Correlation Coefficient between Dynamic Single Valued Neutrosophic Multisets and Its Multiple Attribute...

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1 See discussios, stats, ad author profiles for this publicatio at: Correlatio Coefficiet betwee Dyamic Sigle Valued Neutrosophic Multisets ad Its Multiple Attribute... Article i Iformatio Switzerlad) April 207 DOI: /ifo CITATIONS 0 READS 62 author: Ju Ye Shaoxig Uiversity 47 PUBLICATIONS 3,086 CITATIONS SEE PROFILE Some of the authors of this publicatio are also workig o these related projects: The project of eutrosophic theory, decisio makig, ad applicatios sposored by the Natioal Natural Sciece Foudatio of P.R. Chia No ). View project All cotet followig this page was uploaded by Ju Ye o 8 April 207. The user has requested ehacemet of the dowloaded file.

2 iformatio Article Correlatio Coefficiet betwee Dyamic Sigle Valued Neutrosophic Multisets ad Its Multiple Attribute Decisio-Makig Method Ju Ye Departmet of Electrical ad Iformatio Egieerig, Shaoxig Uiversity, 508 Huacheg West Road, Shaoxig 32000, Chia; Tel.: Academic Editor: Willy Susilo Received: 7 March 207; Accepted: 5 April 207; Published: 7 April 207 Abstract: Based o dyamic iformatio collected from differet time itervals i some real situatios, this paper firstly proposes a dyamic sigle valued eutrosophic multiset DSVNM) to express dyamic iformatio ad operatioal relatios of DSVNMs. The, a correlatio coefficiet betwee DSVNMs ad a weighted correlatio coefficiet betwee DSVNMs are preseted to measure the correlatio degrees betwee DSVNMs, ad their properties are ivestigated. Based o the weighted correlatio coefficiet of DSVNMs, a multiple attribute decisio-makig method is established uder a DSVNM eviromet, i which the evaluatio values of alteratives with respect to attributes are collected from differet time itervals ad are represeted by the form of DSVNMs. The rakig order of alteratives is performed through the weighted correlatio coefficiet betwee a alterative ad the ideal alterative, which is cosidered by the attribute weights ad the time weights, ad thus the best oes) ca also be determied. Fially, a practical example shows the applicatio of the proposed method. Keywords: dyamic sigle valued eutrosophic multiset; correlatio coefficiet; decisio-makig. Itroductio The theory of eutrosophic sets preseted by Smaradache [] is a powerful techique to hadle icomplete, idetermiate ad icosistet iformatio i the real world. As the geeralizatio of a classic set, fuzzy set [2], ituitioistic fuzzy set [3], ad iterval-valued ituitioistic fuzzy set [4], a eutrosophic set ca idepedetly express a truth-membership degree, a idetermiacy-membership degree, ad a falsity-membership degree. All the factors described by the eutrosophic set are very suitable for huma thikig due to the imperfectio of kowledge that humas receive or observe from the exteral world. For example, cosider the give propositio Movie X would be a hit. I this situatio, the huma brai certaily caot geerate precise aswers i terms of yes or o, because idetermiacy is the sector of uawareess of a propositio s value betwee truth ad falsehood. Obviously, the eutrosophic compoets are very suitable for the represetatio of idetermiate ad icosistet iformatio. A eutrosophic set A i a uiversal set X is characterized by a truth-membership fuctio µ A x), a idetermiacy-membership fuctio τ A x) ad a falsity-membership fuctio ν A x). The fuctios µ A x), τ A x), ν A x) i X are real stadard or ostadard subsets of ] 0, + [, such that µ A x): X ] 0, + [, τ A x): X ] 0, + [, ad ν A x): X ] 0, + [. The, the sum of µ A x), τ A x) ad ν A x) is o restrictio, i.e., 0 sup µ A x) + sup τ A x) + sup ν A x) 3 +. Sice the fuctios µ A x), τ A x) ad ν A x) are defied i the ostadard iterval ] 0, + [, it is difficult to apply the eutrosophic set to sciece ad egieerig fields. So, we ca costrai the fuctios µ A x), τ A x) ad ν A x) i the real stadard iterval [0, ] to easily apply to real situatios. Thus, Wag et al. [5,6] defied a sigle Iformatio 207, 8, 4; doi:0.3390/ifo

