RETRACTED: Elementary Operations on L-R Fuzzy Number

doi:10.4236/apm.2015.53016

1

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Title of retracted article: Elementary Operations on L-R Fuzzy Number

Author(s): Abdul Alim*, Fatema Tuj Johora, Shohel Babu, Abeda Sultana

* Corresponding author. abdulalim@buft.edu.bd

Journal: Advances in Pure Mathematics

Year: 2015

Volume: 5

Number: 3

Pages (from - to): 131 - 136

DOI (to PDF): http://dx.doi.org/10.4236/apm.2015.53016

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Advances in Pure Mathematics, 2015, 5, 131-136

Published Online March 2015 in SciRes. http://www.scirp.org/journal/apm

http://dx.doi.org/10.4236/apm.2015.53016

How to cite this paper: Alim, A., Johora, F.T., Babu, S. and Sultana, A. (2015) Elementary Operations on L-R Fuzzy Number.

Advances in Pure Mathematics, 5, 131-136. http://dx.doi.org/10.4236/apm.2015.53016

Elementary Operations on L-R Fuzzy

Number

Abdul Alim1, Fatema Tuj Johora2, Shohel Babu2, Abeda Sultana3

1Mathematics, BGMEA University of Fashion and Technology, Dhaka, Bangladesh

2Mathematics, IUBAT—International University of Business Agriculture and Technology, Dhaka, Bangladesh

3Department of Mathematics, Jahanginagar University, Dhaka, Bangladesh

Email: abdulalim@buft.edu.bd

Received 26 February 2015; accepted 13 March 2015; published 19 March 2015

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativ ecommon s.org/l icenses/by/4. 0/

Abstract

The aim of this paper is to find the formula for the elementary operations on L-R fuzzy number. In

this paper we suggest and describe addition, subtraction, multiplication and division of two L-R

fuzzy numbers in a brief.

Keywords

Fuzzy Number, L-R Fuzzy Number, Membership Function

1. Introduction

A fuzzy set [1] A on



, set of real numbers is called a fuzzy number [2] which satisfies at least the following

three properties:

1)

A

must be a normal fuzzy set [3].

2)

A

α

must be a closed interval for every

(

]

0, 1

α

∈

.

3) The support [1] of A,

0

A

+

must be bounded.

The fundamental idea of the L-R representation of fuzzy numbers is to split the membership function

( )

i

pi

x

µ



of a fuzzy number

i

p



into two curves

( )

i

li

x

µ

and

( )

i

ri

x

µ

, left and right of the modal value

i

x

.

The member ship fu nction

( )

i

pi

x

µ



can be expressed through parameterized reference functions or shape func-

tion L and R in t he form

( )

for

i

ii

lii i

i

pi

ii

rii i

i

xx

x Lxx

xxx

x Rxx

µα

µ

µβ



−

= <



 

=

−

= ≥









(1)

Retracted

A. Alim et al.

132

where

i

x

is the mod al val ue o f the membership f u ncti o n and

i

α

and

i

β

are the sp re a ds cor re spo nd in g to the

left-hand and right-hand cur ve of the membership function [4] respectively.

As an abbreviated notation, we can de fine an L -R fuzz y nu mber

i

p



with the member ship f unctio n

( )

i

pi

x

µ



in (1) by

,

,,

i iii

LR

px

αβ

=



(2)

where the subscripts L and R specify the reference functions [5].

2. Operations on L-R Fuzzy Number

In this section, the formulas for the elementary operations (addition, subtraction, multiplication, division) [5]

between L-R fuzzy numbers [5] will be presented.

2.1. Addition of L-R Fuzzy Number

Suppose two fuzzy numbers

1

p



and

2

p



, represented as L-R fuzzy numbers of the form

111122 2 2

,,

an,, ,,d

LR LR

px px

αβ αβ

= =



(3)

The sum

( )

121 2

,

a

E ppqpp= =+

 

is agai n an L-R fuzzy number of the form

,

,,

LR

qz

αβ

=



(4)

with the moda l value

12

zx x= +

(5)

and the spreads

12 12

and

αααββ β

=+=+

(6)

In short we can write

1112 2 2121212

,, ,

,,,,, ,

LR LRLR

x xxx

αβαβα αββ

+ =+++

(7)

The left-hand reference functions of both fuzzy numbers

1

p



and

2

p



have to be given by L, and the right-

hand reference functions by R.

