Hybrid vector similarity measure of single valued refined neutrosophic sets to multi-attribute decision making problems

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1 Hybrd vecor smlary measure of sgle valued refed eurosohc ses o mul-arbue decso makg roblems Suraa ramak *, arha ram Dey, Bbhas C Gr 3 Dearme of Mahemacs, Nadalal Ghosh B College, aur, O- Narayaur, Dsrc Norh 4 argaas, code- 7436,Wes Begal, da,,3 Dearme of Mahemacs, Jadavur Uversy, Kolkaa-70003, Wes Begal, da * Corresodg Auhor: E-mal: sura_a@yahooco Absrac: hs aer rooses hybrd vecor smlary measures uder sgle valued refed eurosohc ses ad roves some of s basc roeres he roosed smlary measure s he aled for solvg mulle arbue decso makg roblems Lasly, a umercal examle of medcal dagoss s gve o he bass of he roosed hybrd smlary measures ad he resuls are comared h he resuls of oher exsg mehods o valdae he alcably, smlcy ad effecveess of he roosed mehod Keyords: Sgle valued eurosohc ses; Sgle valued refed eurosohc ses; Hybrd vecor smlary measures; Mul-arbue decso makg roduco Smaradache [] orgaed he heory of eurosohc ses NSs hch s characered by a ruh membersh A x, a deermacy membersh A x ad a falsy membersh A x o coe h deermae, comlee ad cosse formao Hoever, sgle valued eurosohoc ses SVNSs defed by Wag e al [] s useful ool for raccal decso makg uroses MADM uder SVNSs araced may researchers ad may mehods have bee roosed for MADM roblems such as OSS [3], grey relaoal aalyss [4, 5, 6, 7], ourakg aroach [8], maxmg devao mehod [9], hybrd vecor smlary measure [0], ec

2 Haafy e al [] roosed a mehod o deerme he correlao coeffce of NSs by usg cerod mehod Ye [] defed correlao of SVNSs, correlao coeffce of SVNSs, ad eghed correlao coeffce of SVNSs he, a mul-crera decso makg mehod MCDM as roosed based o eghed correlao coeffce or he eghed cose smlary measure Ye [3] develoed aoher form of correlao coeffce beee SVNSs ad reseed a MADM mehod Broum ad Smaradache [4] roosed a e mehod called exeded Hausdroff dsace for SVNSs ad a e seres of smlary measures ere develoed o fd he smlary of SVNSs Maumdar ad Samaa [5] roduced he coce of dsace of o SVNSs ad dscussed s roeres hey also reseed some smlary measures beee SVNSs based o dsace, a machg fuco, membersh grades ad defed he oo of eroy measure for SVNSs Ye [6] roosed cross eroy of SVNSs ad solved a MCDM based o he cross eroy of SVNSs Ye ad Zhag [7] formulaed hree smlary measures beee SVNSs by ulg maxmum ad mmum oeraors ad vesgaed her characerscs he develoed eghed smlary measures ere he emloyed for solvg MADM roblems uder sgle valued eurosohc seg Ye [8] suggesed hree smlary measures beee smlfed NSs as a exeso of he Jaccard, Dce ad cose smlary measures vecor sace for solvg MCDM roblems ramak e al [0] vesgaed a e hybrd vecor smlary measure uder boh sgle valued eurosohc ad erval eurosohc assessmes by exedg he oo of varao coeffce smlary mehod [9] h eurosohc formao ad roved some of s fudameal roeres Smaradache [0] geeraled he coveoal eurosohc logc ad defed he mos - symbol or umercal valued refed eurosohc logc Each eurosohc eleme,, ca be refed o,,, m, ad,,,, ad,,, q, resecvely, here m,, q are egers ad m + + q = Broum ad Smaradache [] roosed cose smlary measure for refed eurosohc ses due o cose smlary measure based Bhaacharya s dsace [] ad a mroved cose smlary measure for SVNSs roosed by Ye [3] Ye ad Ye [4] roduced he dea of sgle valued eurosohc mul ses SVNMSs refed ses by combg SVNSs alog h he heory of mulses [5] ad reseed several oeraoal relaos of SVNMSs Ye ad Ye [4] roosed Dce smlary measure ad eghed Dce smlary measure for SVNMSs ad vesgaed her roeres Chaeree e al

