Correlation coefficients of simplified neutrosophic sets and their. multiple attribute decision-making method

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1 Mauscrpt Clck here to ve lked Refereces Correlato coeffcets of smplfed eutrosophc sets ad ther multple attrbute decso-makg method Ju Ye Departmet of Electrcal ad formato Egeerg Shaog Uversty 508 Huacheg West Road Shaog Zheag Provce 3000 P.R. Cha bstract he paper presets to e correlato coeffcets of smplfed eutrosophc sets SSs as the further eteso of the correlato coeffcet of sgle valued eutrosophc sets SVSs ad vestgates ther propertes. he a multple attrbute decso-makg method s proposed based o the eghted correlato coeffcets of SSs hch the evaluato formato for alteratves th respect to attrbutes s represeted by the form of smplfed eutrosophc values uder smplfed eutrosophc evromet. We utlze the eghted correlato coeffcets betee each alteratve ad the deal alteratve to rak the alteratves ad to determe the best oes. ally a llustratve eample demostrates the applcato ad effectveess of the proposed decso-makg method. Keyords: eutrosophc set; Smplfed eutrosophc set; Correlato coeffcet; Multple attrbute decso makg el.: E-mal address: yehu@alyu.com Ju Ye

2 . troducto eutrosophc set [] hch as proposed by Smaradache 999 s a poerful geeral formal frameork hch geeralzes the cocept of the classc set fuzzy set tutostc fuzzy set terval valued fuzzy set terval valued tutostc fuzzy set paracosstet set dalethest set paradost set tautologcal set []. he t ca hadle ot oly complete formato but also the determate formato ad cosstet formato hch est commoly real stuatos. a eutrosophc set X a truth-membershp fucto a determacy-membershp fucto ad a falsty-membershp fucto ca be epressed depedetly. he fuctos ad the eutrosophc set are real stadard or ostadard subsets of ] 0 + [.e. ] 0 + [ ] 0 + [ ad ] 0 + [. here s o restrcto o the sum of ad.e Hoever the eutrosophc set geeralzes the above metoed sets from phlosophcal pot of ve. rom scetfc or egeerg pot of ve t s dffcult to apply real scetfc ad egeerg areas. herefore Wag et al. [ 3] proposed a terval eutrosophc set S ad a sgle valued eutrosophc set SVS respectvely hch are a stace of eutrosophc set ad provded the set-theoretc operators ad varous propertes of SVSs ad Ss. SVSs ad Ss ca be used for the scetfc ad egeerg applcatos because the SVS theory ad the S theory are valuable hadlg ucerta mprecso ad cosstet formato ad easly reflect the ambguous ature of subectve udgmets. fter that Ye [4] preseted the correlato coeffcet

3 of SVSs based o the eteso of the correlato of tutostc fuzzy sets ad proved that the cose smlarty measure of SVSs s a specal case of the correlato coeffcet of SVSs ad the appled t to decso-makg problems th sgle valued eutrosophc formato. Ye [5] proposed a cross-etropy measure for SVSs ad appled t to decso-makg problems uder sgle valued eutrosophc evromet. O the other had Ye [6] also troduced the Hammg ad Eucldea dstaces betee Ss ad ther smlarty measures ad the appled them to decso-makg problems terval eutrosophc settg. urthermore Ye [7] preseted a cocept of a smplfed eutrosophc set SS hch s a subclass of the eutrosophc set ad ecompasses that of a SVS ad a S as specal cases of a SS ad defed some operatos of SSs ad the developed a smplfed eutrosophc eghted averagg SW operator a smplfed eutrosophc eghted geometrc SWG operator ad a multcrtera decso-makg method based o the SW ad SWG operators ad the cose measure of SSs uder smplfed eutrosophc evromet. s metoed above SSs are the eteso of SVSs ad Ss ad sutable for capturg mprecse ucerta ad cosstet formato multple attrbute decso makg. he correlato coeffcets are oe of mportat tools may scetfc ad egeerg applcatos. herefore motvated by [4] the purposes of ths paper are to propose to correlato coeffcets of SSs as a further geeralzato of the correlato coeffcet of SVSs proposed by Ye [4] ad to develop a multple attrbute decso makg method usg the proposed correlato coeffcets of SSs uder smplfed eutrosophc evromet. llustratve eample demostrates the applcato ad effectveess of the proposed decso-makg method. he rest of the paper s orgazed as follos. Secto brefly descrbes some cocepts of 3

