TELK
OMNIKA
T
elecommunication,
Computing,
Electr
onics
and
Contr
ol
V
ol.
19,
No.
2,
April
2021,
pp.
583
598
ISSN:
1693-6930,
accredited
First
Grade
by
K
emenristekdikti,
No:
21/E/KPT/2018
DOI:
10.12928/TELK
OMNIKA.v19i2.18023
r
583
A
modification
of
fuzzy
arithmetic
operators
f
or
solving
near
-zer
o
fully
fuzzy
matrix
equation
W
.
S.
W
.
Daud
1
,
N.
Ahmad
2
,
G.
Malkawi
3
1
Institute
of
Engineering
Mathematics,
Uni
v
ersiti
Malaysia
Perlis,
Perlis,
Malaysia
1,2
School
of
Quantitati
v
e
Sciences,
Uni
v
ersiti
Utara
Malaysia,
K
edah,
Malaysia
3
Higher
Colle
ges
of
T
echnology
,
Ab
u
Dhabi
AlAin
Men's
Colle
ge,
17155,
United
Arab
Emirates
Article
Inf
o
Article
history:
Recei
v
ed
Jun
26,
2020
Re
vised
Sep
8,
2020
Accepted
Sep
16,
2020
K
eyw
ords:
Control
system
engineering
Fully
fuzzy
matrix
equation
Fully
fuzzy
linear
system
Fuzzy
arithmetic
operators
Near
-zero
fuzzy
numbers
ABSTRA
CT
Matrix
equations
ha
v
e
its
o
wn
important
in
the
field
of
control
system
engineering
particularly
in
the
stability
analysis
of
linear
control
systems
and
the
reduction
of
nonlinear
control
system
models.
There
a
re
certain
conditions
where
the
classical
matrix
equation
are
not
well
equipped
to
handle
the
uncertainty
problems
such
as
during
the
process
of
stability
analysis
and
reduction
in
control
system
engineering.
In
this
study
,
an
algorithm
is
de
v
eloped
for
solving
fully
fuzzy
matrix
equation
particularly
for
~
A
~
X
~
B
~
X
=
~
C
,
where
the
coef
ficients
of
the
equation
are
in
near
-zero
fuzzy
numbers.
By
modifying
the
e
xisting
fuzzy
multiplication
arithmetic
operators,
the
proposed
algorithm
e
xceeds
the
positi
v
e
restriction
to
allo
w
the
near
-zero
fuzzy
numbers
as
the
coef
ficients.
Besides
that,
a
ne
w
fuzzy
subtraction
arithmetic
operator
has
a
lso
been
proposed
as
the
e
xisting
operator
f
ailed
to
satisfy
the
both
sides
of
the
near
-zero
fully
fuzzy
matri
x
equation.
Subsequently
,
Kroneck
er
product
and
V
ec
-operator
are
adapted
with
the
modified
fuzzy
arithmetic
operator
in
order
to
transform
the
fully
fuzzy
matrix
equation
to
a
fully
fuzzy
linear
system.
On
top
of
that,
a
ne
w
associated
linear
system
is
de
v
eloped
to
obtain
the
final
solution.
A
numerical
e
xample
and
the
v
erification
of
the
solution
are
pres
ented
to
demonstrate
the
proposed
algorithm.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
W
.
S.
W
.
Daud
Institute
of
Engineering
Mathematics
Uni
v
ersiti
Malaysia
Perlis
Perlis,
Malaysia
Email:
wsuhana@unimap.edu.my
1.
INTR
ODUCTION
There
are
man
y
types
of
matrix
equations
that
ha
v
e
been
modelled
in
v
arious
applications
[1]
particularly
in
control
system
engineering
[2,
3].
Basically
,
control
system
engineering
is
used
to
design
the
feedback
loops
system
[4].
The
e
xample
applications
that
related
to
the
feedback
loops
systems
are
medical
imaging
acquisition
system
[5],
image
restoration
[6],
model
reduction
[7],
signal
processing
[8]
and
stochastic
control
[9].
According
to
[10],
matrix
equation
plays
the
role
as
an
equation
solv
er
for
t
he
control
system
model.
In
dealing
wit
h
an
y
real
applications,
it
is
possible
that
an
y
uncertainty
conditions
could
occur
,
for
e
xample,
if
there
e
xist
an
y
conflicting
requirements
and
instability
of
the
en
vironmental
conditions
during
the
system
process.
If
there
is
an
y
e
xistence
of
noise
or
unnecessary
elements
during
the
process,
it
w
ould
also
J
ournal
homepage:
http://journal.uad.ac.id/inde
x.php/TELK
OMNIKA
Evaluation Warning : The document was created with Spire.PDF for Python.
584
r
ISSN:
1693-6930
distract
the
system
[11].
In
this
case,
the
e
xisting
matrix
equations
sometimes
are
not
well
equipped
to
handle
those
conditions.
Therefore,
one
of
the
approaches
that
can
be
tak
en
is
to
adapt
the
fuzzy
numbers
as
the
coef
ficients
of
the
matrix
equation
[12].
In
the
past
fe
w
years,
man
y
researchers
proposed
their
algorithms
in
solving
matrix
equations
with
parameters
in
fuzzy
numbers.
