Cosine Similarity Measure Of Rough Neutrosophic Sets And Its Application In Medical Diagnosis

Transcription

1 Vol- ssue- -0 : ose mlarty Measure Of ough eutrosoph ets d ts pplato Medal Dagoss urapat ramak Departmet of Mathemats adalal Ghosh ollege apur West egal da Kalya Modal ragar Hgh hool H ragar aaghat West egal da ths paper we defe a rough ose smlarty measure betwee two rough eutrosoph sets he otos of rough eutrosoph sets wll be used as vetor represetatos 3D-vetor spae he ratg of all elemets s epressed wth the upper ad lower appromato operator ad the par of eutrosoph sets whh are haraterzed by truth-membershp degree determay-membershp degree ad falsty-membershp degree umeral eample of the medal dagoss s provded to show the effetveess ad fleblty of the proposed method Geeral erms ough eutrosoph set ose smlarty measure Keywords ough ose smlarty measure rough sets eutrosoph sets rough eutrosoph sets determay-membershp degree 3D-vetor spae ODUO he oept of eutrosoph sets [] was orgated from the ew brah of phlosophy alled eutrosophy [] he eutrosoph set geeralzes the lassal set or rsp set proposed by ator fuzzy set proposed by Zadeh [] terval valued fuzzy set proposed depedetly by Zadeh [3] Gratta-Guess [4] ad Jah [5] vague set proposed by Gau ad uehrer [6] grey set [7 8] tutost fuzzy set proposed by taassov [9] ad terval valued tutost fuzzy set proposed by taassov ad Gargov [0] Wag et al [] trodued sgle valued eutrosoph set V to deal realst problem t has bee studed ad appled dfferet felds suh as medal dagoss problem [] deso makg problems [3] [4] [5] [6] [7] soal problems [8] [9] Eduatoal problems [0] [] oflt resoluto [] ad so o he oto of rough set theory [3] was proposed by awlak he oept of rough set theory [] s a eteso of the rsp set theory for the study of tellget systems haraterzed by eat uerta or suffet formato t s a useful tool for dealg wth uertaty or mpreso formato Lterature survey reflets that the rough set theory has aught a great deal of atteto ad terest amog the researhers he oept of rough eutrosoph sets [4 5] s reetly proposed ad very terestg Whle the oept of eutrosoph sets s a powerful log to hadle determate ad osstet stuato the theory of rough eutrosoph sets s also a powerful mathematal log to hadle ompleteess he ratg of all alteratves s epressed wth the upper ad lower appromato operator ad the par of eutrosoph sets whh are haraterzed by truth-membershp degree determay-membershp degree ad falsty-membershp degree a g e 3 0 J a u a r y 0 5 w w w g j a r o r g

2 Vol- ssue : o measure the degree of smlarty betwee eutrosoph sets may methods have bee proposed the lterature roum ad maradahe [6] studed the Hausdorff dstae betwee eutrosoph sets ad some smlarty measures based o the dstae set theoret approah ad mathg futo to alulate the smlarty degree betwee eutrosoph sets 03 roum ad maradahe [7] also proposed the orrelato oeffet betwee terval eutrosph sets Majumdar ad mata [8] studed several smlarty measures of sgle valued eutrosoph sets based o dstaes a mathg futo membershp grades ad the proposed a etropy measure for a V e [9] proposed the dstae-based smlarty measure of Vs ad appled t to the group deso makg problems wth sgle valued eutrosoph formato e [30] also proposed three vetor smlarty measure for s a stae of V ad terval valued eutrosoph set ludg the Jaard De ad ose smlarty ad appled them to mult-rtera deso-makg problems wth smplfed eutrosoph formato eetly swas et al [3] studed ose smlarty measure based mult-attrbute deso-makg wth trapezodal fuzzy eutrosoph umbers roum ad maradahe [3] proposed a ose smlarty measure of terval valued eutrosoph sets Lterature revew reflets that oly oe study related to rough eutrosoph deso makg s doe by Modal ad ramak [7] here s o vestgato o the smlarty measure of rough eutrosoph sets eause of more omplety ad uertaty ature of the problems t s eessary to utlze more fleble method whh a deal uerta stuato easly ths stuato rough eutrosoph log s very useful ths paper we propose rough eutrosoph smlarty measure ad establsh some of ts propertes o demostrate the applablty ad effetveess of the proposed smlarty measure a umeral eample of medal dagoss s provded hs paper s orgazed as follow: seto some bas deftos of eutrosoph set sgle valued eutrosoph set ose smlarty are preseted brefly seto 3 rough eutrosoph ose smlarty measure of rough eutrosoph sets ad ther bas propertes are trodued seto 4 the proposed smlarty measure s appled to deal wth the problem related to medal dagoss eto 5 presets the oludg remarks MHEML ELME Deftos o eutrosoph set [] Defto : Let H be a spae of pots objets wth geer elemet H deoted by he a eutrosoph set H s haraterzed by a truth membershp futo a determay membershp futo ad a falsty membershp futo 0 H 0 ; : he futos ad are real stadard or o-stadard subsets of H 0-0 that s : H ; : t should be oted that there s o restrto o the sum of + 0 sup + sup +sup 3 Defto : omplemet he omplemet of a eutrosoph set s deoted by ad s defed by ; e Defto 3: otamet eutrosoph set s otaed the other eutrosoph set f ad oly f the followg results hold f sup sup f f sup sup f f sup sup f for all H Defto 4: gle-valued eutrosoph set Let H be a uversal spae of pots objets wth a geer elemet of H deoted by sgle valued eutrosoph set [] s haraterzed by a truth membershp futo ad determay futo wth y [ ] 0 for all H a falsty membershp futo 3 a g e 3 0 J a u a r y 0 5 w w w g j a r o r g

