Neutrosophc Sets ad Systems Vol 2 28 3 Uversty of New Meco Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased Kalya Modal Surapat Pramak 2 ad bhas Gr 3 Departmet of Mathematcs Jadavpur Uversty Kolkata: 732 West egal da E mal:kalyamathematc@gmalcom ²Departmet of Mathematcs Nadalal Ghosh ollege Papur P O - Narayapur ad Dstrct: North 24 Pargaas P ode: 74326 West egal da Emal: sura_pat@yahooco 3 Departmet of Mathematcs Jadavpur Uversty Kolkata: 732 West egal da Emal: bbhascgr@jadavpuruversty bstract: ths paper we troduce ew type of smlarty measures for sgle valued eutrosophc sets based o hyperbolc se fucto he ew smlarty measures are amely sgle valued eutrosophc hyperbolc se smlarty measure ad weghted sgle valued eutrosophc hyperbolc se smlarty measure We prove the basc propertes of the proposed smlarty measures We also develop a mult-attrbute decsomakg strategy for sgle valued eutrosophc set based o the proposed weghted smlarty measure We preset a umercal eample to verfy the practcablty of the proposed strategy ally we preset a comparso of the proposed strategy wth the estg strateges to ehbt the effectveess ad practcalty of the proposed strategy Keywords: Sgle valued eutrosophc set Hyperbolc se fucto Smlarty measure MDM ompromse fucto troducto Smaradache [] troduced the cocept of eutrosophc set NS to deal wth mprecse ad determate data the cocept of NS truth-membershp determacymembershp ad falsty-membershp are depedet determacy plays a mportat role may real world decso-makg problems NS geeralzes the ator set dscovered by Smth [2] 874 ad troduced by Germa mathematca ator [3] 883 fuzzy set troduced by Zadeh [4] tutostc fuzzy set proposed by taassov [5] Wag et al [6] troduced the cocept of sgle valued eutrosophc set SVNS that s the subclass of a eutrosophc set SVNS s capable to represet mprecse complete ad cosstet formato that mafest the real world Neutrosophc sets ad ts varous etesos have bee studed ad appled dfferet felds such as medcal dagoss [7 8 9] decso makg problems [ 2 3 4] socal problems [5 6] educatoal problem [7 8] coflct resoluto [9] mage processg [ 2 2 22] etc he cocept of smlarty s very mportat studyg almost every scetfc feld May strateges have bee proposed for measurg the degree of smlarty betwee fuzzy sets studed by he [23] he et al [24] Hyug et al [25] Papps ad Karacaplds [26] Pramak ad oy [27] etc Several strateges have bee proposed for measurg the degree of smlarty betwee tutostc fuzzy sets studed by Xu [28] Papakostas et al [29] swas ad Pramak [3] Modal ad Pramak [3] etc However these strateges are ot capable of dealg wth the smlarty measures volvg determacy SVNS ca hadle ths stuato the lterature few studes have addressed smlarty measures for eutrosophc sets ad sgle valued eutrosophc sets [32 33 34 35] Ye [36] proposed a MDM method wth completely ukow weghts based o smlarty measures uder SVNS evromet Ye [37] proposed vector smlarty measures of smplfed eutrosophc sets ad appled t mult-crtera decso makg problems Ye [38] developed mproved cose smlarty measures of smplfed eutrosophc sets for medcal dagoss Ye [39] also proposed epoetal smlarty measure of eutrosophc umbers for fault dagoses of steam turbe Ye [4] developed clusterg algorthms based o smlarty measures for SVNSs Ye ad Ye [4] proposed Dce smlarty measure betwee sgle valued eutrosophc multsets Ye et al [42] proposed dstacebased smlarty measures of sgle valued eutrosophc multsets for medcal dagoss Ye ad u [43] developed a sgle valued eutrosophc smlarty measure based o taget fucto for mult-perod medcal dagoss hybrd evromet Pramak ad Modal [44] proposed cose smlarty measure of rough eutrosophc sets ad provded ts applcato medcal dagoss Pramak ad Modal [45] also proposed cotaget Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure based
