Article Cosine Measures of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making

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1 Article Cosie Measures of Neutrosophic Cubic Sets for Multiple Attribute Decisio-Makig Zhikag Lu ad Ju Ye * Departmet of Electrical ad Iformatio Egieerig, Shaoxig Uiversity, 508 Huacheg West Road, Shaoxig 32000, Chia; luzhikag@usx.edu.c * Correspodece: yeju@usx.edu.c Received: 26 Jue 207; Accepted: July 207; Published: 8 July 207 Abstract: The eutrosophic cubic set ca cotai much more iformatio to express its iterval eutrosophic umbers ad sigle-valued eutrosophic umbers simultaeously i idetermiate eviromets. Hece, it is a usual tool for expressig much more iformatio i complex decisio-makig problems. Ufortuately, there has bee o research o similarity measures of eutrosophic cubic sets so far. Sice the similarity measure is a importat mathematical tool i decisio-makig problems, this paper proposes three cosie measures betwee eutrosophic cubic sets based o the icluded agle cosie of two vectors, distace, ad cosie fuctios, ad ivestigates their properties. The, we develop a cosie measures-based multiple attribute decisio-makig method uder a eutrosophic cubic eviromet i which, from the cosie measure betwee each alterative (each evaluated eutrosophic cubic set) ad the ideal alterative (the ideal eutrosophic cubic set), the rakig order of alteratives ad the best optio ca be obtaied, correspodig to the cosie measure values i the decisio-makig process. Fially, a illustrative example about the selectio problem of ivestmet alteratives is provided to illustrate the applicatio ad feasibility of the developed decisio-makig method. Keywords: eutrosophic cubic set; decisio-makig; similarity measure; cosie measure; iterval eutrosophic set; sigle-valued eutrosophic set. Itroductio The classic fuzzy set, as preseted by Zadeh [], is oly described by the membership degree i the uit iterval [0, ]. I the real world, it is ofte difficult to express the value of a membership fuctio by a exact value i a fuzzy set. I such cases, it may be easier to describe vagueess ad ucertaity i the real world usig both a iterval value ad a exact value, rather tha uique iterval/exact values. Thus, the hybrid form of a iterval value ad a exact value may be a very useful expressio for a perso to describe certaity ad ucertaity due to his/her hesitat judgmet i complex decisio-makig problems. For this purpose, Ju et al. [2] itroduced the cocept of (fuzzy) cubic sets, icludig iteral cubic sets ad exteral cubic sets, by the combiatio of both a iterval-valued fuzzy umber (IVFN) ad a fuzzy value, ad defied some logic operatios of cubic sets, such as the P-uio, P-itersectio, R-uio, ad R-itersectio of cubic sets. Also, Ju ad Lee [3] ad Ju et al. [4 6] applied the cocept of cubic sets to BCK/BCI-algebras ad itroduced the cocepts of cubic subalgebras/ideals, cubic o-subalgebras ad closed cubic ideals i BCK/BCI-algebras. However, the cubic set is described by two parts simultaeously, where oe represets the membership degree rage by the iterval value ad the other represets the membership degree by a fuzzy value. Hece, a cubic set is the hybrid set combied by both a IVFN ad a fuzzy value. Obviously, the advatage of the cubic set is that it ca cotai much more iformatio to express the IVFN ad fuzzy value simultaeously. Symmetry 207, 9, 2; doi:0.3390/sym

