Arithmetic of Triangular Fuzzy Variable from Credibility Theory

Transcription

1 Vol., Issue 3, August 0 Arithmetic of Triagular Fuzzy Variable from Credibility Theory Ritupara Chutia (Correspodig Author) Departmet of Mathematics Gauhati Uiversity, Guwahati, Assam, Idia. Rituparachutia7@rediffmail.com Supahi Mahata Departmet of Statistics Gauhati Uiversity, Guwahati, Assam, Idia. supahi_mahata@rediffmail.com D. Datta Health Physics Divisio, BARC, Mumbai, Idia ddatta@barc.gov.i Abstract I this article, it has bee show that the credibility distributio of triagular fuzzy variable leads us to fid a very simple alterative method of fidig the membership fuctio for fuctios of triagular fuzzy variables. This cocept of credibility theory also leads us i fidig a alterative method of computig basic arithmetic operatios of triagular fuzzy variables ad geeralised membership fuctio for the root of triagular fuzzy variable. The method has bee demostrated with the help of some eamples. Keywords: Credibility distributio, Credibility desity fuctio, Credibility measure, Membership fuctio.. Itroductio The basic arithmetic operatio of fuzzy umber has bee developed well through the years. Ideed Chou [4, 5] has actually developed a method of fidig the membership fuctio of the square root ad cube root of a triagular fuzzy umber. Although there are may arithmetical operatio approaches, oe of the approaches presets a geeralised th method for fidig the root of the fuzzy umber. Zadeh [7] proposed the cocept of possibility measure ad thereafter this has bee widely used i solvig fuzzy problems. The ecessity measure is defied as a dual part of possibility measure. However, both possibility measure ad ecessity measure are ot self-dual. I order to defie a self-dual measure, Liu ad Liu [] preset the cocept of credibility measure. I this paper the triagular fuzzy variable are cosidered ad we try to develop a geeralised method for fidig the membership fuctio for fuctios of triagular fuzzy variables, ad very well applied i dealig the basic arithmetical operatios of triagular fuzzy variables, based o the cocept of credibility distributio. 9

2 Vol., Issue 3, August, 0. Prelimiaries.. Triagular Fuzzy Variable A fuzzy variable determied by the triplet [ ad whose membership fuctio is give by of crisp umber with a b c a, if a b b a c, if b c (..) c b is called a triagular fuzzy variable... Credibility Measure Let be a o-empty set, ad P the power set of ad A P, Liu ad Liu [] defied the credibility measure as Cr{ A} ( Pos{ A} Nec{ A}) (..) Furthermore, for ay A P we have Pos{ A} mi( Cr{ A},) (..) Li ad Liu [8] defied that a set fuctio Cr is a credibility measure if it holds the followig. Cr { }.. Cr{ A} Cr{ B} wheever A B C 3. Cr{ A} Cr{ A } for ay evet A. 4. Cr A } sup Cr{ A } for ay evets A } with sup Cr { } { i i i i { i i A i Let be a fuzzy variable defied o the credibility space (, P, Cr). The its membership fuctio is defied from the credibility measure by mi(cr{ },). (..3).3. Credibility Distributio Liu [] defied credibility distributio as : R [0, ] of ay fuzzy variable as Cr{ : ( ) }. 0

3 Vol., Issue 3, August 0 That is the credibility that the fuzzy variable takes a value less tha or equal to. If the fuzzy variable is give by a membership fuctio, the its credibility distributio is determied by {sup y ( sup y ( }, R..4. Credibility Distributio of Triagular Fuzzy Variable The credibility distributio of a triagular fuzzy variable [ is give by 0, if a a, if a b Cr ( b a) ( ) { } (.4.) c b, if b c ( c b), if c.5. Credibility Desity Fuctio The credibility desity fuctio defied by Liu [3] as : R [0, ) of ay fuzzy variable is a fuctio such that ( dy, R..6. Credibility Desity Fuctio of Triagular Fuzzy Variable, The credibility desity fuctio of a triagular fuzzy variable [ is give by f, if a b ( b a) g ( ) ( ), if b c (.6.) ( c b) 0, otherwise 3. The Membership Fuctio for Fuctios of Triagular Fuzzy Variable Let [,( c 0), be a triagular fuzzy variable. Let F( ) [ F( a), F( b), F( c)], be the fuzzy variable of the fuctio F (). Suppose the membership fuctio of is give by (..). The credibility distributio fuctio is as give i (.4.) ad the credibility desity fuctio is as give i (.6.).

