Alternative Algebra for Trapezoidal Fuzzy Number and Comparison

with Various Other Algebra

Mashadi

and Helmi Kholida

Department of Mathematics, Universitas Riau, Pekanbaru, Indonesia

Keywords: Trapezoidal Fuzzy Number, Fully Fuzzy Linear System, Arithmetic of Fuzzy Number, Cramer Method.

Abstract: In this article will be given a new algebra for trapezoidal fuzzy number, and then the new algebra we got will

be compared with other algebra by privious outhor. Finally will be given a fully fuzzy linear system which

will be solved by author algebra and other algebra by privious outhor, will be shown the new algebra which

the outhor suggest its better.

1 INTRODUCTION

Fuzzy logic is a branch of mathematical science first

introduced by LA Zadeh, a professor from UC

Berkeley's electrical engineering, computer science

department in 1965. LA Zadeh thinks fuzzy logic can

bridge precision machine language into human

language that emphasizes meaning or significance

(Zadeh, 1965).

Many methods to solve fully fuzzy linear system,

but almost all methods of partitioning are in solution

and in partitioning each matiks partition must be of

positive or negative value (there is no mixed matrix).

In this article we will give an example of solving a

fully fuzzy linear system using the Cramer method,

with 𝑥





det𝐴





det𝐴





. In this case even though the

matrix 𝐴



is a mixture of each 𝑖 -th element, it can still

be solved. First the fuzzy number identity value is

given so that it can be used to obtain the inverse value

of the fuzzy number so that 𝑥



det𝐴





⊗

det𝐴







can be completed.

Previously (Vijayalakshmi, 2011) had defined

𝑢







𝑚,𝑛,𝛼,𝛽





with 𝑛 is negative is



𝑚



,𝑛



,𝑛𝑚



𝛽,𝑛𝑛



𝛼



in this case there is a

weakness in the symbol so that it can confuse the

reader in interpreting 𝑛 as a point on a fuzzy number

or 𝑛 as a rank in the fuzzy number. Furthermore

(Vijayalakshmi, 2011) also gives the definition 𝑢







𝑚,𝑛,𝛼,𝛽







𝑚



,𝑛



,𝑛𝑚



𝛽,𝑛𝑛



𝛼



with 𝑛 is

positive , in this case we can assume 𝑢



𝑢⊗𝑢

using the given multiplication formula but gives

different results when the value 𝑛2 is substituted

into 𝑢



Furthermore (Jafarian, 2016) only defines fuzzy

numbers said to be not negative if given fuzzy

numbers 𝑢𝑚,𝑛,𝛼,𝛽 if 𝑚𝛼0 and does not

provide a definition of fuzzy numbers said to be

positive or negative.

Finally author will compare the arithmetic that the

writer obtained with the arithmetic given by other

writers especially on the inverse value that the writer

obtained and the inverse value given (Vijayalakshmi,

2011) in solving the fully fuzzy linear system.

2 PRELIMINARIES

Some basic definitions of trapezoidal fuzzy number

set theory are reviewed (Jafarian, 2016),

(Radhakrishnan et al., 2012), (Vijayalakshmi, 2011).

A fuzzy number 𝑠̃

ℎ,𝑘,𝜌,𝜏 is said to be a

trapezoidal fuzzy numberif its membership function

is given by

𝜇





𝑥





⎩

⎪

⎨

⎪

⎧

1

ℎ 𝑥

𝜌

, ℎ𝜌𝑥ℎ

1, ℎ 𝑥𝑘

1

𝑥𝑘

𝜏

, 𝑘𝑥𝑘𝜏

0, otherwise

The trapezoidal fuzzy number can be written in

the parametric form

Mashadi, . and Kholida, H.

Alternative Algebra for Trapezoidal Fuzzy Number and Comparison with Various Other Algebra.

DOI: 10.5220/0010139800002775

In Proceedings of the 1st International MIPAnet Conference on Science and Mathematics (IMC-SciMath 2019), pages 237-241

ISBN: 978-989-758-556-2

237

𝑢

𝑟ℎ 



1𝑟



𝜌,

𝑢

𝑟𝑘



1𝑟



𝜏.

Definition 2.1. Fuzzy subset 𝑢 is defined with 𝑠̃

𝑥,𝜇





𝑥



. In pairs 𝑥,𝜇





𝑥



, 𝑥 is a member of the

set 𝑠̃ and 𝜇





𝑥



the value on interval [0, 1] which is

called the membership function.

