Interval Valued Bipolar Fuzzy Weighted Neutrosophic Sets and Their Application

Transcription

1 Iterval Valued polar Fuzzy Weghted Neutrosophc Sets ad Ther pplcato Irfa Del Muallm fat Faculty of Educato Kls 7 ralk Uversty Kls Turkey rfadel@kls.edu.tr Yusuf uba Muallm fat Faculty of Educato Kls 7 ralk Uversty Kls Turkey ysubas@kls.edu.tr Floret Smaradache Uversty of Ne Mexco 705 Gurley ve. Gallup Ne Mexco 870 US. fsmaradache@gmal.com Mumtaz l Departmet of Mathematcs Quad--zam Uversty Islamabad Paksta mumtazal788@gmal.com bstract Iterval valued bpolar fuzzy eghted eutrosophc set(ivfwn-set) s a e geeralzato of fuzzy set bpolar fuzzy set eutrosophc set ad bpolar eutrosophc set so that t ca hadle ucerta formato more flexbly the process of decso makg. Therefore ths paper e propose cocept of IVFWN-set ad ts operatos. lso e gve the IVFWN-set eghted average operator ad IVFWN-set eghted geometrc operator to aggregate the IVFWN-sets hch ca be cosdered as the geeralzatos of some exstg oes uder fuzzy eutrosophc evromets ad so o. Fally a decso makg algorthm uder IVFWN evromet s gve based o the gve aggregato operators ad a real example s used to demostrate the effectveess of the method. Keyords Neutrosophc set terval valued eutrosophc set IVFWN-set average ad geometrc operator mult-crtera decso makg. I. INTODUTION To overcome cotag varous kds of ucertaty the cocept of fuzzy sets [8] has bee troduced by Zadeh. fter Zadeh may studes o mathematcal modelg have bee developed. For example; to model determate ad cosstet formato Smaradache [] troduced the cocept of eutrosophc set hch s depedetly characterzed by three fuctos called truth-membershp fucto determacy-membershp fucto ad falsty membershp fucto. ecetly studes o eutrosophc sets are made rapdly []. polar fuzzy sets hch are a geeralzato of Zadeh s fuzzy sets [8] ere orgally proposed by ee [9]. osc ad Pvert [] sad that polarty refers to the propesty of the huma md to reaso ad make decsos o the bass of postve ad egatve effects. Postve formato states hat s possble satsfactory permtted desred or cosdered as beg acceptable. O the other had egatve statemets express hat s mpossble reected or forbdde. Negatve prefereces correspod to costrats sce they specfy hch values or obects have to be reected (.e. those that do ot satsfy the costrats) hle postve prefereces correspod to shes as they specfy hch obects are more desrable tha others (.e. satsfy user shes) thout reectg those that do ot meet the shes. Presetly orks o bpolar fuzzy sets are progressg rapdly [8-7]. lso bpolar eutrosophc set(n-set) ad ts operatos s gve [7]. I ths study to hadlg some ucertates fuzzy sets ad eutrosophc sets the extesos of fuzzy sets[8] bpolar fuzzy sets[9] eutrosophc sets[] ad bpolar eutrosophc sets[7] terval valued bpolar fuzzy eghted eutrosophc sets th applcato are troduced. II. PEIMINIES I the secto e gve some cocepts related to bpolar fuzzy sets eutrosophc sets terval valued eutrosophc set ad bpolar eutrosophc sets. Defto.. [] et X be a uverse of dscourse. The a sgle valued eutrosophc set s defed as: NS I F : x X hch s characterzed by a truth-membershp T : X 0 a determacy-membershp fucto fucto ( ): 0 ( ): 0. I x X ad a falsty-membershp fucto F x X There s ot restrcto o the sum of T I ( x ) ad F so 0 T I F. Defto.. [5] et X be a space of pots (obects) th geerc elemets X deoted by x. terval valued eutrosophc set (for short IVNS) X s characterzed by truth-membershp fucto T detemacy-membershp fucto I ( x ) ad falsty /6/$.00 c 06 IEEE 60

