Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision-Making

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1 48 Neutrosophc ets d ystems Vol gle Vlued Neutrosophc mlrty Mesures for Multple ttrbute Decso-Mkg Ju Ye d Qsheg Zhg 2 Deprtmet of Electrcl d formto Egeerg hog Uversty 508 Hucheg West Rod hog Zheg 2000 P.R. h. E-ml: yehu@ lyu.com 2 chool of formtcs Gugdog Uversty of oreg tudes Gugzhou P.R. h. E-ml: zhqsh0@gdufs.edu.c bstrct. mlrty mesures ply mportt role dt mg ptter recogto decso mkg mche lerg mge process etc. he sgle vlued eutrosophc sets VNs c descrbe d hdle the determte d cosstet formto whch fuzzy sets d tutostc fuzzy sets cot descrbe d del wth. herefore the pper proposes ew smlrty mes-ures betwee VNs bsed o the mmum d m-mum opertors. he multple ttrbute decso-mkg method bsed o the weghted smlrty mesure of VNs s estblshed whch ttrbute vlues for ltertves re represeted by the form of sgle vlued eutrosophc vlues VNVs d the ttrbute weghts d the weghts of the three depedet elemets.e. truthmembershp degree determcy-membershp degree d flsty-membershp degree VNV re cosdered the decso-mkg method. the decso mkg we utlze the sgle-vlued eutrosophc weghted smlrty mesure betwee the del ltertve d ltertve to rk the ltertves correspodg to the mesure vlues d to select the most desrble oes. lly two prctcl emples re provded to demostrte the pplctos d effectveess of the sgle vlued eutrosophc multple ttrbute decso-mkg method. Keywords: Neutrosophc set sgle vlued eutrosophc set smlrty mesure decso mkg. troducto ce fuzzy sets [] tutostc fuzzy sets s [2] tervl-vlued tutostc fuzzy sets Vs [] were troduced they hve bee wdely ppled dt mg ptter recogto formto retrevl decso mkg mche lerg mge process d so o. lthough they re very successful ther respectve doms fuzzy sets s d Vs cot descrbe d del wth the determte d cosstet formto tht ests rel world. o hdle ucertty mprecse complete d cosstet formto mrdche [4] proposed the cocept of eutrosophc set. he eutrosophc set s powerful geerl forml frmework whch geerlzes the cocepts of the clssc set fuzzy set V etc. [4]. the eutrosophc set truthmembershp determcy-membershp d flstymembershp re represeted depedetly. However the eutrosophc set geerlzes the bove metoed sets from phlosophcl pot of vew d ts fuctos d re rel stdrd or ostdrd subsets of ] 0 +[.e. : X ] 0 +[ : X ] 0 +[ d : X ] 0 +[. hus t s dffcult to pply rel scetfc d egeerg res. herefore Wg et l. [5 6] troduced sgle vlued eutrosophc set VN d tervl eutrosophc set N whch re the subclss of eutrosophc set. hey c descrbe d hdle determte formto d cosstet formto whch fuzzy sets s d Vs cot descrbe d del wth. Recetly Ye [7-9] proposed the correlto coeffcets of VNs d the cross-etropy mesure of VNs d ppled them to sgle vlued eutrosophc decso-mkg problems. he Ye [0] troduced smlrty mesures bsed o the dstces betwee Ns d ppled them to multcrter decso-mkg problems wth tervl eutrosophc formto. h d Lu [] proposed eteded OP method for the multple ttrbute decso mkg problems wth tervl eutrosophc formto. urthermore Ye [2] troduced the cocept of smplfed eutrosophc sets d preseted smplfed eutrosophc weghted ggregto opertors d the he ppled them to multcrter decso-mkg problems wth smplfed eutrosophc formto. Mumdr d mt [] troduced severl smlrty mesures betwee VNs bsed o dstces mtchg fucto membershp grdes d the proposed etropy mesure for VN. roum d mrdche [4] defed the dstce betwee eutrosophc sets o the bss of the Husdorff dstce d some smlrty Ju Ye Qsheg Zhg gle Vlued Neutrosophc 48 mlrty Mesures for Multple ttrbute Decso-Mkg