4 Neutrosophc Sets ad Systems Vol 2 28 smlarty measure of rough eutrosophc sets ad ts applcato to medcal dagoss esearch gap: MDM strategy usg smlarty measure based o hyperbolc se fucto uder sgle valued eutrosophc evromet s yet to appear esearch questos: s t possble to defe a ew smlarty measure betwee sgle valued eutrosophc sets usg hyperbolc se fucto? s t possble to develop a ew MDM strategy based o the proposed smlarty measures sgle valued eutrosophc evromet? Havg motvated from the above researches o eutrosophc smlarty measures we have troduced the cocept of hyperbolc se smlarty measure for SVNS evromet he ew smlarty measures called sgle valued eutrosophc hyperbolc se smlarty measure SVNHSSM ad sgle valued eutrosophc weghted hyperbolc se smlarty measure SVNWHSSM he propertes of hyperbolc se smlarty are establshed We have developed a MDM model usg the proposed SVNWHSSM he proposed hyperbolc se smlarty measure s appled to mult-attrbute decso makg he objectves of the paper: o defe hyperbolc se smlarty measures for SVNS evromet ad prove some of t s basc propertes o defe copromse fucto for determg ukow weght of attrbutes o develop a mult-attrbute decso makg model based o proposed smlarty measures o preset a umercal eample for the effcecy ad effectveess of the proposed strategy est of the paper s structured as follows Secto 2 presets prelmares of eutrosophc sets ad sgle valued eutrosophc sets Secto 3 s devoted to troduce hyperbolc se smlarty measure for SVNSs ad some of ts propertes Secto 4 presets a method to determe ukow attrbute weghts Secto 5 presets a ovel decso makg strategy based o proposed eutrosophc hyperbolc se smlarty measure Secto 6 presets a llustratve eample for the applcato of the proposed method Secto 7 presets a comparso aalyss for the applcablty of the proposed strategy Secto 8 presets the ma cotrbutos of the proposed strategy ally secto 9 presets cocludg remarks ad scope of future research 2 Neutrosophc prelmares 2 Neutrosophc set NS Defto 2 [] Let U be a uverse of dscourse he the eutrosophc set P ca be preseted of the form: P = {< : P P P> U} where the fuctos : U ] + [ defe respectvely the degree of membershp the degree of determacy ad the degree of o-membershp of the elemet U to the set P satsfyg the followg the codto sup P + sup P + sup P 3 + 22 Sgle valued eutrosophc set SVNS Defto 22 [6] Let X be a space of pots wth geerc elemets X deoted by SVNS P X s characterzed by a truth-membershp fucto P a determacy-membershp fucto P ad a falsty membershp fucto P for each pot X P P P [ ] Whe X s cotuous a SVNS P ca be wrtte as follows: P P P P : X X Whe X s dscrete a SVNS P ca be wrtte as follows: P P P P : X or two SVNSs P SVNS = {<: P P P > X} ad Q SVNS = {< Q Q Q> X } the two relatos are defed as follows: P SVNS Q SVNS f ad oly f P Q P Q P Q 2 P SVNS = Q SVNS f ad oly f P = Q P = Q P = Q for ay X 3 Hyperbolc se smlarty measures for SVNSs Let = < > ad = < > be two SVNSs Now hyperbolc se smlarty fucto whch measures the smlarty betwee two SVNSs ca be preseted as follows see Eq : SVNHSSM sh heorem he defed hyperbolc se smlarty measure SVNHSSM betwee SVNSs ad satsfes the followg propertes: Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased
Neutrosophc Sets ad Systems Vol 2 28 5 Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased SVNHSSM 2 SVNHSSM = f ad oly f = 3 SVNHSSM = SVNHSSM 4 f s a SVNS X ad the SVNHSSM SVNHSSM ad SVNHSSM SVNHSSM Proofs: or two eutrosophc sets ad 3 sh Hece SVNHSSM 2 or ay two SVNSs ad f = = = = Hece SVNHSSM = oversely SVNHSSM = hs mples = = = Hece = 3 Sce We ca wrte SVNHSSM = SVNHSSM 4 for X Now we have the followg equaltes: ; ; hus SVNHSSM SVNHSSM ad SVNHSSM SVNHSSM 3 Weghted hyperbolc se smlarty measures for SVNSs Let = < > ad = < > be two SVNSs Now weghted hyperbolc se smlarty fucto whch measures the smlarty betwee two SVNSs ca be preseted as follows see Eq 2: w sh SVN WHSSM 2 Here w w heorem 2 he defed weghted hyperbolc se smlarty measure SVNWHSSM betwee SVNSs ad satsfes the followg propertes: SVNWHSSM 2 SVNWHSSM = f ad oly f = 3 SVNWHSSM = SVNWHSSM 4 f s a SVNS X ad the SVNWHSSM SVNWHSSM ad SVNWHSSM SVNWHSSM Proofs: or two eutrosophc sets ad 3 sh ga w w Hece SVNWHSSM 2 or ay two SVNSs ad f =
6 Neutrosophc Sets ad Systems Vol 2 28 = = = Hece SVNWHSSM = oversely SVNWHSSM = hs mples = = = Hece = 3 Sce We ca wrte SVNWHSSM = SVNWHSSM 4 for X Now we have the followg equaltes: ; ; hus SVNWHSSM SVNWHSSM ad SVNWHSSM SVNWHSSM 4 Determato of ukow attrbute weghts Whe attrbute weghts are completely ukow to decso makers the etropy measure [46] ca be used to calculate attrbute weghts swas et al [47] employed etropy measure for MDM problems to determe completely ukow attrbute weghts of SVNSs 4 ompromse fucto he compromse fucto of a SVNS = j j = 2 m; j = 2 s defed as follows see Eq 3: j m j j j 2 3 3 j he weght of j-th attrbute s defed as follows see Eq 4 w j Here j j j w j j heorem 3 he compromse fucto j satsfes the followg propertes: P j f j j j P2 j f j j j P3 j E j f j j ad j j j j Proofs P j j j m j 3 3 m m m P2 j j j m j 3 m P3 j j m m 2 j j j 3 2 j j j 3 m m j j Sce j j ad j j j j Hece j j 5 Decso makg procedure Let 2 m be a dscrete set of alteratves 2 be the set of attrbutes of each alteratve he values assocated wth the alteratves = 2 m agast the attrbute j j = 2 for MDM problem s preseted a SVNS based decso matr he steps of decso-makg see gure 2 based o sgle valued eutrosophc weghted hyperbolc se smlarty measure SVNWHSSM are preseted usg the followg steps Step : Determato of the relato betwee alteratves ad attrbutes he relato betwee alteratves = 2 m ad the attrbute j j = 2 s preseted the Eq 5 4 Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased
Neutrosophc Sets ad Systems Vol 2 28 7 D[ ] 2 m Here 2 m j 2 m j j 2 m 2 22 m2 2 2 22 m2 2 22 m2 2 m 2 m 2 m 5 = 2 m; j = 2 be SVNS assessmet value Step 2: Determe the weghts of attrbutes Usg the Eq 3 ad 4 decso-maker calculates the weght of the attrbute j j = 2 Step 3: Determe deal soluto Geerally the evaluato attrbute ca be categorzed to two types: beeft type attrbute ad cost type attrbute the proposed decso-makg method a deal alteratve ca be detfed by usg a mamum operator for the beeft type attrbutes ad a mmum operator for the cost type attrbutes to determe the best value of each attrbute amog all the alteratves herefore we defe a deal alteratve as follows: * = { * 2* m*} Here beeft attrbute * j ca be preseted as follows: * ma m m j j j 6 j for j = 2 * Smlarly the cost attrbute j ca be preseted as follows: * m ma ma j j j 7 j for j = 2 Step 4: Determe the smlarty values Usg Eqs 2 ad 5 calculate SVNWHSSM values for each alteratve betwee postve or egatve deal solutos ad correspodg sgle valued eutrosophc from decso matr D[ ] Step 5: akg the alteratves akg the alteratves s prepared based o the descedg order of smlarty measures Hghest value dcates the best alteratve Step 6: Ed 6 Numercal eample ths secto we llustrate a umercal