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Multi-attribute decision making based on rough neutrosophic variational coefficient similarity measure.

1 Introduction

In 1965, L. A. Zadeh grounded the concept of degree of membership and defined fuzzy set [1] to represent/manipulate data with non-statistical uncertainty. In 1986, K. T. Atanassov [2] introduced the degree of non-membership as independent component and proposed intuitionistic fuzzy set (IFS). F. Smarandache introduced the degree of indeterminacy as independent component and defined the neutrosophic set [3, 4, 5]. For purpose of solving practical problems, Wang et al. [6] restricted the concept of neutrosophic set to single valued neutrosophic set (SVNS), since single value is an instance of set value. SVNS is a subclass of the neutrosophic set. SVNS consists of the three independent components namely, truth-membership, indeterminacy-membership and falsit-ymembership functions.

The concept of rough set theory proposed by Z. Pawlak [7] is an extension of the crisp set theory for the study of intelligent systems characterized by inexact, uncertain or insufficient information. The hybridization of rough set theory and neutrosophic set theory produces the rough neutrosophic set theory [8, 9], which was proposed by Broumi, Dhar and Smarandache [8, 9]. Rough neutrosophic set theory is also a powerful mathematical tool to deal with incompleteness.

Literature review reflects that similarity measures play an important role in the analysis and research of clustering analysis, decision making, medical diagnosis, pattern recognition, etc. Various similarity measures [10, 11, 12, 13, 14, 15, 16, 17, 18] of SVNSs and hybrid SVNSs are available in the literature. The concept of similarity measures in rough neutrosophic environment [19, 20, 21] has been recently proposed.

Pramanik and Mondal [19] proposed cotangent similarity measure of rough neutrosophic sets. In the same study [19],Pramanik and Mondal established its basic properties and provided its application to medical diagnosis. Pramanik and Mondal [20] also proposed cosine similarity measure of rough neutrosophic sets and its application in medical diagnosis. The same authors [21] also studied Jaccard similarity measure and Dice similarity measures in rough neutrosophic environment and provided their applications to multi attribute decision making. Mondal and Pramanik [22] presented tri-complex rough neutrosophic similarity measure and its application in multi-attribute decision making. Together with F. Smarandache and S. Pramnik, K. Mondal [23] presented hypercomplex rough neutrosophic similarity measure and its application in multi-attribute decision making. Mondal, Pramanik, and Smarandache [24] presented several trigonometric Hamming similarity measures of rough neutrosophic sets and their applications in multi attribute decision making problems.

Different methods for multiattribute decision making (MADM) and multicriteria decision making (MCDM) problems are available in the literature in different environment such as crisp environment [25, 26, 27, 28, 29], fuzzy environment [30, 31], intuitionistic fuzzy environment [32, 33, 34, 35, 36, 37, 38, 39, 40], neutrosophic environment [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62], interval neutrosophic environment [63, 65, 66, 67, 68], neutrosophic soft expert environment [69], neutrosophic bipolar environment [70, 71], neutrosophic soft environment [72, 73, 74, 75, 76], neutrosophic hesitant fuzzy environment [77, 78, 79]. rough neutrolsophic environment [80, 81], etc. In neutrosophic environment Biswas, Pramanik and Giri [82] studied hybrid vector similarity measure and its application in multi-attribute decision making. Getting motivation from the work of Biswas, Pramanik and Giri [82], for hybrid vector similarity measure in neutrosophic envionment, we extend the concept in rough neutrosophic environment.

In this paper, a new similarity measures namely, rough variational coefficient similarity measure under rough neutrosophic environment is proposed. A numerical example is also provided.

Rest of the paper is structured as follows. Section 2 presents neutrosophic and rough neutrosophic preliminaries. Section 3 discusses various similarity measures and varional coefficient similarity measure in crisp environment. Section 4 presents various similarity measures and variational similarity measure for single valued neutrosophic sets. Section 5 presents variational coefficient similarity measure and weighted variational coefficient similarity measure for rough neutrosophic sets and establishes their basic properties. Section 6 is devoted to present multi attribute decision making based on rough neutrosophic variational coefficient similarity measure. Section 7 demonstrates the application of rough variational coefficient similarity measures to investment problem Finally, section 8 concludes the paper with stating the future scope of research.

2 Neutrosophic preliminaries

Definition 2.1 [3, 4, 5] Neutrosophic set

Let X be a space of points (objects) with generic element in X denoted by x. Then a neutrosophic set A in X is denoted by A = {x([T.sub.A] (x), [I.sub.A] (x), [F.sub.A] (x)): x [member of] X} where, [T.sub.A] (x) is the truth membership function, [I.sub.A] (x) is the indeterminacy membership function and [F.sub.A] (x) is the falsity membership function. The functions [T.sub.A] (x), [I.sub.A] (x) and [F.sub.A] (x) are real standard or nonstandard subsets of ][sup.-] 0,[1.sup.+][. There is no restriction on the sum of [T.sub.A] (x), [I.sub.A] (x) and [F.sub.A] (x) i.e. [sup.-]0 <[T.sub.A] (x) + [I.sub.A] (x) + [F.sub.A] (x) [less than or equal to] [3.sup.+].

Definition 2.2 [6] (Single-valued neutrosophic set).

Let X be a universal space of points (objects), with a generic element x [member of] X. A single-valued neutrosophic set (SVNS) N [subset] X is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], when X is continuous; N = [[summation].sup.m.sub.i=1](tn(x),[I.sub.N](x),[F.sub.N](x))/x, [for all] x [member of] X, when X is discrete.