4 Vol., Issue 3, August, 0 Let y F() or, ( d or, d ( ( ) m(, say. Replacig by ( dy dy f () ad g (), we obtai f ( ad g (, say. The the credibility distributio fuctio of F ( ) is 0,, F ( a) if ( m( dy, F ( c) F( a) if F( a) F( b) ( m( dy, if F( b) F( c) if F( c) Thus the membership fuctio of the fuzzy variable F () is give by ( ), if F( a) F( b) ) ( ), if F( b) F( c) F ( )( i 3.. Iverse of a Triagular Fuzzy Variable Cosider a triagular fuzzy variable [,( c 0), with membership fuctio give i (..) ad let [ c, b, a ]. The credibility distributio fuctio ad the credibility desity fuctio are as give i (.4.) ad (.6.) respectively. Let y d so that. The credibility distributio of is dy y 0, if c dy, if c b c ( c b) y dy, if b a a ( b a) y, if a The membership fuctio of is c if c b ( c b) a ( ), if b a ( b a) 3.. Negative of a Triagular Fuzzy Variable Cosider a triagular fuzzy variable [,( c 0), with membership fuctio as give i (..) ad let [ c, a]. The credibility distributio fuctio ad the credibility desity fuctio are as give i (.4.) ad (.6.) respectively.

5 Vol., Issue 3, August 0 Let d y, so that. The credibility distributio of is dy The the membership fuctio of 0, if c c, if c b ( c b) a b, if b a ( b a), if a ( ) is c, if c b c b a, if b a a b 3.3. Square Root of a Triagular Fuzzy Variable Cosider a triagular fuzzy variable [,( c 0), with membership fuctio is as give i (..), ad let [. Accordigly the credibility distributio ad the credibility desity fuctio are as give i (.4.) ad (.6.) respectively. d Let y so that y. The credibility distributio fuctio for, is dy 0, if a a, if a b ) ( b a) c b, if b c ( c b), if c ( The the membership fuctio of the fuzzy variable is a, if a b a c, if b c b b c 3.4. th Root of a Triagular Fuzzy Variable Cosider a triagular fuzzy variable [,( c 0), with membership fuctio as give i (..), ad let [. Accordigly the credibility 3

6 Vol., Issue 3, August, 0 distributio ad the credibility desity fuctio are as give i (.4.) ad (.6.) respectively. Let d y so that y. The credibility distributio for is dy The the membership fuctio of 0, if a a, if a b ( b a) c b, if b c ( c b), if c th root of the fuzzy variable is a, if a b a c, if a c b 4. Arithmetic of Triagular Fuzzy Variables 4.. Additio of Fuzzy Variable Cosider the fuzzy variables [ ad [ p, r]. Suppose Z [ a p, b c r] be the fuzzy umber of. Let the membership fuctio of ad be () ad (, where, L if a R if b b ( ), ), c (4..) L y if p y R y if q y q ( ( ), ), r (4..) Let the credibility distributio of the triagular fuzzy variables (4..) ad (4..) are 0, if a, if a b ( ), if b c, if c (4..3) 0, if y p ( (, if p y q (, if q y r, if y c (4..4) b b 4

7 Vol., Issue 3, August 0 Let the credibility desity fuctio of the credibility distributio fuctio (4..3) is f if a b ( ), (4..5) g( ), if b c We start with equatig ( ) with ( ad ( ) with (. Ad so, we obtai y ad y respectively. Let z y, so we have z ad z, so that ad, say. The credibility desity fuctio (4..5) should be trasformed i terms of the fuctio of z by replacig by ad z z, say. i f ad g, so that ad Now let, d d dz dz distributio fuctio of is 0, a, f g z m z, ad z m z ( m ( dz, p cr d d. The credibility dz dz if a p if a p b q ( m( dz, if b q c r if c r Thus the membership fuctio of additio of the fuzzy variables ca be easily foud from the credibility distributio fuctio (). 4.. Subtractio of Fuzzy Variable Cosider the triagular fuzzy variables [ ad [ p, r]. Suppose Z, the the membership fuctio of Z is give by Z ( ) Multiplicatio of Fuzzy Variables Cosider the triagular fuzzy variables [,( c 0) ad [ p, r],( p, r 0). Suppose Z. [ a. p, b. c. r] be the fuzzy variable of.. Let the membership fuctio of ad be () ad ( as metioed i (4..) ad (4..) respectively. Let the credibility distributio of (4..) ad (4..) are as metioed i (4..3) ad (4..4) respectively. Let the credibility desity fuctio of the credibility distributio fuctio (4..3) is (4..5). We start with equatig ( ) with ( ad ( ) with (. Ad so, we obtai y ad y respectively. Let z. y, so we have z. ad z., so that ad, say. The credibility desity fuctio (4..5) should be trasformed i terms of the fuctio of z by replacig by ad i z z, say. f ad g, so that ad f g 5

8 Vol., Issue 3, August, 0 d d d d Now let, z m z ad z mz. The credibility dz dz dz dz distributio fuctio of., is give by 0, if a. p ( m ( dz, if a. p b. q a. p. ( m( dz if b. q c. r c.. r, if c. r Thus the membership fuctio of multiplicatio of the fuzzy variables ca be easily foud from the credibility distributio fuctio ( ) Divisio of Fuzzy Variables. Cosider the triagular fuzzy variables [,( c 0) ad [ p, r],( p, r 0). Suppose Z the membership fuctio of Z is give by Z.. 5. Numerical Eamples Eample Cosider two triagular fuzzy variables [,6,8] ad [5,6,8 ] with membership fuctio as give below, ad let Z 3, 8, , if , if y 5, if 5 y 6 8 y (, if 6 y 8 The credibility distributio fuctio ad credibility desity fuctio of () are as below respectively. 0, if, if 6 8 4, if 6 8 4, if 8 (, if z 6 8 ( ( ) z, if 6 z 8 4 6