2 Neutrosophc ets d ystems Vol mesures bsed o the dstces set theoretc pproch Defto [6]. he complemet of VN s d mtchg fucto to clculte the smlrty degree deoted by c d s defed s c = betwee eutrosophc sets. c = c = for y X. ecuse the cocept of smlrty s fudmetlly he t c be deoted by mportt lmost every scetfc feld d VNs c c descrbe d hdle the determte d cosstet X. formto ths pper proposes ew smlrty mesures betwee VNs bsed o the mmum d mmum Defto 4 [6]. VN s coted the opertors d estblshes multple ttrbute decsomkg other VN f d oly f method bsed o the weghted smlrty mesure for y X. of VNs uder sgle vlued eutrosophc evromet. o do so the rest of the rtcle s orgzed s follows. Defto 5 [6]. wo VNs d re equl ecto 2 troduces some bsc cocepts of VNs..e. = f d oly f d. ecto proposes ew smlrty mesures betwee VNs bsed o the mmum d mmum opertors mlrty mesures of VNs d vestgtes ther propertes. ecto 4 sgle vlued eutrosophc decso-mkg pproch s hs secto proposes severl smlrty proposed bsed o the weghted smlrty mesure of mesures of VNs bsed o the mmum d VNs. ecto 5 two prctcl emples re gve to mmum opertors d vestgtes ther demostrte the pplctos d the effectveess of the proposed decso-mkg pproch. oclusos d propertes. further reserch re coted ecto 6. 2 ome bsc cocepts of VNs mrdche [4] orglly troduced the cocept of eutrosophc set from phlosophcl pot of vew whch geerlzes tht of fuzzy set d V etc.. Defto [4]. Let X be spce of pots obects wth geerc elemet X deoted by. eutrosophc set X s chrcterzed by truth-membershp fucto determcy-membershp fucto d flsty-membershp fucto. he fuctos d re rel stdrd or ostdrd subsets of ] 0 + [. ht s : X ] 0 + [ : X ] 0 + [ d : X ] 0 + [. hus there s o restrcto o the sum of d so 0 sup + sup + sup +. Obvously t s dffcult to pply rel scetfc d egeerg res. Hece Wg et l. [6] troduced the defto of VN. Defto 2 [6]. Let X be uversl set. VN X s chrcterzed by truth-membershp fucto determcy-membershp fucto d flstymembershp fucto. he VN c be deoted by X where [0 ] for ech pot X. herefore the sum of d stsfes the codto geerl smlrty mesure betwee two VNs d s fucto defed s : NX 2 [0 ] whch stsfes the followg propertes: P 0 ; P2 = f = ; P = ; P4 d f for VN. Let two VNs d uverse of dscourse X = { l 2 } be X d X where [0 ] for every X. sed o the mmum d mmum opertors we preset the followg smlrty mesure betwee d : m m. m m m m he smlrty mesure hs the followg proposto. Proposto. Let d be two VNs uverse of dscourse X = { 2 }. he sgle vlued eutrosophc smlrty mesure should stsfy the followg propertes: Ju Ye Qsheg Zhg gle Vlued Neutrosophc 49 mlrty Mesures for Multple ttrbute Decso-Mkg

3 = / Eq. 2 reduces 50 Neutrosophc ets d ystems Vol P 0 ; P2 = f = ; P = ; P4 d f for VN. Proof. t s esy to remrk tht stsfes the propertes P-P. hus we must prove the property P4. Let the d for every X. ccordg to these equltes we hve the followg smlrty mesures:. ce there re c d we c obt tht. mlrly there re d. he we c obt tht. hus stsfes the property P4. herefore we fsh the proof. f the mportt dffereces re cosdered the three depedet elemets.e. truth-membershp determcy-membershp d flsty-membershp VN we eed to tke the weghts of the three depedet terms Eq. to ccout. herefore we develop other smlrty mesure betwee VNs: m m m m m m 2 2 where re the weghts of the three depedet elemets.e. truth-membershp determcy-membershp d flstymembershp VN d + + =. Especlly whe = = to Eq.. he the smlrty mesure of 2 lso hs the followg proposto: Proposto 2. Let d be two VNs uverse of dscourse X = { 2 }. he sgle vlued eutrosophc smlrty mesure 2 should stsfy the followg propertes: P 0 2 ; P2 2 = f = ; P 2 = 2 ; P4 2 2 d 2 2 f for VN. y the smlr proof method Proposto we c prove tht the smlrty mesure of 2 lso stsfes the propertes P-P4 omtted. urthermore f the mportt dffereces re cosdered the elemets uverse of dscourse X = { l 2 } we eed to tke the weght of ech elemet = 2 to ccout. herefore we develop weghted smlrty mesure betwee VNs. Let w be the weght for ech elemet = 2 w [0 ] d w d the we hve the followg weghted smlrty mesure: m m m m m m w. mlrly the weghted smlrty mesure of lso hs the followg proposto: Ju Ye Qsheg Zhg gle Vlued Neutrosophc mlrty Mesures for Multple ttrbute Decso-Mkg

