Generalized Neutrosophic Soft Set

Transcription

1 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Generalized Neutrosophi Soft Set Said Broumi Faulty of Arts and Humanities, Hay El Baraka Ben M'sik Casablana B.P. 7951, Hassan II Mohammedia-Casablana University, Moroo Abstrat In this paper we present a new onept alled generalized neutrosophi soft set. This onept inorporates the benefiial properties of both generalized neutrosophi set introdued by A.A.Salama [7] and soft set tehniques proposed by Molodtsov [4]. We also study some properties of this onept. Some definitions and operations have been introdued on generalized neutrosophi soft set. Finally we present an appliation of generalized neuutrosophi soft set in deision making problem. KEYWORDS Soft Sets, Neutrosophi Set, Generalized Neutrosophi Set, Generalized Neutrosophi Soft Set. 1. INTRODUCTION In Many ompliated problems like, engineering problems, soial, eonomi, omputer siene, medial siene et, the data assoiated are not neessarily risp, preise, and deterministi beause of their vague nature. Most of these problem were solved by different theories. One of these theories was the fuzzy set theory disovered by Lotfi, Zadeh in 1965 [1], Later several researhes present a number of results using different diretion of fuzzy set suh as: interval fuzzy set [12], generalized fuzzy set by Atanassov [2]..., all these are suessful to some extent in dealing with the problems arising due to the vagueness present in the real world,but there are also ases where these theories failed to give satisfatory results, possibly due to indeterminate and inonsistent information whih exist in belief system, then in 1995, Smarandahe [3] initiated the theory of neutrosophi set as new mathematial tool for handling problems involving impreise, indeterminay, and inonsistent data. Several researhers dealing with the onept of neutrosophi set suh as M. Bhowmik and M.Pal in [13], A.A.Salama in [7], and H.Wang in [14]. Furthermore, In 1999, a Russian mathematiian ( Molodtsov [4]) introdue a new mathematial tool for dealing with unertainties, alled soft set theory. This new onept is free from the limitation of variety of theories suh as probability theory, Fuzzy sets and rough sets. Soft set theory has no problem of setting the membership funtion, whih makes it very onvenient and easy to apply in pratie. After Molodtsov work, there have been many researhes in ombining fuzzy set with soft set, whih inorporates the benefiial properties of both fuzzy set and soft set tehniques ( see [11] [6] [8]). So in this paper we present a new model whih ombine two onepts: Generalized neutrosophi set proposed by A.A.Salama [7] and soft set proposed by Molodtsov in [4], together by introduing a new onept alled generalized neutrosophi sof set, thus we introdue its operations namely equal, subset, union, and intersetion, Finally we present an appliation of generalized neutrosophi soft set in deision making problem. DOI : /ijseit

2 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 The ontent of the paper is orgonaized as follow: In setion 2, we briefly present some basi definitions and preliminary results are given whih will be used in the rest of the paper. In setion 3, generalized neutrosophi soft set. In setion 4 an appliation of generalized neutrosophi soft set in a deision making problem. Finally setion 5 presents the onlusion of our work. 2.Preliminaries In this setion, we review some definitions with regard to neutrosophi set, generalized neutrosophi set and and soft set. The definitions in this part may be found in referenes [3, 4, 7, 10]. Throughout this paper, let U be a universal set and E be the set of all possible parameters under onsideration with respet to U, usually, parameters are attributes, harateristis, or properties of objets in U. Definition 2.1 (see [3]). Neutrosophi Set Let U be an universe of disourse then the neutrosophi set A is an objet having the form A = {< x: T A(x), I A(x),F A(x) >,x U}, where the funtions T,I,F : U ] 0,1 + [ define respetively the degree of membership, the degree of indeterminay, and the degree of non-membership of the element x X to the set A with the ondition. 0 T A(x) + I A(x) + F A(x) 3 +. From philosophial point of view, the neutrosophi set takes the value from real standard or non-standard subsets of ] 0,1 + [. So instead of ] 0,1 + [ we need to take the interval [0,1] for tehnial appliations, beause ] 0,1 + [ will be diffiult to apply in the real appliations suh as in sientifi and engineering problems. Definition 2.2. (see [10]) A neutrosophi set A is ontained in another neutrosophi set B i.e. A B if x U, T A (x) T B (x), I A (x) I B (x), F A (x) F B (x). Definition 2.3(see [7]). Generalized Neutrosophi Set Let X be a non-empty fixed set. A generalized neutrosophi set (GNS for short) A is an objet having the form A = {< x: T A(x), I A(x), F A(x) >,x U}, Where T A(x), represent the degree of membership funtion, I A(x) represent the degree of indeterminay, and F A(x) represent the degree of non-member ship respetively of eah element x X to the set A where the funtions satisfy the ondition: T A( x ) F A( x ) I A( x ) 0.5 As an illustration, let us onsider the following example. Example2.4. Assume that the universe of disourse U={x 1,x 2,x 3 },where x 1 haraterizes the apability, x 2 haraterizes the trustworthiness and x 3 indiates the pries of the objets. It may be further assumed that the values of x 1, x 2 and x 3 are in [0, 1] and they are obtained from some questionnaires of some experts. The experts may impose their opinion in three omponents viz. 18

