Some single valued neutrosophic correlated aggregation operators and their applications to material selection

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1 Mauscrpt Clck here to dowload Mauscrpt: p2014 NN_Choquet tegral_19.docx Clck here to vew lked Refereces Some sgle valued eutrosophc correlated aggregato operators ad ther applcatos to materal selecto Yabg Ju, Dawe Ju, Weka Zhag, Xa L School of Maagemet ad Ecoomcs, Bejg Isttute of Techology, Bejg, Cha bstract I egeerg desg, the decso to select a optmal materal has become a challegg task for the desgers, ad the evaluato of alteratve materals may be based o some multple attrbute group decso makg (MGDM methods. Moreover, the attrbutes are ofte ter-depedet or correlated the real decso makg process. I ths paper, wth respect to the materal selecto problems whch the attrbute values take the form of sgle valued eutrosophc umbers (SVNNs, a ovel multple attrbute group decso makg method s proposed. Frst, the cocept ad operatoal laws of SVNNs are brefly troduced. The, motvated by the dea of Choquet tegral, two correlated aggregato operators are proposed for aggregatg sgle valued eutrosophc formato based o the operatoal laws of SVNNs, such as the sgle valued eutrosophc correlated average (SVNC operator, the sgle valued eutrosophc correlated geometrc (SVNCG operator, ad the some desrable propertes of these operators ad the relatoshps amog them are vestgated detal. Furthermore, based o the proposed aggregato operators, a ovel multple attrbute group decso makg method s developed to select the most desrable materal(s uder sgle valued eutrosophc evromet. Fally, a umercal example of materal selecto s gve to llustrate the applcato of the proposed method. Keywords: Multple attrbute group decso makg (MGDM; Materal selecto; Sgle valued eutrosophc set (SVNS; Choquet tegral; Sgle valued eutrosophc correlated aggregato operators 1. Itroducto Materal selecto s oe of the most promet actvtes the process of desg ad developmet of products, whch s a task ormally carred out by desg ad materals egeers ad also crtcal for the success ad compettveess of the producers [1, 2]. approprate selecto of materals may result damage or falure of a assembly ad sgfcatly decreases the performace [3], thus egatvely affectg the productvty, proftablty ad reputato of a orgazato [4]. I the process of selectg materals, there s ot always a defte crtero or attrbute, ad the desgers or egeers have to cosder may attrbutes that fluece the selecto of materals for a gve applcato smultaeously. These attrbutes clude ot oly the tradtoal oes such as avalablty, producto ad cost, but also materal mpact o evromet, recyclg ad cultural aspects ad so o, whch may be cotradcted ad eve coflctg wth each other. Therefore, the selecto of the most desrable materal s a multple attrbute decso makg (MDM problem ad may tradtoal MDM methods have bee proposed to deal wth Correspodg author. ddress: School of Maagemet ad Ecoomcs, Bejg Isttute of Techology, Bejg , Cha. Tel.: Fax: E-mal: juyb@bt.edu.c (Y.B. Ju. 1

2 the materal selecto problem, such as shby approach [5], aalytc herarchy process (HP [6], aalytc etwork process (NP [7], techque of order preferece by smlarty to deal soluto (TOPSIS [8], qualty fucto deploymet (QFD-based approach [9], gray relatoal aalyss (GR [10], graph theory ad matrx approach [11], ELECTRE (ELmato Et Chox Tradusat la REalte [12], VIKOR (VIsekrterjumska optmzacja KOmpromso Reseje [13-15], Preferece Rakg Orgazato METHhod for Erchmet Evaluato (PROMETHEE [16], DEMTEL-based NP (DNP [17] ad COPRS (COmplex PRoportoal Ssessmet [18,19]. However, due to the creasg complexty of the materal selecto process ad the vagueess of heret subjectve ature of huma thk, decso makers usually caot express hs/her preferece to materal alteratves by crsp umbers ad some are more sutable to be deoted by fuzzy values. Sce fuzzy set was troduced by Zadeh [20], may extesos of fuzzy set have bee wdely dscussed [21-27]. Recetly, a ew cocept called eutrosophc set (NS has bee troduced by Smaradache [28], where each elemet of the uverse has a degree of truth (T, determacy (I ad falsty (F respectvely ad whch les ]0 -, 1 + [. Dfferet from the tutostc fuzzy set where the corporated ucertaty s depedet of the membershp degree ad the o-membershp degree, the determacy degree eutrosophc set s depedet of the truth ad falsty degrees. Moreover, from the practcal pot of vew, the eutrosophc set eeds to be specfed. Otherwse, t wll be dffcult to use the real applcatos. Therefore, Wag et al. [29] troduced a stace of eutrosophc set kow as sgle valued eutrosophc set (SVNS. By the dea of sgle valued eutrosophc set, we ca utlze the sgle valued eutrosophc umbers to express the decso makers preferece to materals, such as <0.6, 0.2, 0.3>, whch meas that the truth-membershp, determacy-membershp ad falsty-membershp of oe materal alteratve to a gve attrbute are 0.6, 0.2 ad 0.3, respectvely. However, f we use the tutostc fuzzy umbers to express the attrbute preferece formato, we oly cosder the membershp degree ad the o-membershp degree of a elemet to a gve set, ad the determacy membershp formato s lost. Therefore, tutostc fuzzy set (IFS s a stace of eutrosophc set. Sce ts appearace, eutrosophc set has receved more ad more atteto from researchers ad practtoers [29-32]. Majumdar ad Samata [33] defed several smlarty measures betwee two sgle valued eutrosophc sets ad vestgated ther characterstcs as well as a measure of etropy of a sgle valued eutrosophc set was troduced. Ye [34, 35] proposed two ovel multple attrbute decso makg methods based o the correlato coeffcet ad cross-etropy of SVNSs, respectvely, whch the attrbute value s descrbed by truth membershp degree, determacy membershp degree ad falsty membershp degree uder sgle valued eutrosophc evromet. Habay ad Talu [36] proposed a ovel sythetc aperture radar (SR mage segmetato algorthm based o the eutrosophc set ad developed a mproved artfcal bee coloy (I-BC algorthm. I the exstg research o decso makg wth sgle valued eutrosophc set, t s geerally assumed that the attrbutes are depedet of oe aother, whch are characterzed by a depedet axom [37]. However, the real decso makg problems, the attrbutes are ofte ter-depedet or correlated. Choquet tegral, orgally developed by Choquet [38], provdes a type of operators used to process the ter-depedece or correlato amog attrbutes [39-46]. Utl ow, to our best kowledge, there s ot ay method for solvg the problem of materal selecto cosderg the ter-depedece or correlato amog attrbutes uder sgle valued 2

