FUZZY PARAMETERIZED SINGLE VALUED NEUTROSOPHIC SOFT EXPERT SET THEORY AND ITS APPLICATION

FUZZY PARAMETERIZED SINGLE VALUED NEUTROSOPHIC SOFT EXPERT SET THEORY AND ITS APPLICATION IN DECISION MAKING Abstract. In this paper, we propose the theory of fuzzy parameterized single valued neutrosophic soft expert set by giving an importance degree for each element in the set of parameters and define some related concepts pertaining to this notion as well as the basic operations of complement, subset, union, intersection, AND and OR along with illustrative examples. The basic properties and relevant laws pertaining to this concept such as the De Morgan’s laws are proved. A comparison between our proposed method and other methods is made to illustrate the advantages of our proposed method and its ability to handle problems involving imprecise, indeterminacy and inconsistent data, which makes it more accurate and realistic than other methods. Lastly, this concept is applied to a decision making problem and its effectiveness is demonstrated using an illustrative example.

Keywords. fuzzy parameterized soft expert set, single valued neutrosophic soft expert set, single valued neutrosophic set, soft expert set.

1. Introduction Fuzzy set was introduced by Zadeh [1] as a mathematical tool to solve problems and vagueness in everyday life. Since then the fuzzy sets and fuzzy logic have been applied in many real life problems in uncertain, ambiguous environment [2-5]. Later on several researches present a number of results using different direction of fuzzy set such as interval fuzzy set [6], intuitionistic fuzzy set [7] and vague set [8]. However, all these theories have their inherent difficulties and weaknesses. Thus neutrosophic set[9] is defined, as a new mathematical tool for dealing with problems involving incomplete, indeterminacy and inconsistent knowledge. In neutrosophic set, the indeterminacy is quantified explicitly and truth-membership, indeterminacy-membership and falsity-membership are completely independent. The works on neutrosophic set and their hybrid structure in theories and applications have been progressing rapidly [10-12]. From scientific or engineering point of view, neutrosophic set’s operators need to be specified. Otherwise, it will be difficult to apply in the real applications. Therefore, Wang et al.[13] defined a single valued neutrosophic set (SVNS in short) and then provided the set theoretic operations and various properties of SVNS. Molodtsov [14] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. Further, soft set has been developed

rapidly to soft expert set [15], soft multiset theory [16], fuzzy soft set [17], intuitionistic fuzzy soft set [18] and neutrosophic soft set [19]. Cagman et al.[20] introduced the concept of fuzzy parameterized soft sets followed by fuzzy parameterized fuzzy soft set [21]. Bashir and Salleh [22] introduced the notion of fuzzy parameterized soft expert sets while Hazaymeh et al.[23] on the notion of fuzzy parameterized fuzzy soft expert sets, followed by Selvachandran and Salleh [24] on the notion of fuzzy parameterized intuitionistic fuzzy soft expert set as a generalization of Hazaymeh et al.[23] We will further extend the studies on single valued neutrosophic set [13] and fuzzy parameterized soft expert set[22] through the establishment of the notion of fuzzy parameterized single valued neutrosophic soft expert set (denoted as FPSVNSES from now on) which utilizes the concept of single valued neutrosophic set which is superior to fuzzy set [1] and intuitionistic fuzzy set [7] and thus can better reflect the imprecision , uncertainties and vagueness of the data and the associated problem parameters compared to the other generalizations of fuzzy parameterized soft expert set such as fuzzy parameterized fuzzy soft expert sets [23] and fuzzy parameterized intuitionistic fuzzy soft expert set [24]. The FPSVNSES model is also significantly more advantageous compared to fuzzy parameterized soft set [20] and fuzzy parameterized fuzzy soft set [21] as it has the added advantage of allowing the users to know the opinion of all the experts in one model.

Moreover, even after performing any operations, the users can still know the opinion of all the experts. This results in a significantly better and improved generalization of single valued neutrosophic set which would in turn produce more accurate results, especially in problem solving contexts.

In this paper, we first present the basic definitions of single valued neutrosophic set, single valued neutrosophic soft expert set and fuzzy parameterized fuzzy soft expert set that are useful for subsequent discussions. We then propose a novel concept of fuzzy parameterized single valued neutrosophic soft expert set theory and define some operations along with illustrative examples. We also give some related properties with supporting proofs and make a comparison between our proposed method and other methods to illustrate the advantages of our proposed method. Finally we give a decision making method for fuzzy parameterized single valued neutrosophic soft expert set theory and present an application of this concept in solving a decision making problem.

2. Preliminaries In this section, we recall some basic notions of single valued neutrosophic set [13], single valued neutrosophic soft expert set [25] and fuzzy parameterized fuzzy soft expert set [23].

DEFINITION 2.1. (see [9]) A neutrosophic set A on the universe of discourse X is defined as A = {< x; TA (x); IA (x); FA (x) >; x ∈ X} where T ; I; F : X →]− 0; 1+ [ and − 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3+ .