2 Symmetry 207, 9, 2 2 of 0 As the geeralizatio of fuzzy sets [], iterval-valued fuzzy sets (IVFSs) [7], ituitioistic fuzzy sets (IFSs) [8], ad iterval-valued ituitioistic fuzzy sets (IVIFSs) [9], Smaradache [0] iitially itroduced a cocept of eutrosophic sets to express icomplete, idetermiate, ad icosistet iformatio. As simplified forms of eutrosophic sets, Smaradache [0], Wag et al. [,2] ad Ye [3] itroduced sigle-valued eutrosophic sets (SVNSs) ad iterval eutrosophic sets (INSs), ad simplified eutrosophic sets (SNSs) as subclasses of eutrosophic sets for easy egieerig applicatios. Sice the, SVNSs, INSs, ad SNSs have bee widely applied to various areas, such as image processig [4 6], decisio-makig [7 32], clusterig aalyses [33,34], medical diagoses [35,36], ad fault diagoses [37]. Recetly, Ali et al. [38] ad Ju et al. [39] have exteded cubic sets to the eutrosophic sets ad proposed the cocepts of eutrosophic cubic sets (NCSs), icludig iteral NCSs ad exteral NCSs, subsequetly itroducig some logic operatios of NCSs, such as the P-uio, P-itersectio, R-uio, ad R-itersectio of NCSs. Furthermore, Ali et al. [38] itroduced a distace measure betwee NCSs ad applied it to patter recogitio. Subsequetly, Baerjee et al. [40] further preseted a multiple attribute decisio-makig (MADM) method with NCSs based o grey relatioal aalysis, i which they itroduced the Hammig distaces of NCSs for weighted grey relatioal coefficiets ad stadard (ideal) grey relatioal coefficiets, ad the gave the relative closeess coefficiets i order to rak the alteratives. From the above review, we ca see that the existig literature maily focus o the theoretical studies of cubic sets ad NCSs, rather tha the studies o their similarity measures ad their applicatios. O the other had, the NCS cotais much more iformatio tha the geeral eutrosophic set (INS/SVNS) because the NCS is expressed by the combied iformatio of both INS ad SVNS. Hece, NCSs used for attribute evaluatio i decisio makig may show its ratioality ad affectivity sice geeral eutrosophic decisio-makig methods with INSs/SVNSs may lose some useful evaluatio iformatio (either INSs or SVNSs) of attributes, which may affect decisio results, resultig i the distortio pheomeo. Moreover, the similarity measure is a importat mathematical tool i decisio-makig problems. Curretly, sice there is o study o similarity measures of cubic sets ad NCSs uder a eutrosophic cubic eviromet, we eed to develop ew similarity measures for NCSs for MADM problems with eutrosophic cubic iformatio, sice the cubic set is a special case of the NCS. For these reasos, this paper aims to propose three cosie measures betwee NCSs based o the icluded agle cosie of two vectors, distace, ad cosie fuctio, ad their MADM method i a eutrosophic cubic eviromet. The remaider of the article is orgaized as follows. Sectio 2 briefly describes some cocepts of cubic sets ad NCSs. Sectio 3 presets three cosie measures of NCSs ad discusses their properties. I Sectio 4, we develop a MADM approach based o the cosie measures of NCSs uder a eutrosophic cubic eviromet. I Sectio 5, a illustrative example about the selectio problem of ivestmet alteratives is provided to illustrate the applicatio ad feasibility of the developed method. Sectio 6 cotais coclusios ad future research. 2. Some Basic Cocepts of Cubic Sets ad NCSs By the combiatio of a fuzzy value ad a IVFN, Ju et al. [2] defied a (fuzzy) cubic set. A cubic set S i a uiverse of discourse X is costructed as follows [2]: where T( x) [ T ( x), T ( x)] S { x, T( x), ( x) x X}, is a IVFN for x X ad μ is a fuzzy value for x X. The, we call (i) S { x, T( x), ( x) x X} a iteral cubic set if T ( x) ( x) T ( x) for x X; a exteral cubic set if ( x) T ( x), T ( x) (ii) S { x, T( x), ( x) x X} for x X. The, Ali et al. [38] ad Ju et al. [39] proposed a NCS based o the combiatio of a iterval eutrosophic umber (INN) ad a sigle-valued eutrosophic umber (SVNN) as the extesio of the (fuzzy) cubic set. A NCS S i X is costructed as the followig form [38,39]:

3 Symmetry 207, 9, 2 3 of 0 P { x, T( x), U( x), F( x), t( x), u( x), f ( x) x X}, where <T(x), U(x), F(x)> is a INN, ad T( x) [ T ( x), T ( x)] [0, ], U( x) [ U ( x), U ( x)] [0, ], ad F( x) [ F ( x), F ( x)] [0, ] for x X are the truth-iterval, idetermiacy-iterval, ad falsity-iterval, respectively; the <t(x), u(x), f(x)> is a SVNN, ad t(x), u(x), f(x) [0, ] for x X are the truth, idetermiacy, ad falsity degrees, respectively. A NCS P { x, T( x), U( x), F( x), t( x), u( x), f ( x) x X} is said to be [38,39]: (i) A iteral NCS P { x, T( x), U( x), F( x), t( x), u( x), f ( x) x X} if T ( x) t( x) T ( x), U ( x) u( x) U ( x), ad F ( x) f ( x) F ( x) for x X; (ii) A exteral NCS P { x, T( x), U( x), F( x), t( x), u( x), f ( x) x X} if, u( x) U ( x), U ( x), ad f ( x) F ( x), F ( x) t( x) T ( x), T ( x) for x X. For coveiece, a basic elemet ( x, T( x), U( x), F( x), t( x), u( x), f ( x) ) i a NCS P is simply deoted by p = (<T, U, F>, <t, u, f>), which is called a eutrosophic cubic umber (NCN), where T, U, F [0, ] ad t, u, f [0, ], satisfyig 0 T ( x) U ( x) F ( x) 3 ad 0 t + u + f 3. Let p = (<T, U, F>, <t, u, f>) ad p2 = (<T2, U2, F2>, <t2, u2, f2>) be two NCNs. The, there are the followig relatios [38,39]: c () p F, F, U, U, T, T, f, u, t (complemet of p); (2) p p2 if ad oly if T T, U U, F F, t t 2, u u, ad f f (P-order); 2 2 (3) p = p2 if ad oly if p2 p ad p p2, i.e., <T, U, F> = <T2, U2, F2> ad <t, u, f> = <t2, u2, f2>. 3. Cosie Measures of NCSs I this sectio, we propose three cosie measures betwee NCSs. Defiitio. Let X ={x, x2,, x} be a fiite set ad two NCSs be P ={p, p2,, p} ad Q ={q, q2,, q}, where = (<T, U, F>, <t, u, f>) ad = (<T, U, F>, <t, u, f>) for j =, 2,, are two collectios of NCNs. The, three cosie measures of P ad Q are proposed based o the icluded agle cosie of two vectors, distace, ad cosie fuctio, respectively, as follows: () Cosie measure based o the icluded agle cosie of two vectors S ( P, Q) T T T T U U U U F F F F t t u u f f u f t u f ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) j ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) j t () (2) Cosie measure based o distace T T T T U U U U F F F F cos 2 S2( P, Q) 2 j t t u u f f cos 6 (2) (3) Cosie measure based o cosie fuctio

4 Symmetry 207, 9, 2 4 of 0 S ( P, Q) 3 T T T T t t 2 cos 2 cos 8 4 U U U U u u 2 cos 2 3( 2 ) j 8 2 cos 3( 2 ) j 4 F F F F f f 2 cos 8 2 cos 4 Obviously, the three cosie measures Sk(P, Q) (k =, 2, 3) satisfy the followig properties (S) (S3): (3) (S) 0 Sk(P, Q) ; (S2) Sk(P, Q) = Sk(Q, P); (S3) Sk(P, Q) = if P = Q, i.e., <T, U, F>, = <T, U, F> ad <t, u, f> = <t, u, f>. Proof. Firstly, we prove the properties (S) (S3) of S(P, Q). (S) The iequality S(P, Q) 0 is obvious. The, we oly prove S(P, Q). Based o the Cauchy Schwarz iequality: x y x2 y2 x y x x2 x y y2 y, where (x, x2,, x) R ad (y, y2,, y) R, we ca give the followig iequality: x y x2 y2 x y x x2 x y y2 y. Accordig to the above iequality, we have the followig iequality: T T T T U U U U F F F F ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) ( T ) ( T ) ( U ) ( U ) ( F ) ( F ), Hece, there is the followig result: t t uu f f t u f t u f. T T T T U U U U F F F F, ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) j ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) t t u u f f j t u f t u f Based o Equatio (), we have S(P, Q). Hece, 0 S(P, Q) holds. (S2) It is straightforward. (S3) If P = Q, there are <T, U, F> = <T, U, F> ad <t, u, f> = <t, u, f>. Thus T = T, U = U, F = F, t = t, u = u, ad f = f for j =, 2,,. Hece S(P, Q) = holds. Secodly, we prove the properties (S) (S3) of S2(P, Q).