3 eutrosophc evromet. Hece, t s ecessary to develop some ew correlated aggregato operators of sgle valued eutrosophc formato based o Choquet tegral. Ths s the motvato of our study. The purpose of ths paper s to develop a method for solvg materal selecto problem uder sgle valued eutrosophc evromet. Frstly, based o Choquet tegral, two sgle valued eutrosophc correlated aggregato operators are proposed,.e., sgle valued eutrosophc correlated average (SVNC operator ad sgle valued eutrosophc correlated geometrc (SVNCG operator. The, a ovel MGDM method s proposed to solve the materal selecto problems uder sgle valued eutrosophc evromet based o the developed operators. To do so, the remader of ths paper s orgazed as follows: some basc cocepts of eutrosophc set ad Choquet tegral are troduced Secto 2; I Secto 3, some ew correlated aggregato operators are proposed based o Choquet tegral uder sgle valued eutrosophc evromet, ad the some propertes ad specal cases of the proposed operators are examed. Secto 4 develops a ovel multple attrbute group decso makg (MGDM method based o these proposed operators. I Secto 5, a umercal example of materal selecto s gve to llustrate the applcato of the developed method. The paper s cocluded Secto Prelmares To facltate the followg dscusso, some cocepts related to eutrosophc set ad sgle valued eutrosophc set are brefly troduced ths secto Neutrosophc set ad sgle valued eutrosophc set Defto 1. [28]. Let X be a uverse set, wth a geerc elemet X deoted by x. eutrosophc set X s characterzed by a truth-membershp fucto T( x, a determacy-membershp fucto I( x ad a falsty-membershp fucto F ( x. The fuctos T( x, I( x ad F ( x are real stadard or ostadard subsets of ]0 -, 1 + [, that s T ( x : X ]0,1 [, I ( x : X ]0,1 [ ad F ( x : X ]0,1 [. There s o restrcto o the sum of T( x, I( x ad F ( x, so 0 sup T ( x sup I ( x sup F ( x 3. Defto 2. [28]. The complemet of a eutrosophc set s deoted by C ad s defed as c T c c (x {1 } Ө T (x, I ( x {1 } Ө I ( x, ad (x {1 } Ө F ( x for every elemet x X. F Defto 3. [28]. eutrosophc set s cotaed the other eutrosophc set B, B f ad oly f f T ( x f T ( x, sup T ( x sup T ( x, f I ( x f I ( x, sup I ( x sup I ( x, B f F (x f F (x ad sup F ( x sup F ( x for every x X. B B B B B 3

4 Defto 4. [28]. The uo of two eutrosophc sets ad B s a eutrosophc set C, deoted by C B, whose truth-membershp, determacy-membershp ad false-membershp fuctos are related to those of ad B by T ( x T ( x T ( x Ө T ( x T ( x, C B I ( x I ( x I ( x Ө I ( x I ( x ad F ( x F ( x F ( x Ө F ( x F ( x for ay x X. C B B C B Defto 5. [28]. The tersecto of two eutrosophc sets ad B s a eutrosophc set C, deoted by C B, whose truth-membershp, determacy-membershp ad false-membershp fuctos are related to those of ad B by T ( x T ( x T ( x, I ( x I ( x I ( x, ad F ( x F ( x F ( x for ay x X. C B C B B B C B 2.2. Sgle valued eutrosophc set sgle valued eutrosophc set (SVNS s a stace of a eutrosophc set, whch ca be used real scetfc ad egeerg applcatos [35]. Defto 6. [29]. Let X be a uverse set, wth a geerc elemet X deoted by x. sgle valued eutrosophc set (SVNS X s characterzed by truth-membershp fucto T( x, determacy-membershp fucto I( x ad falsty-membershp fucto F ( x. For each elemet x X, T( x, I ( x, F ( x [0,1]. Therefore, a SVNS ca be wrtte as follows [35]: { x, T ( x, I ( x, F ( x x X}. For two SVNSs, B, Wag et al. [29] preseted the followg expressos: (1 B f ad oly f T ( x T ( x, I ( x I ( x, ad F ( x F ( x for every x X. B B (2 B f ad oly f B ad B. (3 c { x, F ( x,1 I ( x, T ( x x X}. SVNS s usually deoted by the smplfed symbol T ( x, I ( x, F ( x for ay x 4 X. For ay two SVNSs ad B, the operatoal relatos are defed by Wag et al. [29]. (1 B max( T ( x, T ( x,m( I ( x, I ( x,m( F ( x, F ( x for every x X. B B B (2 B m( T ( x, T ( x,max( I ( x, I ( x,max( F ( x, F ( x for every x X. B B B (3 B T ( x T ( x T ( x T ( x, I ( x I ( x, F ( x F ( x for every x X. B B B B For a SVNS X, Ye [47] called the trplet T ( x, I ( x, F ( x sgle valued

5 eutrosophc umber (SVNN, whch s deoted by T, I, F. Defto 7. [48]. Let a T, I, F be a SVNN, the the score fucto ad the accuracy fucto of are determed by Eqs. (1 ad (2, respectvely. S( a ( T 1 I 1 F / 3 (1 V( a ( T F 1 I / 3 (2 Theorem 1. [48]. Let a T, I, F ad b T, I, F a a a b b b be two SVNNs, the the comparso laws betwee them are show as follows: If S( a S( b, the a b; If S( a S( b, the a b; If S( a S( b, the: (1 If V ( a V ( b, the a b; (2 If V ( a V ( b, the a b; (3 If V ( a V ( b, the a b. Defto 8 [49]. Let a T, I, F, a1 T1, I1, F1 ad a2 T2, I2, F2 be ay three sgle valued eutrosophc umbers, ad 0, the some operatoal laws of the SVNNs are defed as follows. (1 a1 a2 T1 T2 - T1 T2, I1 I2, F1 F2 ; (2 a1 a2 T1 T2, I1 I2 - I1 I2, F1 F2 -F1 F2 ; a 1 1 T, I, F, 0; (3 (4 a T,1 (1 I,1 (1 F, 0. Obvously, the above operatoal results are stll SVNNs. Some relatoshps ca be further establshed for these operatos o SVNNs Choquet tegral Defto 9. [50]. Let X x, x,..., x } be a fte set ad P(X be the power set of X. The set { fucto : P( X [0,1 ] s called a fuzzy measure satsfyg the followg axoms: (1 ( 0, ( X 1; (2 If, B P( X ad B, the ( ( B ; (3 If F P(X for 1 ad a sequece {F } s mootoe, the lm ( F (lm F. 5