This
equation
is
kno
wn
as
the
fully
fuzzy
matrix
equation
(FFME).
Otadi
and
Mosleh
[13]
are
the
pioneers
in
this
field,
who
has
applied
linear
programming
technique
to
obtain
a
positi
v
e
solution
for
arbitrary
FFME,
~
A
~
X
m
=
~
B
m
.
Apart
from
that,
there
is
a
study
which
has
e
xtended
the
algorithm
used
in
solving
the
fully
fuzzy
linear
system
(FFLS)
to
solv
e
the
FFME
~
A
~
X
~
B
=
~
C
[14].
Subsequently
,
in
2015,
Shang
et
al.
[15]
proposed
their
algorithm
in
solving
fully
fuzzy
Sylv
ester
matrix
equation
(FFSE),
~
A
~
X
+
~
X
~
B
=
~
C
by
applying
the
arithmetic
multiplication
operator
,
which
has
been
pre
viously
proposed
in
Dehghan
et
al.
[16].
On
the
other
hand,
Malka
wi
et
al.
[17]
ha
v
e
proposed
an
algorithm
which
of
fers
f
aster
computational
compared
to
Shang
et
al.
[15].
Whi
le
in
2020,
Elsayed
et
al.
[18]
carried
out
a
study
in
solving
the
FFME
of
~
A
~
X
+
~
X
~
B
=
~
C
,
which
considering
the
entries
of
the
equation
are
in
trapezoidal
fuzzy
numbers.
In
this
paper
,
we
are
propose
an
algorithm
to
solv
e
the
FFME
of
~
A
~
X
~
B
~
X
=
~
C
(1)
considering
the
fuzzy
coef
ficient
~
A
=
(
~
a
ij
)
m
n
or
~
B
=
(
~
b
ij
)
n
n
is
a
near
-zero
fuzzy
number
,
while
~
C
=
(
~
c
ij
)
m
n
is
an
arbitrary
fuzzy
coef
ficient
and
~
X
=
(
~
x
ij
)
m
n
is
the
solution
of
the
FFME.
This
equation
has
been
pre
viously
solv
ed
by
Daud
et
al.
[19]
in
2018.
Unfortunately
the
algorithm
proposed
is
only
limited
to
non-singular
and
positi
v
e
fuzzy
matrices.
This
limitation
has
moti
v
ated
us
to
construct
an
algorithm
to
solv
e
the
(1)
without
an
y
restrictions.
Moreo
v
er
,
in
real-life
applications,
the
coef
ficients
of
the
FFME
can
either
be
positi
v
e,
ne
g
ati
v
e
or
near
-zero
fuzzy
numbers.
In
de
v
eloping
the
algorithm,
the
e
xisting
fuzzy
multiplication
arithmetic
operators
are
modified
as
the
e
xisting
operators
introduced
by
[17]
and
[20]
are
not
applicable
to
perform
the
multiplication
in
v
olving
near
-zero
fuzzy
numbers.
Besides
that,
a
ne
w
fuzzy
subtraction
operation
is
also
de
v
eloped
in
solving
the
FFME,
since
the
e
xisting
operator
is
inadequated
to
subtract
a
near
-zero
fuzzy
number
to
a
positi
v
e
fuzzy
number
.
Subsequently
,
the
modified
fuzzy
arithmetic
operator
is
adapted
with
the
Kroneck
er
product
and
V
ec
-operator
in
con
v
erting
the
FFME
to
a
simpler
form
of
equation,
which
is
a
fully
fuzzy
linear
system
(FFLS).
Later
on,
the
solution
is
obtained
by
means
of
associated
linear
system
(ALS)
which
has
been
established
based
on
the
modified
fuzzy
multiplication
arithmetic
operator
.
The
remaining
part
of
the
paper
proceeds
as
follo
ws.
In
Section
2,
some
preliminaries
on
the
fuzzy
numbers
and
Kroneck
er
product
are
sho
wn.
Then
in
Section
3,
the
theoretical
foundation
which
supports
the
de
v
eloped
algorithm
are
established.
In
Section
4,
the
de
v
eloped
algorithm
for
solving
the
FFME
of
(1)
is
sho
wn.
Mo
ving
on,
a
numerical
e
xample
and
v
erification
of
the
solution
are
illustrated
in
Section
5.
Finally
,
the
conclusion
is
dra
wn
in
Section
6.
2.
PRELIMIN
ARIES
2.1.
Fundamental
concepts
of
matrix
and
set
theory
The
fundamental
concept
of
matrix
theory
is
important
in
order
to
solv
e
the
matrix
equations.
Some
fundamentals
of
matrix
theory
are
defined
in
the
follo
wing:
Definition
1.
[21]
Let
N
be
a
3
3
bloc
k
matrix,
suc
h
that
N
=
0
@
A
B
C
D
E
F
G
H
I
1
A
;
(2)
then,
j
N
j
=
det
A
B
D
E
C
F
I
1
G
H
det
[
I
]
=
det
A
C
I
1
G
B
C
I
1
H
D
F
I
1
G
E
F
I
1
H
det
[
I
]
(3)
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
2,
April
2021
:
583
–
598
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
r
585
Remark
1.