The formula o f the L-R add ition in (7) is motivated by the fo llowing ways :

We first consider the right-hand curves

( )

11rx

µ

and

( )

22r

x

µ

of the L-R fuzzy numbers

1

p



and

2

p



with

( )( )

12

112 2

12

and

rr

xx xx

xR xR

µµ

ββ

 

−−

= =

 

 

(8)

The degree of membership

[ ]

*

0, 1

µ

∈

is take n on for t he argument values

( )( )

*1* *1*

11122 2

andxx RxxR

βµ βµ

−−

=+=+

(9)

This implies

( )

** *1*

1 2 121 2

z xx xxR

ββ µ

−

=+=+++

(10)

and we obtain for the right-hand curve

( )

r

z

µ

of the fuzzy number

q



( )

*

** 121 2

with and

rzz

z Rzxx

µ µβββ

β



−

== =+=+





(11)

The same rea s oning holds for the left -hand curves of

1

p



,

2

p



and

q



, and we get

( )

1212

with and

l

zz

z Lzxx

µ ααα

α

−



= =+=+





(12)

Retracted

A. Alim et al.

133

2.2. Subtraction of L-R Fuzzy Number

Suppose two fuzzy numbers

1

p



and

2

p



, represented as L-R fuzzy numbers of the form

111122 2 2

,,

an,, ,,d

LR LR

px px

αβ αβ

= =



(13)

The opposite

p−

of the L-R fuzzy number is defined as

1,,

,, ,,

LR RL

px x

αβ βα

−=− =−



(14)

Now by using (7) we can deduce the following formula for the subtraction

( )

121 2

,

s

qE pppp== +

 

of the

L-R fuzz y numb er s:

1112 22121212

,, ,

,,,,, ,

LR LRLR

x xxx

αβαβαββ α

−=−+ +

(15)

2.3. Multiplication of L-R Fuzzy Number

Let us consider two positive fuzzy numbers

1

p



and

2

p



of the same L-R type given by the L-R representations

111122 2 2

,,

an,, ,,d

LR LR

px px

αβ αβ

= =



(16)

We can construct the right-hand curve

( )

r

z

µ

of the product

( )

1 212

,

m

qEp ppp= =

 

on the basis of the

right-hand curves

( )( )

12

112 2

12

and

rr

xx xx

xR xR

µµ

ββ

 

−−

= =

 

 

(17)

of L-R fuzzy numbers

1

p



and

2

p



. In accordance with the deduction of the formula for the L-R addition, the

degree of membership

[ ]

0, 1

µ

∗

∈

is take n on for t he argument values

( )( )

11

111 222

andxxRxxR

βµ βµ

∗−∗ ∗−∗

=+=+

(18)

This implies

( )

( )( )

2

* **1*1*

121212 2112

z xx xxxxRR

ββµββµ

−−



==++ +



(19)

Two approximations have been proposed, which is referred to as tangent approximation and secant approxi-

mation in the following:

2.3.1. Tangent Approximation

Let

1

α

and

2

α

are small compared to

1

x

and

2

x

and

*

µ

is in the neighborhood of 1. Then we can neglect

the quadratic term

( )

2

1

R

µ

−∗





in (19) and we obtain for the right-hand curve

( )

r

z

µ

of the approximated

product

t

q



an expr ession of the fo rm

( )

*

** 121 221

with and

r

zz

zRz xxx x

µµββ β

β



−

= ===+





(20)

Usin g the sa me r easonin g for the left-ha nd cur ves of

1

p



,

2

p



and

t

q



, we deduce the following formula for

the multiplication of L-R fuzzy numbers

111 2221212211221

,, ,

,, ,,,,

LR LRLR

xxxx xxxx

αβαβαα ββ

≈ ++⋅

(21)

2.3.2. Secant Approximation

If the spreads are not negligible compared to the modal values

1

x

and

2

x

, the rough shape of the product

12

q pp=

 

can be estimated by approximating quadratic term

( )

2

1*

R

µ

−





in (19) by the linear ter m

( )

1*

R

µ

−





. This gives the right-hand cur ve

( )

r

z

µ

of the approximated product

s

q



in the fo rm

( )

*

** 1212 21 12

with and

r

zz

zRz xxxx

µµβββββ

β



−

====++





(22)

Retracted

A. Alim et al.

134

With t he sa me rea soni ng for t he le ft-hand cur ves of

1

p



,

2

p



and

s

q



, the overall formula for t he multiplica-

tion of L-R fuzzy numbers results in

11122212122112 122112

,, ,

,, ,,,,

LR LRLR

xxxx xxxx

α βαβααααββββ

≈+− +⋅ +

(23)

2.4. Division of L-R Fuzzy Number

An appropriate formulation for the quotient

( )

121 2

,

d

q E pppp= =



of two L-R fuzzy numbers

1

p



and

2

p



can be obtained by reducing the division of the fuzzy numbers

1

p



and

2

p



to the multiplication of the divi-

dend

1

p



with the inverse

1

22

1pp

−

=



of the divisor

2

p



.