3 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 the degree of goodness, the degree of indeterminay and that of poorness to explain the harateristisof the objets. Suppose A is an generalized neutrosophi set ( GNS ) of U, suh that, A = {< x 1, 0.3, 0.5, 0.4 >,< x 2,0.4, 0.2, 0.6 >,< x 3, 0.7, 0.3, 0.5 >}, where the degree of goodness of apability is 0.3, degree of indeterminay of apability is 0.5 and degree of falsity of apability is 0.4 et. Definition 2.5 (see[4]). Soft Set Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Consider a nonempty set A, A E A pair ( F, A ) is alled a soft set over U, where F is a mapping given by F : A P(U). As an illustration, let us onsider the following example. Example 2.6. Suppose that U is the set of houses under onsideration, say U = {h 1, h 2,..., h 5 }. Let E be the set of some attributes of suh houses, say E = {e 1, e 2,..., e 8 }, where e 1, e 2,..., e 8 stand for the attributes expensive, beautiful, wooden, heap, modern, and in bad repair, respetively. In this ase, to define a soft set means to point out expensive houses, beautiful houses, and so on. For example, the soft set (F, A) that desribes the attrativeness of the houses in the opinion of a buyer, say Thomas, may be defined like this: A={e 1, e 2, e 3, e 4, e 5 }; F(e 1 ) = {h 2, h 3, h 5 }, F(e 2 ) = {h 2, h 4 }, F(e 3 ) = {h 1 }, F(e 4 ) = U, F(e 5 ) = {h 3, h 5 }. For more details on the algebra and operations on generalized neutrosophi set and soft set, the reader may refer to [ 5, 6, 8, 7, 9, 11]. 3. Generalized Neutrosophi Soft Set In this setion,we will initiate the study on hybrid struture involving both generalized neutrosophi set and soft set theory. Definition 3.1 Let U be an initial universe set and A E be a set of parameters. Let GNS( U ) denotes the set of all generalized neutrosophi sets of U. The olletion (F,A) is termed to be the soft generalized neutrosophi set over U, where F is a mapping given by F : A GNS(U). Remark 3.2. We will denote the generalized neutrosophi soft set defined over an universe by GNSS. Let us onsider the following example. Example 3.3 Let U be the set of blouses under onsideration and E is the set of parameters (or qualities). Eah parameter is a generalized neutrosophi word or sentene involving generalized neutrosophi words. Consider E = { Bright, Cheap, Costly, very ostly, Colorful, Cotton, Polystyrene, long sleeve, expensive }. In this ase, to define a generalized neutrosophi soft set means to point out 19

4 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Bright blouses, Cheap blouses, Blouses in Cotton and so on. Suppose that, there are five blouses in the universe U given by, U = {b 1, b 2, b 3, b 4, b 5 } and the set of parameters A = {e 1, e 2, e 3, e 4 }, where eah e i is a speifi riterion for blouses: Suppose that, e 1 stands for Bright, e 2 stands for Cheap, e 3 stands for ostly, e 4 stands for Colorful, F(Bright)={<b 1, 0.5, 0.6, 0.3>,<b 2,0.4, 0.7, 0.2>,<b 3,0.6, 0.2, 0.3>,<b 4, 0.7, 0.3, 0.2>,<b 5,0.8, 0.2, 0.3>}. F(Cheap)={<b 1, 0.6, 0.3, 0.5>,<b 2, 0.7, 0.4, 0.3>,<b 3, 0.8, 0.1, 0.2>,<b 4, 0.7, 0.1, 0.3>,<b 5, 0.8, 0.3, 0.4}. F(Costly)={<b 1, 0.7, 0.4, 0.3>,<b 2, 0.6, 0.1, 0.2>,<b 3, 0.7, 0.2, 0.5>,< b 4, 0.5, 0.2, 0.6 >,< b 5, 0.7, 0.3, 0.2 >}. F(Colorful)={<b 1, 0.8, 0.1, 0.4>,<b 2, 0.4, 0.2, 0.6>,<b 3, 0.3, 0.6, 0.4>,<b 4, 0.4,0.8,0.5>,< b 5, 0.3, 0.5, 0.7 >}. The generalized neutrosophi soft set ( GNSS ) ( F, E ) is a parameterized family {F(e i ), i = 1,,10} of all generalized neutrosophi sets of U and desribes a olletion of approximation of an objet. The mapping F here is blouses (.), where dot(.) is to be filled up by a parameter e i E. Therefore, F(e 1 ) means blouses (Bright) whose funtional-value is the generalized neutrosophi set {< b 1, 0.5, 0.6, 0.3 >,< b 2, 0.4, 0.7, 0.2 >, < b 3, 0.6, 0.2, 0.3 >,< b 4, 0.7, 0.3, 0.2 >,< b 5, 0.8, 0.2, 0.3 >}. Thus we an view the generalized neutrosophi soft set ( approximation as below: GNSS) (F,A ) as a olletion of ( F, A ) = { Bright blouses= {< b 1, 0.5, 0.6, 0.3 >,< b 2, 0.4, 0.7, 0.2 >, < b 3, 0.6, 0.2, 0.3 >,< b 4,0.7,0.3,0.2 >,< b 5,0.8,0.2,0.3 >}, Cheap blouses= {< b 1,0.6,0.3,0.5 >,< b 2,0.7,0.4,0.3 >,< b 3,0.8,0.1,0.2 >, < b 4, 0.7,0.1,0.3 >,< b 5, 0.8,0.3,0.4 >}, ostly blouses= {< b 1,0.7,0.4,0.3 >,< b 2, 0.6, 0.1, 0.2 >,< b 3, 0.7, 0.2, 0.5 >,< b 4, 0.5,0.2,0.6 >,< b 5, 0.7, 0.3, 0.2 >}, Colorful blouses= {< b 1, 0.8, 0.1,0.4 >,< b 2, 0.4, 0.2,0.6 >,< b 3, 0.3, 0.6,0.4 >, < b 4, 0.4, 0.8, 0.5>,< b 5, 0.3, 0.5, 0.7 >}}. where eah approximation has two parts: (i) a prediate p, and (ii) an approximate value-set v ( or simply to be alled value-set v ). For example, for the approximation Bright blouses={<b 1,0.5,0.6,0.3>,<b 2,0.4,0.7,0.2 >,<b 3,0.6,0.2,0.3>,<b 4,0.7,0.3,0.2>,<b 5,0.8,0.2,0.3>}. We have (i) the prediate name Bright blouses, and (ii) the approximate value -set is{<b 1,0.5,0.6,0.3>,<b 2,0.4,0.7,0.2>,<b 3,0.6,0.2,0.3>,<b 4,0.7,0.3,0.2>,< b 5,0.8,0.2,0.3 >}. Thus, an generalized neutrosophi soft set ( F, E ) an be viewed as a olletion of approximation like ( F, E ) = {p 1 = v 1,p 2 = v 2,,p 10 = v 10 }. In order to store a generalized neutrosophi soft set in a omputer, we ould represent it in the form of a table as shown below ( orresponding to the generalized neutrosophi soft set in the above example ). In this table, the entries are ij orresponding to the blouse b i and the parameter e j, where ij = (true-membership value of b i, indeterminay-membership value of b i, falsity membership value of b i ) in F(e j ). The table 1 represent the generalized neutrosophi soft set ( F, A ) desribed above. 20