6 To avod the problems wth computatoal complexty ad practcal estmato, λ-fuzzy measure, a specal kd of fuzzy measure, was proposed by Sugeo [51], whch satsfes the followg addtoal propertes [52]: ( B ( ( B ( ( B (3 where ( 1,, for all, B P( X ad B. For the teracto betwee ad B, f 0, the there exsts the multplcatve effect; f 0, the there exsts the substtutve effect. If 0, the ad B are depedet of each other ad the Eq.(3 reduces to the followg addtve measure: ( B ( ( B, for all, BP( X ad B. (4 If the elemets of X are depedet, we have ( ( x, for all P(X. (5 x If X s a fte set, the 1 x X. The λ-fuzzy measure satsfes the followg Eq.(6: 1 (1 ( x 1, 0 1 ( X ( x (6 1 ( x, 0 1 where x x, for all, j 1,2,..., ad j. It ca be oted that x for a subset j ( wth a sgle elemet x s called a fuzzy desty ad ca be deoted as x. ( Defto 10. [53]. Let be a fuzzy measure of (X, P(X, X x, x,..., x } be a fte set. { The Choquet tegral of a fucto h : X [0,1 ] wth respect to the fuzzy measure s expressed as follows: ( ( H ( ( H ( 1 h( x ( 1 hd (7 where ( (1, (2,..., ( s a permutato of ( 1,2,..., such that h x h( x h( x, H ( { x (1, x (2,..., x ( } ad H. (0 ( ( 1 (2 ( 3. Some sgle valued eutrosophc correlated aggregato operators I ths secto, we shall develop some correlated aggregato operators to aggregate sgle valued eutrosophc formato based o the operatos of sgle valued eutrosophc umbers. Defto 11. Let a j = Tj, I j, F j (j=1,2,, be a collecto of SVNNs o X, be a fuzzy 6

7 measure o X, the the sgle valued eutrosophc correlated average (SVNC operator s defed as follows: SVNC ( a, a,, a ( ( H ( H a (8 ( ( 1 ( 1 where ( (1, (2,..., ( s a permutato of ( 1,2,..., such that a (1 a (2 a, ( x s the attrbute correspodg to a ( (, H x k }, for ( { ( k 1, H. (0 Based o the operatoal laws of SVNNs, we get Theorem 2. Theorem 2. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, be a fuzzy measure o X, the ther aggregated value obtaed by the SVNC operator s stll a SVNN, ad SVNC a, a,, a T ( I ( F ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 1 1,, (9 Proof. The frst result follows quckly from Defto 11. I what follows, we prove Eq. (9 usg the mathematcal ducto o. (1 Whe =2, t s easy to coclude that Eq. (9 holds accordg to the operatoal law (1 Defto 8: 2 1 SVNC a, a ( ( H ( H a ( ( 1 ( ( ( H (1 ( H (0 a (1 ( ( H (2 ( H (1 a (2 1 1 T (1, I (1, F (1 ( H (1 ( H (0 ( H (1 ( H (0 ( H (1 ( H (0 ( H ( 2 ( H (1 T (2 I(2, F(2 1 1, ( ( H ( 2 ( H (1 ( H ( 2 ( H ( T ( I ( F ( ( H ( ( H ( 1 ( ( H ( ( H ( 1 ( ( H ( ( H ( 1 1 1,, (2 ssume that Eq. (9 holds for k ( k 2, amely, SVNC a, a,, a k k k T ( I ( F ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 1 1,, Whe k1, we get 7

8 SVNC ( a, a,, a k 1 1 k1 ( ( H ( H a ( ( 1 ( k ( ( H ( H a ( ( H ( H a 1 k k k ( H ( ( H ( 1 11 T (, I(, F( ( ( 1 ( ( k 1 ( k k 1 SVNC ( a, a,, a ( ( H ( H a k ( k 1 ( k k 1 T ( k 1 I ( k 1 F ( k 1 ( H ( ( H ( 1 ( H ( ( H ( ( H ( k1 ( H ( k ( H ( k1 ( H ( k ( H ( k1 ( H ( k 1 1,, a k ( H ( ( H ( 1 k ( H( ( H ( 1 k Let a1 1T (, b1 I (, c1 F ( 1 ( H( k1 ( H( k ( H( k1 ( H( k 1 T k, b2 I ( k 1, c2 F ( k 1 2 ( 1 ccordg to the operatoal law (1 Defto 8, we have SVNC ( a, a,, a k 1 1 a, b, c 1 a, b, c = 1 a a, b b, c c k 1 k 1 k 1 T ( I ( F ( 1 1 ( H( ( H( 1, ( H( k1 ( H( k, ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 1 1,, e., Eq. (9 holds for k 1. ccordg to steps (1 ad (2, we kow that Eq. (9 holds for ay postve teger. Some specal cases of the SVNC operator are cosdered as follows. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, ad be a fuzzy measure o X. (1 If ( H 1for ay H P( x SVNC (,,, =,,, the a1 a2 a T (1 I (1 I. (1 (2 If ( H 0 for ay H P( x ad H X SVNC (,,, =,, a1 a2 a T ( I ( I. ( (3 If the depedet codto (5 holds, the 8, the ( x ( ( H ( ( H ( 1, 1,2,..., (10 I ths case, the SVNC operator reduces to the followg sgle valued eutrosophc weghted average (SVNW operator: ( x ( x ( x a1 a2 a x a T I F (11 SVNW,,, ( 1 1,, I partcular, f ( x, for 1,2,...,, the the SVNC operator Eq.(8 reduces to the sgle valued eutrosophc arthmetrc average (SVN operator.

9 1/ 1/ 1/ a a a x a T I F SVN,,, ( 1 1,, (4 If H (12 ( H w,for all H X (13 1 where H s the umber of the elemets H, the ( H( ( H( 1, 1,2,..., (14 where ( 1, 2,..., T such that 0, 1,2,...,, ad 1 1. I ths case, the the SVNC operator reduces to the followg sgle valued eutrosophc ordered weghted average (SVNOW operator: a1 a2 a a ( T ( I ( F ( SVNOW,,, 1 1,, j I partcular, f H ( H, for all H X, the the SVNC operator Eq.(8 reduces to the SVN operator Eq. (12. (5 If I =0 ad T F 1, the SVNNs a = T, I, F (=1,2,, are reduced to tutostc fuzzy umbers (IFNs, ad we ca obta the followg tutostc fuzzy correlated average (IFC operators proposed by Ta ad che [39, 54]. IFC ( b, b,, b 1 ( ( H ( H b ( ( 1 ( T( F( 1 1, ( ( H ( ( H ( 1 ( ( H ( ( H ( where b = T, F (=1,2,, be a collecto of tutostc fuzzy values o X, ad be a fuzzy measure o X. It ca be proved that the SVNC operator has the followg propertes. Theorem 3. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, be a fuzzy measure o X, the we have the followg propertes. (1 (Idempotecy If a = T, I, F (=1,2,, are equal,.e., a = a Ta, Ia, Fa, the SVNC a, a,, a a. 9

