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

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Appled Mathematcs 0 09-045 do:046/am0844 Publshed Ole August 0 (http://scrporg/joural/am Soluto of the Fuzzy Equato A + X = B Usg the Method of Supermposto Fokrul Alom Mazarbhuya Ajaa Kakot Mahata

1 Appled Mathematcs do:046/am0844 Publshed Ole August 0 ( Soluto of the Fuzzy Equato A + X = B Usg the Method of Supermposto Fokrul Alom Mazarbhuya Ajaa Kakot Mahata Hemata K Baruah Departmet of Computer Scece College of Computer Scece Kg Khald Uersty Abha Saud Araba Departmet of Computer Scece Gauhat Uersty Assam Ida Departmet of Statstcs Gauhat Uersty Assam Ida E-mal: fokrul_005@yahoocom ajaagu@yahooco hemata_bh@yahoocom Receed March 7 0; resed Jue 0; accepted July 0 Abstract Fuzzy equatos ere soled by usg dfferet stadard methods Oe of the ell-ko methods s the method of -cut The method of supermposto of sets has bee used to defe arthmetc operatos of fuzzy umbers I ths artcle t has bee sho that the fuzzy equato A X B A X B are fuzzy umbers ca be soled by usg the method of supermposto of sets It has also bee sho that the method ges same result as the method of -cut Keyords: Fuzzy Number Possblty Dstrbuto Probablty Dstrbuto Sural Fucto Supermposto of Sets Supermposto of Iterals -Cut Method Itroducto Fuzzy equatos ere estgated by Dubos ad Prade [] Sachez [] put forard a soluto of fuzzy equato by usg exteded operatos Accordgly arous researchers hae proposed dfferet methods for solg the fuzzy equatos [see eg Buckley [] Wasosk [4] Baco ad Letter [5] After ths a lot research papers hae appeared proposg solutos of arous types of fuzzy equatos z algebrac fuzzy equatos a system of fuzzy lear equatos smultaeous lear equatos th fuzzy coeffcets etc usg dfferet methods ([see eg Jag [6] Buckley ad Qu [7] Kaaguch ad Da-Te [8] Zhao ad Gobd [9] Wag ad Ha [0] Klr ad Yua [] soled the fuzzy equatos A X B A X ad B are fuzzy umbers by usg the method of -cut Mazarbhuya et al [] defed the arthmetc operatos z addto ad subtracto of fuzzy umbers th out usg the method of -cuts e usg a method called supermposto of sets troduced by Baruah [] I ths artcle e ould put forard a procedure of solg a fuzzy equato A X B thout utlsg the stadard methods Our method s based o the operato of supermposto of sets It ll be sho ths artcle that our method for the soluto of equato A X B ges same result as ge by the method of -cut The paper s orgased as follos I Secto e dscuss about the deftos ad otatos used ths artcle I Secto e dscuss the soluto of fuzzy equato by -cut method I Secto 4 e dscuss about equ-fuzzy teral arthmetc I Secto 5 e dscuss our proposed method of soluto A X B I Secto 6 e ge bref cocluso of the ork ad les for future ork Deftos ad Notatos We frst ree certa stadard deftos Let E be a set ad let x be a elemet E The a fuzzy subset A of E s characterzed by A xax ; x E A x s the grade of membershp of x A A(x s commoly called the fuzzy membershp fucto of the fuzzy set A For a ordary set A(x s ether 0 or hle for a fuzzy set A x 0 A fuzzy set A s sad be ormal f ts membershp fucto Ax s uty for at least oe x E A -cut A of a fuzzy set A s a ordary set of elemets th membershp ot less tha for 0 Ths meas A xe; A x Copyrght 0 ScRes

