Soft Computing Similarity measures between interval neutrosophic sets and their multicriteria decisionmaking

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1 Soft omutg Smlarty measures betwee terval eutrosohc sets ad ther multcrtera decsomakg method --Mauscrt Draft-- Mauscrt Number: ull tle: rtcle ye: Keywords: bstract: SOO-D Smlarty measures betwee terval eutrosohc sets ad ther multcrtera decsomakg method Orgal Research Neutrosohc set; terval eutrosohc set; Hammg dstace; Eucldea dstace; Smlarty measure; Decso-makg terval eutrosohc set s a stace of a eutrosohc set whch ca be used real scetfc ad egeerg alcatos. the aer the Hammg ad Eucldea dstaces betwee terval eutrosohc sets NSs are defed ad the smlarty measures betwee NSs are roosed based o the relatosh betwee smlarty measures ad dstaces. he a multcrtera decso-makg method s establshed terval eutrosohc settg whch crtero values for alteratves are NSs ad the crtero weghts are kow formato. We utlze the smlarty measures betwee each alteratve ad the deal alteratve to rak the alteratves ad to determe the best oe. ally a llustratve eamle demostrates the alcato of the roosed decso-makg method. Powered by Edtoral Maager ad Prert Maager from res Systems ororato

2 Mauscrt lck here to dowload Mauscrt: sms_mcdmm.doc lck here to vew lked Refereces Smlarty measures betwee terval eutrosohc sets ad ther multcrtera decso-makg method Ju Ye aculty of Egeerg Shaog Uversty Shaog Zheag Provce 3000 P.R. ha bstract terval eutrosohc set s a stace of a eutrosohc set whch ca be used real scetfc ad egeerg alcatos. the aer the Hammg ad Eucldea dstaces betwee terval eutrosohc sets NSs are defed ad the smlarty measures betwee NSs are roosed based o the relatosh betwee smlarty measures ad dstaces. he a multcrtera decso-makg method s establshed terval eutrosohc settg whch crtero values for alteratves are NSs ad the crtero weghts are kow formato. We utlze the smlarty measures betwee each alteratve ad the deal alteratve to rak the alteratves ad to determe the best oe. ally a llustratve eamle demostrates the alcato of the roosed decso-makg method. Keywords: Neutrosohc set; terval eutrosohc set; Hammg dstace; Eucldea dstace; Smlarty measure; Decso-makg orresodg author. el.: E-mal address: ye@yahoo.com.c Ju Ye

3 . troducto Neutrosohy s a brach of hlosohy whch studes the org ature ad scoe of eutraltes as well as ther teractos wth dfferet deatoal sectra Smaradache 999. Neutrosohc set s a owerful geeral formal framework whch geeralzes the cocet of the classc set fuzzy set Zadeh 95 terval valued fuzzy set urkse 98 tutostc fuzzy set taassov 98 terval valued tutostc fuzzy set taassov ad Gargov 989 aracosstet set Smaradache 999 dalethest set Smaradache 999 aradost set Smaradache 999 tautologcal set Smaradache 999. eutrosohc set determacy s quatfed elctly ad truth-membersh determacy-membersh ad false-membersh are deedet. hs assumto s very mortat may alcatos such as formato fuso whch the data are combed from dfferet sesors. Recetly eutrosohc sets had maly bee aled to mage rocessg heg ad Guo 008; Guo ad heg 009. tutostc fuzzy sets ad terval valued tutostc fuzzy sets ca oly hadle comlete formato but ot the determate formato ad cosstet formato whch est commoly real stuatos. or eamle whe we ask the oo of a eert about certa statemet he or she may that the ossblty that the statemet s true s betwee 0.5 ad 0.7 ad the statemet s false s betwee 0. ad 0.4 ad the degree that he or she s ot sure s betwee 0. ad 0.3. Here s aother eamle ose there are 0 voters durg a votg rocess. tme t four vote yes three vote o ad three are udecded. or eutrosohc otato t ca be eressed as ; tme t two vote yes three vote o two gve u ad three are udecded