2 membershp fucto F ( x ). For each pot x X e have that T I F ( x ) 0. For to IVNS IVNS x T x I x I x f F sup F : x X ad f ( )sup ( ) f ( )sup ( ) IVNS x T x I x I x f F sup F : x X f ( )sup ( ) f ( )sup ( ) The. IVNS IVNS f ad oly f f T x f T x supt x sup T x f Ix f Ix sup Ix sup Ix sup F x sup F x sup F x sup F x for all x X.. IVNS IVNS. f ad oly f f T x f T x supt x sup T x f Ix f Ix sup I x sup I x sup F x sup F x sup F x sup F x for ay x X. IVNS f ad oly f x f F sup F sup I IVNS f I f T sup T : x X. IVNS IVNS f ad oly f IVNS IVNS xf T f T sup T sup T f I f I sup I sup I f F f F sup F sup F : x X 5. IVNS IVNS f ad oly f IVNS IVNS xf T f T sup T sup T f I f I sup I sup I f F f F sup F sup F : x X Defto.. [9] et X be a o-empty set. The a bpolarvalued fuzzy set deoted by s defed as; F x x x : xx F Where x : X 0 ad x X postve membershp degree x : 0. The deotes the satsfacto degree of a elemet x to the property correspodg to F ad the egatve membershp degree x deotes the satsfacto degree of x to some mplct couter property of. F Defto.. [7] bpolar eutrosophc set X s defed as a obect of the form I F T x I x F : x X here T I F : X 0 ad T I F : X 0 The postve membershp degree T I F deotes the truth membershp determate membershp ad false membershp of a elemet x X correspodg to a bpolar eutrosophc set ad the egatve membershp degree T I F deotes the truth membershp determate membershp ad false membershp of a elemet x X to some mplct couter-property correspodg to a bpolar eutrosophc set.. Defto.5. [7] et I F T x I : xx ad I F T x I : xx be to bpolar eutrosophc sets. I. The f ad oly f T T I I F F ad T T I I F F for all x X. II. The f ad oly f T T I I F F ad T T I I F F for all x X. III. The ther uo s defed as: ( ) I I T x F x max ( ) ( ) m ( ) ( ) I I m T T ma F fo r all x X. 06 IEEE Iteratoal oferece o Fuzzy Systems (FUZZ) 6

3 IV. The ther tersecto s defed as: ( ) I I T x F x m ( ) ( ) max ( ) ( ) I I ma T m F F fo r all x X. c V. The the complemet of s deoted by ad s defed by T { } T c F { } F c ad T { } T c F { } F c for all x X. Defto.6. [7] et I { } I c I { } I c I F T x I x F : xx ad I F T x I x F : xx be to bpolar eutrosophc umber. The the operatos for these umbers are defed as belo; T I F T I a. F b. T I F T I F c. T T T T I I F F T T.. ( I I I. I ) ( F F F. F ) d. T T I I I I F F F F.. ( T T T. T ) I I F F here 0. Defto.7. [7] et a I F T x I : xx be a bpolar eutrosophc umber. The the score fucto sa accuracy fucto a a ad certaty fucto ca of a NN are defed as follos: s a T I F T I F ) 6 a a T F T F c a T F Defto.8. [7] et a T I F T I F... be a famly of bpolar eutrosophc umbers. The a) FW : s called bpolar eutrosophc eghted average operator f t satsfes; F a a... a a W T I F T ( ) I F here s the eght of a (... ) 0 ad. b) HW : s called bpolar eutrosophc eghted geometrc operator f t satsfes; H a a... a a W T I F ( ) T I F here s the eght of a (... ) 0 ad. III. INTEV VUED IPO FUZZY WEIGHTED NEUTOSOPHI SET I ths secto e gve cocept of IVFWN-set ad ts operatos. lso e gve the IVFWN-set eghted average operator ad IVFWN-set eghted geometrc operator th propertes to aggregate the IVFWN-sets based o the study gve [7]. Defto.. terval valued bpolar fuzzy eghted eutrosophc set(ivfwn-set) X s defed as a obect of the form 6 06 IEEE Iteratoal oferece o Fuzzy Systems (FUZZ)