2 040 F A MAZARBHUIYA ET AL A fuzzy set s sad to be coex f all ts -cuts are coex sets (see eg [4] A fuzzy umber s a coex ormal fuzzy set A defed o the real le such that A(x s pecese cotuous The support of a fuzzy set A s deoted by sup p A ad s defed as the set of elemets th membershp ozero e sup p A xe; A x 0 A fuzzy umber A deoted by a trad [abc] such that Aa 0 Ac ad Ab A x for x [ ab ] s called the left referece fucto ad for x [] bc s called rght referece fucto The left referece fucto s rght cotuous mootoe ad o-decreasg as the rght referece fucto s left cotuous mootoe ad o-creasg The aboe defto of a fuzzy umber s called L-R fuzzy umber [5] We ould call a fuzzy set A ( oer the support A equ-fuzzy f all elemets of A ( are th membershp 0 The operato of supermposto S of equ-fuzzy sets A ( ad B ( s defed as [] B A B A SB AAB AB 0 ad the operato + stads for uo of dsjot sets fuzzy or otherse The arthmetc operato usg the method of -cut o to fuzzy umbers A ad B s defed by the formula A* B A* A B are -cuts of A ad B (0] ad * s the arthmetc operato o A ad B I the case of dso 0 B for ay (0] The resultg fuzzy umber A* B s expressed as A* B A* B (see eg[] ( Soluto of the Fuzzy Equato AX B by Usg the Method of -Cut For ay (0] Let A a a B b b ad X x x deote respectely the -cuts of A B ad X the ge equato (see eg Klr ad Yua [] The the ge equato has a soluto f a oly f b a b a for eery (0] ad b a b a b a b a Property esures that the teral equato A X B B has a soluto hch s X b a b a Property esures that the soluto of the teral equatos for ad are ested e f the X X f a soluto X exsts for eery (0] ad property s satsfed the by ( the soluto X of the fuzzy equato s X X ( X x X x [0] 4 Equ-Fuzzy Iteral Arthmetc The usual teral arthmetc ca be geeralzed for equ-fuzzy terals If A [ a b] ad B [ a b] e deote teral addto ad teral subtracto as A( B [ a a b b ] ad A( B a b a b Accordgly ( ( ( A ( B aa b b ( ( ( A ( B ab a b Let o ( ( be the ordered alues of ascedg magtude The a b Sa b ( c d Sc d (/ (/ (/ (/ a a a b b b (/ ( (/ ( ( ( ( ( ( ( c c c d d d (/ ( (/ ( ( ( ( ( ( (/ a( c( a( c ( a b d b d ( ( ( ( (/ a b c d Smlarly c b d ( ( ( ( a b Sa b ( c d Sc d (/ (/ (/ (/ (/ ( (/ a( a ( a( b ( b( b ( ( (/ ( (/ c( c ( c( d ( d( d ( (/ a( d( a( d ( a b c b c ( ( ( ( (/ d b c ( ( ( ( ( ( ( (4 Copyrght 0 ScRes

3 F A MAZARBHUIYA ET AL 04 I the ext secto e shall use ( ad (4 to fd the soluto X of the fuzzy equato A X B 5 Soluto of the Fuzzy Equato AX B by Usg the Method of Supermposto Let a a a are sample realsatos from the uform populato [ u ] ad b b b are sample realsatos from the uform populato [ ] We deote G a b as the supermpostos of equfuzzy terals [ a b ]; th membershp (/ e (/ (/ (/ G a b a b S a b S S a b (/ (/ ( ( ( ( a a a a (( / ( a( a ( a( b ( (/ (/ b( b ( b( b ( (/ b( b ( Ha b (say (5 a a a are ordered alues of a a a ad b b b are ordered alues of b b b ascedg magtude Here [ a b] From (5 e get the membershp fuctos are the combato of emprcal probablty dstrbuto fucto ad complemetary probablty dstrbuto fucto respectely as ad 0 x a( r x a( r x a ( r x a( x b( r x b( r xb ( r 0 x b( It s ko that the Gleko-Catell lemma of Order Statstcs [6] states that the mathematcal expectato of emprcal dstrbuto fucto s the theoretcal probablty dstrbuto fucto ad that of emprcal complemetary probablty dstrbuto the theoretcal sural fucto Thus x E x P u ad E x P x (6 0 x u x u Pu x u x x u s the uform probablty dstrbuto fucto o [ u ] ad 0 x x P x x x s the uform probablty dstrbuto fucto o [ ] From (5 usg (6 e get the membershp grades Ga b hch s othg but H ab ca be estmated by the membershp fucto 0 xu x x u x x A x u x u A [ u ] s a fuzzy umber Aga let x x x are sample realsatos from the uform populato [ u ] ad y y y are sample realsatos from the uform populato [ ] We deote G x y as the supermposto of equfuzzy terals [ x y ] ; th membershp (/ e (/ (/ (/ G x y x y S x y S S x y (/ (/ x( x ( x( x ( (( / ( (/ x( x ( x( y ( y( y ( (/ (/ y( y ( y( y ( Hx y (8 x x x are the ordered alues of x x x ad y y y are the ordered alues of y y y ascedg order of magtude ad here (7 Copyrght 0 ScRes

