Rough Neutrosophic Multisets Relation with Application in Marketing Strategy

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1 Neutrosophc Sets and Systems, Vol. 21, 2018 Unversty of New Mexco 36 Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy Surana Alas 1, Daud Mohamad 2, and Adbah Shub 3 1 Faculty of Computer and Mathematcal Scences, Unverst Teknolog Mara (UTM) Kelantan, Campus Machang, Kelantan, 18500, Malaysa. E-mal: sura588@kelantan.utm.edu.my 2 Faculty of Computer and Mathematcal Scences, Unverst Teknolog Mara (UTM) Shah Alam, Shah Alam, 40450, Malaysa. E-mal: daud@tmsk.utm.edu.my 3 Faculty of Computer and Mathematcal Scences, Unverst Teknolog Mara (UTM) Shah Alam, Shah Alam, 40450, Malaysa. E-mal: adbah@tmsk.utm.edu.my Abstract. The concepts of rough neutrosophc multsets can be easly extended to a relaton, manly snce a relaton s also a set,.e. a subset of a Cartesan product. Therefore, the objectve of ths paper s to defne the defnton of rough neutrosophc multsets relaton of Cartesan product over a unversal set. Some of the relaton propertes of rough neutrosophc multsets such as max, mn, the composton of two rough neutrosophc multsets relaton, nverse rough neutrosophc multsets relaton, and reflexve, symmetrc and transtve rough neutrosophc multsets relaton over the unverse are defned. Subsequently, ther propertes are successfully proven. Fnally, the applcaton of rough neutrosophc multsets relaton for decson makng n marketng strategy s presented. Keywords: Neutrosophc Multsets, relaton, rough set, rough neutrosophc multsets 1 Introducton Imperfect nformaton resulted n an ncomplete, mprecson, nconsstency and uncertanty nformaton whereby all the condton must be overcome to represent the perfect nformaton. A relaton between each nformaton from the same unverse or object s also an mportant crteron of the nformaton to explan the strong relatonshp element between them. Fuzzy sets as defned by Zadeh [1] has been used to model the mperfect nformaton especally for uncertanty types of nformaton by representng the membershp value between [0, 1]. Ths ndcates the human thnkng opnon by replacng the nformaton of lngustc value. Many theores were later ntroduced wth the am of establshng a fuzzy relaton structure [2]. Attanassov ntroduced an ntutonstc fuzzy set by generalzng the theory of fuzzy sets and ntroducng two grades of the membershp functon, namely the degree of membershp functon and degree of non-membershp functon [3]. Ths theory has made the uncertanty decson more nterestng. Meanwhle, Burllo et al. studed the ntutonstc fuzzy relaton wth propertes [4]. There are also another theory ntroduced for solvng uncertanty condton such as rough set [5] and soft set [6]. All these studes have extended to rough relaton [7] and soft set relaton [8]. Smarandache ntroduced a neutrosophc set as a generalzaton of the ntutonstc fuzzy set theory [9]. He beleved that somehow n a lfe stuaton, especally for uncertanty condton, there also exst n-between (ndetermnacy) opnon or unexpected condton that cannot be controlled. Instead of two grades of the membershp functon, neutrosophc set ntroduced n-between (ndetermnacy) functon where there exsts an element whch conssts of a set of truth membershp functon (T), ndetermnacy functon (I) and falsty membershp functon (F). Compared to other uncertanty theores, the neutrosophc set can deal wth ndetermnacy stuaton. The study n neutrosophc relaton wth propertes are also dscussed [10], [11]. Later, Smarandache et al. refned T, I, F to T 1, T 2,, T m and I 1, I 2,, I n and F 1, F 2,, F r was also known as a neutrosophc refned set or neutrosophc multsets [12], [13]. Instead of one-tme occurrng for each element of T, I, F, the neutrosophc refned set allowed an element of T, I, F to occur more than once wth Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

2 Neutrosophc Sets and Systems, Vol. 21, possbly the same or dfferent truth membershp values, ndetermnacy values, and falsty membershp values. The study of the neutrosophc refned set s a generalzaton of a mult fuzzy set [14] and ntutonstc fuzzy multsets [15]. Later, Del et al. have studed the relaton on neutrosophc refned set wth propertes [16], [17]. Latest, Smarandache has dscussed n detal about neutrosophc perspectves n theory and applcaton parts for neutrosophc trplets, neutrosophc duplets, neutrosophc multsets, hybrd operators and modal logc [18]. The successful applcaton of the neutrosophc refned set n mult crtera decson makng problem such as n medcal dagnoss and selecton problem [13], [19] [25] has made ths theory more applcable n decson makng area. Hybrd theores of uncertanty and mprecson condton were ntroduced, especally wth rough set theory, such as rough fuzzy set and fuzzy rough set [26], rough ntutonstc fuzzy set [27], ntutonstc rough fuzzy set [28], rough neutrosophc set [29], neutrosophc rough set [30], nterval rough neutrosophc set [31], rough neutrosophc soft set [32], rough bpolar neutrosophc set [33], sngle valued neutrosophc rough set model [34] and rough neutrosophc multset [35]. Ths s because a rough set theory can handle the mprecson condton from the exstence of a value whch cannot be measured wth sutable precson. Samanta et al. have dscussed the fuzzy rough relaton on unverse set and ther propertes [36]. Then, Xuan Thao et. al have extended that concept by ntroducng the rough fuzzy relatons on the Cartesan product of two unversal sets [37]. The hybrd theory of a rough set also gves a contrbuton for solvng a problem n decson makng area. Some researchers already proved that hybrd theory such as rough neutrosophc set can handle the decson makng problem n order to get the best soluton accordng to three membershp degree (truth, ndetermnate and falsty) [38] [44]. The objectve of ths paper s to defne a rough neutrosophc multsets relaton propertes as a novel noton. Ths study also generalzes relaton propertes of a rough fuzzy relaton, rough ntutonstc fuzzy relaton and rough neutrosophc relaton over unversal. Subsequently, ther propertes are examned. The remanng parts of ths paper are organzed as follows. In secton 2, some mathematcal prelmnary concepts were recalled for a deeper understandng of rough neutrosophc multsets relatons. Secton 3 ntroduces the defnton of rough neutrosophc multsets relaton of Cartesan product on a unverse set wth some examples. Related propertes and operatons are also nvestgated. Secton 3 also defned the composton of two rough neutrosophc multsets relaton, nverse rough neutrosophc multsets relaton and the reflexve, symmetrc and transtve rough neutrosophc multsets relaton. Subsequently, ther propertes are examned. In secton 4, the rough neutrosophc multsets relaton s represented as a marketng strategy by evaluatng the qualty of the product. Fnally, secton 5 concludes the paper. 2 Prelmnares In ths secton, some mathematcal prelmnary concepts were recalled to understandng more about rough neutrosophc multsets relatons. Defnton 2.1 ([10]) Let U be a non-empty set of objects, R s an equvalence relaton on U. Then the space (U, R) s called an approxmaton space. Let X be a fuzzy set on U. We defne the lower and upper approxmaton set and upper approxmaton of X, respectvely R U (X) = {x U: [x] R X}, R U (X) = {x U: [x] R X 0} where T RU (X) = nf y U {T X (y): y [x] R }, T RU (X) = sup y U {T X (y): y [x] R }. The boundary of X, BND(X) = R U (X) R U (X). The fuzzy set X s called the rough fuzzy set f BND(X) 0. Defnton 2.2 ([18]) Let U be a non-empty set of objects, R s an equvalence relaton on U. Then the space (U, R) s called an approxmaton space. Let X be an ntutonstc fuzzy set on U. We defne the lower and upper approxmaton set and upper approxmaton of X, respectvely R U (X) = {x U: [x] R X} R U (X) = {x U: [x] R X 0}, Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

