Bipolar Neutrosophic Sets And Their Application Based On Multi-Criteria Decision Making Problems.

Transcription

1 Bipolar Neutrosophic Sets Ad Their Applicatio Based O Multi-Criteria Decisio Makig Problems Irfa Deli 1, Mumtaz Ali 2 ad Floreti Smaradache 1 Muallim Rıfat Faculty of Educatio, Kilis 7 Aralık Uiversity, Kilis, Turkey, irfadeli@kilis.edu.tr 2 Departmet of Mathematics, Quaid-e-Azam Uiversity Islamabad. mumtazali7288@gmail.com 3 Uiversity of New Mexico, 705 Gurley Ave., Gallup, New Mexico 87301, USA. fsmaradache@gmail.com ABSTRACT I this paper, we itroduce cocept of bipolar eutrosophic set ad its some operatios. Also, we propose score, certaity ad accuracy fuctios to compare the bipolar eutrosophic sets. The, we develop the bipolar eutrosophic weighted average operator (A w ) ad bipolar eutrosophic weighted geometric operator (G w ) to aggregate the bipolar eutrosophic iformatio. Furthermore, based o the A w ad G w operators ad the score, certaity ad accuracy fuctios, we develop a bipolar eutrosophic multiple criteria decisio-makig approach, i which the evaluatio values of alteratives o the attributes take the form of bipolar eutrosophic umbers to select the most desirable oe(s). Fially, a umerical example of the method was give to demostrate the applicatio ad effectiveess of the developed method. Keywords: Neutrosophic set, bipolar eutrosophic set, average operator, geometric operator, score, certaity ad accuracy fuctios, multi-criteria decisio makig. 1. Itroductio To hadle with imprecisio ad ucertaity, cocept of fuzzy sets ad ituitioistic fuzzy sets origially itroduced by Zadeh [26] ad Ataassov [1], respectively. The, Smaradache [17] proposed cocept of eutrosophic set which is geeralizatio of fuzzy set theory ad ituitioistic fuzzy sets. These sets models have bee studied by may authors; o applicatio [4-6,10-12,15,16], theory [18-20,21-25,27,28], ad so o. Bosc ad Pivert [2] said that Bipolarity refers to the propesity of the huma mid to reaso ad make decisios o the basis of positive ad egative affects. Positive iformatio states what is possible, satisfactory, permitted, desired, or cosidered as beig acceptable. O the other had, egative statemets express what is impossible, rejected, or forbidde. Negative prefereces correspod to costraits, sice they specify which values or objects have to be rejected (i.e., those that do ot satisfy the costraits), while positive prefereces correspod to wishes, as they specify 1

2 which objects are more desirable tha others (i.e., satisfy user wishes) without rejectig those that do ot meet the wishes. Therefore, Lee [8,9] itroduced the cocept of bipolar fuzzy sets which is a geeralizatio of the fuzzy sets. Recetly, bipolar fuzzy models have bee studied by may authors o algebraic structures such as; Che et. al. [3] studied of m-polar fuzzy set ad illustrates how may cocepts have bee defied based o bipolar fuzzy sets. The, they examied may results which are related to these cocepts ca be geeralized to the case of m-polar fuzzy sets. They also proposed umerical examples to show how to apply m-polar fuzzy sets i real world problems. Bosc ad Pivert [2] itroduced a study is called bipolar fuzzy relatios where each tuple is associated with a pair of satisfactio degrees. Maemara ad Chellappa [14] gave some applicatios of bipolar fuzzy sets i groups are called the bipolar fuzzy groups, fuzzy d-ideals of groups uder (T-S) orm. They ivestigate some related properties of the groups ad itroduced relatios betwee a bipolar fuzzy group ad bipolar fuzzy d-ideals. Majumder [13] proposed bipolar valued fuzzy subsemigroup, bipolar valued fuzzy bi-ideal, bipolar valued fuzzy (1,2)- ideal ad bipolar valued fuzzy ideal. Zhou ad Li [29] itroduced a ew framework of bipolar fuzzy subsemirigs ad bipolar fuzzy ideals which is a geeralizatio of fuzzy subsemirigs ad bipolar fuzzy ideals i semirigs ad ad bipolar fuzzy ideals, respectively, ad related properties are examied by the authors. I this paper, we itroduced the cocept of bipolar eutrosophic sets which is a extesio of the fuzzy sets, bipolar fuzzy sets, ituitioistic fuzzy sets ad eutrosophic sets. Also, we give some operatios ad operators o the bipolar eutrosophic sets. The operatios ad operators geeralizes the operatios ad operators of fuzzy sets, bipolar fuzzy sets, ituitioistic fuzzy sets ad eutrosophic sets which has bee previously proposed. Therefore, i sectio 2, we itroduce cocept of bipolar eutrosophic set ad its some operatios icludig the score, certaity ad accuracy fuctios to compare the bipolar eutrosophic sets. I the same sectio, we also develop the bipolar eutrosophic weighted average operator (A w ) ad bipolar eutrosophic weighted geometric operator(g w ) operator to aggregate the bipolar eutrosophic iformatio. I sectio 3, based o the A w ad G w operators ad the score, certaity ad accuracy fuctios, we develop a bipolar eutrosophic multiple criteria decisio-makig approach, i which the evaluatio values of alteratives o the attributes take the form of bipolar eutrosophic umbers to select the most desirable oe(s) ad give a umerical example of the to demostrate the applicatio ad effectiveess of the developed method. I last sectio, we coclude the paper. 2. Prelimiaries I the subsectio, we give some cocepts related to eutrosophic sets ad bipolar sets. Defiitio 2.1. [17] Let X be a uiverse of discourse. The a eutrosophic set is defied as: A = { x, F A (x), T A (x), I A (x) : x X}, which is characterized by a truth-membership fuctio T A : X ]0, 1 + [, a idetermiacymembership fuctio I A : X ]0, 1 + [ad a falsity-membership fuctio F A : 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 +. For applicatio i real scietific ad egieerig areas, Wag et al.[18] proposed the cocept of a sigle valued eutrosophic set as follows; 2