DEFINITION 2.2. (see [13]) ( Single Valued Neutrosophic Set) Let X be a space of points (objects), with a generic element in X denoted by x . A single valued neutrosophic

set (SVNS) A in X is characterized by truth-membership function TA , indeterminacymembership function IA and falsity-membership function FA . For each point x in X, TA (X), IA (X), FA (X) ∈ [0, 1].

When X is continuous, a SVNS A can be written as Z

A=

hT (X), I(X), F(X)i/x, x ∈ X.

X

When X is discrete, a SVNS A can be written as n

A = ∑ hT (Xi ), I(Xi ), F(Xi )i/xi , xi ∈ X. i=1

DEFINITION 2.3. (see [13]) (Complement) The complement of a single valued neutrosophic set A is denoted by c(A) and is defined by

Tc(A) (X) = FA (X), Ic(A) (X) = 1 − IA (X), Fc(A) (X) = TA (X), for all x in X. DEFINITION 2.4. (see [13] ) ( Containment) A single valued neutrosophic set A is contained in the other single valued neutrosophic set B, A ⊆ B, if and only if

TA (X) ≤ TB (X), IA (X) ≤ IB (X), FA (X) ≥ FB (X), for all x in X. DEFINITION 2.5. (see [13]) Two single valued neutrosophic sets A and B ar equal, written as A = B, if and only if A ⊆ B and B ⊆ A. DEFINITION 2.6. (see [13]) ( Union ) The union of two single valued neutrosophic sets A and B is a single valued neutrosophic set C , written as C = A ∪ B , whose truthmembership, indeterminacy-membership and falsity-membership functions are related to those of A and B by

TC (X) = max((TA (X), TB (X)), IC (X) = max((IA (X), IB (X)), FC (X) = min((FA (X), FB (X)), for all x in X .

DEFINITION 2.7. (see [13]) ( Intersection ) The intersection of two single valued neutrosophic sets A and B is a single valued neutrosophic set C , written as C = A ∩ B , whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by

TC (X) = min((TA (X), TB (X)), IC (X) = min((IA (X), IB (X)), FC (X) = max((FA (X), FB (X)), for all x in X .

DEFINITION 2.9. (see [25]) Let U = {u1 , u2 , ..., un } be a universal set of elements, E = {e1 , e2 , ..., em } be a universal set of parameters, X = {x1 , x2 , ..., xi } be a set of experts (agents) and O = {1 = agree, 0 = disagree} be a set of opinions . Let Z = {E × X × O} and A ⊆ Z . Then the pair (U, Z) is called a soft universe . Let F : Z → SV NU , where

SV NU denotes the collection of all single valued neutrosophic subsets of U. Suppose F : Z → SV NU be a function defined as:

F(z) = F(z)(ui ), ∀ui ∈ U. Then F(z) is called a single valued neutrosophic soft expert value (SVNSEV in short) over the soft universe (U, Z).

2.10. (see [23]) Let U be a universe set, E be a set of parameters, I E denotes all fuzzy subsets of E, X be a set of experts (agents), and O = {1 = agree, 0 = disagree} a set of opinions. Let Z = Ψ × X × O and A ⊆ Z where Ψ ⊆ I E . Let F be a mapping given by DEFINITION

FΨ : A → IU where IU denotes the power set of . A pair (F, A)Ψ is called a fuzzy parameterized fuzzy soft expert set (FPFSES) over U.

3. Fuzzy Parameterized Single Valued Neutrosophic Soft Expert Set In this section we introduce the concept of fuzzy parameterized single valued neutrosophic soft expert set and define some operations on this concept , namely subset, equality, complement, union, intersection, AND and OR. We also give some properties of this concept.

Now, we propose the definition of a fuzzy parameterized single valued neutrosophic soft expert set and give an illustrative example of it.

Let U be a universe set, E be a set of parameters, I E denotes all fuzzy subsets of E , X be a set of experts, and O = {1 = agree, 0 = disagree} a set of opinions. Let Z = Ψ × X × O and A ⊆ Z where Ψ ⊂ I E .

D EFINITION 3.1. A pair ( f , A)Ψ is called a fuzzy parameterized single valued neutrosophic soft expert set (FPSV NSES) over U, where F is a mapping given by fΨ : A → SV N(U), and SV N(U) denotes the set of all single valued neutrosophic subsets of U. Example 3.2. Suppose that a hotel chain is looking for a construction company to upgrade the hotel to keep pace with globalization and wishes to take the opinion of some experts concerning this matter. Let U = {u1 , u2 , u3 } be a set of construction companies, E = {e1 , e2 , e3 } a set of decision parameters where ei (i = 1, 2, 3) denotes the decisions

“good service”, “quality” and “cheap” respectively, and Ψ = set of

IE

n

e1 e2 e3 0.3 , 0.5 , 0.8

o

a fuzzy sub-

and let X = {p, q} be a set of experts (committee members).

