Applied Mathematics, 2011, 2, 10391045 doi:10.4236/am.2011.28144 Published Online August 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Solution of the Fuzzy Equation A + X = B Using the Method of Superimposition Fokrul Alom Mazarbhuiya1, Anjana Kakoti Mahanta2, Hemanta K. Baruah3 1Department of Com p ut er Sci ence, College of Computer Science, King Khalid University, Abha, Saudi Arabia 2Department of Com p ut er Sci ence, Gauhati University, Assam, India 3Department of Stat i s t i c s, Gauhati University, Assam, India Email: fokrul_2005@yahoo.com, anjanagu@yahoo.co.in, hemanta_bh@yahoo.com Received March 7, 2011; revised June 23, 2011; accepted July 1, 2011 Abstract Fuzzy equations were solved by using different standard methods. One of the wellknown methods is the method of cut. The method of superimposition of sets has been used to define arithmetic operations of fuzzy numbers. In this article, it has been shown that the fuzzy equation AX B , where A, X, B are fuzzy numbers can be solved by using the method of superimposition of sets. It has also been shown that the method gives same result as the method of cut. Keywords: Fuzzy Number, Possibility Distribution, Probability Distribution, Survival Function, Superimposition of Sets, Superimposition of Intervals, Cut Method 1. Introduction Fuzzy equations were investigated by Dubois and Prade [1]. Sanchez [2] p ut forward a solu tion of fuzzy equation by using extended operations. Accordingly various re searchers 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 DaTe [8], Zhao and Gobind [9], Wang and Ha [10]). Klir and Yuan [11] solved the fuzzy equations X where A, X and B are fuzzy numbers, by using the me thod of cut. Mazarbhuiya et al. [12] defined the arithmetic opera tions 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 XB without utilising the standard methods. Ou r method is based on the op era tion of superimposition of sets. It will be shown in this article that our method for the solution of equation XB 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 equifuzzy interval arithmetic. In Section 5, we discuss our proposed method of solution XB . In Section 6, we give brief conclusion of the work and lines for fu ture work. 2. 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 ,; xAxxE where 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 0,1Ax. A fuzzy set A is said be normal if its membership function x is unity for at least one E . An cut A of a fuzzy set A is an ordinary set of elements with membership not less than for 01 . This means ;AxEAx
1040 F. A. MAZARBHUIYA ET AL. 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 pA and is defined as the set of elements with membership nonzero i.e., sup; 0pAx EAx A fuzzy number A, denoted by a triad [a,b,c] such that 0 aA c and 1 b, where x for[,] ab is called the left reference function and for [,] bc is called right reference function. The left reference func tion is right continuous monotone and nondecreasing where as the right reference function is left continuous, monotone and nonincreasing. The above definition of a fuzzy number is called LR fuzzy number [15]. We would call a fuzzy set A( ) over the support A equifuzzy if all elements of A( ) are with membership where 01 . The operation of superimposition S of equifuzzy sets A( ) and B( ) is defined as [13] ASBAABA B BAB where , 0, 1 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 ** BA B where ,are cuts of A and B, B (0,1] and * is the arithmetic operation on A and B. In the case of divi sion for any 0 B(0,1] . The resulting fuzzy number * B is expressed as **AB AB (see e.g.