3 Defiitio 2.1. [18]Let X be a uiverse of discourse. The a sigle valued eutrosophic set is defied as: A NS = { x, F A (x), T A (x), I A (x) : x X}, which is characterized by a truth-membership fuctio T A : X [0,1], a idetermiacymembership fuctio I A : X [0,1] ad a falsity-membership fuctio F A : 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. Set- theoretic operatios, for two sigle valued eutrosophic set, A NS = {<x, T A (x), I A (x), F A (x)> x X } ad B NS = {<x, T B (x), I B (x), F B (x)> x X } are give as; 1. The subset; A NS B NS if ad oly if T A (x) T B (x), I A (x) I B (x), F A (x) F B (x). 2. A NS = B NS if ad oly if, T A (x) =T B (x),i A (x) =I B (x),f A (x) =F B (x) for ayx X. o 3. The complemet of A NS is deoted by A NS 4. The itersectio ad is defied by A o NS = {<x, F A (x), 1 I A (x), T A (x) x X } A NS B NS = {<x, mi{t A (x), T B (x)},max{i A (x), I B (x)}, max{f A (x), F B (x)}>:x X } 5. The uio A NS B NS = {<x, max{t A (x), T B (x)},mi{i A (x), I B (x)}, mi{f A (x), F B (x)}>:x X } A sigle valued eutrosophic umber is deoted by à = T, I, F for coveiece. Defiitio 2.2. [15] Let à 1 = T 1, I 1, F 1 ad à 2 = T 2,I 2, F 2 be two sigle valued eutrosophic umber. The, the operatios for NNs are defied as below; i. λã = 1 (1 T 1 ) λ, I λ λ 1, F 1 ii. à λ 1 = T λ 1, 1 (1 I 1 ) λ, 1 (1 F 1 ) λ iii. à 1 + à 2 = T 1 + T 2 T 1 T 2, I 1 I 2, F 1 F 2 3