3 38 Neutrosophc Sets and Systems, Vol. 21, 2018 where T RU (X) = nf y U {T X (y): y [x] R }, T RU (X) = sup y U {T X (y): y [x] R }, F RU (X) = sup y U {F X (y): y [x] R }, F RU (X) = nf y U {T X (y): y [x] R }. The boundary of X, BND(X) = R U (X) R U (X). The ntutonstc fuzzy set X s called the rough ntutonstc fuzzy set f BND(X) 0. Defnton 2.3 ([6]) Let U be a non-null set and R be an equvalence relaton on U. Let A be neutrosophc set n U wth the membershp functon T A, ndetermnacy functon I A and non-membershp functon F A. The lower and the upper approxmatons of A n the approxmaton (U, R) denoted by N(A) and N(A) are respectvely defned as follows: N(A) = { x, (T N(A) (x), I N(A) (x), F N(A) (x), ) y [x] R, x U}, N(A) = { x, (T N(A) (x), I N(A) (x), F N(A) (x), ) y [x] R, x U} where T N(A) (x) = y [x] R T A (y), I N(A) (x) = y [x] R I A (y), F N(A) (x) = y [x] R F A (y), T N(A) (x) = y [x] R T A (y), I N(A) (x) = y [x] R I A (y), F N(A) (x) = F A (y) such that, T N(A) (x), I N(A) (x), F N(A) (x), T N(A) (x), I N(A) (x), F N(A) (x): A [0, 1], 0 T N(A) (x) + I N(A) (x) + F N(A) (x) 3 and 0 T N(A) (x) + I N(A) (x) + F N(A) (x) 3 y [x] R. Here and denote mn and max operators respectvely, and [x] R s the equvalence class of the x. T A (y), I A (y) and F A (y) are the membershp sequences, ndetermnacy sequences and non-membershp sequences of y wth respect to A. Snce N(A) and N(A) are two neutrosophc sets n U, thus the neutrosophc set mappngs N, N: N(U) N(U) are respectvely referred as lower and upper rough neutrosophc set approxmaton operators, and the par of (N(A), N(A)) s called the rough neutrosophc set n (U, R). Defnton 2.4 ([1]) Let U be a non-null set and R be an equvalence relaton on U. Let A be neutrosophc multsets n U wth the truth-membershp sequence T A, ndetermnacy-membershp sequences I A and falstymembershp sequences F A. The lower and the upper approxmatons of A n the approxmaton (U, R) denoted by Nm(A) and Nm(A) are respectvely defned as follows: Nm = { x, (T Nm(A) (x), I Nm(A) (x), F Nm(A) (x), ) y [x] R, x U}, Nm(A) = { x, (T Nm(A) (x), I Nm(A) (x), F Nm(A) (x), ) y [x] R, x U} where = 1, 2,, p and postve nteger T Nm(A) (x) = y [x] T R A (y), (x) = I A (y) I Nm(A) y [x] R, (x) = F A (y) F Nm(A) T Nm(A) I Nm(A) F Nm(A) (x) = (x) = (x) = y [x] R, T A (y) y [x] R, I A (y) y [x] R, F A (y) y [x] R. Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

4 Neutrosophc Sets and Systems, Vol. 21, Here and denote mn and max operators respectvely, and [x] R s the equvalence class of the x. T A (y), I A (y) and F A (y) are the membershp sequences, ndetermnacy sequences and non-membershp sequences of y wth respect to A. It can be sad that T Nm(A) (x), I Nm(A) (x), F Nm(A) (x) [0, 1] and 0 T Nm(A) (x) + I Nm(A) (x) + (x) 3. Then, Nm(A) s a neutrosophc multsets. Smlarly, we have F Nm(A) T Nm(A) (x) [0, 1] and 0 T Nm(A) (x), I Nm(A) (x), F Nm(A) (x) + I Nm(A) (x) + F Nm(A) (x) 3. Then, Nm(A) s neutrosophc multsets. Snce Nm(A) and Nm(A) are two neutrosophc multsets n U, the neutrosophc multsets mappngs Nm, Nm: Nm(U) Nm(U) are respectvely referred to as lower and upper rough neutrosophc multsets approxmaton operators, and the par of (Nm(A), Nm(A)) s called the rough neutrosophc multsets n (U, R), respectvely. 3 Rough Neutrosophc Multsets Relaton The concept of a rough set can be easly extended to a relaton snce the relaton s also a set,.e. a subset of the Cartesan product. Ths concept s also used to defne the rough neutrosophc multsets relaton over the unverse. In the followng secton, the Cartesan product of two rough neutrosophc multsets s defned wth some examples. We only consdered the case where T, I, F are refned nto the same number of subcomponents 1, 2,, p, and T A, I A and F A are a sngle valued neutrosophc number. Some of the concepts are quoted from [2], [10], [12], [36] Defnton 3.1 ([7]) Let A = (U, R) be an approxmaton space. Let X U. A relaton T on X s sad to be a rough relaton on X f T T, where T and T are a lower and upper approxmaton of T, respectvely defned by; T = {(x, y) U U: [x, y] R X} T = {(x, y) U U: [x, y] R X Defnton 3.2 Let U be a non-empty set and X and Y be the rough neutrosophc multsets n U. Then, Cartesan product of X and Y s rough neutrosophc multsets n U U, denoted by X Y, defned as where X Y = {< (x, y), (T X Y T X Y I X Y F X Y T X Y (x, y)) (I X Y (x, y) = mn{t X (x), T Y (y)}, (x, y) = max{i X (x), I Y (y)}, (x, y) = max{f X (x), F Y (y)},, I X Y, F X Y : U [0, 1], and = 1, 2,, p. (x, y)), ( F X Y (x, y)) >: (x, y) U U} Defnton 3.3 Let U be a non-empty set and X and Y be the rough neutrosophc multsets n U. We call R U U s a rough neutrosophc multsets relaton on U U based on the X Y, where X Y s characterzed by truth-membershp sequence T R, ndetermnacy-membershp sequences R I and falsty-membershp sequences F R, defned as R = {< (x, y), (T R (x, y)) (I R (x, y)), ( F R (x, y)) >: (x, y) U U} wth a condton f t satsfes: (1) ) T R (x, y) = 1 for all (x, y) X Y where X Y = R U (X) R U (Y), ) ) T R (x, y) = 0, for all (x, y) U U X Y where X Y = R U (X) R U (Y), 0 < T R (x, y) < 1, for all (x, y) X Y X Y. Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

