Multi-criteria neutrosophic decision making method based on score and accuracy functions under neutrosophic environment

Transcription

1 Multi-criteria eutrosophic decisio makig method based o score ad accuracy fuctios uder eutrosophic eviromet Rıdva Şahi Departmet of Mathematics, Faculty of Sciece, Ataturk Uiversity, Erzurum, 50, Turkey Keywords: Neutrosophic sets Sigle valued eutrosophic sets Iterval eutrosophic sets Aggregatio operators Accuracy fuctio Score fuctio Multi criteria eutrosophic decisio makig ABSTRACT A eutrosophic set is a more geeral platform, which ca be used to preset ucertaity, imprecise, icomplete ad icosistet. I this paper a score fuctio ad a accuracy fuctio for sigle valued eutrosophic sets is firstly proposed to make the distictio betwee them. The the idea is exteded to iterval eutrosophic sets. A multi-criteria decisio makig method based o the developed score-accuracy fuctios is established i which criterio values for alteratives are sigle valued eutrosophic sets ad iterval eutrosophic sets. I decisio makig process, the eutrosophic weighted aggregatio operators (arithmetic ad geometric average operators) are adopted to aggregate the eutrosophic iformatio related to each alterative. Thus, we ca rak all alteratives ad make the selectio of the best of oe(s) accordig to the score-accuracy fuctios. Fially, some illustrative examples are preseted to verify the developed approach ad to demostrate its practicality ad effectiveess. 1. Itroductio The cocept of eutrosophic set developed by Smaradache ([16], [17]) is a more geeral platform which geeralizes the cocept of the classic set, fuzzy set [3], ituitioistic fuzzy set [1] ad iterval valued ituitioistic fuzzy sets ([],[3]). I cotrast to ituitioistic fuzzy sets ad also iterval valued ituitioistic fuzzy sets, idetermiacy is characterized explicitly i the eutrosophic set. A eutrosophic set has three basic compoets such that truth membership, idetermiacy membership ad falsity membership, ad they are idepedet. However, the eutrosophic set geeralizes the above metioed sets from philosophical poit of view ad its fuctios T A (x), I A (x) ad F A (x) are real stadard or ostadard subsets of ]0, 1 + [ ad are defied by T A (x): X ]0, 1 + [, I A (x): X ]0, 1 + [ ad F A (x): X ]0, 1 + [. That is, its compoets T(x), I(x), F(x) are o-stadard subsets icluded i the uitary ostadard iterval ]0, 1 + [ or stadard subsets icluded i the uitary stadard iterval [0, 1] as i the ituitioistic fuzzy set. Furthermore, the coectors i the ituitioistic fuzzy set are oly defied by T(x) ad F(x) (i.e. truth-membership ad falsity-membership), hece the idetermiacy I(x) is what is left from 1, while i the eutrosophic set, they ca be defied by ay of them (o restrictio) [16]. For example, whe we ask the opiio of a expert about certai statemet, he/she may say that the possibility i which the statemet is true is 0.6 ad the statemet is false is 0.5 ad the degree i which he/she is ot sure is 0.. For eutrosophic otatio, it ca be expressed as 1 x(0.6,0.,0.5). For aother example, suppose there are 10 voters durig a votig process. Five vote aye, two vote blackball ad three are udecided. For eutrosophic otatio, it ca be expressed as x(0.5,0.3,0.). However, these expressios are beyod the scope of the ituitioistic fuzzy set. Therefore, the otio of eutrosophic set is more geeral ad overcomes the aforemetioed issues. But, a eutrosophic set will be difficult to apply i real scietific ad egieerig fields. Therefore, Wag et al. ([5], [6]) proposed the cocepts of iterval eutrosophic set INS ad sigle valued eutrosophic set (SVNS), which are a istace of a eutrosophic set, ad provided the set- theoretic operators ad various properties of INSs ad SVNSs, respectively. The, SVNSs (or INSs) preset ucertaity, imprecise, icosistet ad icomplete iformatio existig i real world. Also, it would be more suitable to hadle idetermiate iformatio ad icosistet iformatio. Majumdar et al. [11] itroduced a measure of etropy of SVNSs. Ye [3] ad proposed the correlatio coefficiets of SVNSs ad developed a decisio-makig method uder sigle valued eutrosophic eviromet. Broumi ad Smaradache [1] exteded this idea i INSs. Ye [33] also itroduced the cocept of simplified eutrosophic sets (SNSs), ad applied the sets i a MCDM method usig the aggregatio operators of SNSs. Peg et al. [] showed that some operatios i Ye [33] may also be urealistic. They defied the ovel operatios ad aggregatio operators ad applied them to MCDM problems. Ye [30,31] proposed the similarity measures betwee SVNSs

