A Note on Complement of Trapezoidal Fuzzy Numbers Using the α-cut Method

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1 Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN - Vol. - A Note on Complement of Trpezoidl Fuzzy Numers Using the α-cut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment of Mthemtics T.B.M.L. College Poryr Indi E-mil: dsdin@rediffmil.com E-mil: kjeevgn@gmil.com Astrct In this pper fuzzy rithmetic involving trpezoidl fuzzy numers nd using the α-cut method is pplied on the complement of fuzzy numers. Some properties of the complement of fuzzy numers re lso studied using the ove discussed rithmetic. Relevnt numericl emples re included to illustrte our results. Keywords: Fuzzy numers Trpezoidl fuzzy numers α-cut method complement of fuzzy numers. Received: July 7 Revised: August Accepted nd pulished: Septemer. Introduction Fuzzy numers re of gret importnce in fuzzy systems. To use fuzzy numers in ny system we must e le to perform rithmetic opertions on these quntities. One of the most sic concepts tht cn e used to generlize rithmetic opertions on fuzzy numers is the etension principle [ 9 ]. A good overview of fuzzy numers ws presented y Duois et l. []. A specilized ook on fuzzy rithmetic with fuzzy numers ws written y Kufmnn nd Gupt [5]. Some theoreticl detils nd pplictions of fuzzy quntities nd especilly of fuzzy numers cn e found in Duois nd Prde

2 Dingr & Jivgn [] s well s in Fuller nd Mesir []. In ddition M et l. [8] present fuzzy numers with new prmetric forms nd provide fuzzy rithmetic sed on this representtion. The crucil point in fuzzy modeling is to ssign memership functions corresponding to fuzzy numers tht represent vgue concepts nd imprecise terms epressed often in nturl lnguge. The prolem of constructing meningful memership functions is difficult one nd numerous methods for their constructions hve een descried in the literture []. In [9] Mhmoud Theri introduced nd investigted the concept of C-fuzzy numer Complement of fuzzy numer nd lso etended the rithmetic opertions on it. Some properties of these opertions were lso studied in his work. In [] Plsh Dutt et l. proved tht the α-cut method is sufficient in generl to del with different types of fuzzy rithmetic. In this pper the fuzzy rithmetic involving trpezoidl fuzzy numer using the α-cut method is introduced nd pplied on the complement of fuzzy numers. In section we present generl introduction on the suject. In section some sic definitions re included. In section some importnt properties on the complement of fuzzy numers re proved y using the α-cut method. In section the memership function of C-fuzzy numers nd its rithmetic opertions re investigted y the α-cut method nd numericl emple is presented. Finlly rief conclusion is given in our lst section 5.. Preliminries. Definition [7]: Let X e the universl set of the discourse. Then fuzzy set à in X is defined in terms of function µ à : X [ ] which ssigns memership grde to ech element of X within the intervl [ ] nd it is known s the

3 A Note on the Complement of Trpezoidl Fuzzy Numers memership function of Ã. The memership grdes correspond to the degrees to which the elements of X re comptile to the concept represent y the fuzzy set.. Definition: A conve nd normlized fuzzy set defined on the set R of rel numers whose memership function is piecewise continuous is clled fuzzy numer. A fuzzy set is clled norml when t lest one of its elements ttins the mimum possile memership grde i.e. there eists R such tht µ à =. Further is clled fuzzy conve i.e. µ à - µ à µ à R.. Definition: An α cut of fuzzy set à is crisp set à α tht contins ll the elements of the universl set X tht hve memership grde in à greter thn or equl to the specified vlue of the rel numer α. Thus à = { X: µ à α }.. Definition: A fuzzy numer à is clled positive fuzzy numer if its memership function is such tht µ à = nd is denoted y Ã>...5 Definition [5]: A Tringulr fuzzy numer à denoted s à = is defined y the memership function: µ A if if if > if It cn e chrcterized y defining the intervl of confidence t level α i.e. à α = [ - α - - α ] for ll This is denoted y Ã> α [ ]..6 Definition [6]: A Trpezoidl fuzzy numer Ã= is defined y the memership function 5

