𝐴
˜ be positive (negative). 𝐴
˜ will be non positive (non negative) and denoted by 𝐴
˜ ≤
0(𝐴
˜ ≥ 0) if each element of 𝐴
˜ be non positive (non negative).
We may represent 𝑛 × 𝑚 fuzzy matrix 𝐴
˜ = (𝑎
˜𝑖𝑗)
𝑛×𝑚
, where 𝑎
˜𝑖𝑗 is a trapezoidal fuzzy
number denoted as, 𝑎
˜𝑖𝑗 = (𝑏𝑖𝑗, 𝑐𝑖𝑗, 𝑑𝑖𝑗, 𝑒𝑖𝑗) and defined as
𝑎
˜𝑖𝑗 =
{
(𝑥 − 𝑏𝑖𝑗)
(𝑐𝑖𝑗 − 𝑏𝑖𝑗)
𝑏𝑖𝑗 < 𝑥 ≤ 𝑐𝑖𝑗
1 𝑐𝑖𝑗 < 𝑥 ≤ 𝑑𝑖𝑗
(𝑒𝑖𝑗 − 𝑥)
(𝑒𝑖𝑗 − 𝑑𝑖𝑗)
𝑑𝑖𝑗 < 𝑥 ≤ 𝑒𝑖𝑗
0 otherwise }
1.2. Operations on fuzzy numbers.
2.2.1. 𝛼-cut / level-cut of fuzzy number. For 0 < 𝛼 ≤ 1, denote the 𝛼-cut as, [𝑢]𝛼
= {𝑥 ∈
𝑅: 𝑢(𝑥) ≥ 𝛼} and [𝑢]0
= {𝑥 ∈ 𝑅; 𝑢(𝑥) > 0
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅}.
Then it is well-known that for each 𝛼 ∈ [0,1], the 𝛼-cut, [𝑢]𝛼
is a bounded closed interval
[ 𝛼
𝑢, 𝛼
𝑢
‾] in 𝑅.
The addition of the fuzzy numbers and the scalar multiplication is defined using 𝛼-cut,
∀𝛼 ∈ [0; 1] as given below, refer [13].

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2.2.2. Addition of two fuzzy numbers. For 𝑢, 𝑣 ∈ 𝑅𝐹, the sum 𝑢 + 𝑣[𝑢 + 𝑣]𝛼
= [𝑢]𝛼
+
[𝑣]𝛼
= [𝑢, 𝑢
‾] + [𝑣, 𝑣
‾] = [𝑢 + 𝑣, 𝑢
‾ + 𝑣
‾].
2.2.3. Scalar multiplied with fuzzy number. For 𝜆 ∈ 𝑅, the product 𝜆 ⋅ 𝑢 is given by [𝜆 ⋅
𝑢]𝛼
= 𝜆[𝑢]𝛼
= 𝜆[𝑢, 𝑢
‾] = [𝜆𝑢, 𝜆𝑢
‾], 𝜆 > 0.
V. System reduction using level-cuts. Consider the fully fuzzy
system in 2dimension,
𝐴
˜𝐱
˜ = 𝐛
˜
i.e.
(
𝑎
˜11 𝑎
˜12
𝑎
˜21 𝑎
˜22
) ⨂ (
𝑥
˜1
𝑥
˜2
) = (
𝑏
˜1
𝑏
˜2
)
Using the 𝛼-cut of the fuzzy elements, we get ∀𝛼 ∈ [0,1],
𝛼𝐴
˜𝛼
𝐱
˜ = 𝛼
𝐛
˜
That is,
(
[ 𝛼
𝑎11, 𝛼
𝑎
‾11] [ 𝛼
𝑎12, 𝛼
𝑎
‾12]
[ 𝛼
𝑎21, 𝛼𝑎
‾21] [ 𝛼
𝑎22, 𝛼𝑎
‾22]
) (
[ 𝛼
𝑥1, 𝛼
𝑥
‾1]
[ 𝛼
𝑥2, 𝛼𝑥
‾2]
) = ([
𝛼
𝑏1, 𝛼𝑏
‾1
[ 𝑏𝑏2, 𝛼𝑏
‾2]
)
which becomes
[ 𝛼
𝑎11, 𝛼
𝑎
‾11] ⨂ [ 𝛼
𝑥1, 𝛼
𝑥
‾1] ⨁ [ 𝛼
𝑎12, 𝛼
𝑎
‾12] ⨂ [ 𝛼
𝑥2, 𝛼
𝑥
‾2] = [ 𝛼
𝑏1, 𝛼
𝑏
‾1]
[ 𝛼
𝑎21, 𝛼
𝑎
‾21] ⨂ [ 𝛼
𝑥1, 𝛼
𝑥
‾1] ⨁ [ 𝛼
𝑎22, 𝛼
𝑎
‾22] ⨂ [ 𝛼
𝑥2, 𝛼
𝑥
‾2] = [ 𝛼
𝑏2, 𝛼
𝑏
‾2]
Using the operations on the cuts the above system can be put into crisp system of linear
equations as,
𝛼𝑎11
𝛼
𝑥1 + 𝛼
𝑎12
𝛼
𝑥2 = 𝛼
𝑏1
𝛼
𝑎21
𝛼
𝑥1 + 𝛼
𝑎22
𝛼
𝑥2 = 𝛼
𝑏2
𝛼𝑎
‾11
𝛼
𝑥
‾1 + 𝛼
𝑎
‾12
𝛼
𝑥
‾2 = 𝛼
𝑏
‾1
𝛼𝑎
‾21
𝛼
𝑥
‾1 + 𝛼
𝑎
‾22
𝛼
𝑥
‾2 = 𝛼
𝑏
‾2
For each 𝛼 ∈ [0,1] the system to be solved is:
we get a crisp system of the form
𝐴 × 𝐱 = 𝐛