2 ad INSs based o the relatioship betwee similarity measures ad distaces. Şahi ad Küçük [15] proposed the cocept of eutrosophic subsethood based o distace measure for SVNSs. We usually eed the decisio makig methods because of the complex ad ucertaity uder the physical ature of the problems. By the multi-criteria decisio makig methods, we ca choose the optimal alterative from multiple alteratives accordig to some criteria. The proposed set theories have provided the differet multi-criteria decisio makig methods. Some authors ([7],[8],[9],[10],[18],[19],[ 3],[7]) studied o multi-criteria fuzzy decisio-makig methods based o ituitioistic fuzzy sets while some authors ([5],[13],[0],[1],[],[8],[9]) proposed the multi-criteria fuzzy decisio-makig methods based o iterval-valued ituitioistic fuzzy eviromet. Xu ad Yager [3] defied some geometric aggregatio operators amed the ituitioistic fuzzy weighted geometric operator, the ituitioistic fuzzy ordered weighted geometric operator ad the ituitioistic fuzzy hybrid weighted geometric operator, ad applied the ituitioistic fuzzy hybrid weighted geometric operator to a multi-criteria decisio makig problem uder ituitioistic fuzzy eviromet. The Xu [19] proposed the arithmetic aggregatio operators which are arithmetic types of above metioed oes. Xu ad Che [0] geeralized the arithmetic aggregatio operators to iterval valued ituitioistic fuzzy such that the iterval-valued ituitioistic fuzzy weighted geometric operator, the iterval-valued ituitioistic fuzzy ordered weighted geometric operator ad the iterval-valued ituitioistic fuzzy hybrid weighted geometric operator, ad applied the aggregatio operators to a multi-criteria decisio makig problems by usig the score fuctio ad accuracy fuctio of iterval-valued ituitioistic fuzzy umbers. The geometric aggregatio operators for iterval valued ituitioistic fuzzy sets are also proposed i [18]. But, util ow there have bee o may studies o multicriteria decisio makig methods based o score-accuracy fuctios i which criterio values for alteratives are sigle valued eutrosophic sets or iterval eutrosophic sets. Ye [30] proposed a multi-criteria decisio makig method for iterval eutrosophic sets by meas of the similarity measure betwee each alterative ad the ideal alterative. Also, Ye [31] preseted the correlatio coefficiet of SVNSs ad the crossetropy measure of SVNSs ad applied them to sigle valued eutrosophic decisio-makig problems. Recetly, Zhag et al. [6] established two iterval eutrosophic aggregatio operators such as iterval eutrosophic weighted arithmetic operator ad iterval eutrosophic weighted geometric operator ad preseted a method for multi-criteria decisio makig problems based o the aggregatio operators. Therefore the mai purposes of this paper were (1) to defie two measuremet fuctios such that score fuctio ad accuracy fuctio to rak sigle valued eutrosophic umbers ad exted the idea i iterval eutrosophic umbers, () to establish a multi-criteria decisio makig method by use of the proposed fuctios ad eutrosophic aggregatio operators for eutrosophic sets, ad (3) to demostrate the applicatio ad effectiveess of the developed methods by some umerical examples. This paper is orgaized as follows. The defiitios of eutrosophic sets, sigle valued eutrosophic sets, iterval eutrosophic sets ad some basic operators o them as well as arithmetic ad geometric aggregatio operators are briefly itroduced i sectio. I sectio 3, the score fuctio ad the accuracy fuctio for sigle valued eutrosophic umbers are itroduced ad studied by givig illustrative properties. Also the cocepts is exteded to iterval eutrosophic sets i sectio. This is followed by applicatios of the proposed this fuctios to multi-criteria decisio makig problems i Sectio 5. The sectio 6 icludes a compariso aalyze. This paper is cocluded i Sectio 7.. Prelimiaries I the followig we give a brief review of some prelimiaries..1 Neutrosophic set Defiitio.1 [16] Let X be a space of poits (objects) ad x X. A eutrosophic set A i X is defied by a truth-membership fuctio T A (x), a idetermiacy-membership fuctio I A (x) ad a falsity-membership fuctio F A (x). T A (x), I A (x) ad F A (x) are real stadard or real ostadard subsets of ]0, 1 + [. That is T A (x): X ]0, 1 + [, T A (x): X ]0, 1 + [ ad T A (x): X ]0, 1 + [. There is ot restrictio o the sum of T A (x), I A (x) ad F A (x), so 0 sup T A (x) sup I A (x) sup F A (x) 3 +. Defiitio. [17] The