3 3 [6] slghly modfed he defo of SVNMSs [4] ad cororaed fe e se-heorec oeraors of SVNMSs ad her roeres Broum ad Del [7] defed correlao measure of eurosohc refed ses ad aled he roosed model o medcal dagoss ad aer recogo roblems Ye e al [8] furher defed geeraled dsace ad s o smlary measures beee SVNMSs ad aled he coce o medcal dagoss roblem Modal ad ramak [9] roosed eurosohc refed smlary measure based o coage fuco ad reseed a alcao o suable educaoal sream seleco roblem Del e al [30] suded several oeraors of eurosohc refed ses such as uo, erseco, covex, srogly covex order o deal h deermae ad cosse formao her aer, Del e al [30] also examed several resuls of eurosohc refed ses usg he roosed oeraors ad defed dsace measure of eurosohc refed ses h roeres Karaasla [3] develoed hree mehods based o smlary measure for sgle valued refed eurosohc ses SVRNSs ad erval eurosohc refed ses by exedg Jaccard, Dce ad Cose smlary measures of SVNSs- ad erval eurosohc ses roosed by Ye [8] Broum ad Smaradache [3] develoed a e smlary measure beee refed erosohc ses based o exeded Housdorff dsace of SVNSs ad roved some of her basc roeres Modal ad ramak [33] dscussed refed age smlary measure for SVNSs ad hey aled he roosed smlary measure o medcal dagoss roblems Jua-ua ad Ja-qag [34] defed several mul-valued eurosohc aggregao oeraors ad esablshed a MCDM mehod based o he roosed oeraors Ye ad u [35] reseed e smlary measure of SVNSs hrough age fuco ad a mul- erod medcal dagoss mehod usg he roosed smlary measure ad he eghed aggregao of mul-erod formao for solvg mul-erod medcal dagoss roblems uder sgle valued eurosohc evrome Ye ad Smaradache [36] reseed a MCDM mehod h sgle valued refed eurosohc formao by exedg he coce of smlary mehod h sgle valued eurosohc formao of Maumdar ad Samaa [5] hs aer, e roose aoher form of cose smlary measures uder SVRNSs by exedg he coce gve [37, 38] ad rove some of s basc roeres We roose hybrd vecor smlary measure h sgle valued refed eurosohc formao by exedg hybrd vecor smlary measure of SVNSs [0] ad rove some of s roeres he 3

4 4 roosed smlary measure s a hybrdao of Dce ad cose smlary measures uder sgle valued refed eurosohc formao Moreover, e esablsh eghed hybrd vecor smlary measure uder sgle valued refed eurosohc evrome ad rove s basc roeres No he arcle s srucured he follog ay Seco reses some mahemacal relmares hch are requred for he cosruco of he aer Seco 3 defes hybrd smlary ad eghed hybrd smlary measures of SVRNSs ad roves some of her roeres Seco 4 s devoed o develo o algorhms for solvg MADM roblems volvg sgle valued refed eurosohc formao A llusrave examle of medcal dagoss s solved o demosrae he alcably of he roosed rocedure Seco 5 Coclusos ad fuure scoe of research are reseed Seco 6 Mahemacal relmares hs Seco, e recall some basc defos cocerg eurosohc ses, sgle valued eurosohc ses, sgle valued refed eurosohc ses Neurosohc se [] Le U be a uversal sace of obecs h a geerc eleme of U deoed by he, a eurosohc se o U s defed as gve belo here,,, = {,,, U} : U ] - 0, + [ sad for he degree of membersh, he degree of deermacy, ad he degree of falsy-membersh resecvely of a o U o he se sasfyg he codo Sgle valued eurosohc ses [] Cosder U be a sace of os h a geerc eleme of U deoed by, he a SVNS s defed as follos: 4