F
or
the
bloc
k
matrix
3
3
suc
h
that
P
=
0
@
A
B
C
D
E
F
G
H
I
1
A
:
(4)
Clearly
that
based
on
Definition
1,
the
determinant
P
is
given
as
follows:
j
P
j
=
det
E
F
H
I
D
G
A
1
B
C
det
[
A
]
=
det
E
D
A
1
B
F
D
A
1
C
H
GA
1
B
I
GA
1
C
det
[
A
]
(5)
Definition
2.
[22]
Let
A
and
B
be
sets.
The
union
of
A
and
B
is
the
set
of
A
[
B
=
f
x
:
x
2
A
or
x
2
B
g
.
2.2.
Theory
of
fuzzy
numbers
The
follo
wing
definition
describing
the
theory
of
fuzzy
numbers
has
been
introduced
since
1965
by
Zadeh
[23].
Definition
3.
Let
X
be
a
nonempty
set.
A
fuzzy
set
~
A
in
X
is
c
har
acterized
by
its
member
ship
function
~
A
:
X
!
[0
;
1]
(6)
and
~
A
(
x
)
r
epr
esents
the
de
gr
ee
of
member
ship
of
element
x
in
fuzzy
set
~
A
for
eac
h
x
2
X
.
In
this
study
,
the
representation
of
fuzzy
numbers
is
based
on
the
triangular
fuzzy
numbers.
2.2.1.
T
riangular
fuzzy
number
Definition
4.
A
fuzzy
number
~
M
=
(
m;
;
)
is
said
to
be
a
triangular
fuzzy
number
(TFN),
if
its
member
ship
function
is
given
by:
~
M
(
x
)
=
8
>
<
>
:
1
m
x
;
m
x
m;
>
0
;
1
x
m
;
m
x
m
+
;
>
0
;
0
;
otherwise
:
(7)
In
this
case,
m
is
the
mean
v
alue
of
~
M
,
whereas
and
are
the
right
and
left
spreads,
respecti
v
ely
.
Definition
5.
A
fuzzy
number
~
M
=
(
m;
;
)
is
called
as
an
arbitr
ary
fuzzy
number
wher
e
it
may
be
positive
,
ne
gative
or
near
zer
o
whic
h
can
be
classified
as
follows:
•
~
M
is
a
positive(ne
gative)
fuzzy
number
if
f
m
0
(
+
m
0)
.
•
~
M
is
a
zer
o
fuzzy
number
if
(
m
=
0
;
;
=
0)
.
•
~
M
is
a
near
zer
o
fuzzy
number
if
f
m
0
+
m
.
The
follo
wing
definitions
describe
some
important
arithmetic
operations
of
TFN
[20].
Definition
6.
The
arithmetic
oper
ations
of
two
TFN,
~
M
=
(
m;
;
)
and
~
N
=
(
n;
;
)
,
ar
e
as
follows:
i.
Addition:
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
m
+
n;
+
;
+
)
:
(8)
ii.
Opposite:
~
M
=
(
m;
;
)
=
(
m;
;
)
:
(9)
iii.
Subtr
action:
(
m;
;
)
(
n;
;
)
=
(
m;
;
)
(
n;
;
)
=
(
m;
;
)
(
n;
;
)
=
(
m
n;
+
;
+
)
:
(10)
A
modification
of
fuzzy
arithmetic
oper
ator
s
for
solving
near
-zer
o...
(W
.
S.
W
.
Daud)
Evaluation Warning : The document was created with Spire.PDF for Python.
586
r
ISSN:
1693-6930
iv
.
Multiplication:
•
If
~
M
>
0
and
~
N
>
0
,
then
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
mn;
m
+
n
;
m
+
n
)
(11)
•
If
~
M
<
0
and
~
N
>
0
,
then
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
mn;
n
m
;
n
m
)
(12)
•
If
~
M
<
0
and
~
N
<
0
,
then
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
mn;
n
m
;
n
m
)
(13)
Based
on
the
multiplication
arithmetic
operator
in
(11)
to
(13),
there
is
no
operator
applicable
for
a
near
-zero
fuzzy
number
.
This
is
because
a
near
-zero
fuzzy
number
cannot
be
defined
in
the
form
of
(
m;
;
)
,
unlik
e
a
positi
v
e
or
ne
g
ati
v
e
fuzzy
number
could.
Therefore,
a
ne
w
form
of
multiplication
arithmetic
operator
has
been
introduced
by
[24]
which
adapted
the
system
of
min-max
function.
Definition
7.
[24]
The
pr
oduct
of
two
fuzzy
number
s
~
M
=
(
m;
;
)
and
~
N
=
(
n;
;
)
,
can
be
defined
as
~
M
~
N
=
(
mn;
f
1
;
f
2
)
(14)
wher
e
f
1
=
mn
Min
((
m
)(
n
)
;
(
m
)(
n
+
))
,
f
2
=
Max
((
m
+
)(
n
)
;
(
m
+
)(
n
+
))
mn:
The
operator
as
gi
v
en
in
(14)
is
basically
has
been
init
iated
based
on
[25]
and
[26].
In
implementing
the
multiplication,
fe
w
times
multiplication
and
comparison
are
needed,
to
obtain
the
minimum
and
maximum
v
alues.
Besi
des
that,
the
opera
tor
is
only
compatible
for
positi
v
e
fuzzy
number
~
N
as
stated
in
the
follo
wing
Theorem
1.
