Iterval Neutrosohc Murhead mea Oerators ad Ther lcato Multle ttrbute Grou Decso Mag ede u ab * Xl You a a School of Maagemet Scece ad Egeerg Shadog versty of Face ad Ecoomcs Ja Shadog 5004 Cha b School of Ecoomcs ad Maagemet Cvl vato versty of Cha Ta 300300 Cha *The corresodg author: ede.lu@gmal.com bstract: I recet years eutrosohc sets (NSs have attracted wdesread attetos ad bee wdely aled to multle attrbute decso-mag (MDM. The terval eutrosohc set (INS s a exteso of NS whch the truth-membersh determacy-membersh ad falsty-membersh degree are exressed by terval values resectvely. Obvously INS ca coveetly descrbe comlex formato. t the same tme Murhead mea (MM ca cature the terrelatoshs amog the mult-ut argumets whch s a geeralzato of some exstg aggregato oerators. I ths aer we exted MM to INS ad develo some terval eutrosohc Murhead mea (INMM oerators ad the we rove ther some roertes ad dscuss some secal cases wth resect to the arameter vector. Moreover we roose two ew methods to deal wth MDM roblems based o the roosed oerators. Fally we verfy the valdty of our methods by a llustratve examle ad aalyze the advatages of our methods by comarg wth other exstg methods. Keywords: terval eutrosohc sets; Murhead mea oerators; multle attrbute decso mag.. Itroducto I real decso mags because of comlexty ad fuzzess of decso mag roblems t s dffcult for decso-maers to exress a referece recsely by crs umbers for multle attrbute decso-mag (MDM ad multle attrbute grou decso-mag (MGDM roblems wth comlete determate ad cosstet formato. der these crcumstaces Zadeh [35] roosed the fuzzy set (FS theory whch s a effectve tool to descrbe fuzzy formato ad the s used to solve MDM ad MGDM roblems [5 30]. Sce FS oly has oe membersh ad t caot hadle some comlcated fuzzy formato. The taassov [] roosed the tutostc fuzzy set (IFS by addg the o-membersh o the bass of Zadeh s FS so t s comosed of truth-membersh T ( x ad falsty-membersh F ( x. I the IFS membersh degree ad o-membersh degree are exressed by real umbers sometmes t s suffcet or adequate to exress more comlex qualtatve formato the Che ad u [7] roosed a cocet called lgustc tutostc fuzzy umbers (IFNs whch the membersh degree ad o-membersh degree are exressed by lgustc terms. However these cocets aforemetoed ca oly rocess comlete formato but ot the determate formato ad cosstet formato. Therefore based o the eutrosohy Smaradache roosed the cocet of the eutrosohc set (NS [5 6] 999 whch added a deedet determacy-membersh o the bass of tutostc fuzzy set. Obvously NS s comosed of truth-membersh determacy-membersh ad falsty-membersh resectvely ad t s a geeralzato of fuzzy set aracosstet set tutostc fuzzy set aradoxst set etc. Nowadays may great achevemets about NS have bee
made. For examle Wag et al. [8 9] roosed a sgle valued eutrosohc set (SVNS wth exteso of NS. Maumdar ad Samat [9] roosed a measure of etroy of SVNSs. d that Ye [3] roosed smlfed eutrosohc sets (SNSs. Wag ad [7] roosed mult-valued eutrosohc sets (MVNSs. I fact sometmes t may be dffcult to exress the degrees of truth-membersh falsty-membersh ad determacy-membersh by real values smlar to terval valued tutostc fuzzy set (IVIFS troduced by taassov [3] Wag et al. [8] roosed the cocet of terval eutrosohc set (INS whch s more coveet to deal wth comlex formato. I recet years formato aggregato oerators [ 4 5 7] have attracted wde attetos of researchers ad have become a mortat research toc MDM or MGDM felds. Because they have more advatages tha some tradtoal aroaches such as TOSIS [0] VIKOR [3] EECTRE [6] ad so o. For stace aggregato oerators ca rovde the comrehesve values of the alteratves ad the do rag o the bass of them whle tradtoal aroaches ca oly gve the rag results. Now may dfferet oerators were develoed for some secal fuctos such as ower average ( oerator [3] whch ca aggregate the ut data by assgg the weghted vector based o the suort degree betwee the ut argumets; Heroa mea [4] ad Boferro mea [6] ca cosder the terrelatosh of the ut argumets. Yu ad Wu [34] exlaed the advatages of HM over BM are that HM ca cosder the correlato betwee a attrbute ad tself ad ca releve the calculato redudacy. However real decso mag because of the comlexty of decso mag roblems there exst the terrelatoshs amog more tha two attrbutes ad BM or HM ca oly cosder the terrelatosh betwee ay two ut argumets. Obvously they are dffcult to deal wth ths stuato. Murhead mea (MM [] s a well-ow aggregato oerator whch ca cosder terrelatoshs amog ay umber of argumets assged by a varable vector ad some exstg oerators such as arthmetc ad geometrc oerators (ot cosderg the terrelatoshs BM oerator ad Maclaur symmetrc mea [] are ts secal cases. Therefore the MM ca offer a flexble ad robust mechasm to rocess the formato fuso roblem ad mae t more adequate to solve MDM roblems. However the orgal MM ca oly deal wth the umerc argumets order to mae the MM oerator to rocess the fuzzy formato Q ad u [3] exteded the MM oerator to rocess the -tule lgustc formato ad roosed some -tule lgustc MM oerators the aled them to solve the MGDM roblems. Due to the creasg comlexty of the decso-mag evromet t s usually dffcult for decso maers to gve the evaluato formato by crs umbers. Because INSs ca better deal wth fuzzy comlete determate ad cosstet formato ad the MM oerator ca cosder terrelatoshs amog ay umber