4 iv. à 1. à 2 = T 1 T 2, I 1 +I 2 I 1 I 2, F 1 + F 2 F 1 F 2 where 0. Defiitio 2.3. [15] Let à 1 = T 1, I 1, F 1 be a sigle valued eutrosophic umber. The, the score fuctio s(ã 1 ), accuracy fuctio a(ã 1 ) ad certaity fuctio c(ã 1 ) of a SNN are defied as follows: i. s(ã 1 ) = (T I F 1 )/3; ii. a(ã 1 ) = T 1 F 1 ; iii. c(ã 1 ) = T 1 Defiitio 2.4. [15] Let à 1 = T 1, I 1, F 1 ad à 2 = T 2,I 2, F 2 be two sigle valued eutrosophic umber. The compariso method ca be defied as follows: i. if s(ã 1 ) > s(ã 2 ), the à 1 is greater tha à 2, that is, à 1 is superior to à 2, deoted by à 1 >à 2 ii. if s(ã 1 ) = s(ã 2 ) ad a(ã 1 ) > a(ã 2 ), the à 1 is greater tha à 2, that is, à 1 is superior to à 2, deoted by à 1 < à 2 ; iii. if s(ã 1 ) = s(ã 2 ), a(ã 1 ) = a(ã 2 ) ad c(ã 1 ) > c(ã 2 ), the à 1 is greater tha à 2, that is, à 1 is superior to à 2, deoted by à 1 >à 2 ; iv. if s(ã 1 ) = s(ã 2 ), a(ã 1 ) = a(ã 2 ) ad c(ã 1 ) = c(ã 2 ), the à 1 is equal to à 2, that is, à 1 is idifferet to à 2, deoted by à 1 =à 2. Defiitio 2.4. [6,14] Let X be a o-empty set. The, a bipolar-valued fuzzy set, deoted by A BF, is dified as; A BF = { x, µ B + (x), µ B (x) : x X} where µ B + : X [0,1] ad µ B : X [0,1]. The positive membership degree µ B + deotes the satisfactio degree of a elemet x to the property correspodig to A BF ad the egative membership degree µ B (x) deotes the satisfactio degree of x to some implicit couter property of A BF. 3. Bipolar Neutrosophic Set I this sectio, we itroduce cocept of bipolar eutrosophic set ad its some operatios icludig the score, certaity ad accuracy fuctios to compare the bipolar eutrosophic sets. We also develop the bipolar eutrosophic weighted average operator (A w ) ad bipolar eutrosophic weighted geometric operator ( G w ) operator to aggregate the bipolar eutrosophic iformatio. Some of it is quoted from [2,6,8,9,14,17,18,20,24,26]. Defiitio 3.1. A bipolar eutrosophic set A i X is defied as a object of the form A x, T, I, F, T x, I x, F : x X, where T, I, F : X 1,0 ad T, I, F : X 1,0. 4

5 The positive membership degree T, I, F deotes the truth membership, idetermiate membership ad false membership of a elemet x X correspodig to a bipolar eutrosophic set A ad the egative membership degree T, I, F deotes the truth membership, idetermiate membership ad false membership of a elemet x X to some implicit couterproperty correspodig to a bipolar eutrosophic set A. Example 3.2. Let X { x1, x2, x3}. The x1,0.5,0.3,0.1, 0.6, 0.4, 0.01, A x2,0.3,0.2,0.7, 0.02, 0.003, 0.5, x3,0.8,0.05,0.4, 0.1, 0.5, 0.06 is a bipolar eutrosophic subset of X. Theorem 3.4. A bipolar eutrosophic set is the geeralizatio of a bipolar fuzzy set. Proof: Suppose that X is a bipolar eutrosophic set. The by settig the positive compoets I, F equals to zero as well as the egative compoets I, F equals to zero reduces the bipolar eutrosophic set to bipolar fuzzy set. Defiitio 3.5. Let A 1 = x,t + 1 (x), I + 1 (x), F + 1 (x), T 1 (x), I 1 (x), F 1 (x) ad A 2 = x,t + 2 (x),i + 2 (x), F + 2 (x), T 2 (x), I 2 (x), F 2 (x) be two bipolar eutrosophic sets. The A1 A2 if ad oly if ad T x T x 1( ) 2 ( ) 1 2 I I, F 1 F2, T 1 T2, I 1 I 2, F1 F2 for all x X. Defiitio 3.6. Let A 1 = x,t + 1 (x), I + 1 (x), F + 1 (x), T 1 (x), I 1 (x), F 1 (x) ad A 2 = x,t + 2 (x),i + 2 (x), F + 2 (x), T 2 (x), I 2 (x), F 2 (x) be two bipolar eutrosophic set. The A 1 = A 2 if ad oly if ad T 1 T2, I 1 I2, F 1 F2, T 1 T2, I 1 I 2, F1 F2 for all x X. Defiitio 3.7. Let A 1 = x,t + 1 (x), I + 1 (x), F + 1 (x), T 1 (x), I 1 (x), F 1 (x) ad A 2 = x,t + 2 (x),i + 2 (x), F + 2 (x), T 2 (x), I 2 (x), F 2 (x) be two bipolar eutrosophic set. The their uio is defied as: I1 I2 I1 I2 ( A1 A2 ) max( T1, T2 ),,mi(( F1, F2 ),mi(t 1, T2 ),,max(( F1, F2 ) 2 2 for all x X. Example 3.8. Let X { x1, x2, x3}. The 5