5 40 Neutrosophc Sets and Systems, Vol. 21, 2018 (2) ) I R (x, y) = 0, for all (x, y) X Y where X Y = R U (X) R U (Y), ) I R (x, y) = 1, for all (x, y) U U X Y where X Y = R U (X) R U (Y), ) 0 < I R (x, y) < 1, for all (x, y) X Y X Y. (3) ) F R (x, y) = 0, for all (x, y) X Y where X Y = R U (X) R U (Y), ) F R (x, y) = 1, for all (x, y) U U X Y where X Y = R U (X) R U (Y), ) 0 < F R (x, y) < 1, for all (x, y) X Y X Y. Remark 3.4: The rough neutrosophc multsets relaton s a relaton on neutrosophc multsets, so we can consder that s a rough neutrosophc multsets relaton over the unverse. The rough neutrosophc multsets relaton follows the condton of relaton on neutrosophc multsets whch s T R (x, y) T X Y (x, y), I R (x, y) I X Y (x, y), F R (x, y) F X Y (x, y) for all (x, y) U U, and 0 T R (x, y) + I R (x, y) + F R (x, y) 3. Therefore, the rough neutrosophc multsets relaton wll generalze the followng relaton: (1) Rough Neutrosophc Set Relaton When = 1 for all element T, I, F n defnton 3.2, we obtan the relaton for rough neutrosophc set over unverse; R = {< (x, y), (T R (x, y))(i R (x, y)), ( F R (x, y)) >: (x, y) U U}. (2) Rough Intutonstc Fuzzy Set Relaton When = 1 for element T and F, and propertes (2) n defnton 3.3 s also omtted, we obtan the relaton for rough ntutonstc fuzzy set over unverse; R = {< (x, y), (T R (x, y)), ( F R (x, y)) >: (x, y) U U}. (3) Rough Fuzzy Set Relaton When = 1 for element T and propertes (2) and (3) n defnton 3.3 s also omtted, we obtan the relaton for rough fuzzy set over unverse; R = {< (x, y), (T R (x, y)) >: (x, y) U U}. The rough neutrosophc multsets relaton can be presented by relatonal tables and matrces, lke a representaton of fuzzy relaton. Snce the trple (T A, I A, F A ) has values wthn the nterval [0, 1], the elements of the neutrosophc matrx also have values wthn [0, 1]. Consder the followng example: Example 3.5: Let U = {u 1, u 2, u 3 } be a unversal set and R U = (x, y): xr U y s equvalent relatons on U. Let X = (1,0.3),(0.4,0.7),(0.6,0.8) u 1 + (0.5,0.7),(0.1,0.3),(0.4,0.5) + (1,0.6),(0.4,0.5),(0.6,0.7) u 2 u 3 Y = (0.4,0.6),(0.3,0.5),(0.1,0.7) u 1 + (0.5,0.4),(0.1,0.7),(0.3,0.8) u 2 + (1,0.7),(0.2,0.5),(0.1,0.7) u 3 are rough neutrosophc multsets on U. Here we can defne a rough neutrosophc multsets relaton R by a matrx. xr U y s composed by R U = {{u 1, u 3 }, {u 2 }. Based on defnton 2.4 and 3.2, we solve for;, and R U (X) = (1,0.3),(0.4,0.7),(0.6,0.8) u 1 + (0.5,0.7),(0.1,0.3),(0.4,0.5) u 2 + (1,0.3),(0.4,0.7),(0.6,0.8) u 3, R U (X) = (1,0.6),(0.4,0.5),(0.6,0.7) u 1 + (0.5,0.7),(0.1,0.3),(0.4,0.5) u 2 + (1,0.6),(0.4,0.5),(0.6,0.7) u 3, R U (Y) = (0.4,0.6),(0.3,0.5),(0.1,0.7) u 1 + (0.5,0.4),(0.1,0.7),(0.3,0.8) u 2 + (0.4,0.6),(0.3,0.5),(0.1,0.7) u 3, R U (Y) = (1,0.7),(0.2,0.5),(0.1,0.7) u 1 + (0.5,0.4),(0.1,0.7),(0.3,0.8) u 2 + (1,0.7),(0.2,0.5),(0.1,0.7) u 3 Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

6 Neutrosophc Sets and Systems, Vol. 21, We have X Y = R U (X) R U (Y) and X Y = R U (X) R U (Y). Then, by satsfed all the condton n defnton 3.3, we defned R U U as a rough neutrosophc multsets relaton on U U based on the X Y by a matrx form: (0.3, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) M(R) = [ (0, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) ] (0.4, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) Now, we consder some propertes of a rough neutrosophc multsets relaton. Proposton 3.6 Let R 1, R 2 be two rough neutrosophc multsets relaton on U U based on the X Y. Then R 1 R 2 where T R1 R 2 (x, y) = mn{ T R1 (x, y)}, I R1 R 2 (x, y) = max{i R1 (x, y)}, F R1 R 2 (x, y) = max{f R1 (x, y), F R2 (x, y)} for all (x, y) U U, s a rough neutrosophc multsets on U U based on the X Y and = 1, 2,, p. Proof: We show that R 1 R 2 satsfy defnton 3.3. (1) ) Snce T R1 ) ) T R1 R 2 Snce T R1 then T R1 R 2 Snce 0 < T R1 0 < T R1 R 2 (2) ) Snce I R1 ) ) I R1 R 2 Snce I R1 then I R1 R 2 (x, y) = T R2 (x, y) = mn{ T R1 (x, y) = T R2 (x, y) = mn{ T R1 (x, y) = mn{ T R1 (x, y) = I R2 (x, y) = 1 for all (x, y) X Y then (x, y)} = 1 for all (x, y) X Y. (x, y) = 0 for all (x, y) U U X Y (x, y)} = 0 for all (x, y) U U X Y. (x, y) < 1, for all (x, y) X Y X Y then (x, y)} < 1 for all (x, y) X Y X Y. (x, y) = 0 for all (x, y) X Y then (x, y) = max{ I R1 (x, y)} = 0 for all (x, y) X Y. (x, y) = I R2 (x, y) = 1 for all (x, y) U U X Y Snce 0 < I R1 0 < I R1 R 2 (x, y) = max{ I R1 (x, y) = max{i R1 (x, y)} = 1 for all (x, y) U U X Y. (x, y) < 1, for all (x, y) X Y X Y then (x, y)} < 1 for all (x, y) X Y X Y. Proof (3) for falsty functon are smlarly to provng (2) for ndetermnate functon. Proposton 3.7 Let R 1, R 2 be two rough neutrosophc multsets relaton on U U based on the X Y. Then R 1 R 2 where T R1 R 2 (x, y) = max{ T R1 (x, y)}, I R1 R 2 (x, y) = mn{i R1 (x, y)}, F R1 R 2 (x, y) = mn{f R1 (x, y), F R2 (x, y)} for all (x, y) U U, s a rough neutrosophc multsets on U U based on the X Y and = 1, 2,, p. Proof: We show that R 1 R 2 satsfy defnton 3.3. (1) ) Snce T R1 ) ) T R1 R 2 Snce T R1 then T R1 R 2 Snce 0 < T R1 0 < T R1 R 2 (x, y) = T R2 (x, y) = max{ T R1 (x, y) = T R2 (x, y) = max{ T R1 (x, y) = max{ T R1 (x, y) = 1 for all (x, y) X Y then (x, y)} = 1 for all (x, y) X Y. (x, y) = 0 for all (x, y) U U X Y (x, y)} = 0 for all (x, y) U U X Y. (x, y) < 1, for all (x, y) X Y X Y then (x, y)} < 1 for all (x, y) X Y X Y. Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

7 42 Neutrosophc Sets and Systems, Vol. 21, 2018 (2) ) Snce I R1 ) ) I R1 R 2 Snce I R1 then I R1 R 2 Snce 0 < I R1 0 < I R1 R 2 (x, y) = I R2 (x, y) = mn{ I R1 (x, y) = I R2 (x, y) = mn{ I R1 (x, y) = mn{i R1 (x, y) = 0 for all (x, y) X Y then (x, y)} = 0 for all (x, y) X Y. (x, y) = 1 for all (x, y) U U X Y (x, y)} = 1 for all (x, y) U U X Y. (x, y) < 1, for all (x, y) X Y X Y then (x, y)} < 1 for all (x, y) X Y X Y. Proof (3) for falsty functon are smlarly to provng (2) for ndetermnate functon. Lemma 3.8: If 0 < x, y < 1, then () 0 < xy < 1, () 0 < x + y xy < 1. Snce 0 < x, y < 1 then x + y 2 xy > 2xy > xy > 0, therefore x + y xy > 0. On the other hand, 1 (x + y xy) = (1 x)(1 y) > 0 then x + y xy < 1. The followng propertes of a rough neutrosophc multsets relaton are obtaned by usng these algebrac results. Proposton 3.9 Let R 1, R 2 be two rough neutrosophc multsets relaton on U U based on the X Y. Then R 1 R 2 where T R1 R 2 (x, y), I R1 R 2 (x, y) + I R2 (x, y) I R1 (x, y). I R 2 (x, y) F R1 R 2 (x, y) = F R1 (x, y) + F R2 (x, y) F R1 (x, y). F R 2 (x, y) for all (x, y) U U, s a rough neutrosophc multsets on U U based on the X Y and = 1, 2,, p. Proof: The relaton R 1 R 2 satsfed defnton 3.3. Indeed: (1) ) Snce T R1 ) ) (2) ) Snce I R1 ) ) (x, y) = T R2 (x, y) = 1 for all (x, y) X Y then T R1 R 2 (x, y) = 1 for all (x, y) X Y. Snce T R1 (x, y) = T R2 (x, y) = 0 for all (x, y) U U X Y then T R1 R 2 (x, y) = 0 for all (x, y) U U X Y. Snce 0 < T R1 (x, y) < 1, for all (x, y) X Y X Y then 0 < T R1 R 2 (x, y) < 1 for all (x, y) X Y X Y (Lemma 3.8 ()). I R1 R 2 Snce I R1 I R1 (x, y) = I R2 (x, y) = 0 for all (x, y) X Y then (x, y) + I R2 (x, y) I R1 (x, y) I R 2 (x, y) = 0 for all (x, y) X Y. (x, y) = I R2 (x, y) = 1 for all (x, y) U U X Y then I R1 R 2 (x, y) = (x, y) I R1 (x, y) I R 2 (x, y) = 1 for all (x, y) U U X Y. (x, y) + I R2 Snce 0 < I R1 (x, y) < 1, for all (x, y) X Y X Y then 0 < I R1 R 2 (x, y) + I R2 (x, y) I R1 (x, y) I R 2 (x, y) < 1 for all (x, y) X Y X Y (Lemma 3.8 ()). Proof (3) for falsty functon are smlarly to provng (2) for ndetermnate functon. Proposton 3.10 Let R 1, R 2 be two rough neutrosophc multsets relaton on U U based on the X Y. Then R 1 R 2 where T R1 R 2 (x, y) + T R 2 (x, y) T R1 (x, y), I R1 R 2 (x, y) I R 2 (x, y), F R1 R 2 (x, y) = F R1 (x, y) F R 2 (x, y) for all (x, y) U U, s a rough neutrosophc multsets on U U based on the X Y and = 1, 2,, p. Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

