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Malaysian Journal of Mathematical Sciences 12(3): 383–400 (2018)
MALAYSIANJOURNALOFMATHEMATICALSCIENCES
Journal homepage: http://einspem.upm.edu.my/journal
Positive Solution of Pair Fully Fuzzy Matrix
Equations
Daud, W.S.W. ∗1,2, Ahmad, N.2, and Malkawi, G.3
1Institute of Engineering Mathematics, Universiti Malaysia Perlis,
Perlis, Malaysia
2School of Quantitative Sciences, Universiti Utara Malaysia,
Kedah, Malaysia
3Higher Colleges of Technology, Abu Dhabi AlAin Men's College,
17155, United Arab Emirates
E-mail: wsuhana@unimap.edu.my
∗ Corresponding author
Received: 29 March 2018
Accepted: 15 August 2018
ABSTRACT
A pair matrix equation is a matrix system that contains two matrix
equations which is solved simultaneously to obtain its solution. In this
study, an algorithm for obtaining the positive fuzzy solution of positive
pair fully fuzzy matrix equation is proposed. The constructed algorithm
utilizes fuzzy Kronecker product and fuzzy Vec-operator to transform
pair fully fuzzy matrix equation into fully fuzzy linear system. Then,
an associated linear system is used to reach the final solution. Necessary
theorems, corollary and numerical example are presented to illustrate the
proposed algorithm.
Keywords: Fully fuzzy pair matrix equation, Fully fuzzy linear system,
Kronecker product, Vec-operator, Associated linear system
Daud, W.S.W., Ahmad, N. and Malkawi, G.
1. Introduction
In manyapplications, there exist situations where the crisp numbers are less
adequate to represent the uncertainty, vagueness and ambiguity of
information. In this case, fuzzy numbers plays a prominent role to model
the fuzzy environment.
The past few decades have seen a growing trend towards the matrix
equations in the fuzzy environment. There are fuzzy matrix equation (FME)
˜ ˜
of AX = B (Guo and Gong, 2010), fully fuzzy matrix equation (FFME)
m m
˜ ˜ ˜
of AX = B (Otadi and Mosleh, 2012), fuzzy Sylvester matrix equation
m m
˜ ˜ ˜
(FSE) of AX + XB = C (Araghi and Hosseinzadeh, 2012, Guo, 2011, Guo
and Bao, 2013, Guo and Shang, 2012, 2013, Salkuyeh, 2010) and also fully
˜ ˜ ˜ ˜ ˜
fuzzy Sylvester matrix equation of AX+XB = C (Malkawi et al., 2015, Shang
˜ ˜ ˜ ˜ ˜
et al., 2015) and AX −XB = C (Daud et al., 2018b, Dookhitram et al., 2015).
This considerable amount of literature have shown that, fuzzy set theory plays
a significant role to model the matrix equations. It is undeniable that, the
previous proposed methods demonstrated various significant contribution in
solving the matrix equation in fuzzy environment. However, there are still
many gaps that can be filled in this area.
Thisstudyaimstoconstructanewalgorithmforsolvingapositivepairfully
fuzzy matrix equation (PFFME). Basically, a pair matrix equation is a matrix
system that contains two matrix equations which are solved simultaneously
to obtain its solution. These equations are important in real application for
example in control theory (Asari and Amirfakhrian, 2016). Previously, a study
wascarriedoutbySadeghietal.(2011), whichproposedasignificantknowledge
˜ ˜ ˜ ˜ ˜
in solving fuzzy pair matrix equation of A X + XB = C and A XB = C ,
1 1 1 2 2 2
˜ ˜
whereA ,B ,A ,B areknowncrispmatrices,C ,C areknownfuzzymatrices
1 1 2 2 1 2
˜
and X is unknown fuzzy matrices.
Contrary to this study, two fully fuzzy matrix equations are solved
simultaneously, which are fully fuzzy continuous-time Sylvester matrix equation
of
˜ ˜ ˜ ˜ ˜
AX+XB=C (1)
and also fully fuzzy discrete-time Sylvester matrix equation of
˜ ˜ ˜ ˜ ˜
AXB−X=C. (2)
Thus, a pair fully fuzzy matrix equation is given by
(˜ ˜ ˜ ˜ ˜
A1X+XB1=C1 (3)
˜ ˜ ˜ ˜ ˜
A XB −X=C
2 2 2
384 Malaysian Journal of Mathematical Sciences
Positive Solution of Pair Fully Fuzzy Matrix Equations
˜ ˜ ˜ ˜
where A1, A2 and B1, B2 represents p × p and q × q positive fuzzy
˜ ˜ ˜
matrices respectively, C1 and C2 are p×q arbitrary fuzzy matrices, and X is a
p×q positive fuzzy solution. This study utilizes fuzzy Kronecker product and
fuzzy Vec-operator in converting the equation into a fully fuzzy linear system.