complemet of a eutrosophic set A is deoted by A c ad is defied as T A c (x) = {1 + } T A (x), I A c (x) = {1 + } I A (x) ad F A c (x) = {1 + } F A (x) for all x X. Defiitio.3 [17] A eutrosophic set A is cotaied i the other eutrosophic set B, A B iff if T A (x) if T B (x), sup T A (x) sup T B (x), if I A (x) if I B (x), sup I A (x) sup I B (x) ad if F A (x) if F B (x), sup F A (x) sup F B (x) for all x X. I the followig, we adopt the represetatios u A (x), w A (x) ad v A (x) istead of T A (x), I A (x) ad F A (x), respectively.. Sigle valued eutrosophic sets A sigle valued eutrosophic set has bee defied i [5] as follows: Defiitio. [5] Let X be a uiverse of discourse. A sigle valued eutrosophic set A over X is a object havig the form A = { x, u A (x), w A (x), v A (x) : x X}

3 where u A (x): X [0,1], w A (x): X [0,1] ad v A (x): X [0,1] with 0 u A (x) + w A (x) + v A (x) 3 for all x X. The itervals u A (x), w A (x) ad v A (x) deote the truth- membership degree, the idetermiacy-membership degree ad the falsity membership degree of x to A, respectively. Defiitio.5 [5] The complemet of a SVNS A is deoted by A c ad is defied as u A c (x) = v(x), w A c (x) = 1 w A (x), ad v A c (x) = u(x) for all x X. That is, A c = { x, v A (x), 1 w A (x), u A (x) : x X}. Defiitio.6 [5] A sigle valued eutrosophic set A is cotaied i the other SVNS B, A B, iff u A (x) u B (x), w A (x) w B (x)ad v A (x) v B (x) for all x X. Defiitio.7 [5] Two SVNSs A ad B are equal, writte as A = B, iff A B ad B A. We will deote the set of all the SVNSs i X by SVNS(X). A SVNS value is deoted by A = (a, b, c) for coveiece. Based o the study give i [6], we defie two weighted aggregatio operators related to SVNSs as follows: Defiitio.8 Let A k (k = 1,,, ) SVNS(X). The sigle valued eutrosophic weighted average operator is defied by F ω = (A 1, A,, A ) = ω k A k = (1 (1 u Ak, (w A k (x))ω k, (v Ak ) (1) where ω k is the weight of A k (k = 1,,, ), ω k [0,1] ad ω k = 1. Especially, assume ω k = 1/ (k = 1,,, ), the F ω is called a arithmetic average operator for SVNSs. Similarly, we ca defie the sigle valued eutrosophic weighted geometric average operator as follows: Defiitio.9 Let A k (k = 1,,, ) SVNS(X). The sigle valued eutrosophic weighted geometric average operator is defied by G ω = (A 1, A,, A ) = A k ω k = ( (u Ak, 1 (1 wa k (x))ω k, 1 (1 v Ak ) () where ω k is the weight of A k (k = 1,,, ), ω k [0,1] ad ω k = 1. Especially, assume ω k = 1/ (k = 1,,, ), the G ω is called a geometric average for SVNSs. 3 The aggregatio results F ω ad G ω are still SVNSs. Obviously, there are differet emphasis poits betwee Defiitios.8 ad.9. The weighted arithmetic average operator idicates the group s ifluece, so it is ot very sesitive to A k (k = 1,,, ) SVNS(X), whereas the weighted geometric average operator idicates the idividual ifluece, so it is more sesitive to A k (k = 1,,, ) SVNS(X). Defiitio.10 Let A be a sigle valued eutrosophic set over X. (i) A sigle valued eutrosophic set over X is empty, deoted by A if u A (x) = 1, w A (x) = 0 ad v A (x) = 0 for all x X. (ii) A sigle valued eutrosophic set over X is absolute, deoted by Φ if u A (x) = 0, w A (x) = 1 ad v A (x) = 1 for all x X..3 Iterval eutrosophic sets A INS is a istace of a eutrosophic set, which ca be used i real scietific ad egieerig applicatios. I the followig, we itroduce the defiitio of a INS. Defiitio.11 [6] Let X be a space of poits (objects) ad It[0,1] be the set of all closed subsets of [0,1]. A INS A i X is defied with the form A = { x, u A (x), w A (x), v A (x) : x X} where u A (x): X it[0,1], w A (x): X it[0,1] ad v A (x): X it[0,1] with 0 sup u A (x) + sup w A (x) + sup v A (x) 3 for all x X. The itervals u A (x), w A (x) ad v A (x) deote the truth-membership degree, the idetermiacymembership degree ad the falsity membership degree of x to A, respectively. For coveiece, if let u A (x) = [u A (x), u A +(x)], w A (x) = [w A (x), w A +(x)] ad v(x) = [v A (x), v A +(x)], the A = { x, [u A (x), u A +(x)], [w A (x), w A +(x)], [v A (x), v A +(x)] : x X} with the coditio, 0 sup u A +(x) + sup w A +(x) + sup v A +(x) 3 for all x X. Here, we oly cosider the subuitary iterval of [0,1]. Therefore, a INS is clearly eutrosophic set. Defiitio.1 [6] The complemet of a INS A is deoted by A c ad is defied as u A c(x) = v(x), (w A ) c (x) = 1 w A +(x), (w A +) c (x) = 1 w A (x) ad v A c(x) = u(x) for all x X. That is, A c = { x, [v A (x), v A +(x)], [1 w A +(x), 1 w A (x)], [u A (x), u A +(x)] : x X}.