4 SSs ad the correlato coeffcet of SVSs. Secto 3 proposes to correlato coeffcets for SSs ad vestgates ther propertes. Secto 4 establshes a decso-makg approach based o the proposed correlato coeffcets of SSs. llustratve eample valdatg our approach ad the comparatve aalyss are gve Secto 5. Secto 6 cotas a cocluso ad future research.. Prelmares.. Smplfed eutrosophc set Smaradache [] preseted the eutrosophc set from phlosophcal pot of ve ad gave the follog defto of a eutrosophc set. Defto []. Let X be a space of pots obects th a geerc elemet X deoted by. eutrosophc set X s characterzed by a truth-membershp fucto a determacy-membershp fucto ad a falsty-membershp fucto. he fuctos ad are real stadard or ostadard subsets of ] 0 + [.e. : X ] 0 + [ : X ] 0 + [ ad : X ] 0 + [. here s o restrcto o the sum of ad so Obvously t s dffcult to apply the eutrosophc set to practcal problems. herefore Ye [7] troduced the cocept of a SS hch s a subclass of the eutrosophc set. Defto [7]. Let X be a space of pots obects th a geerc elemet X deoted by. eutrosophc set X s characterzed by a truth-membershp fucto a determacy-membershp fucto ad a falsty-membershp fucto. f the fuctos ad are sgleto subtervals/subsets the real stadard [0 ] that s : X [0 ] : X [0 ] ad : X [0 ]. he a smplfed eutrosophc set s defed by 4

5 X. t s a subclass of eutrosophc sets ad cludes the cocepts of Ss ad SVSs. Whe e use the SS hose ad values are sgle pots the real stadard [0 ] stead of subtervals/subsets the real stadard [0 ] the SS reduce to the SVS hch as proposed by Wag et al. [3]. hus each SS ca be descrbed by three real umbers the real ut terval [0 ]. herefore the sum of ad [0 ] satsfes the codto ths case e troduce the follog deftos [3 7]. Defto 3. SS s cotaed the other SS f ad oly f ad for every X. Defto 4. he complemet of a SS s deoted by c ad s defed as c = c = c = for ay X. Defto 5. o SSs ad are equal rtte as = f ad oly f ad. Whe e oly cosder three fuctos ad the SS as subutary tervals the real stadard [0 ] the SS reduce to the S hch as proposed by Wag et al. []. hus a SS ca be descrbed by three terval umbers the real ut terval [0 ]. herefore for each pot X there are the three terval pars = [f ] [0 ] = [f ] [0 ] ad = [f ] [0 ] ad ther sum satsfes the codto for ay X. ths case e troduce the follog deftos [ 7]. Defto 6. he complemet of a SS s deoted by c ad s defed as c = = [f ] c = [ f ] c = = [f ] for ay X. 5

6 Defto 7. SS s cotaed the other SS f ad oly f f f f f f f ad for ay X. Defto 8. o SSs ad are equal rtte as = f ad oly f ad. f for a SS the loer ad er ed pots of the three terval pars = [f ] = [f ]ad = [f ] for ay X are detcal the SS reduce to the SVS. Hoever the S ad the SVS belog to the SS. he ths paper oly cosders the SS hose ad values are terval umbers... Correlato coeffcet of SVSs ased o the eteso of the correlato of tutostc fuzzy sets Ye [4] defed the formatoal eergy of a SVS the correlato of to SVSs ad ad the correlato coeffcet of to SVSs ad. or a SVS the uverse of dscourse X = { } the formatoal eergy of the SVS s defed as. or to SVSs ad the uverse of dscourse X = { } the the correlato of the SVSs ad s defed as C. herefore the correlato coeffcet of the SVSs ad s defed by the follog formula: 6

