A New Approach for Ranking of Fuzzy Numbers using the Incentre of Centroids

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1 Intern. J. Fuzzy Mthemticl rchive Vol. 4 No ISSN: 0 4 (P 0 50 (online Published on pril 04.reserchmthsci.org Interntionl Journl of Ne pproch for nking of Fuzzy Numbers using the Incentre of Centroids P. jrjesri nd.shy Sudh Deprtment of Mthemtics Chikknn Government rts College Tirupur-40 Emil: p.rjrjesri9@ gmil.com Deprtment of Mthemtics Nirml College for Women Coimbtore-408 Emil: sudh.dss@yhoo.com eceived 4 pril 04; ccepted pril 04 bstrct. The fuzzy set theory hs been pplied in mny fields such s mngement engineering etc. In modern mngement pplictions rnking using fuzzy numbers re the most importnt spect in decision mking process. This pper proposes ne method on the incentre of centroids nd uses of Eucliden distnce to rnking generlized hegonl fuzzy numbers. We hve used rnking method for ordering fuzzy numbers bsed on res nd eights of generlized fuzzy numbers. Keyords: nking egonl fuzzy numbers Centroid re Eucliden distnce MS Mthemtics Subject Clssifiction (00: F07 0E7. Introduction nking fuzzy number is used minly in decision-mking dt nlysis rtificil intelligence nd vrious other fields of opertions reserch. In fuzzy environment rnking fuzzy numbers is very importnt decision mking procedure. nking fuzzy numbers ere first proposed by Jin [9] for decision mking in fuzzy situtions by representing the ill defined quntity s fuzzy set nd he hs given procedure for multi spect decision mking using fuzzy sets in [0]. Some of these rnking methods hve been compred nd revieed by Bortoln nd Degni [] nd more recently by Chen nd ng [5]. ee nd i [4] proposed the comprison of fuzzy numbers. iou nd Wng [5] presented rnking fuzzy numbers ith intervl vlues. The centroids of fuzzy numbers hve been emined recently nd one of the most commonly used methods under the clss of fuzzy scoring is the centroid point method. Centroid concept in rnking fuzzy number only strted in 980 by Yger []. Yger s the resercher ho contributed the centroid concept in the rnking method nd used the horizontl co-ordinte s nd the verticl y co-ordintes of the centroid point s the rnking inde. Cheng [] used centroid bsed distnce method to rnk fuzzy numbers in 998.Then Chu nd Tso [7] utilized the re beteen the centroid point nd the origin to rnk fuzzy numbers in 00. bbsbndy nd sdy [] suggested sign distnce method for rnking fuzzy numbers in 00. Wng nd ee [] proposed the revised method of rnking fuzzy numbers ith n re 5

2 Ne pproch for nking of Fuzzy Numbers using the Incentre of Centroids beteen the centroid nd originl points in008.since then severl methods hve been proposed by vrious reserchers in [-] nd rnking trpezoidl fuzzy numbers using re compenstion distnce method mimizing nd minimizing set decomposition principle nd signed distnce [44]. For n overvie of fuzzy rithmetic theory one my refer [-] nd the men-vlue of fuzzy number in [8]. In this pper ne method is proposed hich is bsed on incentre of centroids to rnk fuzzy quntities.in hegonl fuzzy number the hegonl is split into three plne figures here the first prt being Tringle nd the second egon nd the third once gin Tringle nd then clculting the centroids of ech plne figure folloed by the incenter of these centroids nd then finding the Euclidin distnce. The proposed pproch is compred ith different eisting pproches.. Preliminries Definition.. et X be nonempty set. fuzzy set in X is chrcterized by its membership function : X [0] here ( is interpreted s the degree of membership of element in fuzzy for ech X. Definition.. fuzzy set is conve if (λ (-λ min µ ( ( } X λ [0 ] { µ Definition.. The fuzzy set Ã is norml if height (. In other ords there eist t lest one X such tht µ ( Definition.4. Fuzzy number is conve normlized fuzzy set on the rel line such tht: There eist t lest one ith µ ( µ ( is pieceise continuous.. egonl Fuzzy Numbers fuzzy number ɶ is hegonl fuzzy number denoted by ɶ ( µ ɶ is given here re rel numbers nd its membership function ( belo. Definition.. fuzzy set ɶ defined on the universl set of rel numbers is sid to be generlized fuzzy number of its membership function hs the folloing chrcteristics function (i P (u is bounded left continuous non decresing function over [00.5] (ii Q (v is bounded left continuous non decresing function over [0.5] (iii Q (v is bounded continuous non incresing function over [ 0.5] (iv P (u is bounded left continuous non incresing function over [0.50]. 5