SVNS is characterized by a true membership function [T.sub.N](x), a falsity membership function [F.sub.N](x) and an indeterminacy function [I.sub.N](x) ith [T.sub.N](x), [F.sub.N](x), [I.sub.N](x) [member of] [0, 1] for all x [member of] X. For each x [member of] X, of a SVNS N 0 [less than or equal to] [T.sub.N](x) + [I.sub.N](x) + [F.sub.N](x) [less than or equal to] 3.

2.1 Some operational rules and properties of SVNSs

Let [N.sub.A]={[[T.sub.A], [I.sub.A], [F.sub.A]) and [N.sub.B]={[T.sub.B], [I.sub.B], [F.sub.B]) be two SVNSs in X. Then the following operations are defined as follows:

I. Complement: n[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

II. Addition: [N.sub.A] [cross product] [N.sub.B] = ([T.sub.A] + [T.sub.B] - [T.sub.A][T.sup.B], [I.sub.A][I.sup.B], [F.sub.A][F.sub.B])

III. Multiplication:

[N.sub.A] [cross product] [N.sub.B] =([T.sub.A][T.sub.B], [I.sub.A] + [I.sub.B] - [I.sub.A][I.sub.B], [F.sub.A] + [F.sub.B] - [F.sub.A][F.sub.B])

IV. Scalar Multiplication:

[lambda][N.sub.A]=<1-[(1-[T.sub.A]).sup.[lambda]],[I.sup.[lambda].sub.A],[F.sup.[lambda].sub.A]>. for [lambda] > 0.

V. [<[N.sub.A]>.sup.[lambda]]=<([[T.sub.A]).sup.[lambda]], 1-[(1-[I.sub.A]).sup.[lambda]], 1-[(1-[F.sub.A]).sup.[lambda]] for [lambda] > 0.

Definition 2.3 [6]

Complement of a SVNS N is denoted by N and is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 2.4 [6]

A SVNS [N.sub.A] is contained in the other SVNS NB, denoted as [N.sub.A] [subset or equal to] [N.sub.B], if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 2.5 [6]

Two SVNSs [N.sub.A] and [N.sub.B] are equal, i.e. [N.sub.A] = [N.sub.B], if and only if [N.sub.A] [contains or equal to] [N.sub.B] and [N.sub.A] [subset or equal to] [N.sub.B]

Definition 2.6 [6]

Union of two SVNSs [N.sub.A] and [N.sub.B] is a SVNS [N.sub.C], written as [N.sub.C] = [N.sub.A] [union] [N.sub.B]. Its truth membership, indeterminacy-membership and falsity membership functions are related to those of [N.sub.A] and [N.sub.B] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.7 [6] Intersection of two SVNSs [N.sub.A] and [N.sub.B] is a SVNS [N.sub.D], written as [N.sub.D] = [N.sub.A][ [intersection] [N.sub.b], whose truth membership, indeterminacy-membership and falsity membership functions are related to those of [N.sub.A] and [N.sub.B] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.8 Rough Neutrosophic Sets [8, 9]

Let Z be a non-null set and R be an equivalence relation on Z. Let P be neutrosophic set in Z with the membership function [T.sub.P] indeterminacy function [I.sub.P] and non-membership function [F.sub.P]. The lower and the upper approximations of P in the approximation (Z, R) denoted by [N.bar](P) and [bar.N](P) are respectively defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [disjunction] and [conjunction] denote "max" and "min" operators respectively. [T.sub.P](z), [I.sub.P](z) and [F.sub.P](z) denote respectively the membership, indeterminacy and non-membership function of z with respect to P. It is easy to see that [N.bar](P) and [bar.N](P) are two neutrosophic sets in Z.

Thus NS mappings [N.bar.], [bar.N]: N(Z) [left arrow] N(Z) are, respectively, referred to as the lower and the upper rough NS approximation operators, and the pair ([N.bar](P), [bar.N](P)) is called the rough neutrosophic set [8, 9] in (Z, R).

From the above definition, it is seen that [N.bar](P) and [bar.N](P) have constant membership on the equivalence classes of R. if [N.bar](P) = [bar.N](P) i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

P is said to be a definable neutrosophic set in the approximation (Z, R). It can be easily proved that zero neutrosophic set ([0.sub.N] = (0, 1, 1)) and unit neutrosophic sets ([1.sub.N] = (1, 0, 0)) are definable neutrosophic sets.

Definition 2.9 [8, 9]

If N(P) = ([N.bar](P),[bar.N](P)) is a rough neutrosophic set in (Z, R), the rough complement [8, 9] of N(P) is the rough neutrosophic set denoted by ~ N(P) = ([N.bar][(P).sup.c],[bar.N][(P.sup.)]c) where [N.bar][(P).sup.c], [bar.N][(P).sup.c] are the complements of neutrosophic sets of ([N.bar](P), [bar.N](P)) respectively.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 2.10 [8, 9]

If N([P.sub.1]) and N([P.sub.2]) are the two rough neutrosophic sets of the neutrosophic set P respectively in Z, then the following definitions [8, 9] hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If N, M, L are the rough neutrosophic sets in (Z, R), then the following proposition are stated from definitions [8, 9].

Proposition 1 [8, 9]

1. ~(~ N) = N

2. N [union] M = M [union] N, N [intersection] M = M [intersection] N

3. (L [union] M) [union] N = L [union] (M [union] N), (L [intersection] M) [intersection] [intersection] = L [intersection] (M [intersection] [intersection])

4. (L [union] M) [intersection] N = (L [union] M) [intersection] (L [union] N), (L [intersection] M) [union] [intersection] = (L [intersection] M) [union] (L [intersection] [intersection])

Proposition 2 [8, 9]

De Morgan's Laws are satisfied for rough neutrosophic sets.

