Elementary Operations on L-R Fuzzy Number

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.


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

R e t r a c t e d
where i x is the modal value of the membership function and i α and i β are the spreads corresponding to the left-hand and right-hand curve of the membership function [4] respectively.As an abbreviated notation, we can define an L-R fuzzy number i p  with the membership function where the subscripts L and R specify the reference functions [5].

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.

Addition of L-R Fuzzy Number
Suppose two fuzzy numbers 1 p  and 2 p  , represented as L-R fuzzy numbers of the form The sum ( ) with the modal value and the spreads In short we can write 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 of the L-R addition in (7) is motivated by the following ways: We first consider the right-hand curves ( ) The degree of membership and we obtain for the right-hand curve ( ) with and The same reasoning holds for the left-hand curves of 1 p  , 2 p  and q  , and we get ( ) with and R e t r a c t e d

Subtraction of L-R Fuzzy Number
Suppose two fuzzy numbers 1 p  and 2 p  , represented as L-R fuzzy numbers of the form The opposite p −  of the L-R fuzzy number is defined as Now by using (7) we can deduce the following formula for the subtraction ( )

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 We can construct the right-hand curve ( ) 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 is taken on for the argument values ( ) ( ) This implies ( ) ( ) ( ) Two approximations have been proposed, which is referred to as tangent approximation and secant approximation in the following:

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 ( ) ) and we obtain for the right-hand curve ( ) r z µ of the approximated product t q  an expression of the form ( ) with and Using the same reasoning for the left-hand curves of 1 p  , 2 p  and t q  , we deduce the following formula for the multiplication of L-R fuzzy numbers

Secant Approximation
If the spreads are not negligible compared to the modal values 1 x and 2 x , the rough shape of the product 1 2 q p p =    can be estimated by approximating quadratic term ( ) . This gives the right-hand curve ( ) r z µ of the approximated product s q  in the form ( ) with and

R e t r a c t e d
With the same reasoning for the left-hand curves of 1 p  , 2 p  and s q  , the overall formula for the multiplica- tion of L-R fuzzy numbers results in

Division of L-R Fuzzy Number
An appropriate formulation for the quotient ( ) and the secant approximation ( ) Using the above mentioned identity   as well as the approximation formulas for the multiplica- tion of L-R fuzzy numbers on one side and those for the inverse of an L-R fuzzy number on the other, a number of different approximated L-R representations for the quotient 1 2 p p   can be formulated.

Example
We consider two L-R fuzzy number Again in the case of secant approximation the result p − is approximated by the triangular L-R fuzzy number ( ) But if we use the secant approximation the inverse 1 2 p − is approximated by the triangular L-R fuzzy number ( ) ( ) ( )

Conclusion
In this paper we have presented exact calculation formulas for addition, subtraction, multiplication and division R e t r a c t e d through parameterized reference functions or shape function L and R in the form r a c t e d