4 Defiitio.13 [6] A iterval eutrosophic set A is cotaied i the other INS B, A B, iff u A (x) u B (x), u A +(x) u B +(x), w A (x) w B (x), w A +(x) w B +(x) ad v A (x) v B (x), v A +(x) v B +(x) for all x X. Defiitio.1 [6] Two INSs A ad B are equal, writte as A = B, iff A B ad B A. We will deote the set of all the INSs i X by INS(X). A INS value is deoted by A = ([a, b], [c, d], [e, f]) for coveiece. Next, we give two weighted aggregatio operators related to INSs. Defiitio.15 [6] Let A k (k = 1,,, ) INS(X). The iterval eutrosophic weighted average operator is defied by F ω = (A 1, A,, A ) = ω k A k = ([1 (1 u A k, 1 (1 u + A k ], [ (w A k, (w + A k ], [ (v A k, (v + A k ]) (3) where ω k is the weight of A k (k = 1,,, ), ω k [0,1] ad ω k = 1. Especially, assume ω k = 1/ (k = 1,,, ), the F ω is called a arithmetic average operator for INSs. Defiitio.16 [6] Let A k (k = 1,,, ) INS(X). The iterval eutrosophic weighted geometric average operator is defied by G ω = (A 1, A,, A ) = A k ω k = ([ (u A k, (u + A k ], [1 (1 w ω k A k(x)), 1 (1 w + A k ], [1 (1 v A k, 1 (1 v + A k ]) () where ω k is the weight of A k (k = 1,,, ), ω k [0,1] ad ω k = 1. Especially, assume ω k = 1/ (k = 1,,, ), the G ω is called a geometric average for INSs. The aggregatio results F ω ad G ω are still INSs. Obviously, there are differet emphasis poits betwee Defiitios.15 ad.16. The weighted arithmetic average operator idicates the group s ifluece, so it is ot very sesitive to A k (k = 1,,, ) INS(X), whereas the weighted geometric average operator idicates the idividual ifluece, so it is more sesitive to A k (k = 1,,, ) INS(X). Defiitio.17 [6] Let A be a iterval eutrosophic set over X. (i) A iterval eutrosophic set over X is empty, deoted by A if u A (x) = [1,1], w A (x) = [0,0] ad v A (x) = [0,0] for all x X. (ii) A iterval eutrosophic set over X is absolute, deoted by Φ if u A (x) = [0,0], w A (x) = [1,1] ad v A (x) = [1,1] for all x X. 3. Rakig by score fuctio I the followig, we itroduce a score fuctio for rakig SVN umbers by takig ito accout the truth-membership degree, idetermiacy-membership degree ad falsity membership degree of SVNSs (ad INSs), ad discuss some basic properties. Defiitio 3.18 Let A = (a, b, c) be a sigle valued eutrosophic umber, a score fuctio K of a sigle valued eutrosophic value, based o the truth-membership degree, idetermiacy-membership degree ad falsity membership degree is defied by K(A) = where K(A) [ 1,1]. 1 + a b c (5) The score fuctio K is reduced the score fuctio proposed by Li ([8]) if b = 0 ad a + c 1. It is clear that if truth-membership degree a is bigger, ad the idetermiacy-membership degree b ad falsity membership degree c are smaller, the the score value of the SVNN A is greater. We give the followig example. Example 3.19 Let A 1 = (0.5,0.,0.6) ad A = (0.6,0.,0.) be two sigle valued eutrosophic values for two alteratives. The, by applyig Defiitio 3.18, we ca obtai K(A 1 ) = = K(A ) = = 0.3. I this case, we ca say that alterative A is better tha A 1. Propositio 3.0 Let A = (a, b, c) be a sigle valued eutrosophic value. The the score fuctio K has some properties as follows: (i) K (A) = 0 if ad oly if a = b + c 1. (ii) K (A) = 1 if ad oly if a = b + c + 1. (iii) K (A) = 1 if ad oly if a = b + c 3. Moreover, we have that K (A ) = 1, which A is the absolute sigle valued eutrosophic value, ad K (Φ) = 1, which Φ is the ull sigle valued eutrosophic value.

5 Theorem 3.1 Let A 1 = (a 1, b 1, c 1 ) ad A = (a, b, c ) be two sigle valued eutrosophic sets. If A 1 A, the K(A 1 ) K(A ). Proof. By Defiitio 3.18, we have that K(A 1 ) = 1+a 1 b 1 c 1 ad K(A ) = 1+a b c. Now, K(A ) K(A 1 ) = ((a a 1 ) + (b 1 b ) + (c 1 c ))/. Sice A 1 A, a 1 a, b 1 b, c 1 c ad hece (a a 1 ) 0, (b 1 b ) 0 ad (c 1 c ) 0. The it follows that K(A ) K(A 1 ) 0. Now, we defie a score fuctio for the rakig order of the iterval eutrosophic umbers (INSs). Defiitio 3. Let A = ([a, b], [c, d], [e, f]) be a iterval eutrosophic umber, a score fuctio L of a iterval eutrosophic value, based o the truth-membership degree, idetermiacy-membership degree ad falsity membership degree is defied by L(A ) = + a + b c d e f where L(A ) [ 1,1]. We give the followig example. (6) Example 3.3 Let A 1 = ([0.6,0.], [0.3,0.1], [0.1,0.3]) ad A = ([0.1,0.6], [0.,0.3], [0.1, 0.]) be two iterval eutrosophic values for two alteratives. The, by applyig Defiitio 3., we ca obtai L(A 1) = L(A ) = = 0.5, = 0.3. I this case we ca say that alterative A 1 is better tha A. Propositio 3. Let A = ([a, b], [c, d], [e, f]) be a iterval eutrosophic value. The the score fuctio L has some properties as follows: (i) L (A ) = 0 if ad oly if a + b = b + d + e + f. (ii) L (A ) = 1 if ad oly if