4 Neutrosophc ets d ystems Vol Proposto. Let d be two VNs uverse of Hece there re d dscourse X = { 2 }. he the sgle vlued. eutrosophc smlrty mesure should stsfy 4 Decsos mkg method usg the weghted the followg propertes: smlrty mesure of VNs P 0 ; P2 = f = ; P = ; P4 d f for VN. mlr to the proof method Proposto we c prove tht the weghted smlrty mesure of lso stsfes the propertes P P4 omtted. f w = / / / the Eq. reduces to Eq. 2. or Emple ssume tht we hve the followg three VNs uverse of dscourse X = { l 2 }: = {< > < >} = {< > < >} = {< > < >}. he there re wth d for ech X = { 2 }. y usg Eq. the smlrty mesures betwee the VNs re s follows: = = 0.60 d = hus there re d. f the weght vlues of the three depedet elemets.e. truth-membershp degree determcymembershp degree d flsty-membershp degree VN re = 0.25 = 0.5 d = 0.4 by pplyg Eq. 2 the smlrty mesures betwee the VNs re s follows: 2 = = d 2 = he there re 2 2 d 2 2. ssume tht the weght vector of the two ttrbutes s w = d the weght vlues of the three depedet elemets.e. truth-membershp degree determcy-membershp degree d flstymembershp degree VN re = 0.25 = 0.5 d = 0.4. y pplyg Eq. the weghted smlrty mesures betwee the VNs re s follows: = = 0.48 d = ths secto we propose multple ttrbute decso-mkg method bsed o the weghted smlrty mesures betwee VNs uder sgle vlued eutrosophc evromet. Let = { 2 m } be set of ltertves d = { 2 } be set of ttrbutes. ssume tht the weght of the ttrbute = 2 s w wth w [0 ] d the weghts of the three depedet elemets.e. truth-membershp determcy-membershp d flsty-membershp VN re d d + + = whch re etered by the decso-mker. ths cse the chrcterstc of the ltertve = 2 m o ttrbute = 2 s represeted by VN form: { } Ju Ye Qsheg Zhg gle Vlued Neutrosophc 5 mlrty Mesures for Multple ttrbute Decso-Mkg where [0 ] d = 2 d = 2 m. for or coveece the three elemets the VN re deoted by sgle vlued eutrosophc vlue VNV = t f = 2 m; = 2 whch s usully derved from the evluto of ltertve wth respect to ttrbute by the epert or decso mker. Hece we c estblsh sgle vlued eutrosophc decso mtr D = m : D m m2. 2 m multple ttrbute decso mkg evromets the cocept of del pot hs bee used to help detfy the best ltertve the decso set [7 8]. Geerlly the evluto ttrbutes c be ctegorzed to two kds: beeft ttrbutes d cost ttrbutes. Let H be collecto of beeft ttrbutes d L be collecto of cost ttrbutes. the preseted decso-mkg method del ltertve c be detfed by usg mmum opertor for the beeft ttrbutes d mmum opertor for the cost ttrbutes to determe the best vlue of ech

5 52 Neutrosophc ets d ystems Vol ttrbute mog ll ltertves. herefore we defe del VNV for beeft ttrbute the del ltertve s t f m t m m f for H; whle for cost ttrbute we defe del VNV the del ltertve s t f m t m m f for L. hus by pplyg Eq. the weghted smlrty mesure betwee ltertve d the del ltertve re wrtte s f m m t t t t f m 4 w m 4 m m f f whch provdes the globl evluto for ech ltertve regrdg ll ttrbutes. ccordg to the weghted smlrty mesure betwee ech ltertve d the del ltertve the bgger the mesure vlue 4 = 2 4 the better the ltertve. Hece the rkg order of ll ltertves c be determed d the best oe c be esly selected s well. 5 Prctcl emples hs secto provdes two prctcl emples for multple ttrbute decso-mkg problems wth sgle vlued eutrosophc formto to demostrte the pplctos d effectveess of the proposed decsomkg method. Emple. Let us cosder the decso-mkg problem dpted from [7 8]. here s vestmet compy whch wts to vest sum of moey the best opto. here s pel wth four possble ltertves to vest the moey: s cr compy; 2 2 s food