1. ~ (N([P.sub.1]) [union] N([P.sub.2])) = (~(N([P.sub.1])) [intersection] (~ N([P.sub.2]))

2. ~ (N([P.sub.1]) [intersection] N([P.sub.2])) = (~ (N([P.sub.1]) [union] (~N([P.sub.2]))

Proposition 3 [8, 9]

If [P.sub.1] and [P.sub.2] are two neutrosophic sets in U such that [P.sub.1] [subset or equal to] [P.sub.2] then N([P.sub.1]) [subset or equal to] N([P.sub.2])

1. N([P.sub.1] [intersection] [P.sub.2]) [subset or equal to] N([P.sub.1]) [intersection] N([P.sub.2])

2. N([P.sub.1] [union] [P.sub.2]) [contains or equal to] N([P.sub.1]) [union] N(p)

Proposition 4 [8, 9]

1. [N.bar](P) = ~[bar.N](~P)

2. [bar.N](P) = ~[N.bar](~P)

3. [N.bar](P) [subset or equal to] [bar](P)

3 Similarity measures and variational coefficient similarity measure in crisp environment

The vector similarity measure is one of the important tools for the degree of similarity between objects. However, the Jaccard, Dice, and cosine similarity measures are often used for this purpose. Jaccard [83], Dice [84], and cosine [85] similarity measures between two vectors are stated below.

Let X = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and Y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) be two ndimensional vectors with positive co-ordinates.

Definition 3.1 [83]

Jeccard index of two vectors (measuring the "similarity" of these vectors) can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the Euclidean norm of X and Y, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the inner product of the vector X and Y.

Proposition 5 [83]

Jaccard index satisfies the following properties:

1. 0 [less than or equal to] J(X, Y) [less than or equal to] 1

2. J(X, Y) = J(Y, X)

3. J(X, Y) = 1, forX = Yi.e, [x.sub.i] = [y.sub.i](i = 1, 2, ..., n) for every [x.sub.i] [member of] X and [y.sub.i] [member of] Y

Definition 3.2 [84]

The Dice similarity measure can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Proposition 6 [84]

The Dice similarity measure satisfies the following properties:

1.0 [less than or equal to] E(X, Y) [less than or equal to] 1

2. E(X, Y) = E(Y, X)

3. J(X, Y) = 1, for X = Y i.e, [x.sub.i] = [y.sub.i](i = 1, 2, ..., n) for every [x.sub.i] [member of] X and [y.sub.i] [member of] Y.

Definition 3.3 [85]

The cosine similarity measure between two vectors X and Y is the inner product of these two vectors divided by the product of their lengths and can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Proposition 7 [85]

The cosine similarity measure satisfies the following properties

1. 0 [less than or equal to] C(X, Y) [less than or equal to] 1

2. C(X, Y) = C(Y, X)

3. C(X, Y) = 1, forX = Y i.e, [x.sub.i] = [y.sub.i](i = 1, 2, ..., n) for every [x.sub.i] [member of] X and [y.sub.i] [member of] Y.

These three formulas are similar in the sense that they take values in the interval [0, 1]. Jaccard and Dice similarity measures are undefined when [x.sub.i] = 0, and [y.sub.i] = 0 for i = 1, 2, ..., n and cosine similarity measure is undefined when [x.sub.i] = 0 or [y.sub.i] = 0 for i = 1, 2, ..., n.

Definition 3.4 [86]

Variational co-efficient similarity measure can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Proposition 8 [86]

Variational co-efficient similarity measure satisfies the following properties:

1. 0 [less than or equal to] V(X, Y) [less than or equal to] 1

2. V(X, Y) = V(Y, X)

3. V(X, Y) = 1, for X = Y i.e, [x.sub.i] = [y.sub.i] (i = 1, 2, ..., n) for every [x.sub.i] [member of] X and [y.sub.i] [member of] Y.

4. Various similarity measures for single valued neutrosophic sets.

Assume [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> be two SVNSs in a universe of discourse X = ([x.sub.1], [x.sub.2], ..., [x.sub.n]). [T.sub.A]],[I.sub.A],[F.sub.A] [member of] [0,1] for any [x.sub.i] [member of] X in [N.sub.A] or [T.sub.B]],[I.sub.B],[F.sub.B] [member of] [0,1] for any [x.sub.i] [member of] X in [N.sub.B] can be considered as a vector representation with three elements. Let [w.sub.i] [member of] [0,1] be the weight of each element [x.sub.i] for i = 1, 2, ..., n such that [[summation].sup.n.sub.i=1] [w.sub.i] = 1, then Jaccard, Dice and cosine similarity measures can be presented as follows:

Definition 4.1[10] Jaccard similarity measure between [N.sub.A] = <[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Proposition 9 [10]

Jaccard similarity measure satiefies the following properites:

1. 0 [less than or equal to] Jac([N.sub.A],[N.sub.B]) [less than or equal to] 1;

2. Jac([N.sub.A],[N.sub.B]) = Jac([N.sub.B],[N.sub.A]);

3. Jac([N.sub.A],[N.sub.B]) = 1; if [N.sub.A]=[N.sub.B] i.e., [T.sub.A]([x.sub.i]) = [T.sub.B]([x.sub.i]), [I.sub.A] ([x.sub.i]) = [I.sub.B] ([x.sub.i]), and [F.sub.A] ([x.sub.i]) = [F.sub.B] ([x.sub.i]), for every [x.sub.i] (i = 1, 2, ..., n) in X.