8 Neutrosophc Sets and Systems, Vol. 21, Proof: The relaton R 1 R 2 satsfed defnton 3.3. Indeed: (1) ) Snce T R1 ) ) T R1 R 2 (x, y) = T R2 (x, y) = 1 for all (x, y) X Y then (x, y) + T R 2 (x, y) T R1 (x, y) = 1 for all (x, y) X Y. (x, y) = T R2 (x, y) = 0 for all (x, y) U U X Y then T R1 R 2 (x, y) = (x, y) T R1 (x, y) = 0 for all (x, y) U U X Y. Snce T R1 T R1 (x, y) + T R 2 Snce 0 < T R1 0 < T R1 R 2 (2) ) Snce I R1 ) ) (x, y) < 1, for all (x, y) X Y X Y then (x, y) + T R 2 (x, y) T R1 (x, y) < 1 for all (x, y) X Y X Y (Lemma 3.8 ()). (x, y) = I R2 (x, y) = 0 for all (x, y) X Y then (x, y) I R 2 (x, y) = 0 for all (x, y) X Y. (x, y) = I R2 (x, y) = 1 for all (x, y) U U X Y then I R1 R 2 (x, y) = 1 for all (x, y) U U X Y. I R1 R 2 Snce I R1 I R 2 Snce 0 < I R1 (x, y) (x, y) < 1, for all (x, y) X Y X Y then 0 < I R1 R 2 (x, y) I R 2 (x, y) < 1 for all (x, y) X Y X Y (Lemma 3.8 ()). Proof (3) for falsty functon are smlarly to provng (2) for ndetermnate functon. 3.1 Composton of Two Rough Neutrosophc Multsets Relaton The composton of relaton s mportant for applcatons because f a relaton on X and Y s known and f a relaton on Y and Z s known, then the relaton on X and Z could be computed over a unverse wth the useful sgnfcance. Defnton Let U be a non-empty set and X, Y and Z are the rough neutrosophc multsets n U. Let R 1, R 2 are two rough neutrosophc multsets relatons on U U, based on X Y, Y Z, respectvely. The composton of R 1, R 2 denote as R 1 R 2 whch defned on U U based on X Z where T R1 R 2 I R1 R 2 (x, z) = max y U {mn[t R1 (x, z) = mn y U {max[i R1 F R1 R 2 (x, z) = mn y U {max[f R1 for all (x, z) U U and = 1, 2,, p. (y, z)]}, (y, z)]}, (x, y), F R2 (y, z)]}. Proposton R 1 R 2 s a rough neutrosophc multsets relaton on U U based on X Z. Proof: Snce R 1, R 2 are two rough neutrosophc multsets relatons on U U based on X Y, Y Z respectvely; (1) ) Then T R1 ) ) T R1 R 2 (x, z) = 1 = T R2 (x, z) for all (x, z) X Z. Let (x, z) X Z, now (x, z) = max y U {mn[t R1 (y, z)]} = 1. Ths holds for all (x, z) X Z. Let (x, z) U U X Z. So, T R1 Then T R1 R 2 (2) ) Then I R1 (x, z) = max y U {mn[t R1 (x, z) = 0 = T R2 (x, z) for all (x, z) U U X Z. (y, z)]} = 0 for all (x, z) U U X Z. Agan, snce 0 < T R1 (x, z), T R2 (x, z) < 1, for all (x, z) X Z X Z, then 0 < max y U {mn[t R1 (y, z)]} < 1 such that 0 < T R1 R 2 (x, z) < 1 for all (x, z) X Z X Z. I R1 R 2 (x, z) = 0 = I R2 (x, z) for all (x, z) X Z. Let (x, z) X Z, now (x, z) = mn y U {max[i R1 (y, z)]} = 0. Ths holds for all (x, z) X Z. Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

9 44 Neutrosophc Sets and Systems, Vol. 21, 2018 ) ) Let (x, z) U U X Z. So, I R1 I R1 R 2 (x, z) = mn y U {max[i R1 (x, z) = 1 = I R2 (x, z) for all (x, z) U U X Z. Then (y, z)]} = 1 for all (x, z) U U X Z. Agan, snce 0 < I R1 (x, z), I R2 (x, z) < 1, for all (x, z) X Z X Z, then 0 < mn y U {max[i R1 (y, z)]} < 1 such that 0 < I R1 R 2 (x, z) < 1 for all (x, z) X Z X Z. Proof (3) for falsty functon are smlarly to provng (2) for ndetermnate functon. Proposton Let U be a non-empty set. R 1, R 2, R 3 are rough neutrosophc multsets relatons on U U based on X Y, Y Z, Z Z, respectvely. Then (R 1 R 2 ) R 3 = R 1 (R 2 R 3 ) Proof: We only proof for truth functon. For all x, y, z, t U we have T R1 (R 2 R 3 ) (x, t) = max y U {mn[t R1 (x, y), T (R2 R 3 )(y, t)]} = max y U {mn[t R1 (x, y), max z U {mn[t R2 (y, z), T R3 (z, t)]]}} = max z U {mn{max y U {mn[t R1 (y, z)], T R3 (z, t)]}}} = max z U {mn[t (R1 R 2 )(x, y), T R3 (y, t)]} = T (R1 R 2 ) R 3 (x, t); Smlarly proof for ndetermnate functon and falsty functon. Note that R 1 R 2 R 2 R 1, snce the composton of two rough neutrosophc multsets relatons R 1, R 2 exsts, the composton of two rough neutrosophc multsets relatons R 2, R 1 does not necessarly exst. 3.2 Inverse Rough Neutrosophc Multsets Relaton Defnton Let U be a non-empty set and X and Y be the rough neutrosophc multsets n U. R U U s a rough neutrosophc multsets relaton on U U based on the X Y. Then, we defne R 1 U U s the rough neutrosophc multsets relaton on U U based on Y X as follows: R 1 = {< (y, x), (T R 1 (y, x)) (I R 1 (y, x)), ( F R 1(y, x)) >: (y, x) U U} where T R 1(y, x) = T R (x, y), I R 1(y, x) = I R (x, y), F R 1(y, x) = F R (x, y) for all (y, x) U U and = 1, 2,, p. Defnton The relaton R 1 s called the nverse rough neutrosophc multsets relaton of R. Proposton (R 1 ) 1 = R. Proof: From defnton 3.3; (1) ) T (R 1 ) 1 ) ) T (R 1 ) 1 0 < T (R 1 ) 1 (2) ) I (R 1 ) 1 ) ) I (R 1 ) 1 0 < I (R 1 ) 1 (x, y) = T R 1(y, x) = T R (x, y) = 1 for all (x, y) X Y (x, y) = T R 1(y, x) = T R (x, y) = 0 for all (x, y) U U X Y (x, y) = T R 1(y, x) = T R (x, y) < 1 for all (x, y) X Y X Y (x, y) = I R 1(y, x) = I R (x, y) = 0 for all (x, y) X Y (x, y) = I R 1(y, x) = I R (x, y) = 1 for all (x, y) U U X Y (x, y) = I R 1(y, x) = I R (x, y) < 1 for all (x, y) X Y X Y Proof (3) for falsty functon are smlarly to provng (2) for ndetermnate functon. Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