5 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Table 1: Tabular form of the GNSS ( F, A ). U bright heap ostly olorful b 1 ( 0.5, 0.6, 0.3 ) ( 0.6, 0.3, 0.5 ) ( 0.7, 0.4, 0.3 ) ( 0.8, 0.1, 0.4 ) b 2 ( 0.4, 0.7, 0.2 ) ( 0.7, 0.4, 0.3 ) ( 0.6, 0.1, 0.2 ) ( 0.4, 0.2, 0.6 ) b 3 ( 0.6, 0.2, 0.3 ) ( 0.8, 0.1, 0.2 ) ( 0.7, 0.2, 0.5 ) ( 0.3, 0.6, 0.4 ) b 4 ( 0.7, 0.3, 0.2 ) ( 0.7, 0.1, 0.3 ) ( 0.5, 0.2, 0.6 ) ( 0.4, 0.8, 0.5 ) b 5 ( 0.8, 0.2, 0.3 ) ( 0.8, 0.3, 0.4 ) ( 0.7, 0.3, 0.2 ) ( 0.3, 0.5, 0.7 ) Remark 3.4. A generalized neutrosophi soft set is not a generalized neutrosophi set but a parametrized family of a generalized neutrosophi subsets. Definition 3.5 For two generalized neutrosophi soft sets ( F, A ) and ( G, B ) over the ommon universe U. We say that ( F, A ) is a generalized neutrosophi soft subset of ( G, B ) iff: (i) A B. (ii) F(e) is a generalized neutrosophi subset of G(e). Or T F(e) (x) T G(e) (x), I F(e) (x) I G(e) (x), F F(e) (x) F G(e) (x), e A, x U. We denote this relationship by ( F, A ) ( G, B ). ( F, A ) is said to be generalized neutrosophi soft super set of ( G, B ) if ( G, B ) is a generalized neutrosophi soft subset of ( F, A ). We denote it by ( F, A ) ( G, B ). Example 3.6 Let (F, A) and (G, B) be two GNSS over the same universe U = {o 1, o 2, o 3, o 4, o 5 }. The GNSS (F, A) desribes the sizes of the objets whereas the GNSS ( G, B ) desribes its surfae textures. Consider the tabular representation of the GNSS ( F, A ) is as follows. Table 2. Tabular form of the GNSS ( F, A ). U small large olorful O 1 ( 0.4, 0.4, 0.6 ) ( 0.3, 0.2, 0.7 ) ( 0.4, 0.7, 0.5 ) O 2 ( 0.3, 0.5, 0.4 ) ( 0.4, 0.7, 0.8 ) ( 0.6, 0.3, 0.4 ) O 3 ( 0.6, 0.3, 0.5 ) ( 0.3, 0.2, 0.6 ) ( 0.4, 0.4, 0.8 ) O 4 ( 0.5, 0.1, 0.6 ) ( 0.1, 0.6, 0.7 ) ( 0.3, 0.5, 0.8 ) O 5 ( 0.3, 0.4, 0.4 ) ( 0.3, 0.1, 0.6 ) ( 0.5, 0.4, 0.4 ) The tabular representation of the GNSS ( G, B ) is given by table 3 Table 3. Tabular form of the GNSS ( G, B ). U Small Large Colorful Very smooth O1 (0.6, 0.3, 0.3 ) ( 0.7, 0.1, 0.5 ) ( 0.5, 0.1, 0.4 ) ( 0.1, 0.5, 0.4 ) O2 ( 0.7, 0.1, 0.2 ) ( 0.4, 0.2, 0.3 ) ( 0.7, 0.3, 0.2 ) ( 0.5, 0.2, 0.3 ) O3 ( 0.6, 0.2, 0.5 ) ( 0.7, 0.1, 0.4 ) ( 0.6, 0.3, 0.3 ) ( 0.2, 0.5, 0.4 ) O4 ( 0.8, 0.1, 0.4 ) ( 0.3, 0.5, 0.4 ) ( 0.4, 0.3, 0.7 ) ( 0.4, 0.4, 0.5 ) O5 ( 0.5, 0.2, 0.2 ) ( 0.4, 0.1, 0.5 ) ( 0.6, 0.2, 0.3 ) ( 0.5, 0.6, 0.3 ) 21