10 (2 (Boudedess Let Tm m{ T }, Tmax max{ T }, Im m{ I }, Imax max{ I }, F m m{ F }, Fmax max{ F }. The we ca obta 1 1 T, I, F SVNC ( a, a,, a T, I, F. (15 m max max max m m (3 (Mootocty If T T, I I ad F F for all, the SVNC ( a, a,, a SVNC ( a, a,, a. (4 (Commutatvty If a = T, I, F (=1,2,, s ay permutato of a = T, I, F (=1,2,,, the SVNC ( a, a,, a =SVNC ( a, a,, a Proof. (1 Sce a T, I, F a a a for all, we have SVNC a, a,, a ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 Ta Ia Fa 1 1,, ( ( H ( ( H ( 1 ( ( H ( ( H ( 1 ( ( H ( ( H ( 1 a 1 a 1 a 1 T I F 1 1,, T, I, F a a a SVNC 1, 2,, a a a a. Thus, we have (2 Sce Tm T Tmax, Im I Imax, Fm F Fmax for all, the we have ( H( ( H( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 T ( Tm T m 1 Tm , ( H( ( H( 1 ( H ( ( H ( 1 ( ( H ( ( H ( 1 T ( Tmax T max 1 Tmax ,.e., ( ( H( ( H( 1 m 1 1 ( max 1 T T T. (16 Smlarly, we have ( ( H( ( H( 1 m ( max 1 I I I, (17 ( ( H( ( H( 1 m ( max 1 F F F. (18 Let SVNC,,,,, a1 a2 a Ta Ia Fa a, Tmax, Im, Fm a * ad Tm, Imax, Fmax a*, the Eqs. (16, (17 ad (18 are trasformed to the followg forms, respectvely: T T T (19 m a max 10

11 Im Ia Imax (20 F F F (21 m a max Thus, we have 1 1 * S( a Ta 1 Ia 1 Fa Tmax 1 Im 1 Fm S( a. 3 3 * <1> If S( a S( a, the by Theorem 1, we have SVNC ( a, a,, a < T, I, F max m m. (22 * <2> If S( a S( a, the by the followg codtos: Ta Tmax, 1 Ia 1 Im, ad 1 Fa 1 Fm, we have Ta Tmax, 1 Ia 1 Im, ad 1 Fa 1 Fm, thus, 1 1 * V ( a Ta Fa 1 Ia Tmax Fm 1 Im V ( a. 3 3 I ths case, by Theorem 1, we have SVNC ( a, a,, a = T, I, F max m m. (23 From Eqs.(22 ad (23, we have SVNC ( a, a,, a T, I, F. max m m (24 Smlarly, we have Tm Imax Fmax a1 a2 a,, SVNC (,,,. (25 From Eqs.(24 ad (25, we kow that Eq.(15 always holds,.e., T, I, F SVNC ( a, a,, a T, I, F. m max max max m m (3. Sce T T, I I ad F F for all, the we have T T, ( ( I I ad ( ( F F. ( ( So, ( H ( H 1 T( 1 T( ( ( 1 ( H( ( H( 1, ( H ( H I( I( ( ( 1 ( H( ( H( 1 ( H ( H F( F(, ( ( 1 ( H( ( H( 1 Furthermore, we have. ( H( ( H( 1 ( H( ( H( 1 T( T(, ( ( H( ( H( 1 ( H( ( H( 1 I( I(, (

12 ( H( ( H( 1 ( H( ( H( 1 F( F(. ( Therefore, we have T ( I ( F ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( T ( I ( F ( Let ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 1. (29 a1 a2 a Ta Ia Fa a ad a a a SVNC,,,,, the Eq. (29 s trasformed to the followg forms: S a ( S( a SVNC ( a, a,, a = T, I, F a, <1> If S( a S( a, the by Theorem 1, we have SVNC ( a1, a2,, a<svnc ( a1, a2,, a. (30 <2> If S a ( S( a, the by Eqs.(26 to (28, we have ( H( ( H( 1 ( H( ( H( 1 T( T(, ( H( ( H( 1 ( H( ( H( 1 I( I(, ( H( ( H( 1 ( H( ( H( 1 F( F( Thus, we have.e., T ( F ( I ( ( H ( H ( H ( H ( H ( H ( ( ( ( ( ( V a ( V( a T ( F ( I ( ( H ( H ( H ( H ( H ( H ( ( ( ( ( ( I ths case, by Theorem 1, we have SVNC ( a1, a2,, a=svnc ( a1, a2,, a (31 Therefore, from Eqs.(30 ad (31, we have SVNC ( a, a,, a SVNC ( a, a,, a. (4. Sce a = T, I, F (=1,2,, s a permutato of a = T, I, F (=1,2,,, we have a a ( (, for all =1,2,,. The, based o Defto 11, we obta SVNC ( a, a,, a =SVNC ( a, a,, a. 12

13 Theorem 4. Let a = T, I, F (=1,2,, be a set of SVNNs o X, be a fuzzy measure o X. If s= Ts, Is, Fs s a SVNN o X, the SVNC ( a s, a s,, a s=svnc ( a, a,, a s Proof. ccordg to the operatoal law (1 Defto 8, for all =1,2,,, we have a s T T - T T, I I, F F 1 (1 T (1- T, I I, F F. s s s s s s s ccordg to Theorem 2, we have SVNC ( a s, a s,, a s T ( Ts I ( Is F ( Fs ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 1 (1 (1,, ( ( H ( ( H ( 1 1 s 1 ( H ( ( H ( 1 1 (1 T 1,(, ( ( H ( ( H ( 1 1 s 1 ( F ( H ( ( H ( 1 ( ( H ( ( H ( 1 1 T ( Is I ( ( H ( ( H ( 1 ( F 1 ( H ( ( H ( 1 ( H ( ( H ( 1 ( ( ( ( H ( ( H ( 1 s s s (1 T 1 T, I I, F F. O the other had, accordg to the operatoal law (1 Defto 8, we have SVNC ( a, a,, a s ( ( ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( T, I, F T, I, F s s s ( ( H ( ( H ( 1 ( ( H ( ( H ( 1 ( ( Fs F ( 1 (1 T 1 T, I I Thus, s s 1 1 SVNC ( a s, a s,, a s=svnc ( a, a,, a s. 1 ( ( H ( ( H ( 1,. Theorem 5. Let a = T, I, F (=1,2,, be a set of SVNNs o X, be a fuzzy measure o X. If r 0, the SVNC ( ra, ra,, ra = r SVNC ( a, a,, a Proof. ccordg to the operatoal law (3 Defto 8, for all (=1,2,, ad r 0, we have r r r ra 1 1 T, I, F. ccordg to Theorem 2, we have 13