5 Symmetry 207, 9, 2 5 of 0 (S) Let x T T T T U U U U F F F F x t /3 2 t u u f f /6 ad. It is obvious that there exist 0 x ad 0 x2. Thus, there are 0 cos(xπ/2) ad 0 cos(x2π /2). Hece, 0 S2(P, Q) holds. (S2) It is straightforward. (S3) If P = Q, there are <T, U, F> = <T, U, F> ad <t, u, f> = <t, u, f>. Thus T = T, U = U, F = F, t = t, u = u, ad f = f for j =, 2,,. Hece, S2(P, Q) = holds. Thirdly, we prove the properties (S) (S3) of S3(P, Q). (S) Let y ( T T T T ) / 2, y2 ( U U U U) / 2, y3 ( F F F F ) / 2, y t t 4, y u u 5, ad y f f 6. Obviously, there exists yk + for k =, 2,..., 6. Thus, 22 cos(ykπ/4), ad the there exists 0 S3(P, Q). (S2) It is straightforward. (S3) If P = Q, there are <T, U, F> = <T, U, F> ad <t, u, f> = <t, u, f>. Thus T = T, U = U, F = F, t = t, u = u, ad f = f for j =, 2,,. Hece, S3(P, Q) = holds. Whe the weight of the elemets ad (j =, 2,, ) is take ito accout, w = {w, w2,, w} is give as the weight vector of the elemets ad (j =, 2,, ) with wj [0, ] ad wj. The, we have the followig three weighted cosie measures betwee P ad Q, respectively: j S w T T T T U U U U F F F F wj j ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) , ( P, Q) ( T ) ( T ) ( U ) ( U ) ( F ) ( F ) 2 t t u u f f w j j t u f t u f T T T T U U U U F F F F cos 2, Sw2( P, Q) wj 2 j t t u u f f cos 6 T T T T t t 2 cos 2 cos 8 4. U U U U u u Sw3( P, Q) wj 2 cos 2 3( 2 ) 8 2 cos j 4 F F F F f f 2 cos 8 2 cos 4 (4) (5) (6) It is obvious that the three cosie measures Swk(P, Q) (k=, 2, 3) also satisfy the followig properties (S)-(S3): (S) 0 Swk(P, Q) ; (S2) Swk(P, Q) = Swk(Q, P); (S3) Swk(P, Q) = if P = Q, i.e., <T, U, F> = <T, U, F> ad <t, u, f> = <t, u, f>. By similar proof ways, we ca prove the properties (S) (S3) for Swk(P, Q) (k =, 2, 3). Their proofs are omitted here.