3 Vol- ssue : Whe H s otuous a V a be wrtte as follows: H ad whe H s dsrete a V a be wrtte as follows: H t should be observed that for a V sup sup 3 H 0 sup Defto5: he omplemet of a sgle valued eutrosoph set s deoted by ad s defed by ; ; Defto 6: V s otaed the other V deoted as ff ; H Defto 7: wo sgle valued eutrosoph sets ad are equal e = ff ad Defto 8: Uo he uo of two Vs ad s a V 3 wrtte as 3 ; ts truth membershp determay-membershp ad falsty membershp futos are related to ad by the followg equato ma ma ; m ; 3 3 for all H 3 Defto 9: terseto he terseto of two Vs ad s a V 3 wrtte as 3 ts truth membershp determay membershp ad falsty membershp futos are related to a by the followg equato m ; 3 ; ma 3 3 H ma Dstae betwee two eutrosoph sets he geeral V a be preseted the follow form : H te Vs a be represeted as follows: m m m m H Defto 0: Let 3 be two sgle-valued eutrosoph sets the the Hammg dstae [8]betwee two V ad s defed as follows: d ad ormalzed Hammg dstae betwee two Vs ad s defed as follows: 4 4 a g e 3 0 J a u a r y 0 5 w w w g j a r o r g

4 Vol- ssue : d wth the followg propertes 0 d 3 0 d Deftos o rough eutrosoph set ough set theory ossts of two bas ompoets amely rsp set ad equvalee relato whh are the mathematal bass of s he bas dea of rough set s based o the appromato of sets by a ouple of sets kow as the lower appromato ad the upper appromato of a set Here the lower ad upper appromato operators are based o equvalee relato ough eutrosoph sets [5 6] are the geeralzato of rough fuzzy sets [33] [34] [35] ad rough tutost fuzzy sets [36] Defto : Let be a o-ull set ad be a equvalee relato o Let be eutrosoph set wth the membershp futo determay futo ad o-membershp futo he lower ad the upper appromatos of the appromato deoted by ad are respetvely defed as follows: / / 9 Where z z z o Where ad mea ma ad m operators respetvely ad ad ad o-membershp of wth respet to t s easy to see that hus mappg s two eutrosoph sets are the membershp determay : are respetvely referred to as the lower ad upper rough appromato operators ad the par s alled the rough eutrosoph set rom the above defto t s see that ad have ostat membershp o the equvalee lasses of f ; e for ay belogs to s sad to be a defable eutrosoph set the appromato t a be easly proved that Zero eutrosoph set 0 ad ut eutrosoph sets are defable eutrosoph sets 5 a g e 3 0 J a u a r y 0 5 w w w g j a r o r g

5 Vol- ssue : a g e 3 0 J a u a r y 0 5 w w w g j a r o r g Defto f = s a rough eutrosoph set the rough omplemet of s the rough eutrosoph set deoted ~ where are the omplemets of eutrosoph sets of respetvely / ad 0 / Defto 3 f ad are two rough eutrosoph sets of the eutrosoph sets respetvely the the followg deftos holds [5 6]: f are rough eutrosoph sets the the followg proposto are stated from deftos roposto : ~ ~ 3 4 roposto : De Morga s Laws are satsfed for rough eutrosoph sets ~ ~ ~ ~ ~ ~ roposto 3: f ad are two rough eutrosoph sets U suh that the roposto 4: ~ ~ ~ ~