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where, the coefficient matrix, 𝐴 is of dimension 2𝑛 × 2𝑛, 𝐱 and 𝐛 are column vectors of
dimension 2𝑛. We know that invertibility of 𝐴 is necessary for uniqueness of the solution
of (2). Thus the necessary condition for obtaining the fuzzy solution for the fully fuzzy
system is that the coefficient matrix of converted crisp system (2) is invertible ∀𝛼 ∈ [0,1].
Also observe that (2) can be partitioned into two systems as:
That is, 𝐴 × 𝐱 = 𝐛 and 𝐴
‾ × 𝐱
̅ = 𝐛
̅ . The solution of these crisp systems determines 𝛼
𝑥𝑖 and
𝛼
𝑥
‾𝑖, which are used to reconstruct the components of fuzzy 𝐱
˜ as shown by the lemma
below. The constructed components of the unknown vector 𝐱
˜ are fuzzy numbers
provided they satisfy the conditions given in the main result.
Lemma 3.1. The 𝑖𝑡ℎ
component of the fuzzy solution vector, 𝑥
˜ of the fully fuzzy system (1)
can be reconstructed form the components 𝛼
𝑥𝑖 and 𝛼
𝑥
‾𝑖 of the crisp system (2) given as
𝑥
̃𝑖 = 𝛼𝛼∈[0,1] 𝛼𝑥
̃𝑖
where, 𝛼𝑥
̃𝑖 = 𝛼 ⋅ 𝛼𝑥
̃𝑖 and 𝛼
𝑥
̃𝑖 = [ 𝛼
𝑥𝑖, 𝛼𝑥𝑖
̅].
Proof. For each particular 𝑦 ∈ 𝑅, let 𝑎 = 𝑥
̃𝑖(𝑦).
Then
For each 𝛼 ∈ (𝑎, 1], we have 𝑥
̃𝑖(𝑦) = 𝑎 < 𝛼 and, therefore 𝛼𝑥
̃𝑖(𝑦) = 0. On the other hand
for each 𝛼 ∈ [0, 𝑎], we have 𝑥
̃𝑖(𝑦) = 𝑎 ≥ 𝛼, therefore 𝛼𝑥
̃𝑖(𝑦) = 𝛼. Hence,
(
⋃
𝛼
𝑥
̃𝑖
𝛼 ∈ [0,1]
) (𝑦) = sup
𝛼∈[0,1]
𝛼 = 𝑎 = 𝑥
̃𝑖(𝑦)
Theorem 3.2. (Existence and Uniqueness)
The components of the solution vector 𝐱 of system (2) can determine the unique fuzzy
solution for system (1) if the parameters for the fuzzy system (1) satisfy the conditions:
(I) ∀𝛼 ∈ [0,1]
(ii) ∀𝛼, 𝛽 ∈ [0,1], 𝛼 ≤ 𝛽

