Theory of Complex Fuzzy Soft Set and its Applications

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1 IJIRST Intrntionl Jornl for Innovtiv Rsrc in Scinc & Tcnology Volm Iss 0 Mrc 07 ISSN onlin: Tory of ompl Fzzy Soft St nd its pplictions P. Tirnvkrs R. Srs ssistnt Profssor ssistnt Profssor Dprtmnt of Mtmtics Dprtmnt of Mtmtics Priyr E.V.R ollg tonomos Tircirpplli 60 0 Tmilnd Indi. V. sokkmr ssocit Profssor Dprtmnt of Mtmtics Sins ollg of Tcnology Sltnt of Omn bstrct Kings ollg of Enginring Pnlklm Pdkkotti Dt 6 0 Tmilnd Indi. T objctiv of tis ppr is to invstigt t frtr dvlopmnt of tory of soft compl fzzy st. onsqntly mjor prt of tis work is ddictd to discssion of t intitiv intrprttion of ggrgtion oprtion in soft compl fzzy st. W giv n mpl of possibl pplictions wic dmonstrt t pplictions of ggrgtion oprtions tt t mtod cn b sccssflly pplid to mny problms tt contins ncrtintis nd priodicitis. Kywords: compl fzzy st soft fzzy st soft compl fzzy st ggrgtions of soft compl fzzy st I. INTRODUTION ompl fzzy st FS []-[] is nw dvlopmnt in t tory of fzzy systms []. T concpt of FS is n tnsion of fzzy st by wic t mmbrsip for c lmnt of compl fzzy st is tndd to compl-vld stt. Soft st tory is gnrliztion of fzzy st tory wic ws proposd by Molodtsov [8] in 999 to dl wit ncrtinty in non-prmtric mnnr. On of t most importnt stps for t tory of soft sts ws to dfin mppings on soft sts; tis ws civd in 009 by mtmticin tr Krl tog t rslts wr pblisd in 0. Soft sts v lso bn pplid to t problm of mdicl dignosis for s in mdicl prt systms. Fzzy soft sts v lso bn introdcd in [0]. Mppings on fzzy soft sts wr dfind nd stdid in t grond brking work of Krl nd md. Soft compl fzzy sts wic is dfind by [] Tis ppr writtn for inspird from [90] wrs ll t concpts in soft sts wr rplcd by soft compl fzzy sts. T ppr is orgnizd s follows: Sction rviws t notions of soft sts; compl fzzy st nd rlvnt dfinitions sd in t proposd work nd lso discssd t innovtiv concpt of soft compl fzzy sts wit mpls. In sction w introdc t ggrgtion oprtion on soft compl fzzy st nd its proprtis. In sction pplictions of soft compl fzzy st wit mpl providd. W lso dmonstrt sccssfl ppliction of soft compl fzzy st sing ggrgtion oprtion. Finlly w concld t ppr in sction 5. Dfinition.. II. PRELIMINRIES Lt U b n initil nivrs PU b t powr st of U E b t st of ll prmtrs nd [8] F ovr U is st dfind by fnction f rprsnting mpping f : E PU sc tt f = if. E. Tn soft st s dfind in Hr f is clld pproimt fnction of t soft st F nd t vl f is st clld -lmnt of t soft st for ll E. It is wort noting tt t sts f my b rbitrry. Som of tm my b mpty som my v nonmpty intrsction. Ts soft st F ovr U cn b rprsntd by t st of ordrd pirs F f : E f P U Not tt t st of ll soft sts ovr U will b dnotd by S U. Dfinition.. [8] pir F E is clld soft st ovr U if nd only if F is mpping of E into t st of ll sbsts of t st U. ll rigts rsrvd by