14 SVNC ( ra, ra,, ra r r r T ( I ( F ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 1 (1, (, ( r( ( H ( ( H ( 1 r( ( H ( ( H ( 1 T ( I ( F ( 1 1,, r( ( H ( ( H ( 1 O the other had, accordg to the operatoal law (3 Defto 8, we have r SVNC ( a, a,, a ( ( ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 r 1 1 T, I, F ( ( 1 ( ( 1 T ( I ( F ( r( ( H ( H r( ( H ( H r( ( H ( ( H ( 1 1 1,, Thus, 1 1 SVNC ( ra, ra,, ra = r SVNC ( a, a,, a. 1. Defto 12. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, be a fuzzy measure o X, the the sgle valued eutrosophc correlated geometrc (SVNCG operator s defed as follows: ( 1 ( H( ( H( 1 SVNCG ( a, a,, a a (32 where ( (1, (2,..., ( s a permutato of ( 1,2,..., such that a (1 a (2 a, ( x s the attrbute correspodg to a ( (, H x k }, for ( { ( k 1, H. (0 Based o the operatoal laws of SVNNs, we get Theorem 6. Theorem 6. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, be a fuzzy measure o X, the ther aggregated value obtaed by the SVNCG operator s stll a SVNN, ad SVNCG a, a,, a T (,1 1 I (,1 1 F ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( Ths theorem ca be proved smlar to Theorem 2. (33 Some specal cases of the SVNCG operator are cosdered as follows. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, ad be a fuzzy measure o X. (1 If ( H 1for ay H P( x, the 14

15 SVNCG (,,, =,, a1 a2 a T (1 I (1 I. (1 (2 If ( H 0 for ay H P( x ad H X SVNCG (,,, =,, a1 a2 a T ( I ( I. (, the (3 If Eqs. (5 ad (10 hold, the the SVNCG operator reduces to the followg sgle valued eutrosophc weghted geometrc (SVNWG operator: ( x ( x ( x ( x a1 a2 a a T I F (34 SVNWG,,,,1 1, I partcular, f ( x, for 1,2,...,, the the SVNCG operator Eq.(32 reduces to the sgle valued eutrosophc geometrc average (SVNG operator. 1/ 1/ 1/ 1/ a1 a2 a a T I F (35 SVNG,,,,1 1, (4 If Eqs. (13 ad (14 hold, the the SVNCG operator reduces to the followg sgle valued eutrosophc ordered weghted geometrc (SVNOWG operator: a1 a2 a a ( T ( I ( F ( SVNOWG,,,,1 1, I partcular, f H ( H, for all H X, the the SVNC operator Eq.(8 reduces to the SVNG operator Eq. (33. (5 If I =0 ad T F 1, the SVNNs a = T, I, F (=1,2,, are reduced to tutostc fuzzy umbers (IFNs, ad we ca obta the followg tutostc fuzzy correlated geometrc (IFCG operators proposed by Xu [54]. IFCG ( a, a,, a ( H ( ( H ( 1 1 a ( T(,1 1 F( ( H ( ( H ( 1 ( H ( ( H ( where a = T, F (=1,2,, be a collecto of tutostc fuzzy values o X, ad be a fuzzy measure o X. Smlar to the SVNC operator, we ca prove that the SVNCG operator has the followg propertes. Theorem 7. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, be a fuzzy measure o X, the we have the followg propertes. 15

16 (1 If a = T, I, F (=1,2,, are equal,.e., a = a Ta, Ia, Fa, the SVNCG a, a,, a a. (2 (Boudedess Let Tm m{ T }, Tmax max{ T }, Im m{ I }, Imax max{ I }, F m m{ F }, Fmax max{ F }. The we ca obta 1 1 T, I, F SVNCG ( a, a,, a T, I, F. m max max max m m (3 (Mootocty If T T, I I ad F F for all, the SVNCG ( a, a,, a SVNCG ( a, a,, a. (4 (Commutatvty If a = T, I, F (=1,2,, s ay permutato of a = T, I, F (=1,2,,, the SVNCG ( a, a,, a =SVNCG ( a, a,, a Theorem 8. Let a = T, I, F (=1,2,, be a set of SVNNs o X, be a fuzzy measure o X. If s= Ts, Is, Fs s a SVNN o X, the SVNCG ( a s, a s,, a s=svncg ( a, a,, a s Proof. ccordg to the operatoal law (2 Defto 8, for all =1,2,,, we have a s T T, I I I I, F F F F T T,1 (1 I (1- I,1 (1 F (1- F. s s s s s s s s ccordg to Theorem 6, we have SVNCG ( a s, a s,, a s T ( Ts,1 (1 I ( (1 Is,1 (1 F ( (1 Fs ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( ( H ( ( H ( s T ( Is 1 1 ( T ( H ( ( H ( 1 (,1 (1 1 I, ( ( H ( ( H ( 1 1 s 1 1 (1 F 1 F ( H ( ( H ( 1 ( ( H ( ( H ( 1 ( H ( ( H ( 1 (,1 (1 1 ( Fs F ( ( H ( ( H ( 1 s s T T I I O the other had, accordg to the operatoal law (2 Defto 8, we have,1 (

17 SVNCG ( a, a,, a s ( ( ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 T,1 1 I,1 1 F T, I, F s s s ( H ( ( H ( 1 ( H ( ( H ( 1 (,1 (1 1 ( Fs F ( T T I I Thus, s s 1 1 SVNCG ( a s, a s,, a s=svncg ( a, a,, a s. 1 ( H ( ( H ( 1,1 (1 1. Theorem 9. Let a = T, I, F (=1,2,, be a set of SVNNs o X, be a fuzzy measure o X. If r 0, the r r r SVNCG ( a, a,, a = SVNCG ( a, a,, a r Proof. ccordg to the operatoal law (4 Defto 8, for all (=1,2,, ad r 0, we have r r r a T,1 1 I,1 1 F. ccordg to Theorem 6, we have SVNCG ( a, a,, a r r r r r r r ( T (,1 (1 I (,1 (1 F ( r( ( H ( ( H ( 1 r( ( H ( ( H ( 1 T(,1 1 I( F ( ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( r( ( H ( ( H ( 1,1 1. O the other had, accordg to the operatoal law (4 Defto 8, we have SVNCG ( a, a,, a r T (,1 1 I (,1 1 F ( r( ( H ( ( H ( 1 r( ( H ( ( H ( 1 r( ( H ( ( H ( Thus, r r r SVNCG ( a, a,, a = SVNCG ( a, a,, a r Lemma 1. [55]. Let a j 0, wj 0, j 1,2,..., ad j1 w 1, the j j1 wj j wja j j1 a (36 wth equalty f ad oly f a1 a2 a. To compare the aggregated values betwee the SVNC ad SVNCG operators, we gve the 17

18 followg theorem. Theorem 10. Let a = T, I, F (=1,2,, be a collecto of SVNNs o X, be a fuzzy measure o X, the SVNCG a, a,, a SVNC a, a,, a. (37 Proof. ccordg to Lemma1, we have 1 1 1T 1 1 ( ( H ( ( H ( 1 ( ( H ( H 1T ( ( 1 ( 1 ( ( H ( H T 1 T ( ( ( 1 ( Thus, we have ( H ( ( H ( 1 ( H ( ( H ( 1 ( H ( ( H ( 1 T( T( ( Smlarly, we have ( H ( ( H ( 1 ( H ( ( H ( 1 I( 1 1 I( ( ( H ( ( H ( 1 ( H ( ( H ( 1 F( 1 1 F(. ( Let SVNC a1, a2,, a Ta, Ia, Fa a, SVNCG a, a,, a Tb, Ib, Fb b, the Eqs. (38, (39 ad (40 are trasformed to the followg forms, respectvely: T T I a a F a b I, b F. b Thus, we have 1 1 S( b Tb 1 Ib 1 Fb Ta 1 Ia 1 Fa S( a. 3 3 ccordg to Theorem 1, we have SVNCG a, a,, a SVNC a, a,, a. (41 (42 (43 4. multple attrbute group decso makg method to materal selecto uder sgle valued eutrosophc evromet I ths secto, we apply the SVNC (SVNCG operator to solve materal selecto problems 18