3 Iformatio 207, 8, 4 2 of 9 valued eutrosophic set SVNS) ad a iterval eutrosophic set INS), which are the subclasses of the eutrosophic set. SVNSs ad INSs are more suitable for the represetatio of idetermiate ad icosistet iformatio i sciece ad egieerig applicatios. Recetly, eutrosophic sets have become research topics i egieerig areas, such as decisio-makig [7 7], clusterig aalysis [8,9], ad image processig [20 22]. By meas of combiig eutrosophic sets with other sets, some extesios of them have bee recetly developed, such as eutrosophic soft sets [23 25], sigle valued eutrosophic hesitat fuzzy sets [26,27], iterval eutrosophic hesitat sets [28], iterval eutrosophic liguistic sets [29], sigle valued eutrosophic liguistic sets [30]. Moreover, these have bee successfully applied i decisio-makig. Furthermore, multi-valued eutrosophic sets for medical diagosis [3] ad decisio-makig [32,33] were proposed uder multi-valued eutrosophic eviromets. Neutrosophic refied sets ad bipolar eutrosophic refied sets were also developed for solvig medical diagosis problems [34 37]. Normal eutrosophic fuzzy umbers were preseted to hadle multiple attribute decisio-makig problems with ormal eutrosophic fuzzy iformatio [38,39]. I geeral, all of the above studies have sigificatly advaced i the theory ad applicatios of various eutrosophic sets. However, the aforemetioed eutrosophic iformatio is collected all at oce, at the same time, which is also called static iformatio. But, as for some complex problems i real situatios, such as some complex decisio-makig problems, movig image processig problems, complex medical diagosis problems, ad persoel dyamic examiatio, we have to cosider these dyamic problems i differet time itervals. I these cases, how ca we express the dyamic problems? Oe solutio is to express dyamic iformatio collected from differet time itervals by dyamic sigle valued eutrosophic multisets. To do so, the mai aims of this paper are: ) to propose a dyamic sigle valued eutrosophic multiset DSVNM) as a better tool for expressig dyamic iformatio of dyamic problems; 2) to develop correlatio coefficiets betwee DSVNMs for measurig the correlatio degree betwee two DSVNMs; ad 3) to apply the correlatio coefficiet to multiple attribute decisio-makig problems with DSVNM iformatio. The rest of the article is orgaized as follows. Sectio 2 itroduces some cocepts ad basic operatios of SVNSs ad a correlatio coefficiet betwee SVNSs. Sectio 3 presets a DSVNM ad its basic operatioal relatios. A correlatio coefficiet betwee DSVNMs ad a weighted correlatio coefficiet betwee DSVNMs are proposed ad their properties are ivestigated i Sectio 4. Sectio 5 develops a multiple attribute decisio-makig method usig the weighted correlatio coefficiet of DSVNMs uder DSVNM eviromet. I Sectio 6, a practical example of a decisio-makig problem cocerig ivestmet alteratives is provided to demostrate the applicatios of the proposed decisio-makig method uder DSVNM eviromet. Coclusios ad further research are give i Sectio Some Cocepts of SVNSs Smaradache [] origially preseted the cocept of a eutrosophic set from a philosophical poit of view. To easily use it i real applicatios, Wag et al. [6] itroduced the cocept of SVNS as a subclass of the eutrosophic set ad gave the followig defiitio. Defiitio [6]. Let X be a uiversal set. A SVNS A i X is characterized by a truth-membership fuctio µ A x), a idetermiacy-membership fuctio τ A x) ad a falsity-membership fuctio ν A x). The, a SVNS A ca be deoted by the followig form: A = { x, µ A x), τ A x), ν A x) x X} where µ A x), τ A x), ν A x) [0, ] for each x i X. Therefore, the sum of µ A x), τ A x) ad ν A x) satisfies the coditio 0 µ A x) + τ A x) + ν A x) 3.

4 Iformatio 207, 8, 4 3 of 9 For two SVNSs A = { x, µ A x), τ A x), ν A x) x X} ad B = { x, µ B x), τ B x), ν B x) x X}, there are the followig relatios [6]: ) Complemet: A c = { x, ν A x), τ A x), µ A x) x X}; 2) Iclusio: A B if ad oly if µ A x) µ B x), τ A x) τ B x), ν A x) ν B x) for ay x i X; 3) Equality: A = B if ad oly if A B ad B A; 4) Uio: A B = { x, µ A x) µ B x), τ A x) τ B x), ν A x) ν B x) x X}; 5) Itersectio: A B = { x, µ A x) µ B x), τ A x) τ B x), ν A x) ν B x) x X}; 6) Additio: A + B = { x, µ A x) + µ B x) µ A x)µ B x), τ A x)τ B x), ν A x)ν B x) x X}; 7) Multiplicatio: A B = { x, µ A x)µ B x), τ A x) + τ B x) τ A x)τ B x), ν A x) + ν B x) ν A x)ν B x) x X}. The, Ye [8] defied a correlatio coefficiet betwee A ad B as follows: RA, B) = [ µa x j )µ B x j ) + τ A x j )τ B x j ) + ν A x j )ν B x j ) ] [ µa x j ) ) 2 + τa x j ) ) 2 + νa x j ) ) ] 2 [ µ B x j )) 2 + τ B x j ) ) 2 + νb x j ) ) ] ) 2 The correlatio coefficiet betwee A ad B satisfies the followig properties [8]: P) 0 RA, B) ; P2) RA, B) = if A = B; P3) RA, B) = RB, A). 