10 Neutrosophc Sets and Systems, Vol. 21, It means (R 1 ) 1 = R. Proposton Let R 1, R 2 be two rough neutrosophc multsets relatons on U U, based on X Y, Y Z, respectvely. Then (R 1 R 2 ) 1 = R 2 1 R 1 1. Proof: For all x, y, z U, we have T (R1 R 2 ) 1 (z, x) = T R1 R 2 (x, z) = max y U {mn[t R1 = max y U {mn [T (R2 ) 1 = T (R2 ) 1 (R 1 ) 1 (z, x); (y, z)]} = max y U {mn [T (R1 ) 1 (z, y), T (R1 ) 1 (y, x)]} Smlarly, proof for ndetermnate functon and falsty functon. That means (R 1 R 2 ) 1 = R 2 1 R 1 1. (y, x), T (R2 ) 1 (z, y)]} The representaton of nverse rough neutrosophc multsets relaton R 1 can be represented by rough neutrosophc multsets relaton R by usng a matrx M(R) t, t s the transposton of a matrx M(R). Example 3.2.5: Consder the rough neutrosophc multsets relaton M(R) n example 3.5; (0.3, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) M(R) = [ (0, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) ] (0.4, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) Then the nverse rough neutrosophc multsets relaton R 1 (0.3, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0.4, 0), (1, 1), (1, 1) M(R 1 ) = M(R) t = [ (0, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) ] (0.9, 0), (1, 1), (1, 1) (0, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) 3.3 The Reflexve, Symmetrc, Transtve Rough Neutrosophc Multsets Relaton In ths secton, we consder some propertes of rough neutrosophc multsets on unverse, such as reflexve, symmetrc and transtve propertes. Let (U, R) be a crsp approxmaton space and X s a rough neutrosophc multsets on (U, R). From here onwards, the rough neutrosophc multsets relaton R s called a rough neutrosophc multsets relaton on (U, R) based on the rough neutrosophc multsets X. Defnton The rough neutrosophc multsets relaton R s sad to be reflexve rough neutrosophc multsets relaton f T R (x, x) = 1, I R (x, x) = F R (x, x) = 0 and for all (x, x) U U, = 1, 2,, p. Proposton Let R 1, R 2 be two rough neutrosophc multsets relaton on U based X. If R 1, R 2 are the reflexve rough neutrosophc multsets relatons then R 1 R 2, R 1 R 2, R 1 R 2, R 1 R 2 and R 1 R 2 s also reflexve. Proof: If R 1, R 2 are reflexve rough neutrosophc multsets relaton, then T R1 (x, x) = T R2 U U. We have ) T R1 R 2 I R1 R 2 F R1 R 2 (x, x) = 1, I R1 (x, x) = mn{ T R1 (x, x) = max{i R1 (x, x) = I R2 (x, x) = max{f R1 (x, x), F R2 (x, x), T R2 (x, x)} = 1; (x, x), I R2 (x, x)} = 0; and (x, x)} = 0 (x, x) = 0 and F R1 (x, x) = F R2 (x, x) = 0 for all (x, x) Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

11 46 Neutrosophc Sets and Systems, Vol. 21, 2018 for all (x, x) U U and R 1 R 2 s reflexve rough neutrosophc multsets relaton. ) T R1 R 2 I R1 R 2 F R1 R 2 (x, x) = max{ T R1 (x, x) = mn{i R1 (x, x), T R2 (x, x)} = 1; (x, x), I R2 (x, x)} = 0; and (x, x) = mn{f R1 (x, x), F R2 (x, x)} = 0 for all (x, x) U U and R 1 R 2 s reflexve rough neutrosophc multsets relaton. ) T R1 R 2 I R1 R 2 F R1 R 2 (x, x) = T R1 (x, x) T R 2 (x, x) = 1; (x, x) = I R1 (x, x) + I R2 (x, x) I R1 (x, x) = F R1 (x, x) I R 2 (x, x) = 0; and (x, x) = 0 (x, x) + F R2 (x, x) F R1 (x, x) F R 2 for all (x, x) U U and R 1 R 2 s reflexve rough neutrosophc multsets relaton. v) T R1 R 2 (x, x) = T R1 I R1 R 2 (x, x) = I R1 (x, x) I R 2 F R1 R 2 (x, x) = F R1 (x, x) F R 2 (x, x) + T R 2 (x, x) = 0; and (x, x) = 0 (x, x) T R1 (x, x) T R 2 (x, x) = 1; for all (x, x) U U and R 1 R 2 s reflexve rough neutrosophc multsets relaton. v) T R1 R 2 (x, x) = max y U {mn[t R1 (x, x), T R2 (x, x)]} = 1; (y, x)]} = max y U {mn[t R1 I R1 R 2 (x, x) = mn y U {max[i R1 (y, x)]} = mn y U {max[i R1 (x, x), I R2 (x, x)]} = 0; and F R1 R 2 (x, x) = mn y U {max[f R1 (x, y), F R2 (y, x)]} = mn y U {max[f R1 (x, x), F R2 (x, x)]} = 0 for all (x, x) U U, X Y and R 1 R 2 s reflexve rough neutrosophc multsets relaton. Defnton The rough neutrosophc multsets relaton R s sad to be symmetrc rough neutrosophc multsets relaton f T R (x, y) = T R (y, x), I R (x, y) = I R (y, x) and F R (x, y) = F R (y, x) for all (x, y) U U, = 1, 2,, p. We note that f R s a symmetrc rough neutrosophc multsets relaton then the matrx M(R) s a symmetrc matrx. Proposton If R s sad to be symmetrc rough neutrosophc multsets relaton, then R 1 s also symmetrc. Proof: Assume that R s symmetrc rough neutrosophc multsets relaton, then we have T R (x, y) = T R (y, x), I R (x, y) = I R (y, x) and F R (x, y) = F R (y, x). Also, f R 1 s an nverse rough neutrosophc multsets relaton, then we have T R 1(x, y) = T R (y, x), I R 1(x, y) = I R (y, x) and F R 1(x, y) = F R (y, x) for all (x, y) U U. To prove R 1 s symmetrc, we must prove that T R 1(x, y) = T R 1(y, x), I R 1(x, y) = I R 1(y, x) and F R 1(x, y) = F R 1(y, x) for all (x, y) U U. Therefore, T R 1(x, y) = T R (y, x) = T R (x, y) = T R 1(y, x); I R 1(x, y) = I R (y, x) = I R (x, y) = I R 1(y, x); and (x, y) = F R (y, x) = F R (x, y) = F R 1(y, x) F R 1 Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