eample as a applcato of the proposed approach We cosder a decso-makg problem stated as follows Suppose a perso who wats to purchase a SM card for hs/her moble coecto herefore t s ecessary to select sutable SM card for hs/her moble coecto fter tal screeg there are four possble alteratves SM cards for moble coecto he alteratves SM cards are preseted as follows: : rtel 2: Vodafoe 3: SNL 4: elace Jo he perso must take a decso based o the followg fve attrbutes of SM cards: : Servce qualty 2: ost 3: tal talk tme 4: all rate per secod 5: teret ad other facltes he decso-makg strategy s preseted usg the followg steps Step : Determe the relato betwee alteratves ad attrbutes he relato betwee alteratves 2 3 ad 4 ad the attrbutes 2 3 4 5 s preseted the Eq 8 D[ 2 3 4 2 3 7 3 3 5 3 8 2 2 6 3 4 ] 5 2 6 4 3 7 3 6 4 3 5 2 3 8 7 3 6 6 3 4 5 4 4 6 7 3 5 2 Step 2: Determe the weghts of attrbutes 5 5 3 2 5 2 3 5 3 4 9 Usg the Eq 3 ad 4 we calculate the weght of the attrbutes 2 3 4 5 as follows: [w w 2 w 3 w 4 w 5 ] = [223 97 278 29 973] Step 3: Determe deal soluto ths problem attrbutes 3 4 5 are beeft type attrbutes ad 2 s the cost type attrbute * = {8 5 4 3 8 7 9 } Step 4: Determe the weghted smlarty values Usg Eq 2 ad Eq 8 we calculate smlarty measure values for each alteratve as follows SVNWHSSM * = 92422 SVNWHSSM * 2 = 95629 SVNWHSSM * 3 = 97866 8 Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased
8 Neutrosophc Sets ad Systems Vol 2 28 SVNWHSSM * 4 = 96795 Step 5: akg the alteratves akg the alteratves s prepared based o the descedg order of smlarty measures see gure Now the fal rakg order wll be as follows 3 4 2 Hghest value dcates the best alteratve Step 6: Ed Weghted smlarty measure values 8 6 4 2 2 3 4 lteratves GUE : Graphcal represetato of alteratves versus weghted smlarty measures 7 omparso aalyss he rakg results calculated from proposed strategy ad dfferet estg strateges [38 48 49 5] are furshed able We observe that the rakg results obtaed from proposed ad estg strateges the lterature dffer he proposed strategy reflects that the optmal alteratve s 3 he rakg result obtaed from Ye [38] s smlar to the proposed strategy he rakg results obtaed from Ye ad Zhag [48] ad Modal ad Pramak [49] dffer from the optmal result of the proposed strategy Ye [5] the rakg order dffers but the best alteratve s the same to the proposed strategy able he rakg results of estg strateges Strateges akg results Ye ad Zhag[48] 4 2 3 Modal ad Pramak [49] 4 3 2 Ye [38] 3 4 2 Ye [5] 3 2 4 Proposed strategy 3 4 2 2 We have proposed compromse fucto for calculatg ukow weghts structure of attrbutes SVNS evromet 3 We develop a decso makg strategy based o the proposed weghted smlarty measure SVNWHSSM 4 Steps ad calculatos of the proposed strategy are easy to use 5 We have solved a umercal eample to show the feasblty applcablty ad effectveess of the proposed strategy 9 ocluso the paper we have proposed hyperbolc se smlarty measure ad weghted hyperbolc se smlarty measures for SVNSs ad proved ther basc propertes We have proposed compromse fucto to determe ukow weghts of the attrbutes SVNS evromet We have developed a ovel MDM strategy based o the proposed weghted smlarty measure to solve decso problems We have solved a umercal problem ad compared the obtaed result wth other estg strateges to demostrate the effectveess of the proposed MDM strategy he proposed MDM strategy ca be appled other decso-makg problem such as suppler selecto patter recogto cluster aalyss medcal dagoss weaver selecto [5-53] fault dagoss [54] brck selecto [55-56] data mg [57] logstc cetre locato selecto [58-6] teacher selecto [6 62] etc 8 otrbutos of the proposed strategy SVNHSSM ad SVNWHSSM SVNS evromet are frstly defed the lterature We have also proved ther basc propertes Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased
Neutrosophc Sets ad Systems Vol 2 28 9 Mult attrbute decso makg problem Decso makg aalyss phase Determato of the relato betwee alteratves ad attrbutes Step- Determe the weghts of attrbutes Step- 2 Determe deal soluto Step- 3 Determe the smlarty values Step-4 akg the alteratves Step-5 Ed Step- 6 GUE 2: Phase dagram of the proposed decso makg strategy efereces [] Smaradache ufyg feld logcs eutrosophy: eutrosophc probablty set ad logc ehoboth merca esearch Press 998 [2] H J S Smth O the tegrato of dscotuous fuctos Proceedgs of the Lodo Mathematcal Socety Seres 6 874 4 53 [3] G ator Über uedlche leare Puktmagfaltgkete V O fte lear potmafolds sets Mathematsche ale 2 883 545 59 [4] L Zadeh uzzy sets formato ad otrol 8965 338 353 [5] K taassov tutostc fuzzy sets uzzy Sets ad Systems 2986 87-96 [6] H Wag Smaradache Y Q Zhag ad Suderrama Sgle valued eutrosophc sets Multspace ad Multstructure 42 4 43 [7] S Ye ad J Ye Dce smlarty measure betwee sgle valued eutrosophc multsets ad ts applcato medcal dagoss Neutrosophc Sets ad Systems 624 49 54 Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased
Neutrosophc Sets ad Systems Vol 2 28 [8] S Ye J u ad J Ye Medcal dagoss sg dstacebased smlarty measures of sgle valued eutrosophc multsets Neutrosophc Sets ad Systems 724 47 52 [9] Q sar swas ad S ggarwal Proposal for applcablty of eutrosophc set theory medcal teratoal Joural of omputer pplcatos 275 2 5 [] J Ye Sgle valued eutrosophc cross etropy for multcrtera decso makg problems ppled Mathematcal Modelg 3824 7 75 [] J Ye Vector smlarty measures of smplfed eutrosophc sets ad ther applcato multcrtera decso makg teratoal Joural of uzzy Systems 62 24 24 25 [2] J Ye Multple attrbute group decso-makg method wth completely ukow weghts based o smlarty measures uder sgle valued eutrosophc evromet Joural of tellget ad uzzy Systems 27 24 2927 2935 [3] P swas S Pramak ad Gr Etropy based grey relatoal aalyss method for mult-attrbute decsomakg uder sgle valued eutrosophc assessmets Neutrosophc Sets ad Systems 224 2 [4] P swas S Pramak ad Gr ew methodology for eutrosophc mult-attrbute decso makg wth ukow weght formato Neutrosophc Sets ad Systems 324 42 52 [5] S Pramak ad S N hackrabart study o problems of costructo workers West egal based o eutrosophc cogtve maps teratoal Joural of ovatve esearch Scece Egeerg ad echology 2 23 6387 6394 [6] K Modal ad S Pramak study o problems of Hjras West egal based o eutrosophc cogtve maps Neutrosophc Sets ad Systems 524 2-26 [7] K Modal ad S Pramak Mult-crtera group decso makg approach for teacher recrutmet hgher educato uder smplfed eutrosophc evromet Neutrosophc Sets ad Systems 624 28 34 [8] K Modal ad S Pramak Neutrosophc decso makg model of school choce Neutrosophc Sets ad Systems 7 25 62 68 [9] S Pramak ad K oy Neutrosophc game theoretc approach to do-pak coflct over Jammu-Kashmr Neutrosophc Sets ad Systems 2 24 82 [2] H D heg ad Y Guo ew eutrosophc approach to mage thresholdg New Mathematcs ad Natural omputato 43 28 29 38 [2] Y Guo ad H D heg New eutrosophc approach to mage segmetato Patter ecogto 42 29 587 595 [22] M Zhag L Zhag ad H D heg eutrosophc approach to mage segmetato based o watershed method Sgal Processg 95 2 5 57 [23] S M he ew approach to hadlg fuzzy decso makg problems EEE rasactos o Systems Ma ad yberetcs 8 988 2 6 [24] S M he S M