3. Dyamic Sigle Valued Neutrosophic Multiset This sectio proposes a dyamic sigle valued eutrosophic multiset ad its operatioal relatios. Defiitio 2. Let X be a oempty set with geeric elemets i X deoted by x, ad t = {t, t 2,..., t q } be a time sequece. A dyamic sigle valued eutrosophic multiset DSVNM) At) collected from X ad t is characterized by a truth-membership time sequece µ A t, x), µ A t 2, x),..., µ A t q, x)), a idetermiacy-membership time sequece τ A t, x), τ A t 2, x),..., τ A t q, x)), ad a falsity-membership time sequece ν A t, x), ν A t 2, x),..., ν A t q, x)) such that µ A t k, x): X R, τ A t k, x): X R, ν A t k, x): X R for t k t ad x X, where R is the set of all real umbers i the real uit iterval [0, ]. The, a DSVNM At) is deoted by: x, µa t, x), µ A t 2, x),..., µ A t q, x)), At) = τ A t, x), τ A t 2, x),..., τ A t q, x)), ν A t, x), ν A t 2, x),..., ν A t q, x)) t k t, x X Obviously, the sum of µ A t k, x), τ A t k, x), ν A t k, x) [0, ] satisfies the coditio 0 µ A t k, x) + τ A t k, x) + ν A t k, x) 3 for t k t ad x X ad k =, 2,..., q. For coveiece, a DSVNM At) ca be deoted by the followig simplified form: At) = { x, µ A t k, x), τ A t k, x), ν A t k, x) t k t, x X} For example, a DSVNM i the time sequece t = {t, t 2, t 3 } ad a uiversal set X = {x, x 2 } is give as: At) = {<x, 0.3, 0.2, 0.5), 0.4, 0.3, 0.5), 0.6, 0.8, 0.9)>, <x 2, 0.4, 0.4, 0.3), 0., 0.2, 0.), 0.3, 0.5, 0.4)>}

5 Iformatio 207, 8, 4 4 of 9 Defiitio 3. Let At) = { x, µ A t k, x), τ A t k, x), ν A t k, x) t k t, x X} ad Bt) = { x, µ B t k, x), τ B t k, x), ν B t k, x) t k t, x X} be ay two DSVNMs i t = {t, t 2,..., t q } ad X. The, there are the followig relatios: ) Iclusio: At) Bt) if ad oly if µ A t k, x) µ B t k, x), τ A t k, x) τ B t k, x), ν A t k, x) ν B t k, x) for k =, 2,..., q ad x X; 2) Equality: At) = Bt) if ad oly if At) Bt) ad Bt) At); 3) Complemet: A c t) = { x, ν A t k, x), τ A t k, x)), µ A t k, x) t k t, x X} ; 4) Uio: At) Bt) = { x, µ A t k, x) µ B t k, x), τ A t k, x) τ B t k, x), ν A t k, x) ν B t k, x) t k t, x X}; 5) Itersectio: At) Bt) = { x, µ A t k, x) µ B t k, x), τ A t k, x) τ B t k, x), ν A t k, x) ν B t k, x) t k t, x X}. For coveiece, we ca use at) = µt, x), µt 2, x),..., µt q, x)), τt, x), τt 2, x),..., τt q, x)), νt, x), νt 2, x),..., νt q, x)) to represet a basic elemet i a DSVNM At) ad call it a dyamic sigle valued eutrosophic multiset elemet DSVNME). Defiitio 4. Let a t) = µ t, x), µ t 2, x),..., µ t q, x)), τ t, x), τ t 2, x),..., τ t q, x)), ν t, x), ν t 2, x),..., ν t q, x)) ad a 2 t) = µ 2 t, x), µ 2 t 2, x),..., µ 2 t q, x)), τ 2 t, x), τ 2 t 2, x),..., τ 2 t q, x)), ν 2 t, x), ν 2 t 2, x),..., ν 2 t q, x)) be two DSVNMEs ad λ 0, the the operatioal rules of DSVNMEs are defied as follows: a t) a 2 t) = µ t, x) + µ 2 t, x) µ t, x)µ 2 t, x), µ t 2, x) + µ 2 t 2, x) µ t 2, x)µ 2 t 2, x),..., µ t q, x) + µ 2 t q, x) µ t q, x)µ 2 t q, x)), τ t, x)τ 2 t, x), τ t 2, x)τ 2 t 2, x),..., τ t q, x)τ 2 t q, x)), ν t, x)ν 2 t, x), ν t 2, x)ν 2 t 2, x),..., ν t q, x)ν 2 t q, x)) µ t, x)µ 2 t, x), µ t 2, x)µ 2 t 2, x),..., µ t q, x)µ 2 t q, x)), τ t, x) + τ 2 t, x) τ t, x)τ 2 t, x), τ t 2, x) + τ 2 t 2, x) τ t 2, x)τ 2 t 2, x), a t) a 2 t) =..., τ t q, x) + τ 2 t q, x) τ t q, x)τ 2 t q, x)), ν t, x) + ν 2 t, x) ν t, x)ν 2 t, x), ν t 2, x) + ν 2 t 2, x) ν t 2, x)ν 2 t 2, x),..., ν t q, x) + ν 2 t q, x) ν t q, x)ν 2 t q, x)) µ λa t) = t, x)) λ, µ t 2, x)) λ,..., µ t q, x) ) ) λ, τ λ t, x), τ λt 2, x),..., τ λt q, x) ), ν λt, x), ν λt 2, x),..., ν λt q, x) ) a λ t) = µ λ t, x), µ λt 2, x),..., µ λt q, x) ), τ t, x)) λ, τ t 2, x)) λ,..., τ t q, x) ) ) λ, ν t, x)) λ, ν t 2, x)) λ,..., ν t q, x) ) ) λ 4. Correlatio Coefficiet of DSVNMs Correlatio coefficiets are usually used i sciece ad egieerig applicatios. They play a importat role i decisio-makig, patter recogitio, clusterig aalysis, ad so o. I regards to this, this sectio proposes a correlatio coefficiet of DSVNMs ad a weighted correlatio coefficiet of DSVNMs.

