A Revisit to NC-VIKOR Based MAGDM Strategy in Neutrosophic Cubic Set Environment

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1 Neutrosophc Sets nd Systems, Vol., 08 Unversty of New exco Revst to NC-VIKOR Bsed GD Strtegy n Neutrosophc Cubc Set Envronment Shyml Dlpt, Surpt Prmnk, Deprtment of themtcs, Indn Insttute of Engneerng Scence nd Technology, Shbpur, P.O.-Botnc Grden, Howrh- 70, West Bengl, Ind. E-ml: dlptshyml0@gml.com Deprtment of themtcs, Nndll Ghosh B.T. College, Pnpur, P.O.-Nrynpur, Dstrct North Prgns, Pn code- 76, West Bengl, Ind. E-ml: sur_pt@yhoo.co.n bstrct. ult ttrbute group decson mkng wth VIKOR (VlseKrterjusk Optmzcj I Komoromsno Resenje) strtegy hs been wdely ppled to solvng rel-world problems. Recently, Prmnk et l. [S. Prmnk, S. Dlpt, S. lm, nd T. K. Roy. NC- VIKOR bsed GD strtegy under neutrosophc cubc set envronment, Neutrosophc Sets nd Systems, 0 (08), 95-08] proposed VIKOR strtegy for solvng GD, where compromse solutons re not dentfed n neutrosophc cubc envronment. To overcome the shortcomngs of the pper, we further modfy the VIKOR strtegy by ncorportng compromse soluton n neutrosophc cubc set envronment. Fnlly, we solve n GD problem usng the modfed NC-VIKOR strtegy to show the fesblty, pplcblty nd effectveness of the proposed strtegy. Further, we present senstvty nlyss to show the mpct of dfferent vlues of the decson mkng mechnsm coeffcent on rnkng order of the lterntves. Keywords: GD, NCS, NC-VIKOR strtegy.. Introducton Neutrosophc set [] s derved from Neutrosophy [], new brnch of phlosophy. It s chrcterzed by the three ndependent functons, nmely, truth membershp functon, ndetermncy functon nd flsty membershp functon s ndependent components. Ech of three ndependent components of NS belongs to [ - 0, + ]. Wng et l. [] ntroduced sngle vlued neutrosophc set (SVNS) where ech of truth, ndetermncy nd flsty membershp functon belongs to [0, ]. pplctons of NSs nd SVNSs re found n vrous res of reserch such s conflct resoluton [5], clusterng nlyss [6-9], decson mkng [0-9], eductonl problem [0, ], mge processng [-5], medcl dgnoss [6, 7], socl problem [8, 9], etc. Wng et l. [50] proposed ntervl neutrosophc set (INS). ondl et l. [5] defned tngent functon of ntervl neutrosophc set nd develop strtegy for mult ttrbute decson mkng (D) problems. Dlpt et l. [5] defned new cross entropy mesure for ntervl neutrosophc set nd developed mult ttrbute group decson mkng (GD) strtegy. By combnng SVNS nd INS, l et l. [5] proposed neutrosophc cubc set (NCS). Zhn et l. [5] presented two weghted verge opertors on NCSs nd employed the opertors for D problems. Bnerjee et l. [55] ntroduced the grey reltonl nlyss bsed D strtegy n NCS envronment. Lu nd Ye [56] proposed three cosne mesures between NCSs nd presented D strtegy n NCS envronment. Prmnk et l. [57] defned smlrty mesure for NCSs nd proved ts bsc