4 the t ca be eressed as hat s beyod the scoe of the tutostc fuzzy set. So the oto of eutrosohc set s more geeral ad overcomes the aforemetoed ssues. he eutrosohc set geeralzes the above metoed sets from hlosohcal ot of vew. rom scetfc or egeerg ot of vew the eutrosohc set ad set-theoretc oerators eed to be secfed. Otherwse t wll be dffcult to aly the real alcatos Wag et al 005. herefore Wag et al 005 roosed the set-theoretc oerators o a stace of eutrosohc set called terval eutrosohc set NS. he terval eutrosohc set ca rereset ucerta mrecse comlete ad cosstet formato whch est real world. However to the best of our kowledge the estg lterature does ot deal wth smlarty measures betwee NSs ad the decso-makg roblems terval eutrosohc settg. herefore the Hammg ad Eucldea dstaces betwee NSs are defed ad the dstaces-based smlarty measures for NSs are roosed ths aer whch ca be used real scetfc ad egeerg alcatos. hus a multcrtera decso-makg method s establshed based o the roosed smlarty measures. hrough the smlarty measures betwee each alteratve ad the deal alteratve the rakg order of all alteratves ca be determed ad the best oe ca be easly detfed as well. llustratve eamle demostrates the alcato of the roosed decso-makg method. he rest of aer s orgazed as follows. Secto troduces the some cocets of eutrosohc sets Smaradache 999 ad NSs Wag et al 005. he Hammg ad Eucldea dstaces betwee NSs are defed ad a smlarty measure based o the Hammg dstace ad a smlarty measure based o the Eucldea dstace are roosed accordg to the relatosh of smlarty measures ad dstaces Secto 3. decso-makg method s establshed terval eutrosohc settg by meas of the smlarty measure betwee each alteratve ad the deal 3

5 alteratve Secto 4. Secto 5 a llustratve eamle s reseted to llustrate the develoed aroach. ally some fal remarks of the smlarty measures betwee NSs ad the roosed decso-makg method are gve Secto.. Some cocets of eutrosohc sets hs secto gves a bref overvew of cocets of eutrosohc sets Smaradache 999 ad terval eutrosohc sets Wag et al Neutrosohc sets Neutrosohc set s a art of eutrosohy whch studes the org ature ad scoe of eutraltes as well as ther teractos wth dfferet deatoal sectra Smaradache 999 ad s a owerful geeral formal framework whch geeralzes the above metoed sets from hlosohcal ot of vew. he relatosh of eutrosohc set ad other sets s llustrated g. Wag et al 005. Smaradache 999 gave the followg defto of a eutrosohc set. Defto Smaradache 999 Let X be a sace of ots obects wth a geerc elemet X deoted by. eutrosohc set X s characterzed by a truth-membersh fucto a determacy-membersh fucto ad a falsty-membersh fucto. ad are real stadard or ostadard subsets of ]0 + [. hat s : X ]0 + [ : X ]0 + [ ad : X ]0 + [. here s o restrcto o the sum of ad so

6 Defto Smaradache 999 he comlemet of a eutrosohc set s deoted by c ad s defed as c = { + } c = {+} ad c = {+} for every X. Defto 3 Smaradache 999 eutrosohc set s cotaed the other eutrosohc set f ad oly f f f f f f f ad for every X. Neutrosohc set terval eutrosohc set terval valued tutostc fuzzy set terval valued aracosstet set tutostc fuzzy set terval valued fuzzy set Paracosstet set uzzy set lassc set g.. Relatosh of eutrosohc set ad other sets.. terval eutrosohc sets 5

7 NS s a stace of a eutrosohc set whch ca be used real scetfc ad egeerg alcatos. the followg we troduce the defto of a NS Wag et al Defto 4 Wag et al 005 Let X be a sace of ots obects wth geerc elemets X deoted by. NS X s characterzed by a truth-membersh fucto a determacy-membersh fucto ad a falsty-membersh fucto. or each ot X we have that [0 ]. We call t terval because t s the subclass of a eutrosohc set that s we oly cosder the subutary terval of [0 ]. herefore ll NSs are clearly eutrosohc sets. NS R s llustrated g. Wag et al 005. f f f 0 X g.. llustrato of a NS R Defto 5 Wag et al 005 NS s emty f ad oly f ts f = = 0 f

8 = = ad f = = 0 for ay X. Defto Wag et al 005 he comlemet of a NS s deoted by c ad s defed as c = f c = c = f c = for ay X. Let 0 = <0 > ad = < 0 0>. he 0 c = < 0 0> ad c = <0 >. Defto 7 Wag et al 005 terval eutrosohc set s cotaed the other NS f ad oly f f f f f f f ad for ay X. Defto 8 Wag et al 005 wo NSs ad are equal wrtte as = f ad oly f ad. 3. Smlarty measures betwee NSs ths secto we reset the deftos of the Hammg ad Eucldea dstaces betwee NSs ad the smlarty measures betwee NSs based o the dstaces whch ca be used real scetfc ad egeerg alcatos. or coveece two NSs ad X = { } are deoted by X ad X where [0 ] for every X. he we defe the followg dstaces for ad. he Hammg dstace: d f f f f f f he ormalzed Hammg dstace: 7