4 x T T I I F ( ) ( ) x T T I I F F p : x X here T T I I F F : X 0 ad T T I I F F : X 0. lso p: X 0 fuzzy eghted dex of the elemet x X. Example.. et X { x x x}. The x x x s a IVFWN subset of X. Theorem.. IVFWN-set s the geeralzato of a bpolar fuzzy set ad bpolar eutrosophc set. Proof: Straghtforard. Defto.. et T I I F F T T I I F F p : x X ad x T T I I F F T T I I F F p : x X be to IVFWN-sets.. The f ad oly f T T I I T T I I ad p p for all x X. T T F F T T F F. The f ad oly f T T I I T T T T F F T T I I F F ad p p for all x X. I I F F I I F F I I F F I I F F. The ther uo s defed as: ( ) T x T x max ( ) ( ) max ( ) ( ) ( ) ( ) ( ) I x I x I x I ( x ) F x F x T x T x m ( ) ( ) m ( ) ( ) m ( ) ( ) m ( ) ( ) ( ) ( ) ( ) I x I x I x I ( x ) F x F x max ( ) ( ) max ( ) ( ) for all x X.. The ther tersecto s defed as: ( ) T x T x m ( ) ( ) m ( ) ( ) ( ) ( ) ( ) I x I x I x I ( x ) F x F x T x T x max ( ) ( ) max ( ) ( ) max ( ) ( ) max ( ) ( ) ( ) ( ) ( ) I x I x I x I ( x ) F x F x m ( ) ( ) m ( ) ( ) for all x X. c 5. The the complemet of s deoted by s defed by F F I I T F F I I T T p : xx Example.5. et X { x x x}. The x x x ad x x x are to IVFWN-sets X. 06 IEEE Iteratoal oferece o Fuzzy Systems (FUZZ) 6

5 The ther uo s gve as follos: x x x The ther tersecto s gve as follos: x x x Defto.6. et T T I I F F ad T T I I F F p T T I I F F T T I I F F p be to IVFWN-umbers.. The the operatos for IVFWN-umbers are defed as belo;. T T I I F F T T I I F F. T T I I T I I F F T F F T T T. T T T T. T. I I II FF FF T. T T. T ( I I I. I) ( I I I. I) ( F F F. F) ( F F F. F ) v. TT TT I I I. I I I I. I F F F. F F F F. F ( T T T. T) ( T T T. T) II II FF FF here 0. Defto.7. et a T T I I F F T T I I F F be a IVFWN-umber. The the score fucto Sa accuracy fucto a ad certaty fucto a of a NN are defed as follos: px ( ) Sa T T I I F F T T I I F F ( )( ) a p T F F T T F F ( )( ) a p T F F The comparso method ca be defed as follos: the. If Sa Sa a s greater tha a that s a s superor to a deoted by a a ; S a S a ad a a the a s greater tha a that s a s superor to a deoted by a a. If ;. If Sa Sa a a ad a a s greater tha a a the a that s s superor to a ; ad a the a that s a s dfferet to a ; v. If Sa Sa a a a a s equal to Defto.8. et a T T I I F F T T I I F F p... be a famly of IVFWN-umbers. mappg p : s called IVFWN-eghted average operator f t satsfes a a... a p a p T T 6 06 IEEE Iteratoal oferece o Fuzzy Systems (FUZZ)

6 I I F F ( T ) ( T ) I I F F p 0 here s the eght of a (... ) ad. Theorem.9. et a T T I I F F T T I I F F... be a famly of IVFWN-umbers. The. If a a for all... the W a a... a a.. m a a a... a max a W * a a If for all... the * * * a a a a a a W W ( T ) ( T ) I I F F 0 here s the eght of a (... ) ad. Theorem.. et a T T I I F F T T I I F F... be a famly of IVFWN-umbers. The. If a a for all... the GW a a... a a.. If m a G a a... a max a W * a a for all... the * * * GW a a a GW a a a Note that the aggregato results are stll NNs Defto.0. et a T T I I F F T T I I F F... be a famly of IVFWN-umbers. mappg GW : s called IVFWN-eghted geometrc operator f t satsfes G a a... a a W T T I I F F IV. NN- DEISION MKING METHOD I ths secto e develop a approach based o the W (or G W ) operator ad the above rakg method to deal th multple crtera decso makg problems th IVFWNformato.... m... s the set of alteratves ad crteros or attrbutes respectvely. T et... be the eght vector of attrbutes such that 0... ad refers to the Suppose that ad eght of attrbute. alteratve o crteros s evaluated by the decso maker ad the evaluato values are represeted by the form of IVFWN-umbers. ssume that a T T I I F F T T m I I F F m 06 IEEE Iteratoal oferece o Fuzzy Systems (FUZZ) 65