6 ad x1, 0.5, 0.3, 0.1, 0.6, 0.4, 0.01, A1 x2, 0.3, 0.2, 0.7, 0.02, 0.003, 0.5, x3, 0.8, 0.05, 0.4, 0.1, 0.5, 0.06 x1, 0.4, 0.6, 0.3, 0.3, 0.5, 0.1, A2 x2, 0.5, 0.1, 0.4, 0.2, 0.3, 0.3, x3, 0.2, 0.5, 0.6, 0.4, 0.6, 0.7 are two bipolar eutrosophic sets i X. The their uio is give as follows: x1, 0.5, 0.45, 0.1, 0.6, 0.5, 0.1, x2, 0.5, 0.15, 0.7, 0.2, , 0.5, A1A2 x3, 0.8, 0.47, 0.6, 0.4, 0.55, 0.7 Defiitio 3.9. Let A 1 = x,t 1 + (x), I 1 + (x), F 1 + (x), T 1 (x), I 1 (x), F 1 (x) ad A 2 = x,t 2 + (x),i 2 + (x), F 2 + (x), T 2 (x), I 2 (x), F 2 (x) be two bipolar eutrosophic set. The their itersectio is defied as: I1 I2 I1 I2 ( A1 A2 ) mi( T1, T2 ),,max(( F1, F2 ),max(t 1, T2 ),,mi(( F1, F2 ) 2 2 for all x X. Defiitio Let A x, T, I, F, T x, I x, F : x X be a bipolar eutrosophic set i X. The the complemet of A is deoted by ad c A ad is defied by T {1 } T, I {1 } I, F {1 } F A c A A c A A c A T {1 } T, I {1 } I, F {1 } F, A c A for all x X. Example Let X { x1, x2, x3}. The A c A A c A x1,0.5,0.3,0.1, 0.6, 0.4, 0.01, A x2,0.3,0.2,0.7, 0.02, 0.003, 0.5, x3,0.8,0.05,0.4, 0.1, 0.5, 0.06 is a bipolar eutrosophic set i X. The the complemet of A is give as follows: A c x1, 0.5, 0.7, 0.9, 0.4, 0.6, 0.99, x2, 0.7, 0.8, 0.3, 0.08, 0.997, 0.5,. x3, 0.2, 0.95, 0.6, 0.9, 0.5, 0.94 We will deote the set of all the bipolar eutrosophic sets (NBSs) i X by Q. A bipolar eutrosophic umber (NBN) is deoted by a = T, I, F, T, I, F for coveiece. 6