Yeh ad PH Hsao comparso of smlarty measures of fuzzy values uzzy Sets ad Systems 72995 79 89 [25] L K Hyug Y S Sog ad K M Lee Smlarty measure betwee fuzzy sets ad betwee elemets uzzy Sets ad Systems 62 994 29 293 [26] P Papps ad N Karacaplds comparatve assessmet of measures of smlarty of fuzzy values uzzy Sets ad Systems 56993 7 74 [27] S Pramak ad K Modal Weghted fuzzy smlarty measure based o taget fucto ad ts applcato to medcal dagoss teratoal Joural of ovatve esearch Scece Egeerg ad echology 42 25 58 64 [28] Z Xu Some smlarty measures of tutostc fuzzy sets ad ther applcatos to multple attrbute decso makg uzzy Optmzato ad Decso Makg 62 27 9 2 [29] G Papakostas G Hatzmchalds ad V G Kaburlasos 23 Dstace ad smlarty measures betwee tutostc fuzzy sets: comparatve aalyss from a patter recogto pot of vew Patter ecogto Letters 344 69 622 [3] P swas S Pramak ad Gr study o formato techology professoals health problem based o tutostc fuzzy cose smlarty measure Swss Joural of Statstcal ad ppled Mathematcs 2 24 44-5 [3] K Modal ad S Pramak 25 tutostc fuzzy smlarty measure based o taget fucto ad ts applcato to mult-attrbute decso makg Global Joural of dvaced esearch 22 464-47 [32] S roum ad Smaradache Several smlarty measures of eutrosophc sets Neutrosophc Sets ad Systems 23 54 62 [33] P Majumder ad S K Samata O smlarty ad etropy of eutrosophc sets Joural of tellget ad uzzy Systems 26 24 245 252 [34] J Ye ad Q Zhag Sgle valued eutrosophc smlarty measures for multple attrbute decso-makg Neutrosophc Sets ad System 222 48 54 [35] P swas S Pramak ad Gr ose smlarty measure based mult-attrbute decso-makg wth trapezodal fuzzy eutrosophc umbers Neutrosophc Sets ad System 825 47 58 [36] J Ye Multple attrbute group decso-makg method wth completely ukow weghts based o smlarty measures uder sgle valued eutrosophc evromet Joural of tellget ad uzzy Systems 276 24 2927 2935 [37] J Ye Vector smlarty measures of smplfed eutrosophc sets ad ther applcato multcrtera decso makg teratoal Joural of uzzy Systems 62 24 24 2 [38] J Ye cose smlarty measures of smplfed eutrosophc sets for medcal dagoss rtfcal tellgece Medce 633 25 7 79 [39] J Ye ault dagoses of steam turbe usg the epoetal smlarty measure of eutrosophc umbers Joural of tellget ad uzzy Systems 34 26 927 934 [4] J Ye Sgle valued eutrosophc clusterg algorthms based o smlarty measures Joural of lassfcato 34 27 48 62 Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased
Neutrosophc Sets ad Systems Vol 2 28 [4] S Ye ad J Ye Dce smlarty measure betwee sgle valued eutrosophc multsets ad ts applcato medcal dagoss Neutrosophc Sets ad Systems 6 24 48 53 [42] S Ye J u ad J Ye Medcal dagoss usg dstacebased smlarty measures of sgle valued eutrosophc multsets Neutrosophc Sets ad Systems 7 25 47 52 [43] J Ye ad J u Mult-perod medcal dagoss method usg a sgle valued eutrosophc smlarty measure based o taget fucto omputer Methods ad Programs omedce 23 26 42 49 [44] S Pramak ad K Modal ose smlarty measure of rough eutrosophc sets ad ts applcato medcal dagoss Global Joural of dvaced esearch 2 25 22 22 [45] S Pramak ad K Modal otaget smlarty measure of rough eutrosophc sets ad ts applcato to medcal dagoss Joural of New heory 4 25 464 47 [46] P Majumdar ad S K Samata O smlarty ad etropy of eutrosophc sets Joural of tellgece ad uzzy Systems 26 24 245 252 [47] P swas S Pramak ad Gr Etropy based grey relatoal aalyss method for mult-attrbute decsomakg uder sgle valued eutrosophc assessmets Neutrosophc Sets ad Systems 2 24 2 [48] J Ye ad Q S Zhag Sgle valued eutrosophc smlarty measures for multple attrbute