12 Neutrosophc Sets and Systems, Vol. 21, for all (x, y) U U and R s sad to be symmetrc rough neutrosophc multsets relaton, then R 1 s also symmetrc rough neutrosophc multsets relaton. Proposton Let R 1, R 2 be two rough neutrosophc multsets relatons on U based rough neutrosophc multsets. If R 1, R 2 are the symmetrc rough neutrosophc multsets relatons then R 1 R 2, R 1 R 2, R 1 R 2 and R 1 R 2 also symmetrc. Proof: Snce R 1 s symmetrc, then we have; T R1 (y, x), I R1 (y, x) and F R1 (x, y) = F R1 (y, x) Smlarly, R 2 s symmetrc, then we have; T R2 (x, y) = T R2 (y, x), I R2 (x, y) = I R2 Therefore; ) T R1 R 2 (x, y) = mn{ T R1 (y, x), T R2 (y, x) and F R2 (x, y)} = mn{ T R1 (y, x)} = T R1 R 2 (y, x); I R1 R 2 (x, y) = max{i R1 (x, y)} = max{ I R1 (y, x), I R2 (y, x)} = I R1 R 2 (y, x); and F R1 R 2 (x, y) = max{f R1 (x, y), F R2 (x, y)} = max{ F R1 (y, x), F R2 (y, x)} = F R1 R 2 (y, x) (x, y) = F R2 (y, x) for all (x, y) U U and R 1 R 2 s symmetrc rough neutrosophc multsets relaton. ) T R1 R 2 (x, y) = max{ T R1 (y, x), T R2 (x, y)} = max{ T R1 (y, x)} = T R1 R 2 (y, x); I R1 R 2 (x, y) = mn{i R1 (x, y)} = mn{i R1 (y, x), I R2 (y, x)} = I R1 R 2 (y, x); and F R1 R 2 (x, y) = mn{f R1 (x, y), F R2 (x, y)} = mn{f R1 (y, x), F R2 (y, x)} = F R1 R 2 (y, x) for all (x, y) U U and R 1 R 2 s symmetrc rough neutrosophc multsets relaton. ) T R1 R 2 (y, x) T R 2 (y, x) = T R1 R 2 (y, x); I R1 R 2 (x, y) + I R2 (x, y) I R1 (x, y) I R 2 (x, y) = I R1 (y, x) + I R2 (y, x) I R1 (y, x) I R 2 (y, x) = I R1 R 2 (y, x); and F R1 R 2 (x, y) = F R1 (x, y) + F R2 (x, y) F R1 (x, y) F R 2 (x, y) = F R1 (y, x) + F R2 (y, x) F R1 (y, x) F R 2 (y, x) = F R1 R 2 (y, x) for all (x, y) U U and R 1 R 2 s symmetrc rough neutrosophc multsets relaton. v) T R1 R 2 (x, y) + T R 2 (x, y) T R1 (x, y) = T R1 (y, x) + T R 2 (y, x) T R1 (y, x) T R 2 (y, x) = T R1 R 2 (y, x); I R1 R 2 (x, y) I R 2 (y, x) I R 2 (y, x) = I R1 R 2 (y, x); and F R1 R 2 (x, y) = F R1 (x, y) F R 2 (x, y) = F R1 (y, x) F R 2 (y, x) = F R1 R 2 (y, x) for all (x, y) U U and R 1 R 2 s symmetrc rough neutrosophc multsets relaton. Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

13 48 Neutrosophc Sets and Systems, Vol. 21, 2018 Remark 3.3.6: R 1 R 2 n general s not symmetrc, as T R1 R 2 (x, z) = max y U {mn[t R1 (y, x), T R2 = max y U {mn[t R1 (y, z)]} (z, x)]} T R1 R 2 (z, x) The proof s smlarly for ndetermnate functon and falsty functon. But, R 1 R 2 s symmetrc f R 1 R 2 = R 2 R 1, for R 1 and R 2 are symmetrc relatons. T R1 R 2 (x, z) = max y U {mn[t R1 (y, z)]} = max y U {mn[t R1 (y, x), T R2 (z, y)]} = max y U {mn[t R2 (y, x), T R2 (z, y)]} = max y U {mn[t R2 (z, y), T R2 (y, x)]} = max y U {mn[t R1 (z, y), T R2 (y, x)]} = T R1 R 2 (z, x) for all (x, z) U U and y U. The proof s smlarly for ndetermnate functon and falsty functon. Defnton The rough neutrosophc multsets relaton R s sad to be transtve rough neutrosophc multsets relaton f R R R such that T R (y, x) T R R (y, x), I R (y, x) I R R (y, x) and F R (y, x) (y, x) for all x, y U. F R R Defnton The rough neutrosophc multsets relaton R on U based on the neutrosophc multsets X s called a rough neutrosophc multsets equvalence relaton f t s reflexve, symmetrc and transtve rough neutrosophc multsets relaton. Proposton If R s transtve rough neutrosophc multsets relaton, then R 1 s also transtve. Proof: R s transtve rough neutrosophc multsets relaton f R R R, hence f R 1 R 1 R 1, then R 1 s transtve. Consder; T R 1(x, y) = T R (y, x) T R R (y, x) = max z U {mn[t R (y, z), T R (z, x)]} = max z U {mn[t R 1(z, y), T R 1(x, z)]} = max z U {mn[t R 1(x, z), T R 1(z, y)]} = T R 1 R 1 (x, y); The proof s smlarly for ndetermnate functon and falsty functon. Hence, the proof s vald. Proposton Let R 1, R 2 be two rough neutrosophc multsets relatons on U based rough neutrosophc multsets. If R 1 are the transtve rough neutrosophc multsets relaton, then R 1 R 2 s also transtve. Proof: As R 1 and R 2 are transtve rough neutrosophc multsets relaton, R 1 R 1 R 1 and R 2 R 2 R 2. Also (x, y) T (R1 R 2 ) (R 1 R 2 )(x, y); (x, y) I (R1 R 2 ) (R 1 R 2 )(x, y); and (x, y) F (R1 R 2 ) (R 1 R 2 )(x, y) T R1 R 2 I R1 R 2 F R1 R 2 mples that (R 1 R 2 ) (R 1 R 2 ) R 1 R 2, hence R 1 R 2 s transtve. Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

14 Neutrosophc Sets and Systems, Vol. 21, Proposton If R 1 and R 2 are transtve rough neutrosophc multsets relatons, then R 1 R 2, R 1 R 2 and R 1 R 2 are not transtve. Proof: ) As T R1 R 2 I R1 R 2 F R1 R 2 (x, y) = max{ T R1 (x, y) = mn{i R1 (x, y) = mn{f R1 (x, y)}, (x, y)}, and (x, y), F R2 (x, y)} and, T R1 R 2 (x, y) T (R1 R 2 ) (R 1 R 2 )(x, y); I R1 R 2 (x, y) I (R1 R 2 ) (R 1 R 2 )(x, y); and (x, y) F (R1 R 2 ) (R 1 R 2 )(x, y) ) As F R1 R 2 T R1 R 2 (x, y), I R1 R 2 (x, y) + I R2 (x, y) I R1 (x, y) I R 2 (x, y), and F R1 R 2 (x, y) = F R1 (x, y) + F R2 (x, y) F R1 (x, y) F R 2 (x, y) and, T R1 R 2 (x, y) T (R1 R 2 ) (R 1 R 2 )(x, y); I R1 R 2 (x, y) I (R1 R 2 ) (R 1 R 2 )(x, y); and F R1 R 2 (x, y) F (R1 R 2 ) (R 1 R 2 )(x, y) () As T R1 R 2 (x, y) + T R 2 (x, y) T R1 (x, y), I R1 R 2 (x, y) I R 2 (x, y), and F R1 R 2 (x, y) = F R1 (x, y) F R 2 (x, y) and T R1 R 2 (x, y) T (R1 R 2 ) (R 1 R 2 )(x, y); I R1 R 2 (x, y) I (R1 R 2 ) (R 1 R 2 )(x, y); and F R1 R 2 (x, y) F (R1 R 2 ) (R 1 R 2 )(x, y) Hence, R 1 R 2, R 1 R 2 and R 1 R 2 are not transtve. 4 An applcaton to marketng strategy The ams of mult crtera decson makng (MCDM) are to solve the problem nvolvng mult decson by many expert opnons and many alternatves gven and MCDM also try to get the best alternatve soluton based on the mult crtera evaluate by many experts. The study of MCDM wth the neutrosophc envronment s well establshed n [38] [44]. Ths secton gves a stuaton of solvng a real applcaton of the rough neutrosophc multsets relaton n marketng strategy. Assume J = {j 1, j 2, j 3 } denotes for three jeans showed avalable to be purchased n a shop G. Let R J be a relaton defned on the J as ar J b f and only f a, b comng from the same contnent about qualty of the jeans. ar J b s composed by R J = {j 1, j 2, j 3 }. The relaton R J s explans the effect of the qualty of jeans n shop Z. We now try to get the opnon from two ndependent customers about the qualty of jeans consderng whether the jeans are comprsed of "good texture", a level of ndetermnacy wth respect to the customers whch s no comment and whether they feel that the jean s comprsed of "a not all that great texture". In the customers opnon, rough neutrosophc multsets, A and B can be defned as follows: A = {< j 1, (0.9, 0.2), (0.3, 0.6), (0.5,0.7) >, < j 2, (0.4, 0.6), (0.2, 0.4), (0.5, 0.6) >, Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