6 Iformatio 207, 8, 4 5 of 9 Based o DSVNMs costructed by dyamic truth-membership degrees, dyamic idetermiacymembership degrees, ad dyamic falsity-membership degrees correspodig to t = {t, t 2,..., t q }, we ca give the followig defiitio of a correlatio coefficiet betwee DSVNMs. Defiitio 5. Let At)= { x, µ A t k, x), τ A t k, x), ν A t k, x) t k t, x X} ad Bt) = { x, µ B t k, x), τ B t k, x), ν B t k, x) t k t, x X} be ay two DSVNMs i t = {t, t 2,..., t q } ad X = x, x 2,..., x ). The, a correlatio coefficiet betwee At) ad Bt) is defied as: ρat), Bt)) = q q k= [µ A t k,x j )µ B t k,x j )+τ A t k,x j )τ B t k,x j )+ν A t k,x j )ν B t k,x j )] [ µa t k, x j ) ) 2 + τa t k, x j ) ) 2 + νa t k, x j ) ) ] 2 [µ B t k, x j )) 2 + τ B t k, x j ) ) 2 + νb t k, x j ) ) ] 2 2) Theorem. The correlatio coefficiet betwee A ad B satisfies the followig properties: P) 0 ρat), Bt)) ; P2) ρat), Bt)) = if At) = Bt); P3) ρat), Bt)) = ρbt), At)). Proof. P) The iequality ρat), Bt)) 0 is obvious. The, let us prove ρat), Bt)). Accordig to the Cauchy Schwarz iequality: x y + x 2 y x y ) 2 ) ) x 2 + x2 2 + x2 y 2 + y2 2 + y2 where x, x 2,..., x ) R ad y, y 2,..., y ) R. The, we ca obtai the followig iequality: x y + x 2 y x y ) x 2 + x2 2 + x2 ) Accordig to the above iequality, there is the followig iequality: µ A t k, x j ) µ B t k, x j ) + τ A t k, x j ) µa t k, x j ) ) 2 + τa t k, x j ) ) 2 + νa t k, x j ) ) 2 Hece, there is: τ B t k, x j ) + µ B t k, x j )) 2 + y 2 + y2 2 + ) y2 ν A t k, x j ) ν B t k, x j ) τb t k, x j ) ) 2 + νb t k, x j ) ) 2 [ µa t k, x j )µ B t k, x j ) + τ A t k, x j )τ B t k, x j ) + ν A t k, x j )ν B t k, x j ) ] [ µa t k, x j ) ) 2 + τa t k, x j ) ) 2 + νa t k, x j ) ) ] 2 [µ B t k, x j )) 2 + τ B t k, x j ) ) 2 + νb t k, x j ) ) ] 2 From Equatio 2), we have ρat), Bt)). Thus, 0 ρat), Bt)). P2) At) = Bt) µ A t k, x j ) = µ B t k, x j ), τ A t k, x j ) = τ B t k, x j ), ad ν A t k, x j ) = ν B t k, x j ) for t k t ad x j X ρat), Bt)) =. P3) It is straightforward. I practical applicatios, we should cosider differet weights for each elemet x j j =, 2,..., ) i X ad each time t k k =, 2,..., q) i t. The, let w = w, w 2,..., w ) T be the weightig vector of x j j =, 2,..., ) with w j 0 ad w j = ad ωt) = ωt ), ωt 2 ),..., ωt q )) T be the weightig

7 Iformatio 207, 8, 4 6 of 9 vector of t k k =, 2,..., q) with ωt k ) 0 ad q k= ωt k) =. Hece, we further exted the correlatio coefficiet of Equatio 2) to the followig weighted correlatio coefficiet: ρ w At), Bt)) = q k= ωt k ) w j[µ A t k,x j )µ B t k,x j )+τ A t k,x j )τ B t k,x j )+ν A t k,x j )ν B t k,x j )] [ µa w j t k, x j ) ) 2 + τa t k, x j ) ) 2 + νa t k, x j ) ) ] 2 [ w j µ B t k, x j )) 2 + τ B t k, x j ) ) 2 + νb t k, x j ) ) ] 2 3) whe w j = / j =, 2,..., ) ad ωt k ) = /q k =, 2,..., q), Equatio 3) reduces to Equatio 2). Theorem 2. The correlatio coefficiet ρ w At), Bt)) also satisfies the followig three properties: P) 0 ρ w At), Bt)) ; P2) ρ w At), Bt)) = if At) = Bt); P3) ρ w At), Bt)) = ρ w Bt), At)). By the previous similar proof method i Theorem, we ca prove the properties P) P3) omitted). 