compy; s computer compy; 4 4 s rms compy. he vestmet compy must tke decso ccordg to the three ttrbutes: s the rsk; 2 2 s the growth; s the evrometl mpct where d 2 re beeft ttrbutes d s cost ttrbute. he weght vector of the three ttrbutes s gve by w = he four possble ltertves re to be evluted uder the bove three ttrbutes by the form of VNVs. or the evluto of ltertve = 2 4 wth respect to ttrbute = 2 t s obted from the questore of dom epert. or emple whe we sk the opo of epert bout ltertve wth respect to ttrbute he/she my sy tht the possblty whch the sttemet s good s 0.4 d the sttemet s poor s 0. d the degree whch he/she s ot sure s 0.2. or the eutrosophc otto t c be epressed s = hus whe the four possble ltertves wth respect to the bove three ttrbutes re evluted by the epert we c obt the followg sgle vlued eutrosophc decso mtr D: D Wthout loss of geerlty let the weght vlues of the three depedet elemets.e. truth-membershp degree determcymembershp degree d flsty-membershp degree VNV be = = = /. he we utlze the developed pproch to obt the most desrble ltertves. rstly from the sgle vlued eutrosophc decso mtr we c yeld the followg del ltertve: }. { 2 he by usg Eq. 4 we c obt the vlues of the weghted smlrty mesure 4 = 2 4: 4 = = = d 4 4 = hus the rkg order of the four ltertves s 4 2. herefore the ltertve 4 s the best choce mog the four ltertves. rom the bove results we c see tht the rkg order of the ltertves d best choce re greemet wth the results.e. the rkg order s 4 2 d the best choce s 4. Ye s method [8] but ot greemet wth the results.e. the rkg order s 2 4 d the best choce s 2. Ye s method [7]. he reso s tht dfferet mesure methods my yeld dfferet rkg orders of the ltertves the decso-mkg process. Ju Ye Qsheg Zhg gle Vlued Neutrosophc 52 mlrty Mesures for Multple ttrbute Decso-Mkg

6 Neutrosophc ets d ystems Vol Emple 2. mult-crter decso mkg problem rom the bove two emples we c see dopted from Ye [9] s cocered wth mufcturg tht the proposed sgle vlued eutrosophc compy whch wts to select the best globl suppler multple ttrbute decso-mkg method s ccordg to the core competeces of supplers. Now more sutble for rel scetfc d egeerg suppose tht there re set of four supplers = { 2 pplctos becuse t c hdle ot oly 4 } whose core competeces re evluted by mes complete formto but lso the determte of the four ttrbutes 2 4 : the level of formto d cosstet formto whch techology ovto 2 the cotrol blty of flow est commoly rel stutos. Especlly 2 the blty of mgemet 4 the level of the proposed decso-mkg method we servce 4 whch re ll beeft ttrbutes. he the cosder the mportt dffereces the three weght vector for the four ttrbutes s w = depedet elemets.e. truth-membershp 0.2. he four possble ltertves re to be evluted degree determcy-membershp degree d uder the bove four ttrbutes by the form of VNVs. flsty-membershp degree VNV d c or the evluto of ltertve = 2 4 dust the weght vlues of the three depedet wth respect to ttrbute = 2 4 by the smlr elemets. hus the proposed sgle vlued evluto method Emple t s obted from the eutrosophc decso-mkg method s more fleble d prctcl th the estg decsomkg methods [7-9]. he techque proposed questore of dom epert. or emple whe we sk the opo of epert bout ltertve wth ths pper eteds the estg decso-mkg respect to ttrbute he/she my sy tht the methods d provdes ew wy for decsomkers. possblty whch the sttemet s good s 0.5 d the sttemet s poor s 0. d the degree whch he/she s ot sure s 0.. or the eutrosophc otto t c be 6 ocluso epressed s = hus whe the four possble ltertves wth respect to the bove four ttrbutes re evluted by the smlr method from the epert we c estblsh the followg sgle vlued eutrosophc decso mtr D: D Wthout loss of geerlty let the weght vlues of the three depedet elemets.e. truth-membershp degree determcy-membershp degree d flstymembershp degree VNV be = = = /. he the proposed decso-mkg method s ppled to solve ths problem for selectg supplers. rom the sgle vlued eutrosophc decso mtr we c yeld the followg del ltertve: { } y pplyg Eq. 4 the weghted smlrty mesure vlues betwee ltertve = 2 4 d the del ltertve re s follows: 4 = = = d 4 4 = ccordg to the mesure vlues the rkg order of the four supplers s 2 4. Hece the best suppler s. rom the results we c see tht the rkg order of the ltertves d best choce re greemet wth the results Ye s method [9]. 