6 Vol- ssue : a g e 3 0 J a u a r y 0 5 w w w g j a r o r g 3 3 ose smlarty futo Defto 3 fudametal agle-based smlarty measure betwee two vetors of dmesos usg the ose of the agle betwee them s kow as ose smlarty measure t alulates the smlarty betwee two vetors based o the dreto egletg the mpat of the dstae betwee them Gve two attrbute vetors X = ad = y y y the ose smlarty osθ s preseted as follows: y y os ose smlarty measure based o hattaharya s dstae [37] betwee two fuzzy set ad are defed as follows: D vetor spae a ose smlarty measure betwee two tutost fuzzy sets proposed by e [38] s as follows: Where all vetors les betwee 0 ad 3 ose smlarty measure of rough eutrosoph sets he ose smlarty measure s alulated as the er produt of two vetors dvded by the produt of ther legths t s the ose of the agle betwee the vetor represetatos of two rough eutrosoph sets he ose smlarty measure s a fudametal measure used formato tehology Estg ose smlarty measures does ot deal wth rough eutrosoph sets tll ow herefore a ew ose smlarty measure betwee rough eutrosoph sets s proposed 3-D vetor spae Defto 3: ssume that there are two rough eutrosoph sets ad X = { ose smlarty measure betwee rough eutrosoph sets ad s proposed as follows: Where roposto 5 Let ad be rough eutrosoph sets the 0 3 = ff = 4 f s a ad the ad roofs : t s obvous beause all postve values of ose futo are wth 0 ad t s obvous that the proposto s true 3 Whe = the obvously = O the other had f = the e hs mples that =

7 Vol- ssue : f the we a wrte he ose futo s dereasg futo wth the terval 0 Hee we a wrte ad f we osder the weghts of eah elemet a weghted rough ose smlarty measure betwee rough eutrosoph sets ad a be defed as follows: W w w [0] = ad w f we take w = the W = he weghted rough ose smlarty measure W betwee two rough eutrosoph sets ad also satsfes the followg propertes: 0 W W W 3 W = ff = 4 f s a W ad the W W ad W W roof : he proofs of above propertes are smlar proofs of the propostos 5 4 EXMLE O MEDL DGO We osder a medal dagoss problem from pratal pot of vew for llustrato of the proposed approah Medal dagoss omprses of uertates ad reased volume of formato avalable to physas from ew medal tehologes he proess of lassfyg dfferet set of symptoms uder a sgle ame of a dsease s very dffult task some pratal stuatos there ests possblty of eah elemet wth a lower ad a upper appromato of eutrosoph sets t a deal wth the medal dagoss volvg more determay tually ths approah s more fleble ad easy to use he proposed smlarty measure amog the patets versus symptoms ad symptoms versus dseases wll provde the proper medal dagoss he ma feature of ths proposed approah s that t osders truth membershp determate ad false membershp of eah elemet betwee two appromatos of eutrosoph sets by takg oe tme speto for dagoss ow a eample of a medal dagoss s preseted Let = {₁ ₂ ₃} be a set of patets D = {Vral ever Malara tomah problem hest problem} be a set of dseases ad = {emperature Headahe tomah pa ough hest pa} be a set of symptoms Our soluto s to eame the patet ad to determe the dsease of the patet rough eutrosoph evromet able : elato- he relato betwee atets ad ymptoms elato- emperature Headahe tomah pa ough hest pa a g e 3 0 J a u a r y 0 5 w w w g j a r o r g

8 Vol- ssue : able : elato- he relato amog ymptoms ad Dseases elato- Vral ever Malara tomah problem hest problem emperature Headahe tomah pa ough hest pa able 3: he orrelato Measure betwee elato- ad elato- ough ose smlarty measure Vral ever Malara tomah problem hest problem he hghest orrelato measure see the able 3 reflets the proper medal dagoss herefore all three patets ₁ ₂ ₃ suffer from vral fever 5 OLUO ths paper we have proposed rough ose smlarty measure of rough eutrosoph sets ad proved some of ther bas propertes We have preseted a applato of rough ose smlarty measure of rough eutrosoph sets medal dagoss problems he authors hope that the proposed oept a be appled solvg realst mult-rtera deso makg problems EEEE [] maradahe 998 ufyg feld logs: eutrosoph log eutrosophy eutrosoph set eutrosoph probablty ad eutrosoph statsts ehoboth: mera esearh ress [] Zadeh L 965 uzzy sets formato ad otrol [3] Zadeh L 975 he oept of a lgust varable ad ts applato to appromate reasog formato ees [4] Gratta-Guess 975 uzzy membershp mapped oto terval ad may-valued quattes Z Math Logk Grudlade Math [5] Jah K U 975 tervall-wertge Mege Math ah [6] Gau W L ad uehrer D J Vague sets EEE rasatos o ystems Ma ad yberets [7] Deg J L 98 otrol problems of grey system ystem ad otrol Letters [8] aga M ad amaguh D 004 Grey theory ad egeerg applato method Kyortsu publsher [9] taassov K 986 tutost fuzzy sets uzzy ets ad ystems [0] taassov K ad Gargov G 989 terval valued tutost fuzzy sets uzzy ets ad ystems [] Wag H maradahe Zhag Q ad uderrama 00 gle valued eutrosoph sets Multspae ad Mult struture [] Kharal 03 eutrosoph multrtera deso makg method ew Mathemats ad atural omputato reghto Uversty U [3] e J 03 Multrtera deso-makg method usg the orrelato oeffet uder sgle-valued eutrosoph evromet teratoal Joural of Geeral ystems a g e 3 0 J a u a r y 0 5 w w w g j a r o r g