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Proof. (Existence:)
Suppose condition (I) is satisfied that is ∀𝛼 ∈ [0,1]
That is,
( 𝛼
𝐴)
−1
⋅ 𝛼
𝐛 ≤ ( 𝛼
𝐴
‾)−1
⋅ 𝛼
𝐛
̅
which means for 𝑖 = 1,2, we have
𝛼
𝑥𝑖 ≤𝛼
𝑥
‾𝑖
This implies that the interval equation 𝛼
𝐴
˜𝛼
𝐗
˜ = 𝛼
𝐛
˜ has a solution, which is
𝛼
𝐱
˜ = [( 𝛼
𝐴)
−1
⋅ 𝛼
𝐛, ( 𝛼
𝐴
‾)−1
⋅ 𝛼
𝐛
̅ ]
Satisfying of condition (ii) ∀𝛼, 𝛽 ∈ [0,1], 𝛼 ≤ 𝛽
implies ∀𝛼, 𝛽 ∈ [0,1], 𝛼
𝑥𝑖 ≤𝛽
𝑥𝑖 ≤𝛽
𝑥
‾𝑖 ≤𝛼
𝑥
‾𝑖 for 𝑖 = 1,2.
This ensures that the solutions of the interval equations for 𝛼 and 𝛽 are nested; that is if
𝛼 ≤ 𝛽, then 𝛽𝑥
˜𝑖 ⊆ 𝛼𝑥
˜𝑖 i.e. 𝑖𝑡ℎ
component 𝑥
˜𝑖 of the fuzzy number 𝑥
˜, is convex.
Hence, ∀𝛼 ∈ [0,1], 𝑥
˜𝑖 = (𝛼𝑥𝑥𝑖, 𝛼𝑥
‾𝑖), for 𝑖 = 1,2. Where,
 𝛼
𝑥𝑖 is a bounded left continuous non-decreasing function over [0,1].
 𝛼𝑥
‾𝑖 is a bounded left continuous non-increasing function over [0,1].
Thus, the solution of system (2) satisfying the above conditions would indeed generate
the components of fuzzy solution vector 𝐱
˜ for the system (1).
2. (Uniqueness:)
Let 𝑥
̃ and 𝑦
̃ be two solutions of (1). Then ∀𝛼 ∈ [0,1], 𝛼
𝐴
˜𝛼
𝐱
˜ = 𝛼
𝐛
˜ and 𝛼
𝐴
˜𝛼
𝐲
˜ = 𝛼
𝐛
˜
Therefore,
𝛼
𝐴
˜( 𝛼
𝐱
˜ − 𝛼
𝐲
˜) = 0