Theor
em
1.
[24]
Consider
an
arbitr
ary
fuzzy
number
~
M
=
(
m;
;
)
and
a
positive
fuzzy
number
~
N
=
(
n;
;
)
,
i.
If
~
M
is
positive
,
then
the
following
inequalities
ar
e
satisfied:
0
(
m
)(
n
)
(
m
)(
n
+
)
;
(15)
0
(
m
+
)(
n
)
(
m
+
)(
n
+
)
(16)
ii.
If
~
M
is
ne
gative
,
then
the
following
inequalities
ar
e
satisfied:
0
(
m
)(
n
)
(
m
)(
n
+
)
;
(17)
0
(
m
+
)(
n
)
(
m
+
)(
n
+
)
(18)
iii.
If
~
M
is
near
zer
o,
then
the
inequalities
in
(16)
and
(17)
ar
e
satisfied.
2.3.
Fundamental
concepts
of
fuzzy
Kr
oneck
er
pr
oducts
and
fuzzy
V
ec
-operator
Kroneck
er
products
and
V
ec
-operator
are
the
important
tools
in
solving
matrix
equations.
The
definitions
and
theorems
of
the
fuzzy
Kroneck
er
products
and
fuzzy
V
ec
-operator
,
are
pro
vided
as
follo
ws:
Definition
8.
[17]
Let
~
A
=
(
~
a
ij
)
m
n
and
~
B
=
(
~
b
ij
)
p
q
be
fuzzy
matrices.
Fuzzy
Kr
onec
k
er
pr
oduct
is
r
epr
esented
as
~
A
k
~
B
,
wher
e
~
A
k
~
B
=
0
B
B
B
@
~
a
11
~
B
~
a
12
~
B
:
:
:
~
a
1
n
~
B
~
a
21
~
B
~
a
22
~
B
:
:
:
~
a
2
n
~
B
.
.
.
.
.
.
.
.
.
.
.
.
~
a
m
1
~
B
~
a
m
2
~
B
:
:
:
~
a
mn
~
B
1
C
C
C
A
=
[
~
a
ij
~
B
]
(
mp
)
(
nq
)
(19)
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
2,
April
2021
:
583
–
598
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
r
587
Definition
9.
[17]
V
ec
-oper
ator
of
a
fuzzy
matrix
is
a
linear
tr
ansformation
that
con
verts
the
fuzzy
matrix
of
~
C
=
(
~
c
1
;
~
c
2
;
:::;
~
c
n
)
into
a
column
vector
as
V
ec
(
~
C
)
=
0
B
B
B
@
~
c
1
~
c
2
.
.
.
~
c
n
1
C
C
C
A
:
(20)
Theor
em
2.
[17]
If
~
A
=
(
~
a
ij
)
m
m
is
a
fuzzy
matrix,
and
~
U
=
(
~
u
ij
)
p
p
is
a
unitary
fuzzy
matrix
defined
as
~
U
=
0
B
B
B
@
(1
;
0
;
0)
(0
;
0
;
0)
:
:
:
(0
;
0
;
0)
(0
;
0
;
0)
(1
;
0
;
0)
:
:
:
(0
;
0
;
0)
.
.
.
.
.
.
.
.
.
.
.
.
(0
;
0
;
0)
(0
;
0
;
0)
:
:
:
(1
;
0
;
0)
1
C
C
C
A
;
(21)
then
i.
~
A
~
U
=
~
U
~
A
=
~
A
ii.
~
U
T
=
~
U
.
Definition
10.
[17]
Let
A
=
(
a
ij
)
m
m
,
B
=
(
b
ij
)
n
n
and
X
=
(
x
ij
)
m
n
,
then
i.
V
ec
[
~
A
~
X
]
=
[
~
U
n
k
~
A
]
V
ec
(
~
X
)
ii.
V
ec
[
~
X
~
B
]
=
[
~
B
T
k
~
U
m
]
V
ec
(
~
X
)
iii.
V
ec
[
~
A
~
X
~
B
]
=
[
~
B
T
k
~
A
]
V
ec
(
~
X
)
iv
.
V
ec
(
~
X
)
=
[
~
U
]
V
ec
(
~
X
)
3.
THEORETICAL
DEVELOPMENT
This
section
demonstrates
the
establishment
of
the
theoretical
foundations
which
in
v
olv
ed
some
theorems,
definitions
and
corallaries.
There
are
four
sections
presented,
consist
of
the
introduction
of
a
ne
w
near
-zero
positi
v
e
subtraction
operator
,
a
modification
of
arithmetic
multiplication
operator
,
some
related
properties
of
FFME
~
A
~
X
~
B
~
X
=
~
C
and
also
the
construction
of
an
associated
linear
systems.
3.1.
Near
-zer
o
positi
v
e
subtraction
operator
Theor
em
3.
Let
~
M
=
(
m;
;
)
be
a
near
-zer
o
fuzzy
number
and
~
N
=
(
n;
;
)
is
a
positive
fuzzy
number
.
The
subtr
action
of
~
M
and
~
N
is
given
by
~
M
np
~
N
=
(
m
n;
+
;
)
:
(22)
wher
e
>
.
Pr
oof.
Let
<
,
then
<
0
,
which
means
that
the
spread
v
alue
of
is
ne
g
ati
v
e.