5. Correlatio Coefficiet for Multiple Attribute Decisio-Makig I this sectio, we apply the weighted correlatio coefficiet of DSVNMs to multiple attribute decisio-makig problems with DSVNM iformatio. For a multiple attribute decisio-makig problem with DSVNM iformatio, let G = {g, g 2,..., g m } be a discrete set of alteratives, X = {x, x 2,..., x } be a set of attributes, ad t = {t, t 2,..., t q } be a time sequece. If the decisio makers provide a evaluatio value for the alterative g i i =, 2,..., m) regardig the attribute x j j =, 2,..., ) at a time t k k =, 2,..., q), the evaluatio value ca be represeted by the form of a DSVNME d ij t) = µ i t, x j ), µ i t 2, x j ),..., µ i t q, x j )), τ i t, x j ), τ i t 2, x j ),..., τ i t q, x j )), ν i t, x j ), ν i t 2, x j ),..., ν i t q, x j )) j =, 2,..., ; i =, 2,..., m). Therefore, we ca elicit a DSVNM decisio matrix Dt) = d ij t)) m, where d ij t) i =, 2,..., m; j =, 2,..., ) is i a DSVNME form. I multiple attribute decisio-makig problems, the ideal alterative is used to help the idetificatio of the best alterative i the decisio set. Therefore, we defie each ideal DSVNME d j *t) = µ*t, x j ), µ*t 2, x j ),..., µ*t q, x j )), τ*t, x j ), τ*t 2, x j ),..., τ*t q, x j )), ν*t, x j ), ν*t 2, x j ),..., ν*t q, x j )) =,,..., ), 0, 0,..., 0), 0, 0,..., 0) j =, 2,..., ) i the ideal solutio ideal alterative) } g t) = { x j, d j t) t k t, x j X k =, 2,..., q; j =, 2,..., ). Assume that the weightig vector of attributes for the differet importace of each attribute x j j =, 2,..., ) is give by w = w, w 2,..., w ) T with w j 0, w j = ad the time weightig vector for the differet importace of each time t k k =, 2,..., q) is give by ωt) = ωt ), ωt 2 ),..., ωt q )) T with ωt k ) 0 ad q k= ωt k) =. The, we utilize the weighted correlatio coefficiet for multiple attribute decisio-makig problems with DSVNM iformatio. The weighted correlatio coefficiet betwee a alterative g i t) i =, 2,..., m) ad the ideal solutio g*t) is calculated by use of the followig formula: ρ w g i t), g t)) = q k= ωt k ) [ w j µi t k, x j ) ] [ µi w j t k, x j ) ) 2 + τi t k, x j ) ) 2 + νi t k, x j ) ) ] 4) 2 The bigger the value of ρ w g i t), g*t)), the better the alterative g i. The, we rak the alteratives ad select the best oes) accordig to the values of weighted correlatio coefficiets.