7 7 C K /. 3 he correlato coeffcet K satsfes the follog propertes [4]: K = K ; 0 K ; 3 K = f =. 3. Correlato coeffcets of SSs SSs are a subclass of a eutrosophc set ad a geeralzato of fuzzy sets ad tutostc fuzzy sets terval valued tutostc fuzzy sets SVSs ad Ss. o eted the correlato coeffcet of SVSs [4] to SSs e defe the formatoal eergy of a SS the correlato of to SSs ad the correlato coeffcet of to SSs hch ca be used real scetfc ad egeerg applcatos the follog. Defto 9. Let ay SS be X the uverse of dscourse X = { } here [0 ] for every X. he the formatoal eergy of the SS s defed as E f f f. 4 Defto 0. or to SSs ad the uverse of dscourse X = { } the correlato of the SSs ad s defed as

8 8 f f f f f f. 5 t s obvous that the correlato of the SSs ad satsfes the follog propertes: = E =. ccordg to Deftos 9 ad 0 e ca derve the correlato coeffcet for SSs. Defto. or to SSs ad the uverse of dscourse X = { } the correlato coeffcet betee to SSs ad s gve by / / f f f f f f f f f f f f M. 6 hus e ca derve the follog heorem from the correlato coeffcet betee to SSs ad. heorem. or to SVSs ad the uverse of dscourse X = { } the correlato coeffcet M satsfes the follog propertes: 4 M = M ; 5 0 M ; 6 M = f =. Proof : t s straghtforard.

9 9 he equalty M 0 s obvous. elo let us prove M : ] f f f f f f f f [f ] f f f f f f f f [f ] f f f f f [f. Usg the Cauchy-Scharz equalty: y y y y y y here R ad y y y R e obta f f f f f f ] f f [f ] f f [f ] f f [f ] f f [f ] f f [f ] f f [f. herefore / /. hus 0 M. 3 = f = f = f = f = f = f ad = for ay X M =. Especally he both the loer ad er ed pots of the terval umbers of ad the SS ad the loer ad er ed pots of the terval umbers of ad the SS are detcal for ay X there are the three real umbers of

10 0 [0 ] ad the three real umbers of [0 ]. hus Eq. 6 reduces to Eq. 3. herefore the correlato coeffcet of SVSs s a specal case of the correlato coeffcet of SSs. s a geeralzato of the correlato coeffcet used terval tutostc fuzzy sets [8] e gve aother formula of the correlato coeffcet of SSs. Defto. or to SSs ad the uverse of dscourse X = { } the correlato coeffcet betee the to SSs ad s defed by M f f f f f f ma f f f f f f ma 7 heorem. he correlato coeffcet M follos the same propertes lsted heorem as follos: M = M ; 0 M ; 3 M = f =. Proof : he process to prove the propertes ad 3 s aalogous to that heorem omtted. he equalty M 0 s obvous. o e oly prove M. ased o the proof process of heorem e have / /

11 ad the ma. hus 0 M. Especally he both the loer ad er ed pots of the terval umbers of ad the SS ad the loer ad er ed pots of the terval umbers of ad the SS are detcal for ay X there are the three real umbers of [0 ] ad the three real umbers of [0 ]. hus Eq. 7 reduces to the follog formula: C M 3 ma ma. 8 Obvously t s aother formula of the correlato coeffcet betee the SVSs ad hch s a specal case of the correlato coeffcet betee the SSs ad. Hoever the dffereces of mportace are cosdered the elemets the uverse. herefore e eed to take the eghts of the elemets = to accout. the follog e develop to eghted correlato coeffcets betee SSs. Let be the eght for each elemet = [0 ] ad the e have the follog to eghted correlato coeffcets betee the SSs ad respectvely as follos:

12 / / 4 f f f f f f f f f f f f M 9 M 5 f f f f f f ma f f f f f f ma 0 f = / / / the Eqs. 9 ad 0 reduce to Eqs. 6 ad 7 respectvely. ote that both M 4 ad M 5 also satsfy the three propertes of heorem. heorem 3. Let be the eght for each elemet = [0 ] ad the the eghted correlato coeffcet M 4 defed Eq. 9 satsfes the follog propertes: M 4 = M 4 ; 0 4 M ; 3 M 4 = f =. Sce the process to prove these propertes s smlar to that heorem e do ot repeat t here. heorem 4. Let be the eght for each elemet = [0 ] ad