a + b = b + d + e + f +. (iii) L (A ) = 1 if ad oly if a + b = b + d + e + f 6. Moreover, we have that L (A ) = 1, which A is the absolute iterval eutrosophic value, ad L (Φ) = 1, which Φ is the ull iterval eutrosophic value. Theorem 3.5 Let A 1 = ([a 1, b 1 ], [c 1, d 1 ], [e 1, f 1 ]) ad A = ([a, b ], [c, d ], [e, f ]) be two iterval eutrosophic sets. If A 1 A, the L(A 1) L(A ). Proof. By Defiitio 3., we have L(A 1) = +a 1 +b 1 c 1 d 1 e 1 f 1 Now, ad L(A ) = +a +b c d e f. L(A ) L(A 1) = (a a 1 ) + (b b 1 ) + (c 1 c ) + (d 1 d ) + (e 1 e ) + (d 1 d ). Sice A 1 A, a 1 a, b 1 b, c 1 c, d 1 d ad e 1 e, f 1 f ad hece (a a 1 ) 0, (b b 1 ) 0, (c 1 c ) 0, (d 1 d ) 0, (e 1 e ) 0ad (f 1 f ) 0. The it follows that L(A ) L(A 1) 0.. Rakig by accuracy fuctio Defiitio.6 Let A = (a, b, c) be a sigle valued eutrosophic umber, a accuracy fuctio M of a sigle valued eutrosophic value, based o the truth-membership degree, idetermiacy-membership degree ad falsity membership degree is defied by M(A) = a b(1 a) c(1 b) (7) where M(A) [ 1,1]. Example.7 Let A 1 = (0.5,0.,0.6) ad A = (0.6,0.,0.) be two sigle valued eutrosophic values for two alteratives. The, by applyig Defiitio.6, we ca obtai M(A 1 ) = 0.08 ad M(A ) = 0,3. I this case, we ca say that alterative A is better tha A 1. Now, we exted the cocept of accuracy fuctio to iterval eutrosophic umbers. Defiitio.8 Let A = ([a, b], [c, d], [e, f]) be a iterval eutrosophic umber. The a accuracy fuctio N of a iterval eutrosophic value, based o the truth-membership degree, idetermiacy-membership degree ad falsity membership degree is defied by N(A) = 1 (a + b d(1 b) c(1 a) f(1 c) e(1 d)) (8) where L(A) [ 1,1]. The accuracy fuctio N is reduced the accuracy fuctio proposed by Nayagam et al. ([13]) if c, d = 0 ad b + f 1. Example.9 Let A 1 = ([0.6,0.], [0.3,0.1], [0.1,0.3]) ad A = ([0.1,0.6], [0.,0.3], [0.1, 0.]) be two iterval eutrosophic values for two alteratives. The, by applyig Defiitio.8, we ca obtai M(A 1 ) = 0,6 ad M(A ) = 0,3. I this case we ca say that alterative A is better tha A 1. Accordig to score ad accuracy fuctios for SVNNs, we ca obtai the followig defiitios. 5

6 Defiitio.30 Suppose that A 1 = (a 1, b 1, c 1 ) ad A = (a, b, c ) are two sigle valued eutrosophic umber. The we defie the rakig method as follows: (i) If K(A 1 ) > K(A ), the A 1 > A. (ii) If K(A 1 ) = K(A ) ad L(A 1 ) > L(A ), the A 1 > A. Defiitio.31 Suppose that A 1 = ([a 1, b 1 ], [c 1, d 1 ], [e 1, f 1 ]) ad A = ([a, b ], [c, d ], [e, f ]) are two iterval eutrosophic sets The we defie the rakig method as follows: (i) If K(A 1) > K(A ), the A 1 > A. (ii) If K(A 1) = K(A ) ad L(A 1) > L(A ), the A 1 > A. Example.3 Let A 1 = (0.5,0.,0.6) ad A = (0.6,0.,0.) be two sigle valued eutrosophic values for two alteratives. The, by applyig Defiitio 3.18, we ca obtai K(A 1 ) = K(A ) = 0.6 ad L(A 1 ) = 0.6, L(A ) = The it implies that A 1 > A. From the above aalysis, we develop a method based o the score fuctio K ad the accuracy fuctio L for multi criteria decisio makig problem, which are criterio values for alteratives are the sigle valued eutrosophic value ad the iterval eutrosophic value, ad defie it as follows. 5. Multi-criteria eutrosophic decisio-makig method based o the score-accuracy fuctio Here, we propose a method for multi-criteria eutrosophic decisio makig problems with weights. Suppose that A = {A 1, A,..., A m } be the set of alteratives ad C = {C 1, C,..., C } be a set of criteria. Suppose that the weight of the criterio C s (s = 1,,, ), stated by the decisio-maker, is ω s, ω s [0,1] ad s=1 ω s =1. Thus, the characteristic of the alterative A k (k = 1,,, m) is itroduced by the followig SVNS ad INS, respectively: Method 1 A k = { C s, u Ak (C s ), w Ak (C s ), v Ak (C s ) : C s C} where 0 u Ak (C s ) + w Ak (C s ) + v Ak (C s ) 3, u Ak (C s ) 0, w Ak (C s ) 0, v Ak (C s ) 0, s = 1,,, ad k = 1,,, m. The SVNS value that is the triple of values for C s is deoted by α ks = (a ks, b ks, c ks ), where a ks idicates the degree that the alterative A k satisfies the criterio C s ad b ks idicates the degree that the alterative A k is idetermiacy o the criterio C s, where as c ks idicates the degree that the alterative A k does ot satisfy the criterio C s give by the decisio-maker. So we ca express a decisio matrix = (α ks ) m. The aggregatig sigle valued eutrosophic umber α k for A k (k = 1,,, m) is α k = (a k, b k, c k ) = F kω (A k1, A k,, A k ) or α k = (a k, b k, c k ) = G kω (A k1, A k,, A), which is obtaied by applyig Defiitio.8 or Defiitio.9 accordig to each row i the decisio matrix. We ca summarize the procedure of proposed method as follows: Step (1) Obtai the weighted arithmetic average values by usig