# Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Existing SMs between Simplified Neutrosophic Asymmetry Measures and Insufficiency

_{Y}(z): Z → [0, 1], β

_{Y}(z): Z → [0, 1], and γ

_{Y}(z): Z → [0, 1], which are depicted by the truth, indeterminacy, and falsity membership degrees, with either 0 ≤ sup α

_{Y}(z) + sup β

_{Y}(z) + sup γ

_{Y}(z) ≤ 3 for INS or 0 ≤ α

_{Y}(z) + β

_{Y}(z) + γ

_{Y}(z) ≤ 3 for SVNS and z ∈ Z. Then an element $\langle z,{\alpha}_{Y}(z),{\beta}_{Y}(z),{\gamma}_{Y}(z)\rangle $ in the SNS Y is denoted by the simplified neutrosophic number (SNN) $y=\langle {\alpha}_{y},{\beta}_{y},{\gamma}_{y}\rangle $ for short.

_{1}, x

_{2}, …, x

_{n}} and Y = {y

_{1}, y

_{2}, …, y

_{n}} are two SNSs, where ${x}_{j}=\langle {\alpha}_{xj},{\beta}_{xj},{\gamma}_{xj}\rangle $ and ${y}_{j}=\langle {\alpha}_{yj},{\beta}_{yj},{\gamma}_{yj}\rangle $ is the j-th single-valued neutrosophic numbers (SVNNs) (j = 1, 2, …, n) and ${x}_{j}=\langle [{\alpha}_{xj}^{-},{\alpha}_{xj}^{+}],[{\beta}_{xj}^{-},{\beta}_{xj}^{+}],[{\gamma}_{xj}^{-},{\gamma}_{xj}^{+}]\rangle $ and ${y}_{j}=\langle [{\alpha}_{yj}^{-},{\alpha}_{yj}^{+}],[{\beta}_{yj}^{-},{\beta}_{yj}^{+}],[{\gamma}_{yj}^{-},{\gamma}_{yj}^{+}]\rangle $ is the j-th interval neutrosophic numbers (INNs) (j = 1, 2, …, n). Then asymmetry measures between X and Y are defined as follows [32]:

## 3. Improved Normalized SMs of SNSs

_{1}, x

_{2}, …, x

_{n}} and Y = {y

_{1}, y

_{2}, …, y

_{n}} are the two SNSs, including the SVNNs ${x}_{j}=\langle {\alpha}_{xj},{\beta}_{xj},{\gamma}_{xj}\rangle $ and ${y}_{j}=\langle {\alpha}_{yj},{\beta}_{yj},{\gamma}_{yj}\rangle $ (j = 1, 2, …, n) and the INNs ${x}_{j}=\langle [{\alpha}_{xj}^{-},{\alpha}_{xj}^{+}],[{\beta}_{xj}^{-},{\beta}_{xj}^{+}],[{\gamma}_{xj}^{-},{\gamma}_{xj}^{+}]\rangle $ and ${y}_{j}=\langle [{\alpha}_{yj}^{-},{\alpha}_{yj}^{+}],[{\beta}_{yj}^{-},{\beta}_{yj}^{+}],[{\gamma}_{yj}^{-},{\gamma}_{yj}^{+}]\rangle $ (j = 1, 2, …, n) in X and Y.

_{k}(X, Y) = H

_{k}(Y, X) and 0 ≤ H

_{k}(X, Y) ≤ 1 for k = 1, 2. Then the improved SMs of SVNSs and INSs can overcome the insufficiency of the existing SMs of SVNSs and INSs because the improved SMs do not imply the aforementioned undefined/unmeaningful situation, and also show simpler algorithms in comparison to Equations (6) and (7).

_{j}or y

_{j}(j = 1, 2, …, n) is considered in X and Y by w

_{j}, with w

_{j}∈ [0, 1] and ${\sum}_{j=1}^{n}{w}_{j}}=1$, the improved weighted SM between asymmetry measures of SNSs can be presented by:

_{k}(X, Y) = W

_{k}(Y, X) and 0 ≤ W

_{k}(X, Y) ≤ 1 for k = 1, 2.

## 4. Simplified Neutrosophic Sine Entropy

**Definition**

**1.**

_{1}, y

_{2}, …, y

_{n}}be an SNS, where${y}_{j}=\langle {\alpha}_{yj},{\beta}_{yj},{\gamma}_{yj}\rangle $is the j-th SVNN and${y}_{j}=\langle [{\alpha}_{yj}^{-},{\alpha}_{yj}^{+}],[{\beta}_{yj}^{-},{\beta}_{yj}^{+}],[{\gamma}_{yj}^{-},{\gamma}_{yj}^{+}]\rangle $is the j-th INN (j = 1, 2, …, n). Then the sine entropy measures of Y are defined as follows:

**Theorem**

**1.**

_{j}= <0.5, 0.5, 0.5> or the fuzziest INN be a

_{j}= <[0.5, 0.5], [0.5, 0.5], [0.5, 0.5]> (j = 1, 2, …, n) in the fuzziest SNS A = {a

_{1}, a

_{2}, …, a

_{n}}. Then, the sine entropy measure S

_{Ek}(Y) (k = 1, 2) satisfies the following properties:

- (E1)
- S
_{Ek}(Y) = 0 if Y = {y_{1}, y_{2}, …, y_{n}} is a crisp set, i.e., y_{j}= <1, 0, 0> or y_{j}= <0, 0, 1> for SVNN and y_{j}= <[1, 1], [0, 0], [0, 0]> or y_{j}= <[0, 0], [0, 0], [1, 1]> (j = 1, 2, …, n) for INS; - (E2)
- S
_{Ek}(Y) = 1 if and only if y_{j}= a_{j}(j = 1, 2, …, n); - (E3)
- If the closer an SNS Y is to the fuzziest SNS A than an SNS X, the fuzzier Y is than X, then S
_{Ek}(X) ≤ S_{Ek}(Y); - (E4)
- S
_{Ek}(Y) = S_{Ek}(Y^{c}) if Y^{c}is the complement of Y.

**Proof:**

- (E1)
- For a crisp set Y = {y
_{1}, y_{2}, …, y_{n}}, i.e., y_{j}= <1, 0, 0> or y_{j}= <0, 0, 1> for SVNN and y_{j}= <[1, 1], [0, 0], [0, 0]> or y_{j}= <[0, 0], [0, 0], [1, 1]> (j = 1, 2, …, n) for INN, by use of Equation (14) we obtain the following result:$${S}_{E1}(Y)=\frac{1}{3n}{\displaystyle \sum _{j=1}^{n}[\mathrm{sin}({\alpha}_{j}\pi )+\mathrm{sin}({\beta}_{j}\pi )+\mathrm{sin}({\gamma}_{j}\pi )]}=\frac{n}{3n}[\mathrm{sin}(1\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(0\times \pi )]=0,$$$${S}_{E1}(Y)=\frac{1}{3n}{\displaystyle \sum _{j=1}^{n}[\mathrm{sin}({\alpha}_{j}\pi )+\mathrm{sin}({\beta}_{j}\pi )+\mathrm{sin}({\gamma}_{j}\pi )]}=\frac{n}{3n}[\mathrm{sin}(0\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(1\times \pi )]=0,$$$$\begin{array}{l}{S}_{E2}(Y)=\frac{1}{6n}{\displaystyle \sum _{j=1}^{n}[\mathrm{sin}({\alpha}_{j}^{-}\pi )+\mathrm{sin}({\beta}_{j}^{-}\pi )+\mathrm{sin}({\gamma}_{j}^{-}\pi )+\mathrm{sin}({\alpha}_{j}^{+}\pi )+\mathrm{sin}({\beta}_{j}^{+}\pi )+\mathrm{sin}({\gamma}_{j}^{+}\pi )]}\\ =\frac{n}{6n}[\mathrm{sin}(1\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(1\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(0\times \pi )]=0,\end{array}$$$$\begin{array}{l}{S}_{E2}(Y)=\frac{1}{6n}{\displaystyle \sum _{j=1}^{n}[\mathrm{sin}({\alpha}_{j}^{-}\pi )+\mathrm{sin}({\beta}_{j}^{-}\pi )+\mathrm{sin}({\gamma}_{j}^{-}\pi )+\mathrm{sin}({\alpha}_{j}^{+}\pi )+\mathrm{sin}({\beta}_{j}^{+}\pi )+\mathrm{sin}({\gamma}_{j}^{+}\pi )]}\\ =\frac{n}{6n}[\mathrm{sin}(0\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(1\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(0\times \pi )+\mathrm{sin}(1\times \pi )]=0.\end{array}$$ - (E2)
- Let the sine function be f(z
_{j}) = sin(z_{j}π) for z_{j}∈ [0, 1] (j = 1, 2, …, n). By differentiating f(z_{j}) with respect to z_{j}and equating to zero, there exist the following results:$$\frac{\partial f({z}_{j})}{\partial {z}_{j}}=\pi \mathrm{cos}({z}_{j}\pi ),$$$$\frac{\partial f({z}_{j})}{\partial {z}_{j}}=\pi \mathrm{cos}({z}_{j}\pi )=0.$$

_{j}is z

_{j}= 0.5.

_{j}, we obtain:

_{j}) for z

_{j}∈ [0, 1] implies a concave function with the global maximum f(z

_{j}) = 1 at z

_{j}= 0.5. Then, the sine entropy measures of an SNS Y can be expressed as the following two forms:

_{Ek}(Y) = 1 (k = 1, 2) ⇔ y

_{j}= a

_{j}= < 0.5, 0.5, 0.5> or y

_{j}= a

_{j}= < [0.5, 0.5], [0.5, 0.5], [0.5, 0.5]> (j = 1, 2, …, n).

- (E3)
- According to Equation (16), f(z
_{j}) for z_{j}∈ [0, 1] (j = 1, 2, …, n) is an increasing function if z_{j}< 0.5 and then it is a decreasing function if z_{j}> 0.5.

_{Ek}(X) ≤ S

_{Ek}(Y) (k = 1, 2).

- (E4)
- Since the complement of the SVNN ${y}_{j}=\langle {\alpha}_{yj},{\beta}_{yj},{\gamma}_{yj}\rangle $ in Y is ${y}_{j}^{c}=\langle {\gamma}_{yj},1-{\beta}_{yj},{\alpha}_{yj}\rangle $, i.e., (α
_{yj})^{c}= γ_{yj}and (β_{yj})^{c}= 1 − β_{yj}(j = 1, 2,…, n) and the complement of the INN ${y}_{j}=\langle [{\alpha}_{yj}^{-},{\alpha}_{yj}^{+}],[{\beta}_{yj}^{-},{\beta}_{yj}^{+}],[{\gamma}_{yj}^{-},{\gamma}_{yj}^{+}]\rangle $ in Y is ${y}_{j}^{c}=\langle [{\gamma}_{yj}^{-},{\gamma}_{yj}^{+}],[1-{\beta}_{yj}^{+},1-{\beta}_{yj}^{-}],[{\alpha}_{yj}^{-},{\alpha}_{yj}^{+}]\rangle $, i.e., ${[{\alpha}_{yj}^{-},{\alpha}_{yj}^{+}]}^{c}=[{\gamma}_{yj}^{-},{\gamma}_{yj}^{+}]$ and ${[{\beta}_{yj}^{-},{\beta}_{yj}^{+}]}^{c}=[1-{\beta}_{yj}^{+},1-{\beta}_{yj}^{-}]$ (j = 1, 2,…, n). Then, there is S_{Ek}(Y^{c}) = S_{Ek}(Y) (k = 1, 2) by using Equations (14) and (15).

## 5. Decision-Making Method Using the Improved Weighted SMs of SNSs

_{1}, Y

_{2}, …, Y

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

_{1}, R

_{2}, …, R

_{n}}. Thus, we propose the MCDM method based on the improved weighted SMs of SNSs and the sine entropy weights of SNSs in SVNS and INS setting, which is called the improved MCDM method in the following.

_{j}(j = 1, 2, …, n) over criteria R

_{i}(i = 1, 2, …, m) are represented by an SNS ${Y}_{i}=\{{y}_{i1},{y}_{i2},\dots ,{y}_{in}\}$, where y

_{ij}= <α

_{ij}, β

_{ij}, γ

_{ij}> is an SVNN for α

_{ij}, β

_{ij}, γ

_{ij}∈ [0, 1] and 0 ≤ α

_{ij}+ β

_{ij}+ γ

_{ij}≤ 3 or ${y}_{ij}=\langle [{\alpha}_{ij}^{-},{\alpha}_{ij}^{+}],[{\beta}_{ij}^{-},{\beta}_{ij}^{+}],[{\gamma}_{ij}^{-},{\gamma}_{ij}^{+}]\rangle $ is an INN for α

_{ij}, β

_{ij}, γ

_{ij}⊆ [0, 1] and $0\le {\alpha}_{ij}^{+}+{\beta}_{ij}^{+}+{\gamma}_{ij}^{+}\le 3$. Thus, the decision matrix of SNSs M = (y

_{ij})

_{m}

_{×n}can be established in SVNS or INS setting. Thus, the improved MCDM method is indicated by the following steps:

**Step****1.**- Based on the concept of an ideal solution (alternative), we can determine the ideal solution ${Y}_{}^{*}=\{{y}_{1}^{*},{y}_{2}^{*},\dots ,{y}_{n}^{*}\}$ where ${y}_{j}^{*}=<{\alpha}_{j}^{*},{\beta}_{j}^{*},{\gamma}_{j}^{*}>=<\underset{i}{\mathrm{max}}({\alpha}_{ij}),\underset{i}{\mathrm{min}}({\beta}_{ij}),\underset{i}{\mathrm{min}}({\gamma}_{ij})>$ is an ideal SVNN or ${y}_{j}^{*}=<{\alpha}_{j}^{*},{\beta}_{j}^{*},{\gamma}_{j}^{*}>=<[\underset{i}{\mathrm{max}}({\alpha}_{ij}^{-}),\underset{i}{\mathrm{max}}({\alpha}_{ij}^{+})],[\underset{i}{\mathrm{min}}({\beta}_{ij}^{-}),\underset{i}{\mathrm{min}}({\beta}_{ij}^{+})],[\underset{i}{\mathrm{min}}({\gamma}_{ij}^{-}),\underset{i}{\mathrm{min}}({\gamma}_{ij}^{+})>$ is an ideal INN (j = 1, 2, …, n; i = 1, 2, …, m).
**Step****2.**- Since the fuzziness/uncertainty of a criterion evaluation increases, the criterion weight should decrease. So, based on the sine entropy measure formula Equation (14) or (15) we can calculate unknown weights of each criterion by the following sine entropy weight model:$${w}_{j}=\frac{1-{S}_{Ek}({y}_{ij})}{n-{\displaystyle {\sum}_{j=1}^{n}{S}_{Ek}({y}_{ij})}},$$
**Step****3.**- By use of Equation (12) for SVNSs or Equation (13) for INSs, the improved weighted SM between Y
_{i}(i = 1, 2, …, m) and Y* is given by:$${W}_{1}({Y}_{i},{Y}^{*})=1-\frac{\left|{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}({\alpha}_{ij}^{2}+{\beta}_{ij}^{2}+{\gamma}_{ij}^{2})}-{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}[{({\alpha}_{j}^{*})}^{2}+{({\beta}_{j}^{*})}^{2}+{({\gamma}_{j}^{*})}^{2}]}\right|}{{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}({\alpha}_{ij}^{2}+{\beta}_{ij}^{2}+{\gamma}_{ij}^{2})}+{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}[{({\alpha}_{j}^{*})}^{2}+{({\beta}_{j}^{*})}^{2}+{({\gamma}_{j}^{*})}^{2}]}},$$$${W}_{2}({Y}_{i},{Y}^{*})=1-\frac{\left|\begin{array}{l}{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}[{({\alpha}_{ij}^{-})}^{2}+{({\alpha}_{ij}^{+})}^{2}+{({\beta}_{ij}^{-})}^{2}+{({\beta}_{ij}^{+})}^{2}+{({\gamma}_{ij}^{-})}^{2}+{({\gamma}_{ij}^{+})}^{2}]}\\ -{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}[{({\alpha}_{j}^{*-})}^{2}+{({\alpha}_{j}^{*+})}^{2}+{({\beta}_{j}^{*-})}^{2}+{({\beta}_{j}^{*+})}^{2}+{({\gamma}_{j}^{*-})}^{2}+{({\gamma}_{j}^{*+})}^{2}]}\end{array}\right|}{\left\{\begin{array}{l}{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}[{({\alpha}_{ij}^{-})}^{2}+{({\alpha}_{ij}^{+})}^{2}+{({\beta}_{ij}^{-})}^{2}+{({\beta}_{ij}^{+})}^{2}+{({\gamma}_{ij}^{-})}^{2}+{({\gamma}_{ij}^{+})}^{2}]}\\ +{\displaystyle \sum _{j=1}^{n}{w}_{j}^{2}[{({\alpha}_{j}^{*-})}^{2}+{({\alpha}_{j}^{*+})}^{2}+{({\beta}_{j}^{*-})}^{2}+{({\beta}_{j}^{*+})}^{2}+{({\gamma}_{j}^{*-})}^{2}+{({\gamma}_{j}^{*+})}^{2}]}\end{array}\right\}}.$$ **Step****4.**- According to the improved weighted SM values of W
_{1}(Y_{i}, Y*) or W_{2}(Y_{i}, Y*), we can rank alternatives and choose the best one. **Step****5.**- End.