8 j= j= j= The bigger The bigger the The value bigger the value of the ρwgit), value of ρwgit), g*t)), of ρwgit), g*t)), the better g*t)), the better the the alterative better the alterative the gi. alterative The, gi. The, we gi. rak The, we the rak we alteratives the rak alteratives the alteratives ad select ad select the ad best select the oes) best the oes) accordig best oes) accordig to accordig the to values the to values of the weighted values of weighted of correlatio weighted correlatio coefficiets. correlatio coefficiets. coefficiets. 6. Practical 6. Iformatio Practical Example 6. 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The ivestmet The ivestmet compay There compay must is compay take must ivestmet a take must decisio a take decisio compay, accordig a decisio accordig which to accordig the wats to three the to three ivest the three attributes: attributes: a sum ) attributes: of x is ) moey the x risk ) the ix factor; the risk the best factor; 2) risk optio. x2 factor; is 2) the x2 There is growth 2) the x2 is growth isfactor; the a pael growth factor; 3) with x3 factor; is 3) the four x3 is evirometal 3) possible the x3 is evirometal the alteratives evirometal factor. factor. Let us ivest factor. Let us the Let us cosider cosider moey: the cosider evaluatios the ) gevaluatios the is aevaluatios of car the compay; of alteratives the of alteratives the 2) alteratives o g 2 the is ao attributes food the o attributes compay; the give attributes give by 3) decisio g 3 give by decisio a computer by makers decisio makers or compay; experts makers or experts i or 4) experts i g 4 is i the time the asequece time arms the sequece compay. time t = sequece {t, t t2, = The {t, t3}. t ivestmet t2, Assume = {t, t3}. t2, Assume t3}. that compay Assume the that weightig the that must weightig the take vector weightig a decisio vector of the vector of attributes accordig the of attributes the is attributes to give the is give three by is w give attributes: = by w = by w = 0.35, 0.35, 0.25, ) x 0.40) 0.35, 0.25, is the T 0.40) ad 0.25, risk T the 0.40) factor; ad weightig T the ad 2) weightig xthe 2 isvector weightig the growth vector of times vector of factor; times is give of 3) times is xgive by 3 is is ωt) the give by evirometal = ωt) 0.25, by = ωt) 0.25, 0.35, = 0.40) 0.25, 0.35, factor. T. 0.40) 0.35, The Let T. four 0.40) us The cosider T. four The four possible possible the alteratives evaluatios possible alteratives alteratives of of gi the i of = alteratives gi, i 2, of = 3, gi, 4) i 2, regardig = 3,, o 4) 2, the regardig 3, 4) attributes the regardig three the give attributes three the attributes three by decisio of attributes xj j of = makers xj, j 2, of = 3) xj, are or j 2, = experts 3) evaluated, are 2, 3) evaluated i are the evaluated time by decisio by sequece decisio by makers, decisio t makers, = {t ad, tmakers, 2 the, ad t 3 }. Assume the ad evaluatio the that evaluatio the the values evaluatio weightig values are represeted values are vector represeted are of represeted the by usig attributes by usig DSVNMEs, by isusig givedsvnmes, which by w which = are 0.35, which are 0.25, are give give as 0.40) the T give as followig ad the as the followig the weightig DSVNM followig DSVNM decisio vector DSVNM decisio matrix of decisio times matrix Dt): matrix give Dt): by Dt): ωt) = 0.25, 0.35, 0.40) T. The four possible alteratives of g i i =, 2, 3, 4) regardig the three attributes of x j j =, 2, 3) are evaluated by decisio 0.4,0.5,0.3),0.,0.2,0.3),0.3,0.2,0.3) makers, ad the the evaluatio values are represeted by usig DSVNMEs, which are give as the followig DSVNM decisio matrix Dt): 0.6,0.4,0.5),0.,0.,0.2),0.,0.2,0.3) Dt ) = Dt ) = Dt ) = 0.4, 0.2, 0.4, 0.3),0.2, 0.2, 0.4, 0.3),0.2, 0.2, 0., 0.3),0.2, 0.2),0.2, 0., 0.2),0.2, 0., 0.3, 0.2),0.2, 0.3) 0.3, 0.3) 0.3, 0.3) 0.4, 0.5, 0.3), 0., 0.2, 0.3), 0.3, 0.2, 0.3) 0.6, 0.7,0.6,0.8),0.,0.0,0.0),0.,0.,0.) 0.4, 0.5), 0., 0., 0.2), 0., 0.2, 0.3) Dt) = 0.4, 0.4,0.5,0.3),0.,0.2,0 0.2, 0.3), 0.2, 0., 0.2),.2),0.2,0.3,0.3) 0.3, 0.3) 0.5, 0.6, 0.5, 0.7, 0.7),0., 0.6, 0.5, 0.6, 0.7),0., 0.6, 0.8), 0., 0.7),0., 0.),0., 0., 0.0, 0.),0., 0.0), 0., 0.2, 0.),0., 0.2) 0.2, 0., 0.2) 0.) 