Suppose that the hotel chain has distributed a questionnaire to the two experts to make decisions on the construction companies, and we get the following information:

n

e1 u1 0.3 , p, 1 = h0.3,0.2,0.3i , o u3 h0.2,0.1,0.5i n

u1 e2 , p, 1 = h0.9,0.4,0.3i f 0.5 , o u3 h0.7,0.1,0.1i n

e3 u1 , , p, 1 = h0.4,0.8,0.6i f 0.8 o u3 h0.1,0.2,0.6i

n e1 u1 , f 0.3 , p, 0 = h0.3,0.8,0.3i o u3 h0.5,0.9,0.2i n

u1 e2 , p, 0 = h0.3,0.6,0.9i f 0.5 , o u3 h0.1,0.9,0.7i

n e3 u1 f 0.8 , p, 0 = h0.6,0.2,0.4i , o u3 h0.6,0.8,0.1i

f

u3 u2 h0.4,0.5,0.1i , h0.7,0.1,0.1i

o n e1 u2 u1 , f 0.3 , q, 1 = h0.6,0.5,0.4i , h0.7,0.1,0.1i ,

u3 u2 h0.4,0.5,0.6i , h0.1,0.6,0.9i

n o u1 e2 u2 , q, 1 = h0.6,0.2,0.4i , f 0.5 , h0.8,0.5,0.1i ,

u3 u2 h0.2,0.1,0.7i , h0.5,0.7,0.8i

o n e3 u2 u1 , f 0.8 , h0.2,0.8,0.9i , , q, 1 = h0.7,0.8,0.1i

u3 u2 h0.1,0.5,0.4i , h0.1,0.9,0.7i

o n e1 u2 u1 , f 0.3 , h0.1,0.9,0.7i , , q, 0 = h0.4,0.5,0.6i

u3 u2 h0.6,0.5,0.4i , h0.9,0.4,0.1i

o n e2 u1 u2 , f 0.5 , q, 0 = h0.4,0.8,0.6i , h0.1,0.5,0.8i ,

u3 u2 h0.7,0.9,0.2i , h0.8,0.3,0.5i

o n e3 u1 u2 , f 0.8 , q, 0 = h0.1,0.2,0.7i , h0.9,0.2,0.2i ,

Then we can view the FPSVNSES ( f , A)ψ as consisting of the following collection of approximations: ( f , A)Ψ = nn

n oo n n u3 u1 u2 e1 u1 , h0.3,0.2,0.3i , h0.4,0.5,0.1i , h0.7,0.1,0.1i , 0.3 , q, 1 , h0.6,0.5,0.4i ,

e1 0.3 , p, 1

u3 u2 h0.7,0.1,0.1i , h0.2,0.1,0.5i

n

n oo n n u3 e3 u1 u2 u1 , h0.6,0.2,0.4i , h0.8,0.5,0.1i , h0.7,0.1,0.1i , , p, 1 , h0.4,0.8,0.6i , 0.8

e2 0.5 , q, 1

u3 u2 h0.2,0.1,0.7i , h0.5,0.7,0.8i

n

oo n n oo e3 u3 u1 u2 , , 0.8 , q, 1 , h0.7,0.8,0.1i , h0.2,0.8,0.9i , h0.1,0.2,0.6i

n oo n n u3 u1 u2 e1 u1 , h0.3,0.8,0.3i , h0.1,0.5,0.4i , h0.1,0.9,0.7i , 0.3 , q, 0 , h0.4,0.5,0.6i ,

e1 0.3 , p, 0

u3 u2 h0.1,0.9,0.7i , h0.5,0.9,0.2i

n

oo n n oo u3 e2 u1 u2 , , p, 1 , , , , 0.5 h0.9,0.4,0.3i h0.4,0.5,0.6i h0.1,0.6,0.9i

oo n n oo u3 e2 u1 u2 , , 0.5 , p, 0 , h0.3,0.6,0.9i , h0.6,0.5,0.4i , h0.9,0.4,0.1i

n oo n n u3 e3 u1 u2 u1 , h0.4,0.8,0.6i , h0.1,0.5,0.8i , h0.1,0.9,0.7i , , p, 0 , h0.6,0.2,0.4i , 0.8

e2 0.5 , q, 0

u3 u2 h0.7,0.9,0.2i , h0.8,0.3,0.5i

oo n n ooo e3 u3 u1 u2 , , q, 0 , , , .

0.8 h0.1,0.2,0.7i h0.9,0.2,0.2i h0.6,0.8,0.1i

In the following, we introduce the concept of the subset of two FPSVNSESs and the equality of two FPSVNSESs with an illustrative example on the subset operation.

D EFINITION 3.3. For two FPSVNSESs ( f , A)Ψ and (G, B)ϒ over U , ( f , A)Ψ is called a FPSVNSE subset of (G, B)ϒ if 1. A ⊆ B, 2. ∀ε ∈ A, fΨ (ε) is single valued neutrosophic subset of Gϒ (ε).

D EFINITION 3.4. Two FPSVNSESs ( f , A)Ψ and (G, B)ϒ over U, are said to be equal if ( f , A)Ψ is a FPSVNSE subset of (G, B)ϒ and (G, B)ϒ is a FPSVNSE subset of ( f , A)Ψ .

