# Multiple Attribute Decision-Making Method Using Correlation Coefficients of Normal Neutrosophic Sets

## Abstract

**:**

## 1. Introduction

## 2. Some Basic Concepts of NIFNs and NNSs

_{A}(x), v

_{A}(x)>, where its membership function is expressed as

_{A}and v

_{A}are a membership degree and a non-membership degree in an IFN and satisfy t

_{A}, v

_{A}∈ [0,1], and 0 ≤ t

_{A}+ v

_{A}≤ 1.

_{B}(x), u

_{B}(x), and v

_{B}(x) in real standard interval [0,1] or nonstandard interval ]

^{−}0, 1

^{+}[, such that t

_{B}(x): X → ]

^{−}0, 1

^{+}[, u

_{B}(x): X → ]

^{−}0, 1

^{+}[, v

_{B}(x): U → ]

^{−}0, 1

^{+}[, and

^{−}0 ≤ sup t

_{B}(x) + sup u

_{B}(x) + sup v

_{B}(x) ≤ 3

^{+}for x ∈ X.

^{−}0, 1

^{+}[, the neutrosophic set shows the difficulty of its actual applications. Thus, Smarandache [9] and Wang et al. [10] introduced the concept of an SVNS as a subclass of the neutrosophic set when the three membership functions in the neutrosophic set are constrained in the real standard interval [0,1].

**Definition**

**1.**

_{S}(x), u

_{S}(x), and v

_{S}(x), where t

_{S}(x), u

_{S}(x), v

_{S}(x) ∈ [0,1], and 0 ≤ t

_{S}(x) + u

_{S}(x) + v

_{S}(x) ≤ 3 for x ∈ X. Then, the SVNS S can be denoted as. $S=\left\{\langle x,{t}_{S}(x),{u}_{S}(x),{v}_{S}(x)\rangle :x\in X\right\}$.

**Definition**

**2.**

_{P}(x), u

_{P}(x), and v

_{P}(x) for x ∈ X satisfy the following properties:

_{P}(x) + u

_{P}(x) + v

_{P}(x) ≤ 1

_{p}, u

_{p}, and v

_{p}are the truth, indeterminacy, and falsity degrees in the SVNN, respectively, and satisfy t

_{p}, u

_{p}, and v

_{p}∈ [0,1] and 0 ≤ t

_{p}+ u

_{p}+ v

_{p}≤ 3.

**Definition**

**3.**

## 3. Correlation Coefficients between NNSs

**Definition**

**4.**

_{1}, p

_{2}, …, p

_{n}} and Q = {q

_{1}, q

_{2}, …, q

_{n}}, where p

_{j}= <N(μ

_{pj}, σ

_{pj}), (t

_{pj}, u

_{pj}, v

_{pj})> and q

_{j}= <N(μ

_{qj}, σ

_{qj}), (t

_{qj}, u

_{qj}, v

_{qj})> for j = 1, 2, …, n are NNNs in P and Q. The correlation between two NNSs P and Q is defined as

**Definition**

**5.**

_{1}, p

_{2}, …, p

_{n}} and Q = {q

_{1}, q

_{2}, …, q

_{n}}, where p

_{j}= <N(μ

_{pj}, σ

_{pj}), (t

_{pj}, u

_{pj}, v

_{pj})> and q

_{j}= <N(μ

_{qj}, σ

_{qj}), (t

_{qj}, u

_{qj}, v

_{qj})> for j = 1, 2, …, n are NNNs in P and Q. The correlation coefficients between two NNSs P and Q are defined as

**Proposition**

**1.**

_{k}(P, Q) (k = 1, 2) satisfy the following properties:

- 0 ≤ ρ
_{k}(P, Q) ≤ 1; - ρ
_{k}(P, Q) = 1 if P = Q, i.e., N(μ_{pj}, σ_{pj}) = N(μ_{qj}, σ_{qj}) and (t_{pj}, u_{pj}, v_{pj}) = (t_{qj}, u_{qj}, v_{qj}); - ρ
_{k}(P, Q) = ρ_{k}(Q, P).

**Proof.**

_{1}(P, Q) satisfies the properties (1)–(3).

_{1}(P, Q) ≥ 0 is obvious. Then, we only prove ρ

_{1}(P, Q) ≤ 1.

_{1}, x

_{2}, …, x

_{n}) ∈ R

^{n}and (y

_{1}, y

_{2}, …, y

_{n}) ∈ R

^{n}, we can yield the following inequality:

_{1}(P, Q) ≤ 1. Hence, 0 ≤ ρ

_{1}(P, Q) ≤ 1 holds.

_{pj}, σ

_{pj}) = N(μ

_{qj}, σ

_{qj}) and (t

_{pj}, u

_{pj}, v

_{pj}) = (t

_{qj}, u

_{qj}, v

_{qj}) ⇒ μ

_{pj}= μ

_{qj}, σ

_{pj}= σ

_{qj}, t

_{pj}= t

_{qj}, u

_{pj}= u

_{qj}, and v

_{pj}= v

_{qj}for j = 1, 2, …, n ⇒ ρ

_{1}(P, Q) = 1.

_{2}(P, Q) satisfies the properties (1)–(3).

_{1}(P, Q), we can prove the properties (1)–(3) of ρ

_{2}(P, Q). It is not repeated here.