6 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Clearly, by definition 3.5 we have ( F, A ) ( G, B ). Definition 3.7 Two GNSS ( F, A ) and ( G, B ) over the ommon universe U are said to be generalized neutrosophi soft equal if ( F, A ) is generalized neutrosophi soft subset of ( G, B ) and ( G, B ) is generalized neutrosophi soft subset of ( F, A ) whih an be written as ( F, A )= ( G, B ). Definition 3.8 Let E = {e 1,e 2,,e n } be a set of parameters. The NOT set of E is denoted by E is defined by E ={ e 1, e 2,, e n }, where e i = not e i, i ( it may be noted that and are different operators ). Example 3.9 Consider the example 3.3. Here E = { not bright, not heap, not ostly, not olorful }. Definition 3.10 The omplement of generalized neutrosophi soft set ( F, A ) is denoted by (F,A) and is defined by (F,A) = (F, A), where F : A N(U) is a mapping given by F (α) = generalized neutrosophi soft omplement with T F (x) = F F(x), I F (x) = I F(x) and F F (x) = T F(x). Example 3.11 As an illustration onsider the example presented in the example 3.2. the omplement (F,A) desribes the not attrativeness of the blouses. Is given below. F( not bright) = {< b 1, 0.3, 0.6, 0.5 >,< b 2, 0.2, 0.7, 0.4 >,< b 3, 0.3, 0.2, 0.6 >, < b 4, 0.2, 0.3, 0.7 >< b 5, 0.3, 0.2, 0.8 >}. F( not heap ) = {< b 1, 0.5, 0.3, 0.6 >,< b 2, 0.3, 0.4, 0.7 >,< b 3, 0.2, 0.1, 0.8 >, < b 4, 0.3, 0.1, 0.7 >,< b 5, 0.4, 0.3, 0.8 >}. F( not ostly ) = {< b 1, 0.3, 0.4, 0.7 >,< b 2, 0.2, 0.1, 0.6 >,< b 3, 0.5, 0.2, 0.7 >, < b 4, 0.6, 0.2, 0.5 >, < b 5, 0.2, 0.3, 0.7 >}. F( not olorful ) = {< b 1, 0.4, 0.1, 0.8 >, < b 2, 0.6, 0.2, 0.4 >,< b 3, 0.4, 0.6, 0.3 >, < b 4, 0.5, 0.8, 0.4 >< b 5, 0.7, 0.5, 0.3 >}. Definition 3.12 The generalized neutrosophi soft set ( F,A) over U is said to be empty or null generalized neutrosophi soft (with respet to the set of parameters) denoted by Φ A or (Φ,A) if T F(e) (m) = 0,F F(e) (m) = 0 and I F(e) (m) = 0, m U, e A. Example 3.13 Let U = {b 1,b 2,b 3,b 4,b 5 }, the set of five blouses be onsidered as the universal set and A = { Bright, Cheap, Colorful } be the set of parameters that haraterizes the blouses. Consider the generalized neutrosophi soft set ( F, A) whih desribes the ost of the blouses and F(bright)={< b 1, 0, 0, 0 >,< b 2,0, 0, 0 >,< b 3, 0, 0, 0 >,< b 4, 0,0, 0 >, < b 5, 0, 0, 0 >}, 22

7 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 F(heap)={< b 1, 0, 0, 0 >,< b 2,0, 0, 0 >,< b 3, 0, 0, 0 >,< b 4, 0, 0, 0 >, < b 5, 0, 0, 0 >}, F(olorful)={< b 1, 0, 0, 0 >,< b 2, 0, 0,0 >,< b 3, 0, 0, 0 >, < b 4, 0, 0, 0 >,< b 5, 0, 0, 0 >}. Here the NGNSS ( F, A ) is the null generalized neutrosophi soft set. Definition Union of Two Generalized Neutrosophi Soft Sets. Let (F, A) and (G, B) be two GNSS over the same universe U. Then the union of (F, A) and (G, B) is denoted by (F, A) (G, B) and is defined by (F, A) (G, B) = (K, C), where C = A B and the truth-membership, indeterminay-membership and falsity-membership of ( K, C) are as follows: T K(e) (m) = T F(e) (m), if e A B = T G(e) (m), if e B A = max (T F(e) (m), T G(e) (m)), if e A B. I K(e) (m) = I F(e) (m), if e A B = I G(e) (m), if e B A = min (I F(e) (m), I G(e) (m)), if e A B. F K(e) (m) = F F(e) (m), if e A B = F G(e) (m), if e B A = min (F F(e) (m), F G(e) (m)), if e A B. Example Let ( F, A ) and ( G, B ) be two GNSS over the ommon universe U. Consider the tabular representation of the GNSS ( F, A ) is as follow: Table 4.Tabular form of the GNSS ( F, A ). Bright Cheap Colorful b 1 ( 0.6, 0.3, 0.5 ) ( 0.7, 0.3, 0.4 ) ( 0.4, 0.2, 0.6 ) b 2 ( 0.5, 0.1, 0.8 ) ( 0.6, 0.1, 0.3 ) ( 0.6, 0.4, 0.4 ) b 3 ( 0.7, 0.4, 0.3 ) ( 0.8, 0.3, 0.5 ) ( 0.5, 0.7, 0.2 ) b 4 ( 0.8, 0.4, 0.1 ) ( 0.6, 0.3, 0.2 ) ( 0.8, 0.2, 0.3 b 5 ( 0.6, 0.3, 0.2 ) ( 0.7, 0.3, 0.5 ) ( 0.3, 0.6, 0.5 The tabular representation of the GNSS ( G, B ) is as follow: Table 5. Tabular form of the GNSS ( G, B ). U Costly Colorful b 1 ( 0.6, 0.2, 0.3) ( 0.4, 0.6, 0.2 ) b 2 ( 0.2, 0.7, 0.2 ) ( 0.2, 0.8, 0.3 ) b 3 ( 0.3, 0.6, 0.5 ) ( 0.6, 0.3, 0.4 ) b 4 ( 0.8, 0.4, 0.1 ) ( 0.2, 0.8, 0.3 ) b 5 ( 0.7, 0.1, 0.4 ) ( 0.5, 0.6, 0.4 ) Using definition 3.12 the union of two GNSS (F, A ) and ( G, B ) is ( K, C ) an be represented as follow. 23

