[Prakash* et al., 5(8): August, 2016] ISSN: IC Value: 3.00 Impact Factor: 4.116
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1 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 IJESRT INTERNTIONL JOURNL OF ENGINEERING SIENES & RESERH TEHNOLOGY SOME PROPERTIES ND THEOREM ON FUZZY SU-TRIDENT DISTNE Prveen Prksh* M Geeth Lkshmi Professor Deprtment of Mthemtis Hindustn University Pdur henni - 6 ssistnt Professor Deprtment of Mthemtis KG ollege of Tehnology Krpkkm henni DOI: 58/zenodo68 STRT This pper introdues some simple properties nd theorem sed on Fuzzy Su-dent Distne long with the help of Trpezoidl Fuzzy Numers The results re ussed with suitle numeril exmple KEYWORDS: Trpezoidl Fuzzy Numer Su-dent Distne Positive Negtive MS Mthemtis Sujet lssifition: E7 9D5 5 INTRODUTION Fuzzy Set Theory is introdued y LotfiZdeh in the yer 965 []Lter Liem Trn nd Luien Dukstein gve the omprison of fuzzy numers using fuzzy tne mesure in the yer [] Lter Shnhuo hen nd hienhung Wng introdued the Fuzzy Distne of Trpezoidl Fuzzy Numers in the yer 8 [] In the yer Ngoorgni [5] gve new opertion on ngulr Fuzzy numer for solving Fuzzy Liner Progrmming Prolem New Method for Rnk Mode Divergene nd spred on Generlized Exponentil Trpezoidl Fuzzy Numers is given y Slim Rezvni in the yer [6] rithmeti Opertions on Generlized Trpezoidl Fuzzy Numer nd its pplitions is given y Snhit nerjee nd Tpn Kumr Roy in the yer [7] In the yer Prdhsrdhi nd Rvi Shnkr gve n ide on Fuzzy Distne Mesure [8] In this Pper Some simple properties nd theorem sed on Fuzzy Su-dent Distne long with the help of Trpezoidl Fuzzy Numers re given This Pper onsists of five setions The preliminries in the first setion Defining Trpezoidl Positive Trpezoidl Negtive Trpezoidl Fuzzy Numers in the seond setion Fuzzy Su-dent Distne in the third setion Properties nd Theorem sed on Fuzzy Su-dent Distne in the fourth setion nd finlly the results re ussed with suitle numeril exmples PRELIMINRIES Definition The hrteristi funtion of risp set X ssigns vlue either or to eh memer in X This funtion n e generlized to funtion suh tht the vlue ssigned to the element of the universl set X fll within speified rnge ie : X The ssigned vlue indites the memership grde of the element in the set is lled the memership funtion nd the set x x ; is lled fuzzy set [] The funtion x X defined y x for eh x X http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [9]
2 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 Definition fuzzy set defined on the universl set of is sid to e fuzzy numer if its memership funtion hs the following hrteristis: i is onvex ie min x x x x x ii iii x x is norml ie x suh tht is pieewise ontinuous[] x Representtion of Generlized Trpezoidl Fuzzy Numer In generl generlized fuzzy numer funtion x stisfies the following onditions: x x is ontinuous mpping from to [] x L x is stritly inresing on [] x x w x x R is stritly deresing on [d] x x d x where w is desried t ny fuzzy suset of the rel line whose memership nd nd d rel numers We denote this type of generlized fuzzy numers s d; w When w= this type of generlized fuzzy numer d LR When Lx nd Rx re stright line then is denoted y d [6] Positive Trpezoidl Fuzzy Numer: LR is Trpezoidl fuzzy numer nd it Positive Trpezoidl Fuzzy Numer is denoted s where ll Negtive Trpezoidl Fuzzy Numer: i s i Negtive Trpezoidl Fuzzy Numer is denoted s where ll s i Fuzzy Su-dent Distne The tne etween the two fuzzy numers re lulted y using the new tehnique lled the Fuzzy Su-dent Distne s follows: Let nd then the Fuzzy Su-dent Distne is given y tn e i http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [9]
