Improved cosine similarity measures of simplified intuitionistic sets for. medicine diagnoses

Transcription

1 *Mauscript lick here to dowload Mauscript: cossimm_sss.doc lick here to view liked Refereces mproved cosie similarity measures of simplified ituitioistic sets for medicie diagoses Ju Ye.* Departmet of Electrical ad formatio Egieerig, Shaoig Uiversity, 508 Huacheg West Road, Shaoig, Zheiag 3000, P.R. hia bstract: Similarity measures are a importat tool i patter recogitio ad medical diagosis. o overcome some disadvatages of eistig cosie similarity measures for simplified eutrosophic sets SNSs i vector space, this paper proposes improved cosie similarity measures for SNSs based o the cosie fuctio, icludig sigle valued eutrosophic cosie similarity measures ad iterval eutrosophic cosie similarity measures. he, the weighted cosie similarity measures of SNSs are itroduced by cosiderig the importace of each elemet. Moreover, compared with eistig cosie similarity measures of SNSs by umerical eamples, the improved cosie similarity measures of SNSs demostrate their effectiveess ad ratioality ad ca overcome some disadvatages of eistig cosie similarity measures of SNSs i some cases. ially, the medical diagosis problems are give to show the applicatios ad effectiveess of the improved cosie similarity measures. Keywords: Simplified eutrosophic set; Sigle valued eutrosophic set; terval eutrosophic set; osie similarity measure; Medical diagosis * orrespodig author: Ju Ye el.: ; yehu@aliyu.com

2 . troductio Due to the icreased volume of ormatio available to physicias from modem medical techologies, medicie diagosis cotais a lot of icomplete, ucertaity, ad icosistet ormatio, which is importat ormatio of medical diagosis problems. symptom usually implies a lot of icomplete, ucertaity, ad icosistet ormatio for a disease, hece the icomplete, ucertaity ad icosistet ormatio characterizes a relatio betwee symptoms ad diseases. hus we work with the ucertaities ad icosistecies to lead us to proper decisio makig i medicie. most of the medical diagosis problems, there eist some patters, ad the eperts make decisio based o the similarity betwee ukow sample ad the basic diagosis patters. some practical situatios, there is the possibility of each elemet havig differet truth-membership, idetermiacy-membership, ad falsity-membership fuctios. herefore, Smaradache [] origially proposed the cocept of a eutrosophic set from philosophical poit of view. eutrosophic set i a uiversal set X is characterized idepedetly by a truth-membership fuctio, a idetermiacy-membership fuctio, ad a falsity-membership fuctio. he fuctios,, i X are real stadard or ostadard subsets of ] 0, [, i.e., : X ] 0, [, : X ] 0, [, ad : X ] 0, [. However, the domai of the defiitio ad rage of the fuctios, ad i a eutrosophic set is the o-stadard uit iterval ] 0, [, it is oly used for philosophical applicatios, especially whe distictio is required betwee absolute ad relative truth/falsehood/idetermiacy. o easily use i techical applicatios of the eutrosophic set, the domai of the defiitio ad rage of, ad ca be restraied to the ormal stadard real uit iterval [0, ]. s a simplified form of

3 the eutrosophic set, a simplified eutrosophic set [] is the appropriate choice as it is easily epresses ad deals with icomplete, ucertaity, ad icosistet ormatio i real sciece ad egieerig fields. Simplified eutrosophic sets iclude sigle valued eutrosophic sets SVNSs ad iterval eutrosophic sets NSs ad a geeralizatio of classic sets, fuzzy sets Ss [3], ituitioistic fuzzy sets Ss [4] ad iterval-valued ituitioistic fuzzy sets VSs [5]. However, Ss, Ss ad VSs caot represet ad hadle ucertaity ad icosistet ormatio []. he, similarity measures are ot oly a importat tool i patter recogitio, medicie diagosis, ad decisio makig but also a importat research topic i the eutrosophic theory. Various similarity measures have bee proposed by some researchers. roumi ad Smaradache [6] defied the Hausdorff distace betwee eutrosophic sets ad some similarity measures based o the distace, set theoretic approach, ad matchig fuctio to calculate the similarity degree betwee eutrosophic sets. Maumdar ad Samata [7] itroduced several similarity measures of sigle valued eutrosophic sets SVNSs based o distaces, a matchig fuctio, membership grades, ad the proposed a etropy measure for a SVNS. Ye [8] also preseted the Hammig ad Euclidea distaces betwee iterval eutrosophic sets NSs ad their similarity measures ad applied them to multiple attribute decisio-makig problems with iterval eutrosophic ormatio. Ye [9] further proposed the distace-based similarity measure of SVNSs ad applied it to group decisio makig problems with sigle valued eutrosophic ormatio. urthermore, Ye [] proposed three vector similarity measures for SNSs, icludig the Jaccard, Dice, ad cosie similarity measures for SVNSs ad NSs, ad applied them to multicriteria decisio-makig problems with simplified eutrosophic ormatio. ill ow, eistig similarity measures for eutrosophic sets are scarcely applied to medical diagosis problems. However, the cosie similarity measures defied i vector

4 space [] have some drawbacks i some situatios. or istace, they may produce o defied umeaigful pheomea or some results calculated by the cosie similarity measures are ureasoable i some real cases details give i Sectios 3. herefore, i the situatios, it is difficult to apply them to patter recogitio ad medicie diagosis. o overcome some drawbacks of eistig cosie measures i [], this paper aims to propose improved cosie similarity measures for SNSs ad apply them to medicie diagosis. o do so, the rest of the article is orgaized as follows. Sectio, we briefly itroduce some basic cocepts of SNSs. Sectio 3 reviews eistig cosie similarity measures of SNSs i vector space ad their drawbacks. Sectios 4 proposes the improved cosie similarity measures of SNSs based o the cosie fuctio, icludig sigle valued eutrosophic cosie similarity measures ad iterval eutrosophic cosie similarity measures, ad ivestigates their properties. Sectio 5, by two umerical eamples we give the comparative aalysis betwee the improved cosie similarity measures ad eistig cosie similarity measures for SNSs to show the effectiveess ad ratioality of the improved cosie measures. Sectio 6, the cosie similarity measures are applied to medicie diagosis problems. oclusios ad further research are cotaied i Sectio 7.. Some basic cocepts of SNSs Smaradache [] origially preseted the cocept of a eutrosophic set from philosophical poit of view. a eutrosophic set i a uiversal set X, its characteristic fuctios are epressed by a truth-membership fuctio, a idetermiacy-membership fuctio, ad a falsity-membership fuctio, respectively. he fuctios,, i X are real stadard or ostadard subsets of ] 0, [, i.e., : X ] 0, [, : X ] 0, [, ad : X

5 ] 0, [. he, the sum of, ad is o restrictio, i.e o apply a eutrosophic set to sciece ad egieerig areas, Ye [] itroduced SNS, which is a subclass of the eutrosophic set, ad gave the followig defiitio of a SNS. Defiitio []: Let X be a space of poits obects, with a geeric elemet i X deoted by. eutrosophic set i X is characterized by a truth-membership fuctio, a idetermiacy-membership fuctio, ad a falsity-membership fuctio. f the fuctios, ad are sigleto subitervals/subsets i the real stadard [0, ], such that : X [0, ], : X [0, ], ad : X [0, ]. he, a simplificatio of the eutrosophic set is deoted by {,, X},, which is called a SNS. t is a subclass of the eutrosophic set ad icludes the cocepts of NS ad SVNS. O the oe had, if we oly use the SNS whose, ad values are sigle poits i the real stadard [0, ] istead of subitervals/subsets i the real stadard [0, ], the SNS ca be described by three real umbers i the real uit iterval [0, ]. herefore, the sum of,, [0, ] satisfies the coditio 0 3. this case, the SNS reduces to the SVNS. or two SVNSs {,,, X} ad {,,, X}, there are the followig relatios [0]: c omplemet: {,, X }, ; clusio: if ad oly if,, for ay i X;

6 3 Equality: if ad oly if ad. O the other had, if we oly cosider three membership degrees i a SNS as the subuit iterval of the real uit iterval [0, ], the SNS ca be described by three iterval umbers i the real uit iterval [0, ]. or each poit i X, we have that [, ], [, ], [, ] [0, ] ad 0 3 for ay X. this case, the SNS reduces to the NS. or two NSs {,,, X} ad {,,, X}, there are the followig relatios []: omplemet: c { [, ],[, ],[, X }, ; clusio: if ad oly if,,,,,, for ay i X; 3 Equality: if ad oly if ad. Especially whe he upper ad lower eds of three iterval umbers,, i are equal, the NS degrade to the SVNS. herefore, the SVNS is a special case of the NS, ad also both are the special cases of the SNS. 3. Eistig cosie similarity measures of SNSs ad their drawbacks this sectio, we itroduce eistig cosie similarity measures for SNSs i the literature [] ad review their drawbacks. he, similarity measures have the followig defiitio.

7 Defiitio. real-valued fuctio S: SNSX SNSX [0, ] is called a similarity measure o SNSX if it satisfies the followig aiomatic requiremets for,, SNSX: S 0 S, ; S S, if ad oly if ; S3 S, S, ; S4 f, the S, S, ad S, S,. 3. Eistig cosie similarity measure for SVNSs ad its drawbacks this sectio, we oly use SVNSs i SNSs. ssume that there are two SVNSs {,,, X} ad {,,, X} i the uiverse of discourse X {,,, }, where,, [0, ] for ay X i ad,, [0, ] for ay X i. he, Ye [] preseted the cosie similarity measure of SVNSs i vector space as follows:,, However, oe ca fid some drawbacks of Eq. as follows: or two SVNSs ad, if 0 ad/or 0 for ay i X,,,, Eq. is udefied or umeaigful. this case, oe caot utilize it to calculate the cosie similarity measure betwee ad. f,, ad or,, ad for ay i X,,,. y applyig Eq., we have,.

8 Sice, the measure value of Eq. is equal to. his meas that it oly satisfies the ecessary coditio of the property S i Defiitio, but ot the sufficiet coditio. herefore, i this case, it is ureasoable to apply it to patter recogitio ad medicie diagosis. 3. Eistig cosie similarity measure for NSs ad its drawbacks this sectio, we oly use NSs i SNSs. ssume that there are two NSs {,,, X} ad {,,, X} i the uiverse of discourse X {,,, }, where [, ], [, ], [, ] [0, ] for ay X i ad [, ], [, ], [, ] [0, ] for ay X i. he, Ye [] preseted the cosie similarity measure of NSs i vector space as follows:, [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Similarly, oe ca fid some drawbacks of Eq. as follows:. or two NSs ad, if [0, 0] ad/or [0, 0] for ay i X,,,, Eq. is udefied or umeaigful. this case, oe caot calculate the cosie similarity measure betwee ad. f [, ], [, ], ad [, ] or [, ], [, ], ad

9 [, ] for ay i X,,,. y usig Eq., we have [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ],. Sice, the measure value of Eq. is equal to. his meas that it oly satisfies the ecessary coditio of the property S i Defiitio, but ot the sufficiet coditio. herefore, i this case, the cosie similarity measure is ureasoable i the applicatio of patter recogitio ad medicie diagosis. order to overcome the above metioed disadvatages, we shall improve the cosie similarity measures of SNSs i the followig sectio. 4. mproved cosie similarity measures for SNSs 4. mproved cosie similarity measures for SVNSs ased o the cosie fuctio, we propose two improved cosie similarity measures betwee SVNSs ad ivestigate their properties. Let {,,, X} ad {,,, X} be ay two SVNSs i X {,,, }, where,, [0, ] for ay X i ad,

10 , [0, ] for ay X i. he, based o the cosie fuctio, we propose two improved cosie similarity measures betwee ad, respectively, as follows: S, S, π cos, 3 π cos, 4 6 where the symbol is maimum operatio. he, the two improved cosie similarity measures satisfy the aiomatic requiremets of similarity measures. Propositio. or two SVNSs ad i X {,,, }, the cosie similarity measure S k, k, should satisfy the followig properties S-S4: S 0 S k, ; S S k, if ad oly if ; S3 S k, S k, ; S4 f is a SVNS i X ad, the S k, S k, ad S k, S k,. Proof: S Sice the truth-membership degree, idetermiacy-membership degree, ad falsity-membership degree i SVNS ad the value of the cosie fuctio are withi [0, ], the similarity measure based o the cosie fuctio also is withi [0, ]. Hece 0 S k, for k,. S or ay two SVNSs ad, if, this implies,, for,,, ad X. Hece 0, 0, ad 0. hus S k, for k,. f S k, for k,, this implies 0, 0, ad 0 sice cos0. he, these equalities idicate,

11 , for,,, ad X. Hece. S3 Proof is straightforward. S4 f, the there are,, ad for,,, ad X. he, we have the followig iequalities:,,,,,. Hece, S k, S k, ad S k, S k, for k, sice the cosie fuctio is a decreasig fuctio withi the iterval [0, π/]. herefore, we complete the proofs of these properties. Usually, oe takes the weight of each elemet for X ito accout ad assumes that the weight of a elemet is w,,, with w [0, ] ad w. hus we ca itroduce the followig weighted cosie similarity measures betwee SVNSs: w WS cos, π, 5 w WS 6 cos, π, 6 Especially whe w / for,,,, Eqs. 5 ad 6 reduce to Eqs. 3 ad mproved cosie similarity measures for NSs Similarly, we propose two improved cosie similarity measures betwee NSs ad ivestigate their properties. Let {,,, X} ad {,,, X} be ay two

12 NSs i X {,,, }, where [, ], [, ], [, ] [0, ] for ay X i ad [, ], [, ], [, ] [0, ] for ay X i. he, based o the cosie fuctio, we propose two improved cosie similarity measures betwee ad, respectively, as follows: S 3 4 cos, π, 7 S 4 cos, π, 8 where the symbol is maimum operatio. he, the two improved cosie similarity measures of NSs satisfy the aiomatic requiremets i Defiitio. Propositio. or two NSs ad i X {,,, }, the cosie similarity measure S k, k 3, 4 should satisfy the followig properties S-S4: S 0 S k, ; S S k, if ad oly if ; S3 S k, S k, ; S4 f is a NS i X ad, the S k, S k, ad S k, S k,. Proof: S Sice the truth-membership degree, idetermiacy-membership degree, ad falsity-membership degree i a NS ad the value of the cosie fuctio are withi [0, ], the similarity measure value

13 based o the cosie fuctio also is withi [0, ]. hus 0 S k, for k 3, 4. S or ay two NSs ad, if, this implies,, for,,, ad X. Hece 0, 0, 0, 0, 0, ad 0. hus S k, for k 3, 4. f S k, for k 3, 4, this implies 0, 0, 0, 0, 0, ad 0 sice cos0. he, these equalities idicate,, for,,, ad X. Hece. S3 Proof is straightforward. S4 f, the there are,,,,, ad for,,, ad X. he, we have the followig iequalities:,,,,,

14 ,,,,,,. Sice the cosie fuctio is a decreasig fuctio withi the iterval [0, π/], hece S k, S k, ad S k, S k, for k 3, 4. hus, we complete the proofs of these properties. Whe oe takes the weight of each elemet for X ito accout ad assumes that the weight of a elemet is w,,, with w [0, ] ad w, we ca itroduce the followig weighted cosie similarity measures betwee NSs ad : w WS 3 4 cos, π, 9 w WS 4 cos, π. 0 Especially whe w / for,,,, Eqs. 9 ad 0 reduce to Eqs. 7 ad 8. he, whe,, ad for ay X i ad,,

15 for ay X i, the NSs ad reduce to the SVNSs ad, ad the Eqs. 7-0 reduce to Eqs. 3-6, respectively. 5. omparative aalyses of various cosie similarity measures o compare the improved cosie measures with eistig cosie measures [] i simplified eutrosophic settig, we provide two umerical eamples to demostrate the effectiveess ad ratioality of the improved cosie similarity measures of SNSs. Eample. We cosider two SVNSs ad i X {} ad compare the improved cosie similarity measures with eistig cosie similarity measure i []. y applyig Eqs., 3 ad 4 the compariso of patter recogitios is illustrated for the umerical eample. hese similarity measure results are show i able. able. Similarity measure values of Eqs., 3 ad 4 ase ase ase 3 ase 4 ase 5,0.,0.3,0.4,0.3,0.,0.4,,0,0,,0,0,0.4,0.,0.6,0.,0.3,0.4,0.4,0.,0.3,0,,,0,0,0,0.,0., 0.3, [] ull S, S, Eample. Let us cosider two NSs ad i X {} ad compare the improved cosie similarity measures with eistig cosie similarity measure i []. he compariso of patter recogitios for the umerical eample is demostrated by usig Eqs., 7, 8. hese similarity measure results are show i able. able. Similarity measure values of Eqs., 7 ad 8

16 ase ase ase 3 ase 4 ase 5,[0.3,0.5], [0.,0.4], [0,0.],[0.3,0.5], [0.,0.4], [0.4, 0.5],[,], [0,0], [0,0],[,], [0,0], [0,0],[0.3,0.4], [0.,0.3], [0.4,0.5],[0.3,0.5], [0.,0.4], [0,0.],[0.4,0.5], [0.,0.4], [0.3, 0.5],[0,0], [,], [,],[0,0], [0,0], [0,0],[0.6,0.8], [0.4,0.6], [0.8,], [] ull S 3, S 4, he results of ables ad show that the eistig cosie similarity measure [] ot oly caot carry out the recogitio betwee ase ad ase 5 but also produces a ureasoable pheomeo for ase 5 ad a udefied umeaigful pheomeo for ase 4. his will get the decisio maker ito trouble i practical applicatios. However, the improved cosie similarity measure S caot also carry out the recogitio betwee ase 3 ad ase 4, but does ot produces a udefied umeaigful pheomeo. he, the improved cosie similarity measure S demostrates stroger discrimiatio amog them. Obviously, the improved cosie similarity measures are erior to the eistig cosie similarity measure i []. he, the cosie similarity measure S is erior to the cosie similarity measure S. he two eamples all demostrate that i some cases the improved cosie similarity measures of SNSs based o the cosie fuctio ca overcome the disadvatages of the eistig cosie similarity measures betwee two vectors. 6. Medicie diagoses Due to the icreased volume of ormatio available to physicias from modem medical techologies, medicie diagosis cotais a lot of icomplete, ucertaity, ad icosistet ormatio. some practical situatios, there is the possibility of each elemet havig differet

17 truth-membership, idetermiacy-membership, ad falsity-membership degrees, by which a SNS is epressed. Hece, similarity measures for SNSs are a suitable tool to cope with it. herefore, we apply the improved cosie similarity measures of SNSs to medicie diagosis. this sectio, we shall discuss the medical diagosis problems adapted from []. Let us cosider a set of diagoses Q {Q Viral fever, Q Malaria, Q 3 yphoid, Q 4 Stomach problem, Q 5 hest problem}, ad a set of symptoms S {s emperature, s Headache, s 3 Stomach pai, s 4 ough, s 5 hest pai}. he each diagosis Q i i,, 3, 4, 5 ca be idicated by SNSs with respect to all the symptoms as follows: Q Viral fever { s, 0.4, 0.6, 0.0, s, 0.3, 0., 0.5, s 3, 0., 0.3, 0.7, s 4, 0.4, 0.3, 0.3, s 5, 0., 0., 0.7 }, Q Malaria { s, 0.7, 0.3, 0.0, s, 0., 0., 0.6, s 3, 0.0, 0., 0.9, s 4, 0.7, 0.3, 0.0 }, s 5, 0., 0., 0.8 }, Q 3 yphoid { s, 0.3, 0.4, 0.3, s, 0.6, 0.3, 0., s 3, 0., 0., 0.7, s 4, 0., 0., 0.6, s 5, 0., 0.0, 0.9 }, Q 4 Stomach problem { s, 0., 0., 0.7, s, 0., 0.4, 0.4, s 3, 0.8, 0., 0.0, s 4, 0., 0., 0.7, s 5, 0., 0., 0.7 }, Q 5 hest problem { s, 0., 0., 0.8, s, 0.0, 0., 0.8, s 3, 0., 0.0, 0.8, s 4, 0., 0.0, 0.8, s 5, 0.8, 0., 0. }. Suppose a patiet P with all the symptoms ca be represeted by the followig SVNS ormatio: P Patiet { s, 0.8, 0., 0., s, 0.6, 0.3, 0., s 3, 0., 0., 0.8, s 4, 0.6, 0.5, 0., s 5, 0.,

18 0.4, 0.6 }. o fid a proper diagosis, we ca calculate the cosie measure S k P, Q i for k or ad i,, 3, 4, 5. he proper diagosis Q i* for the patiet P is derived by i * arg ma{ S P, Q }. i5 or coveiet compariso, we utilize the eistig cosie measure [] ad the two improved k i cosie measures to hadle the diagosis problem. y applyig Eqs., 3 ad 4, we ca obtai the results of the three similarity measures betwee the patiet P ad the cosidered disease Q i i,, 3, 4, 5, as show i able 3. able 3. Various similarity measure values for SVNS ormatio Viral fever Q Malaria Q yphoid Q 3 Stomach hest problem problem Q 4 Q 5 P, Q i [] S P, Q i S P, Q i able 3, the largest similarity measure idicates the proper diagosis. herefore, Patiet P suffers from malaria. We ca see that the medicie diagoses usig various similarity measures idicate the same diagosis results ad demostrate the effectiveess of these diagoses. However, as metioed above, the improved cosie measures ca overcome some drawbacks of the eistig cosie measure i [] i some cases. Hece, the improved cosie measures are erior to the eistig cosie measure. ompared with the diagosis results i [], the diagosis results of P are differet. he reaso is that the diagosis method i [] is o the basis of the cosie measure of Ss, while the diagosis methods i this paper are based o the improved cosie measures of SVNSs. herefore differet measure methods with differet kids of ormatio represeted by Ss ad SVNSs may give differet diagosis results. urthermore, the diagosis method i [] caot hadle the diagosis

19 problem with sigle valued eutrosophic ormatio, while the diagosis methods i this paper ca deal with the diagosis problems with ituitioistic fuzzy ormatio ad simplified eutrosophic ormatio. Hece, the improved cosie measures of SVNSs are erior to the cosie measure of Ss []. However, by oly takig oe time ispectio, we woder whether oe ca obtai a coclusio from a particular perso with a particular decease or ot. Hece, we have to eamie the patiet at differet time itervals e.g. two or three times a day ad ca obtai that data draw from multiple time ispectios for the patiet are iterval values rather tha sigle values. this case, the improved cosie measures of NSs are a better tool to fid a proper disease diagosis. Suppose a patiet P with all the symptoms ca be represeted by the followig NS ormatio: P Patiet { s, [0.3, 0.5], [0., 0.3], [0.4, 0.5], s, [0.7, 0.9], [0., 0.], [0., 0.], s 3, [0.4, 0.6], [0., 0.3], [0.3, 0.4], s 4, [0.3, 0.6], [0., 0.3], [0.4, 0.7], s 5, [0.5, 0.8], [0., 0.4], [0., 0.3] }. Similarly, we utilize the eistig cosie measure [] ad the two improved cosie measures of NSs to hadle the diagosis problem. y applyig Eqs., 7 ad 8, we ca obtai the results of various similarity measures betwee the patiet P ad the cosidered disease Q i i,, 3, 4, 5, as show i able 4. able 4. Various similarity measure values for NS ormatio Viral fever Q Malaria Q yphoid Q 3 Stomach hest problem problem Q 4 Q 5 P, Q i [] S 3 P, Q i S 4 P, Q i able 4, the largest similarity measure idicates the proper diagosis. herefore, Patiet P suffers from typhoid. We ca see that the medicie diagoses usig various similarity measures

20 idicate the same diagosis results ad demostrate the effectiveess of these diagoses. However, as metioed above, the improved cosie measures ca overcome some drawbacks of the eistig cosie measure i [] i some cases. Hece, the improved cosie measures are erior to the eistig cosie measure. 7. oclusio his paper proposed the improved cosie similarity measures for SNSs based o the cosie fuctio, icludig sigle valued eutrosophic cosie similarity measures ad iterval eutrosophic cosie similarity measures. he, the weighted cosie similarity measures of SNSs are proposed by cosiderig the importace of each elemet. ompared with eistig cosie similarity measures uder simplified eutrosophic eviromet, the improved cosie measures of SNSs demostrate their effectiveess ad ratioality ad ca overcome some drawbacks of eistig cosie similarity measures of SNSs. ially, medical diagosis problems with simplified eutrosophic ormatio are provided to demostrate the applicatios ad effectiveess of the improved cosie similarity measures of SNSs. further work, it is ecessary to apply the cosie similarity measures of SNSs to other areas such as decisio makig, mage processig, ad clusterig aalysis. Refereces []. Smaradache, uifyig field i logics. eutrosophy: Neutrosophic probability, set ad logic. Rehoboth: merica Research Press 999. [] J. Ye, Vector similarity measures of simplified eutrosophic sets ad their applicatio i

21 multicriteria decisio makig, teratioal Joural of uzzy Systems [3] L.. Zadeh, uzzy Sets, formatio ad otrol [4] K. taassov, tuitioistic fuzzy sets, uzzy Sets ad Systems [5] K. taassov ad G. Gargov, terval valued ituitioistic fuzzy sets, uzzy Sets ad Systems [6] S. roumi ad. Smaradache, Several similarity measures of eutrosophic sets, Neutrosophic Sets ad Systems [7] P. Maumdar ad S.K. Samata, O similarity ad etropy of eutrosophic sets, Joural of telliget ad uzzy Systems [8] J. Ye, Similarity measures betwee iterval eutrosophic sets ad their applicatios i multicriteria decisio-makig, Joural of telliget ad uzzy Systems [9] J. Ye, Multiple attribute group decisio-makig method with completely ukow weights based o similarity measures uder sigle valued eutrosophic eviromet, Joural of telliget ad uzzy Systems 04 DO: 0.333/S-45 [0] H. Wag,. Smaradache, Y.Q. Zhag, R. Suderrama, Sigle valued eutrosophic sets, Multispace ad Multistructure [] H. Wag,. Smaradache, Y.Q. Zhag, R. Suderrama, terval eutrosophic sets ad logic: heory ad applicatios i computig, Heis, Phoei, Z 005. [] J. Ye, osie similarity measures for ituitioistic fuzzy sets ad their applicatios, Mathematical ad omputer Modellig

22 able able. Similarity measure values of Eqs., 3 ad 4 ase ase ase 3 ase 4 ase 5,0.,0.3,0.4,0.3,0.,0.4,,0,0,,0,0,0.4,0.,0.6,0.,0.3,0.4,0.4,0.,0.3,0,,,0,0,0,0.,0., 0.3, [] ull S, S,

23 able able. Similarity measure values of Eqs., 7 ad 8 ase ase ase 3 ase 4 ase 5,[0.3,0.5], [0.,0.4], [0,0.],[0.3,0.5], [0.,0.4],,[0.3,0.5], [0.,0.4], [0.4, 0.5],[0.4,0.5], [0.,0.4],,[,], [0,0], [0,0],[0,0], [,],,[,], [0,0], [0,0],[0,0], [0,0],,[0.3,0.4], [0.,0.3], [0.4,0.5],[0.6,0.8], [0.4,0.6], [0,0.] [0.3, 0.5] [,] [0,0] [0.8,], [] ull S 3, S 4,

24 able3 able 3. Various similarity measure values for SVNS ormatio Viral fever Q Malaria Q yphoid Q 3 Stomach hest problem problem Q 4 Q 5 P, Q i [] S P, Q i S P, Q i

25 able4 able 4. Various similarity measure values for NS ormatio Viral fever Q Malaria Q yphoid Q 3 Stomach hest problem problem Q 4 Q 5 P, Q i [] S 3 P, Q i S 4 P, Q i