7 Defiitio Let a 1 = T 1 +, I 1 +, F 1 +, T 1, I 1, F 1 ad a 2 = T 2 +, I 2 +, F 2 +, T 2, I 2, F 2 be two bipolar eutrosophic umber. The the operatios for NNs are defied as below; i. λa 1 = 1 (1 T 1 + ) λ, (I 1 + ) λ, (F 1 + ) λ, ( T 1 ) λ, ( I 1 ) λ, (1 (1 ( F 1 )) λ ) ii. a 1λ = (T + 1 ) λ, 1 (1 I + 1 ) λ, 1 (1 F + 1 ) λ, (1 (1 ( T 1 )) λ ), ( I 1 ) λ, ( F 1 ) λ iii. a 1 + a 2 = T + 1 +T + 2 T + 1 T + 2, I + 1 I + 2, F + 1 F + 2, T 1 T 2, ( I 1 I 2 I 1 I 2 ), ( F 1 F 2 F 1 F 2 ) iv. a 1. a 2 = T + 1 T + 2, I + 1 +I + 2 I + 1 I + 2, F + 1 +F + 2 F + 1 F + 2, ( T 1 T 2 T 2 T 2 ), I 1 I 2, F 1 F 2 where 0. Defiitio Let a 1 = T + 1, I + 1, F + 1, T 1, I 1, F 1 be a bipolar eutrosophic umber. The, the score fuctio s(a 1), accuracy fuctio a(a 1) ad certaity fuctio c(a 1) of a NBN are defied as follows: i. s (a 1)= (T I F T 1 I 1 F 1 )/6 ii. a (a 1) = T + 1 F T 1 F 1 iii. c (a 1) = T + 1 F 1 Defiitio a 1 = T 1 +, I 1 +, F 1 +, T 1, I 1, F 1 ad a 2 = T 2 +,I 2 +, F 2 +, T 2, I 2, F 2 be two bipolar eutrosophic umber. The compariso method ca be defied as follows: i. if s (a 1) > s (a 2), the a 1 is greater tha a 2, that is, a 1 is superior to a 2, deoted by a 1 >a 2 ii. s (a 1) = s (a 2) ad a (a 1) > a (a 2), the a 1 is greater tha a 2, that is, a 1 is superior to a 2, deoted by a 1 < a 2; iii. if s (a 1) = s (a 2), a (a 1) = a (a 1) ad c (a 1) > c (a 2), the a 1 is greater tha a 2, that is, a 1 is superior to a 2, deoted by a 1>a 2; iv. if s (a 1) = s (a 2), a (a 1) = a (a 2)) ad c (a 1) = c (a 2), the a 1 is equal to a 2, that is, a 1 is idifferet to a 2, deoted by a 1=a 2. Based o the study give i [15,20] we defie some weighted aggregatio operators related to bipolar eutrosophic sets as follows; Defiitio Let a j = T,,,,, j I j F j T j I j F (j = 1,2,, ) be a family of bipolar j eutrosophic umbers. A mappig A ω : Q Q is called bipolar eutrosophic weighted average operator if it satisfies A w (a 1, a 2,, a )= j=1 ω j a j = 1 (1 T + j ) ω j + j=1, I ω j + j=1 j, F ω j j=1 j, ( T j ) ω j j=1 (1 ( I ω j j=1 j )) ), (1 (1 ( F j )) ω j j=1 ), (1 7

8 where ω j is the weight of a j (j = 1,2,, ), ω j [0,1] ad j=1 ω j = 1. Based o the study give i [15,20] we give the theorem related to bipolar eutrosophic sets as follows; Theorem Let a j = T,,,,, j I j F j T j I j F (j = 1,2,, ) be a family of bipolar j eutrosophic umbers. The, i. If a j = a for all j = 1,2,, the, A w (a 1, a 2,, a ) = a ii. mi j=1,2,, {a j} A w (a 1, a 2,, a ) max j=1,2,, {a j} iii. If a j a j for all j = 1,2,, the, A w (a 1, a 2,, a ) A w (a 1, a 2,, a ) Based o the study give i [15,20] we defie some weighted aggregatio operators related to bipolar eutrosophic sets as follows; Defiitio Let a j = T,,,,, j I j F j T j I j F (j = 1,2,, ) be a family of bipolar j eutrosophic umbers. A mappig G ω : Q Q is called bipolar eutrosophic weighted geometric operator if it satisfies G w ( a 1, a 2,, a )= j + j=1 a jω = T ω j j=1 j, 1 (1 I + ω j j=1 j ), 1 (1 F + j ) ωj j=1, (1 (1 ( T j )) ω j j=1, ( I j ) ω j ω j=1, ( F j j=1 j ) where ω j is the weight of a j (j = 1,2,, ), ω j [0,1] ad j=1 ω j = 1. Based o the study give i [15,20] we give the theorem related to bipolar eutrosophic sets as follows; Theorem Let a j = T,,,,, j I j F j T j I j F (j = 1,2,, ) be a family of bipolar j eutrosophic umbers. The, i. If a j = a for all j = 1,2,, the, G w (a 1, a 2,, a ) = a ii. mi j=1,2,, {a j} G w (a 1, a 2,, a ) max j=1,2,, {a j} iii. If a j a j for all j = 1,2,, the, G w (a 1, a 2,, a ) G w (a 1, a 2,, a ) Note that the aggregatio results are still NBNs 8