Definition 2.2. Fuzzy number is a fuzzy set 𝑠̃:ℝ→

0,1 which satisfies the following:

1. 𝑠̃ is upper semicontinuous.

2. 𝑠̃0 outside the interval 

ℎ 𝜌,𝑘𝜏.

3. There exist real number 

ℎ,𝑘 in interval ℎ 

𝜌,𝑘𝜏 such that,

i. 𝑠̃ monotonic increasing in 

ℎ 𝜌,ℎ.

ii. 𝑠̃ monotonic decreasing in 𝑘,𝑘 𝜏.

iii. 𝑠̃1 for

ℎ𝑥𝑘.

The alternative definition of other fuzzy numbers

that are often used by authors is as follows.

Definition 2.3. Fuzzy number 𝑢 in ℝ is defined as a

function pair 𝑠̃𝑠



𝑟



,𝑠



𝑟



 which satisfy the

following:

1. 𝑠



𝑟



is a bounded left continuous non decreasing

function over 0,1.

2. 𝑠



𝑟



is a bounded right continuous non increasing

function over 0,1.

3. 𝑠



𝑟



𝑠



𝑟



,0𝑟1.

Definition 2.4. A trapezoidal fuzzy number 𝑠̃

ℎ,𝑘,𝜌,𝜏 is said to be zero trapezoidal fuzzy number

if and only if ℎ0,𝑘0,𝜌0 and 𝜏0.

Definition 2.5. Two fuzzy number 𝑠̃ℎ,𝑘,𝜌,𝜏

and 𝑡

𝑝,𝑞,𝜎,𝜔 are said to be equal if and only if

ℎ𝑝,𝑘𝑞 and are said pure same if and only if

ℎ𝑝,𝑘𝑞,𝜌𝜎,𝜏𝜔.

Here's the algebra that other author give:

Definition 2.6. Given 𝑠̃



ℎ,𝑘,𝜌,𝜏



𝑠

,𝑠



ℎ



1𝑟



𝜌,𝑘



1𝑟



𝜏



, 𝑡





𝑝,𝑞,𝜎,𝜔





𝑡

,𝑡



𝑝



1𝑟



𝜎,𝑞 



1𝑟



𝜔



and 𝑛 is real

a. Addition

𝑠̃𝑡

𝑠𝑡

,𝑠 𝑡 

𝑠̃𝑡





ℎ  𝑝 ,𝑘 𝑞 ,𝜌  𝜎 ,𝜏 𝜔



b. Subtraction

𝑠̃𝑡

𝑠

𝑡 ,𝑠 𝑡

𝑠̃𝑡





ℎ𝑞 ,𝑘𝑝,𝜌𝜔 ,𝜏𝜎



c. Negative number

𝑠̃



𝑘,ℎ,𝜏,𝜌



d. Scalar multiplication

𝑛⊗𝑠̃𝑘⊗



ℎ,𝑘,𝜌,𝜏







𝑛ℎ,𝑛𝑘,𝑛𝜌,𝑛𝜏



, 𝑛0



𝑛𝑘,𝑛ℎ,𝑛𝜏,𝑛𝜌



, 𝑛0

e. Multiplication

If 𝑠̃0 and 𝑡

0, so

𝑠̃⊗𝑡

ℎ𝑝,𝑘𝑞,



ℎ𝜎  𝑝𝜌





𝑘𝜔 𝑞𝜏





In this case (Jafarian, 2016) and (Vijayalakshmi,

2011) do not provide alternative algebra for other

fuzzy numbers such as multiplication of positive

fuzzy numbers - negative fuzzy numbers, negative

fuzzy numbers - positive fuzzy numbers and negative

fuzzy numbers - negative fuzzy numbers. And didnt

give definition of fuzzy numbers said to be positive

or fuzzy numbers said to be negative.

3 ALGEBRA OF FUZZY

NUMBER

Fuzzy numbers that are said to be positive fuzzy or

negative fuzzy by determining the area of the x-axis,

then we will be given the identity of fuzzy numbers,

inverse fuzzy numbers and alternative divisions of

fuzzy numbers.