15 50 Neutrosophc Sets and Systems, Vol. 21, 2018 < j 3, (1.0, 0.6), (0.4,0.5), (0.6, 0.7) >} and B = {< j 1, (0.5, 0.7), (0.4, 0.6), (0.2, 0.8) >, < j 2, (0.6, 0.7), (0.2, 0.8), (0.4, 0.5) >, < j 3, (1.0, 0.8), (0.3, 0.6), (0.2, 0.8) >} By satsfed all the condton n defnton 3.3, we wll defne the relaton of rough neutrosophc multsets R on qualtes of jeans J J based on customers opnon A B as follows: Step 1: Compute lower and upper approxmaton values for rough neutrosophc multsets. R J (A) = {< j 1, (0.4, 0.2), (0.4, 0.6), (0.6,0.7) >, < j 2, (0.4, 0.2), (0.4, 0.6), (0.6,0.7) >, < j 3, (0.4, 0.2), (0.4, 0.6), (0.6,0.7) >} R J (B) = {< j 1, (0.5, 0.7), (0.4, 0.8), (0.4,0.8) >, < j 2, (0.5, 0.7), (0.4, 0.8), (0.4,0.8) >, < j 3, (0.5, 0.7), (0.4, 0.8), (0.4,0.8) >} R J (A) = {< j 1, (1.0, 0.6), (0.2, 0.4), (0.5,0.6) >, < j 2, (1.0, 0.6), (0.2, 0.4), (0.5,0.6) >, < j 3, (1.0, 0.6), (0.2, 0.4), (0.5,0.6) >} R J (B) = {< j 1, (1.0, 0.8), (0.2, 0.6), (0.2,0.5) >, < j 2, (1.0, 0.8), (0.2, 0.6), (0.2,0.5) >, < j 3, (1.0, 0.8), (0.2, 0.6), (0.2,0.5) >} Step 2: Construct the relaton of A B = R J (A) R J (B), relaton of A B = R J (A) R J (B), and relaton of J J. All the relaton was represented n the Table 1, Table 2 and Table 3, respectvely. A B j 1 j 2 j 3 j 1 (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) j 2 (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) j 3 (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) Table 1: Relaton of A B (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) (0.4, 0.2), (0.4, 0.8), (0.6, 0.8) A B j 1 j 2 j 3 j 1 (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) j 2 (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) j 3 (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) Table 2: Relaton of A B (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) (1.0, 0.6), (0.4, 0.6), (0.5, 0.6) Note that T R (a, b) = 1, I R (a, b) = 0 and F R (a, b) = 0 for all (a, b) A B. Therefore, the relaton of J J s Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

16 Neutrosophc Sets and Systems, Vol. 21, J J j 1 j 2 j 3 j 1 (0.0, 0.0), (0.0, 0.0) (0.0, 0.0), (0.0, 0.0) j 2 (0.0, 0.0), (0.0, 0.0) j 3 (0.0, 0.0), (0.0, 0.0) Table 3: Relaton of J J (0.0, 0.0), (0.0, 0.0) (0.0, 0.0), (0.0, 0.0) (0.0, 0.0), (0.0, 0.0) (0.0, 0.0), (0.0, 0.0) (0.0, 0.0), (0.0, 0.0) Step 3: Construct a rough neutrosophc multsets relaton R. Note that, T R (a, b) = 0, I R (a, b) = 1 and F R (a, b) = 1 for all (a, b) J J A B. Table 4 represent the rough neutrosophc multsets relaton R. R j 1 j 2 j 3 j 1 (t1, 0.0), (1.0, 1.0) (t2, 0.0), (1.0, 1.0) j 2 (t4, 0.0), (1.0, 1.0) j 3 (t7, 0.0), (1.0, 1.0) (t5, 0.0), (1.0, 1.0) (t8, 0.0), (1.0, 1.0) Table 4: Rough neutrosophc mult relaton R. (t3, 0.0), (1.0, 1.0) (t6, 0.0), (1.0, 1.0) (t9, 0.0), (1.0, 1.0) Step 4: Compute the values for t 1 untl t 9 n Table 4. Note that t 1, t 2,.., t 9 [0, 1] and 0 < T R (a, b) < 1, for all (a, b) A B A B and neutrosophc mult relaton of T R (a, b) T A B (a, b) (a, b) J J. () () () t 1 = T R (j 1, j 1 ) = T A B (j 1, j 1 ) T A B (j 1, j 1 ) = = 0.6, T 1 1 R (j 1, j 1 ) T A B (j 1, j 1 ) where Therefore, the possble values of t 1 s 0.9, 0.8, 0.7 and 0.6. t 2 = T R (j 1, j 2 ) = T A B (j 1, j 2 ) T A B (j 1, j 2 ) = = 0.6, T 1 1 R (j 1, j 2 ) T A B (j 1, j 2 ) where Therefore, the possble values of t 2 s 0.9, 0.8, 0.7 and 0.6. The same calculaton was used for t 3 untl t 9. Therefore, the possble values for t 1 untl t 9 s represent n Table 5. t n, n Possble values t n, n Possble values = 1, 2,, 9 = 1, 2,, 9 t 1 0.6, 0.7, 0.8, 0.9 t 6 0.6, 0.7, 0.8, 0.9 t 2 0.6, 0.7, 0.8, 0.9 t 7 0.6, 0.7, 0.8, 0.9 t 3 0.6, 0.7, 0.8, 0.9 t 8 0.6, 0.7, 0.8, 0.9 t 4 0.6, 0.7, 0.8, 0.9 t 9 0.6, 0.7, 0.8, 0.9 t 5 0.6, 0.7, 0.8, 0.9 Table 5: Possble values for t 1 untl t 9 Step 5: We defned R J J as a rough neutrosophc multsets relaton on J J based on the A B by a matrx form. We can have dfferent values for t 1 untl t 9 as t s true for all possble values n Table 5. We try to get some pattern of the rough neutrosophc multsets relaton matrx of our study by three possble cases. Case 1: (j 1, j n ) > (j 2, j n ) > (j 3, j n ) for all n, and unknown value. Therefore, there are two rough neutrosophc multsets relaton matrx resulted for ths case, represented as M(R 1 ) and M(R 2 ), respectvely. (0.9, 0), (1, 1), (1, 1) (0.9,0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) M(R 1 ) = [(0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) ] (0.7, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