In addition, the associated linear system based on (Malkawi et al., 2014) is
adapted in obtaining the final solution. Overall, this study provides valuable
contribution in finding the solution of PFFME, and, at the same time advances
the understanding in theory of fuzzy sets and matrices.
The remaining part of the paper proceeds as follows. In Section 2, the
fundamental concept of fuzzy set theory and Kronecker operation are provided.
In Section 3, the algorithm for solving the PFFME is shown. Later on, a
numerical example is illustrated in Section 4 followed by the conclusion in
Section 5.
2. Preliminaries
In this section, some definitions and theorems used in this study are recalled.
Definition 2.1. (Zadeh, 1965) A fuzzy number is a function such as
u:R→[0,1] satisfying the following properties:
1. u is normal, that is, there exist an x0 ∈ R such that u(x0) = 1;
2. u is fuzzy convex, that is u(λx + (1 − λ)y) ≥ min{u(x),u(y)} for any
x,y ∈ R, λ ∈ [0,1];
3. u is upper semicontinuous;
4. supp u = {x ∈ R|u(x) > 0} is the support of u, and its closure
cl(supp u) is compact.
˜
Definition 2.2. A fuzzy number M = (m,α,β) is said to be a triangular fuzzy
number (TFN), if its membership function is given by:
m−x
1− , m−α≤x≤m,α>0,
α
µ˜(x)= 1−x−m, m≤x≤m+β,β>0, (4)
M β
0, otherwise.
˜
In this case, m is the mean value of M, whereas α and β are right and left
spreads, respectively.
Malaysian Journal of Mathematical Sciences 385
Daud, W.S.W., Ahmad, N. and Malkawi, G.
Definition 2.3. (Dubois and Prade, 1978) The arithmetic operations of two
˜ ˜
fuzzy numbers M = (m,α,β) and N = (n,γ,δ), are as follows:
1. Addition:
˜ ˜
M⊕N=(m,α,β)⊕(n,γ,δ)=(m+n,α+γ,β+δ) (5)
2. Opposite:
˜
−M=−(m,α,β)=(−m,β,α) (6)
3. Subtraction:
˜ ˜
M⊖N=(m,α,β)⊖(n,γ,δ)=(m−n,α+δ,β+γ) (7)
4. Multiplication:
˜ ˜ ∼
M⊗N=(m,α,β)⊗(n,γ,δ)=(mn,mγ+nα,mδ+nβ) (8)
Definition 2.4. (Dehghan et al., 2006) An n × n fully fuzzy linear system
(FFLS) is defined as follows.
˜
a˜ x˜ +a˜ x˜ +...+a˜ x˜ = b
11 1 12 2 1n n 1
˜
a˜ x˜ +a˜ x˜ +...+a˜ x˜ = b
21 1 22 2 2n n 2 (9)
.
.
.
˜
a˜ x˜ +a˜ x˜ +...+a˜ x˜ =b
m1 1 m2 2 mn n m
which can also be written in a matrix form of
˜
a˜ a˜ . . . a˜ x˜ b1
11 12 1n 1
a˜ a˜ . . . a˜ x˜ ˜
21 22 2n 2 b2, (10)
. . . . .=.
. . .. . . .
. . . . .
a˜ a˜ . . . a˜ x˜ ˜
m1 m2 mn n bm
and it is usually denoted in a form of
˜ ˜ ˜
AX=B. (11)
˜
Definition 2.5. (Dehghan et al., 2006) A positive fuzzy number X = (x,y,z)
˜ ˜ ˜ ˜
where x,y,z ≥ 0 be the solution of FFLS, AX = B, which A = (A,M,N) ≥ 0
˜
and B = (b,h,g) ≥ 0 iff
Ax=b
(12)
Ay+Mx=h
Az+Nx=g.
386 Malaysian Journal of Mathematical Sciences
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