9 4. NBN- Decisio Makig Method I this sectio, we develop a approach based o the A w (or G w ) operator ad the above rakig method to deal with multiple criteria decisio makig problems with bipolar eutrosophic iformatio. Suppose that A = {A 1, A 2,, A m } ad C = {C 1, C 2,, C } is the set of alteratives ad criterios or attributes, respectively. Let ω = (ω 1, ω 2,, ω ) T be the weight vector of attributes, such that j=1 ω j = 1, ω j 0 (j = 1,2,, ) ad ω j refers to the weight of attribute C j. A alterative o criterios is evaluated by the decisio maker, ad the evaluatio values are represeted by the form of bipolar eutrosophic umbers. Assume that (a ij ) = (,,,,, m T ij I ij F ij T ij I ij F ij ) m the decisio matrix provided by the decisio maker; a ij is a bipolar eutrosophic umber for T ij I ij alterative A i associated with the criterios C j. We have the coditios,, F [0,1] such that 0 ij ij ij ij ij ij ij T I F T I F F T I,, ij ij ij is ad 6 for i = 1,2,, m ad j = 1,2,,. Now, we ca develop a algorithm as follows; Algorithm Step1. Costruct the decisio matrix provided by the decisio maker as; (a ij ) = (,,,,, m T ij I ij F ij T ij I ij F ij ) m Step 2. Compute a i = A w (a i1, a i2,, a i ) (or G w (a i1, a i2,, a i )) for each i = 1,2,, m. Step 3. Calculate the score values of s (a 1 ) (i = 1,2,, m.) for the collective overall bipolar eutrosophic umber of a i (i = 1,2,, m.) Step 4. Rak all the software systems of a i (i = 1,2,, m.) accordig to the score values Now, we give a umerical example as follows; Example 4.1. Let us cosider decisio makig problem adapted from Xu ad Cia [20]. A customer who iteds to buy a car. Four types of cars (alteratives) A i (i = 1,2,3,4) are available. The customer takes ito accout four attributes to evaluate the alteratives; C 1 = Fuel ecoomy; C 2 =Aerod; C 3 =Comfort; C 4 =Safety ad use the bipolar eutrosophic values to evaluate the four possible alteratives A i (i = 1, 2, 3, 4) uder the above four attributes. Also, the weight vector of the attributes C j (j = 1,2,3,4) is ω = ( 1 2, 1 4, 1 8, 1 8 )T. The, 9

10 Algorithm Step1. Costruct the decisio matrix provided by the customer as; Table 1: Decisio matrix give by customer C 1 C 2 C 3 C 4 A 1 0.5,0.7,0.2, 0.7, 0.3, ,0.4,0.5, 0.7, 0.8, ,0.7,0.5, 0.8, 0.7, ,0.5,0.7, 0.5, 0.2, 0.8 A 2 0.9,0.7,0.5, 0.7, 0.7, ,0.6,0.8, 0.7, 0.5, ,0.4,0.6, 0.1, 0.7, ,0.2,0.7, 0.5, 0.1, 0.9 A 3 0.3,0.4,0.2, 0.6, 0.3, ,0.2,0.2, 0.4, 0.7, ,0.5,0.5, 0.6, 0.5, ,0.5,0.3, 0.4, 0.2, 0.2 A 4 0.9,0.7,0.2, 0.8, 0.6, ,0.5,0.2, 0.5, 0.5, ,0.4,0.5, 0.1, 0.7, ,0.2,0.8, 0.5, 0.5, 0.6 Step 2. Compute ã i = A w (ã i1, ã i2, ã i3, ã i4 ) for each i = 1,2,3,4 as; ã 1 = 0.471,0.583,0.329, 0.682, 0.531, ã 2 = 0.839,0.536,0.600, 0.526, 0.608, ã 3 = 0.489,0.355,0.235, 0.515, 0.447, ã 4 = 0.751,0.513,0.266, 0.517, 0.580, Step 3. Calculate the score values of s ( ã 1 ) ( i = 1,2,3,4) for the collective overall bipolar eutrosophic umber of ã i (i = 1,2,, m.) as; s (ã 1 )=0.50 s (ã 2 )=0.52 s (ã 3 )=0.56 s (ã 4 )=0.54 Step 4. Rak all the software systems of A i (i = 1,2,3,4.) accordig to the score values as; A 3 A 4 A 2 A 1 ad thus A 3 is the most desirable alterative. 5. Coclusios This paper preseted a bipolar eutrosophic set ad its score, certaity ad accuracy fuctios. The, the A w ad G w operators were proposed to aggregate the bipolar eutrosophic iformatio. Furthermore, based o the A w ad G w operators ad the score, certaity ad accuracy fuctios, we have developed a bipolar eutrosophic multiple criteria decisio-makig approach, i which the evaluatio values of alteratives o the attributes take the form of bipolar eutrosophic umbers. The A w ad G w operators are utilized to aggregate the bipolar eutrosophic iformatio correspodig to each alterative to obtai the collective overall values of the alteratives, ad the the alteratives are raked accordig to the values of the score, certaity ad accuracy fuctios to 10

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