[11]) (1) 3. Solution of the Fuzzy Equation XB by Using the Method of Cut For any(0,1] . Let12 , aa , and 12 ,Bbb 1 ,2 xx denote, respectively, the cuts of 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 11 2 baba 2(0,1] and 2) 1111222 babababa 2 Property 1) ensures that the interval equation XB has a solution, which is 112 2 , baba . Property 2) ensures that the solution of the interval equations for and are nested i.e. if then X . if a solution exists for every (0,1] and property 2) is satisfied, then by (2.1) the solution X of the fuzzy equation is [0,1] X X (2) where . xX x 4. EquiFuzzy Interval Arithmetic The usual interval arithmetic can be generalized for equifuzzy intervals. If 11 [,] ab and 22 [,]Bab , we denote interval addition and interval subtraction as 1212 () [,] Baabb and 1221 () , Babab Accordingly, () ()() 1212 () ,ABaabb () ()() 1221 () ,ABabab Let now, (1), (2) be the ordered values of 1, 2 in ascending magnitude, Then (1/2) (1/2)(1/2)(1/2) 112 21 122 (1/2)(1)(1/2) (1)(2)(2)(1)(1)(2) (1/2) (1)(1/2) (1)(2)(2) (1)(1) (2) (1/2) (1)(1)(2)(2) ,,(),, ,,, () ,,, , ab SabcdScd aaabbb cccd dd acac a (1) (2)(2)(1)(1) (1/2) (1)(1) (2)(2) , , cbd bdbd (3) where 2 1, ii iab , 2 1, ii icd Similarly, (1/2)(1/2)(1/2) (1/2) 112 21 122 (1/2)(1) (1/2) (1)(2)(2)(1)(1) (2) (1/2)(1)(1/2) (1)(2)(2)(1)(1)(2) (1/2) (1)(2)(2)(1) ,,(),, ,,, () ,,, , ab SabcdScd aaabbb cccd dd adad a (1) (2)(1)(1)(2) (1/2) (1)(2) (2)(1) , , db c bcbc (4) Copyright © 2011 SciRes. AM
F. A. MAZARBHUIYA ET AL. 1041 In the next section, we shall use (3) and (4) to find the solution X of the fuzzy equation XB . 5. Solution of the Fuzzy Equation XB by Using the Method of Superimposition Let 12 are sample realisations from the uniform population 11 and 12 ,,, n aa a [,uv],,, n bb b are sample realisa tions from the uniform population . 11 We denoteas the superimpositions of equi fuzzy intervals [, ; with membership (1/n) i.e. [, ]vw , n n Gab i ab] i1, 2,,i (1/ ) (1/ )(1/ ) 112 2 (1/) (2/) (1)(2) (2)(3) ((1)/ )(1) (1)()() (1) (11/ )(2/ ) (1)(2)(2)( 1) (1/ ) (1) () ,, ,, ,, ,, ,..., , n nn nn nn nn nn n n nn n nn GababSa bSSa b aa aa aa ab bbb b bb Ha ,(say)b (5) where are ordered values of 12 ,,, n aa a 12 ,,, n aa a 12 ,,, n bb b and are ordered values of 12 ,,, n bb b in ascending magnitude. Here n 1[,] ii iab From (5), we get the membership functions are the combination of empirical probability distribution function and complementary probability distribution function re spectively as (1) 1(1) () 0, 1, 1, rr n xa r() axa n xa and (1) 2(1) () 1, 1 1, 0, rr n xb r() bxb n xb It is known that the GlivenkoCantelli lemma of Order Statistics [16] states that the mathematical expectation of empirical distribution function is the theoretical probabil ity distribution function and that of empirical comple mentary probability distribution the theoretical survival function. Thus 11 ,ExPu and 2 1,Ex Pv 1 x (6) where 1 1 11 11 1 0, ,, 1, xu xu Pu xuxv vu xv 1 is the uniform probability distribution function on . and 11 [,]uv 1 1 11 11 1 0, ,, 1, xv xv Pv xvxw wv xw 1 is the uniform probability distribution function on . 11 From (5) using (6) we get the membership grades in [, ]vw ,Gabwhich is nothing but , ab can be estimated by the membership function 11 111 11 111 11 0, , , 1, xuxw xu Axux v vu xv vxw wv (7) where 111 [,, ] uvw is a fuzzy number. Again let 12 n ,,, xx are sample realisations from the uniform population 22 [, and 12 ]uv,,, n yy y [,v are sample realisations from the uniform population ]. 22 We denote w ,yGx as the superimposition of equi fuzzy intervals [, ] ii y;1,2,,in with membership (1/n) i.e. (1/ ) (1/) (1/) 1122 (1/ )(2/ ) (1)(2) (2)(3) (( 1)/)(1)(11/) (1) ()()(1)(1)(2) (2/) (1/) (2) (1)(1)() ,, ,, ,,... ,,, ... ,, n nn nn nn nn n nn n nn nn nn Gxyx ySxySSxy xx xx xxxy yy yy yy Hx ,y (8) where 12 ,,, n xx , n are the ordered values of 12 ,, xx yy and 12n ,,,y are the ordered values of 12 ,,, n yyy in ascending order of magnitude and here x Copyright © 2011 SciRes. AM