5 5 = {,,, U} here, x, x, x : U [0, ] deoe he degree of membersh, he degree of deermacy, ad he degree of falsy-membersh resecvely of a o U o he se sasfyg he codo ad 0 x + x + x 3 for each o U 3 Sgle valued eurosohc refed ses [4] A SVNRS R he uverse U = {,,, } s defed as follos: R = {,,,,,,,,,,,, R R sr R R sr R R sr U} here R, R,, sr :U [0, ], R, R,, sr : U [0, ], R, R,, sr : U [0, ] such ha 0 R + R + R 3for =,,, s here, s s sad o be he dmeso of R Defo [4]: Le R ad R be o SVRNSs U, here R = {,,,,,,,,,,,, R R sr R R sr R R sr U}, R = {,,,,,,,,,,,, R R sr R R sr R R sr U}, he he relaos beee R ad R are reseed as follos: Coame: R R, f ad oly f,, R R R R R R for =,,, s Equaly: R= R, f ad oly f =, =, = R R R R R R for =,,, s 3 Uo: R R = {,, U} for =,,, s 5, R R R R R R

6 6 6 4 erseco: R R = {,,, R R R R R R U} for =,,, s 3 Hybrd vecor smlary measures of SVRNSs Defo 3 [37]: Le = {,,, U} ad = {,,, U} be o SVNSs o-refed he uverse of dscourse U he, he Dce smlary measure of SVNSs s defed as follos Dce, = ad f [0, ] be he egh of for =,,, such ha =, he he eghed Dce smlary measure of SVNSs ca be defed as follos Dce, = Defo 3 [38]: Le = {,,, U} ad = {,,, U} be o SVNSs o-refed he uverse of dscourse U = {,,, } he, he cose smlary measure of SVNSs s defed as gve belo Cos, = 3 ad f [0, ] be he egh of for =,,, sasfyg =, he he eghed cose smlary measure of SVNSs ca be defed as follos Cos, = 4

7 7 7 Defo 33 [0]: Hybrd vecor smlary measure of SVNSs Cosder = {,,, U} ad = {,,, U} be o SVNSs U he, he hybrd vecor smlary measure of ad s defed as follos: Hyb, = 5 here [0, ] Defo 34 [0]: Weghed hybrd vecor smlary measure of SVNSs he eghed hybrd vecor smlary measure of = {,,, U} ad = {,,, U} ca be defed as follos: WHyb, = 6 here [0, ] be he egh of for =,,, such ha =, ad [0, ] Defo 35 [4]: Dce smlary measure beee o SVNRSs, s defed as follos DceSVRNS, = 7 Defo 36 [4]: Weghed Dce smlary measure beee o SVNRSs, s reseed as follos

8 8 8 WDceSVRNS, = 8 Defo 37: Cose smlary measure beee o SVNRSs, ca be defed he follog ay: CosSVRNS, = 9 rooso 3 he defed cose smlary measure CosSVNRS, beee SVRNSs ad sasfes he follog roeres: 0 CosSVRNS, CosSVRNS, =, f ad oly f = 3 CosSVRNS, = CosSVRNS, roof : Accordg o Cauchy-Schar equaly:, here,,, ad,,,, e have

9 9 9 herefore,, So, CosSVRNS, =, Obvously, CosSVRNS, 0, hus 0 CosSVRNS, : f =, he,, ad for =,,, ; =,,, herefore, CosSVRNS, = = 3: CosSVRNS, = = = CosSVRNS, Defo 38: Weghed cose smlary measure beee SVNRSs, ca be defed as follos: WCosSVRNS, = 0

10 0 0 rooso 3 he defed eghed cose smlary measure WCosSVNRS, beee SVRNSs ad sasfes he follog roeres: 0 WCosSVRNS, WCosSVRNS, =, f ad oly f = 3 WCosSVRNS, = CosSVRNS, roof : rom Cauchy-Schar equaly, e have So,, [0, ] ad = WCosSVRNS, =, here [0, ] be he egh of for =,,, such ha = Obvously, WCosSVRNS, 0, ad herefore 0 WCosSVRNS, : f =, he,, ad for =,,, ; =,,,

11 WCosSVRNS, = = 3: WCosSVRNS, = = = WCosSVRNS, Nex, e have defed hybrd vecor smlary mehods beee SVRNSs by exedg he coce of ramak e al [0] as gve belo Defo 39: Hybrd vecor smlary measure beee SVNRSs, ca be defed as follos: HybSVRNS, = here [0, ] rooso 33 he defed sgle valued refed hybrd vecor smlary measure HybSVNRS, beee o SVRNSs ad sasfes he follog roeres: 0 HybSVRNS,