3.1 Positive and Negative Fuzzy Number

Definition 3.1. The fuzzy number 𝑢 is said to be

positive (negative) fuzzy denoted 𝑠̃0,



𝑠̃0



using the area rules in the x-axis, that is:

1. If the fuzzy region is exactly one of the 𝑥 -axes

then fuzzy 𝑠̃ is said to be positive (negative) if ℎ

𝜌0



𝑘𝜏0



2. If the fuzzy region is both of the 𝑥 -axes so:

a. If ℎ0, 𝑘0 and 𝑘 𝜏0 𝑠̃ said positive

fuzzy number is ℎ𝑘



















0, and

𝑠̃ said to be negative fuzzy number is ℎ𝑘



















0.

b. If ℎ0 and 𝑘0 𝑠̃ said positive fuzzy

number is 𝑘ℎ











0, and 𝑠̃ said to be

negative fuzzy number is 𝑘ℎ











0.

c. If ℎ0 and 𝑘0, 𝑠̃ said positive fuzzy

number is 𝑘ℎ



















0, and 𝑢 said

to be negative fuzzy number is 𝑘ℎ



















0.

3.2 Arithmetic Trapezoidal Fuzzy Number

Given 𝑠̃



ℎ,𝑘,𝜌,𝜏



with parametric function such

as:

𝑠̃𝑠

𝑟,𝑠𝑟



ℎ



1𝑟



𝜌,𝑘



1𝑟



𝜏



IMC-SciMath 2019 - The International MIPAnet Conference on Science and Mathematics (IMC-SciMath)

238

and

𝑡

𝑡

𝑟,𝑡𝑟



𝑝



1𝑟



𝜎,𝑞 



1𝑟



𝜔



Theorem 3.1. If 𝑠̃𝑠



𝑟



,𝑠



𝑟



 and 𝑡

𝑡



𝑟



,𝑡



𝑟





is two positive trapezoidal fuzzy number so 𝑤𝑠̃⊗

𝑡





𝑤



𝑟



,𝑤



𝑟





with

𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





For 𝑟∈



0,1



is a positive trapezoidal fuzzy

number.

Based on theorem 1, for two trapezoidal fuzzy

number 𝑠̃𝑠



𝑟



,𝑠



𝑟



 and 𝑡

𝑡



𝑟



,𝑡



𝑟





following this conditions:

i. If 𝑠̃ positive and 𝑡

negative, so 𝑤𝑠̃⊗



𝑡





negative,

ii. If 𝑠̃ negative and 𝑡

positive, so 𝑤



𝑠̃



⊗𝑡



negative,

iii. If 𝑠̃ negative and 𝑡

negative, so 𝑤



𝑠̃



⊗



𝑡



 positive.

Based on theorem 1 and multiplication (i-iii), so

multiplication on two trapezoidal fuzzy number 𝑠̃

𝑠



𝑟



,𝑠



𝑟



 and 𝑡

𝑡



𝑟



,𝑡



𝑟



 and 𝑟 ∈



0,1



obtained:

i. If 𝑢 positive and 𝑣 negative, s

𝑤

𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





ii. If 𝑢 negative and 𝑣 positive, so

𝑤

𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





iii. If 𝑢 negative and 𝑣 negative, so

𝑤

𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





𝑤



𝑟



𝑠



𝑟



𝑡





𝑠





𝑡



𝑟



𝑠





𝑡





By following the description we get the fuzzy

number multiplication formula as follows:

a. Multiplication of Fuzzy

I. If 𝑠̃0 and 𝑡

0 so:

𝑤



𝑤



𝑟



,𝑤



𝑟





𝑤ℎ𝑝,𝑘𝑞,



ℎ𝜎  𝑝𝜌





𝑘𝜔 𝑞𝜏





II. If 𝑠̃0 and 𝑡

0 so:

𝑤



𝑤



𝑟



,𝑤



𝑟





𝑤𝑘𝑝,ℎ𝑞,



𝑘𝜎 𝑝𝜏





ℎ𝜔  𝑞𝜌





III. If 𝑠̃0 and 𝑡

0 so:

𝑤



𝑤



𝑟



,𝑤



𝑟





𝑤ℎ𝑞,𝑘𝑝,



𝑞𝜌  ℎ𝜔





𝑝𝜏 𝑘𝜎





IV. If 𝑠̃0 and 𝑡

0 so:

𝑤



𝑤



𝑟



,𝑤



𝑟





𝑤𝑘𝑞,ℎ𝑝,



𝑞𝜏  𝑘𝜔



,



ℎ𝜎  𝑝𝜌





b. Fuzzy Number Identity

Given 𝑠̃



ℎ,𝑘,𝜌,𝜏



and 𝐼



𝑝,𝑞,𝜎,𝜔, so 𝐼



said

to be identity if

𝑠̃⊗𝐼



𝑠̃



ℎ,𝑘,𝜌,𝜏



⊗



𝑝,𝑞,𝜎,𝜔







ℎ,𝑘,𝜌,𝜏





ℎ𝑝,𝑘𝑞,ℎ𝜎  𝑝𝜌,𝑘𝜔 𝑞𝜏







ℎ,𝑘,𝜌,𝜏



and

ℎ𝑝ℎ 𝑘𝑞𝑘

𝑝1 𝑞1

ℎ𝜎  𝑝𝜌𝜌 𝑘𝜔 𝑞𝜏𝜏

ℎ𝜎  𝜌𝜌 𝑘𝜔 𝜏𝜏

ℎ𝜎0 𝑘𝜔0

𝜎0 𝑖𝑓 ℎ0 𝜔0 𝑖𝑓 𝑘0

So obtained 𝚤̃



1,1,0,0 is called pure identity, but

it will be difficult to obtain it so given 𝚤̃1,1,𝜀



,𝜀





is called identity.

c. Invers of Fuzzy

Given 𝑠̃ℎ,𝑘,𝜌,𝜏 and 𝑣𝑝,𝑞,𝜎,𝜔, so 𝑡

said

to be inverse of 𝑠̃ if 𝑠̃⊗𝑡





1,1,0,0



with 𝚤̃





1,1,0,0.

𝑠̃⊗𝑡





1,1,0,0



ℎ,𝑘,𝜌,𝜏 ⊗𝑝,𝑞,𝜎,𝜔



1,1,0,0





ℎ𝑝,𝑘𝑞,ℎ𝜎  𝑝𝜌,𝑘𝜔 𝑞𝜏



1,1,0,0

So obtained

ℎ𝑝1 𝑘𝑞1

𝑝

ℎ



𝑞

𝑘



ℎ𝜎  𝑝𝜌0 𝑘𝜔 𝑞𝜏0

𝜎

𝜌

ℎ





𝜔

𝜏

𝑘





𝑡



𝑠̃



ℎ

𝑘

𝜌

ℎ



𝜏

𝑘





Alternative Algebra for Trapezoidal Fuzzy Number and Comparison with Various Other Algebra

239

d. Division of Fuzzy

Given 𝑠̃



ℎ,𝑘,𝜌,𝜏



and 𝑡

𝑝,𝑞,𝜎,𝜔,so

𝑢

𝑣





ℎ,𝑘,𝜌,𝜏



𝑝,𝑞,𝜎,𝜔



⎩

⎪

⎨

⎪

⎧



ℎ

𝑝

𝑘

𝑞

ℎ𝜎  𝑝𝜌

𝑝



𝑘𝜔  𝑞𝛽

𝑞



,𝑠̃0 and 𝑡

0



𝑘

𝑞

ℎ

𝑝

𝑘𝜎 𝑝𝜏

𝑞



ℎ𝜔  𝑞𝜌

𝑞



,𝑠̃0 and 𝑡

0



ℎ

𝑞

𝑘

𝑝

𝑞𝜌  ℎ𝜔

𝑞



𝑝𝜏 𝑘𝜎

𝑝



,𝑠̃0 and 𝑡

0



𝑘

𝑞

ℎ

𝑝

𝑘𝜔 𝑞𝜏

𝑞



ℎ𝜎  𝑝𝜌

𝑝



,𝑠̃0 and 𝑡

0

Furthermore, the inverse value that the writer

obtained will be compared with the inverse value by

Vijayalakshmi. Vijayalakshmi (2011), defines

𝑠̃







ℎ,𝑘,𝜌,𝜏









ℎ



,𝑘



,𝑛ℎ



𝜏,𝑛𝑘



𝜌



for positive 𝑛



ℎ



,𝑘



,𝑛ℎ



𝜏,𝑛𝑘



𝜌



for negative 𝑛

From the equation given by Vijayalakshmi an inverse

value was obtained

𝑠̃





ℎ

𝑘

𝜏

ℎ



𝜌

𝑘



.