_{j}and q

_{j}(j = 1, 2, … , n) is taken into account,

**w**= {w

_{1}, w

_{2}, … , w

_{n}} is given as the weight vector of the elements p

_{j}and q

_{j}(j = 1, 2, … , n) with w

_{j}∈ [0,1] and ${\sum}_{j=1}^{n}{w}_{j}=1$. Then, we have the following weighted correlation coefficients of NNSs:

**Proposition**

**2.**

_{kw}(P, Q) (k = 1, 2) also satisfy the following properties:

- 0 ≤ ρ
_{kw}(P, Q) ≤ 1; - ρ
_{kw}(P, Q) = 1 if and only if P = Q, i.e., N(μ_{pj}, σ_{pj}) = N(μ_{qj}, σ_{qj}) and (t_{pj}, u_{pj}, v_{pj}) = (t_{qj}, u_{qj}, v_{qj}); - ρ
_{kw}(P, Q) = ρ_{kw}(Q, P).

## 4. The MADM Method Using the Correlation Coefficients of NNSs

_{1}, P

_{2}, …, P

_{m}} is a set of m alternatives and R = {R

_{1}, R

_{2}, …, R

_{n}} is a set of n attributes. The weight vector of the attributes is given as

**w**= (w

_{1}, w

_{2}, …, w

_{n}) satisfying w

_{j}∈ [0,1] and ${\sum}_{j=1}^{n}{w}_{j}=1$. Then, the average value μ

_{ij}and standard derivation σ

_{ij}in the normal distribution N(μ

_{ij}, σ

_{ij}) are obtained by the statistical analysis of data corresponding to the alternative P

_{i}(i = 1, 2, …, m) over the attribute R

_{j}(j = 1, 2, …, n), while the evaluation values of SVNNs corresponding to the alternative P

_{i}(i = 1, 2, …, m) over the attribute R

_{j}(j = 1, 2, …, n) are given by decision-makers. Based on the obtained NNNs p

_{ij}= <N(μ

_{ij}, σ

_{ij}), (t

_{ij}, u

_{ij}, v

_{ij})> (i = 1, 2, …, m; j = 1, 2, …, n), we can yield the normal neutrosophic decision matrix M(p

_{ij})

_{m}

_{×}

_{n}:

_{i}(i = 1, 2, …, m) and the ideal solution P

^{*}by using Equation (7) or Equation (8) and obtain the values of ρ

_{1w}(P

_{i}, P

^{*}) or ρ

_{2w}(P

_{i}, P

^{*}) (i = 1, 2, …, m).

_{1w}(P

_{i}, P

^{*}) or ρ

_{2w}(P

_{i}, P

^{*}).

## 5. Illustrative Example

_{1}is a car company; (2) P

_{2}is a food company; (3) P

_{3}is a computer company; (4) P

_{4}is an arms company. In the decision-making process, the four possible alternatives must satisfy the requirements of the three attributes: (1) R

_{1}is the risk; (2) R

_{2}is the growth; (3) R

_{3}is the environment, where the attributes R

_{1}and R

_{2}are benefit types and the attribute R

_{3}is a cost type. Assume that the weighting vector of the attributes is given by

**w**= (0.35, 0.25, 0.4). By the statistical analysis and the evaluation of investment data regarding the four possible alternatives of P

_{i}(i = 1, 2, 3, 4) over the three attributes of R

_{j}(j = 1, 2, 3), we can establish the following NNN decision matrix [18]:

_{1}and the ideal solution P

^{*}by using Equation (7) as follows:

_{i}(i = 2, 3, 4) and the ideal solution P

^{*}can be given as the following values of ρ

_{1w}(P

_{i}, P

^{*}) (i = 2, 3, 4):

_{1w}(P

_{2}, P

^{*}) = 0.9891, ρ

_{1w}(P

_{3}, P*) = 0.9169, and ρ

_{1w}(P

_{4}, P*) = 0.9875.

_{1w}(P

_{i}, P*) (i = 1, 2, 3, 4), the ranking order of the alternatives is P

_{2}> P

_{4}> P

_{3}> P

_{1}and the best one is P

_{2}. These results are the same as in [18].

_{1}and the ideal solution P* by using Equation (8) as follows:

_{i}(i = 2, 3, 4) and the ideal solution P

^{*}can be given as the following values of ρ

_{2w}(P

_{i}, P

^{*}) (i = 2, 3, 4):

_{2w}(P

_{2}, P

^{*}) = 0.9151, ρ

_{2w}(P

_{3}, P

^{*}) = 0.6575, and ρ

_{2w}(P

_{4}, P

^{*}) = 0.9522.

_{2w}(P

_{i}, P

^{*}) (i = 1, 2, 3, 4), the ranking order of the alternatives is P

_{4}> P

_{2}> P

_{1}> P

_{3}, and the best one is P

_{4}. These results also are the same as in [18].

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Ye, J.
Multiple Attribute Decision-Making Method Using Correlation Coefficients of Normal Neutrosophic Sets. *Symmetry* **2017**, *9*, 80.
https://doi.org/10.3390/sym9060080

**AMA Style**

Ye J.
Multiple Attribute Decision-Making Method Using Correlation Coefficients of Normal Neutrosophic Sets. *Symmetry*. 2017; 9(6):80.
https://doi.org/10.3390/sym9060080

**Chicago/Turabian Style**

Ye, Jun.
2017. "Multiple Attribute Decision-Making Method Using Correlation Coefficients of Normal Neutrosophic Sets" *Symmetry* 9, no. 6: 80.
https://doi.org/10.3390/sym9060080