8 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Table 6. Tabular form of the GNSS ( K, C ). U Bright Cheap Colorful Costly b 1 ( 0.6, 0.3, 0.5 ) ( 0.7, 0.3, 0.4 ) ( 0.4, 0.2, 0.2 ) ( 0.6, 0.2, 0.3 ) b 2 ( 0.5, 0.1, 0.8 ) ( 0.6, 0.1, 0.3 ) ( 0.6, 0.4, 0.3 ) ( 0.2, 0.7, 0.2 ) b 3 ( 0.7, 0.4, 0.3 ) ( 0.8, 0.3, 0.5 ) ( 0.6, 0.3, 0.2 ) ( 0.3, 0.6, 0.5 ) b 4 ( 0.8, 0.4, 0.1 ) ( 0.6, 0.3, 0.2 ) ( 0.8, 0.2, 0.3 ) ( 0.8, 0.4, 0.1 ) b 5 ( 0.6, 0.3, 0.2 ) ( 0.7, 0.3, 0.5 ) ( 0.5, 0.6, 0.4 ) ( 0.7, 0.1, 0.4 ) Definition Intersetion of Two Generalized Neutrosophi Soft Sets. Let ( F,A) and (G,B ) be two GNSS over the same universe U suh that A B 0. Then the intersetion of (F, A) and ( G, B) is denoted by ( F, A) (G, B) and is defined by ( F, A ) ( G, B ) = ( K, C), where C =A B and the truth-membership, indeterminay membership and falsity-membership of ( K, C ) are related to those of (F, A) and (G, B) by: T K(e) (m) = min (T F(e) (m), T G(e) (m)) I K(e) (m) = min (I F(e) (m), I G(e) (m)) F K(e) (m) = max (F F(e) (m), F G(e) (m)), for all e C. Example Consider the above example The intersetion of ( F, A ) and ( G, B ) an be represented into the following table : Table 7. Tabular form of the GNSS ( K, C ). U Colorful b 1 ( 0.4, 0.2, 0.6) b 2 ( 0.2, 0.4, 0.4) b 3 ( 0.6, 0.3, 0.4) b 4 ( 0.8, 0.2, 0.3) b 5 ( 0.3, 0.6, 0.5) Proposition If (F, A) and (G, B ) are two GNSS over U and on the basis of the operations defined above, then: (1) (F, A) (F, A) = (F, A). (F, A) (F, A) = (F, A). (2) (F, A) (G, B) = (G, B) (F, A). (F, A) (G, B) = (G, B) (F, A). (3) (F, A) Φ = (F, A). (4) (F, A) Φ = Φ. (5) [(F, A) ] = (F, A). Proof. The proof of the propositions 1 to 5 are obvious. Proposition If ( F, A ), ( G, B ) and ( K, C ) are three GNSS over U, then: (1) (F, A) [(G, B) (K, C)] = [(F, A) (G, B)] (K, C). 24