12 HybSVRNS, =, f ad oly f = 3 HybSVRNS, = HybSVRNS, roof rom Dce ad cose measures of SVRNSs defed Eq 7 ad Eq 9, e have 0 DceSVRNS,, 0 CosSVRNS, herefore, e have, DceSVRNS, =,for =,,,, CosSVRNS, =, for =,,,, Here, DceSVRNS, + - CosSVRNS, = =, for =,,, herefore, HybSVRNS,

13 3 3 = Also, DceSVRNS,, CosSVRNS, 0, for =,,, Obvously, HybSVRNS, 0 hs roves ha 0 HybSVNRS, or ay o SVNRSs ad, f =, hs mles =, =, =, for =,,, ad =,,, DceSVRNS, = =, for =,,,, ad CosSVRNS, = =, for =,,, Hece, HybSVRNS, = 3 HybSVRNS, =

14 4 4 = = HybSVRNS, Defo 0: Weghed hybrd vecor smlary measure beee SVRNSs ca be defed as follos WHyb, = Here, [0, ] rereses he egh of for =,,, such ha =, here [0, ], ad WHyb, should sasfy he follog roeres rooso 34 0 WHyb, WHyb, =, f ad oly f = 3 WHyb, = WHyb, roof Usg Dce ad cose measures of SVRNSs, e have 0 DceSVRNS,, 0 CosSVRNS,

15 5 5 DceSVRNS, + - CosSVRNS, = =, for =,,, herefore, WHyb, = DceSVRNS,, CosSVRNS, 0, for =,,, Obvously, WHyb, 0, herefore 0 WHyb, f =, he =, =, =, for =,,, ad =,,, DceSVRNS, = =, for =,,,, ad CosSVRNS, =

16 6 6 =, for =,,, Hece, WHyb, = 3 WHyb, = = = WHyb, 4 MADM h sgle valued refed eurosohc formao based o hybrd smlary measure Cosder = {,,, m} m be a dscree se of m caddaes, C = {C, C,, C}, be he se of arbues of each caddaes, ad A = {A, A,, Ak}, k be he se of aleraves of each caddae he decso maker or exer reses he rakg of aleraves h regard o each caddae he rakg rereses he erformaces of, =,,, m agas he arbues C, =,,, ad =,,, be he egh vecor of he arbues C, =,,, h 0 ad = he relao beee caddaes ad arbues, ad he relao beee arbues ad aleraves ca be reseed as follos see able ad able resecvely

17 7 7 able he relao beee caddaes ad re-defed arbues m m m m β β β β β β β β β C C C here β =,, rereses sgle valued eurosohc umbers SVNNs, =,,, m; =,,, ; =,,, s able he relao beee arbues ad aleraves k k k C C C A A A k Here, =,, deoes SVNNs, =,,, ; =,,, k We o develo o algorhms for MADM roblems based o hybrd smlary measure h sgle valued refed eurosohc formao as gve belo Algorhm Se Calculae he sgle valued refed hybrd smlary measures beee able ad able by usg Eq Se Rak he aleraves based o he descedg order of hybrd smlary measures he bgges value reflecs he bes alerave

18 8 Se 3 So Algorhm Se Comue he sgle valued refed eghed hybrd smlary measure beee able ad able by meas of Eq Se he aleraves are raked descedg order of he eghed hybrd smlary measures ad he bgger value meas he beer alerave Se 3 So 5 Alcao of he roosed mehod o medcal dagoss roblem We cosder he llusrave examle of medcal dagoss h sgle valued refed eurosohc formao suded [33] Medcal dagoss has o deal h a large amou of uceraes ad huge amou of formao avalable o he medcal racoers usg e ad advaced echologes he rocedure of classfyg dssmlar se of symoms uder a sgle ame of dseases s o easy [] Also, s ossble ha every obec has dffere ruh, deermae ad false membersh fucos ad he roosed smlary measures amog he aes versus symoms ad symoms versus dseases ll rovde he arorae medcal dagoss raccal suao, here may occur errors dagoss f e cosder daa from oe me observao ad herefore mul me seco, by cosderg he samles of same ae a dffere mes ll rovde bes medcal dagoss [39] Cosder = {,, 3, 4} be he se of four aes, C = {vral fever, malara, yhod, somach roblem, ches roblem} be he se of fve dseases, A = {emeraure, headache, somach a, cough, ches a} be he se of sx symoms No our obecve s o exame he ae a dffere me ervals ad e ll oba dffere ruh, deermae ad false membersh fucos for every ae Le hree observaos are ake a day: 7 am, m ad 6 m see he able 3 [33] 8