19 wth sgle valued eutrosophc formato. For a materal selecto problem, let m {,,..., } ( m 2 be a fte set of feasble materal alteratves amog whch decso makers (DMs have to choose, C { C1, C2,..., C } ( 2 be a fte set of attrbutes wth whch alteratve performace s measured, DM { DM1, DM 2,..., DM t }( t 2 be a set of DMs, ad be the weght vector of DMs, such that 0, k=1,2,,t, T ( 1, 2,..., t k t ad 1. Suppose that k1 k ( k ( k R ( rj m s a sgle valued eutrosophc decso matrx gve by the kth DM, where r ( T, I, F ( k ( k ( k ( k j j j j s the assessmet value o the materal alteratve wth respect to the attrbute C j C provded by the kth DM, ( k T j dcates the degree to whch the materal alteratve satsfes the attrbute C provded by j the kth DM, I dcates the determacy degree to whch the materal alteratve ( k j satsfes the attrbute C provded by the kth DM, ad j F dcates the degree to whch the ( k j materal alteratve does ot satsfy the attrbute C j provded by the kth DM. The proposed operators are utlzed to develop a multple attrbute group decso makg method for materal selecto wth sgle valued eutrosophc formato by the followg steps: Step 1. ggregate all dvdual sgle valued eutrosophc decso matrces R ( r ( k 1,2,..., t to a collectve sgle valued eutrosophc decso matrx ( k ( k j m R ( rj m based o SVNW (SVNWG operator as follows: r T, I, F j j j j =SVNW r, r,, r (1 (2 ( k j j j t t t ( k ( k ( k j j j k 1 k 1 k T k, I k, F k, 1,2,..., m, j 1,2,..., r T, I, F j j j j =SVNWG r, r,, r (1 (2 ( k j j j t t t ( k ( k ( k j j j k 1 k 1 k 1 k k k T,1 1 I,1 1 F, 1,2,..., m, j 1,2,..., (44 (45 where s the weght vector of DMs. T ( 1, 2,..., t 19

20 Step 2. Cofrm the fuzzy measures of the attrbutes C j (j=1,2,, ad the attrbute sets of C. The λ-fuzzy measure s used to calculate the fuzzy measure of crtera sets. Frstly, accordg to Eq. (6, the value of s obtaed, ad the the fuzzy measure of crtera sets of C { C1, C2,..., C } are calculated by Eq. (6. Step 3. Utlze the SVNC (or SVNCG operator to aggregate all assessmet values r j of the alteratve (=1,2,,m uder all attrbutes C j (j=1,2,, ad get the overall assessmet values r of alteratves (=1,2,,m by Eq.(46 or (47. r T, I, F or r SVNC r, r,, r ( j ( j ( j ( H ( j ( H ( j1 ( H ( j ( H ( j1 ( H ( j ( H ( j1 1 1 T, I, F, 1,2,..., m, T, I, F j1 j1 j1 SVNCG r, r,, r ( j ( j ( j ( H ( H ( H ( H ( H ( H ( j ( j ( j ( j ( j ( j T,1 1 I,1 1 F, 1,2,..., m, j1 j1 j1 (46 (47 where ( j ( j ( j ( j r T, T, T ( j 1,2,..., s a permutato of r T, T, T ( j 1,2,..., such j j j j that r (1 r (2 r, ( x s the attrbute correspodg to r ( ( j, H ( j { x ( k k j}, for 1, H (0. Step 4. Calculate the score values Vr ( of the overall assessmet values r (=1,2,,m. The score values of the alteratves (=1,2,,m ca be calculated by Eq. (48. S( r ( T 1 I 1 F / 3, 1,2,..., m. (48 If there s o dfferece betwee two score values Sr ( ad Sr (, the we eed to calculate the accuracy values Vr ( ad Vr ( l of the alteratves ad l (,l=1,2,,m, respectvely, accordg to Eq. (49. V( r ( T F 1 I / 3, 1,2,..., m. (49 Step 5. Rak all feasble alteratves (=1,2,,m accordg to Theorem 1 ad select the most desrable alteratve(s. Step 6. Ed. l 5. Numercal example 20

21 I ths secto, a materal selecto problem adopted from Vekata Rao [56] whch the alteratves are the materal alteratves to be selected ad the crtera are the attrbutes uder cosderato a MGDM problem s used to llustrate the applcato of the proposed method wth sgle valued eutrosophc formato proposed Secto 4, ad to demostrate ts feasblty ad effectveess a realstc scearo. compay wats to select a sutable work materal for a product operated a hgh-temperature evromet. fter prelmary screeg, there are four possble materal alteratves 1, 2, 3 ad 4 to be selected, accordg to the followg four attrbutes: (1 C 1 s the tesle stregth (MPa; (2 C 2 s the youg s modulus (GPa; (3 C 3 s the desty (gm/cm 3 ; (4 C 4 s the corroso resstace. commttee of three decso makers D k (k=1,2,3 whose weght vector s (0.34,0.28,0.38 T s vted to evaluate the materal alteratves (=1,2,3,4 wth respect to the attrbutes C j (j=1,2,3,4 ad three dvdual sgle ( k ( k valued eutrosophc decso matrces Rj ( rj 4 4 (k=1,2,3 are costructed, whch are as show Tables 1-3. Table 1 Sgle valued eutrosophc decso matrx gve by DM 1. C 1 C 2 C 3 C 4 1 <0.30, 0.40, 0.52> <0.50, 0.65, 0.20> <0.80, 0.24, 0.15> <0.45, 0.32, 0.15> 2 <0.42, > <0.15, 0.45, 0.50> <0.80, 0.21, 0.20> <0.50, 0.36, 0.13> 3 <0.62, 0.34, 0.40> <0.24, 0.22, 0.72> <0.90, 0.35, 0.15> <0.35, 0.40, 0.25> 4 <0.81, 0.23, 0.40> <0.45, 0.42, 0.10> <0.21, 0.52, 0.25> <0.60, 0.40, 0.70> Table 2 Sgle valued eutrosophc decso matrx gve by DM 2. C 1 C 2 C 3 C 4 1 <0.57, 0.20, 0.41> <0.25, 0.30, 0.40> <0.35, 0.25, 0.10> <0.75, 0.20, 0.10> 2 <0.67, 0.40, 0.20> <0.40, 0.15, 0.10> <0.28, 0.45, 0.50> <0.50, 0.15, 0.35> 3 <0.45, 0.31, 0.32> <0.70, 0.10, 0.05> <0.55, 0.15, 0.35> <0.53, 0.30, 0.20> 4 <0.45, 0.05, 0.30> <0.70, 0.20, 0.15> <0.90, 0.10, 0.35> <0.52, 0.30, 0.25> Table 3 Sgle valued eutrosophc decso matrx gve by DM 3. C 1 C 2 C 3 C 4 1 <0.40, 0.15, 0.32> <0.50, 0.12, 0.40> <0.80, 0.12, 0.15> <0.53, 0.20, 0.15> 2 <0.65, 0.30, 0.15> <0.25, 0.43, 0.15> <0.85, 0.10, 0.25> <0.80, 0.10, 0.05> 3 <0.60, 0.32, 0.38> <0.35, 0,25, 0.20> <0.53, 0.20, 0.12> <0.77, 0.30, 0.20> 4 <0.52, 0.20, 0.45> <0.60, 0.24, 0.31> <0.72, 0.05, 0.10> <0.72, 0.13, 0,24> I what follows, the proposed method wth sgle valued eutrosophc formato s utlzed to get the most desrable materal alteratve(s, whch volves the followg steps: Step 1. Utlze the dvdual sgle valued eutrosophc decso matrx R ( r ( k 1,2,3 ad the SVNW operator to derve the collectve sgle valued ( k ( k j 44 21