17 52 Neutrosophc Sets and Systems, Vol. 21, 2018 (0.8, 0), (1, 1), (1, 1) (0.8,0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) M(R 2 ) = [(0.7, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) ] (0.6, 0), (1, 1), (1, 1) (0.6, 0), (1, 1), (1, 1) (0.6, 0), (1, 1), (1, 1) Case 2: (j 1, j n ) < (j 2, j n ) < (j 3, j n ) for all n, and unknown value. Therefore, there are two rough neutrosophc multsets relaton matrx resulted for ths case, represented as M(R 3 ) and M(R 4 ), respectvely. (0.7, 0), (1, 1), (1, 1) (0.7,0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) M(R 3 ) = [(0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) ] (0.9, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) (0.6, 0), (1, 1), (1, 1) (0.6,0), (1, 1), (1, 1) (0.6, 0), (1, 1), (1, 1) M(R 4 ) = [(0.7, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) ] (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) Case 3: (j 1, j n ) = (j 2, j n ) = (j 3, j n ) for all n, and unknown value. Therefor the rough neutrosophc multsets relaton matrx resulted as M(R 5 ). (0.8, 0), (1, 1), (1, 1) (0.8,0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) M(R 5 ) = [(0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) ] (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) Case 4: Random possble value for all unknown. Therefore, the rough neutrosophc multsets relaton matrx resulted as M(R 6 ). (0.9, 0), (1, 1), (1, 1) (0.8,0), (1, 1), (1, 1) (0.9, 0), (1, 1), (1, 1) M(R 6 ) = [(0.8, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) (0.8, 0), (1, 1), (1, 1) ] (0.7, 0), (1, 1), (1, 1) (0.6, 0), (1, 1), (1, 1) (0.7, 0), (1, 1), (1, 1) Step 6: Compute the comparson matrx usng the formula D R = T R + I R F R for all, and select the maxmum value for comparson table. The result s shown n Table 6 and Table 7, respectvely. j 1 j 2 j 3 R 1 j 1 (0.9, 0) (0.9, 0) (0.9, 0) j 2 (0.8, 0) (0.8, 0) (0.8, 0) j 3 (0.7, 0) (0.7, 0) (0.7, 0) R 2 j 1 (0.8, 0) (0.8, 0) (0.8, 0) j 2 (0.7, 0) (0.7, 0) (0.7, 0) j 3 (0.6, 0) (0.6, 0) (0.6, 0) R 3 j 1 (0.7, 0) (0.7, 0) (0.7, 0) j 2 (0.8, 0) (0.8, 0) (0.8, 0) j 3 (0.9, 0) (0.9, 0) (0.9, 0) R 4 j 1 (0.6, 0) (0.6, 0) (0.6, 0) j 2 (0.7, 0) (0.7, 0) (0.7, 0) j 3 (0.8, 0) (0.8, 0) (0.8, 0) R 5 j 1 (0.8, 0) (0.8, 0) (0.8, 0) j 2 (0.8, 0) (0.8, 0) (0.8, 0) j 3 (0.8, 0) (0.8, 0) (0.8, 0) R 6 j 1 (0.9, 0) (0.8, 0) (0.9, 0) j 2 (0.8, 0) (0.7, 0) (0.8, 0) j 3 (0.7, 0) (0.6, 0) (0.7, 0) Table 6: Comparson matrx of rough neutrosophc mult relaton R. Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

18 Neutrosophc Sets and Systems, Vol. 21, J j 1 j 2 j 3 J 1 j j j J 2 j j j J 3 j j j J 4 j j j J 5 j j j J 6 j j j Table 7: Comparson table for rough neutrosophc multsets, J Step 7: Next we compute the row-sum, column-sum, and the score for all cases as shown n Table 8. J Row sum Column sum Score J 1 j j j J 2 j j j J 3 j j j J 4 j j j J 5 j j j J 6 j j j Table 8: Score of three jeans for all cases. The relaton of qualty of jeans n shop G s successfully approxmate by usng rough neutrosophc multsets relaton. Jean type j 1 has the hghest score of 0.3 for case 1, jean type j 3 has the hghest score of 0.3 for case 2, nether choose a jean or not for case 3, and jeans type j 1 and j 2 have a hghest score of 0.2 for case 4. The dfferent selecton of jeans has resulted n dfferent cases. From the scorng perspectve, the hghest value for Surana Alas, Daud Mohamad and Adbah Shub, Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy

19 54 Neutrosophc Sets and Systems, Vol. 21, 2018 each unknown wll resulted n the hghest possblty to select the subject. Besdes that, t cannot take the same possble values for each unknown at the same tme because the score result wll be equal to zero (case 3). Based on the result, the customers should purchase the jeans of type j 1 n the shop G and the manager should sell more jeans of type j 1. Concluson The successful dscusson of rough neutrosophc multsets relaton wth applcaton n marketng strategy s obtaned n ths paper. Frstly, ths paper s defned the rough neutrosophc multsets relaton wth ther propertes and operatons such as max, mn, the composton of two rough neutrosophc multsets, nverse rough neutrosophc multsets, and symmetry, reflexve and transtve of rough neutrosophc multsets. The approxmaton set boundary of rough neutrosophc multsets was appled for rough neutrosophc multsets relaton. Ths relaton theory s useful to apply n marketng strategy problem by gettng the real relaton of goods sold n the market. Decson matrx analyss s further conducted to get the best result. For further work, the relaton of two unverse sets can be derved as a rough neutrosophc multsets relaton of two unverse sets. References [1] L. A. Zadeh, Fuzzy sets, Inf. Control, 8(3), (1965), [2] M. Hussan, Fuzzy relaton, Master Thess Bleknge Insttute of Technology School of Engneerng, Department of Mathematcs and Scence, (2010). [3] K. T. Atanassov, Intutonstc Fuzzy Set, Fuzzy Sets and Systems, 20, (1986), [4] P. Burllo, H. Bustnce, and C. Arrosad, Intutonstc fuzzy relatons. (Part I), Mathw. Soft Comput., 2, (1995), [5] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sc., 11(5), (1982), [6] P. K. Maj, R. Bswas, and A. R. Roy, Soft set theory, Comput. Math. wth Appl., 45(4), (2003), [7] M. Ncolett, J. Uchoa, and M. Baptstn, Rough relaton propertes, Int. J. Appl., 11(3), (2001), [8] K. V. Babtha and J. J. Sunl, Soft set relatons and functons, Comput. Math. wth Appl., 60(7), (2010), [9] F. Smarandache, A Unfyng Feld n Logcs: Neutrosophc Logc, Neutrosophy, Neutrosophc set, Neutrosophc Probablty", Amercan Research Press, (1999), [10] H. Yang, Z. Guo, and X. Lao, On sngle valued neutrosophc relatons, J. Intell. Fuzzy Syst., 30(2), (2016), [11] A. A. Salama, M. Esa, and M. M. Abdelmoghny, Neutrosophc Relatons Database, Int. J. Inf. Sc. Intell. Syst., 3(2), (2014), [12] R. Chatterjee, P. Majumdar, and S. K. Samanta, Sngle valued neutrosophc multsets, Ann. Fuzzy Math. Informatcs, x(x), (2015), [13] I. Del, S. Broum, and F. Smarandache, On Neutrosophc refned sets and ther applcatons n medcal dagnoss, J. New Theory, (2015), [14] S. Myamoto, Fuzzy multsets and ther generalzatons, Multset Process. Math. Comput. Sc. Mol. Comput. Ponts Vew, 2235, (2001), [15] T. K. Shnoj and S. J. John, Intutonstc Fuzzy Multsets and Its Applcaton n Medcal Dagnoss, World Acad. Sc. Eng. Technol., 6(1), (2012), [16] S. Broum, I. Del, and F. Smarandache, Relatons on neutrosophc mult sets wth propertes, arxv: , (2015), [17] S. Broum, I. Del, and F. Smarandache, Neutrosophc Refned Relatons and Ther Propertes, Neutrosophc Theory Its Appl., (2014), [18] F. Smarandache, Neutrosophc Perspectves: Trplets, Duplets, Multsets, Hybrd Operators, Modal Logc, Hedge Algebras. And Applcatons. Second extended and mproved edton, Pons, Bruxelles, [19] S. Broum and F. Smarandache, Extended Hausdorff Dstance and Smlarty Measures for Neutrosophc Refned Sets and Ther Applcaton n Medcal Dagnoss, Journal of New Theory, 7, (2015), [20] J. Ye and F. Smarandache, Smlarty Measure of Refned Sngle-Valued Neutrosophc Sets and Its Multcrtera Decson Makng Method, Neutrosophc Sets Syst., 12, (2016), [21] S. Ye and J. Ye, Dce smlarty measure between sngle valued neutrosophc multsets and ts applcaton n medcal dagnoss, Neutrosophc Sets Syst., 6, (2014), [22] K. Mondal and S. Pramank, Neutrosophc refned smlarty measure based on cotangent functon and ts applcaton to mult-attrbute decson makng, J. New theory, 8, (2015), [23] K. Mondal and S. Pramank, Neutrosophc Refned Smlarty Measure Based on Cotangent Functon and ts Applcaton to Mult-attrbute Decson Makng, Glob. J. Advanced Res, (2015), [24] S. Pramank, D. Banerjee, and B. C. Gr, Mult Crtera Group Decson Makng Model n Neutrosophc Refned Set and Its Applcaton, Glob. J. Eng. Sc. Res. Manag., 3(6), (2016), Surana Alas, Daud Mohamad and Adbah Shub. Rough Neutrosophc Multsets Relaton wth Applcaton n Marketng Strategy