7 s the decso matrx provded by the decso maker; a s a IVFWN-umber for alteratve. assocated th the crteros. We have the codtos T T I I F F ad T T I I F F 0 such that 0 T T I I F F T T I I F F for (... m) ad (... ). No e ca develop a algorthm as follos; lgorthm Step. ostruct the decso matrx provded by the decso maker as; a T T I I F F T T m I I F F m W or GW a a Step. ompute a a a... a... a )) for each a (... m) ( ( Step. alculate the score values of Sa for the (... m) collectve overall IVFWN-umber of a (... m) Step. ak all the softare systems of a (... m) accordg to the score values No e gve a umercal example as follos; Example.. et us cosder decso makg problem adapted from Ye [6]. There s a vestmet compay hch ats to vest a sum of moey the best opto. There s a pael th the set of the four alteratves s deoted by car compay food compay computer compay arms compay to vest the moey. The vestmet compay must take a decso accordg to the set of the four attrbutes s deoted by rsk groth evrometal mpact performace. lso the eght vector of the attrbutes ( ) s T ( ). The the accordg to ths algorthm e have Step. ostruct the decso matrx provded by the customer as; Table : Decso matrx gve by customer for each Step. ompute a W a a a a as; a a a a for the collectve overall IVFWN-umber of a (... m) as; Step. alculate the score values of Sa Sa Sa Sa S a Step. ak all the softare systems of accordg to the score values as; IEEE Iteratoal oferece o Fuzzy Systems (FUZZ)

8 ad thus s the most desrable alteratve. ONUSION Ths paper preseted a terval-valued bpolar eutrosophc set ad ts score certaty ad accuracy fuctos. I the future e shall further study more aggregato operators for terval-valued bpolar eutrosophc set ad apply them to solve practcal applcatos group decso makg expert system formato fuso system game theory ad so o. EFEENES [] M. l ad F. Smaradache omplex Neutrosophc Set Neural omputg ad pplcatos DOI: 0.007/s y. [] M. l I. Del F. Smaradache The Theory of Neutrosophc ubc Sets ad Ther pplcatos Patter ecogto Joural of Itellget ad Fuzzy Systems DOI:0./IFS [] P. osc O. Pvert O a fuzzy bpolar relatoal algebra Iformato Sceces 9 (0) 6. [] J. he S. S. Ma ad X. Wag -Polar Fuzzy Sets: Exteso of polar Fuzzy Sets The Scetfc World Joural (0) /0/650. [5] I. Del Iterval-valued eutrosophc soft sets ad ts decso makg Iteratoal Joural of Mache earg ad yberetcs DOI: 0.007/s [6] I. Del S. roum F. Smaradache O eutrosophc refed sets ad ther applcatos medcal dagoss Joural of Ne Theory 6 (05) [7] I. Del M. l ad F. Smaradache polar Neutrosophc Sets ad Ther pplcato ased o Mult-rtera Decso Makg Problems Proceedgs of the 05 Iteratoal oferece o dvaced Mechatroc Systems - ugust 05 eg ha. [8] M. K. Kag ad J. G. Kag polar fuzzy set theory appled to subsemgroups th operators semgroups. J. Korea Soc. Math. Educ. Ser. Pure ppl. Math. 9/ (0) -5. [9] K. M. ee polar-valued fuzzy sets ad ther operatos. Proc. It. of. o Itellget Techologes agkok Thalad (000) 07-. [0] K. J. ee polar fuzzy subalgebras ad bpolar fuzzy deals of K/I-algebras ull. Malays. Math. Sc. Soc. / (009) 6-7. [] S.K.Maumder polar Valued Fuzzy Sets -Semgroups Mathematca etera / (0) 0. [] S.V. Maemara. hellappa Structures o polar Fuzzy Groups ad polar Fuzzy D-Ideals uder (T S) Norms Iteratoal Joural of omputer pplcatos 9/ 7-0. [] F. Smaradache Ufyg Feld ogcs. Neutrosophy : Neutrosophc Probablty Set ad ogc ehoboth: merca esearch Press999. [] H. Wag F. Smaradache Y.Q. Zhag ad. Suderrama Sgle valued eutrosophc sets Multspace ad Multstructure (00) 0-. [5] H. Wag F. Smaradache Y.Q. Zhag ad. Suderrama Iterval eutrosophc sets ad logc: theory ad applcatos computg (005) Hexs rzoa. [6] J. Ye Vector Smlarty Measures of Smplfed Neutrosophc Sets ad Ther pplcato Multcrtera Decso Makg Iteratoal Joural of Fuzzy Systems 6/ (0) 0-. [7] M. Zhou S. I pplcato of polar Fuzzy Sets Semrgs Joural of Mathematcal esearch th pplcatos Vol. / (0) 6-7. [8].. Zadeh Fuzzy sets If. otrol 8 (965) IEEE Iteratoal oferece o Fuzzy Systems (FUZZ) 67