6 Symmetry 207, 9, 2 6 of 0 4. Decisio-Makig Method Usig Cosie Measures I this sectio, we propose a MADM method by usig oe of three cosie measures to solve decisio-makig problems with eutrosophic cubic iformatio. I a MADM problem, let P = {P, P2,, Pm} be a set of m alteratives ad R = {R, R2,, R} be a set of attributes. The evaluatio value of a attribute Rj (j =, 2,, ) with respect to a alterative Pi (i =, 2,, m) is expressed by a NCN pij = (<Tij, Uij, Fij>, <tij, uj, fij>) (j =, 2,, ; i =, 2,, m), where T, U, F [0,] ij ij ij ad t, u, f [0,] ij ij ij. Therefore, all the evaluatio values expressed by NCNs ca be costructed as the eutrosophic cubic decisio matrix P = (pij)m. The, the weight vector of the attributes Rj (j =, 2,, ) is cosidered as w = (w, w2,, w), satisfyig wj [0, ] ad j w j. I this case, the proposed decisio steps are described as follows: Step : Establish a ideal solutio (ideal alterative) * * * * P { p, p,..., p } 2 by the ideal NCN * p max( T ), max( T ), mi( U ), mi( U ), mi( F ), mi( F ), max( t ), mi( u ), mi( f ) j ij ij ij ij ij ij ij ij ij i i i i i i i i i correspodig to the beefit type of attributes ad * p mi( T ), mi( T ), max( U ), max( U ), max( F ), max( F ), mi( t ), max( u ), max( f ) j ij ij ij ij ij ij ij ij ij i i i i i i i i i correspodig to the cost type of attributes. Step 2: Calculate the weighted cosie measure values betwee a alterative Pi (i =, 2,, m) ad the ideal solutio P * by usig Equatio (4) or Equatio (5) or Equatio (6) ad get the values of Sw(Pi, P * ) or Sw2(Pi, P * ) or Sw3(Pi, P * ) (i =, 2,, m). Step 3: Rak the alteratives i descedig order correspodig to the weighted cosie measure values ad select the best oe(s) accordig to the bigger value of Sw(Pi, P * ) or Sw2(Pi, P * ) or Sw3(Pi, P * ). Step 4: Ed. 5. Illustrative Example ad Compariso Aalysis I this sectio, a illustrative example of the selectio problem of ivestmet alteratives is provided i order to demostrate the applicatio of the proposed MADM method with eutrosophic cubic iformatio. 5.. Illustrative Example A ivestmet compay wats to ivest a sum of moey for oe of four potetial alteratives: (a) P is a textile compay; (b) P2 is a automobile compay; (c) P3 is a computer compay; (d) P4 is a software compay. The evaluatio requiremets of the four alteratives are o the basis of three attributes: (a) R is the risk; (b) R2 is the growth; (c) R3 is the evirometal impact; where the attributes R ad R2 are beefit types, ad the attribute R3 is a cost type. The weight vector of the three attributes is w = (0.32, 0.38, 0.3). Whe the expert or decisio maker is requested to evaluate the four potetial alteratives o the basis of the above three attributes usig the form of NCNs. Thus, we ca costruct the followig eutrosophic cubic decisio matrix: P [0.5,0.6],[0.,0.3],[0.2,0.4], 0.6,0.2,0.3 [0.5,0.6],[0.,0.3],[0.2,0.4], 0.6,0.2,0.3 [0.6,0.8],[0.2,0.3],[0.,0.2], 0.7,0.2,0.. [0.6,0.8],[0.,0.2],[0.2,0.3], 0.7,0.,0.2 [0.6,0.7],[0.,0.2],[0.2,0.3], 0.6, 0.,0.2 [0.6,0.7],[0.3,0.4],[0.,0.2], 0.7,0.4,0. [0.4,0.6],[0.2,0.3],[0.,0.3], 0.6,0.2,0.2 [0.5,0.6],[0.2,0.3],[0.3,0.4], 0.6,0.3,0.4 [0.5,0.7],[0.2,0.3],[0.3,0.4], 0.6,0.2,0.3 [0.7,0.8],[0.,0.2],[0.,0.2], 0.8,0.,0.2 [0.6,0.7],[0.,0.2],[0.,0.3], 0.7,0.,0.2 [0.6,0.7],[0.3,0.4],[0.2,0.3], 0.7,0.3,0.2 Hece, the proposed MADM method ca be applied to this decisio-makig problem with NCSs by the followig steps: Firstly, correspodig to the beefit attributes R, R2, ad the cost attribute R3, we establish a ideal solutio (ideal alterative):