Example 3.5. Consider Example 3.2 and suppose that the hotel chain takes the opinion of the o experts once again after the n hotel chain o has been opened. Let Ψ = n e1 e2 e3 e1 e2 e3 0.4 , 0.1 , 0.2 be a fuzzy subset of E , ϒ = 0.5 , 0.5 , 0.7 be another fuzzy subset over E. Suppose, AΨ =

n

Bϒ =

n

o e3 e2 , 0.1 , q, 0 , 0.2 , q, 1 ,

e1 0.4 , p, 1

e1 0.5 , p, 1

o e3 e2 e1 , 0.5 , q, 0 , 0.7 , q, 1 , 0.5 , q, 0 .

Since Ψ is a fuzzy subset of ϒ , clearly AΨ ⊂ Bϒ . Let ( f , A)Ψ and (G, B)ϒ be defined as follows: nn n oo n n u3 e1 u1 e2 u1 u2 ( f , A)Ψ = , p, 1 , , , , , q, 0 , h0.6,0.5,0.3i , 0.4 0.1 h0.1,0.2,0.3i h0.4,0.1,0.5i h0.7,0.2,0.7i u3 u2 h0.3,0.1,0.4i , h0.2,0.1,0.5i

oo n n ooo e3 u3 u1 u2 , q, 1 , , , , , 0.2 h0.1,0.5,0.9i h0.2,0.1,0.6i h0.4,0.2,0.7i

and (G, B)ϒ =

nn

u3 u2 h0.7,0.2,0.3i , h0.2,0.1,0.5i

n

n oo n n u3 u1 e2 u1 u2 , h0.3,0.5,0.2i , h0.6,0.5,0.1i , h0.9,0.4,0.1i , , q, 0 , h0.7,0.8,0.2i , 0.5

e1 0.5 , p, 1

oo n n oo n

e3 u3 u1 u2 e1 , , q, 1 , , , , q, 0 , 0.7 0.5 h0.9,0.5,0.3i h0.4,0.5,0.1i h0.5,0.6,0.4i

u3 u1 u2 h0.1,0.5,0.7i , h0.2,0.8,0.6i , h0.8,0.1,0.3i

oo .

Therefore ( f , A)Ψ ⊆ (G, B)ϒ .

In the following, we propose the definition of the complement of a FPSVNSES along with an illustrative example and give a proposition on the complement of a FPSVNSES. D EFINITION 3.6. The complement of a FPSVNSES ( f , A)Ψ is denoted by ( f , A)cΨ and is defined by ( f , A)cΨ = ( f c , ¬A)Ψ where fΨc : ¬A → SV N(U) is a mapping given by fΨc (ε) = c( fΨ (¬ε)), ∀ε ∈ ¬A where c is a single valued neutrosophic complement and ¬A ⊂ {Ψc × X × O}.

Example 3.7. Consider Example 3.2. by using the basic fuzzy complement and single valued neutrosophic complement, we have ( f , A)cΨ = nn n oo n n u3 e1 u1 u2 e1 u1 , 0.7 , q, 1 , h0.4,0.5,0.6i , 0.7 , p, 1 , h0.3,0.8,0.3i , h0.1,0.5,0.4i , h0.1,0.9,0.7i u3 u2 h0.1,0.9,0.7i , h0.5,0.9,0.2i

n

n oo n n e3 u3 u1 u1 u2 , h0.4,0.8,0.6i , , p, 1 , h0.6,0.2,0.4i , h0.1,0.5,0.8i , h0.1,0.9,0.7i , 0.2

e2 0.5 , q, 1

u3 u2 h0.7,0.9,0.2i , h0.8,0.3,0.5i

n

oo n oo n u3 e2 u1 u2 , , 0.5 , p, 1 , h0.3,0.6,0.9i , h0.6,0.5,0.4i , h0.9,0.4,0.1i

oo n n oo e3 u3 u1 u2 , , q, 1 , , , , 0.2 h0.1,0.2,0.7i h0.9,0.2,0.2i h0.6,0.8,0.1i

n oo n n u3 u1 u2 e1 u1 , h0.3,0.2,0.3i , h0.4,0.5,0.1i , h0.7,0.1,0.1i , , q, 0 , h0.6,0.5,0.4i , 0.7 oo n oo n u3 u3 e2 u2 u2 u1 , , 0.5 , p, 0 , h0.9,0.4,0.3i , h0.4,0.5,0.6i , h0.1,0.6,0.9i h0.7,0.1,0.1i , h0.2,0.1,0.5i n n oo n n u3 e3 e2 u1 u2 u1 , 0.5 , q, 0 , h0.6,0.2,0.4i , h0.8,0.5,0.1i , h0.7,0.1,0.1i 0.2 , p, 0 , h0.4,0.8,0.6i , e1 0.7 , p, 0

u3 u2 h0.2,0.1,0.7i , h0.5,0.7,0.8i

oo n ooo n e3 u3 u2 u1 , q, 0 , , , , .