propertes. In the sme study, Prmnk et l. [57] presented new GD strtegy wth lngustc vrbles n NCS envronment. Prmnk et l. [58] proposed the score nd ccurcy functons for NCSs nd prove ther bsc propertes. In the sme study, Prmnk et l. [58] developed strtegy for rnkng of neutrosophc cubc numbers (NCNs) bsed on the score nd ccurcy functons. In the sme study, Prmnk et l. [58] frst developed TODI (Tomd de decso ntertv e multcrtévo), clled the NC-TODI nd presented new NC-TODI [58] strtegy for solvng GD n NCS envronment. Sh nd Ye [59] ntroduced Domb ggregton opertors of NCSs nd ppled them for D problem. Prmnk et l. [60] proposed n extended technque for order preference by smlrty to del soluton (TOPSIS) strtegy n NCS envronment for solvng D problem. Ye [6] present opertons nd ggregton method of neutrosophc cubc numbers for D. Prmnk et l. [6] presented some opertons nd propertes of neutrosophc cubc soft set. Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment

2 Neutrosophc Sets nd Systems, Vol., 08 Oprcovc [6] proposed the VIKOR strtegy for mult crter decson mkng (CD) problem wth conflctng crter [6-65]. In 05, Busys nd Zvdsks [66] extended the VIKOR strtegy to INS envronment nd ppled t to solve CD problem. Further, Hung et l. [67] proposed VIKOR strtegy for ntervl neutrosophc GD. Pouresmel et l. [68] proposed n GD strtegy bsed on TOPSIS nd VIKOR n SVNS envronment. Lu nd Zhng [69] extended VIKOR strtyegy n neutrosophc hestnt fuzzy set envronment. Hu et l. [70] proposed ntervl neutrosophc projecton bsed VIKOR strtegy nd employed t for doctor selecton. Selvkumr et l. [7] proposed VIKOR strtegy for decson mkng problem usng octgonl neutrosophc soft mtrx. Prmnk et l. [7] proposed VIKOR bsed GD strtegy under bpolr neutrosophc set envronment. The remnder of the pper s orgnzed s follows: In the secton, we revew some bsc concepts nd opertons relted to NS, SVNS, NCS. In Secton, we present modfed NC-VIKOR strtegy to solve the GD problems n NCS envronment. In Secton, we solve n llustrtve exmple usng the modfed NC- VIKOR n NCS envronment. Then, n Secton 5, we present the senstvty nlyss. In Secton 6, we present conlcuson nd future scope reserch.. Prelmnres Defnton. Sngle vlued neutrosophc set Let X be spce of ponts (objects) wth generc element n X denoted by x. sngle vlued neutrosophc set [] B n X s expressed s: B = {< x: ( T B(x), I B (x), F B (x) )>: xx}, where T B(x), I B (x), F B(x) [0, ]. For ech xx, T B(x), I B (x), F B (x) [0, ] nd 0 T B(x) + I B (x) + F B(x). Defnton. Intervl neutrosophc set n ntervl neutrosophc set [50] (x) of nonempty set X s expressed by truth-membershp functon T (x), the ndetermncy membershp functon I (x) nd flsty membershp functon F (x). For ech xx, T (x), I (x), (x) F (x) [0, ] nd defned s follows: = {< x, [T (x),t (x)], [I (x),i (x)], I, (x), I (x) Here, we consder, F (x) F (x) : X ] 0, [ nd T (x), T (x) Defnton. Neutrosophc cubc set, I (x), I (x), [F (x),f (x)] x X}. Here,, T (x), T (x). 