Given 𝑠̃ℎ,𝑘,𝜌,𝜏 will be show that 𝑠̃⊗𝑠̃



𝐼



a) Using the inverse value given by Vijayalakshmi

. 𝑠̃⊗𝑠̃



ℎ,𝑘,𝜌,𝜏⊗

























ℎ

ℎ

,𝑘

𝑘

,ℎ



𝜏

ℎ





𝜌

ℎ

,𝑘



𝜌

𝑘





𝜏

𝑘



1,1,

𝜏𝜌

ℎ

𝜌𝜏

𝑘





1,1,0,0



𝚤̃



b) Using the inverse value that the author obtained

𝑠̃⊗𝑠̃



ℎ,𝑘,𝜌,𝜏⊗

ℎ

𝑘

𝜌

ℎ



𝜏

𝑘









ℎ

ℎ

,𝑘

𝑘

,ℎ



𝜌

ℎ







ℎ



𝜌



,𝑘



𝜏

𝑘







𝑘



𝜏









1,1,0,0



𝚤̃



As an illustration given an example of the following

fully fuzzy linear equation, then it will be solved

using the Cramer method. And the end will be using

two of inverse that author sugest and invers from

Vijayalakshmi.



1,2,3,5



𝑥







2,3,4,5



𝑥



7,26,37,88



1,3,4,5



𝑥







2,4,6,7



𝑥



7,36,44,115

The equation can be changed in the form of a matrix

as follows

𝐴



⊗𝑥𝑏







1,2,3,5



2,3,4,5





1,3,4,5



2,4,6,7



⊗

𝑥



,𝑦



,𝜌



,𝜏



𝑥



,𝑦



,𝜌



,𝜏







7,26,37,88

7,36,44,115



Obtained

det𝐴







7,6,42,46



𝐴













7,26,37,88



2,3,4,5





7,36,44,115



2,4,6,7





det𝐴













94,90,641,650



𝐴













1,2,3,5



7,26,37,88





1,3,4,5



7,36,44,115





det𝐴













71,65,459,475



Will be solve with two of inverse value such as:

a) Using the inverse value that the author obtained

𝑥





det𝐴









det𝐴



det𝐴









⊗det𝐴











94,90,641,650



⊗

1

42

46







13.43,15,11,6.67



𝑥





det𝐴









det𝐴



det𝐴









⊗det𝐴











71,65,459,475



⊗

1

42

46







10.14,10.83,4.71,3.89



𝑥



13.43,15,11,6.67





10.14,10.83,4.71,3.89





b) Using the inverse value given by Vijayalakshmi

𝑥





det𝐴









det𝐴



det𝐴









⊗det𝐴











94,90,641,650



⊗

1







13.43,15,179.82,213.33



𝑥





det𝐴









det𝐴



det𝐴









⊗det𝐴











71,65,459,475



⊗

1







10.14,10.83,132.22,155



𝑥



13.43,15,179.82,213.33





10.14,10.83,132.22,155





From the two solutions above it can be seen that the

center values of a and b are the same, but the inverse

value that the author obtained provides smaller right

and left area.

IMC-SciMath 2019 - The International MIPAnet Conference on Science and Mathematics (IMC-SciMath)

240

4 CONCLUSIONS

From the previous explanation it can be concluded

that the element of identity must be distinguished

between pure identity



1,1,0,0



with identity

1,1,𝜀



,𝜀



 as well as for the similarity of fuzzy

numbers. Furthermore, by defining the positivity

and negativity of fuzzy numbers in multiplication

cases will give a better result.

REFERENCES

Jafarian, A. (2016). New decomposition method for solving

dual fully fuzzy linear system. International Journal

Fuzzy Computation and Modelling, 2, 76–85.

Radhakrishnan, S., Sattanathan, R., & Gajivaradhan, P.

(2012). LU decomposition method for solving fully

fuzzy linear system with trapezoidal fuzzy numbers.

Bonfring International Journal of Man Machine

Interface, 2, 1–3.

Vijayalakshmi, V. (2011). ST decomposition method for

solving fully fuzzy linear system using Gauss-Jordan

for trapezoidal fuzzy matrices. International

Mathematical Forum, 6, 2245–2254.

Zadeh, L. A. (1965). Fuzzy sets. Information and Control,

8, 338–353.

Alternative Algebra for Trapezoidal Fuzzy Number and Comparison with Various Other Algebra

241