Possibility neutrosophic soft sets with applications in decision making and similarity measure

Possibility neutrosophic soft sets with applications in decision making and similarity measure

arXiv:1407.3211v1 [cs.AI] 9 Jul 2014

Faruk Karaaslan Department of Mathematics, Faculty of Sciences, C ¸ ankırı Karatekin University, 18100, C ¸ ankırı, Turkey

Abstract In this paper, concept of possibility neutrosophic soft set and its operations are defined, and their properties are studied. An application of this theory in decision making is investigated. Also a similarity measure of two possibility neutrosophic soft sets is introduced and discussed. Finally an application of this similarity measure in personal selection for a firm.

Keywords: Soft set, neutrosophic soft set, possibility neutrosophic soft set, similarity measure, decision making 1. Introduction In this physical world problems in engineering, medical sciences, economics and social sciences the information involved are uncertainty in nature. To cope with these problems, researchers proposed some theories such as the theory of fuzzy set [35], the theory of intuitionistic fuzzy set [4], the theory of rough set [26], the theory of vague set [18]. In 1999, Molodtsov [20] initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties as different from these theories. A wide range of applications of soft sets have been developed in many different fields, including the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory.

Maji et al. [21, 22] applied soft set theory to decision making problem and in 2003, they introduced some new operations of soft sets. After Maji’s work, Email address: fkaraaslan@karatekin.edu.tr (Faruk Karaaslan)

Preprint submitted to Elsevier

July 14, 2014

works on soft set theory and its applications have been progressing rapidly. see [1, 2, 9, 10, 14, 15, 16, 24, 25, 29, 33, 34, 36].

Neutrosophy has been introduced by Smarandache [30, 31] as a new branch of philosophy and generalization of fuzzy logic, intuitionistic fuzzy logic, paraconsistent logic. Fuzzy sets and intuitionistic fuzzy sets are characterized by membership functions, membership and non-membership functions, respectively. In some real life problems for proper description of an object in uncertain and ambiguous environment, we need to handle the indeterminate and incomplete information. But fuzzy sets and intuitionistic fuzzy sets don’t handle the indeterminant and inconsistent information. Thus neutrosophic set is defined by Samarandache [31], as a new mathematical tool for dealing with problems involving incomplete, indeterminacy, inconsistent knowledge.

Maji [23] introduced concept of neutrosophic soft set and some operations of neutrosophic soft sets. Karaaslan [19] redefined concept and operations of neutrosophic soft sets as different from Maji’s neutrosophic soft set definition and operations. Recently, the properties and applications on the neutrosophic soft sets have been studied increasingly [6, 7, 8, 12, 13, 28].

Alkhazaleh et al [3] were firstly introduced concept of possibility fuzzy soft set and its operations. They gave applications of this theory in solving a decision making problem and they also introduced a similarity measure of two possibility fuzzy soft sets and discussed their application in a medical diagnosis problem. In 2012, Bashir et al. [5] introduced concept of possibility intuitionistic fuzzy soft set and its operations and discussed similarity measure of two possibility intuitionistic fuzzy sets. They also gave an application of this similarity measure.

This paper is organized as follow: in Section 2, some basic definitions and operations are given regarding neutrosophic soft set required in this paper.

In Section 3, possibility neutrosophic soft set is defined as a generalization of possibility fuzzy soft set and possibility intuitionistic fuzzy soft set introduced

by Alkhazaleh [3] and Bashir [5], respectively. In Section 4, a decision making method is given using the possibility neutrosophic soft sets. In Section 5, similarity measure is introduced between two possibility neutrosophic soft sets and in Section 6, an application related personal selection for a firm is given as regarding this similarity measure method.

2

2. Preliminary In this paper, we recall some definitions, operation and their properties related to the neutrosophic soft set [19, 23] required in this paper.

Definition 1. A neutrosophic soft set (or namely ns-set) f over U is a neutrosophic set valued function from E to N (U). It can be written as n o f = (e, {hu, tf (e)(u), if (e) (u), ff (e) (u)i : u ∈ U}) : e ∈ E where, N (U) denotes set of all neutrosophic sets over U. Note that if f (e) = {hu, 0, 1, 1i : u ∈ U}, the element (e, f (e)) is not appeared in the neutrosophic soft set f . Set of all ns-sets over U is denoted by NS. Definition 2. [19] Let f, g ∈ NS. f is said to be neutrosophic soft subset of g, if tf (e) (u) ≤ tg(e) (u), if (e) (u) ≥ ig(e) (u) ff (e) (u) ≥ fg(e) (u), ∀e ∈ E, ∀u ∈ U. We denote it by f ⊑ g. f is said to be neutrosophic soft super set of g if g is a neutrosophic soft subset of f . We denote it by f ⊒ g.

If f is neutrosophic soft subset of g and g is neutrosophic soft subset of f . We denote it f = g Definition 3. [19] Let f ∈ NS. If tf (e) (u) = 0 and if (e) (u) = ff (e) (u) = 1 for all e ∈ E and for all u ∈ U, then f is called null ns-set and denoted by ˜ Φ.

Definition 4. [19] Let f ∈ NS. If tf (e) (u) = 1 and if (e) (u) = ff (e) (u) = 0 for all e ∈ E and for all u ∈ U, then f is called universal ns-set and denoted ˜

by U. Definition 5. [19] Let f, g ∈ NS. Then union and intersection of ns-sets f and g denoted by f ⊔ g and f ⊓ g respectively, are defined by as follow n f ⊔ g = (e, {hu, tf (e) (u) ∨ tg(e) (u), if (e) (u) ∧ ig(e) (u), o ff (e) (u) ∧ fg(e) (u)i : u ∈ U}) : e ∈ E . and ns-intersection of f and g is defined as n f ⊓ g = (e, {hu, tf (e) (u) ∧ tg(e) (u), if (e) (u) ∨ ig(e) (u), o ff (e) (u) ∨ fg(e) (u)i : u ∈ U}) : e ∈ E .

3

Definition 6. [19] Let f, g ∈ NS. Then complement of ns-set f , denoted by f c˜, is defined as follow n o c˜ f = (e, {hu, ff (e) (u), 1 − if (e) (u), tf (e) (u)i : u ∈ U}) : e ∈ E .

Proposition 1. [19] Let f, g, h ∈ NS. Then, ˜ ⊑f i. Φ ˜ ii. f ⊑ U iii. f ⊑ f iv. f ⊑ g and g ⊑ h ⇒ f ⊑ h Proposition 2. [19] Let f ∈ NS. Then ˜ c˜ = U˜ i. Φ ˜ ii. U˜ c˜ = Φ iii. (f c˜)c˜ = f .

Proposition 3. [19] Let f, g, h ∈ NS. Then, i. f ⊓ f = f and f ⊔ f = f ii. f ⊓ g = g ⊓ f and f ⊔ g = g ⊔ f ˜ =Φ ˜ and f ⊓ U˜ = f iii. f ⊓ Φ ˜ = f and f ⊔ U˜ = U˜ iv. f ⊔ Φ v. f ⊓ (g ⊓ h) = (f ⊓ g) ⊓ h and f ⊔ (g ⊔ h) = (f ⊔ g) ⊔ h vi. f ⊓ (g ⊔ h) = (f ⊓ g) ⊔ (f ⊓ h) and f ⊔ (g ⊓ h) = (f ⊔ g) ⊓ (f ⊔ h).

Proof. The proof is clear from definition and operations of neutrosophic soft sets.

Theorem 1. [19] Let f, g ∈ NS. Then, De Morgan’s law is valid. i. (f ⊔ g)c˜ = f c˜ ⊓ g c˜ 4

ii. (f ⊔ g)c˜ = f c˜ ⊓ g c˜ Definition 7. [19] Let f, g ∈ NS. Then ’OR’ product of ns-sets f and g denoted by f ∧ g, is defined as follow n _ f g = ((e, e′ ), {hu, tf (e)(u) ∨ tg(e) (u), if (e) (u) ∧ ig(e) (u), o ′ ff (e) (u) ∧ fg(e) (u)i : u ∈ U}) : (e, e ) ∈ E × E .

Definition 8. [19] Let f, g ∈ NS. Then ’AND’ product of ns-sets f and g denoted by f ∨ g, is defined as follow n ^ f g = ((e, e′ ), {hu, tf (e) (x) ∧ tg(e) (u), if (e) (u) ∨ ig(e) (u), o ff (e) (u) ∨ fg(e) (u)i : u ∈ U}) : (e, e′ ) ∈ E × E .

Proposition 4. [19] Let f, g ∈ NS. Then, W V 1. (f g)c˜ = f c˜ g c˜ V W 2. (f g)c˜ = f c˜ g c˜

Proof. The proof is clear from Definition 7 and 8.

Definition 9. [3] Let U = {u1 , u2, ..., un } be the universal set of elements and E = {e1 , e2 , ..., em } be the universal set of parameters. The pair (U, E) will be called a soft universe. Let F : E → I U and µ be a fuzzy subset of E, that is µ : E → I U , where I U is the collection of all fuzzy subsets of U. Let Fµ : E → I U × I U be a function defined as follows: Fµ (e) = (F (e)(u), µ(e)(u)), ∀u ∈ U. Then Fµ is called a possibility fuzzy soft set (PFSS in short) over the soft universe (U, E). For each parameter ei , Fµ (ei ) = (F (ei )(u), µ(ei )(u)) indicates not only the degree of belongingness of the elements of U in F (ei ), but also the degree of possibility of belongingness of the elements of U in F (ei ), which is represented by µ(ei ).

Definition 10. [5] Let U = {u1 , u2, ..., un } be the universal set of elements and E = {e1 , e2 , ..., em } be the universal set of parameters. The pair (U, E) will be called a soft universe. Let F : E → (I × I)U × I U where (I × I)U is 5

the collection of all intuitionistic fuzzy subsets of U and I U is the collection of all intuitionistic fuzzy subsets of U. Let p be a fuzzy subset of E, that is, p : E → I U and let Fp : E → (I × I)U × I U be a function defined as follows: Fp (e) = (F (e)(u), p(e)(u)), F (e)(u) = (µ(u), ν(u))∀u ∈ U. Then Fp is called a possibility intuitionistic fuzzy soft set (PIFSS in short) over the soft universe (U, E). For each parameter ei , Fp (ei ) = (F (ei )(u), p(ei )(u)) indicates not only the degree of belongingness of the elements of U in F (ei ), but also the degree of possibility of belongingness of the elements of U in F (ei ), which is represented by p(ei ).

3. Possibility neutrosophic soft sets In this section, we introduced the concept of possibility neutrosophic soft set, possibility neutrosophic soft subset, possibility neutrosophic soft null set, possibility neutrosophic soft universal set and possibility neutrosophic soft set operations such as union, intersection and complement.

Throughout paper U is an initial universe, E is a set of parameters and Λ is an index set.

Definition 11. Let U be an initial universe, E be a parameter set, N (U) be the collection of all neutrosophic sets of U and I U is collection of all fuzzy subset of U. A possibility neutrosophic soft set (P NS-set) fµ over U is defined by the set of ordered pairs fµ = {(ei , {(

uj , µ(ei )(uj )) : uj ∈ U}) : ei ∈ E} f (ei )(uj )

where, i, j ∈ Λ, f is a mapping given by f : E → N (U) and µ(ei ) is a fuzzy set such that µ : E → I U . Here, f˜µ is a mapping defined by fµ : E → N (U) × I U .

For each parameter ei ∈ E, f (ei ) = {huj , tf (ei ) (uj ), if (ei ) (uj ), ff (ei ) (uj )i : uj ∈ U} indicates neutrosophic value set of parameter ei and where t, i, f : U → [0, 1] are the membership functions of truth, indeterminacy and falsity respectively of the element uj ∈ U. For each uj ∈ U and ei ∈ E, 0 ≤ tf (ei ) (uj ) + if (ei ) (uj ) + ff (ei ) (uj ) ≤ 3. Also µ(ei ), degrees of possibility of belongingness of elements of U in f (ei ). So we can write fµ (ei ) =

n

o u2 un u1 , µ(ei )(u1 ) , , µ(ei )(u2 ) , ..., , µ(ei )(un ) f (ei )(u1 ) f (ei )(u2 ) f (ei )(un )

6

From now on, we will show set of all possibility neutrosophic soft sets over U with PS(U, E) such that E is parameter set.

Example 1. Let U = {u1, u2 , u3 } be a set of three cars. Let E = {e1 , e2 , e3 } be a set of qualities where e1 =cheap, e2 =equipment, e3 =fuel consumption and let µ : E → I U . We define a function fµ : E → N (U) × I U as follows:  n

o  u1 u2 u3   f (e ) = , 0.8 , , 0.4 , , 0.7     µ 1  (0.7,0.3,0.5) (0.4,0.5,0.8) o  n (0.5,0.2,0.6) u2 u3 u1 fµ = , 0.6 , (0.5,0.7,0.2) , 0.8 , (0.7,0.3,0.9) , 0.4 fµ (e2 ) = 

o  n (0.8,0.4,0.5)     u2 u3 u1  fµ (e3 ) =  , 0.2 , , 0.6 , , 0.5 (0.6,0.7,0.5) (0.5,0.3,0.7) (0.6,0.5,0.4) also we can define a function gν : E → N (U) × I U as follows:  n

o u1 u2 u3  g (e ) = , 0.4 , , 0.7), , 0.8  ν 1   (0.6,0.5,0.5) (0.2,0.6,0.4) o n (0.6,0.3,0.8) u2 u3 u1 gν = , 0.3 , , 0.6 , , 0.8 gν (e2 ) =  (0.4,0.6,0.5) (0.7,0.2,0.5) o n (0.5,0.4,0.3)   u2 u3 u1  gν (e3 ) = , 0.8 , (0.4,0.4,0.6) , 0.5 , (0.8,0.5,0.3) , 0.6 (0.7,0.5,0.3)

        

For the purpose of storing a possibility neutrosophic soft set in a computer, we can use matrix notation of possibility neutrosophic soft set fµ .

For example, matrix notation of possibility neutrosophic soft set fµ can be written as follows: for m, n ∈ Λ,   (h0.5, 0.2, 0.6i, 0.8) (h0.7, 0.3, 0.5i, 0.4) (h0.4, 0.5, 0.8i, 0.7) fµ =  (h0.8, 0.4, 0.5i, 0.6) (h0.5, 0.7, 0.2i, 0.8) (h0.7, 0.3, 0.9i, 0.4)  (h0.6, 0.7, 0.5i, 0.2) (h0.5, 0.3, 0.7i, 0.6) (h0.6, 0.5, 0.4i, 0.5) where the m−th row vector shows f (em ) and n−th column vector shows un .

Definition 12. Let fµ , gν ∈ PS(U, E). Then, fµ is said to be a possibility neutrosophic soft subset (P NS-subset) of gν , and denoted by fµ ⊆ gν , if i. µ(e) is a fuzzy subset of ν(e), for all e ∈ E ii. f is a neutrosophic subset of g, Example 2. Let U = {u1, u2 , u3 } be a set of tree houses, and let E = {e1 , e2 , e3 } be a set of parameters where e1 =modern, e2 =big and e3 =cheap.

Let fµ be a P NS-set defined as follows:

7

 n

o u1 u2 u3  f (e ) = , 0.8 , , 0.4 , , 0.7  µ 1   (0.7,0.3,0.5) (0.4,0.5,0.9) o n (0.5,0.2,0.6) u2 u3 u1 , 0.6 , (0.5,0.7,0.2) , 0.8 , (0.7,0.3,0.9) , 0.4 fµ = fµ (e2 ) = 

o n (0.8,0.4,0.5)   u2 u3 u1  , 0.2 , , 0.6 , , 0.5 fµ (e3 ) = (0.6,0.7,0.5) (0.5,0.3,0.8) (0.6,0.5,0.4)

gν : E → N (U) × I U be another P NS-set defined as follows:  n

o u1 u2 u3  g (e ) = , 0.9 , , 0.6 , , 0.8  ν 1   (0.8,0.2,0.3) (0.7,0.5,0.8) o n (0.6,0.1,0.5) u2 u3 u1 gν = , 0.7 , (0.9,0.5,0.1) , 0.9 , (0.8,0.1,0.9) , 0.5 gν (e2 ) = 

o n (0.9,0.2,0.4)   u2 u3 u1  gν (e3 ) = , 0.4 , , 0.9 , , 0.7 (0.6,0.5,0.4) (0.7,0.1,0.7) (0.8,0.2,0.4) it is clear that fµ is P NS − subset of gν

                 

Definition 13. Let fµ , gν ∈ PN (U, E). Then, fµ and gν are called possibility neutrosophic soft equal set and denoted by fµ = gν , if fµ ⊆ gν and fµ ⊇ gν .

Definition 14. Let fµ ∈ PN (U, E). Then, fµ is said to be possibility neutrosophic soft null set denoted by φµ , if ∀e ∈ E, φµ : E → N (U) × I U such u , µ(e)(u)) : u ∈ U}, where φ(e) = {hu, 0, 1, 1i : u ∈ U} that φµ (e) = {( φ(e)(u) and µ(e) = {(u, 0) : u ∈ U}).

Definition 15. Let fµ ∈ PN (U, E). Then, fµ is said to be possibility neutrosophic soft universal set denoted by Uµ , if ∀e ∈ E, Uµ : E → N (U) × I U such u that Uµ (e) = {( U (e)(u) , µ(e)(u)) : u ∈ U}, where U(e) = {hu, 1, 0, 0i : u ∈ U} and µ(e) = {(u, 1) : u ∈ U}).

Proposition 5. Let fµ , gν and hδ ∈ PN (U, E). Then, i. φµ ⊆ fµ ii. fµ ⊆ Uµ iii. fµ ⊆ fµ iv. fµ ⊆ gν and gν ⊆ hδ ⇒ fµ ⊆ hδ Proof. It is clear from Definition 13, 14 and 15.

Definition 16. Let fµ ∈ PN (U), where fµ (ei ) = {(f (ei )(uj ), µ(ei )(uj )) : ei ∈ E, uj ∈ U} and f (ei ) = {hu, tf (ei ) (uj ), if (ei ) (uj ), ff (ei ) (uj )i} for all ei ∈ E, u ∈ U. Then for ei ∈ E and uj ∈ U, 8

1. fµt is said to be truth-membership part of fµ , fµt = {(fijt (ei ), µij (ei ))} and fijt (ei ) = {(uj , tf (ei ) (uj ))}, µij (ei ) = {(uj , µ(ei )(uj ))} 2. fµi is said to be indeterminacy-membership part of fµ , fµi = {(fiji (ei ), µij (ei ))} and fiji (ei ) = {(uj , if (ei ) (uj ))}, µij (ei ) = {(uj , µ(ei )(uj ))} 3. fµf is said to be truth-membership part of fµ , fµf = {(fijf (ei ), µij (ei ))} and fijf (ei ) = {(uj , ff (ei ) (uj ))}, µij (ei ) = {(uj , µ(ei )(uj ))} We can write a possibility neutrosophic soft set in form fµ = (fµt , fµi , fµf ).

If considered the possibility neutrosophic soft set fµ in Example 1, fµ can be expressed in matrix form as follow:   (0.5, 0.8) (0.7, 0.4) (0.4, 0.7) fµt =  (0.8, 0.6) (0.5, 0.8) (0.7, 0.4)  (0.6, 0.2) (0.5, 0.6) (0.6, 0.5)   (0.2, 0.8) (0.3, 0.4) (0.5, 0.7) fµi =  (0.4, 0.6) (0.7, 0.8) (0.3, 0.4)  (0.7, 0.2) (0.3, 0.6) (0.5, 0.5)   (0.6, 0.8) (0.5, 0.4) (0.8, 0.7) fµf =  (0.5, 0.6) (0.2, 0.8) (0.9, 0.4)  (0.5, 0.2) (0.7, 0.6) (0.4, 0.5) Definition 17. [27] A binary operation ⊗ : [0, 1]×[0, 1] → [0, 1] is continuous t−norm if ⊗ satisfies the following conditions i. ⊗ is commutative and associative, ii. ⊗ is continuous, iii. a ⊗ 1 = a, ∀a ∈ [0, 1], iv. a ⊗ b ≤ c ⊗ d whenever a ≤ c, b ≤ d and a, b, c, d ∈ [0, 1].

Definition 18. [27] A binary operation ⊕ : [0, 1]×[0, 1] → [0, 1] is continuous t−conorm (s-norm) if ⊕ satisfies the following conditions i. ⊕ is commutative and associative, ii. ⊕ is continuous, 9

iii. a ⊕ 0 = a, ∀a ∈ [0, 1], iv. a ⊕ b ≤ c ⊕ d whenever a ≤ c, b ≤ d and a, b, c, d ∈ [0, 1].

Definition 19. Let I 3 = [0, 1] × [0, 1] × [0, 1] and N(I 3 ) = {(a, b, c) : a, b, c ∈ [0, 1]}. Then (N(I 3 ), ⊕, ⊗) be a lattices together with partial ordered relation , where order relation on N(I 3 ) can be defined by for (a, b, c), (d, e, f ) ∈ N(I 3 ) (a, b, c) (e, f, g) ⇔ a ≤ e, b ≥ f, c ≥ g Definition 20. A binary operation

2 ˜ : [0, 1] × [0, 1] × [0, 1] → [0, 1] × [0, 1] × [0, 1] ⊗ ˜ satisfies the following conditions is continuous n−norm if ⊗ ˜ is commutative and associative, i. ⊗ ˜ is continuous, ii. ⊗ ˜ = 0, a⊗1 ˜ = a, ∀a ∈ [0, 1] × [0, 1] × [0, 1], (1 = (1, 0, 0)) and (0 = iii. a⊗0 (0, 1, 1)) ˜ ≤ c⊗d ˜ whenever a c, b d and a, b, c, d ∈ [0, 1] × [0, 1] × [0, 1]. iv. a⊗b Here, ˜ = ⊗(ht(a), ˜ a⊗b i(a), f (a)i, ht(b), i(b), f (b)i) = ht(a)⊗t(b), i(a)⊕i(b), f (a)⊕f (b) Definition 21. A binary operation

2 ˜ : [0, 1] × [0, 1] × [0, 1] → [0, 1] × [0, 1] × [0, 1] ⊕ ˜ satisfies the following conditions is continuous n−conorm if ⊕ ˜ is commutative and associative, i. ⊕ ˜ is continuous, ii. ⊕ ˜ = a, a⊕1 ˜ = 1, ∀a ∈ [0, 1] × [0, 1] × [0, 1], (1 = (1, 0, 0)) and (0 = iii. a⊕0 (0, 1, 1)) 10

Ëœ ≤ c⊕d Ëœ whenever a ≤ c, b ≤ d and a, b, c, d ∈ [0, 1] Ă— [0, 1] Ă— [0, 1]. iv. a⊕b Here, Ëœ = ⊕(ht(a), Ëœ a⊕b i(a), f (a)i, ht(b), i(b), f (b)i) = ht(a)⊕t(b), i(a)⊗i(b), f (a)⊗f (b) Definition 22. Let fÂľ , gν ∈ PN (U, E). The union of two possibility neutrosophic soft sets fÂľ and gν over U, denoted by fÂľ âˆŞ gν is defined by as follow: fÂľ âˆŞgν =

ei ,

uj t (e ) ⊕ g t (e ), f i (e ) ⊗ (fij i ij i ij i

i (e ), f f (e ) gij i ij i

⊗

f gij (ei ))

, Âľij (ei )⊕νij (ei ) : uj ∈ U : ei ∈ E

Definition 23. Let fÂľ , gν ∈ PN (U, E). The intersection of two possibility neutrosophic soft sets fÂľ and gν over U, denoted by fÂľ ∊ gν is defined by as follow: fÂľ ∊gν =

ei ,

uj t (e ) ⊗ g t (e ), f i (e ) ⊕ g i (e ), f f (e ) ⊕ g f (e )) (fij i ij i ij i ij i ij i ij i

, Âľij (ei )⊗νij (ei ) : uj ∈ U : ei ∈ E

Example 3. Let us consider the possibility neutrosophic soft sets fÂľ and gν defined as in Example 1. Let us suppose that t−norm is defined by a ⊗ b = min{a, b} and the t−conorm is defined by a ⊕ b = max{a, b} for a, b ∈ [0, 1].

Then, 

o  n u2 u3 u1  , 0.8 , (0.7,0.3,0.5) , 0.7 , (0.4,0.5,0.4) , 0.8 (fÂľ âˆŞ gν )(e1 ) =    n (0.6,0.2,0.6)

o u1 u2 u3 (fÂľ âˆŞ gν )(e2 ) = , 0.6 , , 0.8 , , 0.8 fÂľ âˆŞgν =  (0.5,0.6,0.2) (0.7,0.2,0.5) o n (0.8,0.4,0.3)   u2 u3 u1  (fÂľ âˆŞ gν )(e3 ) = , 0.8 , , 0.6 , (0.8,0.5,0.3) , 0.6 (0.7,0.3,0.3) (0.5,0.3,0.6)

and



o n u2 u3 u1  , 0.4 , (0.6,0.5,0.5) , 0.4 , (0.2,0.6,0.8) , 0.7 (fÂľ ∊ gν )(e1 ) =  (0.5,0.3,0.8)  

o n u2 u3 u1 , 0.3 , , 0.6 , , 0.4 (fÂľ ∊ gν )(e2 ) = fÂľ ∊gν =  (0.4,0.7,0.5) (0.7,0.3,0.9) o n (0.5,0.4,0.5)   u2 u3 u1  (fÂľ ∊ gν )(e3 ) = , 0.2 , , 0.5 , (0.6,0.5,0.4) , 0.5 (0.6,0.7,0.5) (0.4,0.5,0.7)

Proposition 6. Let fÂľ , gν , hδ ∈ PN (U, E). Then, i. fÂľ ∊ fÂľ = fÂľ and fÂľ âˆŞ fÂľ = fÂľ ii. fÂľ ∊ gν = gν ∊ fÂľ and fÂľ âˆŞ gν = gν âˆŞ fÂľ iii. fÂľ ∊ φ¾ = φ¾ and fÂľ ∊ UÂľ = fÂľ iv. fÂľ âˆŞ φ = fÂľ and fÂľ âˆŞ UÂľ = UÂľ 11

                

v. fµ ∩ (gν ∩ hδ ) = (fµ ∩ gν ) ∩ hδ and fµ ∪ (gν ∪ hδ ) = (fµ ∪ gν ) ∪ hδ vi. fµ ∩(gν ∪hδ ) = (fµ ∩gν )∪(fµ ∩hδ ) and fµ ∪(gν ∩hδ ) = (fµ ∪gν )∩(fµ ∪hδ ).

Proof. The proof can be obtained from Definitions 22. and 23.

Definition 24. [17, 32] A function N : [0, 1] → [0, 1] is called a negation if N(0) = 1, N(1) = 0 and N is non-increasing (x ≤ y ⇒ N(x) ≥ N(y)). A negation is called a strict negation if it is strictly decreasing (x < y ⇒ N(x) > N(y)) and continuous. A strict negation is said to be a strong negation if it is also involutive, i.e. N(N(x)) = x Definition 25. [30] A function nN : [0, 1]×[0, 1]×[0, 1] → [0, 1]×[0, 1]×[0, 1] is called a negation if nN (0) = 1, nN (1) = 0 and nN is non-increasing (x y ⇒ nN (x) nN (y)). A negation is called a strict negation if it is strictly decreasing (x ≺ y ⇒ N(x) ≻ N(y)) and continuous.

Definition 26. Let fµ ∈ PN (U, E). Complement of possibility neutrosophic soft set fµ , denoted by fµc is defined as follow: n

o uj c fµ = e, {( , N(µij (ei )(uj )) : uj ∈ U} : e ∈ E nN (f (ei ))

where (nN (fij (ei )) = (N(fijt (ei )), N(fiji (ei )), N(fijf (ei )) for all i, j ∈ Λ

Example 4. Let us consider the possibility neutrosophic soft set fµ define in Example 1. Suppose that the negation is defined by N(fijt (ei )) = fijf (ei ), N(fijf (ei )) = fijt (ei ), N(fiji (ei )) = 1 − fiji (ei ) and N(µij (ei )) = 1 − µij (ei ), respectively. Then, fµc is defined as follow:  n

o  u1 u2 u3 c   f (e ) = , 0.2 , , 0.6 , , 0.3   1    µ (0.5,0.7,0.7) (0.8,0.5,0.4) o  n (0.6,0.8,0.5) u2 u3 u1 c c , 0.4 , (0.2,0.3,0.5) , 0.2 , (0.9,0.7,0.7) , 0.6 fµ = fµ (e2 ) = 

o  n (0.5,0.6,0.8)     u2 u3 u1   f c (e3 ) = , 0.8 , , 0.4 , , 0.5 µ (0.5,0.3,0.6) (0.7,0.7,0.5) (0.4,0.5,0.6) Proposition 7. Let fµ ∈ PN (U, E). Then, i. φcµ = Uµ ii. Uµc = φµ iii. (fµc )c = fµ .

12

Proof. It is clear from Definition 26 Proposition 8. Let fÂľ , gν â&#x2C6;&#x2C6; PN (U, E). Then, De Morganâ&#x20AC;&#x2122;s law is valid. i. (fÂľ â&#x2C6;Ş gν )c = fÂľc â&#x2C6;Š gνc ii. (fÂľ â&#x2C6;Š gν )c = fÂľc â&#x2C6;Ş gνc Proof.

i. Let i, j â&#x2C6;&#x2C6; Î&#x203A; (fÂľ â&#x2C6;Ş gν )

c

=

=

=

=

â&#x2C6;Š

=

â&#x2C6;Š



 uj ei , ,  t (e ) â&#x160;&#x2022; g t (e ), f i (e ) â&#x160;&#x2014; g i (e ), f f (e ) â&#x160;&#x2014; g f (e )) (fij i ij i ij i ij i ij i ij i c 

 Âľij (ei ) â&#x160;&#x2022; νij (ei ) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E 



 uj ei , , f f  i (e ) â&#x160;&#x2014; g i (e )), f t (e ) â&#x160;&#x2022; g t (e )) (fij (ei ) â&#x160;&#x2014; gij (ei ), N(fij i ij i ij i ij i  

 N(Âľij (ei ) â&#x160;&#x2022; νij (ei )) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E 



 uj ei , , f f i (e )) â&#x160;&#x2022; N(g i (e ))), f t (e ) â&#x160;&#x2022; g t (e ))  (fij (ei ) â&#x160;&#x2014; gij (ei ), N(fij i ij i ij i ij i  

 N(Âľij (ei )) â&#x160;&#x2014; N(νij (ei )) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E 

  

  uj ei , , N(Âľij (ei )) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E f i (e )), f t (e )   (fij (ei ), N(fij i ij i   

   uj ei , , N(νij (ei )) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E f  i t   gij (ei ), N(gij (ei )), gij (ei )) c  

  uj ei , , Âľij (ei ) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E   t (e ), N(f i (e )), f f (e ) (fij i i i ij ij c  

  uj ei , , νij (ei ) : uj â&#x2C6;&#x2C6; U  : ei â&#x2C6;&#x2C6; E f   g t (e ), g i (e ), g (e )) ij

=

c

i

ij

i

ij

i

c

fÂľ â&#x2C6;Š gν

ii. The proof can be made with similar way.

Definition 27. Let fÂľ and gν â&#x2C6;&#x2C6; PN (U, E). Then â&#x20AC;&#x2122;ANDâ&#x20AC;&#x2122; product of PNS-set fÂľ and gν denoted by fÂľ â&#x2C6;§ gν , is defined as follow: fÂľ â&#x2C6;§ gν

=

n f f t t i i (ek , el ), (fkj (ek ) â&#x2C6;§ glj (el ), fkj (ek ) â&#x2C6;¨ glj (el ), fkj (ek ) â&#x2C6;¨ glj (el )),

o Âľkj (ek ) â&#x2C6;§ νlj (el ) : (ek , el ) â&#x2C6;&#x2C6; E Ă&#x2014; E, j, k, l â&#x2C6;&#x2C6; Î&#x203A;

13

Definition 28. Let fµ and gν ∈ PN (U, E). Then ’OR’ product of PNS-set fµ and gν denoted by fµ ∨ gν , is defined as follow: fµ ∨ gν

=

n f f t t i i (ek , el ), (fkj (ek ) ∨ glj (el ), fkj (ek ) ∧ glj (el ), fkj (ek ) ∧ glj (el )),

o µkj (ek ) ∨ νlj (el ) : (ek , el ) ∈ E × E, j, k, l ∈ Λ

4. Decision making method In this section we will construct a decision making method over the possibility neutrosophic soft set. Firstly, we will define some notions that necessary to construct algorithm of decision making method. And then we will present an application of possibility neutrosophic soft set theory in a decision making problem.

Definition 29. Let fµ ∈ PN (U, E), fµt , fµi and fµf be the truth, indeterminacy and falsity matrices of ∧-product matrix, respectively. Then, weighted matrices of fµt , fµi and fµf are denoted by ∧t , ∧i and ∧f , respectively and computed by as follows: ∧t (eij , uk ) = t(fµ ∧gν )(eij ) (uk )+(µik (ei )∧νjk (ej ))−t(fµ ∧gν )(eij ) (uk )×(µik (ei )∧νjk (ej )) ∧i (eij , uk ) = i(fµ ∧gν )(eij ) (uk ) × (µik (ei ) ∧ νjk (ej )) ∧f (eij , uk ) = f(fµ ∧gν )(eij ) (uk ) × (µik (ei ) ∧ νjk (ej ))

for i, j, k ∈ Λ

Definition 30. Let fµ ∈ PN (U, E), ∧t , ∧i and ∧f be the weighed matrices of fµt , fµi and fµf , respectively. Then, for all ut ∈ U such that t ∈ Λ, scores of ut is in the weighted matrices ∧t , ∧i and ∧f are denoted by st (uk ), si (uk ) and sf (uk ), and computed by as follow, respectively X st (ut ) = δijt (ut ) i,j∈Λ

si (ut ) =

X

i,j∈Λ

14

δiji (ut )

sf (ut ) =

X

δijf (ut )

i,j∈Λ

here δijt (ut ) δiji (ut ) δijf (ut )

=

∧t (eij , ut), ∧t (eij , ut ) = max{∧t (eij , uk ) : uk ∈ U} 0, otherwise

=

∧i (eij , ut ), ∧i (eij , ut ) = max{∧i (eij , uk ) : uk ∈ U} 0, otherwise

=

∧f (eij , ut ), ∧f (eij , ut ) = max{∧f (eij , uk ) : uk ∈ U} 0, otherwise

Definition 31. Let st (ut ), si (ut ) and sf (ut ) be scores of ut ∈ U in the weighted matrices ∧t , ∧i and ∧f . Then, decision score of ut ∈ U, denoted by ds(ut ), is computed as follow: ds(ut ) = st (ut ) − si (ut ) − sf (ut ) Now, we construct a possibility neutrosophic soft set decision making method by the following algorithm; Algorithm Step 1: Input the possibility neutrosophic soft set fµ , Step 2: Construct the matrix ∧-product Step 3: Construct the matrices fµt , fµi and fµf Step 4: Construct the weighted matrices ∧t , ∧i and ∧f , Step 5: Compute score of ut ∈ U, for each of the weighted matrices, Step 6: Compute decision score, for all ut ∈ U, Step 7: The optimal decision is to select ut = maxds(ui ).

15

Example 5. Assume that U = {u1 , u2, u3 } is a set of houses and E = {e1 , e2 , e3 } = {cheap, large, moderate} is a set of parameters which is attractiveness of houses. Suppose that Mr.X want to buy one the most suitable house according to to himself depending on three of the parameters only.

Step 1:Based on the choice parameters of Mr.X, let there be two observations f two experts defined µ and gν byn as follows: o  u1 u2 u3  f (e ) = , 0.6 , (0.6,0.2,0.5) , 0.2 , (0.7,0.6,0.5) , 0.4    µ 1

o n (0.5,0.3,0.7) u2 u3 u1 fµ = , 0.4 , , 0.5 , , 0.6 fµ (e2 ) =  (0.7,0.8,0.3) (0.2,0.4,0.4) o n (0.35,0.2,0.6)   u2 u3 u1  fµ (e3 ) = , 0.5 , , 0.3 , , 0.2 (0.7,0.2,0.5) (0.4,0.5,0.2) (0.5,0.3,0.6)

 n

o u1 u2 u3  g (e ) = , 0.2 , , 0.5 , , 0.3    ν 1 n (0.3,0.4,0.5) (0.7,0.3,0.4) (0.4,0.5,0.2) o u1 u2 u3 gν = gν (e2 ) = , 0.3 , , 0.7 , , 0.8  (0.2,0.5,0.3) (0.4,0.6,0.2) o n (0.4,0.6,0.2)   u u3 u 2 1  gν (e3 ) = , 0.7 , , 0.4 , , 0.4 (0.2,0.1,0.6) (0.8,0.4,0.5) (0.6,0.4,0.3)

       

        

Step 2: Let us consider possibility neutrosophic soft set ∧-product which

is the mapping ∧ : E × E → N (U) × I U given as follows:                

∧ u1 , µ e11 (h0.3, 0.4, 0.7i, 0.2) e12 (h0.4, 0.6, 0.7i, 0.3) e13 (h0.2, 0.3, 0.7i, 0.6) e21 (h0.3, 0.4, 0.6i, 0.2) e22 (h0.35, 0.6, 0.6i, 0.3) e23 (h0.2, 0.2, 0.6i, 0.4) e31 (h0.3, 0.4, 0.5i, 0.2) e32 (h0.4, 0.6, 0.5i, 0.3) e33 (h0.2, 0.2, 0.6i, 0.5)

u2 , µ (h0.6, 0.3, 0.5i, 0.2) (h0.2, 0.5, 0.5i, 0.2) (h0.6, 0.4, 0.5i, 0.2) (h0.7, 0.8, 0.4i, 0.5) (h0.2, 0.8, 0.3i, 0.5) (h0.7, 0.8, 0.5i, 0.4) (h0.4, 0.5, 0.4i, 0.3) (h0.2, 0.5, 0.3i, 0.3) (h0.4, 0.5, 0.5i, 0.3)

u3 , µ (h0.4, 0.6, 0.5i, 0.3) (h0.4, 0.6, 0.5i, 0.4) (h0.6, 0.6, 0.5i, 0.4) (h0.2, 0.5, 0.5i, 0.3) (h0.2, 0.6, 0.5i, 0.6) (h0.2, 0.4, 0.5i, 0.4) (h0.4, 0.5, 0.6i, 0.2) (h0.4, 0.6, 0.6i, 0.2) (h0.5, 0.4, 0.6i, 0.2)

Matrix representation of ∧-product

16

               

Step 3: We construct matrices fµt , fµi and fµf as follows:                

∧ u1 , µ e11 (0.3, 0.2) e12 (0.4, 0.3) e13 (0.2, 0.6) e21 (0.3, 0.2) e22 (0.35, 0.3) e23 (0.2, 0.4) e31 (0.3, 0.2) e32 (0.4, 0.3) e33 (0.2, 0.5)

u2 , µ (0.6, 0.2) (0.2, 0.2) (0.6, 0.2) (0.7, 0.5) (0.2, 0.5) (0.7, 0.4) (0.4, 0.3) (0.2, 0.3) (0.4, 0.3)

u3 , µ (0.4, 0.3) (0.4, 0.4) (0.6, 0.4) (0.2, 0.3) (0.2, 0.6) (0.2, 0.4) (0.4, 0.2) (0.4, 0.2) (0.5, 0.2)

               

Matrix fµt of ∧-product

               

∧ e11 e12 e13 e21 e22 e23 e31 e32 e33

u1 , µ (0.4, 0.2) (0.6, 0.3) (0.3, 0.6) (0.4, 0.2) (0.6, 0.3) (0.2, 0.4) (0.4, 0.2) (0.6, 0.3) (0.2, 0.5)

u2 , µ (0.3, 0.2) (0.5, 0.2) (0.4, 0.2) (0.8, 0.5) (0.8, 0.5) (0.8, 0.4) (0.5, 0.3) (0.5, 0.3) (0.5, 0.3)

u3 , µ (0.6, 0.3) (0.6, 0.4) (0.6, 0.4) (0.5, 0.3) (0.6, 0.6) (0.4, 0.4) (0.5, 0.2) (0.6, 0.2) (0.4, 0.2)

               

Matrix fµi of ∧-product

               

∧ e11 e12 e13 e21 e22 e23 e31 e32 e33

u1 , µ (0.7, 0.2) (0.7, 0.3) (0.7, 0.6) (0.6, 0.2) (0.6, 0.3) (0.6, 0.4) (0.5, 0.2) (0.5, 0.3) (0.6, 0.5)

u2 , µ (0.5, 0.2) (0.5, 0.2) (0.5, 0.2) (0.4, 0.5) (0.3, 0.5) (0.5, 0.4) (0.4, 0.3) (0.3, 0.3) (0.5, 0.3) 17

u3 , µ (0.5, 0.3) (0.5, 0.4) (0.5, 0.4) (0.5, 0.3) (0.5, 0.6) (0.5, 0.4) (0.6, 0.2) (0.6, 0.2) (0.6, 0.2)

               

Matrix fµf of ∧-product

Step 4: We obtain weighted matrices ∧t , ∧i and ∧f using Definition 29 as follows:                

∧t e11 e12 e13 e21 e22 e23 e31 e32 e33

u1 0.44 0.58 0.68 0.44 0.55 0.52 0.44 0.58 0.60

u2 0.64 0.36 0.68 0.85 0.60 0.82 0.58 0.44 0.58

u3 0.58 0.64 0.76 0.44 0.68 0.48 0.52 0.52 0.60



       ,       

               

∧i e11 e12 e13 e21 e22 e23 e31 e32 e33

u1 0.08 0.18 0.18 0.08 0.18 0.08 0.08 0.18 0.10

u2 0.16 0.10 0.08 0.40 0.40 0.32 0.15 0.15 0.15

u3 0.18 0.24 0.24 0.15 0.36 0.16 0.10 0.12 0.08



       ,       

               

∧f e11 e12 e13 e21 e22 e23 e31 e32 e33

u1 0.14 0.21 0.42 0.12 0.18 0.24 0.10 0.15 0.30

u2 0.10 0.10 0.10 0.20 0.15 0.20 0.12 0.09 0.15

u3 0.15 0.20 0.20 0.15 0.30 0.20 0.12 0.12 0.12

Weighed matrices of fµt , fµi and fµf from left to right, respectively.

Step 5: For all u ∈ U, we find scores using Definition 30 as follow: st (u1 ) = 1, 18,

st (u2 ) = 2, 89,

st (u3 ) = 2, 68

si (u1 ) = 0, 18 si (u2 ) = 1, 42 si (u3) = 0, 66 sf (u1 ) = 1, 32 sf (u2 ) = 0, 32 sf (u3 ) = 0, 57 Step 5: For all u ∈ U, we find scores using Definition 30 as follows: ds(u1) = 1, 18 − 0, 18 − 1, 32 = −0, 32 ds(u2) = 2, 89 − 1, 42 − 0, 32 = 0, 90 ds(u3) = 2, 68 − 0, 66 − 0, 57 = 1, 45 Step 5: Then the optimal selection for Mr.X is u3 .

5. Similarity measure of possibility neutrosophic soft sets In this section, we introduce a measure of similarity between two P NSsets.

18

               

Definition 32. Similarity between two P NS-sets fµ and gν , denoted by S(fµ , gν ), is defined as follows: S(fµ , gν ) = M(f (e), g(e)).M(µ(e), ν(e)) such that n

1 1X M(f (e), g(e)) = Mi (f (e), g(e)), M(µ, ν) = M(µ(ei ), ν(ei )), n n i=1 where v u n X 1 u p t (φf (e ) (uj ) − φgν (ei ) (uj ))p , 1 ≤ p ≤ ∞ Mi (f (e), g(e)) = 1 − √ p n i=1 µ i such that and gijt (ei ) + giji (ei ) + gijf (ei ) fijt (ei ) + fiji (ei ) + fijf (ei ) , φgµ (ei ) (uj ) = , φfµ (ei ) (uj ) = 3 3 Pn j=1 |µij (ei ) − νij (ei )| M(µ(ei ), ν(ei )) = 1 − Pn j=1 |µij (ei ) + νij (ei )|

Definition 33. Let fµ and gν be two PNS-sets over U. We say that fµ and gν are significantly similar if S(fµ , gν ) ≥ 12

Proposition 9. Let fµ , gν ∈ PN (U, E). Then, i. S(fµ , gν ) = S(gµ , fν ) ii. 0 ≤ S(fµ , gν ) ≤ 1 iii. fµ = gν ⇒ S(fµ , gν ) = 1

Proof. The proof is straightforward and follows from Definition 32.

Example 6. Let us consider PNS-sets fµ and gν in Example 1 given as follows:

19

 n

o u1 u2 u3  f (e ) = , 0.8 , , 0.4 , , 0.7  µ 1   (0.7,0.3,0.5) (0.4,0.5,0.8) o n (0.5,0.2,0.6) u2 u3 u1 , 0.6 , (0.5,0.7,0.2) , 0.8 , (0.7,0.3,0.9) , 0.4 fµ (e2 ) = 

o n (0.8,0.4,0.5)   u2 u3 u1  , 0.2 , , 0.6 , , 0.5 fµ (e3 ) = (0.6,0.7,0.5) (0.5,0.3,0.7) (0.6,0.5,0.4) and  n

o u1 u2 u3  gν (e1 ) = , 0.4 , (0.6,0.5,0.5) , 0.7), (0.2,0.6,0.4) , 0.8   

o n (0.6,0.3,0.8) u2 u3 u1 , 0.3 , , 0.6 , , 0.8 gν (e2 ) =  (0.4,0.6,0.5) (0.7,0.2,0.5) o n (0.5,0.4,0.3)   u u3 u 2 1  gν (e3 ) = , 0.8 , , 0.5 , , 0.6 (0.7,0.5,0.3) (0.4,0.4,0.6) (0.8,0.5,0.3) then,

P3

j=1

M(µ(e1 ), ν(e1 )) = 1 − P3

            

|µ1j (e1 ) − ν1j (e1 )|

j=1 |µ1j (e1 )

= 1−

    

+ ν1j (e1 )|

|0.8 − 0.4| + |0.4 − 0.7| + |0.7 − 0.8| = 0.79 |0.8 + 0.4| + |0.4 + 0.7| + |0.7 + 0.8|

Similarly we get M(µ(e2 ), ν(e2 )) = 0.74 and M(µ(e3 ), ν(e3 )) = 0.75, then 1 M(µ, ν) = (0.79 + 0.75 + 0.74) = 0.76 3 v u n X 1 u p t M1 (f (e), g(e)) = 1 − √ (φf (e ) (uj ) − φgν (ei ) (uj ))p p n i=1 µ i

1 p = 1− √ (0.43 − 0.57)2 + (0, 50 − 0.53)2 + (0, 57 − 0, 40)2 = 0.73 3 M2 (f (e), g(e)) = 0.86 M3 (f (e), g(e)) = 0.94 1 M(f (e), g(e)) = (0.73 + 0.86 + 0.94) = 0.84 3 and S(fµ , gν ) = 0.84 × 0.76 = 0.64 20

6. Decision-making method based on the similarity measure In this section, we give a decision making problem involving possibility neutrosophic soft sets by means of the similarity measure between the possibility neutrosophic soft sets.

Let our universal set contain only two elements ”yes” and ”no”, that is U = y, n. Assume that P = {p1 , p2 , p3 , p4 , p5 } are five candidates who fill in a form in order to apply formally for the position. There is a decision maker committee. They want to interview the candidates by model possibility neutrosophic soft set determined by committee. So they want to test similarity of each of candidate to model possibility neutrosophic soft set.

Let E = {e1 , e2 , e3 , e4 , e5 , e6 , e7 } be the set of parameters, where e1 =experience, e2 =computer knowledge, e3 =training, e4 =young age, e5 =higher education, e6 =marriage status and e7 =good health.

Our model possibility neutrosophic soft set determined by committee for suitable candidates properties fµ is given in Table 1. fµ y n

e1 , µ (h1, 0, 0i, 1) (h0, 1, 1i, 1) fµ y n

e2 , µ (h1, 0, 0i, 1) (h1, 0, 0i, 1)

e5 , µ (h1, 0, 0i, 1) (h0, 1, 1i, 1)

e3 , µ (h0, 1, 1i, 1) (h0, 1, 1i, 1)

e6 , µ (h0, 1, 1i, 1) (h1, 0, 0i, 1)

e4 , µ (h0, 1, 1i, 1) (h1, 0, 0i, 1)

e7 , µ (h1, 0, 0i, 1) (h0, 1, 1i, 1)

Table 1: The tabular representation of model possibility neutrosophic soft set gν y n

e1 , ν (h0.7, 0.2, 0.5i, 0.4) (h0.3, 0.7, 0.1i, 0.3) gν y n

e2 , ν (h0.5, 0.4, 0.6i, 0.2) (h0.7, 0.3, 0.5i, 0.4)

e5 , ν (h0.2, 0.4, 0.3i, 0.5) (h0.1, 0.5, 0.2i, 0.6)

e3 , ν (h0.2, 0.3, 0.4i, 0.5) (h0.6, 0.5, 0.3i, 0.2)

e6 , ν (h0, 1, 1i, 0.3) (h1, 0, 0i, 0.5)

e4 , ν (h0.8, 0.4, 0.6i, 0.3) (h0.2, 0.1, 0.5i, 0.4)

e7 , ν (h0.1, 0.4, 0.7i, 0.2) (h0.3, 0.5, 0.1i, 0.4)

Table 2: The tabular representation of possibility neutrosophic soft set for p1 hδ y n

e1 , δ (h0.8, 0.2, 0.1i, 0.3) (h0.2, 0.4, 0.3i, 0.5) hδ y n

e2 , δ (h0.4, 0.2, 0.6i, 0.1) (h0.6, 0.3, 0.2i, 0.3)

e5 , δ (h0.5, 0.2, 0.4i, 0.5) (h0.4, 0.5, 0.6i, 0.2)

e3 , δ (h0.7, 0.2, 0.4i, 0.2) (h0.4, 0.3, 0.2i, 0.1)

e6 , δ (h0, 1, 1i, 0.5) (h1, 0, 0i, 0.2)

21

e4 , δ (h0.3, 0.2, 0.7i, 0.6) (h0.8, 0.1, 0.3i, 0.3)

e7 , δ (h0.3, 0.2, 0.5i, 0.4) (h0.7, 0.3, 0.4i, 0.2)

Table 3: The tabular representation of possibility neutrosophic soft set for p2

rθ y n

e1 , θ (h0.3, 0.2, 0.5i, 0.4) (h0.1, 0.7, 0.6i, 0.3) rθ y n

e2 , θ (h0.7, 0.1, 0.5i, 0.6) (h0.4, 0.2, 0.3i, 0.7)

e5 , θ (h0.6, 0.4, 0.3i, 0.2) (h0.4, 0.5, 0.9i, 0.1)

e3 , θ (h0.6, 0.5, 0.3i, 0.2) (h0.7, 0.4, 0.3i, 0.5)

e6 , θ (h0, 1, 1i, 0.3) (h1, 0, 0i, 0.3)

e4 , θ (h0.3, 0.1, 0.4i, 0.5) (h0.7, 0.1, 0.2i, 0.1)

e7 , θ (h0.9, 0.1, 0.1i, 0.5) (h0.2, 0.1, 0.7i, 0.6)

Table 4: The tabular representation of possibility neutrosophic soft set for p3

sα y n

e1 , α (h0.2, 0.1, 0.4i, 0.5) (h0.6, 0.5, 0.1i, 0.1) sα y n

e2 , α (h0.7, 0.5, 0.4i, 0.8) (h0.3, 0.7, 0.2i, 0.2)

e5 , α (h0.3, 0.2, 0.5i, 0.8) (h0.2, 0.1, 0.5i, 0.3)

e3 , α (h0.8, 0.1, 0.2i, 0.4) (h0.7, 0.5, 0.1i, 0.7)

e6 , α (h1, 0, 0i, 0.7) (h0, 1, 1i, 0.2)

e4 , α (h0.5, 0.4, 0.5i, 0.4) (h0.1, 0.3, 0.7i, 0.5)

e7 , α (h0.1, 0.8, 0.9i, 0.7) (h0.5, 0.1, 0.4i, 0.1)

Table 5: The tabular representation of possibility neutrosophic soft set for p4

mγ y n

e1 , γ (h0.1, 0.2, 0.1i, 0.3) (h0.4, 0.5, 0.3i, 0.2) mγ y n

e2 , γ (h0.2, 0.3, 0.5i, 0.8) (h0.7, 0.6, 0.1i, 0.3)

e5 , γ (h0.4, 0.2, 0.8i, 0.1) (h0.5, 0.4, 0.7i, 0.2)

e3 , γ (h0.4, 0.1, 0.3i, 0.9) (h0.2, 0.3, 0.4i, 0.5)

e6 , γ (h1, 0, 0i, 0.5) (h0, 1, 1i, 0.5)

e4 , γ (h0.7, 0.3, 0.2i, 0.3) (h0.5, 0.2, 0.3i, 0.6)

e7 , γ (h0.3, 0.2, 0.1i, 0.7) (h0.3, 0.2, 0.1i, 0.9)

Table 6: The tabular representation of possibility neutrosophic soft set for p5

Now we find the similarity between the model possibility neutrosophic soft set and possibility neutrosophic soft set of each person as follow S(fµ , gν ) ∼ = 0, 49 < 12 , S(fµ , hδ ) ∼ = 0, 47 < 21 , S(fµ , rθ ) ∼ = 0, 51 > 12 , 1 1 S(fµ , sα ) ∼ = 0, 54 > 2 , S(fµ , mγ ) ∼ = 0, 57 > 2 , Consequently, p5 is should be selected by the committee.

22

7. Conclusion In this paper we have introduced the concept of possibility neutrosophic soft set and studied some of the related properties. Applications of this theory have been given to solve a decision making problem. We also presented a new method to find out the similarity measure of two possibility neutrosophic soft sets and we applied to a decision making problem. In future, these seem to have natural applications and algebraic structure.

References [1] M.I. Ali, F. Feng, X. Liu, W.K. Min, On some new operations in soft set theory, Comput. Math. Appl. 57 (9) (2009) 15471553.

[2] H. Akta¸s and N. C ¸ agman, Soft sets and soft groups. Information Sciences, 1(77) (2007) 2726-2735.

[3] S. Alkhazaleh, A.R. Salleh and N. Hassan, Possibility fuzzy soft set, Advances in Decision Science, doi:10.1155/2011/479756.

[4] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96.

[5] M. Bsahir, A.R. Salleh and S. Alkhazaleh, Possibility intuitionistic fuzzy soft set, Advances in Decision Science, doi:10.1155/2012/404325.

[6] S. Broumi, Generalized Neutrosophic Soft Set International Journal of Computer Science, Engineering and Information Technology, 3/2 (2013) 17-30.

[7] S. Broumi, F. Smarandache, Intuitionistic Neutrosophic Soft Set, Journal of Information and Computing Science, 8/2 (2013) 130-140.

[8] S. Broumi, I. Deli and F. Smarandache, Neutrosophic Parametrized Soft Set Theory and Its Decision Making, International Frontier Science Letters, 1 (1) (2014) 1-11.

[9] N. C ¸ a˘gman and S. Engino˘glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848-855.

[10] N. C ¸ a˘gman, Contributions to the theory of soft sets, Journal of New Results in Science, 4 (2014) 33-41.

23

[11] N. C ¸ a˘gman, Contributions to the Theory of Soft Sets, Journal of New Result in Science, 4 (2014) 33-41.

[12] I. Deli,Interval-valued neutrosophic soft sets ant its decision making, arXiv:1402.3130 [13] I. Deli, S. Broumi, Neutrosophic soft sets and neutrosophic soft matrices based on decision making, arXiv:1402.0673 [14] F. Feng, C. Li, B. Davvaz, M.I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing, 14(9) (2010) 899-911.

[15] F. Feng, Y.M. Li, N. ¸Ca˘gman, Generalized uni-int decision making schemes based on choice value soft sets, European Journal of Operational Research 220 (2012) 162-170.

[16] F. Feng, Y.M. Li, Soft subsets and soft product operations, Information Sciences, 232 (2013) 44-57.

[17] J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multiciriteria Decision Support. Kluwer (1994).

[18] W.L. Gau, D.J. Buehrer, Vague sets, IEEE Transactions on Systems, Man and Cybernetics 23 (2) (1993) 610-614.

[19] F. Karaaslan, Neutrosophic soft sets with applications in decision making, arXiv:1405.7964v2 [cs.AI] 2 Jun 2014 Submitted.

[20] D. Molodtsov, Soft set theory first results, Computers and Mathematics with Applications, 37 (1999) 19-31.

[21] P.K. Maji, A.R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077-1083.

[22] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl.

45 (2003) 555562.

[23] P.K. Maji, Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics, 5/ 1 (2013) 157-168.

24

[24] T. J. Neog, D.M. Sut, A New Approach to the Theory of Soft Sets, International Journal of Computer Applications 32/2 (2011) 0975-8887.

[25] D. Pei, D. Miao, From soft sets to information systems, in: X. Hu, Q. Liu, A. Skowron, T.Y. Lin, R.R. Yager, B. Zhang (Eds.), Proceedings of Granular Computing, vol. 2, pp. 617-621, 2005, IEEE. [26] Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci. 11 (1982) 341356.

[27] B. Schweirer, A. Sklar, Statistical metric space, Pasific Journal of Mathematics 10, 314-334 (1960).

[28] R. S¸ahin, A. K¨ uu¸ u ¨ k, Generalized neutrosophic soft set and its integration to decision making problem, Applied Mathematics and Information Sciences, 8(6) 1-9 (2014).

[29] A. Sezgin, A. O. Atagun, On operations of soft sets, Computers and Mathematics with Applications 61 (2011) 1457-1467.

[30] F. Smarandache, An unifying field in logics: Neutrosophic Logic, Neutrosophy , Neutrosophic set, Neutrosophic probability and Statistics.

American Research Press Rehoboth, (2005).

[31] F. Smarandache, Neutrosophic set - a generalization of the intuitionistic fuzzy set, International Journal of Pure and Applied Mathematics, 24/3 (2005) 287-297.

[32] E. Trillas, Sobre funciones de negacion en la teoria de conjuntos difusos.

Stochastica 3,47-60 (1979).

[33] Y. Xia, L. Zuhua, Some new operations of soft sets, Uncertainty Reasoning and Knowledge Engineering (URKE), 2nd International Conference, pp. 290 - 295, 14-15 Aug. 2012, IEEE. [34] C.F. Yang, A note on Soft Set Theory [Comput. Math. Appl. 45 (45) (2003) 555562], Comput. Math. Appl., 56 (2008) 1899-1900.

[35] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353.

[36] P. Zhu, Q. Wen, Operations on Soft Sets Revisited, Journal of Applied Mathematics 2013 (2013) 1-7.

25