0.2 h0.7,0.8,0.1i h0.2,0.8,0.9i h0.1,0.2,0.6i

Proposition 3.8 If ( f , A)Ψ is a FPSVNSES over U , then ( f , A)cΨ

c

= ( f , A)Ψ .

Proof. By using Definition 3.6 we have fΨc : ¬A → SV N(U) is a mapping given by fΨc (ε) = c( fΨ (¬ε)), ∀ε ∈ ¬A and ¬A ⊂ {Ψc × X × O}.

Now, ( fΨc )c : ¬(¬A) → SV N(U) is a mapping given by ( fΨc )c (ε) = c ( fΨc )c (¬(¬ε)) , ∀ε ∈ ¬(¬A) and ¬(¬A) ⊂ {(Ψc )c × X × O}, and since (Ψc )c = Ψ and ( f c )c = f , the proposition is proved.

Now, we put forward the definition of an agree-FPSVNSES and the definition of a disagree- FPSVNSES. D EFINITION 3.9. An agree- FPSVNSES ( f , A)Ψ1 over U is a FPSVNSE subset of ( f , A)Ψ where the opinions of all experts are agree and is defined as follows:

( f , A)Ψ1 = FΨ (ε) : ε ∈ Ψ × X × {1} .

D EFINITION 3.10. A disagree- FPSVNSES ( f , A)Ψ0 over U is a FPSVNSE subset of ( f , A)Ψ where the opinions of all experts are disagree and is defined as follows:

( f , A)Ψ0 = FΨ (ε) : ε ∈ Ψ × X × {0} .

We introduce the definition of union operation along with an illustrative example and give a proposition on union operation.

D EFINITION 3.11. The union of two FPSVNSESs ( f , A)Ψ and (G, B)ϒ over U, e (G, B)ϒ is the FPSVNSES (H,C)Φ such that C = Φ × X × O, denoted by ( f , A)Ψ ∪ where Φ = Ψ ∪ ϒ and ∀ε ∈ C,

e Gϒ (ε), HΦ (ε) = fΨ (ε) ∪ e is the single valued neutrosophic union. where ∪ n o e1 e2 e3 Example 3.12. Consider Example 3.2. Let Ψ = 0.9 , 0.4 , 0.1 be a fuzzy subset of n o e1 e2 e3 E, and ϒ = 0.3 , 0.7 , 0.8 be another fuzzy subset over E. AΨ =

n

Bϒ =

n

o e3 e2 , 0.4 , q, 0 , 0.1 , q, 1 ,

e1 0.9 , p, 1

o e3 e1 , 0.8 , q, 1 , 0.3 , q, 0 .

e2 0.7 , q, 0

Suppose ( f , A)Ψ and (G, B)ϒ are two FPSVNSESs over the same U given by nn n oo n n u3 e1 u1 e2 u1 u2 ( f , A)Ψ = , p, 1 , , , q, 0 , h0.6,0.5,0.4i , , , 0.9 0.4 h0.3,0.7,0.9i h0.4,0.5,0.6i h0.7,0.2,0.7i ooo n ooo n e3 u3 u2 u1 , , q, 1 , , , , 0.1 h0.8,0.1,0.3i h0.6,0.8,0.1i h0.2,0.5,0.6i

u3 u2 h0.3,0.1,0.8i , h0.4,0.5,0.6i

and (G, B)ϒ =

n

n oo n n u3 e3 u1 u1 u2 , h0.2,0.1,0.9i , q, 1 , h0.9,0.5,0.3i , h0.2,0.8,0.5i , h0.6,0.8,0.5i , , 0.8

e2 0.7 , q, 0

u3 u2 h0.8,0.5,0.6i , h0.4,0.8,0.6i

oo n n oo u3 e1 u1 u2 , , q, 0 , , , .

0.3 h0.8,0.6,0.7i h0.7,0.2,0.3i h0.4,0.8,0.9i

By using the single valued neutrosophic union and the basic fuzzy union (maximum) we have e (G, B)ϒ = ( f , A)Ψ ∪ nn n oo n n u3 e1 u1 u2 e2 u1 , p, 1 , , , , , q, 0 , h0.6,0.5,0.4i , 0.9 0.7 h0.3,0.7,0.9i h0.4,0.5,0.6i h0.7,0.2,0.7i u3 u2 h0.3,0.8,0.5i , h0.6,0.8,0.5i

ooo n n oo e3 u3 u1 u2 , , 0.8 , q, 1 , h0.9,0.5,0.3i , h0.8,0.8,0.1i , h0.4,0.8,0.6i

n

n ooo u3 u1 u2 , h0.8,0.6,0.7i , h0.7,0.2,0.3i , h0.4,0.8,0.9i .

e1 0.3 , q, 0

Proposition 3.13 Let ( f , A)Ψ , (G, B)ϒ and (H,C)Φ be any three FPSVNSESs over a universe U, then the following properties hold true: e (G, B)ϒ = (G, B)ϒ ∪ e ( f , A)Ψ 1. ( f , A)Ψ ∪

e (G, B)ϒ ∪ e (H,C)Φ = ( f , A)Ψ ∪ e (G, B)ϒ ∪ e (H,C)Φ .