Definition 4.1.1 [10] Weighted Jaccard similarity measure between [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Proposition 10 [10]

Weighted Jaccard similarity measure satisfies the following properties:

1. 0 [less than or equal to] [Jac.sub.w]([N.sub.A],[N.sub.B]) [less than or equal to] 1;

2. [Jac.sub.w]([N.sub.A],[N.sub.B]) = [Jac.sub.w]([N.sub.B],[N.sub.A]);

3. [Jac.sub.w]([N.sub.A],[N.sub.B]) = 1; if [N.sub.A] = [N.sub.B] i.e., [T.sub.A]([x.sub.i]) = [T.sub.B]([x.sub.i]),

[I.sub.A] ([x.sub.i]) = [I.sub.B] ([x.sub.i]), and [F.sub.A] ([x.sub.i]) = [F.sub.B] ([x.sub.i]), for every x, (i = 1, 2, ..., n) in X.

Definition 4.2 [11]

Dice similarity measure between [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Proposition 11 [11]

Dice similarity measure satisfies the following properties:

1. 0 [less than or equal to] Dic([N.sub.A],[N.sub.B]) [less than or equal to] 1

2. Dic([N.sub.A],[N.sub.B]) = Dic([N.sub.B],[N.sub.A]);

3. Dic([N.sub.A],[N.sub.B]) = 1; if [N.sub.A]=[N.sub.B] i.e., [T.sub.A]([x.sub.i]) = [T.sub.B]([x.sub.i]), [I.sub.A] (x) = [I.sub.B] ([x.sub.i]), and [F.sub.A] ([x.sub.i]) = [F.sub.B] ([x.sub.i]), for every [x.sub.i] (i = 1, 2, ..., n) in X.

Definition 4.2.1 [11]

Weighted Dice similarity measure between [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Proposition 12 [11]

Weighted Dice similarity measure

1. 0 [less than or equal to] [Dic.sub.w]([N.sub.A],[N.sub.B]) [less than or equal to] 1;

2. [Dic.sub.w] ([N.sub.A], [N.sub.B]) = [Dic.sub.w] ([N.sub.B], [N.sub.A]);

3. [Dic.sub.w] ([N.sub.A], [N.sub.B]) = 1; if [N.sub.A] = [N.sub.B] i.e., [T.sub.A]([x.sub.i]) = [T.sub.B] ([x.sub.i]),

[I.sub.A]([x.sub.i]) = [I.sub.B]([x.sub.i]), and [F.sub.A]([x.sub.i]) = [F.sub.B]([x.sub.i]), for every x, (i = 1, 2, ..., n) in X.

Definition 4.3 [12]

Cosine similarity measure between [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Proposition 13 [12]

Cosine similarity measure satisfies the following properties:

1. 0 [less than or equal to] [Cos.sub.w]([N.sub.A],[N.sub.B]) [less than or equal to] 1;

2. [Cos.sub.w]([N.sub.A],[N.sub.B]) = [Cos.sub.w]([N.sub.B],[N.sub.A])

3. [Cos.sub.w]([N.sub.A],[N.sub.B]) = 1; if [N.sub.A] = [N.sub.B] i.e., [T.sub.A]([x.sub.i]) = [T.sub.B]([x.sub.i]), [I.sub.A]([x.sub.i]) = [I.sub.B] ([x.sub.i]), and [F.sub.A] ([x.sub.i]) = [F.sub.B] ([x.sub.i]), for every [x.sub.i] (i = 1, 2, ..., n) in X.

Definition 4.3.1 [12]

Weighted cosine similarity measure between [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Proposition 14 [12]

Weighted cosine similarity measure satisfies the following properties:

1. 0 [less than or equal to] [Cos.sub.w] ([N.sub.A],[N.sub.B]) [less than or equal to] 1;

2. [Cos.sub.w](N.sub.A],[N.sub.B]) = [Cos.sub.w]([N.sub.B],[N.sub.A])

3. [Cos.sub.w]([N.sub.A],[N.sub.B])=1; if [N.sub.A] = [N.sub.B] i.e., [T.sub.A]([x.sub.i]) = [T.sub.B]([x.sub.i]),

[I.sub.A] ([x.sub.i]) = [I.sub.B]([x.sub.i]), and [F.sub.A]([x.sub.i]) = [F.sub.B]([x.sub.i]), for every [x.sub.i] (i = 1, 2, ..., n) in X.

Jaccard and Dice similarity measures between two neutrosophic sets [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B]=<[T.sub.B]],[I.sub.B],[F.sub.B]> are undefined when [T.sub.A] ([x.sub.i]) = [I.sub.A] ([x.sub.i]) = [F.sub.A] ([x.sub.i]) = 0 and [T.sub.b] ([x.sub.i]) = [I.sub.B]([x.sub.i]) = [F.sub.B]([x.sub.i]) = 0 for all i = 1, 2, ..., n. Similarly the cosine formula for two neutrosophic sets [N.sub.A]=<[T.sub.A]],[I.sub.A],[F.sub.A]> and [N.sub.B] = <[T.sub.B]],[I.sub.B],[F.sub.B]> is undefined when [T.sub.A] ([x.sub.i]) = [I.sub.A] ([x.sub.i]) = [F.sub.A] ([x.sub.i]) = 0 or [T.sub.B] ([x.sub.i]) = [I.sub.B] ([x.sub.i]) = [F.sub.B] ([x.sub.i]) = 0 for all i = 1, 2, ..., n.