0 sup T (x) sup I (x) sup F (x), F (x) F (x) : X [0, ] for rel pplctons. neutrosophc cubc set [5] n non-empty set X s defned s N = {< x, (x), (x) >: xx}, where ~ nd re the ntervl neutrosophc set nd neutrosophc set n X respectvely. For convenence, we cn smply use N = < ~, > to represent n element N n neutrosophc cubc set nd the element N cn be clled neutrosophc cubc number (NCN). Some opertons of neutrosophc cubc sets: [5]. Unon of ny two neutrosophc cubc sets Let N (x), (x) nd N (x), (x) be ny two neutrosophc cubc sets n non-empty set H. Then the unon of N nd N denoted by N N s defned s follows: N N (x) (x), (x) (x), xx, where, Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment

3 Neutrosophc Sets nd Systems, Vol., 08 (x) = {< x, [mx{ (x) { I (x), I (x) }], [mn { F (x), T (x), mx { T (x), T (x) }, mn { I (x) T (x) },mx { T (x),. Intersecton of ny two neutrosophc cubc sets T (x) }], [mn { (x) I, I (x) }, mn F (x) }, mn { F (x), F (x) }]>: xx} nd (x) (x) = {< x,, I (x) }, mn { F (x), F (x) }>: xx}. Intersecton of N nd N denoted by N N s defned s follows: N N = (x) (x), (x) (x) xx, where (x) (x) = {< x, [mn { T (x), T (x) }, mn { T (x), T (x) { F (x), }], [mx { (x) I, I (x) }, mx { I (x), I (x) }], [mx { F (x), F (x) }]>: xx} nd (x) (x) = {< x, mn { T (x), T (x) mx { F (x), F (x) }>: xx}.. Complement of neutrosophc cubc set Let N (x), (x) be n NCS n X. Then complment of N x, c ~ (x), Here, c (x)>: xx}. c ~ = {< x, [ where, T (x) = {} - c F (x) = {} - c F c(x) = {} - v. Contnment F (x), F (x). T (x), c T (x), F (x) c T (x) ], [ c T (x) = {} - c = {} - I (x), c T (x), F (x), nd I (x) ], [ c I (x) c F }, mx { (x) I (x), (x) s denoted by F (x) c = {} - T (x) = {} - c Let ~ N, = {< x, [ T (x), T (x) ], [I (x),i (x)] ~ N, = {< x, [ T (x), T (x) ], [I (x),i (x)] ny two neutrosophc cubc sets n non-empty set X, then, () N N f nd only f F (x) F (x), F (x) (x) T (x) T (x),, I (x), F (x) c I (x) = {} - c T (x), c I (x) = {} -,, T (x) T (x), (x) }, mx, I (x) }, c N = {< ]>: x X}, I (x), I (x), T (x),i (x),f (x) >: xx} nd T (x),i (x),f (x) >: x X} be I (x) (x) I, I (x) I (x), F, nd T (x) T (x), I (x) I (x), F (x) F (x) for ll xx. Defnton. Dstnce between two NCNs Let N = < [, ], [b, b ], [c, c ], (, b, c) > nd N = < [d, d ], [e, e ], [f, f ], (d, e, f) > be ny two NCnumbers, then dstnce [58] between them s defned by H (N, N ) = [ d d b e b e c f c f d b e c f ] () 9 Defnton 5. Procedure of normlzton In generl, beneft type ttrbutes nd cost type ttrbutes cn exst smultneously n GD problem. Therefore the decson mtrx must be normlzed. Let j be n NC-number to express the rtng vlue of -th lterntve wth respect to j-th ttrbute ( j ). When ttrbute j C or Ψ j G (where C nd G be the set of Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment.