13 the the eghted correlato coeffcet M 5 defed Eq. 0 satsfes the follog propertes: M 5 = M 4 ; 0 M 5 ; 3 M 5 = f =. Sce the process to prove these propertes s smlar to that heorem e do ot repeat t here. 4. Decso-makg method based o correlato coeffcets ths secto e propose a multple attrbute decso-makg method based o to correlato coeffcets betee SSs uder smplfed eutrosophc evromet. Let = { m } be a set of alteratves ad C = {C C C } be a set of attrbutes. ssume that the eght of a attrbute C = etered by the decso-maker s [0 ] ad. ths case the characterstc of a alteratve = m o a attrbute C = s represeted by the follog SS: { C C C C C C}. Here e oly cosder that the three terval pars C = [f C C ] C = [f C C ] C = [f C C ] [0 ] are gve a SS here 0 C + C + C 3 for C C = ad = m because a SS s reduced to a SVS he C = f C = C C = f C = C ad C = f C = C are three real umbers 3

14 the real ut terval [0 ]. or coveece the terval pars C = [f C C ] C = [f C C ] C = [f C C ] [0 ] are deoted by a smplfed eutrosophc value SV = [a b ] [c d ] [e f ] = m; = hch s usually derved from the evaluato of a alteratve th respect to a crtero C by the epert or decso maker. hus e ca elct a smplfed eutrosophc decso matr D = m. multple attrbute decso makg problems the cocept of deal pot has bee used to help detfy the best alteratve the decso set. lthough the deal alteratve does ot est real orld t does provde a useful theoretcal costruct agast hch to evaluate alteratves [6]. Geerally the evaluato attrbutes ca be categorzed to to kds: beeft attrbutes ad cost attrbutes. Let H be a collecto of beeft attrbutes ad L be a collecto of cost attrbutes. the decso-makg method a deal alteratve ca be detfed by usg a mamum operator for the beeft attrbutes ad a mmum operator for the cost attrbutes to determe the best value of each attrbute amog all alteratves. herefore e defe a deal SV for a beeft attrbute the deal alteratve as a b c d e f ma a ma b m c m d m e m f for H; hle for a cost attrbutes e defe a deal SV the deal alteratve by a b c d e f m a m b ma c ma d ma e ma f for L. Hece by applyg Eq. 9 the eghted correlato coeffcet betee a alteratve = m ad the deal alteratve s gve by 4

15 5 f e d c b a f e d c b a f f e e d d c c b b a a M 4. Or by applyg Eq. 0 the eghted correlato coeffcet betee a alteratve = m ad the deal alteratve s gve by f e d c b a f e d c b a f f e e d d c c b b a a M 5 ma hrough the correlato coeffcet M k k = 4 or 5; = m e ca obta the rakg order of all alteratves ad the best oes. 5. llustratve eample ad comparatve aalyss 5. llustratve eample ths subsecto a llustratve eample for the multple attrbute decso-makg problem of vestmet alteratves s gve to demostrate the applcato ad effectveess of the proposed decso-makg method. Let us cosder the decso-makg problem adapted from [6]. here s a vestmet compay hch ats to vest a sum of moey the best opto. here s a pael th four possble alteratves to vest the moey: s a car compay; s a food compay; 3 3 s a computer compay; 4 4 s a arms compay. he vestmet compay must take a decso accordg to the three attrbutes: C s the rsk; C s the groth; 3 C 3 s the evrometal mpact here C ad C are beeft attrbutes ad C 3 s a cost attrbute. he eght vector of the attrbutes s gve by = [6]. he four possble alteratves are to be evaluated

16 uder the above three attrbutes by the form of SVs as sho the follog smplfed eutrosophc decso matr D: [0.40.5][0.0.3][0.30.4] [0.60.7][0.0.][0.0.3] D [0.30.6][0.0.3][0.30.4] [0.70.8][0.00.][0.0.] [0.40.6][0.0.3][0.0.4] [0.60.7][0.0.][0.0.3] [0.50.6][0.0.3][0.30.4] [0.60.7][0.0.][0.0.3] [0.70.9][0.0.3][0.40.5] [0.30.6][0.30.5][0.80.9]. [0.40.5][0.0.4][0.70.9] [0.60.7][0.30.4][0.80.9] he e utlze the developed approach to obta the most desrable alteratves. rom the smplfed eutrosophc decso matr D e ca obta the follog deal alteratve: [0.70.8][0.00.][0.0.] [0.60.7][0.0.][0.0.3] [0.30.5][0.30.5][0.80.9]. he by usg Eq. e ca obta the values of the correlato coeffcet M 4 = 3 4: M 4 = M 4 = M 4 3 = ad M 4 4 = hus the rakg order of the four alteratves s 4 3. herefore the alteratve s the best choce amog the four alteratves. Or by usg Eq. e ca also obta the values of the correlato coeffcet M 5 = 3 4: M 5 = M 5 = M 5 3 = ad M 5 4 = herefore the rakg order of the four alteratves s 4 3. Obvously the alteratve s also the best choce amog the four alteratves. rom the above results e ca see that the same rakg order of the four alteratves ad the same best choce are obtaed by use of dfferet correlato coeffcets hch are agreemet th the results of Ye s methods [6]. he above eample clearly dcates that the proposed 6