This
is
violated
since
it
is
al
w
ays
positi
v
e,
as
mentioned
in
Definition
4
.
Thus,
>
.
This
ne
w
operator
is
kno
wn
as
a
Near
-zero
positi
v
e
subtraction
operator
,
denoted
as
np
.
3.2.
Modification
of
multiplication
arithmetic
operators
In
this
study
,
fuzzy
arithmetic
multiplication
operator
as
stated
in
Definition
(7)
is
modified.
The
modified
multiplication
operator
pro
vides
simpler
and
direct
computation
as
compared
to
the
pre
vious
operators.
A
modification
of
fuzzy
arithmetic
oper
ator
s
for
solving
near
-zer
o...
(W
.
S.
W
.
Daud)
Evaluation Warning : The document was created with Spire.PDF for Python.
588
r
ISSN:
1693-6930
Theor
em
4.
Let
~
M
=
(
m;
;
)
be
a
positive
,
ne
gative
or
near
-zer
o
fuzzy
number
,
and
~
N
=
(
n;
;
)
be
a
positive
fuzzy
number
.
Then,
the
min
and
max
in
(14)
ar
e
given
by:
Min
[(
m
)(
n
)
;
(
m
)(
n
+
)]
=
(
(
m
)(
n
)
if
~
M
0
(
m
)(
n
+
)
if
otherwise
(23)
Max
[(
m
+
)(
n
)
;
(
m
+
)(
n
+
)]
=
(
(
m
+
)(
n
)
if
~
M
<
0
(
m
+
)(
n
+
)
if
otherwise
(24)
Pr
oof.
Based
on
Theorem
1
and
realize
that
(
n
)
<
(
n
+
)
,
then
ob
viously:
•
If
~
M
is
positi
v
e
which
is
(
m
)
0
,
both
multiplications
of
(
m
)(
n
)
and
(
m
+
)(
n
)
are
minimum
compared
to
the
multiplications
of
(
m
)(
n
+
)
and
(
m
+
)(
n
+
)
respecti
v
ely
.
Ho
we
v
er
,
if
(
m
)
<
(
m
+
)
,
then
(
m
)(
n
)
is
minimum.
On
the
other
hand,
since
b
ot
h
multiplications
of
(
m
)(
n
+
)
and
(
m
+
)(
n
+
)
are
maximum
compared
to
the
multiplication
of
(
m
)(
n
)
and
(
m
+
)(
n
)
respecti
v
ely
,
b
ut
since
(
m
+
)
>
(
m
)
,
thus
the
maximum
v
alue
is
(
m
+
)(
n
+
)
.
•
If
~
M
is
ne
g
ati
v
e
which
is
(
m
)
<
0
,
both
multiplications
of
(
m
)(
n
+
)
and
(
m
+
)(
n
+
)
are
minimum
compared
to
the
multiplication
of
(
m
)(
n
)
and
(
m
+
)(
n
)
respecti
v
ely
.
From
that,
since
(
m
)
<
(
m
+
)
,
then
(
m
)(
n
+
)
is
minimum.
On
the
other
hand,
since
both
(
m
)(
n
)
and
(
m
+
)(
n
)
are
maximum
compared
to
the
multiplication
of
(
m
)(
n
+
)
and
(
m
+
)(
n
+
)
respecti
v
ely
,
b
ut
(
m
+
)
>
(
m
)
thus
the
maximum
v
alue
is
(
m
+
)(
n
)
.
•
If
~
M
is
near
-zero
which
is
(
m
)
0
(
+
m
)
,
based
on
the
inequilities
in
(16)
and
(17),
then
ob
viously
(
m
)(
n
+
)
is
minimum,
whereas
(
m
+
)(
n
+
)
is
maximum.
From
Theorem
4
and
(14),
the
modified
multiplication
arithmetic
operators
are
defined
in
the
follo
wing
theorem.
Theor
em
5.
Let
~
M
=
(
m;
;
)
be
a
positive
,
ne
gative
or
near
-zer
o
fuzzy
number
,
and
~
N
=
(
n;
;
)
be
a
positive
fuzzy
number
,
then
the
multiplication
of
~
M
~
N
is
defined
as
follows:
1.
If
~
M
is
positive
,
then
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
mn;
n
+
(
m
)
;
n
+
(
m
+
)
)
(25)
2.
If
~
M
is
ne
gative
,
then
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
mn;
n
(
m
)
;
n
(
m
+
)
)
(26)
3.
If
~
M
is
near
-zer
o,
then
~
M
~
N
=
(
m;
;
)
(
n;
;
)
=
(
mn;
n
(
m
)
;
n
+
(
m
+
)
)
(27)
Pr
oof.
By
considering
the
Corollary
4,
and
applying
it
to
(14),
thus:
1.
F
or
~
M
is
positi
v
e,
~
M
~
N
=
(
mn;
mn
(
m
)(
n
)
;
(
m
+
)(
n
+
)
mn
)
=
(
mn;
mn
mn
+
m
+
n
;
mn
+
m
+
n
+
mn
)
=
(
mn;
m
+
n
;
m
+
n
+
)
=
(
mn;
n
+
(
m
)
;
n
+
(
m
+
)
)
(28)
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
2,
April
2021
:
583
–
598
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
r
589
2.
F
or
~
M
is
ne
g
ati
v
e,
~
M
~
N
=
(
mn;
mn
(
m
)(
n
+
)
;
(
m
+
)(
n
)
mn
)
=
(
mn;
mn
mn
m
+
n
+
;
mn
m
+
n
mn
)
=
(
mn;
m
+
n
+
;
m
+
n
)
=
(
mn;
n
(
m
)
;
n
(
m
+
)
)
(29)
3.
F
or
~
M
is
near
-zero,
~
M
~
N
=
(
mn;
mn
(
m
)(
n
+
)
;
(
m
+
)(
n
+
)
mn
)
=
(
mn;
mn
mn
m
+
n
+
;
mn
+
m
+
n
+
mn
)
=
(
mn;
m
+
n
+
;
m
+
n
+
)
=
(
mn;
n
(
m
)
;
n
+
(
m
+
)
)
(30)
Since
(25)
to
(27)
are
sho
wn,
hence
the
theorem
is
pro
v
ed.
Cor
ollary
1.
Let
~
M
=
(
m;
;
)
be
a
positive
,
ne
gative
or
near
-zer
o
fuzzy
number
,
and
~
N
=
(
n;
;
)
be
a
positive
fuzzy
number:
1.
If
~
M
is
positive
,
then
the
multiplication
of
~
M
~
N
is
positive
,
suc
h
that
mn
(
n
+
(
m
)
)
>
0
.
2.
If
~
M
is
ne
gative
,
then
the
multiplication
of
~
M
~
N
is
ne
gative
,
suc
h
that
(
n
(
m
+
)
)
+
mn
<
0
.
3.
If
~
M
is
near
-zer
o,
then
the
multiplication
of
~
M
~
N
is
near
-zer
o,
suc
h
that
mn
(
n
(
m
)
)
<
0
<
(
n
+
(
m
+
)
)
+
mn
.
Pr
oof.
The
multiplication
of
~
M
~
N
in
Theorem
5
must
satisfy
the
Definition
5,
where
1.
If
~
M
is
positi
v
e,
then
mn
(
n
+
(
m
)
)
>
0
which
is
mn
(
n
+
(
m
)
)
=
mn
n
m
+
=
(
m
)
n
(
m
)
=
(
m
)(
n
)
(31)
Since
both
(
m
)
and
(
n
)
are
>
0
,
then
mn
(
n
+
(
m
)
)
>
0
.
2.
If
~
M
is
ne
g
ati
v
e,
then
(
n
(
m
+
)
)
+
mn
<
0
which
is
(
n
(
m
+
)
)
+
mn
=
n
m
+
mn
=
(
m
+
)
n
(
m
+
)
=
(
m
+
)(
n
)
(32)
Since
(
m
+
)
<
0
and
(
n
)
>
0
,
then
mn
+
(
n
(
m
+
)
)
<
0
.
3.
If
~
M
is
near
-zero,
then
mn
(
n
(
m
)
)
<
0
<
(
n
+
(
m
+
)
)
+
mn
which
is
mn
(
n
(
m
)
)
=
mn
n
+
m
=
(
m
)
n
+
(
m
)
=
(
m
)(
n
+
)
(33)
Since
(
m
)
<
0
and
(
n
+
)
>
0
,
then
mn
(
n
(
m
)
)
<
0
.
On
the
other
hand,
(
n
+
(
m
+
)
)
+
mn
=
n
+
m
+
+
mn
=
(
m
+
)
n
+
(
m
+
)
=
(
m
+
)(
n
+
)
(34)
Since
(
m
+
)
>
0
and
(
n
+
)
>
0
,
then
(
n
+
(
m
+
)
)
+
mn
>
0
.
After
all
the
conditions
are
satisfied,
then
the
corollary
is
pro
v
ed.
A
modification
of
fuzzy
arithmetic
oper
ator
s
for
solving
near
-zer
o...
(W
.
S.
W
.
Daud)
Evaluation Warning : The document was created with Spire.PDF for Python.
590
r
ISSN:
1693-6930
3.3.
Related
pr
operties
of
FFME
~
A
~
X
~
B
~
X
=
~
C
The
definition
of
FFME
~
A
~
X
~
B
~
X
=
~
C
is
gi
v
en
as
follo
ws:
Definition
11.
The
matrix
equation
0
B
B
B
@
~
a
11
~
a
12
:
:
:
~
a
1
m
~
a
21
~
a
22
:
:
:
~
a
2
m
.
.
.
.
.
.
.
.
.
.
.
.
~
a
m
1
~
a
m
2
:
:
:
~
a
mm
1
C
C
C
A
0
B
B
B
@
~
x
11
~
x
12
:
:
:
~
x
1
n
~
x
21
~
x
22
:
:
:
~
x
2
n
.
.
.
.
.
.
.
.
.
.
.
.
~
x
m
1
~
x
m
2
:
:
:
~
x
mn
1
C
C
C
A
0
B
B
B
@
~
b
11
~
b
12
:
:
:
~
b
1
n
~
b
21
~
b
22
:
:
:
~
b
2
n
.
.
.
.
.
.
.
.
.
.
.
.
~
b
n
1
~
b
n
2
:
:
:
~
b
nm
1
C
C
C
A
0
B
B
B
@
~
x
11
~
x
12
:
:
:
~
x
1
n
~
x
21
~
x
22
:
:
:
~
x
2
n
.
.
.
.
.
.
.
.
.
.
.
.
~
x
m
1
~
x
m
2
:
:
:
~
x
mn
1
C
C
C
A
=
0
B
B
B
@
~
c
11
~
c
12
:
:
:
~
c
1
n
~
c
21
~
c
22
:
:
:
~
c
2
n
.
.
.
.
.
.
.
.
.
.
.
.
~
c
m
1
~
c
m
2
:
:
:
~
c
mn
1
C
C
C
A
(35)
can
also
be
represented
as
~
A
~
X
~
B
~
X
=
~
C
(36)
where
~
A
=
(
a
ij
)
,
1
i;
j
n
,
~
B
=
(
b
ij
)
,
1
i;
j
m
,
the
right
hand
side
matrix
~
C
=
(
c
ij
)
,
1
i
n;
1
j
m
is
the
fuzzy
matrices,
and
~
X
=
(
x
ij
)
,
1
i
n;
1
j
m
is
an
unkno
wn
fuzzy
matrix.
There
is
a
special
criterion
related
to
the
order
of
matrix
coef
ficients
for
FFME
~
A
~
X
~
B
~
X
=
~
C
.
Remark
2.
Let
~
A
~
X
~
B
~
X
=
~
C
be
an
FFME,
wher
e
the
fuzzy
coef
ficient
of
~
A
and
~
B
must
be
any
squar
e
matrices.
Example
1.
If
~
A
and
~
B
ar
e
non-squar
e
matrices
with
any
appr
opriate
or
der
s
of
~
A
r
p
and
~
B
q
s
,
and
the
solution
is
~
X
p
q
,
then
~
A
r
p
~
X
p
q
~
B
q
s
~
X
p
q
=
~
A
~
X
~
B
r
s
~
X
p
q
:
Howe
ver
,
the
subtr
action
of
~
A
~
X
~
B
r
s
and
~
X
p
q
is
not
possible
due
to
the
dif
fer
ent
or
der
.
Thus,
in
all
cases,
~
A
and
~
B
in
FFME
~
A
~
X
~
B
~
X
=
~
C
must
be
squar
e
matrices.
3.4.
Construction
of
an
associated
linear
system
Definition
12.
Consider
a
fully
fuzzy
linear
system
(FFLS)
in
the
form
of
~
S
~
X
=
~
C
(37)
wher
e
~
S
=
(
m;
;
)
,
~
X
=
(
n;
;
)
and
~
C
=
(
C
;
G;
H
)
,
whic
h
is
equivalent
to
n
X
j
=1
;:::;n
(
m
ij
;
ij
;
ij
)
(
n
j
;
j
;
j
)
=
(
C
i
;
G
i
;
H
i
)
:
(38)
According
to
the
ne
w
multiplication
arithmetic
operators
stated
in
Theorem
5,
the
FFLS
can
be
transformed
in
a
form
of
a
crisp
linear
system,
called
as
the
ALS.
Definition
13.
Let
~
S
=
(
m;
;
)
be
a
positive
,
ne
gative
or
near
-zer
o
fuzzy
number
,
~
X
=
(
n;
;
)
be
a
positive
fuzzy
number
and
~
C
=
(
C
;
G;
H
)
be
any
form
of
fuzzy
number
s,
based
on
the
multiplication
arithmetic
oper
ator
s
in
Theor
em
5.
Then,
thr
ee
forms
of
ALS
ar
e
obtained,
suc
h
that:
•
If
~
S
is
positive
,
8
>
<
>
:
mn
=
C
n
+
(
m
)
=
G
n
+
(
m
+
)
=
H
0
@
m
0
0
(
m
)
0
0
(
m
+
)
1
A
0
@
n
1
A
=
0
@
C
G
H
1
A
(39)
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
2,
April
2021
:
583
–
598
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
r
591
•
If
~
S
is
ne
gative
,
8
>
<
>
:
mn
=
C
n
(
m
)
=
G
n
(
m
+
)
=
H
whic
h
can
be
r
epr
esented
as
0
@
m
0
0
0
(
m
)
(
m
+
)
0
1
A
0
@
n
1
A
=
0
@
C
G
H
1
A
(40)
•
If
~
S
is
near
-zer
o,
8
>
<
>
:
mn
=
C
n
(
m
)
=
G
n
+
(
m
+
)
=
H
whic
h
can
be
r
epr
esented
as
0
@
m
0
0
0
(
m
)
0
(
m
+
)
1
A
0
@
n
1
A
=
0
@
C
G
H
1
A
(41)
By
applying
the
concept
of
union
sets
as
stated
in
Definition
2,
these
three
ALS
block
matrices
in
(39),
(40)
and
(41)
can
be
combined
into
a
single
ALS
as
illustrated
in
Definition
14.
Definition
14.
Let
~
S
~
X
=
~
C
be
a
FFLS,
wher
e
the
fuzzy
coef
ficients
~
S
and
~
C
ar
e
arbitr
ary
fuzzy
number
s
and
~
X
be
a
positive
fuzzy
solution.
ALS
is
r
epr
esented
as
8
>
<
>
:
mn
=
C
n
+
(
m
)
(
m
)
=
G
n
(
m
+
)
+
(
m
+
)
=
H
(42)
whic
h
can
be
written
in
the
matrix
form
of
0
@
m
0
0
(
m
)
(
m
)
(
m
+
)
(
m
+
)
1
A
0
@
n
1
A
=
0
@
C
G
H
1
A
(43)
wher
e
m
=
(
m
ij
)
m
n
=
0
B
@
m
11
:::
m
1
n
.
.
.
.
.
.
.
.
.
m
m
1
:::
m
mn
1
C
A
;
=
(
ij
)
m
n
=
0
B
@
11
:::
1
n
.
.
.
.
.
.
.
.
.
m
1
:::
mn
1
C
A
;
=
(
ij
)
m
n
=
0
B
@
11
:::
1
n
.
.
.
.
.
.
.
.
.
m
1
:::
mn
1
C
A
;
n
=
0
B
@
n
1
.
.
.
n
n
1
C
A
;
=
0
B
@
1
.
.
.
n
1
C
A
;
=
0
B
@
1
.
.
.
n
1
C
A
;
A
modification
of
fuzzy
arithmetic
oper
ator
s
for
solving
near
-zer
o...
(W
.
S.
W
.
Daud)
Evaluation Warning : The document was created with Spire.PDF for Python.
592
r
ISSN:
1693-6930
and
C
=
0
B
@
C
1
.
.
.
C
m
1
C
A
;
G
=
0
B
@
G
1
.
.
.
G
m
1
C
A
;
H
=
0
B
@
H
1
.
.
.
H
m
1
C
A
:
This
ALS
can
be
denoted
as
S
X
=
C
:
(44)
Ho
we
v
er
,
the
matrix
S
in
(43)
is
al
w
ays
inconsistent
since
j
S
j
=
0
,
which
is
pro
v
ed
in
the
follo
wing
theorem:
Theor
em
6.
Let
S
be
a
coef
ficient
of
an
ALS.
The
matrix
S
is
singular
or
j
S
j
=
0
,
when
j
m
j
=
0
or
(
m
)
(
m
)
(
m
+
)
(
m
+
)
=
0
:
Pr
oof.
Let
S
=
0
@
m
0
0
(
m
)
(
m
)
(
m
+
)
(
m
+
)
1
A
The
singularity
of
S
can
be
determined
from
the
follo
wing
procedure,
which
is
based
on
Remark
1.
j
S
j
=
det
(
m
)
(
m
)
1
(0)
(
m
)
(
m
)
1
(0)
(
m
+
)
(
m
)
1
(0)
(
m
+
)
(
m
)
1
(0)
det
[
m
]
=
det
(
m
)
(
m
)
(
m
+
)
(
m
+
)
det
[
m
]
From
this,
if
j
m
j
=
0
,
then
ob
viously
matrix
S
is
singular
.
On
the
other
hand,
if
j
m
j
6
=
0
,
b
ut
(
m
)
(
m
)
(
m
+
)
(
m
+
)
=
0
,
hence,
matrix
S
is
singular
.
Remark
3.
Ther
e
ar
e
two
possibilities
that
mak
e
the
determinant
of
(
m
)
(
m
)
(
m
+
)
(
m
+
)
=
0
,
whic
h
ar
e:
i.
At
least
one
bloc
k
matrix
in
both
dia
gonal
and
anti-dia
gonal
have
all
zer
oes
in
a
r
ow
,
suc
h
that:
0
B
B
@
0
0
a
b
0
0
c
d
0
0
e
f
0
0
g
h
1
C
C
A
ii.
The
i
th
r
ow
or
j
th
column
of
a
matrix
is
a
multiple
of
another
r
ow
or
column,
suc
h
that:
0
B
B
@
a
b
a
b
c
d
c
d
e
f
e
f
g
h
g
h
1
C
C
A
In
order
to
a
v
oid
the
inconsistenc
y
of
the
solution,
the
ALS
in
(43)
has
been
impro
vised
to
be
in
the
follo
wing
form
as
stated
in
the
ne
xt
theorem.
Definition
15.
Let
~
S
~
X
=
~
C
be
a
FFLS
suc
h
that
~
S
=
(
m;
;
)
,
~
X
=
(
n;
;
)
and
~
C
=
(
C
;
G;
H
)
,
with
solution
~
X
as
a
positive
fuzzy
number
.
Then
the
ALS
of
S
X
=
C
is
written
as:
0
@
m
0
0
(
m
)
+
(
m
)
(
m
+
)
(
m
+
)
+
1
A
0
@
n
1
A
=
0
@
C
G
H
1
A
(45)
wher
e
(
m
)
+
and
(
m
+
)
+
contain
the
positive
elements
of
(
m
)
and
(
m
+
)
r
espectively
,
while
the
ne
gati
ve
elements
ar
e
r
eplaced
by
zer
o
values.
Similarly
,
(
m
)
and
(
m
+
)
contain
the
ne
gative
elements
of
(
m
)
and
(
m
+
)
r
espectively
,
while
the
positive
elements
ar
e
r
eplaced
by
zer
o
values.
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
19,
No.
2,
April
2021
:
583
–
598
Evaluation Warning : The document was created with Spire.PDF for Python.