4 Neutrosophc Sets nd Systems, Vol., 08 cost type ttrbutes nd set of beneft type ttrbutes respectvely), the normlzed vlues for cost type ttrbute nd beneft type ttrbute re clculted by usng the followng expresson (). f G * j j j () j f j C where j s the performnce rtng of th lterntve for ttrbute j.. VIKOR strtegy for solvng GD problem n NCS envronment In ths secton, we propose modfed NC-VIKOR strtegy fro n GD strtegy n NCS envronment. ssume tht Φ {Φ, Φ, Φ,..., Φ r } be set of r lterntves nd {,,,..., s } be set of s ttrbutes. ssume tht W {w, w, w,..., w } be the weght vector of the ttrbutes, where w 0 s k nd w k. ssume tht s E {E, E, E,..., E } be the set of decson mkers nd ζ {ζ,ζ,ζ,...,ζ } be the set of weght vector of decson mkers, where p 0 nd p. p The proposed GD strtegy conssts of the followng steps: Step:. Constructon of the decson mtrx Let D p p = ( j ) (p =,,,, t) be the p-th decson mtrx, where nformton bout the lterntve r s Φ provded by the decson mker or expert E wth respect to ttrbute j (j =,,,, s). The p-th decson mtrx denoted by p D (See Equton ()) s constructed s follows:... s p p p p... s D p p p s..... p p p.... r r r rs Here p =,,,, ; =,,,, r; j =,,,, s. Step:. Normlzton of the decson mtrx p We use Equton () for normlzng the cost type ttrbutes nd beneft type ttrbutes. fter normlzton, the normlzed decson mtrx (Equton ()) s represented s follows (see Equton ):. r * p r * p r * s p rs * p * p * p s p * p * p * p D s () Here, p =,,,, ; =,,,, r; j =,,,, s. Step:. ggregted decson mtrx p For group decson, we ggregte ll the ndvdul decson mtrces ( D, p,,..., ) to n ggregted decson mtrx (D) usng the neutrosophc cubc numbers weghted ggregton (NCNW) [7] opertor s follows: k () Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment

5 Neutrosophc Sets nd Systems, Vol., 08 5 j ζ j j j j j j j NCNW (,,..., ) (ζ ζ ζ... ζ ) = [ ζ p j, ζ p j ],[ ζ p j, ζ T T I pij ], p p p p [ ζ p j, ζ p j ],( ζ p j, ζ p j, ζ p j ] F F T I F p p p p p (5) Therefore, the ggregted decson mtrx s defned s follows: Ψ Ψ.....Ψs Φ... s D Φ s.. Φr r r... rs Here, =,,,, r; j =,,,, s; p =,,.,. (6) Step:. Defne the postve del soluton nd negtve del soluton j [mx j, mx t tj],[mn j, mn j ],[mn f j, mn j ],(mx t j,mn f j, mn f j) (7) j [mn j, mn t tj],[mx j,mx j ],[mx f j, mx j ],(mn t j, mx f j, mx f j) (8) Step: 5. Compute nd nd Γ Z Z represent the verge nd worst group scores for the lterntve respectvely wth the reltons * s wj D( j, j) (9) j D( j, j ) * w j D(j,j ) Z mx (0) j D(j,j ) Here, w j s the weght of Ψ j. The smller vlues of nd respectvely. Z correspond to the better verge nd worse group scores for lterntve, Step: 6. Clculte the vlues of ( =,,,, r) (Γ Γ ) (Z Z ) φ γ ( γ) (Γ Γ ) (Z Z ) Here, Γ mn Γ, Γ mxγ, Z mn Z, Z mx Z () nd depcts the decson mkng mechnsm coeffcent. If 0. 5, t s for the mxmum group utlty ; If 0.5, t s the mnmum regret, nd t s both f γ 0.5. Step: 7. Rnk the prorty of lterntves Rnk the lterntves by, nd Z ccordng to the rule of trdtonl VIKOR strtegy. The smller vlue reflects the better lterntve. () Shyml Dlpt, Surpt Prmnk, Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment.