17 decso-makg method s applcable ad effectve uder smplfed eutrosophc evromet. 5. Comparsos to relatve methods s metoed above the SS clude the SVS ad the S hch are specal cases of the SS. herefore the to correlato coeffcets of SSs proposed ths paper are the further eteso of the correlato coeffcet of SVSs proposed [4]. O the oe had compared th the decso makg methods [4-6] the decso-makg method ths paper uses the smplfed eutrosophc formato hle the decso makg methods [4-6] uses the sgle valued eutrosophc formato [4 5] ad the terval eutrosophc formato [6]. urthermore the smplfed eutrosophc decso makg method proposed ths paper s a further geeralzato of the sgle valued eutrosophc decso-makg method proposed by Ye [4]. he later s a specal case of the former. herefore the decso-makg method proposed ths paper ca deal th ot oly sgle valued eutrosophc decso makg problems but also terval eutrosophc decso-makg problems. o some etet the proposed smplfed eutrosophc decso-makg method s more geeral ad more practcal tha estg decso-makg methods [4-6]. O the other had compared th the decso makg method [7] although the decso makg methods ths paper ad [7] all use smplfed eutrosophc formato the decso-makg method proposed ths paper s more smple ad more coveet tha the decso-makg method [7] sce the decso-makg process the former uses relatvely smple calculatos ad steps ad the the later uses relatvely comple calculatos ad steps. 6. Cocluso 7

18 hs paper has developed to correlato coeffcets betee SSs as a geeralzato of the sgle eutrosophc correlato coeffcet. he a multcrtera decso-makg method has bee establshed based the proposed to correlato coeffcets of SSs uder smplfed eutrosophc evromet. hrough the correlato coeffcets betee each alteratve ad the deal alteratve e ca obta the rakg order of all alteratves ad the best alteratve. ally a llustratve eample demostrated the applcato ad effectveess of the developed decso-makg approach. he proposed decso-makg method s sutable for decso makg problems th the complete determate ad cosstet formato hch est commoly real stuatos. urthermore the techques proposed ths paper eted estg decso-makg methods [4-6] ad ca provde a useful ad smple method for decso-makers. the future e shall cotue orkg the applcato of the correlato coeffcets betee SSs to other domas such as patter recogtos ad medcal dagoses. Refereces []. Smaradache ufyg feld logcs. eutrosophy: eutrosophc probablty set ad logc Rehoboth: merca Research Press999. [] H. Wag. Smaradache Y.Q. Zhag R. Suderrama terval eutrosophc sets ad logc: heory ad applcatos computg Hes Phoe Z 005 [3] H. Wag. Smaradache Y.Q. Zhag R. Suderrama Sgle valued eutrosophc sets Multspace ad Multstructure [4] J. Ye Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet teratoal Joural of Geeral Systems

19 [5] J. Ye Sgle valued eutrosophc cross-etropy for multcrtera decso makg problems ppled Mathematcal Modellg 03 do: 0.06/.apm [6] J. Ye Smlarty measures betee terval eutrosophc sets ad ther applcatos multcrtera decso-makg. Joural of tellget ad uzzy Systems 03 do: 0.333/S-074. [7] J. Ye multcrtera decso-makg method usg aggregato operators for smplfed eutrosophc sets Joural of tellget ad uzzy Systems 03 do: 0.333/S [8] Z.S. Xu J. Che J.J. Wu Clusterg algorthm for tutostc fuzzy sets form. Sc