2. ( f , A)Ψ ∪ e (G, B)ϒ = (H,C)Φ , then by Definition 3.11 for all ε ∈ C, Proof (1). Let ( f , A)Ψ ∪ e Gϒ (ε). However, since the union of fuzzy we have Φ = Ψ ∪ ϒ and HΦ (ε) = fΨ (ε)∪ sets and single valued neutrosophic sets is commutative then Φ = Ψ ∪ ϒ = ϒ ∪ Ψ and e Gϒ (ε) = Gϒ (ε)∪ e fΨ (ε). Therefore (H,C)Φ = (G, B)ϒ ∪ e ( f , A)Ψ . Thus HΦ (ε) = fΨ (ε)∪ the union of two FPSVNSESs is commutative.

(2). The proof is similar to that in part (1) and therefore is omitted .

We introduce the definition of intersection operation along with an illustrative example and give a proposition on intersection operation.

D EFINITION 3.14. The intersection of two FPSVNSESs ( f , A)Ψ and (G, B)ϒ over e (G, B)ϒ is the FPSVNSES (H,C)Φ such that C = Φ × X × O, U, denoted by ( f , A)Ψ ∩ where Φ = Ψ ∩ ϒ and ∀ε ∈ C, e Gϒ (ε), HΦ (ε) = fΨ (ε) ∩ e is the single valued neutrosophic intersection. where ∩ Example 3.15. Consider Example 3.12, then by using the basic single valued neutrosophic intersection and the basic fuzzy intersection (minimum) we have e (G, B)ϒ = ( f , A)Ψ ∩ nn n oo n n u3 e1 u1 u2 e2 u1 , 0.9 , p, 1 , h0.3,0.7,0.9i , h0.4,0.5,0.6i , h0.7,0.2,0.7i 0.4 , q, 0 , h0.2,0.1,0.9i , u3 u2 h0.2,0.1,0.8i , h0.4,0.5,0.6i

n

ooo n n oo e3 u3 u1 u2 , , 0.1 , q, 1 , h0.8,0.1,0.3i , h0.6,0.5,0.6i , h0.2,0.5,0.6i

n ooo u3 u1 u2 , h0.8,0.6,0.7i , h0.7,0.2,0.3i , h0.4,0.8,0.9i .

e1 0.3 , q, 0

Proposition 3.16 Let ( f , A)Ψ , (G, B)ϒ and (H,C)Φ be any three FPSVNSESs over a universe U, then the following properties hold true: e (G, B)ϒ = (G, B)ϒ ∩ e ( f , A)Ψ , 1. ( f , A)Ψ ∩

e (G, B)ϒ ∩ e (H,C)Φ = ( f , A)Ψ ∩ e (G, B)ϒ ∩ e (H,C)Φ .

2. ( f , A)Ψ ∩ Proof (1). The proof is similar to that in Proposition 3.13 (1) and therefore is omitted.

(2). The proof is similar to that in part (1) and therefore is omitted.

Proposition 3.17 Let ( f , A)Ψ and (G, B)ϒ be any two FPSVNSESs over a universe U . Then the following De Morgan’s laws hold true:

e (G, B)ϒ c = ( f , A)Ψ c ∩ e (G, B)ϒ c , 1. ( f , A)Ψ ∪

e (G, B)ϒ c = ( f , A)Ψ c ∪ e (G, B)ϒ c .

2. ( f , A)Ψ ∩ Proof (1). Let ( f , A)Ψ and (G, B)ϒ be two FPSVNSESs over a universe U , then ∀ε ∈ ¬A where ¬A ⊂ {Dc × X × O} , it follows that:

( f , A)Ψ

c

e (G, B)ϒ ∩

c

e Gcϒ (ε) = fΨc (ε) ∩

e c Gϒ (¬ε) = c fΨ (¬ε) ∩

e Gϒ (¬ε) = c fΨ (¬ε) ∩

e Gϒ (ε) = c fΨ (ε) ∪

e (G, B)ϒ c . = ( f , A)Ψ ∪ (2). The proof is similar to that in part (1) and therefore is omitted.

Now, we introduce the definitions of AND and OR operations for FPSVNSES and derive their properties.