5 Variational similarity measures for rough neutrosophic sets

The notion of rough neutrosophic set (RNS) is used as vector representations in 3D-vector space. Assume that X= ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and Y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) be two n-dimensional vectors with positive co-ordinates. Jaccard, Dice, cosine and cotangent similarity measures between two vectors are stated as follows.

Definition 5.1 [21] Jaccard similarity measure under rough neutrosophic environment As sume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be any two rough neutrosophic sets. Jacard similarity measure [21] between rough neutrosophic sets A and B can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Proposition 15 [21]

Jaccard similarity measure [21] between A and B satisfies the following properties:

1. 0 [less than or equal to] [Jac.sub.RNS] (A, B) [less than or equal to]1;

2. [Jac.sub.RNS] (A,B) = [Jac.sub.RNS] (B,A);

3. [Jac.sub.RNS] (A, B) =1; iff A = B

4. If C is a RNS in Y and A [subset] B [subset] C then, [Jac.sub.RNS] (A, C) [less than or equal to] [Jac.sub.RNS] (A, B), and [Jac.sub.RNS] (A, C) [less than or equal to] [Jac.sub.RNS] (B, C)

Definition 5.1.1 [21]

If we consider the weights of each element x,, weighted rough Jaccard similarity measure [21] between rough neutrosophic sets A and B can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 16 [21]

The weighted rough Jaccard similarity [21] measure between two rough neutrosophic sets A and B also satisfies the following properties:

1. 0 [less than or equal to] [Jac.sub.WRNS](A,B) [less than or equal to] 1;

2. [Jac.sub.WRNS](A,B) = [Jac.sub.WRNS](B,A) ;

3. [Jac.sub.WRNS](A, B) =1; iff A = B

4. If C is a WRNS in Y and A [subset] B [subset] C then, [Jac.sub.WRNS](A, C)

[less than or equal to] [Jac.sub.w]mSiA, B), and Jucwrns(a, C) [less than or equal to] Jucwrns(B, C) Definition 5.2 [21] Dice similarity measure under rough neutrosophic environment

In this section, Dice similarity measure and the weighted Dice similarity measure for rough neutrosophic sets have been stated due to Pramanik and Mondal [21].

Suppose that

A = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

B = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be any

two rough neutrosophic sets in X = ([x.sub.1], [x.sub.2], ..., [x.sub.n]). Dice similarity measure between rough neutrosophic sets A and B can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Proposition 17 [21]

Dice similarity measure [21] satisfies the following properties.

1. 0 [less than or equal to] [DIC.sub.RNS] (A, B) [less than or equal to]1;

2. [DIC.sub.RNS] (A, B) = [DIC.sub.RNS] (B, A);

3. [DIC.sub.RNS] (A, B)=1; iff A = B

4. If C is a RNS in Y and A [[subset]] B [subset] C then, [DIC.sub.RNS](A, C) [less than or equal to] [DIC.sub.RNS](A, B), and [DIC.sub.RNS](A, C) [less than or equal to] [DIC.sub.RNS](B, C),

For proofs of the above mentioned four properties, see [21].

Definition 5.2.1

If we consider the weights of each element x,, a weighted rough Dice similarity measure between rough neutrosophic sets A and B can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The weighted rough Dice similarity [21] measure between two rough neutrosophic sets A and B also satisfies the following properties:

1. 0 [less than or equal to] [DIC.sub.WRNS](A, B) [less than or equal to] 1;

2. [DIC.sub.WRNS](A, B) = [DIC.sub.WRNS](B, A);

3. [DIC.sub.WRNS](A, B) = 1; iff A = B

4. If C is a RNS in Y and A [subset] B [subset] C then,

[DIC.sub.WRNS](A, C) [less than or equal to] [DIC.sub.WRNS](A, B), and [DIC.sub.WRNS](A, C) [less than or equal to] [DIC.sub.WRNS](B, C).

For proofs of the above mentioned four properties, see [21].

Definition 5.3 [20]

Cosine similarity measure can be defined as the inner product of two vectors divided by the product of their lengths. It is the cosine of the angle between the vector representations of two rough neutrosophic sets. The cosine similarity measure is a fundamental measure used in information technology. Pramanik and Mondal [20] defined cosine similarity measure between rough neutrosophic sets in 3-D vector space.

Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be any rough neutrosophic sets. Pramanik and Mondal [20] defined cosine similarity measure between rough neutrosophic sets A and B as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 19 [20]

Let A and B be rough neutrosophic sets. Cosine similarity measure [20] between A and B satisfies the following properties.

1. 0 [less than or equal to] [C.sub.RNS](A,B) [less than or equal to] 1;

2- [C.sub.RNS] (A, B) = [C.sub.RNS] (B, A);

3. [C.sub.RNS] (A, B) = 1; iff A = B

4. If C is a RNS in Y and A [subset] B [subset] C then, [C.sub.RNS](A, C) [less than or equal to] [C.sub.RNS](A, B), and [C.sub.RNS](A, C) [less than or equal to] [C.sub.RNS](B, C).