decso makg Neutrosophc Sets ad Systems 2 24 48 54 [49] K Modal ad S Pramak Neutrosophc taget smlarty measure ad ts applcato to multple attrbute decso makg Neutrosophc sets ad systems 9 25 8 87 [5] J Ye Sgle-valued eutrosophc smlarty measures based o cotaget fucto ad ther applcato the fault dagoss of steam turbe Soft omputg 23 27 87 825 [5] P P Dey S Pramak ad Gr Mult-crtera group decso makg tutostc fuzzy evromet based o grey relatoal aalyss for weaver selecto khad sttuto Joural of ppled ad Quattatve Methods 4 25-4 [52] P P Dey S Pramak ad Gr eteded grey relatoal aalyss based terval eutrosophc multattrbute decso makg for weaver selecto Joural of New heory 9 25 82-93 [53] P P Dey S Pramak ad Gr Eteded projecto based models for solvg multple attrbute decso makg problems wth terval valued eutrosophc formato Smaradache & S Pramak Eds New reds Neutrosophc heory ad pplcatos Pos Edto russels 26 27-4 [54] L Sh orrelato coeffcet of smplfed eutrosophc sets for bearg fault dagoss Shock ad Vbrato 26 26 rtcle D 54436 do: 55/26/54436 [55] K Modal S Pramak ad Smaradache tutostc fuzzy mult-crtera group decso makg approach to qualty-brck selecto problem Joural of ppled Quattatve Methods 92 24 35-5 [56] K Modal ad S Pramak Neutrosophc decso makg model for clay-brck selecto costructo feld based o grey relatoal aalyss Neutrosophc Sets ad Systems 9 25 72-79 [57] K Modal S Pramak ad Smaradache ole of eutrosophc logc data mg Smaradache & S Pramak Eds New reds Neutrosophc heory ad pplcato Pos Edtos russels 26 5-23 [58] S Pramak S Dalapat ad K oy Logstcs ceter locato selecto approach based o eutrosophc multcrtera decso makg Smaradache & S Pramak Eds New reds Neutrosophc heory ad pplcato Pos Edtos russels 26 6-74 [59] S Pramak S Dalapat ad K oy Neutrosophc mult-attrbute group decso makg strategy for logstc ceter locato selecto Smaradache M asset & V hag Eds Neutrosophc Operatoal esearch Vol Pos sbl russels 28 3-32 [6] S Pramak S Dalapat G based mult crtera decso makg geeralzed eutrosophc soft set evromet Global Joural of Egeerg Scece ad esearch Maagemet 35 2653-69 [6] S Pramak P P Dey ad Gr OPSS for sgle valued eutrosophc soft epert set based mult-attrbute decso makg problems Neutrosophc Sets ad Systems 25 88-95 [62] S Pramak ad D Mukhopadhyaya Grey relatoal aalyss based tutostc fuzzy mult crtera group decso-makg approach for teacher selecto hgher educato teratoal Joural of omputer pplcatos 34 2 2-29 [63] bdel-asset M Mohamed M Smaradache & hag V 28 Neutrosophc ssocato ule Mg lgorthm for g Data alyss Symmetry 4 6 [64] bdel-asset M & Mohamed M 28 he ole of Sgle Valued Neutrosophc Sets ad ough Sets Smart ty: mperfect ad complete formato Systems Measuremet Volume 24 ugust 28 Pages 47-55 [65] bdel-asset M Guasekara M Mohamed M & Smaradache ovel method for solvg the fully eutrosophc lear programmg problems Neural omputg ad pplcatos - [66] bdel-asset M Maogara G Gamal & Smaradache 28 hybrd approach of eutrosophc sets ad DEMEL method for developg suppler selecto crtera Desg utomato for Embedded Systems -22 [67] bdel-asset M Mohamed M & hag V 28 NMD: framework for evaluatg cloud computg servces uture Geerato omputer Systems 86 2-29 [68] bdel-asset M Mohamed M Zhou Y & Hezam 27 Mult-crtera group decso makg based o eutrosophc aalytc herarchy process Joural of tellget & uzzy Systems 336 455-466 [69] bdel-asset M; Mohamed M; Smaradache Eteso of Neutrosophc HP SWO alyss for Strategc Plag ad Decso-Makg Symmetry 28 6 eceved : March 9 28 ccepted : prl 2 28 Kalya Modal Surapat Pramak ad bhas Gr Sgle Valued Neutrosophc Hyperbolc Se Smlarty Measure ased