9 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 (2) (F, A) [(G, B) (K, C)] = [(F, A) (G, B)] (K, C). (3) (F, A) [(G, B) (K, C)] = [(F, A) (G, B)] [(F, A) (K, C)]. (4) (F, A) [(G, B) (K, C)] = [(H, A) (G, B)] [(F, A) (K, C)]. Example Let (F,A) ={ b 1,0.6, 0.3, 0. 1, b 2, 0.4, 0.7, 0. 5),(b 3, 0.4, 0.1, 0.8)}, (G,B) ={ (b 1, 0.2, 0.2, 0.6), (b 2,0.7, 0.2, 0.4), (b 3,0.1, 0.6, 0.7) } and (K,C) ={ (b 1, 0.3, 0.8, 0.2), b 2, 0.4, 0.1, 0.6), b 3,0.9, 0.1, 0.2)} be three GNSS of U, Then: (F, A) (G, B) = { b 1, 0.6, 0.2, 0.1, b 2, 0.7, 0.2,0.4, b 3,0.4, 0.1, 0.7 }. (F, A) (K, C) = { b 1, 0.6, 0.3, 0.1, b 2, 0.4, 0.1, 0.5, b 3,0.9, 0.1, 0.2 }. (G, B) (K, C)] = { b 1, 0.2, 0.2, 0.6, b 2,0.4, 0.1, 0.6, b 3, 0.1, 0.1, 0.7 }. (F, A) [(G, B) (K, C)] = { b 1, 0.6, 0.2, 0.1, b 2,0.4, 0.1, 0.5, b 3,0.4, 0.1, 0.7 }. [(F, A) (G, B)] [(F, A) (K, C)] = { b 1,0.6, 0.2, 0.1, b 2, 0.4, 0.1, 0.5, b 3, 0.4, 0.1,0.7 }. Hene distributive (3) proposition verified. Proof, an be easily proved from definition 3.14.and Definition AND Operation on Two Generalized Neutrosophi Soft Sets. Let ( F, A ) and ( G, B ) be two GNSS over the same universe U. then ( F, A ) AND ( G, B) denoted by ( F, A ) ( G, B ) and is defined by ( F, A ) ( G, B ) = ( K, A B ), where K(α, β)=f(α) B(β) and the truth-membership, indeterminay-membership and falsitymembership of ( K, A B ) are as follows: T K(α,β) (m) = min(t F(α) (m), T G(β) (m)), I K(α,β) (m) = min(i F(α) (m), I G(β) (m)) F K(α,β) (m) = max(f F(α) (m), F G(β) (m)), α A, β B. Example Consider the same example 3.15 above. Then the tabular representation of (F,A) and( G, B ) is as follow: Table 8: Tabular representation of the GNSS ( K, A B). u (bright, ostly) (bright, Colorful) (heap, ostly) b 1 ( 0.6, 0.2, 0.5 ) ( 0.4, 0.3, 0.5 ) ( 0.6, 0.2, 0.4 ) b 2 ( 0.2, 0.1, 0.8 ) ( 0.2, 0.1, 0.8 ) ( 0.2, 0.1, 0.3 ) b 3 ( 0.3, 0.4, 0.5 ) ( 0.6, 0.3, 0.4 ) ( 0.3, 0.3, 0.5 ) b 4 ( 0.8, 0.4, 0.1 ) ( 0.2, 0.4, 0.3 ) ( 0.6, 0.3, 0.2 ) b 5 ( 0.6, 0.1, 0.4 ) ( 0.5, 0.3, 0.4 ) ( 0.7, 0.1, 0.5) u (heap, olorful) (olorful, ostly) (olorful, olorful) b 1 ( 0.4, 0.3, 0.4 ) ( 0.4, 0.2, 0.6 ) ( 0.4, 0.2, 0.6 ) b 2 ( 0.2, 0.1, 0.3 ) ( 0.2, 0.4, 0.4 ) ( 0.2, 0.4, 0.4 ) b 3 ( 0.6, 0.3, 0.5 ) ( 0.3, 0.6, 0.5 ) ( 0.5, 0.3, 0.4 ) b 4 ( 0.2, 0.3, 0.3 ) ( 0.8,0.2, 0.3 ) ( 0.2, 0.2, 0.3 ) b 5 ( 0.5, 0.3, 0.5 ) ( 0.3, 0.1, 0.5 ) ( 0.3, 0.6, 0.5 ) 25

10 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Definition If (F, A) and (G, B) be two GNSS over the ommon universe U then (F, A) OR (G, B) denoted by (F, A) (G, B) is defined by ( F, A) (G, B ) = (O, A B), where, the truth-membership, indeterminay membership and falsity-membership of O( α, β) are given as follows: TO(α,β) (m) = max (T F(α) (m),t G(β) (m)) I O(α,β)(m) = min (I F(α) (m),i G(β) (m)) FO(α,β) (m) = min(f F(α) (m),f G(β) (m)), α A, β B. Example Consider the same example 3.14 above. Then the tabular representation of ( F, A ) OR ( G, B ) is as follow: Table 9: Tabular representation of the GNSS ( O, A B). u (bright, ostly) (bright, olorful) (heap, ostly) b 1 ( 0.6, 0.2, 0.3 ) ( 0.6, 0.3, 0.2 ) ( 0.7, 0.2, 0.3 ) b 2 ( 0.5, 0.1, 0.2 ) ( 0.5, 0.1, 0.3 ) ( 0.6, 0.1, 0.2 ) b 3 ( 0.7, 0.4, 0.3 ) ( 0.7, 0.3, 0.3 ) ( 0.8,0.3, 0.5 ) b 4 ( 0.8, 0.4, 0.1 ) ( 0.8, 0.4, 0.1 ) ( 0.8, 0.3, 0.1 ) b 5 ( 0.7, 0.1, 0.2 ) ( 0.6, 0.3, 0.4 ) ( 0.7, 0.1, 0.4 ) u (heap, olorful) (olorful, ostly) (olorful, olorful) b 1 ( 0.7, 0.3, 0.2 ) ( 0.6, 0.2, 0.3 ) ( 0.4, 0.2, 0.2 ) b 2 ( 0.6, 0.1, 0.3 ) ( 0.6, 0.4, 0.2 ) ( 0.6, 0.4, 0.3 ) b 3 ( 0.8, 0.3, 0.4 ) ( 0.5, 0.6, 0.2 ) ( 0.5, 0.7, 0.2 ) b 4 ( 0.6, 0.3, 0.2 ) ( 0.8, 0.2, 0.1 ) ( 0.8, 0.2, 0.3 ) b 5 ( 0.7, 0.3, 0.4 ) ( 0.7, 0.1, 0.4 ) ( 0.5, 0.6, 0.4) Proposition If ( F, A ) and ( G, B ) are two GNSS over U, then : (1) [(F, A) (G, B)] = (F,A) (G, B) (2) [(F, A) (G, B)] = (F,A) (G, B) Proof 1. Let (F, A)={<b, T F(x) (b), I F(x) (b), F F(x) (b)> b U} and (G, B) = {< b, T G(x) (b), I G(x) (b), F G(x) (b) > b U} be two GNSS over the ommon universe U. Also let (K, A B) = (F, A) (G, B), where, K(α, β) = F(α) G(β) for all (α, β) A B then K(α, β) = {< b, min(t F(α) (b),t G(β) (b)), min(i F(α) (b),i G(β) (b)), max(f F(α) (b),f G(β) (b)) > b U}. Therefore, [(F, A) (G, B)] = (K, A B) = {< b, max(f F(α) (b),f G(β) (b)), min(i F(α) (b),i G(β) (b)), min(t F(α) (b),t G(β) (b)) > b U}. Again 26