0.2, 0.2) 0.4, 0.5, 0.3), 0., 0.2, 0.2), 0.2, 0.3, 0.3) 0.4, 0.5, 0.4, 0.6),0., 0.5, 0.4, 0.6),0., 0.5, 0.2, 0.6),0., 0.2),0.4, 0.2, 0.2),0.4, 0.2, 0.2, 0.2),0.4, 0.3) 0.2, 0.3) 0.2, 0.3) 0.5, 0.6, 0.7), 0., 0., 0.), 0., 0.2, 0.2) 0.6, 0.5, 0.6, 0.4, 0.7),0., 0.5, 0.6, 0.5, 0.7),0., 0.6), 0.5, 0.2, 0.7),0., 0.),0.2, 0.2, 0.2, 0.),0.2, 0.2), 0.2, 0.3, 0.4, 0.),0.2, 0.2) 0.3, 0.2, 0.3) 0.2) 0.3, 0.2) 0.2, 0.3, 0.2, 0.6, 0.2),0.2, 0.3, 0.5, 0.2),0.2, 0.7), 0.3, 0.2, 0.2),0.2, 0., 0.2),0.4, 0.2, 0.2),0.4, 0.), 0.2, 0.3, 0.2, 0.2),0.4, 0.5) 0.3, 0.2) 0.5) 0.3, 0.5) 0.2, 0.3, 0.2), 0.2, 0.2, 0.2), 0.4, 0.3, 0.5) 0.5,0.6,0.7),0.2,0.,0.2),0.,0.2,0.2) 0.5, 0.6, 0.7), 0.2, 0., 0.2), 0., 0.2, 0.2). 0.6, 0.5, 0.6, 0.6),0.3, 0.5, 0.6), 0.2, 0.3, 0.3),0.2, 0.2, 0.3), 0.2, 0.2, 0.2, 0.3).. 0.6, 0.5, 0.6, 0.6),0.3, 0.5, 0.6),0.3, 0.2, 0.3),0.2, 0.2, 0.3),0.2, 0.2, 0.3) 0.2, 0.3) 0.4, 0.5, 0.4,0.5,0.4),0.3,0.2,0.2),0.2,0.,0.2) 0.4), 0.3, 0.2, 0.2), 0.2, 0., 0.2) The, The, the developed The, the developed the approach developed approach is approach utilized is is utilized to is give utilized to the togive give rakig to the give the rakig the order rakig order of order the order of alteratives of the the of alteratives the alteratives ad the ad ad the ad the the best oes). best oes). best oes). By applyig By Byapplyig Equatio applyig Equatio 4), Equatio we 4), ca we we 4), ca obtai ca we obtai obtai ca the the values obtai the values values of the of the values the of weighted the of weighted the correlatio weighted correlatio correlatio coefficiet coefficiet betwee coefficiet betwee betwee each each betwee alterative each alterative each ad ad alterative the the ad ideal ideal the ad alterative ideal the alterative ideal as as alterative follows: as follows: as follows: ρwgt), ρwgt), ρ w g*t)) ρwgt), t), = g*t)) , g*t)) = g*t)) = , ρwg2t), = , ρwg2t), ρg*t)) w ρwg2t), 2 t), = g*t)) 0.922, g*t)) = g*t)) 0.922, = ρwg3t), 0.922, = 0.922, ρwg3t), ρg*t)) w ρwg3t), 3 t), = g*t)) g*t)) = g*t)) ad = = ρwg4t), ad ad ρwg4t), ad g*t)) ρ w ρwg4t), 4 = t), g*t)) g*t)) = g*t)) = = Accordig Accordig to Accordig the to to above the to above values the above values of weighted values of weighted of correlatio weighted correlatio coefficiets, correlatio coefficiets, coefficiets, we ca we we ca give ca give we the give ca the rakig the give rakig rakig the order rakig order order of the order of four the of alteratives: four the alteratives: four alteratives: AA2 2 A4 A2 4 A3 A4 A2 3 A, A3 A4, which A, A3 which is A, i which accordace is i accordace is i with accordace the the with oe oe the with of of [8]. oe the [8]. Therefore, of oe [8]. of [8]. Therefore, Therefore, the alterative the Therefore, alterative the alterative Athe 2 is A2 alterative the is the best A2 best is choice. the A2 choice. best is the choice. best choice. The example The example The clearly example clearly idicates clearly idicates that idicates that the that the the that proposed the proposed decisio-makig decisio-makig method method ismethod simple simple ad is simple effective ad ad effective effective uder uder effective the uder the DSVNM uder DSVNM the eviromet, DSVNM the DSVNM eviromet, based eviromet, based based the weighted the based o weighted the o correlatio weighted the correlatio weighted correlatio coefficiet correlatio coefficiet of coefficiet DSVNMs of coefficiet of for of DSVNMs DSVNMs dealig for DSVNMs dealig with for dealig multiple with for dealig multiple with attribute multiple with attribute multiple decisio-makig attribute attribute decisio-makig problems problems with problems with DSVNM problems DSVNM with iformatio, DSVNM with DSVNM iformatio, sice iformatio, such sice such sice a decisio-makig a such sice a such a decisio-makig method method ca method ca represet method represet ca represet ad ca hadle represet hadle the hadle the dyamic hadle the dyamic evaluatio the dyamic evaluatio data data evaluatio give give data bygive data experts give or decisio makers at differet time itervals, while existig various eutrosophic decisio-makig methods caot do this. 7. Coclusios Based o dyamic iformatio collected from differet time itervals i some real situatios, this paper proposed a DSVNM to express dyamic iformatio ad the operatioal relatios of DSVNMs. The DSVNM is a dyamic set ecompassig a time sequece, where its truth-membership degrees, idetermiacy-membership degrees, ad falsity-membership degrees are represeted by time sequeces. Therefore, DSVNM has desirable characteristics ad advatages of its ow for hadlig.