Definition 5.3.1 [20]

If we consider the weights of each element x,, a weighted rough cosine similarity measure between rough neutrosophic sets A and B can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If we take [w.sub.i] = 1/n, i = 1, 2, ..., n, then [C.sub.WRNS] (A, B) = [C.sub.RNS](A, B)

Proposition 20 [20]

The weighted rough cosine similarity measure [20] between two rough neutrosophic sets A and B also satisfies the following properties:

1. 0 [less than or equal to] [C.sub.WRNS](A, b) [less than or equal to] 1;

2. [C.sub.WRNS](A, B) =[C.sub.WRNS](B, A);

3. [C.sub.WRNS](A, B) =1; iff A = B

4. If C is a WRNS in Y and A [subset] B [subset] C then, [C.sub.WRNS](A, C) [less than or equal to] [C.sub.WRNS](A, B), and [C.sub.WRNS] (A, C) [less than or equal to] [C.sub.WRNS](B, C).

For proofs of the above mentioned four properties, see [20].

Definition 5.4 [19] Cotangent similarity measures of rough neutrosophic sets

Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be any two rough neutrosophic sets. Pramanik and Mondal [19] defined cotangent similarity measure between rough neutrosophic sets A and B as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 21 [19]

Cotangent similarity measure satisfies the following properties:

1. 0 [less than or equal to] [COT.sub.RNS](A, B) [less than or equal to] 1;

2. [COT.sub.RNS](A, B) = [COT.sub.RNS](B, A);

3. [COT.sub.RNS](A,B) =1; iff A = B

4. If C is a RNS in Y and A [subset] B [subset] C then, [COT.sub.RNS](A, C) [less than or equal to] [COT.sub.RNS](A, B), and [COT.sub.RNS](A, C) [less than or equal to] [COT.sub.RNS](B, C).

Definition 5.4.1

If we consider the weights of each element [x.sub.i], a weighted rough cotangent similarity measure [19] between rough neutrosophic sets A and B can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Proposition 22 [19]

The weighted rough cosine similarity measure between two rough neutrosophic sets A and B also satisfies the following properties:

1. 0 [less than or equal to] [COT.sub.WRNS](A, B) [less than or equal to] 1;

2. [COT.sub.WRNS](AA B) = [COT.sub.WRNS](B, A);

3. [COT.sub.WRNS](A, B)=1; iff A = B

4. If C is a WRNS in Y and A [subset] B [subset] C then, [COT.sub.WRNS](A, C) [less than or equal to] [COT.sub.WRNS](A, B), and [COT.sub.WRNS](A, C) [less than or equal to] [COT.sub.WRNS](B, Q

Definition 5.5 (Variational co-efficient similarity measure between rough neutrosophic sets)

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be two rough neutrosophic sets. Variational co-efficient similarity measure between rough neutrosophic sets can be presented as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 23

The variational co-efficient similarity measure [Var.sub.RNS](A, B) between two rough neutrosophic sets A and B, satisfies the following properties:

1. 0 [less than or equal to] [Var.sub.RNS](A,B) [less than or equal to] 1;

2. [Var.sub.RNS] (A,B) = [Var.sub.RNS] (B,A);

3. [Var.sub.RNS] (A, B)=1; if A = B i.e., [delta][T.sub.A]([x.sub.i]) = [delta][T.sub.B] ([x.sub.i]), [delta][I.sub.A]([x.sub.i]) = [delta] [I.sub.B] ([x.sub.i]), and [delta][F.sub.A]([x.sub.i]) = [delta][F.sub.B] ([x.sub.i]), for every [x.sub.i] (i = 1, 2, n) in X.

Proof.

(1.) It is obvious that [Var.sub.RNS](A,B) [greater than or equal to] 0. Thus it is required to prove that [Var.sub.RNS] (A, B) [less than or equal to] 1.

From rough neutrosophic dice similarity measure it can be witten that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

and from rough neutrosophic cosine similarity measure it can be written that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Combining Eq.(20) and Eq.(21), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Thus, 0 [less than or equal to] [Var.sub.RNS](A,B) [less than or equal to] 1;

(2.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These results show the completion of the proofs of the three properties..

Definition 5.6 (Weighted variational co-efficient similarity measure between rough neutrosophic sets)

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be any two rough neutrosophic sets. Rough variational co-efficient similarity measure between rough neutrosophic setsA and B in 3-D vector space can be presented as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Proposition 24

The weighted variational co-efficient similarity measure also satisfies the following properties:

1. 0 [less than or equal to] [Var.sub.WRNS](A,B) [less than or equal to] 1;

2. [Var.sub.WRNS](A,B) = [Var.sub.WRNS](B, A);

3. [Var.sub.WRNS](A, B) = 1; if A = B i.e., [delta][T.sub.A] ([x.sub.i]) = [delta][T.sub.B] ([x.sub.i]), [delta][I.sub.A] ([x.sub.i]) = 8[I.sub.B] ([x.sub.i]), and [delta][F.sub.A] ([x.sub.i]) = [delta][F.sub.B] ([x.sub.i]) for every [x.sub.i](1 = 1, 2, n) in X.

Proof:

(1.) It is obvious that [Var.sub.WRNS](A, B) [greater than or equal to] 0. Thus it is required to prove that [Var.sub.WRNS](A, B) [less than or equal to] 1.

From rough neutrosophic weighted dice similarity measure, it can be written that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

and from rough neutrosophic weighted cosine imilarity measure it can be written that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Combining Eq.(24) and Eq.(25), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Thus, 0 [less than or equal to] [Var.sub.WRNS](A, B) [less than or equal to] 1;

(2.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

These results show the completion of the proofs of the three properties.