of argumets t s meagful to exted the MM oerator to rocess terval eutrosohc formato. So the ams of ths aer are ( to roose some ew terval eutrosohc MM oerators by combg MM oerator ad INS; ( to exlore some desrable roertes ad secal cases of the roosed oerators; (3 to roose a multle-attrbute grou decso mag (MGDM methods based o the roosed oerators; (4 to show the effectveess ad advatages of the roosed methods. So the rest of ths aer s orgazed as follows. I the ext Secto we brefly revew some basc cocets ad oeratoal rules comarso method ad dstace of INNs Murhead mea (MM oerator. I Secto 3 we roose the some terval eutrosohc MM oerators ad study some roertes ad some secal cases of these oerators. I Secto 4 we develo two MDM methods for INSs based o the roosed terval eutrosohc MM oerators. I Secto 5 a llustratve examle
s gve to verfy the valdty of the roosed methods ad to show ther advatages. I Secto 6 we gve some coclusos of ths study.. relmares. The terval eutrosohc set Defto [5]. et X be a sace of ots (obects wth a geerc elemet X deoted by x. eutrosohc set X s deoted by: where T (x I (x ad (x F { x( T ( x I ( x F ( x x X } deotes the truth-membersh fucto the determacy-membersh fucto ad the falsty-membersh fucto of the elemet x X to the set resectvely. For each - + + ot x X we have T ( x I ( x F ( x 0 ad 0 T ( x + I ( x + F ( x 3. The eutrosohc set was maly roosed from the hlosohcal ot of vew t s dffcult to aly to the real alcato. To solve ths roblem Wag et al. [8] further roosed a sgle-valued eutrosohc set from a scetfc or egeerg ersectve whch s a exteso of fuzzy set tutostc fuzzy set aracosstet set aradoxst set etc. The defto of a sgle-valued set s gve as follows. Defto [9]. et X be a sace of ots (obects wth a geerc elemet X deoted by x. sgle valued eutrosohc set (SVNS X s deoted by: { x( T ( x I ( x F ( x x X } where T (x I (x ad (x deotes the truth-membersh fucto the determacy-membersh fucto F ad the falsty-membersh fucto of the elemet x X to the set resectvely. For each ot x X we have T ( x I ( x F( x [ 0] ad 0 T ( x + I ( x + F( x 3. For smlcty we ca use x ( T I F to rereset a elemet x SVNS ad the elemet x s called a sgle valued eutrosohc umber (SVNN. I order to descrbe more comlex formato Wag et al. [7] further defe the cocet of terval valued eutrosohc set show as follows. Defto 3 [8]. et X be a sace of ots (obects wth a geerc elemet X deoted by x. terval eutrosohc set (INS X s deoted by { x( T ( x I ( x F ( x x X } where T (x I (x ad (x deotes the truth-membersh fucto the determacy-membersh fucto ad F the falsty-membersh fucto of the elemet x X to the set resectvely. For each ot x X T x I x F x ad ( T x ( I x ( F x meets ( ( ( [0] 0 su ( + su ( + su ( 3. For smlcty we ca use x ([ T T ][ I I ][ F F ] to rereset a elemet x INS ad t s called terval eutrosohc umber (INN. where T T [ 0] I I [ 0] [ ] F F 0 ad T + I + F 3. Defto 4 [33]. et x ( T T I I F F ad y ( T T I I F F two INNs ad > 0.The oeratos of INNs ca be defed as follows: be 3
( x y ( T + T T T T + T T T I I I I F F F F ( x y ( T T T T I + I I I I + I I I F + F F F F + F F F ( ( K K K K ( (3 x ( T ( T ( I ( I ( F ( F K K K K ( (4 x ( T ( T ( I ( I ( F ( F (3 (4 Examle. et ([ 0.7 0.8] [ 0.0 0.] [ 0. 0.] ad ([ 0.4 0.5] [ 0. 0.3] [ 0.3 0.4] the we ca get the followg oeratoal results. ( ([ 0.9 0.96] [ 0 0.0] [ 0.0 0.04] ( ([ 0.49 0.64] [ 0 0.9] [ 0.9 0.36] (3 B ([ 0.8 0.90] [ 0 0.05] [ 0.03 0.08] (4 B ([ 0.8 0.40] [ 0.0 0.37] [ 0.37 0.5]. B be two INNs ad Theorem. et ( T T I I F F ( T T I I F F ( ad T T I I F F be three INNs the oeratoal rules of INNs have the followg roertes. ( ; (5 ( ; (6 λ( λ λ λ > 0; (7 (3 (4 λ λ λ λ ( > 0; (8 (5 λ λ λ + λ λ > 0 λ 0 ; (9 ( > λ λ λ + λ (6 λ > λ > 0 ; (0 0 ccordg to the score fucto ad accuracy fucto of IFSs [894] the score fucto ad accuracy fucto of a INN ca be defed as follows. Defto 5. et x ( T T I I F F be a INN ad the score fucto sx ( ad accuracy fucto axof ( a INN ca be defed as follows: T + T I + I F + F ( sx ( + + ; ( T + T I + I F + F ( ax ( + +. ( The accordg to score fucto sxad ( accuracy fucto axof ( INNs we ca gve the 4
comarso method for INNs as follows. Defto 6. et x ( T T I I F F ad y ( T T I I F F INNs the we have ( If sx ( > sy ( the x s sueror to y deoted by x y; ( If sx ( sy ( ad ax ( ay ( the x y; (3 If sx ( sy ( ad ax ( ay ( the deoted by x y.. The Murhead mea (MM oerator be two The MM was frstly troduced by Murhead [] 90 whch was defed as follows: Defto 7 []. et α (... be a collecto of oegatve real umbers ad ( R be a vector of arameters. If MM ( α α... α α ϑ (! ϑ S (3 The we call MM the Murhead mea (MM where ϑ ( ( s ay a ermutato of ( ad S s the collecto of all ermutatos of ( I addto From Eq. (3 we ca ow that. (0 0 ( Whe ( 0 0 the MM reduces to MM (... arthmetc averagg oerator. ( Whe ( α α α α whch s the ( the MM reduces to / MM ( α α... α α geometrc averagg oerator. ( 00 0 (3 Whe ( 00 0 the MM reduces to MM (... whch s the BM oerator [6]. (( ((( ((( (((( (4 Whe ( 00 0 MM (( ((( ((( (((( the MM reduces to ( 00 0 ( α α... α (MSM oerator [8]. α... C whch s the α α α αα ( whch s the Maclaur symmetrc mea From the defto 7 ad the secal cases of MM oerator metoed-above we ca ow that the advatage of the MM oerator s that t ca cature the overall terrelatoshs amog the multle ut argumets ad t s a geeralzato of some exstg aggregato oerators. 3.Iterval eutrosohc Murheadmea(INMM oerators Because the tradtoal MM ca oly rocess the crs umber ad INNs ca easly deal 5