4 Dingr & Jivgn µ A if if if It cn e chrcterized y defining the intervl of confidence t level α i.e. Ã α = [ α - α ] for ll α [ ]. Complement of fuzzy numers. Definition: A fuzzy set in R is clled complement fuzzy numer if its complement is fuzz y numer. We denote the set of ll complements of fuzzy numers y F C R EXAMPLE: The fuzzy set A wy from which is chrcterized y the following memership function is complement fuzzy numer. µ A if if It is noted tht fuzzy set close to is complement to fuzzy set wy from clerly it cn e seen y its pictoril representtions.. Definition: Let N F c R nd let f: R R e mp Then we define f N s the complement of f N C. EXAMPLE: i If f = - then N the opposite of N is given y -N= N-. iiif f=λ λ R-{} then λn the sclr multipliction of complement fuzzy numer is given y λn =N λ. 6

5 A Note on the Complement of Trpezoidl Fuzzy Numers. Definition: Let M N F c R nd *: R R R e inry opertion. Then M N is defined y M N = M C N C C.. Theorem [9]: If M N F C *R nd * is continuous incresing decresing inry opertions then M N F C *R. Proof: Since M N F C *R thus M C N C F*R. Then M C N C F*R nd therefore M C N C C F C *R. But from definition. M C N C C = M N.5 Theorem: For ny commuttive opertion * the etended opertion for complement of fuzzy numers in F* c R is commuttive. For ny ssocitive opertion * the etended opertion for complement of fuzzy numers in F* c R is ssocitive. Proof: Let M N F* c R. Then y def. we hve M N = [M N] C C y def. = [M C N C C ] C C = [M C N C ] C complement property = [N C M C ] C commuttive property = N M y def.. Let M N L F* c R. Then y def. we hve L [M N] = L C [M N] C C y def. = L C [M C N C C ] C C = L C [M C N C ] C complement property = [L C M C N C ] C ssocitive property = [L C M C N C ] C = L M N.6 Definition: The complement of the trpezoidl fuzzy numer Ᾱ = is defined y the memership function µ A if if if It cn e chrcterized y defining the intervl of confidence Ᾱ α t level α y 7

6 Dingr & Jivgn Ᾱ α = [ - α - α - ] for ll α [ ].. Arithmetic opertions of the complement of trpezoidl fuzzy numers using the α- cut method. Addition of complements of fuzzy numers Let X = nd Y = e two trpezoidl fuzzy numers. The memership function of complement fuzzy numer of X is µ Similrly for Y µ y X α = [ α - α - ] nd Y α = [ α - α - ] re the α cuts of complement fuzzy numers X nd Y respectively. To clculte the sum of fuzzy numers X nd Y we first dd α cuts of X nd Y using intervl rithmetic: X α Y α = [ α - α - ] [ α - α - = [ α To find the memership function µ y we equte to oth the first nd second component in. which gives X = [ α - - ] nd 8

7 A Note on the Complement of Trpezoidl Fuzzy Numers X = [ α - ]. Now epressing α in terms of X nd setting α = nd α = in. we get X = α - - nd α = Also X = α [ - ] α = nd α = which gives µ y. Sutrction of complements of fuzzy numers Let X = nd Y = e two fuzzy numer. Then X α = [ α - α ] nd Y α = [ α - α ] re the α cuts of complement fuzzy numers X nd Y respectively. Then using intervl rithmetic. we get X α -Y α = [ α - α ] - [ α - α ] = [ α - - α α α - ] = [ -α ] [ α - ]... To find the memership function µ -y we equte to X oth the first nd second component in. which gives X = [ - α ] nd X = [ α ] Now epressing α in terms of X nd Y setting α = nd α = in. we get 9