11 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 (F, A) (G, B) = {< b, max(f F (α) (b)), F G (β) (b)), min(i F (α) (b), I G (β) (b)), min(t F (α) (b), T G (β) (b)) > b U}. = {< b, min(t F(α) (b), T G(β) (b)), min(i F(α) (b), I G(β) (b)), max(f F(α) (b), F G(β) (b)) > b U}. = {< b, max(f F(α) (b), F G(β) (b)), min(i F(α) (b),i G(β) (b)), min(t F(α) (b),t G(β) (b)) > b U}. It follows that [(F, A) (G,B)] = (F, A) (G, B). Proof 2. Let ( F, A ) = {< b, T F(x) (b), I F(x) (b), F F(x) (b) > b U} and (G, B) = {< b, T G(x) (b), I G(x) (b), F G(x) (b) > b U} be two GNSS over the ommon universe U. Also let (O, A B) = (F, A) (G, B), where, O (α, β) = F(α) G(β) for all (α, β) A B. Then O(α, β) = {< b, max(t F(α) (b), T G(β) (b)), min(i F(α) (b), I G(β) (b)), min(f F(α) (b), F G(β) (b)) > b U}. [(F, A) (G, B)] = (O,A B) ={< b, min(f F(α) (b),f G(β) (b)), min(i F(α) (b),i G(β) (b)), max(t F(α) (b),t G(β) (b)) > b U}. Again (H, A) (G, B) = {< b,min(f F (α) (b), F G (β) (b)), min(i F (α) (b), I G (β) (b)), max(t F (α) (b), T G (β) (b)),> b U}. = {< b,max(t F(α) (b), T G(β) (b)), min(i F (α) (b), I G (β) (b)), min(f F(α) (b), F G(β) (b))> b U}. = {< b, min(f F(α) (b), F G(β) (b)), min(i F(α) (b), I G(β) (b)), max(t F(α) (b), T G(β) (b)) > b U}. It follows that [(F, A) (G, B)] = (F, A) (G, B). 4.An Appliation of Generalized Neutrosophi Soft Set in a Deision Making Problem To see an appliation of the onept of generalized neutrosophi soft set: Let us onsider the generalized neutrosophi soft set S = (F,P) (see also Table 10 for its tabular representation), whih desribes the "attrativeness of the blouses" that Mrs. X is going to buy. on the basis of her m number of parameters (e 1, e 2,, e m ) out of n number of blouses (b 1,b 2,,b n ). We also assume that orresponding to the parameter e j (j =1,2,,m) the performane value of the blouse b i (i = 1, 2,, n) is a tuple p ij = (T F(ej) (b i ),I F(ej) (b i ),T F(ej) (b i )), suh that for a fixed i that values p ij (j = 1, 2,,m) represents a generalized neutrosophi soft set of all the n objets. Thus the performane values ould be arranged in the form of a matrix alled the riteria matrix. The more are the riteria values, the more preferability of the orresponding objet is. Our problem is to selet the most suitable objet i.e. The objet whih dominates eah of the objets of the spetrum of the parameters e j. Sine the data are not risp but generalized neutrosophi soft the seletion is not straightforward. Our aim is to find out the most suitable blouse with the hoie parameters for Mrs. X. The blouse whih is suitable for Mrs. X need not be suitable for Mrs. Y or Mrs. Z, as the seletion is dependent on the hoie parameters of eah buyer. We use the tehnique to alulate the sore for the objets Definition: Comparison matrix. The Comparison matrix is a matrix whose rows are labeled by the objet names of the universe suh as b 1, b 2,, b n and the olumns are labeled by the parameters e 1,e 2,, e m. The entries are ij, where ij, is the number of parameters for whih the value of b i exeeds or is equal to the value b j. The entries are alulated by ij = a + d -, where a is the integer alulated as how 27