D EFINITION 3.18. If ( f , A)Ψ and (G, B)ϒ are two FPSVNSESs over U, then “( f , A)Ψ AND (G, B)ϒ ”, denoted by ( f , A)Ψ ∧ (G, B)ϒ , is defined by ( f , A)Ψ ∧ (G, B)ϒ = (H, A × B)Φ , e Gϒ (β ), ∀ (α, β ) ∈ A × B, where Ψ = ϒ × Φ, such that H(α, β )Φ = fΨ (α) ∩ e is the single valued neutrosophic intersection. and ∩ D EFINITION 3.19. If ( f , A)Ψ and (G, B)ϒ are two FPSVNSESs over U, then “( f , A)Ψ OR (G, B)ϒ ”, denoted by ( f , A)Ψ ∨ (G, B)ϒ , is defined by ( f , A)Ψ ∨ (G, B)ϒ = (H, A × B)Φ . e Gϒ (β ), ∀ (α, β ) ∈ A × B, where Ψ = ϒ × Φ, Such that H(α, β )Φ = fΨ (α) ∪ e is the single valued neutrosophic union . and ∪ Proposition 3.20 Let ( f , A)Ψ , (G, B)ϒ and (H,C)Φ be any three FPSVNSESs over a universe U, then the following properties hold true:

1. ( f , A)Ψ ∧ (G, B)ϒ ∧ (H,C)Φ = ( f , A)Ψ ∧ (G, B)ϒ ∧ (H,C)Φ ,

2. ( f , A)Ψ ∨ (G, B)ϒ ∨ (H,C)Φ = ( f , A)Ψ ∨ (G, B)ϒ ∨ (H,C)Φ .

Proof The proofs are straightforward by Propositions 3.13 and 3.16 and therefore are omitted.

4. An Application of Fuzzy Parameterized Single Valued Neutrosophic Soft Expert Set In this section, we present an application of FPSVNSES in a decision making problem.

We consider the following problem.

Example 4.1. Assume that a hotel chain wants to fill a position for the management of the chain. There are three candidates who form the universe . The hiring committee decided to have a set of parameters, where the parameters stand for “computer knowledge”, “experience”, and “language proficiency” respectively, which are important with degree n o e1 e2 e3 0.6, 0.8 and 0.4 respectively. That is, the subset of parameters is Ψ = 0.6 , 0.8 , 0.4 . Let X = {p, q} be a set of experts (committee members). Now we can find a suitable choice for the hotel chain to fill the position. After a serious discussion, the committee evaluates the alternatives from point of view of the goals and the constraints according to a chosen subset Ψ of I E to construct a FPSVNSES . ( f , A)Ψ = n n nn oo n u3 e1 u1 u1 u2 e1 , p, 1 , , q, 1 , h0.9,0.3,0.3i , , , , 0.6 0.6 h0.4,0.2,0.1i h0.8,0.2,0.4i h0.7,0.2,0.3i u3 u2 h0.8,0.2,0.4i , h0.2,0.1,0.5i

n

n oo n n u3 e3 u1 u1 u2 , h0.4,0.2,0.1i , h0.5,0.4,0.3i , h0.7,0.1,0.1i , 0.4 , p, 1 , h0.7,0.1,0.5i ,

e2 0.8 , q, 1

u3 u2 h0.5,0.1,0.1i , h0.5,0.7,0.8i

n

oo n n oo e3 u3 u1 u2 , , 0.4 , q, 1 , h0.6,0.4,0.2i , h0.7,0.4,0.2i , h0.4,0.2,0.6i

n oo n n u3 u1 u2 e1 u1 , h0.1,0.8,0.4i , h0.4,0.8,0.8i , h0.3,0.8,0.7i , 0.6 , q, 0 , h0.3,0.7,0.9i ,

e1 0.6 , p, 0

u3 u2 h0.4,0.8,0.8i , h0.5,0.9,0.8i

n

oo n n oo u3 e2 u1 u2 , , p, 1 , , , , 0.8 h0.5,0.1,0.8i h0.8,0.2,0.3i h0.7,0.1,0.5i

oo n n oo u3 e2 u1 u2 , , p, 0 , , , , 0.8 h0.8,0.9,0.5i h0.3,0.8,0.8i h0.5,0.9,0.7i

n oo n n u3 e3 u1 u2 u1 , h0.1,0.8,0.4i , h0.3,0.6,0.5i , h0.1,0.9,0.7i , , p, 0 , h0.5,0.9,0.7i , 0.4

e2 0.8 , q, 0

u3 u2 h0.1,0.9,0.5i , h0.8,0.3,0.5i

oo n n ooo e3 u3 u1 u2 , , q, 0 , , , .

0.4 h0.2,0.6,0.6i h0.2,0.6,0.7i h0.6,0.8,0.4i

Next the FPSVNSES ( f , A)Ψ is used together with a generalized algorithm to solve the decision making problem stated at the beginning of this section. The algorithm given below is employed by the hiring committee to determine the most suitable candidate to be hired for the position. This algorithm is a generalization of the algorithm introduced

by Alkhazaleh and Salleh [15]. The generalized algorithm is as follows: Algorithm : 1. Input the FPSVNSES ( f , A)Ψ 2. Find the values of ci j = T fΨ (εi ) (u j ) − I fΨ (εi ) (u j ) − FfΨ (εi ) (u j ), ∀u j ∈ U and ∀εi ∈ A, where T fΨ (εi ) (u j ), I fΨ (εi ) (u j ) and FfΨ (εi ) (u j ) are the truth- membership function , indeterminacy - membership function and falsity - membership function ∀u j ∈ U and ∀εi ∈ A, respectively .