4 04 F A MAZARBHUIYA ET AL x y Here the emprcal probablty dstrbuto fucto ad emprcal complemetary dstrbuto fucto are respectely ge by ad ad 0 x x( r x x( r x x ( r x x( 0 x y( r 4 x y( r x y ( r x y( By Gleko Catell lemma of order statstcs e get E x P u x E x P x 4 (9 0 x u x u P u x u x u x s the uform probablty dstrbuto fucto o [u ] Ad 0 x x P x x x s the uform probablty dstrbuto fucto o [ ] From (8 usg (9 e get the membershp grades G(xy hch s othg but H xy ca be estmated by the membershp fucto 0 xu x x u x x X x u x u her e X u [ ] It as assumed that x y s also a fuzzy umber Aga let c c c are sample realsatos from the uform populato [ u ] ad d d d are sample realsatos from the uform populato [ ] We deote G c d as the supermposto of equ- fuzzy terals [ c d ]; th membershp (/ e (/ (/ (/ S c d S c d G c d c d S (/ (/ c ( c( c( c ( (( / ( c( c ( c( d ( (0 c c c are the ordered alues of c c c ad d d d are the ordered alues of d d d ascedg order of magtude ad here c d (/ (/ d( d ( d( d ( d d Hc d ( ( (/ Here the emprcal probablty dstrbuto fucto ad emprcal complemetary dstrbuto fucto are respectely ge by ad 0 x c( r x c x c x c( 5 ( r ( r 0 x d( r x d x d x d( 6 ( r ( r By Gleko Catell lemma of order statstcs e get ad x E x P u 5 E x P 6 x ( Copyrght 0 ScRes

5 F A MAZARBHUIYA ET AL 04 0 x u x u P x u x u x s the uform probablty dstrbuto fucto o [u ] ad 0 x x P x x x s the uform probablty dstrbuto fucto o [ ] From (0 usg ( e get the membershp grades Gc d hch s othg but H cd ca be estmated by the membershp fucto It as assumed that 0 xu x x u B x u x u B [ u ] x x s a fuzzy umber c d The ge equato ca be rtte as Replacg the alues of G a b G x y ad Gc d ad usg the equ-f uzzy teral arthm etc e get e a x a x G a b ( G x y G c d (/ (/ a x a x ( ( ( ( ( ( ( ( a( r x( r a( r x ( r (( / a( x( a( x ( ] ( (/ a( x( b( y( b( y( b( y ( (/ b y b y ( ( ( ( b y b y ( ( ( ( H a x b y H c d (/ (( r/ Usg the equalty of equ-fuzzy terals e get a x c ad b y d ; hch ges x c a ad y d b ; Ths mples (/ (/ x( x ( x( x ( (( r/ (( / x( r x ( r x( x ( ( (/ x( y ( y( y ( (/ (/ y( y ( y( y ( (/ (/ c ( a( c( a( c( a( c( a( (( r/ c( r a( r c( r a ( r (( / c( a( c a( ( ( (/ c d b ( ( d b a d b d b ( ( ( ( ( ( d ( r ( r ( r ( r d b d b ( ( ( ( b (/ (( r/ ( The left sde of the detty ( s Gx y hose membershp fucto X x s estmated by (9 ad from the rght sde e get the emprcal probablty dstrbuto fucto ad sural fucto as ad 0 xc( a( 7 r x c( r a( r xc( r a r x c( a( 0 x d( b( r x d b x d b r x d( b( 8 ( r ( r ( r ( By Gleko Catell Lemma of order Statstcs ad P E x u u x 7 E x P 8 x ( 0 xu u xu u Pu u x u u x u u x s the uform probablty dstrbuto fucto o [ u u ad ] ( Copyrght 0 ScRes