6 6 Neutrosophc Sets nd Systems, Vol., 08 Step: 8. Determne the compromse soluton Obtn lterntve Φ s compromse soluton, whch s rnked s the best by the mesure φ (nmum) f the followng two condtons re stsfed: Condton. cceptble stblty: φ( Φ ) φ( Φ ), where Φ, Φ re the lterntves wth frst nd (r ) second poston n the rnkng lst by φ ; r s the number of lterntves. Condton. cceptble stblty n decson mkng: lterntve Φ must lso be the best rnked by or/nd Z. Ths compromse soluton s stble wthn whole decson mkng process. If one of the condtons s not stsfed, then set of compromse solutons s proposed s follows: lterntves Φ nd Φ re compromse solutons f only condton s not stsfed, or r r Φ, Φ, Φ,, Φ re compromse solutons f condton s not stsfed nd Φ s decded by r constrnt φ( Φ ) φ( Φ ) for mxmum r. (r ). Illustrtve exmple To demonstrte the fesblty, pplcblty nd effectveness of the proposed strtegy, we solve n GD problem dpted from [7]. We ssume tht n nvestment compny wnts to nvest sum of money n the best opton. The nvestment compny forms decson mkng bord comprsng of three members (E, E, E ) who evlute the four lterntves to nvest money. The lterntves re Cr compny ( ), Food compny ( ), Computer compny ( ) nd rms compny ( ). Decson mkers tke decson to evlute lterntves bsed on the ttrbutes nmely, rsk fctor ( ), growth fctor ( ), envronment mpct ( ). We consder three crter s beneft type bsed on Prmnk et l. [58]. ssume tht the weght vector of ttrbutes s T W (0.6, 0.7, 0.7) nd weght vector of decson mkers or experts s the modfed NC-VIKOR strtegy usng the followng steps. T (0.6,0.0,0.). Now, we pply Step:. Constructon of the decson mtrx We construct the decson mtrces s follows: Decson mtrx for D n NCN form < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.6,.8],[.,.],[.,.], (.8,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.6,.8],[.,.],[.,.], (.8,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.,.],[.5,.6],[.5,.7], (.,.6,.7) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > Decson mtrx for D n NCN form < [.,.],[.5,.6],[.5,.7], (.,.6,.7) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > Decson mtrx for D n NC-number form < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.,.5],[.,.5],[.,.5], < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.6,.8],[.,.],[.,.], (.8,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.6,.8],[.,.],[.,.], (.8,.,.) > < [.6,.8],[.,.],[.,.], (.8,.,.) > < [.7,.9],[.,.],[.,.], (.9,.,.) > < [.,.5],[.,.5],[.,.5], (.5,.5,.5) > < [.,.],[.5,.6],[.5,.7], (.,.6,.7) > () () (5) Step:. Normlzton of the decson mtrx Snce ll the crter re consdered s beneft type, we do not need to normlze the decson mtrces (D, D, D ). Shyml Dlpt, Surpt Prmnk, Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment

7 Neutrosophc Sets nd Systems, Vol., 08 7 Step:. ggregted decson mtrx Usng equton eq. (5), the ggregted decson mtrx of (,, 5) s presented below: Ψ Ψ Ψ Φ <[.,.56], [.6,.6], [.6,.5], (.56,.6,.50)> <[.8,.60], [.,.], [.,.], (.60,.,.)> <[.6,.80], [.8,.8], [.8,.8], (.80,.8,.8)> Φ <[.5,.58], [.5,.5], [.5,. 7], (.58,.5,.7)> <[.50,.6], [.0,.0], [.0,.0], (.6,.0,.0)> <[.60,.76], [.0,.0], [.0,.0], (.76,.0,.0)> Φ <[.6,.80], [.8,.8], [.8,.8], (.80,.8,.8)> <[.6,.8], [.6,.6], [.6,.], (.8,.6,.)> <[.7,.60], [.,.], [.,.7], (.60,.,.7)> Φ <[.56,.7], [.,.], [.,.], (.7,.,.)> <[.0,.50], [.0,.50], [.0,.50], (.50,.50,.50)> <[.56,.7], [.,.], [.,.7], (.7,.,.7)> Step:. Defne the postve del soluton nd negtve del soluton The postve del soluton j = Ψ Ψ Ψ <[.6,.80], [.8,.8], [.8,.8], (.80,.8,.8)> <[.6,.8], [.6,.6], [.6,.], (.8,.6,.)> <[.6,.80], [.8,.8], [.8,.8], (.80,.8,.8)> nd the negtve del soluton j = Ψ Ψ Ψ <[.,.56], [.6,.6], [.6,.5], (.56,.6,.50)> <[.0,.50], [.0,.50], [.0,.50], (.50,.50,.50)> <[.7,.60], [.,.], [.,.], (.60,.,.7)> Step: 5. Compute nd Z Usng Equton (9) nd Equton (0), we obtn , , , nd Z mx,, 0., Z mx,, 0., Z mx,, 0., Step: 6. Clculte the vlues of Usng Equtons (), () nd 0. 5, we obtn Z mx,, (0. 0.) (0. 0.) (0. 0.) (0. 0.) φ , , (0. 0.) (0. 0.) ( ) (0.7 0.) , Step 7. Rnk the prorty of lterntves The preference rnkng order of the lterntves s presented n Tble Rnkng order Best lterntve (6) Z ( 0.5) Tble Preference rnkng order nd compromse soluton bsed on, Z nd Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment.