1042 F. A. MAZARBHUIYA ET AL. 1, n ii ixy Here the empirical probability distribution function and empirical complementary distribution function are respectively given by (1) 3(1) () 0, 1, 1, rr n xx r() xxx n xx and (1) 4(1) () 0, 1 1, 1, rr n xy r() yxy n xy By Glivenko Cantelli lemma of order statistics, we get 32 ,ExPu x 2 x and 4 1,Ex Pv (9) where 2 2 22 22 2 0, ,, 1, xu xu Pu xuxv vu xv 2 is the uniform probability distribution function on [u2,v2]. And 2 2 22 22 2 0, ,, 1, xv xv Pv xvxw wv xw 2 is the uniform probability distribution function on . 22 From (8) using (9) we get the membership grades in G(x,y) which is nothing but [, ]vw , xy can be estimated by the membership function 22 222 22 222 22 0, , , 1, xuxw xu Xxuxv vu xv vxw wv e22 2 [,, ] uvw wher is also a fuzzy number. It was assumed that 1, n ii ixy . Again let 12 ,,, n cc c population are sample realisations from the uniform and 33 [,]uv 12 ,,, n dd d pulation [,vw are sample realisationsorm po We denote from the unif33 ]. ,Gcd as the superimposition of equi intervals fuzzy ] i ; 1,2,,in[, i cd with me (1/n) i. mbership e. (10) where (1/ ) ,, n Gcd cdS (1/ ) (1/ ) 112 2 (1/ (2) (1) () / ) (1)(2) (2)(1) (1/ ) (1) () , , ... , ,... , n n nn n nn nn nn n nn Sc dSc d c d d dd Hc ,d )(2/) (1)(2) (3) ((1)/ )(1) ,, n nn cc c c ()(1) , n ccd (1 1(2/ ) ,d d 12 ,,, n cc c , n c are the ordered values of 12 ,,cc and dd 12 ,,, n d are the ordered values of 12 ,,, n dd d in ascend n ing order of magnitude and here 1, ii icd . Here the empirical probability distribution function and empirical complementary distribution function given by are respectively (1) 5(1)() 0, rr xc () 1, n 1, r cxc n xc and (1) 6(1)() () 0, 1 1, 1, rr n xd r dxd n xd antelli lemma of order statistics, we get By Glivenko C 53 ,ExPu x and 63 1,Ex Pv x (11) where Copyright © 2011 SciRes. AM
F. A. MAZARBHUIYA ET AL. 1043 3 3 33 33 3 3 ,, 1, xu Pv xux v vu xv is the uniform probability distribution function on 3 3 0, xu [u,v]. and 3 3 33 33 3 0, ,, 1, xv xv Pv xvxw wv xw 3 is the uniform probability distribution function on [v3,w3]. From (10) using (11) we get the membership grades in ,Gcd which is nothing but , cd can be esti the membership func tion mated by 33 333 33 0, ,xuxw xu xw where is a fuzzy number. It was assumed that 333 33 1, xv v wv ,Bx ux v vu 33 3 [,, ]Buvw 1, n ii icd . The given equation can be written as Replacing the values of , and and using the equif interval arithm we i.e. ,(),,GabGxyGcd ,Gab uzzy ,Gxy etic, ,Gcd get (1/) (2/) (2)) (3)(3) ((1)/ ) ( 1)( 1)()() (1 1/ () , . ,] nn nn nnnn n axax axax a (2/ ) (2)(2)(1)(1) (1/ ) (1) (1)()() , , n nnnn n nnnn byby byby (1)(1) (2)(2)(2 ,axax (( 1)/ ) ( 1)( 1)()() ... ,.. rn rr rr ax ax (1) ) ()(1)(1)(1)(1)(2)(2) ,, . n n x bybyby .. ,, axby Hcd Using the equality of equifuzzy intervals, we get i and i which gives i ii axc ii byd; 1, 2,,in. ii ca and This implies (12) The left side of the identity (12) is whose membership function ii ydb; i1, 2,,in. (1/) (2/) (1) (2)(2) (3) ((1)/)((1)/ ) (1) ()(1) () (1)(1 1/) () (1)(1) (2) (2/) (1/) (2)(1) (1)() (1)(1) (2) ,, ,, ,, ,, , nn rn nn rr nn n n nn nn nn xx xx xx xx xy yy yy yy cac (1/)(2/ ) (2)(2) (2)(3) (3) ((1)/ ) (1)(1)()() ( 1)( 1)() (1)(11/) (1)(1) (1)(1)(2)(2) ) ( , , nn rn rrrr nn n n r acaca caca dbdbdb db (()/ ) 1) ()() (1/ ) ( 1)( 1)()() , , nr n rr n nnnn db dbdb (( 1)/ ) ,nn ca ca ()n ( )() ,, nn ca (1r ,Gxy x 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) 7(1)(1)() () () 0, 1, 1, rr r nn xc a r()r ca xca n xc a and (1) (1) 8(1)(1) () () 0, 1 1, 1, rrr nn xd b r()()r dbxdb n xdb By Glivenko Cantelli Lemma of order Statistics P 731 ,Ex uux and 83 1,Ex Pvv 1 x (13) where 31 31 3131 31 31 31 31 0, ,, 1, xu u xuu Pu uxu uxvv vv uu xv v is the uniform probability distribution function on 1 ] 33 [,uuvv and Copyright © 2011 SciRes. AM