Eq. (1) or the weighted geometric average values by Eq. () Step () Obtai the score (or accuracy) K(A k ) of sigle valued eutrosophic value α k (k = 1,,, m) by usig Eq. (5). Step (3) Rak the alterative A k = (k = 1,,, m) ad choose the best oe(s) accordig to (α k ) (k = 1,,, m). Method A k = { C s, [u A k (C s ), u + A k (C s )], [w A k (C s ), w + A k (C s )], [v A k (C s ), v + A k (C s )] : C s C} where 0 u + A k (C s ) + w + A k (C s ) + v + A k (C s ) 3, u A k (C s ) 0, w A k (C s ) 0, v A k (C s ) 0, s = 1,,, ad k = 1,,, m. The INS value that is the trible of itervals for C s is deoted by α ks = ([a ks, b ks ], [c ks, d ks ], [e ks, f ks ]), where [a ks, b ks ] idicates the degree that the alterative A k satisfies the criterio C s ad [c ks, d ks ] idicates the degree that the alterative A k is idetermiacy o the criterio C s, where as [e ks, f ks ] idicates the degree that the alterative A k does ot satisfy the criterio C s give by the decisio-maker. So we ca express a decisio matrix = (A ks ) m. The aggregatig iterval eutrosophic umber α k for A k (k = 1,,, m) is A k = ([a k, b k ], [c k, d k ], [e k, f k ]) = F kω (A k1, A k,, A k ) or A k = ([a k, b k ], [c k, d k ], [e k, f k ]) = G kω (A k1, A k,, A k ), which is obtaied by applyig Defiitio.15 or Defiitio.16 accordig to each row i the decisio matrix. We ca summarize the procedure of proposed method as follows: Step (1) Obtai the weighted arithmetic average values by usig Eq. (3) or the weighted geometric average values by Eq. (). Step () Obtai the score (or accuracy) L(A k) of iterval eutrosophic value A k (k = 1,,, m) by usig Eq. (6). Step (3) Rak the alterative A k = (k = 1,,, m) ad choose the best oe(s) accordig to (α k) (k = 1,,, m). 6

7 .1. Numerical examples Example 5.3 Let us cosider decisio makig problem adapted from [3]. There is a ivestmet compay, which wats to ivest a sum of moey i the best optio. There is a pael with four possible alteratives to ivest the moey: (1) A 1 is a food compay; () A is a car compay; (3) A 3 is a arms compay; () A is a computer compay. The ivestmet compay must make a decisio accordig to three criteria give below: (1) C 1 is the growth aalysis; () C is the risk aalysis; (3) C 3 is the evirometal impact aalysis. The, the weight vector of the criteria is give by are 0.35, 0.5 ad 0.0. Thus, whe the four possible alteratives with respect to the above three criteria are evaluated by the expert, we ca obtai the followig siglevalued eutrosophic decisio matrix: C 1 C C 3 A 1 (0.,0.,0.3) (0.,0.,0.3) (0.,0.,0.5) A (0.6,0.1,0.) (0.6,0.1,0.) (0.5,0.,0.) A 3 (0.3,0.,0.3) (0.5,0.,0.3) (0.5,0.3,0.) A (0.7,0.0,0.1) (0.6,0.1,0.) (0.,0.3,0.) Suppose that the weights of C 1, C ad C 3 are 0.35, 0.5 ad 0.0. The, we use the approach developed to obtai the most desirable alterative(s). Step (1) We ca compute the weighted arithmetic average value α k for A k = (k = 1,,3,) by usig Eq. (1) as follows: α 1 = (0.368,0.000,0.3680), α = (0.566,0.1319,0.000), α 3 = (0.375,0.35,0.550), α = (0.576,0.0000,0.1569). Step () By usig Eq. (5), we obtai K(α k ) (k = 1,,3,) as K(α 1 ) = 0.79, K(α ) = 0.59, K(α 3 ) = , K(α ) = Step (3) Rak all alteratives accordig to the accuracy degrees of K(α k ) (k = 1,,3,): A > A > A 3 > A 1. Thus the alterative A is the most desirable alterative based weighted arithmetic average operator. Now, assumig the same weights for C 1, C ad C 3, we use the weighted geometric average operator. Step (1) We ca obtai the weighted arithmetic average value α k for A k = (k = 1,,3,) by usig Eq. () as follows: α 1 = (0.97,0.000,0.367), α = (0.510,0.1860,0.161), α 3 = (0.38,0.000,0.60), α = (0.799,0.1555,0.161). 7 Step () By applyig Eq. (5), we obtai K(α k ) (k = 1,,3,) as K(α 1 ) = 0.311, K(α ) = 0.88, K(α 3 ) = 0.378, K(α ) = Step (3) Rak all alteratives accordig to the accuracy degrees of K(α k ) (k = 1,,3,): A > A > A 3 > A 1. Thus the alterative A is also the most desirable alterative based weighted geometric average operator. Example 5.33 Let us cosider decisio makig problem adapted from [30]. Suppose that there is a pael with four possible alteratives to ivest the moey: (1) A 1 is a food compay; () A is a car compay; (3) A 3 is a arms compay; () A is a computer compay. The ivestmet compay must make a decisio accordig to three criteria give below: (1) C 1 is the growth aalysis; () C is the risk aalysis; (3) C 3 is the evirometal impact aalysis. By usig the iterval-valued ituitioistic fuzzy iformatio, the decisio-maker has evaluated the four possible alteratives uder the above three criteria ad has listed i the followig matrix: C 1 C A 1 ([0.,0.5], [0.,0.3], [0.3,0.]) ([0.,0.6], [0.1,0.3], [0.,0.]) A ([0.6,0.7], [0.1,0.], [0.,0.3]) ([0.6,0.7], [0.1,0.], [0.,0.3]) A 