Solution of the Fuzzy Equation A + X = B Using the Method of Superimposition

Fokrul Alom Mazarbhuiya, Anjana Kakoti Mahanta, Hemanta K. Baruah Department of Computer Science, College of Computer Science, King Khalid University, Abha, Saudi Arabia Department of Computer Science, Gauhati University, Assam, India Department of Statistics, Gauhati University, Assam, India E-mail: fokrul_2005@yahoo.com, anjanagu@yahoo.co.in, hemanta_bh@yahoo.com Received March 7, 2011; revised June 23, 2011; accepted July 1, 2011


Introduction
Fuzzy equations were investigated by Dubois and Prade [1].Sanchez [2] put forward a solution of fuzzy equation by using extended operations.Accordingly various researchers have proposed different methods for solving the fuzzy equations [see e.g.Buckley [3], Wasowski [4], Biacino and Lettieri [5].After this a lot research papers have appeared proposing solutions of various types of fuzzy equations viz.algebraic fuzzy equations, a system of fuzzy linear equations, simultaneous linear equations with fuzzy coefficients etc. using different methods ([see e.g.Jiang [6], Buckley and Qu [7], Kawaguchi and Da-Te [8], Zhao and Gobind [9], Wang and Ha [10]).Klir and Yuan [11] solved the fuzzy equations A X B   where A, X and B are fuzzy numbers, by using the method of -cut.
Mazarbhuiya et al. [12] defined the arithmetic operations viz.addition and subtraction of fuzzy numbers with out using the method of -cuts i.e. using a method called superimposition of sets introduced by Baruah [13].
In this article, we would put forward a procedure of solving a fuzzy equation A X B   without utilising the standard methods.Our method is based on the operation of superimposition of sets.It will be shown in this article that our method for the solution of equation A X B   gives same result as given by the method of -cut.
The paper is organised as follows.In Section 2 we discuss about the definitions and notations used in this article.In Section 3, we discuss the solution of fuzzy equation by -cut method.In Section 4, we discuss about equi-fuzzy interval arithmetic.In Section 5, we discuss our proposed method of solution A X B   .In Section 6, we give brief conclusion of the work and lines for future work.

Definitions and Notations
We first review certain standard definitions.Let E be a set, and let x be an element in E. Then a fuzzy subset A of E is characterized by where   A x is the grade of membership of x in A. A(x) is commonly called the fuzzy membership function of the fuzzy set A. For an ordinary set A(x) is either 0 or 1, while for a fuzzy set      .An -cut  A of a fuzzy set A is an ordinary set of elements with membership not less than  for 0 1 A fuzzy set is said to be convex if all its -cuts are convex sets (see e.g.[14]).A fuzzy number is a convex normal fuzzy set A defined on the real line such that A(x) is piecewise continuous.
The support of a fuzzy set A is denoted by sup   p A and is defined as the set of elements with membership nonzero i.e., is called the left reference function and for [ , ] x b c  is called right reference function.The left reference function is right continuous monotone and non-decreasing where as the right reference function is left continuous, monotone and non-increasing.The above definition of a fuzzy number is called L-R fuzzy number [15].
We would call a fuzzy set A () over the support A equi-fuzzy if all elements of A () are with membership  where 0 1    .The operation of superimposition S of equi-fuzzy sets A () and B () is defined as [13]  where , and the operation '+' stands for union of disjoint sets, fuzzy or otherwise.
The arithmetic operation using the method of -cut on two fuzzy numbers A and B is defined by the formula where A, B and X in the given equation (see e.g.Klir and Yuan [11]).Then the given equation has a solution if an only if 1) for every Property 1) ensures that the interval equation Property where

Equi-Fuzzy Interval Arithmetic
The usual interval arithmetic can be generalized for equi-fuzzy intervals.If [ , ] B a b  , we denote interval addition and interval subtraction as (1/ 2) , , , where (1/ 2) , (1/ 2) In the next section, we shall use ( 3) and ( 4) to find the solution X of the fuzzy equation A X B   .

Solution of the Fuzzy Equation   A X B by Using the Method of Superimposition
Let 1 2 are sample realisations from the uniform population 1 1 and 1 2 , , , n a a a  are sample realisations from the uniform population .
[ , ] v w ( , where are ordered values of , , , n a a a   ( 1 ) It is known that the Glivenko-Cantelli lemma of Order Statistics [16] states that the mathematical expectation of empirical distribution function is the theoretical probability distribution function and that of empirical complementary probability distribution the theoretical survival function.Thus where x u is the uniform probability distribution function on .and is the uniform probability distribution function on .
1 1 From ( 5) using ( 6) we get the membership grades in H a b can be estimated by the membership function where We denote as the superimposition of equifuzzy intervals [ , ]   i i  Here the empirical probability distribution function and empirical complementary distribution function are respectively given by   (1)

4
( 1 ) By Glivenko Cantelli lemma of order statistics, we get where From (8) using ( 9) we get the membership grades in G(x,y) which is nothing but wher is also a fuzzy number.
It was assumed that , , , are the ordered values of ( 1 ) ( ) 0, r r x c
From ( 10) using (11) we get the membership grades in where is a fuzzy number.
It was assumed that  .
The given equation can be written as
Using the equality of equi-fuzzy intervals, we get i and i which gives This implies The left side of the identity ( 12) is whose membership function is estim from the right side, we get the em tribution function and survival function as ated by ( 9) and pirical probability dis-  (1) (1) where 13), we get the solution of the equation w the uniform probability distribution function on where From the Equation ( 14), we get   that is similar to the Equation ( 2).Thus, we can conclude that the method of superimposition e result as given by the method 

Conclusion and Lines for Future Works
In new method of solving fuzzy equation gives the sam -cut. of this article, we have presented a A X   B. The method is based on he set superimpo ation.The set superimposiethod has bee t sition oper tion m n used to define the arithmetic operations on fuzzy numbers.It has been found that arithmetic operation based on set superimposition operation gives the same result as given by other standard method viz. the method of -cut.In this article, we have shown that our method of solution of fuzzy equation the A X B   gives the si s milar results a given by other ethods.In future we would like solve other equation namely fuzzy differential equaintegral equation etc. using same method.

From ( 5 )
, we get the membership functions are the combination of empirical probability distribution function and complementary probability distribution function respectively as 2) ensures that the solution of the interval equations for  and  are nested i.e. if   where   