determate ad cosstet formato t s ecessary to exted MM to rocess INNs. I ths secto we wll roose some INMM oerators for the terval eutrosohc formato ad dscuss some roertes of the ew oerators. 3.Iterval eutrosohc Murhead mea (INMM oerator Defto 8. et α ( T T I I F F (... be a collecto of INNs ad ( R be a vector of arameters. If INMM! ( α α... α α ϑ ( (4 ϑ S The we call INMM the terval eutrosohc MM (INMM oerator where ϑ ( ( s ay a ermutato of ( ad S s the collecto of all ermutatos of ( Theorem. et α ( T T I I F F (... aggregato result from Defto 8 s stll a INN ad has. be a collecto of the INNs the the!! INMM ( α α... α ( Tϑ( ( Tϑ( ϑ S ϑ S ( (!! I ϑ ( I ϑ ( ϑ S ϑ S!! ( Fϑ( ( Fϑ( ϑ S ϑ S (5 roof. We eed to rove ( Eq. (5 s rght; ( Eq.(5 s a INN. ( Frstly we rove the Eq. (5 s et. ccordg to the oeratoal laws of INNs we get ( ( T ( T ( I ( I ( F ( F αϑ( ϑ( ϑ( ϑ( ϑ( ϑ( ϑ( ad ( ( ( Fϑ( ( Fϑ( α ( ( ϑ( Tϑ( Tϑ( Iϑ( Iϑ( the 6
αϑ( ( Tϑ( ( Tϑ( ϑ S ϑ S ϑ S ( Iϑ( ( Iϑ( ( Fϑ( ( Fϑ( ϑ S ϑ S ϑ S ϑ S further!!! αϑ( ( Tϑ( ( Tϑ( ϑ S ϑ S ϑ S! Iϑ( I ϑ( ϑ S ϑ S ( (!! ( Fϑ( ( Fϑ( ϑ S ϑ S so!! α ϑ( ( Tϑ( ( Tϑ(! ϑ S ϑ S ϑ S!! ( Iϑ ( ( I ( ϑ ϑ S ϑ S!! ( Fϑ ( ( F ϑ ( ϑ S ϑ S.e. (5 s et. ( The we wll rove that (5 s a INN.!! et T ( Tϑ( T ( Tϑ(! ϑ S ϑ S!! ( ϑ( ( ϑ( ϑ S ϑ S I I I I!! ( ϑ( ( ϑ( ϑ S ϑ S F F F F 7
The we eed rove the followg two codtos. ( T T [ 0 ] I I [ 0 ] F F [ 0] ; ( 0 T + I + F 3. ( Sce [ ] T ϑ ( 0 we ca get ( ( T ϑ [0] ad ( T ϑ ( [0] the ( T ϑ ( [0] ( T ϑ ( [0] ad (! T ϑ ( [0] further ϑ S [0] ϑ S ϑ S!! ( T ϑ ( ad ( T ϑ ( [0] ϑ S.e. 0 T. Smlarly we ca get 0 T 0 I 0 I 0 F 0 F. So codto ( s met. ( Sce 0 T 0 I 0 F the we ca get 0 T + I + F 3. ccordg to ( ad ( we ca ow the aggregato result from (5 s stll a INN. The accordg to ( ad ( Theorem s et. Examle. et x ([ 0.3 0.4] [ 0. 0.3] [ 0.4 0.5] ([ 0.5 0.6] [ 0. 0.4] [ 0. 0.3] ([ 0.4 0.5] [ 0. 0.3] [ 0.3 0.4] y ad z be three INNs ad (.00.50.4 the accordg to (5 we have!! (.00.50.4 INMM ( x y z ( Tϑ( ( Tϑ( ϑ S ϑ S!! ( Iϑ ( ( I ( ϑ ϑ S ϑ S!! ( Fϑ ( ( F ϑ ( ϑ S ϑ S.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4.0+ 0.5+ 0.4 ( ( 0.3 0.5 0.4 ( 0.3 0.4 0.5 ( 0.4 0.5 0.3.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4 ( 0.4 0.3 0.5 ( 0.5 0.3 0.4 ( 0.5 0.4 0.3 3!.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4.0+ 0.5+ 0.4 ( ( 0.4 0.6 0.5 ( 0.4 0.5 0.6 ( 0.5 0.4 0.6.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4 ( 0.5 0.6 0.4 ( 0.6 0.4 0.5 ( 0.6 0.5 0.4 3! 8
((.0 + 0.5 +.0 0.5 0.4 0.4 3! ( 0. ( 0. ( 0. ^ 6 ( ( 0.3 ( 0.4 ( 0.3 ^ ( ( 0.3 ( 0.3 ( 0.4 ^.0 0.5 0.4 ( ( 0.4 ( 0.3 ( 0.3 ^.0 0.5 0.4.0 0.5 0.4 3!.0+ 0.5+ 0.4.0 0.5 0.4.0 0.5 0.4 ( ( 0.4 ( 0. ( 0.3 ( ( 0.4 ( 0.3 ( 0. 3!.0 0.5 0.4.0 0.5 0.4 ( ( 0. ( 0.4 ( 0.3 ( ( 0. ( 0.3 ( 0.4.0 0.5 0.4.0 0.5 0.4 ( ( 0.3 ( 0. ( 0.4 ( ( 0.3 ( 0.4 ( 0..0+ 0.5+ 0.4.0 0.5 0.4.0 0.5 0.4 3! ( ( 0.5 ( 0.3 ( 0.4 ( ( 0.5 ( 0.4 ( 0.3.0 0.5 0.4.0 0.5 0.4 ( ( 0.3 ( 0.4 ( 0.5 ( ( 0.3 ( 0.5 ( 0.4.0 0.5 0.4.0 0.5 0.4 ( ( 0.4 ( 0.5 ( 0.3 ( ( 0.4 ( 0.3 ( 0.5 ([ ] [ ] [ ] 0.3930.495 0.80.335 0.730.404 Next we wll dscuss some roertes of INMM oerator. roerty (Idemotecy. If all (....0+ 0.5+ 0.4 α are equal.e. α ( α T T I I F F the (... INMM α α α α. roof. Sce α α ( T T I I F F based o Theorem we get!! INMM ( αα... α ( T ( T ϑ S ϑ S ( I ϑ S!! ( ϑ S!! ( F ( F ϑ S ϑ S!! ( T ( T ϑ S ϑ S ( I ϑ S! I! ( I ϑ S 9
!! ( F ( F ϑ S ϑ S!!!! ( T ( T!! ( I!! ( I ( F ( F!!!! ( ( T T I I ( ( ( F ( F ( T T I I F F ( T T I I F F ( ( ( (. roerty (Mootocty. et α ( T T I I F F ad α ( T T I I F F (... be two sets of INNs. If T T T T I I I I F F F F for all the INMM ( α α... α INMM ( α α... α. roof. et ( α α α INMM... ( T T I I F F IFMM... ( T T I I F F ( α α α where!! ( ϑ( ( ϑ( T T T T ϑ S ϑ S!! ( ϑ( ( ϑ( T T T T ϑ S ϑ S 0
!! ad I ( Iϑ( I ( I ϑ( ϑ S ϑ S!! ( ϑ( ( ϑ( ϑ S ϑ S I I I I!! ( ϑ( ( ϑ( ϑ S ϑ S F F F F!! ( ϑ( ( ϑ( ϑ S ϑ S F F F F Sce T T T T we ca get ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( ad ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( the ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( ϑ S ϑ S ϑ S ϑ S ad!!!! ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( ϑ S ϑ S ϑ S ϑ S further!!!! ( Tϑ( ( T ϑ( ( Tϑ( ( T ϑ( ϑ S ϑ S ϑ S ϑ S ad!! ( Tϑ( ( T ϑ( ϑ S ϑ S ϑ S ϑ S!! ( Tϑ( ( T ϑ(..e. T T T T.
Smlarly we also have I I I I F F F F. I the followg we wll dscuss three stuatos as follows. ( If T T T T ad I I I I F F F F the INMM ( α α... α INMM ( α α... α > ; (If T T T T ad I I I I F F F F the INMM ( α α... α INMM ( α α... α > ; (3If T T T T ad I I I I F F F F the INMM ( α α... α INMM ( α α... α. So roerty s rght. roerty 3 (Boudedess. et α ( T T I I F F (... ad a ( m( T max( I max( F a + ( max( T m( I m( F (... α INMM α α α α. roof. + Based o roertes ad we have INMM (... INMM (... α α α α α α α ad INMM (... INMM (... + + + + α α α α α α α. So we INMM (... α α α α α. + be a collectos of INNs the I the followg we wll exlore some secal cases of INMM oerator wth resect to the arameter vector. (Whe ( 0 0 the INMM reduces to the terval eutrosohc arthmetc averagg oerator. ( 0 0 INMM ( α α... α α (6 / / / / / / ( T ( T ( I ( I ( F ( F (Whe ( λ0 0 the INMM reduces to the terval eutrosohc geeralzed arthmetc averagg oerator.