2 Tory of ompl Fzzy Soft St nd its pplictions IJIRST/ Volm / Iss 0/ 00 In otr words t soft st is prmtrizd fmily of sbsts of t st U. Evry st F from tis fmily my b considrd s t st of ε- lmnts of t soft st FE or s t st of pproimt lmnts of t soft st. Empl.. Lt U={ } b t st of for oss ndr considrtion nd E={p costly p Btifl p Modrn Tcnology p lrios p 5fcility}b t st of prmtrs nd ={p p p } E. Tn F = {Fp ={ } Fp ={ }Fp ={ }} is t soft st rprsnting t ttrctivnss of t os wic prson is going to by. Dfinition.. Rmot t l. [0] proposd n importnt tnsion of ts ids t ompl Fzzy Sts wr t mmbrsip fnction μ instd of bing rl vld fnction wit t rng [0] is rplcd by compl-vld fnction of t form S μ s = r s jωs ; j= wr r S nd r bot rl vld giving t rng s t nit circl. Howvr tis concpt is diffrnt from fzzy compl nmbr introdcd nd discssd by Bckly [-5] nd Zng. Essntilly s plind in [0] tis still rtins t crctriztion of t ncrtinty trog t mplitd of t grd of mmbrsip ving vl in t rng of [0 ] wilst dding t mmbrsip ps cptrd by fzzy sts. s plind in Rmot t l [0] t ky ftr of compl fzzy sts is t prsnc of ps nd its mmbrsip. Dfinition.5. Lt U b n initil nivrs E b t st of ll prmtrs E nd b fzzy st ovr U or ll E. Tn n Fzzy Soft fs-st ovr U is st dfind by fnction rprsnting mpping :E FU sc tt ; if. Hr is clld fzzy pproimt fnction of t fs-st nd t vl is fzzy st clld -lmnt of t fsst for ll E. Ts n fs-st ovr U cn b rprsntd by t st of ordrd pirs : E F U Not tt t st of ll t fzzy sts ovr U will b dnotd by FU nd from now on t sts of ll fs-sts ovr U will b dnotd by FS U. Empl.6. Lt U={ } b t st of for oss ndr considrtion nd E={p costly p Btifl p Modrn Tcnology p lrios p 5fcility}b t st of prmtrs nd ={p p p } E. Tn F={Fp ={0./ 0./ 0.6/ / } Fp ={0.6/ 0./ 0.6/ / }Fp ={0./ 0./ / 0./ }} is t fzzy soft st rprsnting t ttrctivnss of t os wic prson is going to by. Dfinition.7. Lt U b n initil nivrs E b t st of ll prmtrs E nd b compl fzzy st ovr U for ll E. Tn n Soft compl fzzy st ovr U is st dfind by fnction rprsnting mpping : E U : sc tt if. Hr is clld compl fzzy pproimt fnction of t soft compl fzzy st nd t vl is compl fzzy st clld -lmnt of t soft compl fzzy st for ll E. Ts soft compl fzzy st ovr U cn b rprsntd by t st of ordrd pirs : E U Not tt t st of ll t compl fzzy sts ovr U will b dnotd by U. Oprtions of compl fzzy sts nd soft compl fzzy sts wr dfind in [] rspctivly. Empl.8. [t] Lt U= { Indi strli UK US} b n initil st considr E= { Infltion rt popltion growt Unmploymnt rt sr mrkt ind} b n contry s growt prmtrs st nd E={ } compl fzzy st is dfind s ll rigts rsrvd by

3 Tory of ompl Fzzy Soft St nd its pplictions IJIRST/ Volm / Iss 0/ tn soft compl fzzy st is writtn by 0. j j j 0.5 j j j j 0.9 j 0.8 j 0.8 j j j 0.9 j 0.95 j 0.9 j 0.95 j 0.95 III. GGREGTION OF SOFT OMPLEX FUZZY SET In tis sction w dfin n ggrgtion oprtor on soft compl fzzy st tt prodcs n ggrgt fzzy st from soft compl fzzy st nd its crdinl st. T pproimt fnctions of soft compl fzzy st r fzzy. soft compl fzzy st ggrgtion oprtor on t fzzy sts is n oprtion by wic svrl pproimt fnctions of soft compl fzzy st r combind to prodc singl fzzy st wic is t ggrgt fzzy st of t soft compl fzzy st. Onc n ggrgt fzzy st s bn rrivd t it my b ncssry to coos t bst singl crisp ltrntiv from tis st. Dfinition.. ssm tt U... E... Lt U. following tbl m n nd E tn t cn b prsntd by t n n n m m Wr is t mmbrsip fnction of. m n m If for i... m nd j... tn t soft compl fzzy st is niqly crctrizd by ij mtri ovr U. n i ij mxn Dfinition.. m m n n mn is clld n Lt U. Tn t crdinl st of dnotd by ovr E. T mmbrsip fnction of is dfind by : E 0 m n soft compl fzzy st mtri of t soft compl fzzy st is dfind by / : E U wr U is t crdinlity of nivrs U nd is t sclr crdinlity of fzzy st crdinl sts of t soft compl fzzy sts ovr U will b dnotd by U. It is clr tt U F E is fzzy st. Not tt t st of ll ll rigts rsrvd by 5

4 Dfinition.. Lt U nd U following tbl E. ssm tt E... Tory of ompl Fzzy Soft St nd its pplictions IJIRST/ Volm / Iss 0/ 00 n nd E tn cn b prsntd by t If j= for j= n tn t crdinl st j wic is clld t crdinl mtri of t crdinl st Dfinition.. n n is niqly crctrizd by mtri ovr E. Lt U nd U. Tn soft compl fzzy ggrgtion oprtor dnotd by FS gg FS : U U F U FS gg gg Wr / : U fzzy st is ovr U.. T mmbrsip fnction of is dfind s follows: : U [0] wr E is t crdinlity of E. E Dfinition.5. Lt U nd following tbl i If i= for i= m tn mtri of ovr U. E b its ggrgt fzzy st. ssm ttu j n n is dfind by is clld t ggrgt fzzy st of t soft compl fzzy st m m m is niqly crctrizd by t mtri i m tn t m cn b prsntd by t wic is clld t ggrgt IV. PPLITIONS OF SOFT OMPLEX FUZZY SET It is mc sir to dscrib mn bvior dirctly sowing t st of strtgis wic prson my coos in prticlr sittion spcilly priodic pnomnon. T sittion my b mor complictd in rl world bcs of t fzzy crctrs of t prmtrs [9 0]. In fzzy st t compl fzzy st is tndd to fzzy on; t compl fzzy mmbrsip is sd to dscrib t prmtr pproimt lmnts of soft compl fzzy st. Tis tory lso sd in dt mining pplictions nd dcision mking problm. Tis tory sd in stdy of smootnss of fnctions gm tory oprtions rsrc Rimnn-intgrtion Prron intgrtion probbility Tory of msrmnt tc. Onc n ggrgt fzzy st s bn rrivd t it my b ncssry to coos t bst ltrntiv from tis st. Trfor w cn mk dcision by t following lgoritm. Stp : onstrct n soft compl fzzy st ovr U Stp : Find t crdinl st of for mplitd trm nd ps trm in sprtly Stp : Find t ggrgt fzzy st of lso find for sprtly ll rigts rsrvd by 6