3. Compute the score of each element u j ∈ U by the following formulas : 6

6

Kj =

∑ ∑ ci j (µΨ (ei )), S j = ∑ ∑ ci j (µΨ (ei )) x∈X i=1

x∈X i=1

for the agree-FPSVNSES and disagree-FPSVNSES, where µΨ (ei ) is the corresponding membership function of the fuzzy set Ψ.

4. Find the values of the score r j = K j − S j for each element u j ∈ U. 5. Determine the value of the highest score m = maxu j ∈U {r j }. Then the decision is to choose element u j as the optimal solution to the problem. If there are more than one element with the highest r j score then any one of those elements can be chosen as the optimal solution.

Then, we can conclude that the optimal choice for the hiring committee is to hire candidate u j to fill the vacant position. Table 1 and Table 2 give the values of ci j = T fΨ (εi ) (u j ) − I fΨ (εi ) (u j ) − FfΨ (εi ) (u j ) and the score of each element u j ∈ U for agree-FPSVNSES and disagree-FPSVNSES respectively.

Table 1. Numerical grade for agree-FPSVNSES

U

e1 ,p 0.6 e1 ,q 0.6 e2 , p 0.8 e2 ,q 0.8 e3 ,p 0.4 e3 0.4 , q

u1

u2

u3

0.1

0.2

0.2

0.3

0.2

-0.4

-0.4

0.5

-0.4

0.1

-0.2

0.5

0.1

0.3

-1

0

0.1

-0.4

K1 = 0.04

K2 = 0.64

K3 = −0.6

6

Kj =

∑ ∑ ci j (µΨ (ei ))

x∈X i=1

Let K j and S j , represent the score of each numerical grade for the agree-FPSVNSES and disagree-FPSVNSES , respectively. These values are given in Table 3.

Table 2. Numerical grade for disagree-FPSVNSES

U

e1 ,p 0.6 e1 ,q 0.6 e2 , p 0.8 e2 ,q 0.8 e3 , p 0.4 e3 0.4 , q

u1

u2

u3

-1.1

-1.2

-1.2

-1.3

-1.2

-1.2

-0.6

-1.3

-1.1

-1.1

-0.8

-1.5

-1.1

-1.3

0

-1

-1.1

-0.6

S1 = −3.64

S2 = −4.08

S3 = −3.76

6

Sj =

∑ ∑ ci j (µΨ (ei ))

x∈X i=1

Table 3. The score r j = K j − S j 6

Kj =

6

∑ ∑ ui j (µΨ (ei ))

x∈X i=1

K1 = 0.04 K2 = 0.64 K3 = −0.6

Sj =

∑ ∑ ui j (µΨ (ei ))

rj = Kj − Sj

x∈X i=1

S1 = -3.64 S2 = -4.08 S3 = -3.76

3.68 4.72 3.16

Thus m = maxu j ∈U {r j } = r2 . Therefore, the hiring committee is advised to hire candidate u2 to fill the vacant position.

To illustrate the advantages of our proposed method using FPSVNSES as compared to fuzzy parameterized intuitionistic fuzzy soft expert set as proposed by Selvachandran and Salleh [24] which is a generalization of fuzzy parameterized fuzzy soft expert set, let us consider Example 4.1 above. The fuzzy parameterized intuitionistic fuzzy soft expert set can describe this problem as follows:

( f , A)Ψ =

nn e n oo u1 u2 u3 1 , p, 1 , , , , ...

0.6 h0.4, 0.2i h0.8, 0.2i h0.7, 0.2i

Note that the FPSVNSES is a generalization of fuzzy parameterized intuitionistic fuzzy soft expert set [24] . Thus as shown in Example 4.1 above, the FPSVNSES can explain the universal U in more detail with three membership functions, especially when there are many parameters involved, whereas fuzzy parameterized intuitionistic fuzzy soft expert set [24] can tell us a limited information about the universal U. It can only handle the incomplete information considering both the truth-membership and falsitymembership values, while FPSVNSES can handle problems involving imprecise, indeterminacy and inconsistent data, which makes it more accurate and realistic than fuzzy parameterized intuitionistic fuzzy soft expert set [24] .

5. Conclusion We established the concept of fuzzy parameterized single valued neutrosophic soft expert set by applying the theory of single valued neutrosophic set to fuzzy parameterized

soft expert set [22]. The basic operations on fuzzy parameterized single valued neutrosophic soft expert set, namely complement, subset, union, intersection, AND, and OR operations, were defined. Subsequently, the basic properties of these operations such as De Morgan’s laws and other relevant laws pertaining to the concept of fuzzy parameterized single valued neutrosophic soft expert set were proven . A comparison between our proposed method and other methods was made to illustrate the advantages of our proposed method and its ability to handle problems involving imprecise, indeterminacy and inconsistent data. Finally, a generalized algorithm is introduced and applied to the fuzzy parameterized single valued neutrosophic soft expert set model to solve a hypothetical decision making problem. This new extension will provide a significant addition to existing theories for handling indeterminacy, and spurs more developments of further research and pertinent applications.

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