19 9 08, 0, 0 06, 03, 03 06, 03, 0 00, 08, 0 0, 06, 04 able 3 he relao beee aes ad symoms emeraure Headache Somach a Cough Ches a 06, 0, 03 0, 08, 00 06, 0, 03 0, 06, 03 05, 0, 04 03, 05, 0 04, 04, 04 03, 04, 05 05, 0, 0 0, 03, 04 04, 03, 03 0, 05, 04 0, 06, , 0, 0 06, 04, 0 05, 03, , 0, 03 04, 03, 0 05, 0, 03 04, 04, 0 05, 04, 0 04, 06, 03 08, 0, 0 06, 0, 04 06, 0, 03 05, 04, 0 04, 04, 04 05, 0, 04 06, 0, 03 04, 0, 05 03, 0, 04 00, 06, 04 0, 05, 05 03, 04, 06 03, 04, 03 0, 04, 05 0, 05, 04 0, 07, 0 0, 07, 05 03, 05, 04 0, 07, 0 0, 05, 05 0, 06, 03 07, 0, 0 05, 0, 04 06, 04, 0 0, 08, 0 03, 06, 04 03, 06, 03 00, 05, 05 0, 05, 03 03, 03, 04 03, 04, 03 04, 03, 04 03, 05, 05 he relao beee symoms ad dseases he form sgle valued eurosohc assessmes s gve he able 4 belo able 4 he relao beee symoms ad dseases Vral fever Malara yhod Somach roblem Ches roblem emeraure 06, 03, 03 0, 05, 03 0, 06, 04 0, 06, 06 0, 06, 04 Headache 04, 05, 03 0, 06, 04 0, 05, 04 0, 04, 06 0, 06, 04 Somach 0, 06, 03 00, 06, 04 0, 05, 05 08, 0, 0 0, 07, 0 a Cough 04, 04, 04 04, 0, 05 0, 05, 05 0, 07, 04 04, 05, 04 Ches a 0, 07, 04 0, 06, 03 0, 06, 04 0, 07, 04 08, 0, 0 No by usg Eq, Hybrd vecor refed smlary measures HVRSM by cosderg = 05 beee Relao ad Relao are reseed as gve belo see he able 5 able 5 HVRSM beee Relao ad Relao Vral fever Malara yhod Somach roblem Ches roblem

20 0 he maxmal HVRSM from able 5 deermes he roer medcal dagoss herefore, from able 5, e observe ha, 3, 4 suffer from vral fever, ad suffers from yhod Also, usg Eq, eghed hybrd vecor refed smlary measures WHVRSM h ko egh formao = 03, 0, 05, 0, 05 ad = 05 beee Relao ad Relao are reseed as gve belo see he able 6 able 6 Weghed hybrd vecor refed smlary measure WHVRSM beee Relao ad Relao Vral fever Malara yhod Somach roblem Ches roblem Here, e also see ha, 3, 4 suffer from vral fever, ad suffers from yhod By usg Eqs ad, HVRSMs ad WHVRSMs h dffere values of beee Relao ad Relao are reseed he follog ables 7-4 ad hch ae suffers from hch dsease s dcaed by mark belo he ables able 7 HVRSM beee Relao ad Relao he = 0 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, Somach roblem, 3 Vral fever, 4 Vral fever 0

21 able 8 HVRSM beee Relao ad Relao he = 05 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, Somach roblem, 3 Vral fever, 4 Vral fever able 9 HVRSM beee Relao ad Relao he = 075 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever able 0 HVRSM beee Relao ad Relao he = 090 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever able WHVRSM beee Relao ad Relao he = 0 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever

22 able WHVRSM beee Relao ad Relao he = 05 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever able 3 WHVRSM beee Relao ad Relao he = 075 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever able 4 WHVRSM beee Relao ad Relao he = 090 Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever Noe By usg eurosohc refed age smlary measure, Modal ad ramak [33] obaed he resuls as sho able 5