22 eutrosophc decso matrx R ( rj by Eq.(44, whch s show Table Table 4 Collectve sgle valued eutrosophc decso matrx by usg the SVNW operator. C 1 C 2 C 3 C 4 1 <0.424, 0.227, 0.405> <0.440, 0.276, 0.316> <0.722, 0.187, 0.134> <0.585, 0.235, 0.134> 2 <0.591, 0.472, 0.193> <0.265, 0.325, 0.202> <0.743, 0.196, 0.281> <0.647, 0.173, 0.119> 3 <0.570, 0.324, 0.369> <0.448, 0.185,0.210> <0.726, 0.223, 0.175> <0.600, 0.331, 0.330> 4 <0.636, 0.142, 0.386> <0.589, 0.276, 0.172> <0.701, 0.135, 0.194> <0.632, 0.241, 0.349> Step 2. Suppose that the fuzzy measures of crtera of C are gve as follows: ( C 1 0.2, ( C , ( C , ( C The λ-fuzzy measure s used to calculate the fuzzy measure of crtera sets. Frstly, accordg to Eq. (6, the value of s calculated: , ad the the fuzzy measure of crtera sets of C={C 1,C 2,C 3,C 4 } are calculated by Eq. (6, whch are show as follows: ( C1, C , C1 C3 (, , ( C1, C , ( C2, C , ( C2, C , C3 C4 (, , ( C1, C2, C , ( C1, C2, C , ( C1, C3, C , C2 C3 C4 (,, , ( C1, C2, C3, C4 1. Step 3. Utlze the SVNC operator to calculate the overall assessmets of each materal alteratve. Take 1 for a example: accordg to Theorem 1, we have r1 ( , 0.187, 0.134, r 1 ( , 0.235, 0.134, r1 ( , 0.276, 0.316, r1 ( , 0.227, The, the overall assessmets of the materal alteratve 1 ca be calculated as follows: r SVNC r, r, r, r Smlarly, ( ( ( ( , , , 0.233, r2 (0.596, 0.248, 0.180, r (0.592, 0.262, 0.224, 3 r4 (0.635, 0.205, Step 4. Calculate the score values Sr ( of the overall assessmet values r (=1,2,3,4. ccordg to Eq. (48, the score values of materal alteratves (=1,2,3,4 are obtaed as follows: Sr ( , Sr ( , Sr ( , Sr ( Step 5. Rak all materal alteratves (=1,2,3,4 accordg to the descedg order of correspodg score values Sr ( (=1,2,3,4 ad select the most desrable materal alteratve(s. 22

23 Sce S( r4 S( r2 S( r1 S( r3, the the rakg of all materal alteratves (=1,2,3,4 s show as follows: where the symbol meas superor to. Therefore, the most desrable materal alteratve s Coclusos I ths paper, we study the materal selecto problems whch the attrbute values take the form of sgle valued eutrosophc umbers. Motvated by the dea of Choquet tegral, two correlated aggregato operators are proposed for aggregatg the sgle valued eutrosophc formato based o the operatoal laws of sgle valued eutrosophc umbers, such as the sgle valued eutrosophc correlated average (SVNC operator ad the sgle valued eutrosophc correlated geometrc (SVNCG operator. The promet characterstc of these operators s that the truth degree, determacy degree ad falsty degree of a elemet to a gve set are deoted by a set of three crsp umbers. The, some desrable propertes of the proposed operators ad the relatoshps amog them are vestgated detal. Furthermore, based o the proposed operators, a ovel multple attrbute group decso makg method s developed to solve materal selecto problems uder sgle valued eutrosophc evromet, whch the attrbutes are ofte ter-depedet or correlated. Fally, a umercal example of materal selecto s gve to llustrate the applcato of the proposed method. I future research, we wll focus o the applcato of the proposed method other real decso makg problems, such as persoel evaluato ad emergecy maagemet. ckowledgemets Ths research s supported by Program for New Cetury Excellet Talets Uversty (NCET , Natural Scece Foudato of Cha (No , , ad Bejg Mucpal Natural Scece Foudato (No , Refereces [1] Sapua SM. kowledge-based system for materals selecto mechacal egeerg desg. Mater Des 2001;22: [2] Chatterjee P, Chakraborty S. Materal selecto usg preferetal rakg methods. Mater Des 2012;35: [3] Jaha, Mustapha F, Ismal MY, Sapua SM, Bahramasab M. comprehesve VIKOR method for materal selecto. Mater Des 2011; 32(3: [4] Lu HC, Lu H, Wu J. Materal selecto usg a terval 2-tuple lgustc VIKOR method cosderg subjectve ad objectve weghts. Mater Des 2013; 52: [5] Parate O, Gupta N. Materal selecto for electrostatc mcroactuators usg shby approach. Mater Des 2011;32(3: [6] Mayyas, She Q, Mayyas, abdelhamd M, Sha D, Qattaw, et al. Usg qualty fucto deploymet ad aalytcal herarchy process for materal selecto of Body-I-Whte. Mater Des 2011;32(5:

24 [7] Mla S, Shaa, Lyam C, Scarc T. applcato of the aalytc etwork process multple crtera materal selecto. Mater Des 2013;44(2: [8] Gupta N. Materal selecto for th-flm solar cells usg multple attrbute decso makg approach. Mater Des 2011;32(3: [9] Prasad K, Chakraborty S. qualty fucto deploymet-based model for materals selecto. Mater Des 2013;49(8: [10] Cha JWK, Tog TKL. Mult-crtera materal selectos ad ed-of-lfe product strategy: grey relatoal aalyss approach. Mater Des 2007;28(5: [11] Rao RV. materal selecto model usg graph theory ad matrx approach. Mater Sc Eg 2006;431(1: [12] Shaa, Savadogo O. materal selecto model based o the cocept of multple attrbute decso makg. Mater Des 2006;27(4: [13] Bahramasab M, Jaha. Materal selecto for femoral compoet of total kee replacemet usg comprehesve VIKOR. Mater Des 2011;32(8 9: [14] Cavall C, Gorgett, Ctt P, Ncolae F. Itegral aded method for materal selecto based o qualty fucto deploymet ad comprehesve VIKOR algorthm. Mater Des 2013;47(5: [15] Jaha, Edwards KL. VIKOR method for materal selecto problems wth terval umbers ad target-based crtera. Mater Des 2013;47(5: [16] Peg H, Xao XM. Materal selecto usg PROMETHEE combed wth aalytc etwork process uder hybrd evromet. Mater Des 2013;47(5: [17] Lu HC, You JX, Zhe L, Fa XJ. ovel hybrd multple crtera decso makg model for materal selecto wth target-based crtera. Mater Des 2014;60(8: [18] Chatterjee P, thawale VM, Chakraborty S. Materals selecto usg complex proportoal assessmet ad evaluato of mxed data methods. Mater Des 2011;32(2: [19] Maty SR, Chatterjee P, Chakraborty S. Cuttg tool materal selecto usg grey complex proportoal assessmet method. Mater Des 2012;36(4: [20] Zadeh L. Fuzzy sets. Iform Cotrol 1965; 8(3: [21] Turkse IB. Iterval-valued fuzzy sets based o ormal forms. Fuzzy Sets Syst 1986; 20: [22] taassov K. Itutostc fuzzy sets. Fuzzy Sets Syst 1986; 20: [23] taassov K, Gargov G. Iterval valued tutostc fuzzy sets. Fuzzy Sets Syst 1989; 31: [24] Dubos D, Prade H. Fuzzy sets ad systems: theory ad applcatos. New York: cademc Press; [25] Mzumoto M, Taaka K. Some propertes of fuzzy sets of type 2. If Cotrol 1976; 31: [26] Torra V. Hestat fuzzy sets. I J Itell Syst 2010; 25: [27] Torra V, Narukawa Y. O hestat fuzzy sets ad decso. The 18th IEEE Iteratoal Coferece o Fuzzy Systems, Jeju Islad, Korea, 2009; [28] Smaradache F. ufyg feld logcs. Neutrosophy: Neutrosophc probablty, set & logc. Rehoboth: merca Research; [29] Wag H, Smaradache F, Zhag YQ. Sgle valued eutrosophc sets. Multspace Multstruct 2010; 4:

25 [30] Ye J. Smlarty measures betwee terval eutrosophc sets ad ther applcatos multcrtera decso-makg. J Itell Fuzzy Syst 2014; 26: [31] Ye J. multcrtera decso-makg method usg aggregato operators for smplfed eutrosophc sets. J Itell Fuzzy Syst 2014; 26: [32] Zhag HY, Wag JQ, Che XH. Iterval eutrosophc sets ad ther applcato multcrtera decso makg problems. Sc World J 2014; rtcle ID , [33] Majumdar P, Samata SK. O smlarty ad etropy of eutrosophc sets. J Itell Fuzzy Syst 2014; 26: [34] Ye J. Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet. It J Ge Syst 2013; 42: [35] Ye J. Sgle valued eutrosophc cross-etropy for multcrtera decso makg problems. ppl Math Model 2014; 38: [36] Habay K, Talu MF. Segmetato of SR mages usg mproved artfcal bee coloy algorthm ad eutrosophc set. ppl SoftComput 2014; 21: [37] Wakker P. ddtve represetatos of prefereces, ew foudato of decso aalyss; The algebrac approach. Berl: Sprger, [38] Choquet G. Theory of capactes. ales del Isttut Fourer 1953; 5: [39] Ta CQ, Che XH. Itutostc fuzzy Choquet tegral operator for mult-crtera decso makg. Expert Syst ppl 2010; 37(1: [40] Ta CQ, Che XH. Iduced Choquet ordered averagg operator ad ts applcato to group decso makg. It J Itel Syst 2010; 25: [41] glella S, Greco S, Matarazzo B. No-addtve robust ordal regresso: multple crtera decso model based o the Choquet tegral. Eur J Oper Res 2010; 201(1: [42] Büyüközka G, Rua D. Choquet tegral based aggregato approach to software developmet rsk assessmet. Iform Sceces 2010; 180: [43] Lee WS. Evaluatg ad rakg eergy performace of offce buldg usg fuzzy measure ad fuzzy tegral. Eergy Covers Maage 2010; 51: [44] Hu YC, Tsa JF. Evaluatg classfcato performaces of sgle-layer perceptro wth a Choquet fuzzy tegral-based euro. Expert Syst ppl 2009; 36(2: [45] Mg-Lag T, Chag JH, La LW. Selecto of optmal suppler supply cha maagemet strategy wth aalytc etwork process ad Choquet tegral. Comput Id Eg 2009; 57(1: [46] Ju YB, Yag SH, Lu XY. Some ew dual hestat fuzzy aggregato operators based o Choquet tegral ad ther applcatos to multple attrbute decso makg. J Itell Fuzzy Syst 2014; DOI: /IFS [47] Ye J, Multple attrbute group decso-makg method wth completely ukow weghts based o smlarty measures uder sgle valued eutrosophc evromet, J Itell Fuzzy Syst 2014; DOI: /IFS [48] Ju YB. Some aggregato operators wth sgle valued eutrosophc formato ad ther applcato to multple attrbute decso makg. Techcal Report [49] Ch PP, Lu PD. exteded TOPSIS method for multple attrbute decso makg problems based o terval eutrosophc set. Neutrosophc Sets ad Systems (2013; 1:

26 [50] Wag Z, Klr G. Fuzzy measure theory. New York: Pleum Press, [51] Sugeo M. Theory of fuzzy tegral ad ts applcato. Tokyo: Tokyo Isttute of Techology, PhD dssertato, [52] Yager RR, Flev DP. Essetals of fuzzy modelg ad cotrol. New York: Joh Wlley & Sos Ic, [53] Grabsch M, Murofush T, Sugeo M. Fuzzy measures ad tegrals. New York: Physca-Verlag, [54] Xu ZS. Choquet tegrals of weghted tutostc fuzzy formato. Iform Sc 2010; 180: [55] Xu ZS. O cosstecy of the weghted geometrc mea complex judgmet matrx HP, Eur J Oper Res 2000; 126: [56]Vekata Rao R. decso makg methodology for materal selecto usg a mproved compromse rakg method. Mater Des 2008; 29: