# Symmetry Measures of Simplified Neutrosophic Sets for Multiple Attribute Decision-Making Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−}0, 1

^{+}[, the nonstandard interval shows its difficult application in the real world. As the subclass of NS, Ye [2] presented a simplified neutrosophic set (SNS), where its indeterminacy-membership, truth-membership, and falsity-membership functions are in the real standard interval [0, 1] to conveniently apply in engineering fields. SNS includes an interval neutrosophic set (INS) [3] and a single-valued neutrosophic set (SVNS) [4]. After that, Ye [5] introduced three simplified neutrosophic similarity measures in vector space and their multicriteria decision-making methods. Then, outranking approaches [6,7] were used for simplified neutrosophic and interval neutrosophic decision-making problems. Some researchers proposed correlation coefficients, cross entropy measures, similarity measures for INSs/ SVNSs/SNSs, and their multiple attribute decision-making (MADM) methods [8,9,10,11,12]. Some researchers presented various aggregation operators of SVNSs/INSs/SNSs for decision-making fields [13,14,15,16,17,18,19]. Furthermore, projection and bidirectional projection measures of INSs and SVNSs [20,21] were introduced for their decision-making. TOPSIS method [22] was presented for decision-making with SVNS information, also SVNS graphs [23] were used for decision-making problems. Then, some decision making methods were presented based on the neutrosophic MULTIMOORA, WASPAS-SVNS, and extended TOPSIS and VIKOR methods [24,25,26] under SNS environments.

## 2. Asymmetry Measures of Simplified Neutrosophic Sets

**Definition**

**1**

**.**A SNS is defined as $A=\{\langle x,{u}_{A}(x),{v}_{A}(x),{h}_{A}(x)\rangle |x\in U\}$ in the universe of discourse U, such that u

_{A}(x): U → [0, 1], v

_{A}(x): U → [0,1], and ${h}_{A}(x)$ : U → [0, 1], which are described by the truth, indeterminacy and falsity-membership degrees, satisfying 0 ≤ sup u

_{A}(x) + sup v

_{A}(x) + sup h

_{A}(x) ≤ 3 for INS or 0 ≤ u

_{A}(x) + v

_{A}(x) + h

_{A}(x) ≤ 3 for SVNS and x ∈ U.

_{a}, v

_{a}, h

_{a}), which is called the simplified neutrosophic number (SNN), including a single valued neutrosophic number (SVNN) and an interval neutrosophic number (INN).

**Definition**

**2.**

_{1}, b

_{2}, …, b

_{n}} and A = {a

_{1}, a

_{2}, …, a

_{n}} be two SVNSs, where b

_{j}= (u

_{bj}, v

_{bj}, h

_{bj}) and a

_{j}= (u

_{aj}, v

_{aj}, h

_{aj}) are the j-th SVNNs (j = 1, 2, …, n) of B and A respectively. Then

_{j}or a

_{j}(j = 1, 2, …, n), the weighted asymmetry measure of SNSs can be introduced below.

**Definition**

**3.**

_{1}, b

_{2}, …, b

_{n}} and A = {a

_{1}, a

_{2}, …, a

_{n}} be two SVNSs, where b

_{j}= (u

_{bj}, v

_{bj}, h

_{bj}) and a

_{j}= (u

_{aj}, v

_{aj}, h

_{aj}) are the j-th SVNNs (j = 1,2, …, n) of B and A respectively, and let the weight of an element b

_{j}or a

_{j}be w

_{j}, w

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

**Definition**

**4.**

_{1}, b

_{2}, …, b

_{n}} and A = {a

_{1}, a

_{2}, …, a

_{n}} be two INSs, where${b}_{j}=([{u}_{bj}^{L},{u}_{bj}^{U}],[{v}_{bj}^{L},{v}_{bj}^{U}],[{h}_{bj}^{L},{h}_{bj}^{U}])$and${a}_{j}=([{u}_{aj}^{L},{u}_{aj}^{U}],[{v}_{aj}^{L},{v}_{aj}^{U}],[{h}_{aj}^{L},{h}_{aj}^{U}])$are the j-th INNs (j = 1, 2, …, n) of B and A respectively. Then, two asymmetry measures of B and A are defined as

_{j}or a

_{j}(j = 1, 2, …, n), the weighted asymmetry measures of INSs can be introduced below.

**Definition**

**5.**

_{1}, b

_{2}, …, b

_{n}} and A = {a

_{1}, a

_{2}, …, a

_{n}} be two INSs, where${b}_{j}=([{u}_{bj}^{L},{u}_{bj}^{U}],[{v}_{bj}^{L},{v}_{bj}^{U}],[{h}_{bj}^{L},{h}_{bj}^{U}])$and${a}_{j}=([{u}_{aj}^{L},{u}_{aj}^{U}],[{v}_{aj}^{L},{v}_{aj}^{U}],[{h}_{aj}^{L},{h}_{aj}^{U}])$are the j-th INNs (j = 1, 2, …, n) of B and A respectively, and let the weight of an element a

_{j}or b

_{j}be w

_{j}, w

_{j}∈ [0, 1], and${\sum}_{j=1}^{n}{w}_{j}}=1$. Thus, two weighted asymmetry measures of A on B are defined as

## 3. Normalized Symmetry Measures of Simplified Neutrosophic Sets

**Definition**

**6.**

_{1}, b

_{2}, …, b

_{n}} and A = {a

_{1}, a

_{2}, …, a

_{n}} be two SNSs, where b

_{j}= (u

_{bj}, v

_{bj}, h

_{bj}) and a

_{j}= (u

_{aj}, v

_{aj}, h

_{aj}) are the j-th SNNs (j = 1, 2, …, n) of B and A respectively. Thus

_{j}or a

_{j}(j = 1, 2, …, n).

**Definition**

**7.**

_{1}, b

_{2}, …, b

_{n}} and A = {a

_{1}, a

_{2}, …, a

_{n}} be two SNSs, where b

_{j}= (u

_{bj}, v

_{bj}, h

_{bj}) and a

_{j}= (u

_{aj}, v

_{aj}, h

_{aj}) are the j-th SNNs (j = 1, 2, …, n) of B and A respectively, and let the weight of an element b

_{j}or a

_{j}be w

_{j}, w

_{j}∈ [0, 1], and${\sum}_{j=1}^{n}{w}_{j}}=1$. Thus

## 4. Decision-Making Method Using the Weighted Symmetry Measure

_{1}, S

_{2}, …, S

_{m}} and a set of attributes as A = {A

_{1}, A

_{2}, …, A

_{n}} in a MADM problem. Assume that the weight of the attribute A

_{j}is w

_{j}, w

_{j}∈ [0, 1], and ${\sum}_{j=1}^{n}{w}_{j}}=1$.

#### Decision-Making Method Using the Weighted Symmetry Measure

_{i}(i = 1, 2, …, m) for an attribute A

_{j}(j = 1, 2, …, n) is expressed by an SNS ${S}_{i}=\{{s}_{i1},{s}_{i2},\dots ,{s}_{in}\}$, where s

_{ij}= (u

_{ij}, v

_{ij}, h

_{ij}) satisfies u

_{ij}, v

_{ij}, h

_{ij}∈ [0, 1] and 0 ≤ u

_{ij}+ v

_{ij}+ h

_{ij}≤ 3 for SVNN or u

_{ij}, v

_{ij}, h

_{ij}⊆ [0, 1] and $0\le {u}_{ij}^{U}+{v}_{ij}^{U}+{h}_{ij}^{U}\le 3$ for INN. Thus, the decision matrix of SNSs can be established as D = (s

_{ij})

_{m×n}.

^{*}and S

_{i}(i = 1, 2, …, m) is yielded by

_{w}(S

^{*}, S

_{i}) is, the closer S

_{i}is to S

^{*}, and then the better the alternative S

_{i}is.

## 5. Decision-Making Examples

#### 5.1. Practical Example

_{1}, S

_{2}, S

_{3}, S

_{4}}. They need to satisfy the three attributes: (i) A

_{1}is the improvement in quality; (ii) A

_{2}is the market response; (iii) A

_{3}is the manufacturing cost. In the decision-making problem, the decision maker/expert specifies the weight vector of the attributes as W = (0.36, 0.3, 0.34) corresponding to the importance of the three attributes.

_{i}(i = 1, 2, 3, 4) for an attribute A

_{j}(j = 1, 2, 3) by the evaluation information of SVNNs, and then single-valued neutrosophic decision matrix can be constructed as

^{*}and S

_{i}can be obtained as follows:

_{w}(S

^{*}, S

_{1}) = 0.8945, M

_{w}(S

^{*}, S

_{2}) = 0.9964, M

_{w}(S

^{*}, S

_{3}) = 0. 9717, and M

_{w}(S

^{*}, S

_{4}) = 0.9730

_{w}(S

^{*}, S

_{2}) > M

_{w}(S

^{*}, S

_{4}) > M

_{w}(S

^{*}, S

_{3}) > M

_{w}(S

^{*}, S

_{1}), the four alternatives are ranked as S

_{2}> S

_{4}> S

_{3}> S

_{1}. Obviously, S

_{2}is the best one among the four alternatives.

_{i}(i = 1, 2, 3, 4) for the attributes are expressed by interval neutrosophic information, the interval neutrosophic decision matrix can be constructed as

^{*}and S

_{i}(i = 1, 2, 3, 4) can be obtained as

_{w}(S

^{*}, S

_{1}) = 0.9053, M

_{w}(S

^{*}, S

_{2}) = 0.9423, M

_{w}(S

^{*}, S

_{3}) = 0. 9401, and M

_{w}(S

^{*}, S

_{4}) = 0.9762

_{w}(S

_{4}, S

^{*}) > M

_{w}(S

_{2}, S

^{*}) > M

_{w}(S

_{3}, S

^{*}) > M

_{w}(S

_{1}, S

^{*}), the four alternatives are ranked as S

_{4}> S

_{2}> S

_{3}> S

_{1}. Thus, S

_{4}is the best one among the four alternatives.

_{j}= 1/n = 1/3 for j =1, 2, 3.

^{*}and S

_{i}(i = 1, 2, 3, 4) can be obtained as

_{w}(S

^{*}, S

_{1}) = 0.8933, M

_{w}(S

^{*}, S

_{2}) = 0.9987, M

_{w}(S

^{*}, S

_{3}) = 0.9811, and M

_{w}(S

^{*}, S

_{4}) = 0.9592

^{*}and S

_{i}(i = 1, 2, 3, 4) can be obtained as

_{w}(S

^{*}, S

_{1}) = 0.9207, M

_{w}(S

^{*}, S

_{2}) = 0.9482, M

_{w}(S

^{*}, S

_{3}) = 0.9550, and M

_{w}(S

^{*}, S

_{4}) = 0.9742

_{4}> S

_{3}> S

_{2}> S

_{1}according to the above measure values.

_{2}and S

_{4}in all ranking orders are still identical in these cases.

#### 5.2. Comparative Example with Existing Relative Measures for Single-Valued Neutrosophic Sets

_{1}, S

_{2}, S

_{3}, S

_{4}} needs to satisfy a set of the five attributes A = {A

_{1}, A

_{2}, A

_{3}, A

_{4}, A

_{5}}, where A

_{1}, A

_{2}, A

_{3}, A

_{4}, and A

_{5}are the manufacturing cost, structure complexity, transmission effectiveness, reliability, and maintainability respectively. The SVNS decision matrix of evaluating the four alternatives over the five attributes is adopted from literature [21], which is given as

_{w}(S

^{*}, S

_{1}) = 0.9452, M

_{w}(S

^{*}, S

_{2}) = 0.8390, M

_{w}(S

^{*}, S

_{3}) = 0.8770, and M

_{w}(S

^{*}, S

_{4}) = 0.9803.

_{w}(S

^{*}, S

_{i}) and the various measures like the cosine measures of Cos

_{w}(S

^{*}, S

_{i}) and C

_{w}(S

^{*}, S

_{i}), the Dice measure of D

_{w}(S

^{*}, S

_{i}), the Jaccard measure of J

_{w}(S

^{*}, S

_{i}), the projection measure of Projw

_{s*}(S

_{i}), and the bidirectional projection measure of BProj

_{w}(S

^{*}, S

_{i}) in the literature [21] are shown in Table 1, where the average value (AV) and the standard deviation (SD) of S

_{i}for i = 1, 2, 3, 4 are also given.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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Measure | S_{1} | S_{2} | S_{3} | S_{4} | AV | SD | Ranking Order |
---|---|---|---|---|---|---|---|

Cos_{w}(S^{*}, S_{i}) | 0.9785 | 0.9685 | 0.9870 | 0.9942 | 0.9821 | 0.0096 | S_{4} > S_{3} > S_{1} > S_{2} |

C_{w}(S^{*}, S_{i}) | 0.9798 | 0.9750 | 0.9875 | 0.9929 | 0.9838 | 0.0069 | S_{4} > S_{3} > S_{1} > S_{2} |

D_{w}(S^{*}, S_{i}) | 0.9787 | 0.9696 | 0.9845 | 0.9927 | 0.9814 | 0.0084 | S_{4} > S_{3} > S_{1} > S_{2} |

J_{w}(S^{*}, S_{i}) | 0.9586 | 0.9427 | 0.9694 | 0.9857 | 0.9641 | 0.0157 | S_{4} > S_{3} > S_{1} > S_{2} |

Projw_{s*}(S_{i}) | 0.3933 | 0.3632 | 0.3806 | 0.4158 | 0.3882 | 0.0192 | S_{4} > S_{1} > S_{3} > S_{2} |

BProj_{w}(S^{*}, S_{i}) | 0.9883 | 0.9636 | 0.9728 | 0.9958 | 0.9801 | 0.0126 | S_{4} > S_{1} > S_{3} > S_{2} |

M_{w}(S^{*}, S_{i}) | 0.9452 | 0.8390 | 0.8770 | 0.9803 | 0.9104 | 0.0555 | S_{4} > S_{1} > S_{3} > S_{2} |

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Tu, A.; Ye, J.; Wang, B.
Symmetry Measures of Simplified Neutrosophic Sets for Multiple Attribute Decision-Making Problems. *Symmetry* **2018**, *10*, 144.
https://doi.org/10.3390/sym10050144

**AMA Style**

Tu A, Ye J, Wang B.
Symmetry Measures of Simplified Neutrosophic Sets for Multiple Attribute Decision-Making Problems. *Symmetry*. 2018; 10(5):144.
https://doi.org/10.3390/sym10050144

**Chicago/Turabian Style**

Tu, Angyan, Jun Ye, and Bing Wang.
2018. "Symmetry Measures of Simplified Neutrosophic Sets for Multiple Attribute Decision-Making Problems" *Symmetry* 10, no. 5: 144.
https://doi.org/10.3390/sym10050144