/ / / λ λ λ ( 0 0 λ / λ λ λ / INMM ( α α... α α ( ( T ( ( T / / λ / / λ λ λ ( ( I ( ( I / / λ / / λ ( ( λ λ ( F ( F (7 (3Whe ( 00 0 the INMM reduces to the terval eutrosohc BM oerator. ( ( I I I I / / ( 00 0 ( ( ( ( ( INMM α α... α αα T T T T ( + / / ( ( I + I I I / / ( ( ( ( F F F F F F F F + + (( ((( ((( (((( (4Whe ( 00 0 mea (MSM oerator. (8 the INMM reduces to the terval eutrosohc Maclaur symmetrc INMM (( ((( ((( (((( ( 00 0 ( α α... α α... C / / / C / C T T...... / / / C / C ( I ( I...... / / / C / C ( F ( F...... (9 (5Whe ( the INMM reduces to the terval eutrosohc geometrc averagg oerator. / ( INMM ( α α... α α (0 / / / / / / ( T ( T ( I ( I ( F ( F (6Whe ( the INMM reduces to the terval eutrosohc geometrc averagg 3
oerator. INMM / α α α α ( (... / / / / ( / / ( T ( T ( I ( I ( F ( F Further order to dscuss the mootoc of INMM oerator about the arameter vector R we frstly cted a lemma. emma. [0] et ( ad Q ( q q q be two the arameter vectors f q ( [ ] [ ] q ( where ([][] [ ] s a ermutato of (... ad meets [ ] [ + ] q [ ] q [ + ] for all (... The we ca call that s cotrolled by vector Q exressed by Q. Theorem 3.et α ( T T I I F F (... Q q q q. be a collectos of INNs ad ( ( be two the arameter vectors f Q the INMM Q ( α α... α INMM ( α α... α (3 The roof ths theorem s omtted lease refer to [3]. 3.. The terval eutrosohc weghted MM oerator I actual decso mag the weghts of attrbutes wll drectly fluece the decso-mag results. However INMM oerator caot cosder the attrbute weghts so t s very mortat to tae to accout the weghts of attrbutes for formato aggregato. I ths subsecto we wll roose a weghted INMM oerator as follows. Defto 9. et α ( T T I I F F (... T w ( w w... w be the weght vector of α (... whch satsfes w [ 0] let ( R be a vector of arameters. If INWMM! be a collecto of INNs ad w ad ( α α... α ( wϑ( α ϑ( (4 ϑ S The we call INWMM the terval eutrosohc weghted MM (INWMM where ϑ ( ( s ay a ermutato of ( ad S s the collecto of all ermutatos of ( Theorem 4. et α ( T T I I F F (... Defto 9 s a INN eve. be a collecto of INNs the the result from 4
INWMM ( α α... α!! ( ( ( ( ( w w ϑ ϑ Tϑ( ( Tϑ( ϑ S ϑ S!! wϑ ( wϑ ( ( ( Iϑ ( ( ( Iϑ ( ϑ S ϑ S! wϑ (! wϑ ( ( Fϑ ( ( Fϑ ( ( ( ϑ S ϑ S (5 roof. Because w w ( ( ( ( ( ( ( ( w ( ( ( w ( w ϑ wϑ ϑ ϑ ϑ ϑ( wϑ( αϑ( Tϑ( T ( I ( I ϑ ϑ ϑ( Fϑ( F ϑ( ϑ ( ϑ ( we ca relace T ϑ ( Eq. (5 wth ( Tϑ ( I ϑ ( wth ( Iϑ ( ad ( wth w w F ϑ wϑ ( ϑ ( ( F the we ca get Eq.(5. Because αϑ ( s a INN w ϑ ( ϑ ( α s also a INN. By Eq.(5 we ow (... a INN. I the followg we shall exlore some desrable roertes of INWMM oerator. INWMM α α α s roerty 4 (Mootocty. et α ( T T I I F F ad α ( T T I I F F (... be two sets of INNs. If T T T T I I I I F F F F for all the INWMM ( α α... α INWMM ( α α... α. The roof s smlar to that of INMM oerator t s omtted here. roerty 5 (Boudedess. et α ( T T I I F F ad a ( m( T max( I max( F a + ( max( T m( I m( F (... be a collectos of INNs the ( (... ( T T I I F F INWMM α α α T + T + I + I + F + F + α α α α α α α α α α α α where!! ( w ϑ w ϑ( ( ( m( ( ( m( α α ϑ S ϑ S T T T T 5
!! w ( ϑ w ( ϑ ( max( ( max( a a ϑ S ϑ S I I I I!! w ( ϑ w ( ϑ ( max( ( max( a a ϑ S ϑ S F F F F!! ( w ϑ w ϑ( + ( ( max( + ( ( max( a a ϑ S ϑ S T T T T!! w ( ϑ w ( ϑ + ( m( + ( m( α α ϑ S ϑ S I I I I!! w ( ϑ w ( ϑ + ( m( + ( m( α α ϑ S ϑ S. F F F F roof. ccordg to roerty 4 we have + + + ( α α... α ( α α... α ( α α... α INWMM INWMM INWMM ccordg to Eq. (7 we have INWMM ( a a... a!! ( ( m( ( ( ( m( ( w w ϑ ϑ T T ϑ S ϑ S!! w ( w ( ϑ ϑ ( max( I ( max( I ϑ S ϑ S! wϑ (! w ( ϑ ( max( F ( max( F ϑ S ϑ S ad INWMM + + + ( a a... a!! ( ( max( ( ( w ( max( ( ϑ w ϑ T T ϑ S ϑ S 6
!! w ( w ( ϑ ϑ ( m( I ( m( I ϑ S ϑ S.!! wϑ ( w ( ϑ ( m( F ( m( F ϑ S ϑ S + + + So INWMM ( α α... α INWMM ( α α... α INWMM ( α α... α. Theorem 5.The INMM oerator s a secal case of the INWMM oerator. roof. Whe w!! ( ( ( ( ( w ϑ w ϑ INWMM α α... α ( Tϑ( ( Tϑ( ϑ S ϑ S!! wϑ ( wϑ ( ( ( Iϑ ( ( ( Iϑ ( ϑ S ϑ S! wϑ (! wϑ ( ( ( Fϑ ( ( ( Fϑ ( ϑ S ϑ S!! ( Tϑ( ( Tϑ( ϑ S ϑ S!! ( ( Iϑ ( ( ( Iϑ ( ϑ S ϑ S!! ( Fϑ ( ( F ϑ ( ( ( ϑ S ϑ S!! ( Tϑ( ( Tϑ( ϑ S ϑ S ( ϑ S ( ϑ S!! ( Iϑ ( ( Iϑ (!! ( ( Fϑ( ( ( Fϑ( ϑ S ϑ S INMM ( α α α.... 7