8 Dingr & Jivgn - - X = [ - α - ] nd α = Also X = [ α [ ] α = nd α = which gives µ y Multipliction of complements of fuzzy numers Let X = > nd Y = > e two positive fuzzy numers. Then X α = [ - α α] nd Y α = [ - α α] re the α cuts of complement fuzzy numers X nd Y respectively. To clculte the product of fuzzy numers X nd Y we first multiply the α cuts of X nd Y using intervl rithmetic: X α * Y α = [ - α α] * [ - α α ] X α * Y α = [ - α * - α ][ α * α ]... To find the memership function µ y we equte to oth the first nd second component in. which gives X= - α - α ] α [ = α[ α ] α [ ] X= α[ ] α [ ] Now epressing α in terms of X nd setting α = nd α = in. we get α together with the domin of X: - - -

9 A Note on the Complement of Trpezoidl Fuzzy Numers [ ] = - ] ] [ µ y EXAMPLE: Let X = [ 6 8] nd Y = [ ] e two fuzzy numers. Then X α = [ - α 6 α] nd Y α = [5 - α 7 α] re the α-cuts of the complement fuzzy numers X nd Y respectively. Therefore X α * Y α = [ - α 6 α] * [5 - α 7 α] = [ - α * 5 - α] [6 α * 7 α] = [ - 8α - α α α α α ]= [α - 8α α - 6α ] = [α - 9α α α ].... We tke X = α - 9α nd X = α α Now setting α in terms of X y setting α = nd α = in. we get 8 9 = α = α which implies = y µ. Division of complements of fuzzy numers Let X = nd Y = e two positive fuzzy numers. Then X α = [ - α α ] nd

10 Dingr & Jivgn Y α = [ - α α ] re the α-cuts of complement fuzzy numers X nd Y respectively. We clculte the quotient of fuzzy numers X nd Y using intervl rithmetic: X α = Y α [ -α - - α - ] -α - α - = [ α - α - ] α - α To find the memership function µ /y we equte to oth the first nd second -α - component in. which gives X = nd X = α - α α Now epressing α in terms of X nd setting α = nd α = in. we get - α = α = - - which gives - - µ /y 5. Conclusion We hve descried how linguistic terms such s wy from zero fr from zero etc cn e represented y specil kinds of fuzzy sets of the rel line. We clled these fuzzy sets complements of fuzzy numers. We lso hve investigted the rithmetic opertions on such fuzzy quntities nd their properties using the α- cut method.

11 A Note on the Complement of Trpezoidl Fuzzy Numers References [] Duois D. Kerre E. Mesir R. & Prde H. Fuzzy intervl nlysis. In: D.Duois H Prde Eds Fundmentls of Fuzzy Sets vol. Kluwer Dordrecht [] Duois D. & Prde H. Specil issue on fuzzy numers Fuzzy Sets nd Systems 987. [] Fuller R. & Mesir R. Specil issue on fuzzy rithmetic Fuzzy Sets nd Systems ] Gerl G. &.Scrpti L. Etension principle for fuzzy set theory Informtion Sciences [5] Kufmnn A. & Gupt M. M. Introduction to Fuzzy rithmetic: Theory nd Applictions Vn Nostrnd New York 985. [6] Kufmnn A. Introduction to the Theory of Fuzzy Susets Acdemic Press New York 975. [7] Klir G. J. & Yun B. Fuzzy Sets nd Fuzzy Logic: Theory nd Applictions Prentice Hll Englewood Cliffs 995. [8] M M. Friedmn M. & Kndel A. A new fuzzy rithmetic Fuzzy Sets nd Systems [9] Theri M. S. C-fuzzy numers nd dul of etension principle Informtion sciences [] Nguyen H. T. A note on the etension principle for fuzzy sets Journl of Mthemticl Anlysis nd Applictions [] Dutt P. Bourh H. & Tzid A. Fuzzy rithmetic with nd without using α-cut method: A comprtive study Interntionl Journl of Ltest Trends in Computing [] Zdeh L. A. The concept of linguistic vrile nd its ppliction to pproimte resoning I Informtion Sciences