6 044 F A MAZARBHUIYA ET AL 0 x x Pu u x x x s the uform probablty dstrbuto fucto o [ ] From ( e get the soluto of the equato A X B as X u u (4 0 xu u x xu u X x u u x u u x x Obously u B A X u u u From the Equato (4 e get X u u s a fuzzy umber hose -cut s ge by X u u u u hch s the soluto of A X B Obously [0] X u u u u that s smlar to the Equato ( Thus e ca coclude that the method of supermposto ges the same result as ge by the method of -cut 6 Cocluso ad Les for Future Works I ths artcle e hae preseted a e method of solg fuzzy equato A X B The method s based o the set supermposto operato The set supermpos- has bee used to defe the arthmetc op- to method eratos o fuzzy umbers It has bee foud that the arthmetc operato based o set supermposto operato ges the same result as ge by other stadard method z the method of -cut I ths artcle e hae sho that our method of soluto of fuzzy equato A X B ges the smlar results a s ge by other stadard methods I future e ould lke sole other kd of fuzzy equato amely fuzzy dfferetal equategral equato etc usg same to fuzzy method 7 Refereces [] D Dubos ad H Prade Fuzzy Set Theoretc Dffereces ad Iclusos ad Ther Use The aalyss of Fuzzy Equatos Cotrol Cyber (Warsha Vol 984 pp 9-46 [] E Sachez Soluto of Fuzzy Equatos th Exteded Operatos Fuzzy Sets ad Systems Vol 984 pp 7-48 do:006/065-04( x [] J J Buckley Solg Fuzzy Equatos Fuzzy Sets ad Systems Vol 50 No 99 pp -4 do:006/065-04(99099-e [4] J Wasosk O Solutos to Fuzzy Equatos Cotrol ad Cyber Vol pp [5] L Baco ad A Letter Equato th Fuzzy Numbers Iformato Sceces Vol 47 No 989 pp 6-76 [6] H Jag The Approach to Solg Smultaeous Lear Equatos That Coeffcets Are Fuzzy Numbers Joural of Natoal Uersty of Defece Techology (Chese Vol 986 pp 96-0 [7] J J Buckley ad Y Qu Solg Lear ad Quadratc Equatos Fuzzy Sets ad Systems Vol 8 No 990 pp do: 006/065-04( R [8] M F Kaaguch ad T Da-Te A Calculato Method for Solg Fuzzy Arthmetc Equato th Tragular Norms Proceedgs of d IEEE Iteratoal Coferece o Fuzzy Systems (FUZZY-IEEE Sa Fracsco 99 pp [9] R Zhao ad R God Solutos of Algebrac Equatos Iolg Geeralsed Fuzzy Number Iformato Sceces Vol pp 99-4 do:006/ (9900-o [0] X Wag ad M Ha Solg a System of Fuzzy Lear Equatos I: M Delgado J Kacpryzyk J L Verdegay ad A Vla Eds Fuzzy Optmsato: Recet Adaces Physca-Verlag Heldelberg 994 pp 0-08 [] G J Klr ad B Yua Fuzzy Sets ad Fuzzy Logc Theory ad Applcatos Pretce Hall of Ida Pt Ltd Delh 00 [] F A Mazarrbhuya A K Mahata ad H K Baruah Fuzzy Arthmetc thout Usg the Method of -Cuts Bullet of Pure ad Appled Sceces Vol E No 00 pp [] H K Baruah Set Supermposto ad Its Applcato to the Theory of Fuzzy sets Joural of Assam Scece Copyrght 0 ScRes

7 F A MAZARBHUIYA ET AL 045 Socety Vol 0 No pp 5- [4] G Q Che S C Lee ad S H Yu Ede Applc ato of Fuzzy Set Theory to Ecoomcs I: P P Wag Ed Adaces Fuzzy Sets Possblty Theory ad Applcatos Pleum Press Ne York 98 pp [5] D Dubos ad H Prade Rakg Fuzzy Numbers the Settg of Possblty Theory Iformato Scece Vol 0 No 98 pp 8-4 do:006/ ( [6] M Loee Probablty Theory Sprger Verlag Ne York 977 Copyrght 0 ScRes

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