3.3. The terval eutrosohc weghted dual MM oerator I the theory of aggregato oerator there exst two tyes.e. orgal oerator ad ts dual oerator for examle arthmetc average oerator ad geometrc average oerator. I ths secto we wll roose the dual MM oerator for terval eutrosohc umbers based o the INMM oerator as follows. Defto 0. et α ( T T I I F F (... be a collecto of INNs ad ( R be a vector of arameters. If INDMM ( α α... α ( αϑ ( ϑ S!. (6 The we call INDMM the terval eutrosohc dual MM (INDMM where ϑ ( ( s ay a ermutato of ( ad S s the collecto of all ermutatos of ( Theorem 6. et α ( T T I I F F (.... be a collecto of INNs the the result from Defto 0 s a INN too eve INDMM ( α α... α!! ( Tϑ( ( Tϑ( ϑ S ϑ S!! ( I ϑ ( ( Iϑ ( ϑ S ϑ S!! ( Fϑ( ( F ϑ( ϑ S ϑ S (7 roof. We eed to rove ( Eq. (7 s et; ( Eq. (7 s a INN. ( Frstly we rove the Eq. (7 s et. ccordg to the oeratoal laws of INNs we get ( ( ( ( ( ( ( αϑ( Tϑ( T ( I ( I ϑ ϑ ϑ( Fϑ( F ϑ( ad ( αϑ( Tϑ( Tϑ( ( Iϑ( ( Iϑ( ( Fϑ( ( Fϑ( ( ( 8
the ( αϑ ( ϑ S ( Tϑ( ( Tϑ( ( Iϑ( ( Iϑ( ϑ S ϑ S ϑ S ϑ S ( Fϑ( ( Fϑ( ϑ S ϑ S further so!!! ( α ( ( T ( ( T ( ϑ ϑ ϑ ϑ S ϑ S ϑ S ( Iϑ( ( I ϑ(!! ϑ S ϑ S!! ( Fϑ( ( F ϑ( ϑ S ϑ S ( αϑ ( ϑ S!!! ( T ( ϑ ( Tϑ( ϑ S ϑ S.!! ( I ϑ ( ( Iϑ ( ϑ S ϑ S!! ( Fϑ( ( F ϑ( ϑ S ϑ S.e. (7 s et. ( The we wll rove that (7 s a INN. et!! ϑ( ϑ( ϑ S ϑ S T ( T T ( T!! ( ϑ( ( ϑ( I I I I ϑ S ϑ S 9
!! ( ϑ( ( ϑ( F F F F ϑ S ϑ S The we eed rove the followg two codtos. ( T T [ 0 ] I I [ 0 ] F F [ 0] ; ( 0 T + I + F 3. ( Sce [ ] ( T ϑ ( T ϑ ( 0 we ca get [0] ( ( T ϑ [0] ad ( T ϑ ( [0] the ( [0] further T ϑ (! ( T ϑ ( [0] ad! ( T ϑ ( [0] ϑ S ( [0]! T ϑ ( ϑ S! ( T ϑ ( [0] ϑ S ad! ( T ϑ ( [0]. ϑ S.e. 0 T. Smlarly we ca get 0 T 0 I 0 I 0 F 0 F. So codto ( s met. ( Sce 0 T 0 I 0 F the we ca get 0 T + I + F 3. ccordg to ( ad ( we ca ow the aggregato result from (7 s stll a INN. The ccordg to ( ad ( theorem 6 s et. Examle 3.et x ([ 0.3 0.4] [ 0. 0.3] [ 0.4 0.5] ([ 0.5 0.6] [ 0. 0.4] [ 0. 0.3] ([ 0.4 0.5] [ 0. 0.3] [ 0.3 0.4] y ad z be three INNs ad (.00.50.4 the accordg to (7 we have INDMM (.00.50.4 ( x y z!! ( Tϑ( ( Tϑ( ϑ S ϑ S 0
!! ( Iϑ( ( Iϑ( ϑ S ϑ S!! ( Fϑ( ( F ϑ( ϑ S ϑ S.0 0.5 0.4.0 0.5 0.4 ( ( 0.3 ( 0.5 ( 0.4 ( ( 0.3 ( 0.4 ( 0.5.0 0.5 0.4.0 0.5 0.4 ( ( 0.5 ( 0.4 ( 0.3 ( ( 0.5 ( 0.3 ( 0.4.0 0.5 0.4.0 0.5 0.4 ( ( 0.4 ( 0.3 ( 0.5 ( ( 0.4 ( 0.5 ( 0.3.0 + 0.5 + 0.4 3!.0 0.5 0.4.0 0.5 0.4 ( ( ( 0.4 ( 0.6 ( 0.5 ( 0.4 ( 0.5 ( 0.6.0 + 0.5 + 0.4 3!.0 0.5 0.4.0 0.5 0.4 ( ( 0.6 ( 0.4 ( 0.5 ( ( 0.6 ( 0.5 ( 0.4.0 0.5 0.4.0 0.5 0.4 ( ( 0.5 ( 0.4 ( 0.6 ( ( 0.5 ( 0.6 ( 0.4 ( 0. 0. 0. ^ 6.0 0.5 0.4.0+ 0.5+ 0.4 3! ( (( ( (.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4.0 + 0.5 + 0.4 3! 0.3 0.4 0.3 ^ 0.3 0.3 0.4 ^ 0.4 0.3 0.3 ^ ( ( (.0 + 0.5 +.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4 0.4 3! 0.4 0. 0.3 0.4 0.3 0. 0. 0.3 0.4.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4 ( 0. 0.4 0.3 ( 0.3 0.4 0. ( 0.3 0. 0.4 (.0 0.5 0.5 0.4 ( (.0 + 0.5 + 0.4.0 0.5 0.4.0 0.5 0.4 0.4 3! 0.3 0.5 0.3 0.4 0.4 0.3 0.5.0 0.5 0.4.0 0.5 0.4.0 0.5 0.4 ( 0.4 0.5 0.3 ( 0.3 0.4 0.5 ( 0.3 0.5 0.4 ([ ] [ ] [.393] 0.4040.505 0.0.343 0.360 Next we wll dscuss some roertes of INDMM oerator. roerty 7 (Idemotecy. If all α (... are equal.e. α α ( T I F (... INDMM α α α α. the roerty 8 (Mootocty. et α ( T T I I F F ad α ( T T I I F F (... be two sets of INNs. If T T T T I I I I F F F F for all the INDMM ( α α... α INDMM ( α α... α.
roerty 9 (Boudedess. et α ( T T I I F F (... ad a ( m( T max( I max( F a + ( max( T m( I m( F (... α INDMM α α α α. + be a collectos of INNs the I the followg we wll exlore some secal cases of INDMM oerator wth resect to the arameter vector. (Whe ( 0 0 the INDMM reduces to the terval eutrosohc geometrc averagg oerator. INDMM ( 0 0 ( α α... α / / (8 / / / / ( T ( T ( I ( I ( T ( T (Whe ( λ0 0 the INDMM reduces to the terval eutrosohc geeralzed geometrc averagg oerator. / / λ / / λ ( λ0 0 λ λ INDMM ( α α... α ( ( T ( ( T (9 / λ / λ / λ / λ λ / λ / λ / λ / ( ( I ( ( I ( ( F ( ( F (3Whe ( 00 0 the INDMM reduces to the terval eutrosohc geometrc BM oerator. INDMM ( 00 0 ( α α... α / / ( ( ( ( T T T T T T T T + + ( ( I I ( I I / / ( / / ( ( ( ( F F F IF (30 (( ((( ((( (((( (4Whe ( 00 0 the INDMM reduces to the terval eutrosohc geometrc Maclaur symmetrc mea (MSM oerator. INDMM (( ((( ((( (((( ( 00 0 ( α α... α / / / C / C ( T ( T......