1044 F. A. MAZARBHUIYA ET AL. 31 31 31 31 31 31 31 0, ,, 1, xv v xvv Puu xvvx3 ww vv xw w is ] From (13), we get the solution of the equation w the uniform probability distribution function on 313 1 [,vvww. XB as 31313 1 ,, uuvvww (14) where 313 1 31 31 31 31 313 1 31 31 0, , , 1, xu uxww xuu Xxuux vv vv uu vvxww ww vv Obviously, 31 3 1 xvv 111313131 323 ,,, , Xuvw uuvvww uv wB From the Equation (14) , we g et 31313 1 ,, uuvvww is a fuzzy number whose cut is given by 3131 3 3131 31 Xuuvvuu wwww vv he solution of 1 which is t XB Obviously 3131 31 [0,1] 3131 31 ,Xuuvvuu wwww vv that is similar to the Equation (2). Thus, we can conclude that the method of superimpo sition e result as given by the method 6. Conclusion and Lines for Future Works In new method of solv ing fuzzy equation gives the sam cut. of this article, we have presented a XB. The method is based on he set superimpoation. The set superimposi ethod has bee tsition oper tion mn used to define the arithmetic op erations on fuzzy numbers. It has been found that arithmetic operation based on set superimposition opera tion 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 XB gives the sismilar results a given by other ethods. In future we would like solve other equation namely fuzzy differential equa integral equation etc. using same method. p. 129146. ed p. standard m kind of fuzzy tion, fuzzy 7. References [1] 84, p D. Dubois and H. Prade, “Fuzzy Set Theoretic Differ ences and Inclusions and Their Use in The analysis of Fuzzy Equations,” Control Cybern (Warshaw), Vol. 13, 19 [2] E. Sanchez, “Solution of Fuzzy Equations with Extend Operations,” Fuzzy Sets and Systems, Vol. 12, 1984, p 273248. doi:10.1016/01650114(84)90071X [3] J. J. Buckley, “Solving Fuzzy Equations,” Fuzzy Sets and Systems, Vol. 50, No. 1, 1992, pp. 114. doi:10.1016/01650114(92)90199E [4] J. Wasowski, “On Solutions to Fuzzy Equations,” Con trol and Cybern, Vol. 26, 1997, pp. 653658. [5] L. Biacino and A. Lettieri, “Equation with Fuzzy Num bers,” Information Sciences, Vol. 47, No. 1, 1989, pp. 6376. [6] H. Jiang, “The Approach to Solving Simultaneous Linear ions That Coefficients Are Fuzzy Numbers,” Jour nal of National University of Defence Technology (Chi nese), Vol. 3, 1986, pp. 96102. [7] J. J. Buckley and Y. Qu, “Solving Linear and Quadr Equations,” Fuzzy Sets and Systems, Vol. 38, No.1, 1990 Equat atic , 10.1016/01650114(90)90099Rpp. 4859. doi: nd T. DaTe, “A Calculation Method [8] M. F. Kawaguchi a for Solving Fuzzy Arithmetic Equation with Triangular Norms,” Proceedings of 2nd IEEE International Confer ence on Fuzzy Systems (FUZZYIEEE), San Francisco, 1993, pp. 470476. [9] R. Zhao and R. Govind, “Solutions of Algebraic Equa tions Involving Generalised Fuzzy Number,” Information Sciences, Vol. 56, 1991, pp. 199243. doi:10.1016/00200255(91)90031O [10] X. Wang and M. Ha, “Solving a System of Fuzzy Linear Equations,” In: M. Delgado, J. Kacpryzyk and A. Vila, Eds., Fuzzy Optimisatio, J. L. Verdegay n: Recent Advances, uzzy Logic azarrbhuiya, A. K. Mahanta and H. K. Baruah, ition and Its Application PhysicaVerlag, Heildelberg, 1994, pp. 102108. [11] G. J. Klir and B. Yuan, “Fuzzy Sets and F Theory and Applications,” Prentice Hall of India Pvt. Ltd., Delhi, 2002. [12] F. A. M “Fuzzy Arithmetic without Using the Method of Cuts,” Bulletin of Pure and Applied Sciences, Vol. 22 E, No. 1, 2003, pp. 4554. [13] H. K. Baruah, “Set Superimpos to the Theory of Fuzzy sets,” Journal of Assam Science Copyright © 2011 SciRes. AM
F. A. MAZARBHUIYA ET AL. Copyright © 2011 SciRes. AM 1045 ation 0255(83)900257 Society, Vol. 10, No. 12, 1999, pp. 2531. [14] G. Q. Chen, S. C .Lee and S. H. Yu Eden, “Applic Sett of Fuzzy Set Theory to Economics,” In: P. P. Wang, Ed., Advances in Fuzzy Sets, Possibility Theory, and Applica tions, Plenum Press, New York, 1983, pp. 277305. [15] D. Dubois and H. Prade, “Ranking Fuzzy Numbers in the ing of Possibility Theory,” Information Science, Vol. 30, No. 3, 1983, pp. 183224. doi:10.1016/0020 [16] M. Loeve, “Probability Theory,” Springer Verlag, New York, 1977.