9 Iformatio 207, 8, 4 8 of 9 dyamic problems accordig to a time sequece i real applicatios, whereas existig eutrosophic sets caot deal with them. The, we proposed the correlatio coefficiet of DSVNMs ad weighted correlatio coefficiet of DSVNMs ad ivestigated their properties. Based o the weighted correlatio coefficiet of DSVNMs, the multiple attribute decisio-makig method was proposed uder a DSVNM eviromet, i which the evaluated values of alteratives regardig attributes take the form of DSVNMEs. Through the weighted correlatio coefficiet betwee each alterative ad the ideal alterative, oe ca rak alteratives ad choose the best oes). Fially, a practical example about ivestmet alteratives was give to demostrate the practicality ad effectiveess of the developed approach. The proposed method is simple ad effective uder the DSVNM decisio-makig eviromet. I the future, we shall exted DSVNMs to iterval/bipolar eutrosophic sets ad develop dyamic iterval/bipolar eutrosophic decisio-makig ad medical diagosis methods i differet time itervals. Ackowledgmets: No ). This paper was supported by the Natioal Natural Sciece Foudatio of Chia Coflicts of Iterest: The author declares o coflict of iterest. Refereces. Smaradache, F. A Uifyig Field i Logics: Neutrosophy. Neutrosophic Probability, Set ad Logic; America Research Press: Rehoboth, MA, USA, Zadeh, L.A. Fuzzy Sets. If. Cotrol 965, 8, [CrossRef] 3. Ataassov, K. Ituitioistic fuzzy sets. Fuzzy Sets Syst. 986, 20, [CrossRef] 4. Ataassov, K.; Gargov, G. Iterval valued ituitioistic fuzzy sets. Fuzzy Sets Syst. 989, 3, [CrossRef] 5. Wag, H.; Smaradache, F.; Zhag, Y.Q.; Suderrama, R. Iterval Neutrosophic Sets ad Logic: Theory ad Applicatios i Computig; Hexis: Phoeix, AZ, USA, Wag, H.; Smaradache, F.; Zhag, Y.Q.; Suderrama, R. Sigle valued eutrosophic sets. Multisp. Multistruct. 200, 4, Chi, P.P.; Liu, P.D. A exteded TOPSIS method for the multiple attribute decisio-makig problems based o iterval eutrosophic sets. Neutrosophic Sets Syst. 203,, Ye, J. Multicriteria decisio-makig method usig the correlatio coefficiet uder sigle-valued eutrosophic eviromet. It. J. Ge. Syst. 203, 42, [CrossRef] 9. Liu, P.D.; Chu, Y.C.; Li, Y.W.; Che, Y.B. Some geeralized eutrosophic umber Hamacher aggregatio operators ad their applicatio to group decisio-makig. It. J. Fuzzy Syst. 204, 6, Liu, P.D.; Wag, Y.M. Multiple attribute decisio-makig method based o sigle valued eutrosophic ormalized weighted Boferroi mea. Neural Comput. Appl. 204, 25, [CrossRef]. Liu, P.D.; Tag, G.L. Some power geeralized aggregatio operators based o the iterval eutrosophic umbers ad their applicatio to decisio-makig. J. Itell. Fuzzy Syst. 206, 30, [CrossRef] 2. Liu, P.D.; Wag, Y.M. Iterval eutrosophic prioritized OWA operator ad its applicatio to multiple attribute decisio-makig. J. Syst. Sci. Complex. 206, 29, [CrossRef] 3. Liu, P.D. The aggregatio operators based o Archimedea t-coorm ad t-orm for the sigle valued eutrosophic umbers ad their applicatio to decisio-makig. It. J. Fuzzy Syst. 206, 8, [CrossRef] 4. Şahi, R.; Küçük, A. Subsethood measure for sigle valued eutrosophic sets. J. Itell. Fuzzy Syst. 205, 29, [CrossRef] 5. Şahi, R.; Liu, P.D. Maximizig deviatio method for eutrosophic multiple attribute decisio-makig with icomplete weight iformatio. Neural Comput. Appl. 206, 27, [CrossRef] 6. Şahi, R. Cross-etropy measure o iterval eutrosophic sets ad its applicatios i multicriteria decisio-makig. Neural Comput. Appl [CrossRef] 7. Şahi, R.; Liu, P.D. Possibility-iduced simplified eutrosophic aggregatio operators ad their applicatio to multicriteria group decisio-makig. J. Exp. Theor. Artif. Itell [CrossRef] 8. Ye, J. Sigle valued eutrosophic miimum spaig tree ad its clusterig method. J. Itell. Syst. 204, 23, [CrossRef]

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