8 8 Neutrosophc Sets nd Systems, Vol., 08 Step 8. Determne the compromse soluton The preference rnkng order bsed on n decresng order nd lterntve wth best poston s wth ( ) = 0.0, nd second best poston wth ( ) = 0.. Therefore, ( ) ( ) (snce, r = ; /(r-) = 0.), whch does not stsfy the condton ( φ( Φ ) φ( Φ ) ), but lterntve s the best rnked by, Z, whch stsfes the condton. (r ) Therefore, we obtn the compromse soluton s follows: ( ) ( ) 0. 0., ( ) ( ) 0. 0., ( ) ( ) So,, re compromse solutons. 5. The nfluence of prmeter Tble shows how the rnkng order of lterntves ( ) chnges wth the chnge of the vlue of Tble. Vlues of ( =,,, ) nd rnkng of lterntves for dfferent vlues of. Vlues of Vlues of Preference order of lterntves = 0. = 0., = 0.0, = 0.6, = = 0. = 0., = 0.08, = 0.55, = = 0. = 0.6, = 0., = 0.8, = = 0. = 0.9, = 0.6, = 0., = = 0.5 = 0., = 0., = 0., = = 0.6 = 0., = 0., = 0.8, = = 0.7 = 0.6, = 0.8, = 0., = = 0.8 = 0.9, = 0., = 0., = = 0.9 = 0., = 0.6, = 0.07, = 6. Concluson In ths rtcle, we hve presented modfed NC-VIKOR strtegy to overcome the shortcomngs of obtnng compromse soluton [7]. In the modfed NC-VIKOR strtgey, we hve ncorported the technque of determnng compromse soluton. Fnlly, we solve n GD problem to show the fesblty, pplcblty nd effcency. We present senstvty nlyss to show the mpct of dfferent vlues of the decson mkng mechnsm coeffcent on rnkng order of the lterntves. Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment

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10 0 Neutrosophc Sets nd Systems, Vol., 08 [0] R. Shn, nd. Kucuk. Subsethood mesure for sngle vlued neutrosophc sets. Journl of Intellgent nd Fuzzy System, (0). do:0./ifs-0. [] R. Shn, nd P. Lu. xmzng devton method for neutrosophc multple ttrbute decson mkng wth ncomplete weght nformton. Neurl Computng nd pplctons, (05), do: 0.007/s [] P. Bsws. ult-ttrbute decson mkng n neutrosophc envronment. Ph. D. Thess. Jdvpur Unversty (07), Ind. [] J. Ye. ultcrter decson-mkng method usng the correlton coeffcent under sngle-vlued neutrosophc envronment. Interntonl Journl of Generl Systems, (0), [] J. Ye. Sngle vlued neutrosophc cross-entropy for mult crter decson mkng problems. ppled themtcl odellng, 8 () (0), [5] J. Ye. mult crter decson-mkng method usng ggregton opertors for smplfed neutrosophc sets. Journl of Intellgent nd Fuzzy Systems, 6 (0), [6] F. Smrndche, nd S. Prmnk, (Eds). New trends n neutrosophc theory nd pplctons. Brussels: Pons Edtons, 06. [7] F. Smrndche, nd S. Prmnk, (Eds). New trends n neutrosophc theory nd pplctons. Vol. II. Brussels: Pons Edtons, 08. [8] J. Ye. Projecton nd bdrectonl projecton mesures of sngle vlued neutrosophc sets nd ther decson mkng method for mechncl desgn scheme. Journl of Expermentl