/ / / C / C I I...... / / C / C F F...... / (3 (5Whe ( the IFDMM reduces to the terval eutrosohc arthmetc averagg oerator. INDMM ( ( α α... α / / / / / / (3 ( T ( T ( I ( I ( F ( F (6Whe ( the IFMM reduces to the terval eutrosohc the arthmetc averagg oerator. INDMM ( ( α α... α / / / / / / ( T ( T ( I ( I ( F ( F (33 Theorem 7. et α ( T T I I F F (... Q q q q be a collectos of INNs ad ( ( be two the arameter vectors f Q the INDMM Q ( α α... α INDMM ( α α... α (34 3.4. The terval eutrosohc dual weghted MM oerator Smlar to INWMM oerator we wll roose terval eutrosohc dual weghted MM (INDWMM oerator so as to cosder the attrbute weghts whch s defed as follows. Defto. et α ( T T I I F F (... be a collecto of INNs (... w w w w T be the weght vector of α (... whch satsfes w [ 0] ad w ad let ( R be a vector of arameters. If INDWMM wϑ ( ( α α... α ( αϑ ( ϑ S! (35 The we call INDWMM the terval eutrosohc dual weghted MM (INDWMM where ϑ ( ( s ay a ermutato of ( ad S s the collecto of all ermutatos of (. Theorem 8. et α ( T T I I F F (... 3 be a collecto of INNs the the result from
Defto s a INN too eve INDWMM ( α α... α!! w ( w ϑ ϑ( ( ( T ( ( ( T ( ϑ ϑ ϑ S ϑ S w!! ϑ ( w ϑ ( (36 ( ( Iϑ ( ( ( I ( ϑ ϑ S ϑ S! w! ϑ ( w ϑ ( ( ( Fϑ( ( Fϑ( ( ϑ S ϑ S roof. Because wϑ ( ( ( ( wϑ ( T T ( I ( I ( F ( F w ( w ( w ( w ( w ( α ϑ ϑ ϑ ϑ ϑ ϑ( ϑ( ϑ( ϑ( ϑ( ϑ( ϑ( we ca relace T ( T ( wϑ ϑ ϑ Eq. (7 wth ( Tϑ( ( Tϑ( ( wϑ( Iϑ( Iϑ( wth wϑ( wϑ( ϑ( Iϑ( ϑ( ϑ( ( I ( ad Fϑ( Fϑ( wth ( Fϑ( ( Fϑ( the we ca get Eq. (36. ϑ ( Because α ( s a INN ϑ w ϑ ( α s also a INN. By Eq. (7 we ow (... w w INDWMM α α α s a INN. I the followg we shall exlore some desrable roertes of INDWMM oerator. roerty 0 (Mootocty. et α ( T T I I F F ad α ( T T I I F F (... be two sets of INNs. If T T T T I I I I F F F F for all the INDWMM ( α α... α INDWMM ( α α... α. roerty (Boudedess. et α ( T T I I F F (... ad a ( m( T max( I max( F a + ( max( T m( I m( F be a collectos of INNs The ( (... ( T T I I F F INDWMM α α α T + T + I + I + F + F + α α α α α α α α α α α α. where!! w ( ϑ w ( ϑ ( max( ( max( a a ϑ S ϑ S T T T T 4
!! ( w ϑ w ϑ( ( ( m( ( ( m( α α ϑ S ϑ S I I I I!! ( w ϑ w ϑ( ( ( m( ( ( m( α α ϑ S ϑ S F F F F!! w ( ϑ w ( ϑ + ( m( + ( m( α α ϑ S ϑ S T T T T!! ( w ϑ w ϑ( + ( ( max( + ( ( max( a a ϑ S ϑ S I I I I!! ( w ϑ w ϑ( + ( ( max( + ( ( max( a a ϑ S ϑ S. F F F F Theorem 9.The INDMM oerator s a secal case of the INDWMM oerator. roof. Whe w INDWMM ( α α... α!! w ( w ϑ ϑ( ( ( T ( ( ( T ( ϑ ϑ ϑ S ϑ S w!! ϑ ( w ϑ ( ( ( Iϑ ( ( ( Iϑ ( ϑ S ϑ S! w ϑ (! w ϑ ( ( ( Fϑ( ( Fϑ( ( ϑ S ϑ S!! ( ( ( ( T ( T ( ϑ ϑ ϑ S ϑ S!! ( Iϑ ( ϑ S ( I ( ϑ ϑ S 5
!! ( Fϑ( ( Fϑ( ϑ S ϑ S!! ( Tϑ( ( Tϑ( ϑ S ϑ S! ( Iϑ ( ( Iϑ ( ϑ S ϑ S! ( Fϑ( ( F ϑ( ϑ S ϑ S ( INDMM α α... α.!! 4. The decso mag aroach based o the roosed oerators I ths secto based o the roosed INWMM or INDWMM oerators we wll develo a ovel MDM method whch s descrbed as follows. Suose we eed evaluate m alteratves{ m } wth resect to attrbutes{ C C C} a MDM roblem where the weght vector of the attrbutes s ω ( ω ω ω satsfyg where r ( T I F ω 0( ω. R [ r ] m s the gve decso matrx of ths decso roblem s a INN gve by the decso maer wth resect to alteratve for attrbute C. The the goal s to ra the alteratves. I the followg we wll use the roosed INWMM or INDWMM oerators to solve ths MDM roblem ad the detaled decso stes are show as follows: Ste : Normalzg the attrbute values. I real decso there exst two tyes of the attrbutes whch are cost tye ad beeft tye. It s ecessary to covert them to the same tye so as to gve the rght decso mag. sually we covert cost tye to beeft oe by the followg formula (Note: The coverted attrbute value s stll exressed by r : ( r F F I I T T (37 Ste : ggregatg all attrbute values r ( to the comrehesve value Z by INWMM or INDWMM oerators show as follows: z INWMM ( r r r (38 m or z INDWMM ( r r rm. (39 Ste 3: Rag z ( m based o the score fucto ad accuracy fucto by Defto 6. 6
Ste 4: Rag all the alteratves. The bgger the INN z s the better the alteratve s. 5. llustratve examle I ths secto a examle for the multcrtera decso mag s used to demostrate of the alcato of the roosed decso mag method as well as the effectveess of the roosed method. et us cosder the decso mag roblem adated from [3]. There s a vestmet comay whch wats to vest a sum of moey the best oto. There are four ossble alteratves: ( s a car comay; ( s a food comay; (3 3 s a comuter comay; (4 4 s a arms comay. The vestmet comay must mae a decso accordg to the followg three crtera: ( C s the rs aalyss; ( C s the growth aalyss; (3 C3 s the evrometal mact aalyss where C ad C are beeft crtera ad C3 s a cost crtero. The weght vector of the crtera s w ( 0.350.400.5 T. The four ossble alteratves are evaluated wth resect to the above three crtera by the form of INNs ad terval eutrosohc decso matrx D s lsted table. The goal s to ra alteratves. Table terval eutrosohc decso matrx D C C C 3 ([ 0.4 0.5] [ 0. 0.3] [ 0.3 0.4] ([ 0.4 0.6] [ 0. 0.3] [ 0. 0.4] ([ 0.7 0.9] [ 0. 0.3] [ 0.4 0.5] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.3 0.6] [ 0.3 0.5] [ 0.8 0.9] 3 ([ 0.3 0.6] [ 0. 0.3] [ 0.3 0.4] ([ 0.5 0.6] [ 0. 0.3] [ 0.3 0.4] ([ 0.4 0.5] [ 0. 0.4] [ 0.7 0.9] 4 ([ 0.7 0.8] [ 0.0 0.] [ 0. 0.] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.6 0.7] [ 0.3 0.4] [ 0.8 0.9] 5. The decso mag stes To get the best alteratve(s the stes are show as follows: Ste : Normalzg the attrbute values. Sce C ad C are beeft attrbutes ad C 3 s a cost crtero we use the formulas (37 to get the stadardzed decso matrx whch s show Table. Table stadardzed decso matrx D C C C 3 ([ 0.4 0.5] [ 0. 0.3] [ 0.3 0.4] ([ 0.4 0.6] [ 0. 0.3] [ 0. 0.4] ([ 0.4 0.5] [ 0.8 0.7] [ 0.7 0.9] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.8 0.9] [ 0.7 0.5] [ 0.3 0.6] 3 ([ 0.3 0.6] [ 0. 0.3] [ 0.3 0.4] ([ 0.5 0.6] [ 0. 0.3] [ 0.3 0.4] ([ 0.7 0.9] [ 0.8 0.6] [ 0.4 0.5] 4 ([ 0.7 0.8] [ 0.0 0.] [ 0. 0.] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.8 0.9] [ 0.7 0.6] [ 0.6 0.7] Ste : ggregatg all attrbute values r ( to the comrehesve value Z by INWMM or INDWMM oerators show as follows (suose ( ( For the INWMM oerator we have 7
z ([ 0.393 0.56 ][ 0.450 0.470 ][ 0.44 0.668] ([ 0.640 0.737 ][ 0.36 0.34 ][ 0.4 0.40] z 3 ([ 0.459 0.660 ][ 0.488 0.40 ][ 0.34 0.44] z 4 ([ 0.673 0.769 ][ 0.34 0.336 ][ 0.30 0.447] ( For the INDWMM oerator we have z z ([ 0.4 0.547 ][ 0.39 0.383 ][ 0.333 0.498] ([ 0.686 0.794 ][ 0.83 0.63 ][ 0.4 0.366] z 3 ([ 0.534 0.750 ][ 0.30 0.366 ][ 0.33 0.4] 4 ([ 0.74 0.80 ][ 0.00 0. ][ 0.74 0.335] z z. Ste 3: Calculate the score fucto Sz ( ( 34 of the collectve overall values z ( 34. ( For the INWMM oerator we have Sz (.439 Sz (.009 Sz ( 3.74 Sz ( 4.0 ( For the INDWMM oerator we have Sz (.753 Sz (.3 Sz ( 3.936 Sz ( 4.40. Ste 4: Rag all the alteratves. ccordg to the score fuctos ( ( 34 show as follows 4 3 So the best alteratve s 4. S we ca ra the alteratves{ } z 5. The fluece of the arameter vector o decso mag result of ths examle 3 4 I order to llustrate the fluece of the arameter vector o decso mag of ths examle we set dfferet arameters vector to show the rag results of ths examle. The results are show Table 3 ad Table 4. Table 3 Rag by utlzg the dfferet arameter vector of the INWMM oerator arameter vector The score fucto S( z Rag (00 ( 0 ( (0.5 0.5 0.5 (00 (300 Sz (.640 Sz (.83 Sz (.897 Sz (.37 3 4 Sz (.54 Sz (.078 Sz (.773 Sz (. 3 4 Sz (.439 Sz (.009 Sz (.74 Sz (.0 3 4 Sz (.439 Sz (.009 Sz (.74 Sz (.0 3 4 Sz (.680 Sz (.05 Sz (.93 Sz (.395 3 4 Sz (.73 Sz (.5 Sz (.964 Sz (.40 3 4 4 3 4 3 4 3 4 3 4 3 4 3 8
Table 4 Rag by utlzg the dfferet arameter vector of the INDWMM oerator arameter vector The score fucto S( z Rag (00 ( 0 ( (0.5 0.5 0.5 (00 (300 Sz (.35 Sz (.004 Sz (.70 Sz (.964 3 4 Sz (.668 Sz (.05 Sz ( 3.93 Sz ( 4.395 Sz (.753 Sz (.3 Sz ( 3.936 Sz ( 4.40 Sz (.753 Sz (.3 Sz ( 3.936 Sz ( 4.40 Sz (.47 Sz (.93 Sz ( 3.634 Sz ( 4.8 Sz (.64 Sz (.839 Sz (.569 Sz (.705 3 4 9 4 3 4 3 4 3 4 3 4 3 4 3 s we ca see from Table 3 the score fuctos usg the dfferet arameter vector are dfferet but the rag results are the same. From Table 4 we ca ow the rag results may be dfferet for the dfferet arameters vector whe the arameter vector has oly oe real umber ad the rest are 0 that s whe the INDWMM oerator reduce to the terval eutrosohc geeralzed geometrc averagg oerator ts rag order s 4 3 ; whereas other case the rag results are the same as Table 3. I other words we cosder the terrelatosh of attrbutes the best alteratve s 4 otherwse s. I geeral for the INWMM oerator we ca fd that the more terrelatoshs of attrbutes we cosder that s to say there are fewer 0 the arameter vector the smaller value of score fuctos wll become. The arameter vector have greater cotrol ablty the values of score fucto wll become greater. However for the INDWMM oerator the result s ust the ooste the more terrelatoshs of attrbutes we cosder the greater value of score fuctos wll become. The arameter vector have greater cotrol ablty the values of score fucto wll become small. So dfferet decso maers ca set dfferet arameter vector accordg to dfferet rs referece. 5.3 Comarg wth the other methods To further rove the effectveess of the develoed methods ths aer we solve the same llustratve examle by two exstg MDM methods cludg the smlarty measure roosed by Ye [3] the terval eutrosohc weghted Boferro mea (INWBM oerator exteded from the ormal eutrosohc weghted Boferro mea (NNWBM oerator []. The rag results by these methods are show Table 5 (for the INDWMM oerator there are the same results as the INWMM oerator ad they are omtted. From Table 5 we ca see that these methods roduced the same rag results. Ths shows that the ew methods roosed ths aer are effectve ad feasble. The we gve further aalyss whe (00the INWMM reduces to the terval eutrosohc arthmetc weghted averagg oerator. I other words whe (00 we ca th that the ut argumets are deedet ad the terrelatosh amog ut argumets s ot cosdered ust as the method [3] s based o a smlarty measure. Whe ( 0 or ( 0 the INWMM oerator reduces to the terval eutrosohc weghted Boferro mea oerator whch ca catures terrelatosh of two argumets. So we ca get that the roosed methods ths aer are geeralzato of some exstg
methods. Table 5 Rag results comared wth smlarty measure method ggregato oerator arameter value Rag smlarty measure [3] No 4 3 INWBM [] q 4 3 INWMM ths aer (00 4 3 INWMM ths aer ( 0 4 3 INWMM ths aer ( 0 4 3 INWMM ths aer ( 4 3 INWMM ths aer ( 4 3 However there usually exst the terrelatoshs amog more tha two attrbutes real decso mag BM oerator ca oly cosder the terrelatosh betwee ay two ut argumets. I order to comare the erformace ad advatage of the ew roosed method wth the above exstg methods we revse the truth-membersh values of the alteratve 4 whch are lsted Table 6 ad the fal rag results of the alteratves are show Table 7. Table 6 modfed decso matrx D C C C 3 ([ 0.4 0.5] [ 0. 0.3] [ 0.3 0.4] ([ 0.4 0.6] [ 0. 0.3] [ 0. 0.4] ([ 0.4 0.5] [ 0.8 0.7] [ 0.7 0.9] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.6 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.8 0.9] [ 0.7 0.5] [ 0.3 0.6] 3 ([ 0.3 0.6] [ 0. 0.3] [ 0.3 0.4] ([ 0.5 0.6] [ 0. 0.3] [ 0.3 0.4] ([ 0.7 0.9] [ 0.8 0.6] [ 0.4 0.5] 4 ([ 0.4 0.8] [ 0.0 0.] [ 0. 0.] ([ 0.5 0.7] [ 0. 0.] [ 0. 