9 8 f f f f f f d. he Eucldea dstace: / 3 f f f f f f d 3 V he ormalzed Eucldea dstace: / 4 f f f f f f d.4 Proosto he above defed dstace d k k = 34 betwee NSs ad satsfes the followg roertes D-D4: D d k 0; D d k = 0 f ad oly f = ; D3 d k = d k ; D4 f s a NS X the d k d k ad d k d k. Proof t s easy to see that d k k = 34 satsfes the roertes D D3. herefore we oly rove D4. Let the f f f f f f f f f ad for every X. or = we have f f f f f f f f f f f f f f f f f f f f

10 9 f f f f. Hece f f f f f f f f f f f f f f f f f f f f f f f f. ombg the above equaltes wth the above defed dstace formulas -4 we ca obta that d d k k ad d d k k for k = 3 4. hus the roerty D4 s obtaed. However the dffereces of mortace are cosdered the elemets the uverse. herefore we eed to take the weghts of the elemets = to accout. the followg we develo some weghted dstace measures betwee NSs. Let w = {w w w } s the weght vector of the elemets = the we have the followg the weghted Hammg dstace ad the weghted Eucldea dstace: f f f f f f 5 w d 5 / f f f f f f w d. f w = {/ / /} the Eqs. 5 ad are reduced to the ormalzed Hammg dstace Eq. ad the ormalzed Eucldea dstace Eq. 4 resectvely. t s easy to check that the weghted dstace d k k = 5 betwee NSs ad also satsfy the above roertes D-D4.

11 0 t s well kow that smlarty measures ca be geerated from dstace measures. herefore we may use the roosed dstace measures to defe smlarty measures. ased o the relatosh of smlarty measures ad dstace measures we ca defe some smlarty measures betwee NSs ad as follows: f f f f f f w S 7 / f f f f f f w S 8 ccordg to the above dstace roertes D-D4 t s easy to check that the smlarty measure S k k = has the followg roertes P-P4: P 0 S k ; P S k = f ad oly f = ; P3 S k = S k ; P4 f s a NS X the S k S k ad S k S k. t s clear that the larger the value of S k k = the more the smlarty betwee NSs ad. 4. Decso-makg method based o the smlarty measures ths secto we reset a hadlg method for the multcrtera decso-makg roblem terval eutrosohc settg by meas of the smlarty measures betwee NSs. Let = { m } be a set of alteratves ad let = { } be a set of crtera. ssume that the weght of the crtero = etered by the decso-maker s w w

12 [0 ] ad. ths case the characterstc of the alteratve = m s rereseted by the followg NS: { { [f ][f } ][f ] } where [f ] [f ] [f ] [0 ] 0 3 = ad = m. NS whch s the ar of tervals [f ] [f ] [f ] for s deoted by = [a b ] [c d ] [e f ] for coveece. Here a NS s usually derved from the evaluato of a alteratve wth resect to a crtero by meas of a score law ad data rocessg ractce. herefore we ca elct a terval eutrosohc decso matr D = m. multcrtera decso makg evromets the cocet of deal ot has bee used to hel detfy the best alteratve the decso set. lthough the deal alteratve does ot est real world t does rovde a useful theoretcal costruct agast whch to evaluate alteratves. Geerally the evaluato crtera ca be categorzed to two kds beeft crtera ad cost crtera. Let H be a collecto of beeft crtera ad L be a collecto of cost crtera. he we defe a deal NS for a beeft crtero the deal alteratve * as * * * * * * * a b c d e f for H; whle for a cost crtero we defe a deal NS the deal alteratve * * * * * * * * as a b c d e f 00 for L. hus by alyg Eqs. 7 ad 8 two smlarty measures betwee a alteratve ad the deal alteratve * rereseted by the NSs are defed as follows:

13 * * * * * * * S w a a b b c c d d e e f f 9 * S w * * * * * * a a b b c c d d e e f f /. 0 hrough the smlarty measure S * or S * = m betwee each alteratve ad the deal alteratve the rakg order of all alteratves ca be determed ad the best oe ca be easly detfed as well. 5. llustratve eamle ths secto a eamle for the multcrtera decso-makg roblem of alteratves s used as the demostrato of the alcato of the roosed decso-makg method as well as the effectveess of the roosed method. Let us cosder the decso-makg roblem adated from Ye 009. here s a vestmet comay whch wats to vest a sum of moey the best oto. here s a ael wth four ossble alteratves to vest the moey: s a car comay; s a food comay; 3 3 s a comuter comay; 4 4 s a arms comay. he vestmet comay must take a decso accordg to the followg three crtera: s the rsk aalyss; s the growth aalyss; 3 3 s the evrometal mact aalyss where ad are beeft crtera ad 3 s a cost crtero. he weght vector of the crtera s gve by w = he four ossble alteratves are to be evaluated uder the above three crtera by corresodg to the NSs as show the followg terval eutrosohc decso matr D:

14 [0.40.5][0.0.3][0.30.4] [0.0.7][0.0.][0.0.3] D [0.30.][0.0.3][0.30.4] [0.70.8][0.00.][0.0.] [0.40.][0.0.3][0.0.4] [0.0.7][0.0.][0.0.3] [0.50.][0.0.3][0.30.4] [0.0.7][0.0.][0.0.3] [0.70.9][0.0.3][0.40.5] [0.30.][0.30.5][0.80.9]. [0.40.5][0.0.4][0.70.9] [0.0.7][0.30.4][0.80.9] he we utlze the develoed aroach to obta the most desrable alteratves. y usg Eq. 9 we ca obta the followg smlarty measures of S * * = 3 4: S * * = S * * = S * * 3 = ad S * * 4 = herefore the rakg order of the four alteratves s 4 3 ad. Obvously amogst them 4 s the best alteratve. Or by alyg Eq. 0 we ca gve the smlarty measures of S * * = 3 4 as follows: S * * = S * * = S * * 3 = ad S * * 4 = hus the rakg order of the four alteratves s 4 3 ad obvously amogst them s the best alteratve.. ocluso ths aer we defed the Hammg ad Eucldea dstaces ad roosed the smlarty measures betwee NSs based o the relatosh betwee smlarty measures ad dstaces. he a multcrtera decso-makg method has bee establshed terval eutrosohc settg by meas of the smlarty measure betwee each alteratve ad the deal alteratve. hrough the smlarty measures the rakg order of all alteratves ca be determed ad the best alteratve ca be easly detfed as well. ally a llustratve eamle llustrated the alcato of the develoed aroach. he roosed smlarty measures betwee NSs are more sutable for real scetfc ad 3

15 egeerg alcatos. he the techques roosed ths aer eted estg decso-makg methods ad ca rovde a useful way for decso-makers. the future we shall cotue workg the alcato of the smlarty measures betwee NSs to other domas. Refereces taassov K Gargov G 989 terval valued tutostc fuzzy sets. uzzy Sets ad Systems 3: taassov K 98 tutostc fuzzy sets. uzzy Sets ad Systems 0: 87 9 heg HD Guo Y 008 ew eutrosohc aroach to mage thresholdg. New Mathematcs ad Natural omutato 4 3: Guo Y heg HD 009 ew eutrosohc aroach to mage segmetato. Patter Recogto 4: Smaradache 999 ufyg feld logcs. eutrosohy: Neutrosohc robablty set ad logc merca Research Press Rehoboth Smaradache 005 Neutrosohc set. geeralzato of the tutostc fuzzy set. teratoal Joural of Pure ad led Mathematcs 4: urkse 98 terval valued fuzzy sets based o ormal forms. uzzy Sets ad Systems 0: 9 0 Wag H Smaradache Zhag YQ Suderrama R 005 terval eutrosohc sets ad Logc: heory ad lcatos omutg Hes Phoe Z Ye J 009 Multcrtera fuzzy decso-makg method based o a ovel accuracy fucto uder 4

16 terval-valued tutostc fuzzy evromet. Eert Systems Wth lcatos 3: Zadeh L 95 uzzy Sets. formato ad otrol 8:

17 gure Neutrosohc set terval eutrosohc set terval-valued tutostc fuzzy set terval-valued aracosstet set tutostc fuzzy set terval valued fuzzy set Paracosstet set uzzy set lassc set g.. Relatosh of eutrosohc set ad other sets

18 gure f f f 0 X