9 Vol., Issue 3, August 0 The credibility distributio fuctio of is foud to be d dz 0, if 8 8y, if 8 y 6 4 ( y 7y, if 6 y 5 y, if y 5 Now equatig the credibility distributio fuctios of ad 56 ( z ) m z ad similarly m z distributio fuctio of d dz Z. is of the form. 0, if 3, if 7 4, if ( ), if 056 ( z ) 3/ 3/ 8/ 3 8/ 3 8/ 5, we have 8/ 5. Thus the credibility Thus the membership fuctio of the fuzzy variable. is, Eample. 4 6, if 3/ 8/ 3 8 5, if 8/ 3 8/ 5 Cosider two triagular fuzzy variables [7,4,9] ad [3,5,0] with membership fuctio as give below, ad let Z [3,9,6]. 7, if , if y 3, if 3 y 5 0 y (, if 5 y 0 7

10 Vol., Issue 3, August, 0 The credibility distributio fuctio ad credibility desity fuctio of () are as below respectively. 0, if 7 7, if , if 4 9 0, if 9 (, if 7 z 4 8 ( ( ) z, if 4 z 9 0 The credibility distributio fuctio of is foud to be 0, if 0 y 0, if 0 y 5 0 ( y 7, if 5 y 3 4, if y 3 Now equatig the credibility distributio fuctios of ad d 7 dz fuctio of Z ( ) d 5 dz 7 is of the form m z ad similarly m z ( ) 0, if 3 3, if 3 9 4, if 9 6 4, if 6 Thus the membership fuctio of the fuzzy variable ( ) is Eample ( ) 3, if 3 9 6, if 9 6 7, we have. Thus the credibility distributio Cosider two triagular fuzzy variables [,3,5 ] ad [3,5,6 ] with membership fuctio as give below, ad let Z. [6,5,30], if 3 5, if 3 5 0, otherwise 8

11 Vol., Issue 3, August 0 y 3, if 3 y 5 ( 6 y, if 5 y 6 0, otherwise The credibility distributio fuctios of () ad ( are as below respectively. 0, if, if 3, if 3 5 4, if 5 0, if y 3 y 3, if 3 y 5 ( 4 y 4, if 5 y 6, if y 6 The credibility desity fuctio of the credibility distributio fuctio () (, if z 3 ( ( ) z, if 3 z 5 4 Now equatig the credibility distributio fuctios of ad, we have d dz 8z distributio fuctio of d dz Z. is of the form m z ad similarly m z. 49 8z 0, if 6 7 8, if , if , if 30 Thus the membership fuctio of the fuzzy variable. is. Thus the credibility 7 8, if ( ), if , otherwise These eamples have bee quoted from the book by Bojadziev ad Bojadziev [6], the result tallies with it, where it has bee solved by the method of -cuts. 9

12 Vol., Issue 3, August, 0 6. Coclusio Here we have tried to develop a alterative method of fidig the membership fuctio for fuctios of triagular fuzzy variable from the cocept of credibility theory. A ew method for computatio of basic arithmetical operatios of fuzzy variable is forwarded. This method validity has bee tested evaluatig some eamples which were quoted from [6], where they were solved by the method of alpha-cuts ad are compared with the results obtaied here. The square root of triagular fuzzy variable foud by our proposed method th tallies with the oe defied by Chou []. A geeralised membership fuctio for the root of triagular fuzzy variable has bee forwarded. The method ca be applied i solvig equatios with fuzzy coefficiets. Further the proposed method ca be applied to the ucertaity aalysis of dispersio models ad egieerig problems which ca be take for further research. Refereces [] B. Liu ad. K. Liu, Epected Value of Fuzzy Variable ad Fuzzy Epected Value Models, IEEE Trasactios o Fuzzy Systems, Vol. 0, No. 4, 00, pp [] B. Liu, Theory ad Practice of Ucertai Programmig, Physica-Verlag, Heidelberg, 00. [3] B. Liu, Ucertaity Theory, 3 rd Editio, 008, [4] Chie-Chag Chou, The Cube Roots of Triagular Fuzzy Number, ICIC Epress Letters. Vol. 3, Nos., 009, pp [5] Chie-Chag Chou, The Square Roots of Triagular Fuzzy Number, ICIC Epress Letters. Vol. 3, Nos., 009, pp [6] G. Bojadziev ad M. Bojadziev, Fuzzy Sets, Fuzzy Logic, Applicatios. World Scietific Publishig Co. Pte. Ltd, Sigapore, 995. [7] L. Zadeh, Fuzzy Sets as a basis for a theory of Possibility, Fuzzy Sets ad Systems, Vol., 978, pp-3-8. [8]. Li, ad B. Liu, A Sufficiet ad Necessary Coditio for Credibility Measures, It. Joural of Ucertaity, Fuzziess & Kowledge-Based Systems, Vol. 4, No. 5, 006, pp