0.3] ([ 0.8 0.9] [ 0.7 0.6] [ 0.6 0.7] Table 7 Rag results by dfferet methods ggregato oerator arameter value Rag INWBM [] q 4 3 INWMM ths aer ( 3 4 INWDMM ths aer ( 4 3 From table 7 we ca ow whe we cosder terrelatosh of three argumets the rag results are dfferet wth that roduced by cosderg terrelatosh of two argumets. I a realstc decso mag evromet we eed cosder the terrelatosh for two argumets or multle argumets accordg to the actual decso eed ad the roosed methods ths aer ca cature terrelatosh of ay multle argumets eve do t cosder the terrelatosh by arameter vector. I a word accordg to the comarsos ad aalyss above the roosed methods based o INWMM oerator ad the INDWMM oerator ths aer s better ad more coveet tha the exstg other methods cosderg terrelatosh of attrbutes. 6. Cocluso NS s a geeralzato of fuzzy set aracosstet set tutostc fuzzy set aradoxst set etc. ad the MM oerator has a romet characterstc that t ca cosder the teracto relatoshs amog ay multle attrbutes by a arameter vector. I ths aer we roosed some ew MM aggregato oerators to deal wth MDM roblems uder the terval eutrosohc evromet 30
clude the terval eutrosohc MM (INMM oerator the terval eutrosohc weghted MM (INWMM oerator the terval eutrosohc dual MM (INWMM oerator ad the terval eutrosohc dual weghted MM (INDWMM oerator. The the desrable roertes ad some secal cases were dscussed detal. Moreover we reseted two ew methods based o the roosed aggregato oerators. Fally we used a examle to llustrate the feasblty ad valdty of the ew methods by comarg wth the other exstg methods. I the future we wll research some alcatos of roosed methods real decso mag. Refereces [] K. taassov Itutostc fuzzy sets Fuzzy Sets ad Systems 0 (986 87 96. [] K.T. taassov More o tutostc fuzzy sets Fuzzy Sets ad Systems 33(98937-46. [3] K. taassov G. Gargov Iterval valued tutostc fuzzy sets Fuzzy Sets ad Systems 3(3 (989 343-349. [4]G. Belaov. radera Calvo T. ggregato Fuctos: Gude for racttoers Srger (007. [5] R. Bellma.. Zadeh Decso mag a fuzzy evromet Maagemet Scece 7 (4 (970 4-64. [6] C. Boferro Sulle mede multle d oteze Bolleto Matematca Italaa 5 (3-4 (950 67-70. [7] Z. C. Che. H. u. aroach to multle attrbute grou decso mag based o lgustc tutostc fuzzy umbers Iteratoal Joural of Comutatoal Itellgece Systems 8(4 (05 747-760. [8] S.J. Che C.. Hwag. F. Hwag Fuzzy Multle ttrbute Decso Mag Methods Srger Brl Hedelberg (99 89-486. [9] D.Kahema Tversy. rosect theory: a aalyss of decso uder rs Ecoometrc Socety 47 ( (979 63-9. [0].D. u Mult-attrbute decso-mag method research based o terval vague set ad TOSIS method Techologcal ad Ecoomc Develomet of Ecoomy 5 (009 453-463. [].D. u H.G. Multle attrbute decso-mag method based o some ormal eutrosohc Boferro mea oerators Neural Comutg ad lcatos (05-6. [].D u Z.M u X. Zhag Some tutostc ucerta lgustc Heroa mea oerators ad ther alcato to grou decso mag led Mathematcs ad Comutato 30 (04 570-586. [3].D. u M.H. Wag. exteded VIKOR method for multle attrbute grou decso mag based o geeralzed terval-valued traezodal fuzzy umbers Scetfc Research ad Essays 6 (4 (0 766 776. [4].D. u Y.M. Wag Multle ttrbute Decso-Mag Method Based o Sgle Valued Neutrosohc Normalzed Weghted Boferro Mea Neural Comutg ad lcatos 5(7-8 ( 04 00-00. [5].D. u X.C. Yu -dmeso ucerta lgustc ower geeralzed weghted aggregato oerator ad ts alcato for multle attrbute grou decso mag Kowledge-based systems 57( (04 69-80. [6].D. u X. Zhag Research o the suler selecto of suly cha based o etroy weght ad mroved EECTRE-III method Iteratoal Joural of roducto Research 49 (0 3
637-646. [7].D. u X. Zhag F. J mult-attrbute grou decso-mag method based o terval-valued traezodal fuzzy umbers hybrd harmoc averagg oerators Joural of Itellget ad Fuzzy Systems 3(5( 0 59-68. [8] C. Maclaur secod letter to Mart Foles Esq cocerg the roots of equatos wth demostrato of other rules of algebra hltrasactos 36 (79 59-96. [9]. Maumdar S.K. Samat O smlarty ad etroy of eutrosohc sets Joural of Itellget ad Fuzzy Systems 6 (3 (04 45-5. [0].W. Marshall I. Ol Iequaltes: Theory of Maorzato ad Its lcatos cademc ress (06. [] R.F Murhead Some methods alcable to dettes ad equaltes of symmetrc algebrac fuctos of letters roceedgs of the Edburgh Mathematcal Socety (3 (90 44-6. [] J.D. Q X.W. u aroach to tutostc fuzzy multle attrbute decso mag based o Maclaur symmetrc mea oerators Joural of Itellget ad Fuzzy Systems 7(5 (04 77-90. [3] J.D Q X.W u -tule lgustc Murhead mea oerators for multle attrbute grou decso mag ad ts alcato to suler selecto Kyberetes 45 ( (06-9. [4] G. Shafer Mathematcal Theory of Evdece rceto versty ress (976. [5] F. Smaradache ufyg feld logcs. Neutrosohc logc hlosohy (999-4. [6] F. Smaradache ufyg feld logcs eutrosohc logc. Neutrosohy eutrosohc set eutrosohc robablty ad statstcs Ifte study (005. [7] J.Q. Wag X.E. alcato of the TODIM method wth mult-valued eutrosohc set Cotrol ad Decso 30 (6 (05 39-4. [8] H. Wag F. Smaradache Y. Q. Zhag R. Suderrama Iterval Neutrosohc Sets ad ogc: Theory ad lcatos Comutg Ifte study 5 (005. [9] H. Wag F. Smaradache Y.Q. Zhag R. Suderrama Sgle valued eutrosohc sets Rev r Force 7 (00. [30] R.R. Yager Multle obectve decso-mag usg fuzzy sets Iteratoal Joural of Ma-mache. Studes. 9 (977 375-38. [3] R. R. Yager The ower average oerator Systems Ma ad Cyberetcs art : Systems ad Humas IEEE Trasactoso 3 (6 (00 74 73. [3] J. Ye Smlarty Measures betwee Iterval Neutrosohc Sets ad ther Multcrtera Decso Method Joural of Itellget ad Fuzzy Systems 6 ( (04 65-7. [33] J. Ye Some aggregato oerators of terval eutrosohc lgustc umbers for multle attrbute decso mag Joural of Itellget ad Fuzzy Systems 7(5 (04 3-4. [34] D.J. Yu Y.Y. Wu Iterval-valued tutostc fuzzy Heroa mea oerators ad ther alcato mult-crtera decso mag frca Joural of Busess Maagemet 6 (0 458-468. [35].. Zadeh Fuzzy sets Iformato ad Cotrol 8 (3 (965 338 353. 3