23 3 able 5 he age refed smlary measure beee Relao ad Relao [3] Vral fever Malara yhod Somach roblem Ches roblem Vral fever, yhod, 3 Vral fever, 4 Vral fever rom he above able 5, e foud ha, 3, 4 suffer from vral fever, ad suffers from yhod 5 Cocluso We vesgae hybrd vecor smlary ad eghed hybrd vecor smlary measures h sgle valued refed eurosohc assessmes ad roved some of her basc roeres he, he roosed hybrd smlary measures have bee used o solve a medcal dagoss roblem We comare he obaed resuls h dffere values of he arameer ad h he resuls of oher exsg mehod order o verfy he effecveess of he roosed rocedure We hoe ha he roosed hybrd vecor smlary measure ca be aled o solve decso makg roblems eurosohc evrome such as faul dagoss, cluser aalyss, daa mg, vesme, ec Refereces [] Smaradache A ufyg feld of logcs Neurosohy: Neurosohc robably, se ad logc, Amerca Research ress, Rehoboh, 998 [] H Wag, Smaradache, Y Zhag, ad R Suderrama Sgle valued eurosohc ses, Mul-sace ad Mul-Srucure 4 00, [3] Z Zhag, C Wu A ovel mehod for sgle-valued eurosohc mul-crera decso makg h comlee egh formao, Neurosohc Ses ad Sysems [4] Bsas, S ramak, BC Gr A e mehodology for eurosohc mul-arbue decso makg h uko egh formao, Neurosohc Ses ad Sysems [5] Bsas, S ramak, BC Gr Eroy based grey relaoal aalyss mehod for mul-arbue decso makg uder sgle valued eurosohc assessme, Neurosohc Ses ad Sysems

24 4 [6] K Modal, S ramak Neurosohc Decso makg model of school choce, Neurosohc Ses ad Sysems [7] K Modal, S ramak Neurosohc decso makg model for clay-brck seleco cosruco feld based o grey relaoal aalyss, Neurosohc Ses ad Sysems [8] J eg, J Wag, H Zhag, X Che A ourakg aroach for mul-crera decso-makg roblems h smlfed eurosohc ses, Aled Sof Comug [9] R Şah, Lu Maxmg devao mehod for eurosohc mulle arbue decso makg h comlee egh formao, Neural Comug ad Alcaos 05, DO: 0007/s [0] S ramak, Bsas, BC Gr, Hybrd vecor smlary measures ad her alcaos o mul-arbue decso makg uder eurosohc evrome, Neural Comug ad Alcaos 05, 0007/s [] M Haafy, AA Salama, KM Mahfou Correlao coeffce of eurosohc ses by cerod mehod, eraoal Joural of robably ad Sascs [] J Ye Mulcrera decso makg mehod usg he correlao coeffce uder sgle valued eurosohc evrome, eraoal Joural of Geeral Sysems [3] J Ye Aoher form of correlao coeffce beee sgle valued eurosohc ses ad s mulle arbue decso makg mehod, Neurosohc Ses ad Sysems [4] S Broum, Smaradache Several smlary measures of eurosohc ses, Neurosohc Ses ad Sysems 03, 54-6 [5] Maumder, SK Samaa, O smlary ad eroy of eurosohc ses, Joural of ellge ad uy Sysems [6] J Ye Sgle valued eurosohc cross-eroy for mulcrera decso makg roblems, Aled Mahemacal Modellg [7] J Ye, Zhag Sgle valued eurosohc smlary measures for mulle arbue decso makg, Neurosohc Ses ad Sysems [8] J Ye Vecor smlary measures of smlfed eurosohc ses ad her alcao mulcrera decso makg, eraoal Joural of uy Sysems [9] X Xu, L Zhag, Wa, A varao coeffce smlary measure ad s alcao emergecy grou decso makg, Sysem Egeerg roceda [0] Smaradache -Valued refed eurosohc logc ad s alcaos o hyscs, rogress hyscs [] S Broum, Smaradache Neurosohc refed smlary measure based o cose fuco, Neurosohc Ses ad Sysems [] A Bhaacharya O a measure of dvergece of o mulmoal oulao, SakhyaSer A

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