6. Multi attribute decision making based on rough neutrosophic variational coefficient similarity measure

In this section, a rough variational co-efficient similarity measure is employed to multi-attribute decision making in rough neutrosophic environment. Assume that A = {[A.sub.1], [A.sub.2], ..., [A.sub.m]} be the set of alternatives and C = {C1, [C.sub.2], ..., [C.sub.n]} be the set of attributes in a multi-attribute decision making problem. Assune that [w.sub.j] be the weight of the attribute Cj provided by the decision maker such that each [w.sub.i] [member of] [0,1] and [[summation].sup.n.sub.j=1] [w.sub.j] -1 However, in real situation decision maker may often face difficulty to evaluate alternatives over the attributes due to vague or incomplete information about alternatives in a decision making situation. Rough neutrosophic set can be used in MADM to deal with incomplete information of the alternatives. In this paper, the assessment values of all the alternatives with respect to attributes are considered as the rough neutrosophic values (see Table 1).

Here <[[d.bar].sub.ij],[[bar.d].sub.ij]> is the rough neutrosophic number for the i-th alternative and the j-th attribute.

Definition 6.1: Transforming operator for SVNSs [80]

The rough neutrosophic decision matrix (27) can be transformed to single valued neutrosophic decision matrix whose ij-th element [a.sub.ij] can be presented as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Step 1. Determine the neutrosophic relative positive ideal solution

In multi-criteria decision-making environment, the concept of ideal point has been used to help identify the best alternative in the decision set Definition 6.2 [51]

Let H be the collection of two types of attributes, namely, benefit type attribute (P) and cost type attribute (L) in the MADM problems. The relative positive ideal neutrosophic solution (RPINS) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the solution of the decision matrix [D.sub.S] = [<([delta][T.sub.ij],[delta][I.sub.ij],[delta][F.sub.ij]>.sub.mxn] where, every component of [Q.sup.+.sub.S] has the following form: for benefit type attribute, every component of Q+ has the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

and for cost type attribute, every component of [Q.sup.+.sub.S] has the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Step 2. Determine the weighted variational co-efficient similarity measure between ideal alternative and each alternative.

The variational co-efficient similarity measure between ideal alternative [Q.sup.+.sub.S] and each alternative [A.sub.i] for i = 1, 2, ..., m can be determined by the following equation as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

Step 3. Rank the alternatives.

According to the values obtained from Eq.(31), the ranking order of all the alternatives can be easily determined. Highest value indicates the best alternative. Step 4. End.

7 Numerical example

In this section, rough neutrosophic MADM regarding investment problem is considered to demonstrate the applicability and the effectiveness of the proposed approach. However, investment problem is not easy to solve. It not only requires oodles of patience and discipline, but also a great deal of research and a sound understanding of the market, mathematical tools, among others. Suppose an investment company wants to invest a sum of money in the best option. Assume that there are four possible alternatives to invest the money: (1) [A.sub.1] is a computer company; (2) [A.sub.2] is a garment company; (3) [A.sub.3] is a telecommunication company; and (4) A4 is a food company. The investment company must take a decision based on the following three criteria: (1) [C.sub.1] is the growth factor; (2) [C.sub.2] is the environmental impact; and (3) [C.sub.3] is the risk factor. The four possible alternatives are to be evaluated under the attribute by the rough neutrosophic assessments provided by the decision maker. These assessment values are given in the rough neutrosophic decision matrix (see the table 2).

The known weight information is given as follows:

W = [[[w.sub.1], [w.sub.2], [w.sub.3]].sup.T] = [0.3, 0.3, 0.4] and [[summation].sup.3.sub.i=1] [w.sub.i] = 1.

Step 1. Determine the types of criteria.

First two types i.e. [C.sub.1] and [C.sub.2] of the given criteria are benefit type criteria and the last one criterion i.e. [C.sub.3] is the cost type criteria.

Step 2. Determine the relative neutrosophic positive ideal solution

Using Eq. (29), Eq.(30), the relative positive ideal neutrosophic solution for the given matrix defined in Eq.(32) can be obtained as:

[Q.sup.+.sub.s]=[(0.4,0.2,0.2),(0.7,0.2,0.2),(0.1,0.3,0.3)]

Step 3. Determine the weighted variational similarity measure

The weighted variational co-efficient similarity measure is determined by using Eq.(28), Eq.(31) and Eq.(32). The results obtained for different values of [lambda] have been shown in the Table-3.

Step 4. Rank the alternatives.

According to the different values of [lambda], the results obtained in Table-3 reflects that [A.sub.3] is the best alternative.

8. Comparisons of different rough similarity measure with rough variation similarity measure

In this section four existing rough similarity measures namely: rough cosine similarity measure, rough dice similarity measure, rough cotangent similarity measure and rough Jaccard similarity measure have been compared with proposed rough variational co-efficient similarity measure for different values of [lambda]. The comparison results are listed in the Table 3 and Table 4.

Conclusion

In this paper, we have proposed rough variational coefficient similarity measures. We also proved some of their basic properties. We have presented an application of rough neutrosophic variational coefficient similarity measure for a decision making problem on investment. The concept presented in the paper can be applied to deal with other multi attribute decision making problems in rough neutrosophic environment.