3 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 Properties nd Theorem on Fuzzy Su-dent Distne: The following re the properties sed on Fuzzy Su-dent Distne: Property : Let D tn e tn e e Trpezoidl Fuzzy Numers The Fuzzy Su-dent Distne of nd the Fuzzy Su-dent Distne of tn e D if Property : nd nd D D tn e If the Trpezoidl Fuzzy Numers re Positive then is Positive Property : tn e If the Trpezoidl Fuzzy Numers re Negtive then is Positive The Theorem sed on Fuzzy Su-dent Distne is s follows: Theorem: The Fuzzy Su-dent Distne where onditions hold: i ii iii iv Proof: for i To Prove tn e nd is given y tn e D is given y then re Trpezoidl Fuzzy Numer then the following where retrpezoidlfuzzynumers Let us onsider re Trpezoidl Fuzzy Numers The proof is ovious from the definition of Fuzzy Su-dent Distne is given y Thus for ll vlues of tn e Hene the Proof ii To Prove http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [95]
4 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [96] Let us onsider re Trpezoidl Fuzzy Numers tn tn e e Thus Hene the Proof iii To Prove Let us onsider re Trpezoidl Fuzzy Numers tn e Thus Hene the Proof iv To Prove Let us onsider re Trpezoidl Fuzzy Numers tn e
5 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 Thus Hene the Proof Exmple: Let us onsider the following exmple: nd Let re Trpezoidl Fuzzy Numers ito Prove 87 Hene Proved iito Prove L H S : R H S : From nd LHS = RHS Thus http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [97]
6 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 Hene Proved iiito Prove Let us onsider e two trpezoidl fuzzy numers If then Thus 6 6 onversely If then Thus Hene Proved ivto Prove http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [98]
7 [Prksh* et l 58: ugust 6] ISSN: I Vlue: Impt Ftor: 6 LHS : RHS : From equtions nd LHS RHS Thus Hene Proved ONLUSION The min im of this pper is to introdue new properties nd the theorem sed on Fuzzy Su-dent Distne The dvntge of this pper is simple nd esy to pply nd to solve Trnsporttion Prolems REFERENES [] LotfiZdeh Fuzzy Sets: Informtion nd ontrol vol8 pp [] Kufmnn nd Gupt MM Introdution to Fuzzy rithmetis: Theory nd pplitions Vn Nostrnd Reinhold New York 985 [] Liem Trn nd Luien Dukstein omprison of fuzzy numers using fuzzy tne mesure Fuzzy Sets nd System Vol [] Shnhuo hen nd hienhung Wng Fuzzy Distne of Trpezoidl Fuzzy Numers nd pplition Interntionl Journl of Innovtive omputing Informtion nd ontrol Vol No6 8 [5] Ngoorgni New opertion on ngulr Fuzzy numer for solving Fuzzy Liner Progrmming Prolem pplied Mthemtil Sienes Vol6 No pp55-5 [6] Slim Rezvni New Method for Rnk Mode Divergene nd spred on Generlized Exponentil Trpezoidl Fuzzy Numers Turkish Journl of Fuzzy Systems vol No pp98-7 [7] Snhit nerjee Tpn Kumr Roy rithmeti Opertions on Generlized Trpezoidl Fuzzy Numer nd its pplitions Turkish Journl of Fuzzy Systems Vol No pp6- [8] Ngoorgni nd VN Mohmed Solution of Fuzzy ssignment Prolem y using New Rnking Method Interntionl Journl of Fuzzy Mthemtil rhivevolpp8-6 [9] Prdhsrdhi nd NRvi Shnkr Fuzzy Distne Mesure ShJEngTeh Vol pp98-9 http: // wwwijesrtom Interntionl Journl of Engineering Sienes & Reserh Tehnology [99]
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