nd Theoretcl rtfcl Intellgence, (06). do:0.080/0958x [9] S. Prmnk, S. Dlpt, S. lm, F. Smrndche, nd T. K. Roy. NS-cross entropy-bsed GD under snglevlued neutrosophc set envronment. Informton, 9() (08), 7. [0] K. ondl, nd S. Prmnk. ult-crter group decson mkng pproch for techer recrutment n hgher educton under smplfed Neutrosophc envronment. Neutrosophc Sets nd Systems, 6 (0), 8-. [] K. ondl, nd S. Prmnk. Neutrosophc decson mkng model of school choce. Neutrosophc Sets nd Systems, 7 (05), [] H. D. Cheng, nd Y. Guo. new neutrosophc pproch to mge thresholdng. New themtcs nd Nturl Computton, (008), [] Y. Guo, nd H. D. Cheng. New neutrosophc pproch to mge segmentton. Pttern Recognton, (009), [] Y. Guo,. Sengur, nd J. Ye. novel mge thresholdng lgorthm bsed on neutrosophc smlrty score. esurement, 58 (0), [5]. Zhng, L. Zhng, nd H. D. Cheng. neutrosophc pproch to mge segmentton bsed on wtershed method. Sgnl Processng, 90 (5) (00), [6] J. Ye. Improved cosne smlrty mesures of smplfed neutrosophc sets for medcl dgnoses. rtfcl Intellgence n edcne, 6 (05), [7] S. Ye, nd J. Ye. Dce smlrty mesure between sngle vlued neutrosophc mult sets nd ts pplcton n medcl dgnoss. Neutrosophc Sets nd System, 6 (0), 9-5. [8] K. ondl, nd S. Prmnk. study on problems of Hjrs n West Bengl bsed on neutrosophc cogntve mps. Neutrosophc Sets nd Systems, 5(0), -6. [9] S. Prmnk, nd S. Chkrbrt. study on problems of constructon workers n West Bengl bsed on neutrosophc cogntve mps. Interntonl Journl of Innovtve Reserch n Scence. Engneerng nd Technology, () (0), [50] H. Wng, F. Smrndche, Y. Q. Zhng, nd R. Sunderrmn. Intervl Neutrosophc Sets nd Logc: Theory nd pplctons n Computng; Hexs: Phoenx, Z, US, 005. [5] K. ondl, S. Prmnk, nd B. C. Gr. Intervl neutrosophc tngent smlrty mesure bsed D strtegy nd ts pplcton to D problems. Neutrosophc Sets nd Systems, 9 (08), [5] S. Dlpt, S. Prmnk, S. lm, F. Smrndche, nd T. K. Roy. IN-cross entropy bsed GD strtegy under ntervl neutrosophc set envronment. Neutrosophc Sets nd Systems, 8 (07), -57. [5]. l, I. Del, nd F. Smrndche. The theory of neutrosophc cubc sets nd ther pplctons n pttern recognton. Journl of Intellgent & Fuzzy Systems, 0 () (06), [5] J. Zhn,. Khn,. Gulstn, nd. l. pplctons of neutrosophc cubc sets n mult-crter decson mkng. Interntonl Journl for Uncertnty Quntfcton, 7(5) (07), [55] D. Bnerjee, B. C. Gr, S. Prmnk, nd F. Smrndche. GR for mult ttrbute decson mkng n neutrosophc cubc set envronment. Neutrosophc Sets nd Systems, 5 (07), [56] Z. Lu, nd J. Ye. Cosne mesures of neutrosophc cubc sets for multple ttrbute decson-mkng. Symmetry 9 (7) (07),. [57] S. Prmnk, S. Dlpt, S. lm, nd T. K. Roy, nd F. Smrndche. Neutrosophc cubc CGD method bsed on smlrty mesure. Neutrosophc Sets nd Systems, 6, (07), -56. [58] S. Prmnk, S. Dlpt, S. lm, nd T. K. Roy. NC-TODI-Bsed GD under Neutrosophc Cubc Set Envronment. Informton 8 () (07), 9. [59] L. Sh, nd J. Ye. Domb ggregton opertors of neutrosophc cubc sets for multple ttrbute decson-mkng. lgorthms, (), 9, (08). do:0.90/0009 Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment

11 Neutrosophc Sets nd Systems, Vol., 08 [60] S. Prmnk, P. P. Dey, B. C. Gr, nd F. Smrndche. n extended TOPSIS for mult-ttrbute decson mkng problems wth neutrosophc cubc nformton. Neutrosophc Sets nd Systems, 7 (07), 0-8. [6] J. Ye. Opertons nd ggregton method of neutrosophc cubc numbers for multple ttrbute decson-mkng. [6] S. Prmnk, S. Dlpt, S. lm, nd T. K. Roy. Some opertons nd propertes of neutrosophc cubc soft set. Globl Journl of Reserch nd Revew, () (07). [6] S. Oprcovc. ultcrter Optmzton of Cvl Engneerng Systems; Fculty of Cvl Engneerng: Belgrde, Serb, 998. [6] S. Oprcovc, G. H. Tzeng. Compromse soluton by CD methods: comprtve nlyss of VIKOR nd TOPSIS. Europen journl of opertonl reserch, 56 (00), [65] S. Oprcovc, G. H. Tzeng. Extended VIKOR method n comprson wth outrnkng methods. Europen journl of opertonl reserch, 78 () (007), [66] R. Busys, nd E. K. Zvdsks. ultcrter decson mkng pproch by VIKOR under ntervl neutrosophc set envronment. Economc Computton nd Economc Cybernetcs Studes nd Reserch, (05), -8. [67] Y. Hung, G. We, nd C. We. VIKOR method for ntervl neutrosophc multple ttrbute group decson-mkng. Informton, 8 (07),. do:0.90/nfo800. [68] H. Pouresmel, E. Shvnn, E. Khorrm, nd H. S. Fthbd. n extended method usng TOPSIS nd VIKOR for multple ttrbute decson mkng wth multple decson mkers nd sngle vlued neutrosophc numbers. dvnces nd pplctons n Sttstcs, 50 () (07), 6. [69] P. Lu, L. Zhng. The extended VIKOR method for multple crter decson mkng problem bsed on neutrosophc hestnt fuzzy set, (0). [70] J. Hu, L. Pn, nd X. Chen. n ntervl neutrosophc projecton-bsed VIKOR method for selectng doctors. Cogntve Computton, 9(6) (07), [7] K. Selvkumr, nd.. Prydhrshn. VIKOR method for decson mkng problem usng octgonl neutrosophc soft mtrx. Interntonl Journl of Ltest Engneerng Reserch nd pplctons, (7) (07), -5. [7] S. Prmnk, S. Dlpt, S. lm, nd T. K. Roy. VIKOR bsed GD strtegy under bpolr neutrosophc set envronment. Neutrosophc Sets nd Systems, 9 (08), [7] S. Prmnk, S. Dlpt, S. lm, nd T. K. Roy. NC-VIKOR bsed GD under Neutrosophc Cubc Set Envronment. Neutrosophc Sets nd Systems, 0 (08), [7] J. Ye. Smlrty mesures between ntervl neutrosophc sets nd ther pplctons n mult crter decson-mkng. Journl of Intellgent nd Fuzzy Systems, 6 (0), Receved: ugust 6, 08. ccepted: ugust 9, 08. Shyml Dlpt, Surpt Prmnk. Revst to NC-VIKOR Bsed GD Strtegy under Neutrosophc Cubc Set Envronment.