9 Vol- ssue : [4] e J 04 gle valued eutrosoph ross etropy for multrtera deso makg problems ppled Mathematal Modelg [5] swas ramak ad Gr 04 Etropy based grey relatoal aalyss method for mult-attrbute desomakg uder sgle valued eutrosoph assessmets eutrosoph ets ad ystems 0-0 [6] swas ramak ad Gr 04 ew methodology for eutrosoph mult-attrbute deso makg wth ukow weght formato eutrosoph ets ad ystems [7] Modal K ad ramak 05 ough eutrosoph mult-attrbute deso-makg based o grey relatoal aalyss eutrosoph ets ad ystems press [8] ramak ad hakrabart 03 study o problems of ostruto workers West egal based o eutrosoph ogtve maps teratoal Joural of ovatve esearh ee Egeerg ad ehology [9] Modal K ad ramak 04 study o problems of Hjras West egal based o eutrosoph ogtve maps eutrosoph ets ad ystems 5-6 [0] Modal K ad ramak 04 Mult-rtera group deso makg approah for teaher rerutmet hgher eduato uder smplfed eutrosoph evromet eutrosoph ets ad ystems [] Modal K ad ramak 05 eutrosoph deso makg model of shool hoe eutrosoph ets ad ystems volume 7 ress [] ramak ad oy K 04 eutrosoph game theoret approah to do-ak oflt over Jammu-Kashmr eutrosoph ets ad ystems 8-0 [3] awlak Z 98 ough sets teratoal Joural of formato ad omputer ees [4] roum maradahe ad Dhar M 04 ough eutrosoph sets tala joural of pure ad appled mathemats [5] roum maradahe ad Dhar M ough eutrosoph sets eutrosoph ets ad ystems [6] roum maradahe 03 everal smlarty measures of eutrosoph sets eutrosoph ets ad ystems 54-6 [7] roum maradahe 03 orrelato oeffet of terval eutrosoph se erodal of ppled Mehas ad Materals Vol wth the ttle Egeerg Desos ad etf esearh erospae obots omehas Mehaal Egeerg ad Maufaturg; roeedgs of the teratoal oferee ME uharest Otober 03 [8] Majumder amata K 03 O smlarty ad etropy of eutrosoph sets Joural of tellget ad uzzy ystem do: 0333/-3080 [9] e J 03 Multrtera deso-makg method usg the orrelato oeffet uder sgle-valued eutrosoph evromet teratoal Joural of Geeral ystems [30] e J 04 Vetor smlarty measures of smplfed eutrosoph sets ad ther applato multrtera deso makg teratoal Joural of uzzy ystems [3] swas ramak ad Gr 05 ose smlarty measure based mult-attrbute deso-makg wth trapezodal fuzzy eutrosoph umbers eutrosoph sets ad ystem 8 ress [3] roum ad maradahe 04 ose smlarty measure of terval valued eutrosoph sets eutrosoph ets ad ystems [33] Dubos D ad rade H 990 ough fuzzy sets ad fuzzy rough sets teratoal Joural of Geeral ystem [34] akamura 988 uzzy rough sets ote o Multple-Valued Log Japa [35] ada Majumdar 99 uzzy rough sets uzzy ets ad ystems [36] homas K V ar L 0 ough tutost fuzzy sets a latte teratoal Mathemats orum [37] hattaharya 946 O a measure of dvergee of two multomal populato aakhya er [38] e J 0 ose smlarty measures for tutost fuzzy sets ad ther applatos Mathematal ad omputer Modellg a g e 3 0 J a u a r y 0 5 w w w g j a r o r g