3 µ ɶ ( 0 P. jrjesri nd.shy Sudh f o r < f o r f o r 4 f o r 5 0 f o r > f o r 4 f o r emrk... If then the hegonl fuzzy number is clled norml hegonl fuzzy number. emrk... Membership function µ ɶ ( re continuous functions. Definition.. positive hegonl fuzzy number 54 ɶ is denoted by ɶ ( here ll i s > 0 for ll i 45. Definition.. negtive hegonl fuzzy number ɶ is denoted by ɶ ( here ll i s < 0 for ll i 45. Definition.4. If Then ɶ ( is the hegonl fuzzy number. ɶ ( hich is the symmetric imge of ɶ..5. Ordering of egonl fuzzy number et ɶ ( nd B ɶ (b b b b 4 b 5 b be in F( be the set of ll rel hegonl fuzzy numbers i if nd only if i b i i45 ii if nd only if i b i i45 iii if nd only if i b i i45... nking of egonl Fuzzy Numbers

4 Ne pproch for nking of Fuzzy Numbers using the Incentre of Centroids n efficient pproch for compring the fuzzy numbers is by the use of rnking function : F ( here F ( is set of fuzzy numbers defined on set of rel numbers hich mps ech fuzzy number into rel number here nturl order eists. For ny to hegonl fuzzy numbers (b b b b 4 b 5 b e hve the folloing comprison i ( ii ( ( iii ( ( 4. Proposed Method ɶ ( nd B ɶ Fig. Generlized hegonl fuzzy number The centriod of hegonl fuzzy number is considered to be the blncing point of the hegon (Fig.. Divide the hegonl into three plne figures.these three plne figures re Tringle BQ egon CDEQB nd gin tringle EF respectively. et the centriod of the three plne figures be G G G respectively. The incenter of the centroids G G G is tken s the point of reference to define the rnking of generlized hegonl fuzzy numbers. The reson for selecting this point s point of reference is tht ech centroid points re blncing points of ech individul plne figure nd the incenter of these centroid points is much more blncing point for generlized hegonl fuzzy number. Therefore this point ould be better point thn the centroid point of the hegon. Consider generlize d hegonl fuzzy number ( ;. The centriod of the three plne figures re G respectively. ; G ; G 4 5 Eqution of the line is y nd G does not lie on the line G G. Therefore G G nd G re non- colliner nd they form tringle. 4 5 ; 55

5 P. jrjesri nd.shy Sudh We define the incentre Iɶ ( 0 y0 of the tringle ith vertices G G nd G of the generlized hegonl fuzzy number ( ; s α I ( 0 y0 ( here ( ( 4 β 5 γ α β γ ( 4 5 α ( β ( γ α β γ ( ( 4 5 ( α β γ ( The rnking function of the generlized hegonl fuzzy number ɶ ( ;hich mps the set of ll fuzzy numbers to set of rel numbers ( y 0 0 is defined s: This is the Eucliden distnce from the incentre of the centroids. In sum the rnk of to fuzzy numbers centroids is given in the folloing steps. et ɶ ( ; nd egonl Fuzzy numbers then Step. α β γ α β γ Find B & B B Step. I ( 0 y 0 I ( 0 y 0 Find B & Step. Find numbers ( 0 y0 & ( B 0 y0 i If ( then ii If ( < ( then iii If ( ( then ɶ nd ( ( ( B ɶ bsed on the incentre of the B ɶ (b b b b 4 b 5 b ; be to generlized nd using the folloing for the rnking of fuzzy Emple 5.. et ɶ ( ;0.5 nd B ɶ ( ;0.7 be to generlized fuzzy numbers then Step : α α B ( ( ( (0.4 4(0.7 ( ( (0.5 4(

6 β β B γ Ne pproch for nking of Fuzzy Numbers using the Incentre of Centroids γ B Step : ( 4 5 ( ( ( (0. 4(0.7 ( ( (0. 4( ( 0.4( 0.5( 0.( 0.4( 0.5( I ( 0 y0 ( ( ( 0.4( 0.( 0.9( 0.4( 0.( I ( 0 y0 B ( ( Step : ( 0.4 (0. So > B then 0.44 B > ( 0.7 (0.0 B 0.4 Emple 5.. et ɶ ( ; nd B ɶ ( ; be to generlized fuzzy numbers. Then Step ; α α B β β B γ γ B Step : ( ( ( (0. 4 ( 4 5 ( ( 5 4 ( (0. 4 ( ( (0.4 4 ( ( ( ( 0.5( 0.( 0.47( 0.5( 0.( I ( 0 y0 ( ( ( 0.4( 0.5( 0.44( 0.4( 0.5( I ( 0 y0 B ( ( Step : ( 0.40 ( ( 0.5 (0. B