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This implies, ( 𝛼
𝐱
˜ − 𝛼
𝐲
˜) = 0. Hence 𝑥 = 𝑦. Thus solution if it exists is unique.
Thus, the solution of the fully fuzzy linear systems having the coefficients and/or the
RHS represented by fuzzy numbers exists if they satisfy the conditions given in the above
theorem. The results given in the above theorem can be applied to 𝑛 dimensional
problems, without loss of generality. In the following section we give illustrations in
support of our result, for ease we have considered trapezoidal fuzzy numbers. However,
the results are applicable to the linear systems involving other fuzzy numbers too.
4 - Illustrations. Example-1. (Fuzzy Solution Exists)
Consider fully fuzzy linear system
(3,4,6,7)𝑥1 + (4,5,7,9)𝑥2 = (−10,20,60,97)
(5,6,8,8)𝑥1 + (3,3,5,6)𝑥2 = (−7,18,58,85)
Here, the coefficients and the RHS of the system are represented by the trapezoidal fuzzy
numbers 𝑇 = (𝑎, 𝑏, 𝑐, 𝑑) i.e.
𝑇 = {
(𝑥 − 𝑎)/(𝑏 − 𝑎) 𝑎 < 𝑥 ≤ 𝑏
1 𝑏 < 𝑥 ≤ 𝑐
(𝑑 − 𝑥)/(𝑑 − 𝑐) 𝑐 < 𝑥 ≤ 𝑑
0 otherwise
}
The corresponding level cut system is:
[𝛼 + 3,7 − 𝛼] ⨂ [𝑥1, 𝑥
‾1] ⨁ [𝛼 + 4,9 − 2𝛼] ⨂ [𝑥2, 𝑥
‾2] = [30𝛼 − 10,97 − 37𝛼]
[𝛼 + 5,8] ⨂ [𝑥1, 𝑥
‾1] ⨁ [3,6 − 𝛼] ⨂ [𝑥2, 𝑥
‾2] = [25𝛼 − 7,85 − 27𝛼]
The corresponding crisp system obtained is:
(
𝛼 + 3 𝛼 + 4 0 0
𝛼 + 5 3 0 0
0 0 7 − 𝛼 9 − 2𝛼
0 0 8 6 − 𝛼
) (
𝑥1
𝑥2
𝑥
‾1
𝑥
‾2
) = (
30𝛼 − 10
25𝛼 − 7
97 − 37𝛼
85 − 27𝛼
)
Solution of above crisp system satisfies the condition given the main theorem. Also, it
gives the 𝛼-cuts for the components of 𝑥
˜ as:
𝛼𝑥 = [
25𝛼2
+ 3𝛼 + 2
𝛼2 + 6𝛼 + 11
,
−(17𝛼2
+ 6𝛼 + 183)
17𝛼2 − 13𝛼 − 30
]
and

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𝛼𝑥2
= [
−(22𝛼2
+ 63𝛼 + 29)
𝛼2 + 6𝛼 + 11
,
27𝛼2
+ 22𝛼 − 181
17𝛼2 − 13𝛼 − 30
]
Figure 1. Components of 𝑋
˜
The fuzzy 𝑥
˜1, 𝑥
˜2 are as shown in the figure (1).
Example-2. (Fuzzy Solution Does Not Exist)
Consider another fully fuzzy linear system
(2,3,5,6)𝑥1 + (3,4,6,8)𝑥2 = (40,60,10,17)
(4,5,7,7)𝑥1 + (2,2,4,5)𝑥2 = (38,58,5,7)
The corresponding level cut system is:
[𝛼 + 2,6 − 𝛼] ⨂ [𝑥1, 𝑥
‾1] ⨁ [𝛼 + 3,8 − 2𝛼] ⨂ [𝑥2, 𝑥
‾2] = [10𝛼 + 30,77 − 17𝛼]
[𝛼 + 4,7] ⨂ [𝑥1, 𝑥
‾1] ⨁ [2,5 − 𝛼] ⨂ [𝑥2, 𝑥
‾2] = [5𝛼 + 33,65 − 7𝛼]

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The corresponding crisp system obtained is:
(
𝛼 + 2 𝛼 + 3 0 0
𝛼 + 4 2 0 0
0 0 6 − 𝛼 8 − 2𝛼
0 0 7 5 − 𝛼
) (
𝑥1
𝑥2
𝑥
‾1
𝑥
‾2
) = (
10𝛼 + 30
5𝛼 + 33
77 − 17𝛼
65 − 7𝛼
)
The converted crisp system has solution. But, we can see that the conditions given in the
main theorem are not satisfied as a result the fuzzy solution does not exist, it can be seen
in the figure (2).
Conclusion.
In this article, we extend the result for solution of fuzzy algebraic equations to the
system of fully fuzzy linear equations and give sufficient conditions for the existence
and uniqueness of the solution. The illustrations are given, in support of our result.
The partitioned form in the proposed scheme suggests that
Figure 2. Fuzzy components does not exist 𝑋
˜

10. DISCRETE AND CONTINUOUS Website: www.Karam-ouharou.org
DYNAMICAL SYSTEMS
Supplement 2015 pp. 619 – 627
for the large fuzzy systems even parallel processing can be applied to obtain fuzzy
solutions.
Acknowledgments. I would like to thank Prof. Atraoui Mustapha, Prof. Sima Lev for
their expert comments. I would also like to thank the referees for the vital inputs in
improving the article.