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Received: November 20, 2016. Accepted: December 15, 2016

Kalyan Mondal (1) Surapati, Pramanik (2) and Florentin Smarandache (2)

(1) Department of Mathematics, Jadavpur University, West Bengal, India. Email: kalyanmathematic@gmail.com

(2) Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, PO-Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West Bengal, India. Email: sura_pati@yahoo.co.in,

(3) University of New Mexico. Mathematics & Science Department, 705 Gurley Ave., Gallup, NM 87301, USA. Email: fsmarandache@gmail.com
Table 1: Rough neutrosophic decision matrix

[D.sub.RNS] = [<[[d.bar].sub.ij], [[bar.d].sub.ij]>.sub.mxn] =

               [C.sub.1]           [C.sub.2]           [C.sub.n]

[A.sub.1]  <[[d.bar].sub.11],  <[[d.bar].sub.12],  <[[d.bar].sub.1n],
           [[bar.d].sub.11]>   [[bar.d].sub.12]>   [[bar.d].sub.1n]>

[A.sub.2]  <[[d.bar].sub.21],  <[[d.bar].sub.22],  <[[d.bar].sub.2n],
           [[bar.d].sub.21]>   [[bar.d].sub.22]>   [[bar.d].sub.2n]>

.                 ...                 ...                 ...

.                 ...                 ...                 ...

[A.sub.m]  <[[d.bar].sub.m1],  <[[d.bar].sub.m2],  <[[d.bar].sub.mn],
           [[bar.d].sub.m1]>   [[bar.d].sub.m2]>   [[bar.d].sub.mn]>

(27)

Table 2: Rough neutrosophic decision matrix

D = [<[[N.bar].sub.ij](P), [[bar.N].sub.ij](P)>.sub.mxn] =

              [C.sub.1]        [C.sub.2]        [C.sub.n]

[A.sub.1]   <(0.2,0.2,0.2)   <(0.7,0.2,0.2)   <(0.4,0.1,0.1)
            (0.0,0.4,0.4)>   (0.5,0.4,0.4)>   (0.2,0.3,0.3)>

[A.sub.2]   <(0.5,0.2,0.1)   <(0.7,0.2,0.1)   <(0.2,0.2,0.2)
            (0.3,0.2,0.3)>   (0.5,0.2,0.3)>   (0.0,0.2,0.4)>

[A.sub.3]   <(0.4,0.2,0.3)   <(0.8,0.1,0.1)   <(0.3,0.2,0.3)
            (0.2,0.4,0.3)>   (0.6,0.3,0.3)>   (0.1,0.4,0.3)>
.
[A.sub.4]   <(0.3,0.2,0.2)   <(0.8,0.2,0.3)   <(0.5,0.2,0.1)
            (0.1,0.2,0.2)>   (0.6,0.4,0.3)>   (0.3,0.2,0.3)>   (32)

Table-3. Results of rough variational similarity measure for different
values of [lambda], 0 [less than or equal to] [lambda] [less than
or equal to] 1

Similarity measure   Values of            Measure values
method               [lambda] s

[Var.sub.WRNS]          0.10      0.8769; 0.9741; 0.9917; 0.8107
([Q.sup.+.sub.S],       0.25       0.8740, 0.9739 0.9905 0.8078
[D.sub.S])              0.50      0.8692; 0.9735; 0.9887; 0.8028
                        0.75      0.8643; 0.9730; 0.9868; 0.7979
                        0.90      0.8614; 0.9728; 0.9857; 0.7949

Similarity measure   Values of               Ranking order
method               [lambda] s

[Var.sub.WRNS]          0.10      [A.sub.3] > [A.sub.2] > [A.sub.1] >
([Q.sup.+.sub.S],                   [A.sub.4]
[D.sub.S])              0.25      [A.sub.3] > [A.sub.2] > [A.sub.1] >
                                    [A.sub.4]
                        0.50      [A.sub.3] > [A.sub.2] > [A.sub.1] >
                                    [A.sub.4]
                        0.75      [A.sub.3] > [A.sub.2] > [A.sub.1] >
                                    [A.sub.4]
                        0.90      [A.sub.3] > [A.sub.2] > [A.sub.1] >
                                    [A.sub.4]

Table-4. Results of existing rough neutrosophic similarity measure
methods.

Rough similarity       Values of            Measure values
measure methods        [lambda] s

[JAC.sub.WRNS]            ...       0.7870, 0.9471; 0.9739; 0.6832
  ([Q.sup.+.sub.S],
  [D.sub.S]) [21]
[DIC.sub.WRNS]            ...       0.8595; 0.9726; 0.9873; 0.7929
  ([Q.sup.+.sub.S],
  [D.sub.S]) [21]
[C.sub.WRNS]              ...       0.8788; 0.9738; 0.9920; 0.9132
  ([Q.sup.+.sub.S],
  [D.sub.S]) [20]
[COT.sub.WRNS]            ...       0.8472; 0.9358; 0.9643; 0.8103
  ([Q.sup.+.sub.S],
  [D.sub.S]) [19]

Rough similarity                   Ranking order
measure methods

[JAC.sub.WRNS]         [A.sub.3] > [A.sub.2] > [A.sub.1] >
  ([Q.sup.+.sub.S],      [A.sub.4]
  [D.sub.S]) [21]
[DIC.sub.WRNS]         [A.sub.3] > [A.sub.2] > [A.sub.1] >
  ([Q.sup.+.sub.S],      [A.sub.4]
  [D.sub.S]) [21]
[C.sub.WRNS]           [A.sub.3] > [A.sub.2] > [A.sub.4] >
  ([Q.sup.+.sub.S],      [A.sub.1]
  [D.sub.S]) [20]
[COT.sub.WRNS]         [A.sub.3] > [A.sub.2] > [A.sub.1] >
  ([Q.sup.+.sub.S],      [A.sub.4]
  [D.sub.S]) [19]
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Author:Mondal, Kalyan; Pramanik, Surapati; Smarandache, Florentin
Publication:Neutrosophic Sets and Systems
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Date:Oct 1, 2016
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