7 So < B then B < P. jrjesri nd.shy Sudh Emple I of T.C.Chu nd C.T.Tso pproch: et ɶ ( ;0.5 nd B ɶ ( ;0.7 be to generlized fuzzy numbers then ( y ( ( Similrly f d f d d ( B f d f d d ( f d f d yg dy yg dy yg dy 0 g dy S( 0.4 S( B 0.8 y g ( y dy ( B g dy ( f f d ( yg g dy d dy S ( > S( B then > B Emple II of T.C.Chu nd C.T.Tso: et ɶ ( ; nd B ɶ ( ; be to generlized fuzzy numbers ( f d f d d ( Similrly ( B f d S( 0.S( B 0.8 f d y y d ( ( B ( f d f d ( f f d d 58

8 Ne pproch for nking of Fuzzy Numbers using the Incentre of Centroids S ( > S( B then > B pproches E E Chu T.C. nd Tso.C T.[7] > B > B > B < B Proposed pproch in egonl. Conclusion In this method splitting the generlized hegonl fuzzy numbers into three plne figures nd then clculting the centroid of ech plne figure folloed by the incenter of the centroid nd then finding the Euclidin distnce nd the sme problem s compred ith Chu nd Tso [7] method ith comprtive emples. The min dvntge of the proposed pproch is tht this provides the correct ordering of generlized nd norml hegonl fuzzy numbers nd lso esy to pply in the rel life problems. Presently in rel life there re mny situtions nd prmeters ith more criteri so in order to get comprehensive result this method is very useful. EFEENCES. S.bbsbndy nd B.sdy nking of fuzzy numbers by sign distnce Informtion Sciences 7 ( S.bbsbndy nd T.jjir ne pproch for rnking of trpezoidl numbers Computers nd Mthemtics ith pplictions 57( ( G.Bortoln nd.degni revie of some methods for rnking fuzzy subsets Fuzzy Sets nd Systems 5 ( S..Chen nk in fuzzy numbers ith mimizing set nd minimizing set Fuzzy Sets nd Systems 7( ( S.J.Chen nd C..ng Fuzzy multiple ttribute decision mking Springer Berlin (99.. C..Cheng ne pproch for rnking fuzzy numbers by distnce method Fuzzy Sets nd Systems 95 ( T.C.Chu nd C.T Tso nking fuzzy numbers ith n re beteen the centroid points nd the originl point Computers nd Mthemtics ith pplictions 4 ( D.Dubois nd.prde The men vlue of fuzzy number Fuzzy Sets nd Systems 4( ( Jin Decision mking in the presence of fuzzy vribles IEEE Trnsctions on System Mn nd Cybernetics ( Jin procedure for multispect decision mking using fuzzy sets The Interntionl Journl of System Sciences 8 ( Kuffmnn nd M.Gupt Introduction to Fuzzy rithmetic: Theory nd pplictions Vn Nostrnd einhold Ne York (980.. G.J Klir. Fuzzy rithmetic ith requisite constrints Fuzzy Sets nd Systems 9 (

9 P. jrjesri nd.shy Sudh. G.J Klir Fuzzy Sets: n Overvie of Fundmentls pplictions nd Personl Wies Beijing Norml University Press ( E.S ee nd.j i Comprison of fuzzy numbers bsed on the probbility mesure of fuzzy event Computers nd Mthemtics ith pplictions 5 (0 ( T.S iou nd M.J Wng nking fuzzy numbers ith integrl vlue Fuzzy Sets nd Systems 50 ( Ngoor Gni nd V.N.Mohmed Solution of fuzzy ssignment problem by using nernking method Intern. J. Fuzzy Mthemticl rchive ( P.P.B o nd N.Shnkr nking fuzzy numbers ith distnce method using circumcenter of centroids nd Inde of modlity dvnces in Fuzzy Systems rticle ( S. ezvni ne method for rnking in perimeters of to generlized trpezoidl fuzzy numbers Interntionl Journl of pplied Opertionl eserch ( ( S.ezvni ne method for rnking in res of to generlized trpezoidl fuzzy numbers Interntionl Journl of Fuzzy ogic Systems ( ( S. ezvni nking generlized trpezoidl fuzzy numbers ith Eucliden distnce by the incenter of centroids Mthemtic etern ( ( Y..P Thorni et.l Ordering generlized fuzzy numbers Int. J. Contemp. Mth. Science 7( ( Y.J Wng nd.s ee The revised method of rnking fuzzy numbers ith n re beteen the centroid nd originl points Computers nd Mthemtics ith pplictions 55 ( Yger On generl clss of fuzzy connectives Fuzzy Sets nd Systems 4( ( J.Yo nd K.Wu nking fuzzy numbers bsed on decomposition principle nd signed distnce Fuzzy Sets nd Systems (

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