3 ([0.3,0.6], [0.,0.3], [0.3,0.]) ([0.5,0.6], [0.,0.3], [0.3,0.]) A ([0.7,0.8], [0.0,0.1], [0.1,0.]) ([0.6,0.7], [0.1,0.], [0.1,0.3]) C 3 A 1 ([0.7,0.9], [0.,0.3], [0.,0.5]) A ([0.3,0.6], [0.3,0.5], [0.8,0.9]) A 3 ([0.,0.5], [0.,0.], [0.7,0.9]) A ([0.6,0.7], [0.3,0.], [0.8,0.9]) Suppose that the weights of C 1, C ad C 3 are 0.35, 0.5 ad 0.0. The, we use the approach developed to obtai the most desirable alterative(s). Step (1) We ca compute the weighted arithmetic average value α k for A k = (k = 1,,3,) by usig Eq. () as follows: α 1 = ([0.55,0.7516], [0.1681,0.3000], [0.301,0.373]), α = ([0.996,0.663], [0.1551,0.885], [0.38,0.655]), α 3 = ([0.396,0.566], [0.000,0.3365], [0.10,0.553]), α = ([0.6383,0.7396], [0.0000,0.070], [0.97,0.039]). Step () By usig Eq. (6), we obtai L(α k) (k = 1,,3,) as L(α 1) = 0.08, L(α ) = , L(α 3) = 0.75, L(α ) =

8 Step (3) Rak all alteratives accordig to the accuracy degrees of L(α k) (k = 1,,3,): A > A 1 > A > A 3. Thus the alterative A is the most desirable alterative based weighted arithmetic average operator. Now, assumig the same weights for C 1, C ad C 3, we use the weighted geometric average operator. Step (1) We ca obtai the weighted arithmetic average value α k for A k = (k = 1,,3,) by usig Eq. () as follows: α 1 = ([0.5003,0.660], [0.1760,0.3000], [0.3195,0.]), α = ([0.57,0.6581], [0.1860,0.3371], [0.505,0.6758]), α 3 = ([0.38,0.5578], [0.000,0.318], [0.501,0.7069]), α = ([0.633,0.733], [0.1555,0.569], [0.5068,0.663]). Step () By applyig Eq. (6), we obtai L(α k) (k = 1,,3,) as L(α 1) = 0.361, L(α ) = 0.118, L(α 3) = 0.161, L(α ) = Step (3) Rak all alteratives accordig to the accuracy degrees of L(α k) (k = 1,,3,): A 1 > A > A > A 3. Thus the alterative A 1 is also the most desirable alterative based weighted geometric average operator. Note that we obtai the differet rakigs for sigle valued eutrosophic iformatio ad iterval eutrosophic iformatio. From the examples, we ca see that the proposed eutrosophic decisio-makig method is more suitable for real scietific ad egieerig applicatios because it ca hadle ot oly icomplete iformatio but also the idetermiate iformatio ad icosistet iformatio existig i real situatios. The techique proposed i this paper exteds the existig decisio makig methods ad provides a ew way for decisio makers. 6. Compariso Aalysis ad Discussio I this sectio, we will a compariso aalysis to validate the feasibility of the proposed decisio makig method based o accuracy-score fuctios. To demostrate the relatioships, we utilize the same examples adapted from [3] ad [30]. The score ad accuracy fuctios has extremely importat for process of multi criteria decisio makig. But, util ow there have bee o may studies o multi-criteria decisio makig method based o accuracy-score fuctios, which are criterio values for alteratives are sigle valued eutrosophic sets or iterval eutrosophic sets. Ye [30] defied the similarity measures betwee INSs based o the relatioship betwee similarity measures ad distaces ad proposed the similarity measures betwee each alterative ad the ideal alterative to establish a multi criteria decisio makig method for INSs. After, Zhag et al. [6] preseted a method based o the aggregatio operators for multi criteria decisio makig uder iterval eutrosophic eviromet. By obtaiig the differet results tha give i [30], they showed that the method proposed is more precise ad reliable tha the result produced i [30]. Although the same rakig results with [6] are obtaied i here, the decisio makig method proposed i this paper has less calculatio ad it is more flexible ad more sustaiable for the multi criteria decisio makig with SVN or IVN iformatio. 7. Coclusios At preset, may score-accuracy fuctio techical are applied to the problems based o ituitioistic fuzzy iformatio or iterval valued ituitioistic fuzzy iformatio, but they could ot be used to hadle the problems based o eutrosophic iformatio. So, two measuremet fuctios such that score ad accuracy fuctios for sigle valued eutrosophic umbers ad iterval eutrosophic umbers is proposed i this paper, ad a multi-criteria decisio makig method based o this fuctios is established for eutrosophic iformatio. I decisio makig process, the eutrosophic weighted aggregatio operators (arithmetic ad geometric average operators) are adopted to aggregate the eutrosophic iformatio related to each alterative. Fially, some umerical examples are preseted to illustrate the applicatio of the proposed approaches. Refereces [1] Ataassov, K. (1986). Ituitioistic fuzzy sets. Fuzzy Sets ad Systems, 0, [] Ataassov, K. (199). Operators over iterval-valued ituitioistic fuzzy sets. Fuzzy Sets ad Systems, 