5 Tory of ompl Fzzy Soft St nd its pplictions IJIRST/ Volm / Iss 0/ 00 Stp : Find t bst ltrntiv from tis st tt s t lrgst mmbrsip grd by m modls of Empl.. Sppos bsinss mn wnt to by sr from sr mrkt. Tr r tr sm kind of sr wic form t st of ltrntivs U. T prt committ considr st of prmtrs E. For i = t prmtrs i stnd for crrnt trnd of compny prformnc prticlr compny s stock pric for lst on yr Hom contry infltion rt nd crrnt sittion of prticlr sr s contry sr mrkt rspctivly. nd E compl fzzy sts nd is dfind s Tn Stp.: Soft compl fzzy st is writtn by j 0.5 j j j Stp.: T crdinl is comptd j 0.8 j j j j 0.6 mplitd Trm / 0.6 / / Ps Trm 0.66 / Stp.: T ggrgt fzzy st is fond by following mtod M M mplitd Trm Ps Trm / 0.87 / j j / 0. / 0.5 / / onsidr t modls vl of M 0. / 0.58 / / 0.6 / 0.6 /.0 Tis mns tt t t sr {} s t lrgst mmbrsip grd. Hnc prt committ my b sggstd for by t sr mong otr srs. j 0.85 V. ONLUSION In soft compl fzzy sts t soft st tory is tndd to compl fzzy st; t fzzy mmbrsip is sd to dscrib prmtr pproimt lmnts of compl fzzy soft st. To dvlop t tory in tis work w introdcd ggrgt oprtion of soft compl fzzy st nd its ppliction in dcision mking problms. Finlly w providd n mpl dmonstrting t sccssflly ppliction of tis mtod. It my b pplid to mny filds wit problms tt contin ncrtinty nd priodicity nd it wold b bnficil to tnd t proposd mtod to sbsqnt stdis in compl fzzy st. ll rigts rsrvd by 7

6 Tory of ompl Fzzy Soft St nd its pplictions IJIRST/ Volm / Iss 0/ 00 Ts in ll t soft compl fzzy st sms to b promising nw concpt pving t wy to nmros possibilitis for ftr rsrc. REFERENES [] Gngqn Zng Trm Sing Dillon Ki-Yn i Jn M nd Ji L Oprtion Proprtis nd δ-eqlitis of ompl Fzzy Sts Fzzy sts nd Fzzy Systms. [] Bckly. J.J. 987 Fzzy compl nmbrs in Procdings of ISFK Gngzo in [] Bckly. J.J. 989 Fzzy compl nmbrs Fzzy Sts nd Systms Vol. No.-5 [] Bckly. J.J. 99 Fzzy compl nlysis I: Dfinition Fzzy Sts nd Systms Vol. No [5] Bckly. J.J. 99 Fzzy compl nlysis II: Intgrtion Fzzy Sts nd Systms Vol. 9 No [6] Ibrim..M.O. ysf Dvlopmnt of Soft St Tory mricn Intrntionl Jornl of ontmporry Rsrc Vol. No. 90; pp [7] Kld lzym Sfid bdl Hlim bdl Rzk Sll nd Nsrddin Hssn Soft IFS pplid Mtmticl ScincsVol.6 No. 50 pp [8] Molodtsov. D.. Soft st tory First rslt omptrs Mt. ppl. 7 / [9] Nim gmn Filiz itk Srdr Enginogl FP-Soft st Tory nd its pplictions nnls of Fzzy Mtmtics nd Informtics Vol. No. 0 pp. 9 6 [0] Nim. gmn S. Enginogl nd F. itk Fzzy Soft St Tory nd Its pplictions Irnin Jornl of Fzzy Systms Vol.8 No. 0 pp [] Rmot.D R. Milo M. Fridmn nd. Kndl ompl fzzy sts IEEE Trns. Fzzy Syst. vol. 0 No. 00 pp [] Rmot.D M. Fridmn G. Lngolz nd. Kndl ompl fzzy logic IEEE Trns. Fzzy Syst.vol. no. 00 pp [] Tirnvkrs.P R. Srs nd P. Tmilmni pplictions of ompl Fzzy Sts JP Jornl of pplid Mtmtics Vol.6 Isss &0 pp. 5-. [] Zd. L.. Fzzy Sts Inform. nd ontrol pp. 8-5 ll rigts rsrvd by 8

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