12 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 many times T bi (e j ) exeeds or equal to T bk (e j ), for b i b k, b k U, d is the integer alulated as how many times I bi(ej) exeeds or equal to I bk(ej), for b i b k, b k U and is the integer how many times F bi(ej) exeeds or equal to F bk(e j ), for b i b k, b k U. Definition 4.2. Sore of an objet. The sore of an objet b i is S i and is alulated as : S i = j ij Now the algorithm for most appropriate seletion of an objet will be as follows. Algorithm (1) input the generalized neutrosophi Soft Set ( F, A). (2) input P, the hoie parameters of Mrs. X whih is a subset of A. (3) onsider the GNSS ( F, P) and write it in tabular form. (4) ompute the omparison matrix of the GNSS ( F, P). (5) ompute the sore S i of b i, i. (6) find S k = maxi S i (7) if k has more than one value then any one of b i may be hosen. To illustrate the basi idea of the algorithm, now we apply it to the generalized neutrosophi soft set based deision making problem. Suppose the wishing parameters for Mrs. X where P={ Cheap, Colorful, Polystyreneing, ostly, Bright }. Consider the GNSS ( F, P ) presented into the following table. Table 10. Tabular form of the GNSS (F, P). U Cheap Colorful Polystyreneing ostly Bright b 1 ( 0.6, 0.3, 0.4 ) ( 0.5, 0.2, 0.6 ) ( 0.5, 0.3, 0.4 ) ( 0.8, 0.2, 0.3 ) ( 0.6,0.3, 0.2 ) b 2 ( 0.7, 0.2, 0.5 ) ( 0.6, 0.3, 0.4 ) ( 0.4, 0.2, 0.6 ) ( 0.4, 0.8, 0.3 ) ( 0.8,0.1, 0.2 ) b 3 ( 0.8, 0.3, 0.4 ) ( 0.8, 0.5, 0.1 ) ( 0.3, 0.5, 0.6 ) ( 0.7, 0.2, 0.1 ) ( 0.7,0.2, 0.5 ) b 4 ( 0.7, 0.5, 0.2 ) ( 0.4, 0.8, 0.3 ) ( 0.8, 0.2, 0.4 ) ( 0.8, 0.3, 0.4 ) ( 0.8,0.3, 0.4 ) b 5 ( 0.3, 0.8, 0.4 ) ( 0.3, 0.6, 0.1 ) ( 0.7, 0.3, 0.2 ) ( 0.6,0.2, 0.4 ) ( 0.6,0.4, 0,2 ) The omparison-matrix of the above GNSS ( F, P) is represented as follow: Table 11. Comparison matrix of the GNSS ( F, P ). U Cheap Colorful Polystyreneing ostly Bright b b b b b

13 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Next we ompute the sore for eah b i as shown below: U Sore (S i ) b 1 3 b 2 2 b 3 11 b 4 19 b 5 17 Clearly, the maximum sore is the sore 19, shown in the table above for the blouse b 4. Hene the best deision for Mrs. X is to selet b 4, followed by b CONCLUSIONS In this artile,our main intention was to inorporate the generalized neutrosophi set proposed by A. A. Salama [7] in soft sets introdued by Molodtsov [4] onsidering the fat that the parameters ( whih are words or sentenes ) are mostly generalized neutrosophi set; but both the onepts deal with impreision, We have also defined some operations on GNSS and present an appliation of GNSS in a deision making problem. Finally, we hope that our model opened a new diretion, new path of thinking to engineers, mathematiians, omputer sientist and many other in various tests. ACKNOWLEDGEMENTS Our speial thanks to the anonymous referee and the editor of this journal of their valuable omments and suggestions whih have improved this paper. REFERENCES [1] Zadeh, L. (1965). Fuzzy sets, Inform and Control [2] Atanassov, K. (1986). Generalized fuzzy sets. Fuzzy Sets and Systems [3] Smarandahe, F. (1999). A Unifying Field in Logis. Neutrosophy: Neutrosophi Probability, Set and Logi. Rehoboth: Amerian Researh Press. [4] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) [5] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) [6] M.Irfan Ali, Feng Feng, Xiaoyan Liu, Won Keun Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) [7] A. A. Salama. Generalized Neutrosophi Set and Generalized Neutrosophi Topologial Spaes, Computer Siene and Engineering 2012, 2(7): DOI: /j.omputer [8] P. K. Maji, R. Biswas, and A. R. Roy, Soft Set Theory, Comput. Math. Appl. 45 (2003) [9] P.K.Maji,R.Biswas,andA.R.Roy,An appliation of soft sets in a deision making problem, Comput. Math. Appl. 44 (2002) [10] F. Smarandahe, Neutrosophi set, a generalisation of the generalized fuzzy sets, Inter. J.Pure Appl. Math. 24 (2005) [11] A. R. Roy and P. K. Maji, A fuzzy soft set theoreti approah appliation of soft set in a deision making problem, Comput. Appl. Math. 203 (2007) [12] I. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, vol. 20,pp , [13] M. Bhowmik and M.Pal. Intuitionisti neutrosophi set, Journal of Information and Computing Siene Vol. 4, No. 2, 2009, pp [14] H.Wang, F.Smarandahe, Y.-Q.Zhang, and R.Sunderraman, Interval neutrosophi sets and Logi: Theory and Appliations in Computing,Hexis, Phoenix, AZ

14 International Journal of Computer Siene, Engineering and Information Tehnology (IJCSEIT), Vol.3, No.2,April2013 Authors Broumi Said is an administrator of university of Hassan II-Mohammedia - Casablana. He worked in University for five years. He reeived his Master in industrial automati from the University of Hassan II Ain hok. His researh onentrates on soft set theory, fuzzy theory, intuitionisti fuzzy theory, neutrosophi theory, and ontrol of systems. 30