7 Symmetry 207, 9, 2 7 of 0 [0.7, 0.8],[0., 0.2],[0., 0.2], 0.8, 0., 0.2, * * * * P { p, p2,..., p } [0.6,0.7],[0.,0.2],[0.,0.3], 0.7,0.,0.2,. [0.5, 0.7],[0.3, 0.4],[0.3, 0.4], 0.6, 0.4, 0.3 The, we calculate the weighted cosie measure values betwee a alterative Pi (i =, 2, 3, 4) ad the ideal solutio P * by usig Equatio (4) or Equatio (5) or Equatio (6), get the values of Sw(Pi, P * ) or Sw2(Pi, P * ) or Sw3(Pi, P * ) (i =, 2, 3, 4), ad rak the four alteratives, which are show i Table. Table. All the cosie measure values betwee Pi ad P * ad rakig orders of the four alteratives. Swk(Pi, P * ) Cosie Measure Value Rakig Order The Best Alterative Sw(Pi, P * ) , , , P4 > P2 > P3 > P P4 Sw2(Pi, P * ) , , , P4 > P2 > P3 > P P4 Sw3(Pi, P * ) , , , P4 > P2 > P3 > P P4 From the results of Table, we ca see that all the rakig orders of the four alteratives ad best choice retur the same results correspodig to the three cosie measures i the decisio-makig problem with eutrosophic cubic iformatio. It is obvious that P4 is the best oe Related Compariso For relative compariso, we compare our decisio-makig method with the oly existig related decisio-makig method based o the grey relatioal aalysis uder eutrosophic cubic eviromet [40]. Because the decisio-makig problem/method with CNS weights i [40] is differet from ours, which has exact/crisp weights, we caot compare them uder differet decisio-makig coditios. However, we oly gave the compariso of decisio-makig complexity to show our simple method. The proposed decisio-makig method based o the cosie measures of NCSs directly uses the cosie measures betwee a alterative Pi (i =, 2,, m) ad the ideal alterative (ideal solutio) P * to rak all the alteratives; while the existig decisio-makig method with NCSs itroduced i [40] firstly determies the Hammig distaces of NCSs for weighted grey relatioal coefficiets ad stadard (ideal) grey relatioal coefficiets, ad the derives the relative closeess coefficiets i order to rak the alteratives. It is obvious that our decisio-makig method is simpler ad easier tha the existig decisio-makig method with NCSs itroduced i [40]. But, our decisio-makig method ca oly deal with decisio-makig problems with exact/crisp weights, rather tha NCS weights [40]. Compared with existig related decisio-makig methods with geeral eutrosophic sets (INSs or SVNSs) [7 39], the proposed decisio-makig method with NCSs cotais much more evaluatio iformatio of attributes, which cosists of both INSs ad SVNSs; while the existig decisio-makig methods [7 39] cotai either INS or SVNS iformatio, which may lose some useful evaluatio iformatio of attributes i the decisio-makig process ad affect the decisio results, resultig i the distortio pheomeo. Furthermore, the existig decisio-makig methods [7 39] caot deal with the decisio-makig problem with NCSs Sesitive Aalysis To show the sesitivities of these cosie measures o the decisio results, we ca oly chage the iteral NCS of the alterative P4 ito the exteral NCS ad recostruct the followig eutrosophic cubic decisio matrix:

8 Symmetry 207, 9, 2 8 of 0 P ' [0.5,0.6],[0.,0.3],[0.2,0.4], 0.6,0.2,0.3 [0.5,0.6],[0.,0.3],[0.2,0.4], 0.6,0.2,0.3 [0.6,0.8],[0.2,0.3],[0.,0.2], 0.7,0.2,0.. [0.6,0.8],[0.,0.2],[0.2,0.3], 0.7,0.,0.2 [0.6,0.7],[0.,0.2],[0.2,0.3], 0.6,0.,0.2 [0.6,0.7],[0.3,0.4],[0.,0.2], 0.7,0.4,0. [0.4,0.6],[0.2,0.3],[0.,0.3], 0.6,0.2,0.2 [0.5,0.6],[0.2,0.3],[0.3,0.4], 0.6,0.3,0.4 [0.5,0.7],[0.2,0.3],[0.3,0.4], 0.6,0.2,0.3 [0.7,0.8],[0.,0.2],[0.,0.2 ], 0.9,0.3,0.3 [0.6,0.7],[0.,0.2],[0.,0.3], 0.8,0.3,0.4 [0.6,0.7],[0.3,0.4],[0.2,0.3], 0.8,0.5,0.4 The, the correspodig ideal solutio (ideal alterative) is chaged ito the followig form: [0.7, 0.8],[0., 0.2],[0., 0.2], 0.9, 0., 0.2, *' *' *' *' P { p, p2,..., p } [0.6,0.7],[0.,0.2],[0.,0.3], 0.8,0.,0.2,. [0.5, 0.7],[0.3, 0.4],[0.3, 0.4], 0.6, 0.5, 0.4 Accordig to the results of Table 2, both the cosie measure based o the icluded agle cosie of two vectors Sw ad the cosie measure based o cosie fuctio Sw3 still hold the same rakig orders; while the cosie measure based o distace Sw2 shows aother rakig form. I this case, Sw2 is sesitive to the chage of the evaluatio values, sice its rakig order chages with the chage of the evaluatio values for the alterative P4. Table 2. All the cosie measure values betwee Pi ad P * ad rakig orders of the four alteratives. Swk(Pi, P * ) Cosie Measure Value Rakig Order The Best Alterative Sw(Pi, P * ) 0.945, , , P4 > P2 > P3 > P P4 Sw2(Pi, P * ) , , , P2 > P4 > P3 > P P2 Sw3(Pi, P * ) , , , P4 > P2 > P3 > P P4 Nevertheless, this study provides a ew ad effective method for decisio makers, due to the limited study o similarity measures ad decisio-makig methods with NCSs i the existig literature. I this study, decisio makers ca select oe of three cosie measures of NCSs to apply to MADM problems, accordig to their prefereces ad actual requiremets. 6. Coclusios This paper proposed three cosie measures of NCSs based o the icluded agle cosie of two vectors, distace, ad cosie fuctio, ad discussed their properties. The, we developed a MADM method with eutrosophic cubic iformatio by usig oe of three cosie measures of NCSs. A illustrative example about the selectio problem of ivestmet alteratives was provided to demostrate the applicatios of the proposed MADM method with eutrosophic cubic iformatio. The cosie measures-based MADM method developed i this paper is simpler ad easier tha the existig decisio-makig method with eutrosophic cubic iformatio based o the grey related aalysis, ad shows the mai advatage of its simple ad easy decisio-makig process. However, this study ca oly deal with decisio-makig problems with exact/crisp weights, rather tha NCS weights [40], which is its chief limitatio. Therefore, the three cosie measures of NCSs that were developed, ad their decisio-makig method are the mai cotributios of this paper. The developed MADM method provides a ew ad effective method for decisio makers uder eutrosophic cubic eviromets. I future work, we will further propose some ew similarity measures of NCSs ad their applicatios i other fields, such as image processig, medical diagosis, ad fault diagosis. Ackowledgmets: This paper was supported by the Natioal Natural Sciece Foudatio of Chia (No ). Author Cotributios: Ju Ye proposed three cosie measures of NCSs ad their decisio-makig method; Zhikag Lu provided the illustrative example ad related compariso aalysis; we wrote the paper together. Coflicts of Iterest: The authors declare o coflicts of iterest.

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