New Alternative for Arithmetics Fuzzy Number
Mashadi
1
, Ahmad Syaiful Abidin
1
and Desi Ratna Anta Sari
1
1
Department of Mathematics, Universitas Riau, Pekanbaru, Indonesia
Keywords: Fuzzy Number, Fuzzy Matrix, Fuzzy Matrix Determinant, Inverse.
Abstract: We will discuss about inverse of fuzzy matrix whose elements are fuzzy trapezoidal numbers. The
discussion was prioritized from determining the new concept of positive and negative fuzzy numbers,
namely by using the concept of broad positive areas. Based on the concept, there will be an alternative for
multiplication concept will be discussed inverse fuzzy number matrix so that used directly to solve linear
equation system of fully fuzzy trapezoidal.
1 INTRODUCTION
The famous introducer of the concept of fuzzy
numbers in the world is (Zadeh, 1965) which
explains the fuzzy set. Fuzzy numbers that are often
discussed by researchers are fuzzy trapezoidal
numbers which have some basic arithmetic between
addition, subtraction, multiplication, inverse of
numbers and divisions, and determinants and
inverses of fuzzy matrices.
Some writers who have discussed about
trapezoidal fuzzy numbers include (Kumar et al.,
2010) applies a new method to the fuzzy trapezoidal
number named the Mehar method, (Nasheri &
Gholami, 2011) which resolves linear systems of
fuzzy trapezoidal numbers, (Gemawati et al., 2018)
gave a new algebra using the QR decomposition
method on fuzzy trapezoidal numbers, then solved
the linear equation system on trapezoidal fuzzy
numbers with iterative solutions.
Some authors besides discussing the solution of
fully fuzzy linear system, many of them also discuss
the new arithmetic and new definitions in
determining positive and negative fuzzy numbers
offered in solving problems in fuzzy numbers
including (Sari & Mashadi, 2019) and (Deswita &
Mashadi, 2019) provide new definitions in
determining positive and negative fuzzy numbers
with broad concepts in triangular fuzzy numbers,
(Kholida & Mashadi, 2019) and (Safitri & Mashadi,
2019) also provide new definitions with broad
concepts in determining positive and negative fuzzy
numbers but trapezoidal fuzzy numbers.
On the other hand some authors have discussed
the inverse fuzzy numbers and the inverse fuzzy
matrix inverse, namely (Sari & Mashadi, 2019)
which provides a new definition in determining
inverse fuzzy triangular numbers, other researchers
also discuss methods for finding the rank and
multiplication of inverse fuzzy trapezoid matrices,
however, it does not provide a definition of the
fuzzy matrix identity (Kaur, 2015), whereas
(Mohana & Mani, 2018) provides a note for
determining the adjoining fuzzy trapezoid matrix,
using basic arithmetic which is identical to the same
(Kaur & Kumar, 2017).
In this paper the author will provide and offer
new arithmetic in determining the inverse fuzzy
matrix with the same concept as the concept that has
been given in the previous author's paper, namely
(Safitri & Mashadi, 2019), (Kholida & Mashadi,
2019), (Abidin et al., 2019).
2 PRELIMINARIES
Fuzzy sets and fuzzy number are known in fuzzy,
(Zadeh, 1965) and (Zimmermann, 1996) was given
definition of fuzzy sets.
Definition 2.1. A fuzzy set 𝑀

⊆𝑋 is a characterized
by membership function 𝑓


𝑥 which associates
with each points in 𝑋 real number in the interval

0,1

, with the value of 𝑓


𝑥 at 𝑥 representing the
"grade of membership" of 𝑥 in 𝑀

.
242
Mashadi, ., Syaiful Abidin, A. and Ratna Anta Sari, D.
New Alternative for Arithmetics Fuzzy Number.
DOI: 10.5220/0010139900002775
In Proceedings of the 1st International MIPAnet Conference on Science and Mathematics (IMC-SciMath 2019), pages 242-247
ISBN: 978-989-758-556-2
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

Definition 2.2. Let 𝑋 is a set of object collection
that are denoted in general by 𝑥, the a fuzzy set 𝑀

in
𝑋 is a sequential set of pairs 𝑀



𝑋,𝜇



𝑥
|
𝑥𝜖𝑋

with 𝜇


is a membership function of the fuzzy set 𝑀

which a mapping of the universal set 𝑋 in the
interval 0,1.
Some basic definition and theories related to
fuzzy number has been discussed by (Gemawati et
al., 2018) and (Cong-Xin & Ming, 1991).
Definition 2.3. Fuzzy number is a fuzzy set 𝑢: 𝑅→
0,1 with 𝑢𝑟,𝑠,𝜀

,𝜀

 which satisfies
1. 𝑢 is upper semi continuous;
2. 𝑢

𝑥

0 outside some interval

𝑟𝜀

,𝑠𝜀


;
3. There are real number 𝑟,𝑠 in the interval

𝑟𝜀

,𝑠𝜀


such that
(i) 𝑢

𝑥

monotonic non-decreasing in

𝑟
𝜀

,𝑟

;
(ii) 𝑢

𝑥

monotonic non-increasing in

𝑠,𝑠 
𝜀


;
(iii) 𝑢

𝑥

1, for 𝑟𝑥𝑠.
Definition 2.4. Fuzzy number 𝑢 in 𝑅 are function
pair 𝑢

𝑟

,𝑢

𝑟

, which satisfies the following:
1. 𝑢

𝑟

is a bounded left continuous non decreasing
function over 0,1;
2. 𝑢
𝑟 is a bounded left continuous non increasing
function over

0,1

;
3. 𝑢

𝑟

𝑢

𝑟

,0𝑟1.
(Kumar et al., 2010) provided a definition of
membership functions of trapezoidal fuzzy numbers,
given fuzzy numbers 𝑢𝑎,𝑏,𝛼,𝛽 where 𝑎 and 𝑏
are the center points, 𝛼 distance from the center
point jarak dari titik pusat 𝑎 to the left, and 𝛽
distance from the center point 𝑏 to the right.
Trapezoidal fuzzy numbers function have the
following form :
𝜇



𝑥


⎩
⎪
⎨
⎪
⎧
1
𝑟𝑥
𝜀

, 𝑟  𝜀

𝑥𝑟
1, 𝑟𝑥𝑠
1
𝑥𝑠
𝜀

, 𝑠𝑥𝑠 𝜀

0, otherwise
Trapezoidal fuzzy numbers have the following
parametric form if 𝑢𝑢
𝑟,𝑢𝑟 can be
represented as:
𝑢

𝑟

𝑟

1𝑟

𝜀

𝑢

𝑟

𝑠

1𝑟

𝜀

Some authors have described arithmetic fuzzy
trapezoidal numbers like (Malkawi et al., 2014).
Two fuzzy number 𝑢𝑟,𝑠,𝜀

,𝜀

 and 𝑣
𝑡,𝑢,𝛿

,𝛿

 we call same if only if 𝑟𝑡, 𝑠𝑢,
𝜀

𝛿

and 𝜀

𝛿

.
Arithmetic on trapezoidal fuzzy numbers given
by (Kumar et al., 2011) that is if there are two fuzzy
numbers 𝑢

𝑟,𝑠,𝜀,𝜀

and 𝑣𝑡,𝑢,𝛿,𝛿 then
1. Addition
𝑢⊕𝑣

𝑟𝑡,𝑠𝑢,𝜀𝛿,𝜀𝛿

2. Substraction
𝑢⊖𝑣

𝑟𝑢,𝑠𝑡,𝜀𝛿,𝜀𝛿

3. Multiplication
𝑢⊗ 𝑣








𝑤,







𝑤,
|
𝑠𝛿 𝑢𝜀
|
,
|
𝑠𝛿 𝑠𝜀
|


where
𝑤
𝑔ℎ
2

and
ℎ𝑚𝑖𝑛

𝑟𝑡,𝑟𝑢,𝑠𝑡,𝑠𝑢

𝑔𝑚𝑎𝑥

𝑟𝑡,𝑟𝑢,𝑠𝑡,𝑠𝑢

4. Scalar Mutiplication
𝑘⊗𝑢

𝑘𝑟,𝑘𝑠,𝑘𝜀,𝑘𝜀

, 𝑘0

𝑘𝑠,𝑘𝑟,𝑘𝜀,𝑘𝜀

𝑘0
The weakness of the arithmetic above is that the
definition given only applies to two fuzzy
trapezoidal numbers which have a distance from the
center to the left and to the right of the same value.
Whereas (Kaur, 2015) provide the basic
arithmetic definition of fuzzy trapezoid numbers,
given fuzzy numbers 𝑢𝑚,𝑛,𝑝,𝑞 and 𝑣
𝑟,𝑠,𝑡,𝑢 where 𝑚𝑛𝑝𝑞 and 𝑟𝑠𝑡
𝑢 then:
1. Addition
𝑢⊕𝑣

𝑚𝑟,𝑛𝑠,𝑝𝑡,𝑞𝑢

2. Substraction
𝑢⊖𝑣

𝑚𝑢,𝑛𝑡,𝑝𝑠,𝑞𝑟

3. Multiplication
𝑢⨂𝑣𝑚




,𝑛




,
p




,𝑞





4. Scalar Mutiplication
New Alternative for Arithmetics Fuzzy Number
243

𝑘⊗𝑢

𝑘𝑚,𝑘𝑛,𝑘𝑝,𝑘𝑞

, 𝑘0

𝑘𝑞,𝑘𝑝,𝑘𝑛,𝑘𝑚

𝑘0
5. Division


𝑣





,


,


,



The weakness of this arithmetic is that the
defined product does not give a case if 𝑢 and 𝑣 are
positive or negtive fuzzy numbers, and in division
operation




𝚤̃ or 𝑢⊗



𝚤̃ .
(Mohana & Mani, 2018) defines surgery on the
trapezoidal fuzzy matrix. For example 𝑃

𝑝


where 𝑝

𝑚

,𝑛

,𝑝

,𝑞

 and 𝑄

𝑞

 where
𝑞

𝑟

,𝑠

,𝑡

,𝑢


1. Addition
𝑃

⊕𝑄

𝑝

⊕𝑞


2. Substraction
𝑃

⊖𝑄

𝑝

⊖𝑞


3. Multiplication
𝑃

𝑝



and 𝑄

𝑞



Then 𝑃

⊗𝑄

𝑟̃



where
𝑟̃

𝑝

⊗


𝑞

4. Transpose
𝑃


𝑝


5. Scalar Multiplication
k
𝑃

𝑘𝑝
𝑗𝑖

The weakness of this arithmetic is that the
multiplication that is defined does not give a case if
𝑢 and 𝑣 are positive or negative fuzzy numbers.
3 ALTERNATIVE ARITHMETIC
FOR INVERS TRAPEZOIDAL
FUZZY MATRIX
Now at this article, the basic arithmetic operations of
fuzzy numbers used to determine the inverse matrix
are arithmetic operations with broad concepts that
have been given (Kholida & Mashadi, 2019), (Safitri
& Mashadi, 2019) and (Abidin et al., 2019). Given
two fuzzy numbers 𝑢𝑟,𝑠,𝜀

,𝜀

 and 𝑣
𝑡,𝑢,𝛿

,𝛿

 are equal if only if 𝑟𝑡 and 𝑠𝑢.
Two fuzzy numbers said to be the same pure if and
only if 𝑟𝑡, 𝑠𝑢 and 𝜀

𝛿

, 𝜀

𝛿

.
1. Addition
𝑢⊕𝑣

𝑟𝑡,𝑠𝑢,𝜀

𝛿

,𝜀

𝛿


2. Substraction
𝑢⊖𝑣

𝑟𝑢,𝑠𝑡,𝜀

𝛿

,𝜀

𝛿


3. Scalar Multiplication
𝑘⊗𝑢𝑘⊗
𝑟,𝑠,𝜀
1
,𝜀
2



𝑘𝑟,𝑘𝑠,𝑘𝜀

,𝑘𝜀


, 𝑘0

𝑘𝑠,𝑘𝑟,𝑘𝜀

,𝑘𝜀


𝑘0
4. Multiplication
a. Case 1, if 𝑢 positive and 𝑣 positive, then:
𝑢⨂𝑣𝑟𝑡,𝑠𝑢,

𝑟𝛿

𝑡𝜀


,

𝑠𝛿

𝑢𝜀



b. Case 2, if 𝑢 positive and 𝑣 negative, then:
𝑢⨂𝑣𝑠𝑡,𝑟𝑢,

𝑠𝛿

𝑡𝜀


,

𝑟𝛿

𝑢𝜀



c. Case 3, if 𝑢 negative and 𝑣 positive, then:
𝑢⨂𝑣𝑟𝑢,𝑠𝑡,

𝑢𝜀

𝑟𝛿


,

𝑡𝜀

𝑠𝛿



d. Case 4, if 𝑢 negative and 𝑣 negative, then:
𝑢⨂𝑣𝑠𝑢,𝑟𝑡,

𝑢𝜀

𝑠𝛿


,

𝑟𝛿

𝑡𝜀



5. Identity of Fuzzy Number
The identity for fuzzy numbers is divided into
two, namely pure identity where 𝚤̃



1,1,0,0

and
identity where 𝚤̃1,1,𝜀

,𝜀

.
6. Inverse of Fuzzy Number
𝑢⨂𝑣
𝑟,𝑠,𝜀
1
,𝜀
2
 ⨂𝑡,𝑢,𝛿
1
,𝛿
2

𝑟𝑡,𝑠𝑢,

𝑟𝛿

𝑡𝜀


,

𝑠𝛿

𝑢𝜀



So that inverse of trapezoidal fuzzy number with
𝑟,𝑠 0 are obtain :
𝑣
1
𝑟
,
1
𝑠
,
𝜀

𝑟

,
𝜀

𝑠


7. Inverse of Fuzzy Matrix
Similar to fuzzy numbers, fuzzy matrices also
have two identities namely pure identity is defined
as follows:
𝐼


𝑎


Where
𝑎



0,0,0,0

, 𝑓𝑜𝑟 𝑖𝑗

1,1,0,0

, 𝑓𝑜𝑟 𝑖𝑗
IMC-SciMath 2019 - The International MIPAnet Conference on Science and Mathematics (IMC-SciMath)
244

To get the result of 𝑃

⊗𝑃


𝐼


is very
difficult so that another alternative to the matrix
identity is defined as follows :
𝐼

𝑎


where
𝑎


0,0,𝜀

,𝜀

, 𝑓𝑜𝑟 𝑖𝑗
1,1,𝜀

,𝜀

, 𝑓𝑜𝑟 𝑖𝑗
Let
𝑃

𝑎


where
𝑎

𝑎
𝑖𝑖
,𝑏
𝑖𝑖
,𝛼
𝑖𝑖
,𝛽
𝑖𝑖

𝑎


𝑎

,𝑏

,𝛼

,𝛽


for 𝑖𝑗.
Then it can be partitioned into:
𝑀
𝑎

𝑎

⋯𝑎

𝑎

𝑎

⋯𝑎

⋮
𝑎

⋮
𝑎

⋱ ⋮
⋯𝑎


𝑁
𝑏

𝑏

⋯𝑏

𝑏

𝑏

⋯𝑏

⋮
𝑏

⋮
𝑏

⋱ ⋮
⋯𝑏


𝑃
𝛼

𝛼

⋯𝛼

𝛼

𝛼

⋯𝛼

⋮
𝛼

⋮
𝛼

⋱ ⋮
⋯𝛼


𝑄
𝛽

𝛽

⋯𝛽

𝛽

𝛽

⋯𝛽

⋮
𝛽

⋮
𝛽

⋱ ⋮
⋯𝛽


Then 𝑃

can be written 𝑃

𝑃,𝑄,𝐸

,𝐸

 and
𝑄

𝑅,𝑆,𝐷

,𝐷

, then the fuzzy matrix 𝑃

𝑄

if
𝑃𝑅,𝑄𝑆,𝐸

𝐷

,𝐸

𝐷

. Matrix fuzzy 𝑄

is said to be an inverse of the fuzzy matrix 𝑃

if
𝑃

⊗
𝑄

𝐼

Assuming each element the fuzzy matrix 𝑃

and
𝑄

are positive fuzzy numbers, then

𝑃,𝑄,𝐸

,𝐸


⊗

𝑅,𝑆,𝐷

,𝐷


𝐼,𝐼,𝜀

,𝜀


𝑃𝑅,𝑄𝑆,𝑁𝐺,𝑃𝐷

𝑅𝐸

,𝑄𝐷

𝑆𝐸

𝐼,𝐼,𝜀

,𝜀


was obtain

𝑃𝑅𝐼
𝑄𝑆𝐼
𝑃𝐷

𝑅𝐸

𝜀

𝑄𝐷

𝑆𝐸

𝜀

So
𝑅𝑃

𝑆𝑄

𝐷

𝑃

𝜀

𝑅𝐸


𝐷

𝑄

𝜀

𝑆𝐸


Based on the algebra above 𝑃

and 𝑄

can be
searched directly, so the authors provide the
definition of the matrix 𝑄

or 𝑃


as follow :
𝑃


𝑃

,𝑄

,𝐸


,𝐸



Furthermore, it will be proven that
𝑃

⊗𝑃


𝐼

then

𝑃,𝑄,𝐸

,𝐸


⊗𝑃

,𝑄

,𝐸


,𝐸


𝐼

Assuming each element of the fuzzy matrix is a
positive fuzzy number, arithmetic multiplication of
positive fuzzy number and positive fuzzy number is
obtained:
𝑃

⊗𝑃


𝑃𝑃

,𝑄𝑄

,𝑃

𝐸

𝑃𝐸


,
𝑄

𝐸

𝑄𝐸



 𝐼,𝐼,𝑃

𝐸

𝑃𝐸


,𝑄

𝐸

𝑄𝐸



Assume if 𝑃

𝐸

𝑃𝐸


𝜀

and 𝑄

𝐸


𝑄𝐸


𝜀

.
then
𝑃

⊗𝑃


𝐼,𝐼,𝜀

,𝜀


Thus it is evident that 𝑃

⊗𝑃


have identity results.
Given two trapezoidal fuzzy matrices 𝐴



𝐴,𝐵,𝐶,𝐷

and 𝐵



𝐸,𝐹,𝐺,𝐻

with
𝐴




𝐴

,𝐵

,𝐶

,𝐷


and
𝐵




𝐸

,𝐹

,𝐺

,𝐻


will show that :
𝐴

⊗𝐵



𝐵


⊗𝐴


𝐴

⊗𝐵



𝐴,𝐵,𝐶,𝐷

⊗

𝐸,𝐹,𝐺,𝐻

𝑀𝐾,𝑁𝐿,𝑀𝑅𝐾𝑃,𝑁𝑆𝐿𝑄
𝐴

⊗𝐵





𝑀𝐾


,

𝑁𝐿


,

𝑀𝑅 𝐾𝑃


,

𝑁𝑆𝐿𝑄



𝐾

𝑀

,𝐿

𝑁

,

𝑀𝑅 𝐾𝑃


,

𝑁𝑆𝐿𝑄


and
𝐵


⊗𝐴




𝐾

,𝐿

,𝑅

,𝑆


⊗

𝑀

,𝑁

,𝑃

,𝑄


New Alternative for Arithmetics Fuzzy Number
245

𝐾

𝑀

,𝐿

𝑁

, 𝐾

𝑃

𝑀

𝑅

,𝐿

𝑄

𝑁

𝑆


From the algebra above it is obtained that:
𝑀𝑅 𝐾𝑃

𝑀

𝑅

 𝐾

𝑃

and
𝑁𝑆  𝐿𝑄

𝑁

𝑆

𝐿

𝑄

The next step will be indicated
𝐴

⊗𝐵

⊗𝐴

⊗𝐵



𝐼



𝑀𝐾,𝑁𝐿,𝑀𝑅 𝐾𝑃,𝑁𝑆 𝐿𝑄

⊗

𝑀𝐾


,

𝑁𝐿


,

𝑀𝑅 𝐾𝑃


,

𝑁𝑆 𝐿𝑄



𝑀𝐾

𝑀𝐾


,𝑁𝐿

𝑁𝐿


,𝑀𝐾

𝑀𝑅 𝐾𝑃




𝑀𝐾



𝑀𝑅 𝐾𝑃

,𝑁𝐿𝑁𝑆𝐿𝑄



𝑁𝐿


𝑁𝑆𝐿𝑄
𝐼,𝐼,𝜀

,𝜀

𝐼

So it can be conclude that 𝑃

⊗𝑃


produces
identity, not pure identity.
Numerical Example
Given
𝐴

𝑥𝑏

where
𝐴



5,6,2,3

3,5,4,2


9,11,2,1

3,7,1,2


𝑏


34,56,45,37
54,94,52,32

and will be find 𝑥.
From 𝐴

we get
𝑀
53
93
,𝑁
65
11 7

𝑃
24
21
,𝑄
32
12

and from 𝑏

we get
𝑏
34
54
, 𝑔
56
94
, ℎ
45
52
, 𝑡
37
32

We get
𝐴




1
4
,
7
13
,
1
6
,
1
2

1
4
,
5
13
,
2
3
,
1
2


3
4
,
11
13
,
1
3
,
1
4

5
12
,
6
13
,
1
3
,
3
4


then
𝑥𝐴


𝑏

So we get the results of 𝑥 is
𝑥

5,6,4,1


3,4,1,1


Then to check the truth of the results from 𝐴


will be shown 𝐴

⊗𝐴


𝐼

and 𝐴


⊗𝐴

𝐼

.
𝐴

⊗𝐴



1,1,2
2
3
,1
43
52
 0,0,1
1
6
,
51
52

0,0,7
1
4
,
47
52
1,1,
11
12
,
41
52


𝐼,𝐼,𝜀

,𝜀

𝐼

and
𝐴


⊗𝐴


1,1,2
2
3
,1
7
26
0,0,1
1
6
,1
37
78

0,0,
1
4
,2
17
52
1,1,5
1
12
,
133
156


4 CONCLUSIONS
In this article, arithmetic alternatives to fuzzy
numbers and alternative elements of identity are
pure identity and identity in fuzzy trapezoidal
numbers and fuzzy trapezoidal matrices. After the
arithmetic alternative is obtained, the inverse of a
matrix can be determined. The matrix inverse
obtained can be used to solve system of fully fuzzy
linear equations
𝐴

⊗𝑥𝑏

directly, with 𝑥
𝐴


⊗𝑏

.
REFERENCES
Abidin, A. S., Mashadi, & Gemawati, S. (2019). Algebraic
Modification of Trapezoidal Fuzzy Numbers to
Complete Fully Fuzzy Linear Equations System Using
Gauss-Jacobi Method. IJMFS, 2, 40–46.
Cong-Xin, W., & Ming, M. (1991). Emmbedding Problem
of Fuzzy Number Space: Part I. Fuzzy Set and
Systems, 44, 33–38.
Deswita, Z., & Mashadi. (2019). Alternative Multiplying
Triangular Fuzzy Number and Applied in Fully Fuzzy
Linear System. ASRJETS, 56, 113–123.
Gemawati, S., Nasfianti, I., Mashadi, & Hadi, A. (2018).
A new method for dual fully fuzzy linear system with
trapezoidal fuzzy number by QR decomposition.
Proceeding of International Conference on Science
and Technology, 1116, 1–5.
Kaur, J. (2015). Methods to Find the Rank and
Multiplicative Inverse of Fully Fuzzy Matrices.
Mathematical Sciences an Internasional Journal, 4,
118–122.
Kaur, J., & Kumar, A. (2017). Commentary on
“Calculating fuzzy inverse matrix using fuzzy linear
equation system.” Applied Soft Computing, 58, 324–
327.
Kholida, H., & Mashadi. (2019). Alternative Fuzzy
Algebra for Fuzzy Linear System Using Cramers
Rules on Fuzzy Trapezoidal Number. IJISRT, 4, 494–
500.
IMC-SciMath 2019 - The International MIPAnet Conference on Science and Mathematics (IMC-SciMath)
246

Kumar, A., Babbar, N., & Bansal, A. (2010). A method
for solving fully fuzzy linear system with trapezoidal
fuzzy number. Iranian Journal of Optimization, 2,
359–374.
Kumar, A., Kaur, J., & Singh, P. (2011). A New Method
for Solving Fuzzy Linear Programs with Trapezoidal
Fuzzy Number. Journal of Fuzzy Set Valued Analysis,
1–12.
Malkawi, G., Ahmad, N., & Ibrahim, H. (2014). Solving
fully fuzzy linear system with the necessary an
suficientto have a positive solution. AMIS, 8, 1003–
1019.
Mohana, D. N., & Mani, R. (2018). A Note on Adjoint of
Trapezoidal Fuzzy Number Matrices. IJIRT, 4(9), 32–
37.
Nasheri, S. H., & Gholami, M. (2011). Linear system of
equation with trapezoidal fuzzy numbers. The Journal
of Mathematics and Computer Science, 3, 71–79.
Safitri, Y., & Mashadi. (2019). Alternative Fuzzy Algebra
to Solve Dual Fully Fuzzy Linear System Using ST
Decomposition Method. IOSR-JM, 32–38.
Sari, D. R. A., & Mashadi. (2019). New Arithmetic
Triangular Fuzzy Number for Solving fully Fuzzy
Linear System Using Inverse Matrix. IJSBAR, 46,
169–180.
Zadeh, L. A. (1965). The Concept of a Linguistic Variable
and its Application to Approximate Reasoning-I.
Information Sciences, 8, 199–249.
Zimmermann, H. J. (1996). Fuzzy Set Theory and Its
Applications (Second). Kluwer Academic Publishers.
New Alternative for Arithmetics Fuzzy Number
247