6(), [3] Ataassov, K., & Gargov, G. (1989). Iterval-valued ituitioistic fuzzy sets. Fuzzy Sets ad Systems, 31(3), [] Broumi, S, ad Smaradache, F. (013). Correlatio coefficiet of iterval eutrosophic set, Appl. Mech. Mater [5] Che, S. M., & Lee, L. W. (011c). A ew method for multiattribute decisio makig usig iterval-valued ituitioistic fuzzy values. I Proceedigs of the 011 iteratioal coferece o machie learig ad cyberetics, Guili, Guagxi, Chia (pp ). [6] Zhag, H., Wag, J., ad Che, X. (01). Iterval Neutrosophic Sets ad its Applicatio i Multi-criteria Decisio Makig Problems. The Scietific World Joural, DOI: /01/ [7] Lakshmaa Gomathi Nayagam, V., Vekateshwari, G., & Sivarama, G. (008). Rakig of ituitioistic fuzzy umbers. I Proceedigs of the IEEE iteratioal coferece o fuzzy systems (IEEE FUZZ 008) (pp ). [8] Li, D.-F. (005). Multiattribute decisio makig models ad methods usig ituitioistic fuzzy sets. Joural of Computer ad System Scieces, 70, [9] Li, L., Yua, X.-H., & Xia, Z.-Q. (007). Multicriteria decisiomakig methods based o ituitioistic fuzzy sets.. J. Comput. Syst. Sci. 73,

9 [10] Liu, H. W., & Wag, G. J. (007). Multicriteria decisio-makig methods based o ituitioistic fuzzy sets. Europea Joural of Operatioal Research, 179, [11] Majumdar, P. ad Samata S.K. (01). O similarity ad etropy of eutrosophic sets, J. Itell. Fuzzy Syst. 6 (3) [1] Mitchell, H. B. (00). Rakig ituitioistic fuzzy umbers. Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge Based Systems, 1(3), [13] Nayagam, V. L. G., Muralikrisha, S., & Sivarama, G. (011). Multi-criteria decisio-makig method based o iterval-valued ituitioistic fuzzy sets. Expert Systems with Applicatios, 38(3), [1] Peg J.J., Wag J.Q., Wag J., Zhag H.Y., ad Che X.H. (015). Simplified eutrosophic sets ad their applicatios i multi-criteria group decisio-makig problems, It. J. Syst. Sci.( (i press). [15] Şahi, R. ad Küçük, A. (01). Subsethood measure for sigle valued eutrosophic sets, Joural of Itelliget ad Fuzzy Systems, DOI: /IFS [16] Smaradache, F. (1999). A uifyig field i logics. Neutrosophy: Neutrosophic probability, set ad logic, America Research Press, Rehoboth. [17] Smaradache, F. (005). A geeralizatio of the ituitioistic fuzzy set. Iteratioal joural of Pure ad Applied Mathematics,, [18] Xu, Z. S. (007a). Ituitioistic preferece relatios ad their applicatio i group decisio makig. Iformatio Sciece, 177(11), [19] Xu, Z. S. (007b). Ituitioistic fuzzy aggregatio operators. IEEE Trasactios o Fuzzy Systems, 15(6), [0] Xu, Z. S. (007c). Methods for aggregatig iterval-valued ituitioistic fuzzy iformatio ad their applicatio to decisio makig. Cotrol ad Decisio, (), [1] Xu, Z. S., & Che, J. (007a). A approach to group decisio makig based o iterval-valued ituitioistic fuzzy judgmet matrices. System Egieer Theory ad Practice, 7(), [] Xu, Z. S., & Che, J. (007b). O geometric aggregatio over iterval-valued ituitioistic fuzzy iformatio. I FSKD, th iteratioal coferece o fuzzy systems ad kowledge discovery (FSKD 007) (Vol., pp ). [3] Xu, Z. S., & Yager, R. R. (006). Some geometric aggregatio operators based o ituitioistic fuzzy sets. Iteratioal Joural of Geeral System, 35, [] Wag J.Q., Li J.J. (011) Multi-criteria fuzzy decisio-makig method based o cross etropy ad score fuctios. Expert Syst. Appl. 38: [5] Wag, H., Smaradache, F., Zhag, Y. Q., ad Suderrama, R., (010. Sigle valued eutrosophic sets, Multispace ad Multistructure () [6] Wag, H., Smaradache, F., Zhag, Y. Q. ad Suderrama. (005). Iterval eutrosophic sets ad logic: Theory ad applicatios i computig, Hexis, Phoeix, AZ. [7] Wu JZ, Zhag Q (011) Multicriteria decisio makig method based o ituitioistic fuzzy weighted etropy. Expert Syst. Appl. 38:916 9 [8] Ye, J. (009). Multicriteria fuzzy decisio-makig method based o a ovel accuracy fuctio uder iterval-valued ituitioistic fuzzy eviromet. Expert Systems with Applicatios, 36, [9] Ye, J. (010). Multicriteria fuzzy decisio-makig method usig etropy weights-based correlatio coefficiets of iterval-valued ituitioistic fuzzy sets, Applied Mathematical Modellig, 3 (1) [30] Ye, J. (01). Similarity measures betwee iterval eutrosophic sets ad their applicatios i Multi-criteria decisio-makig. Joural of Itelliget ad Fuzzy Systems [31] Ye, J. (01). Sigle valued eutrosophic cross-etropy for multi-criteria decisio makig problems, Appl. Math. Model. 38 (3) [3] Ye, J. (013). Multicriteria decisio-makig method usig the correlatio coefficiet uder sigle-valued eutrosophic eviromet, Iteratioal Joural of Geeral Systems () [33] Ye, J. (01). A multicriteria decisio-makig method usig aggregatio operators for simplified eutrosophic sets, J. Itell. Fuzzy Syst. 6 (5) [3] Zadeh, L. A. (1965). Fuzzy sets. Iformatio ad Cotrol, 8(3),