When we consider a fuzzy number

p



which is either positive or negative, i.e.,

( )

0suppp∉

, given by the

L-R representa tion

,

,, LR

px

αβ

=



the ta ngent approximation

( )

1

t

p

−



for the inverse

1

p

−



is defined by

( )

11

22

,

1,,

tRL

pp

xxx

βα

−−

= ≈



and the secant approximation

( )

1

s

p

−



by

( )

() ()

11

,

1,,

sRL

pp

xxx xx

βα

−−

= ≈

+−



Usin g t he a b ove menti o ned id e ntit y

1

1 212

p ppp

−

=

 

as well as the approxi mation for mulas for the multiplic a-

tion of L-R fuzzy numbers on one side and those for the inverse of an L-R fu zz y nu mber on the other, a number

of different approximated L-R repr e sentations for the quo tient

12

pp



can be formulated.

3. Example

We consider two L-R fuzzy number

12

,,

2,1,1 and4,2,4

ll ll

pp= =



Then using Equation (7 ) we get

12 ,,

24,1 2,1 46,3,5

ll ll

ppq+==+++ =

 

Also can be written in the form

( )

0; 3

3;3 6

3tfn6,3,5 .

11 ; 611

5

0; 11

x

xx

qxx

x

≤



−

<<



= =

−

≤<



≥





Using (15) we get

12 ,,

24,1 4,1 22,5,3

ll ll

ppq−==−++ =−

 

Also can be written in the form

( )

0; 7

7;7 2

5tfn2,5,3 .

1;21

3

0; 1

x

xx

qxx

x

≤−



+

− <<−



== −

−

−≤<



≥





Retracted

A. Alim et al.

135

If we use the tangent approximation the product

12

q pp=

 

is approximated by the triangular L-R fuzzy

number

( )

,

0; 0

; 08

8

8,8,12tnf 8,8,1220 ; 820

12

0; 20

tll

x

xx

qxx

x

≤





<<



===−

≤<



≥





Again in the case of secant approximation the result

12

q pp=

 

is approximated by

( )

,

0; 2

2;2 8

6

8,6,16tnf 8,6,1624;824

16

0; 24

sll

x

xx

qxx

x

≤



−

<<



===−

≤<



≥





If we use the tangent approximation the inverse 1

2

p− is approximated by the triangular L-R fuzz y numbe r

( )

1

2,,

1 42111

, ,,,

4 16 16448

tll ll

p

−

= =

Thus

( )

1,,,

2

,

11111 11 1

2,1,1,,2,21,21

4484448 4

1

0; 4

4111

;

131 131

,,tfn ,,3 42

242 2421

21 ;1

2

0; 1

tt ll ll ll

ll

pq

p

x

xx

x

=== ++

≤−



+

− <<





===





− ≤<



⋅⋅ ⋅⋅⋅⋅

≥







But if we use the seca nt approximation the inve rse

1

2

p−

is approximated by the triangular L-R fuzzy number

( )

() ( )

1

2,

,

1 42111

, ,,,

4444442484

sll

ll

p

−

= =

++

Thus

1,,,

2

,

11111 111 11

2,1,1,,2,21 1,21 1

48448 484 44

0;1 8

81

; 1812

13133

, ,1tfn, ,132

28 28; 1232

2

0;3 2

ss llll ll

ll

pq

p

x

xx

x

=== +− ++

≤



−

<<





=== 



−



≤<

⋅⋅⋅ ⋅⋅⋅



⋅

≥

⋅









4. Conclusion

In this paper we have present ed exact calculation formulas for a ddition, subtraction, multiplica tion and division

Retracted

A. Alim et al.

136

of two L-R fuzz y n u mbe r s. F i nal l y we ha ve ta ken two L -R fuzzy numbers as an example and obtained results of

addition, subtraction, multiplication and division. We have reviewed some research papers with proper refer-

ences.

References

[1] Klir, G.J. an d Yuan, B. (1995) Fuzzy Sets and Fu zzy Logic, Theory and Applications. Prentice Hall Inc., Upper Saddle

River.

[2] Dubis, D. and Prade, H. (1980) Fuzzy Sets and Systems, Theory and Applications. Academic P ress, New York.

http://www.jstor.org/stable/2030767

[3] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[4] Hanss, M. (2005) Applied Fuzzy Arithmetic—An Introduction with Engineering Applications. Springer-Verlag, Berlin

Heidelberg.

[5] Zimmermann, H.J. (1995) Fuzzy Set Theory and its Applications. Third Edition, Kluwer Academic Publishers, Boston.

Retracted

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