4. hs pper hs developed three smlrty mesures betwee VNs bsed o the mmum d mmum opertors d vestgted ther propertes. he the proposed weghted smlrty mesure of VNs hs bee ppled to multple ttrbute decso-mkg problems uder sgle vlued eutrosophc evromet. he proposed method dffers from prevous pproches for sgle vlued eutrosophc multple ttrbute decso mkg ot oly due to the fct tht the proposed method use the weghted smlrty mesure of VNs but lso due to cosderg the weghts of the truth-membershp determcy-membershp d flsty-membershp VNs whch mkes t hve more fleble d prctcl th estg decso mkg methods [7-9] rel decso mkg problems. hrough the weghted smlrty mesure betwee ech ltertve d the del ltertve we c obt the rkg order of ll ltertves d the best ltertve. lly two prctcl emples demostrted the pplctos d effectveess of the decso-mkg pproch uder sgle vlued eutrosophc evromets. he proposed decso-mkg method c effectvely del wth decso-mkg problems wth the complete determte d cosstet formto whch est commoly rel stutos. urthermore by the smlr method we c esly eted the proposed weghted smlrty mesure of VNs d ts decso-mkg method to tht of Ns. the future we shll vestgte smlrty mesures betwee VNs d betwee Ns the pplctos of other doms such s ptter recogto clusterg lyss mge process d medcl dgoss. Ju Ye Qsheg Zhg gle Vlued Neutrosophc mlrty 5 Mesures for Multple ttrbute Decso-Mkg

7 54 Neutrosophc ets d ystems Vol KNOWLEDGEMEN hs work ws supported by the Ntol ocl cece ud of h GL0 the Gugdog Provce Nturl cece oudto the Humtes d ocl ceces Reserch Youth oudto of Mstry of Educto of h 2YJZH28 d the Ntol ttstcl cece Reserch Plg Proect 202LY59. Refereces [] L.. Zdeh. uzzy ets. formto d otrol [2] K. tssov. tutostc fuzzy sets. uzzy ets d ystems [] K. tssov d G. Grgov. tervl vlued tutostc fuzzy sets. uzzy ets d ystems [4]. mrdche. ufyg feld logcs. eutrosophy: Neutrosophc probblty set d logc. Rehoboth: merc Reserch Press 999 [5] H. Wg. mrdche Y. Q. Zhg d R. uderrm. tervl eutrosophc sets d logc: heory d pplctos computg Hes Phoe Z 2005 [6] H. Wg. mrdche Y. Q. Zhg d R. uderrm. gle vlued eutrosophc sets. Multspce d Multstructure [7] J. Ye. Multcrter decso-mkg method usg the correlto coeffcet uder sgle-vlued eutrosophc evromet. tertol Jourl of Geerl ystems [8] J. Ye. other form of correlto coeffcet betwee sgle vlued eutrosophc sets d ts multple ttrbute decso-mkg method. Neutrosophc ets d ystems [9] J. Ye. gle vlued eutrosophc cross-etropy for multcrter decso mkg problems. ppled Mthemtcl Modellg [0] J. Ye. mlrty mesures betwee tervl eutrosophc sets d ther pplctos multcrter decso-mkg. Jourl of tellget d uzzy ystems [] P. P. h d P.D. Lu. eteded OP method for the multple ttrbute decso mkg problems bsed o tervl eutrosophc sets. Neutrosophc ets d ystems [2] J. Ye. multcrter decso-mkg method usg ggregto opertors for smplfed eutrosophc sets. Jourl of tellget d uzzy ystems 20 do: 0.2/-096. [] P. Mumdr d.k. mt. O smlrty d etropy of eutrosophc sets. Jourl of tellget d uzzy ystems 20 do: 0.2/ [4]. d d. mrdche. everl mlrty Mesures of Neutrosophc ets. Neutrosophc ets d ystems Receved: ebrury 20th 204. ccepted: Mrch rd 204. Ju Ye Qsheg Zhg gle Vlued Neutrosophc 54 mlrty Mesures for Multple ttrbute Decso-Mkg