## 6. Actual Decision Examples and Comparative Analysis

#### 6.1. Actual Decision Example

_{1}, Y

_{2}, Y

_{3}, Y

_{4}}. Then, decision-makers should select the best one, which must satisfy the requirements of the three criteria: the improvement of quality (R

_{1}), the market response (R

_{2}), and the manufacturing cost (R

_{3}). However, the criteria weights are unknown in this decision-making situation.

_{i}(i = 1, 2, 3, 4) over the criteria R

_{j}(j = 1, 2, 3) by the evaluation information of SVNSs, which can be established as the following decision matrix of SVNSs:

**Step****1.**- By ${y}_{j}^{*}=<{\alpha}_{j}^{*},{\beta}_{j}^{*},{\gamma}_{j}^{*}>=<\underset{i}{\mathrm{max}}({\alpha}_{ij}),\underset{i}{\mathrm{min}}({\beta}_{ij}),\underset{i}{\mathrm{min}}({\gamma}_{ij})>$ (j = 1, 2, 3; i = 1, 2, 3, 4), the ideal solution (ideal alternative) of SVNSs is given as:$${Y}_{}^{*}=\{{y}_{1}^{*},{y}_{2}^{*},{y}_{3}^{*}\}=\{<0.8,0.1,0.1>,<0.8,0.15,0.1>,<0.75,0.2,0.1>\}.$$
**Step****2.**- By Equation (18), the criteria weight vector is obtained as follows:W = (w
_{1}, w_{2}, w_{3}) = (0.5682, 0.2952, 0.1366). **Step****3.**- By Equation (19), the improved weighted SM values between Y
_{i}(i = 1, 2, 3, 4) and Y* can be yielded as follows:W_{1}(Y_{1}, Y*) = 0.9757, W_{1}(Y_{2}, Y*) = 0.9977, W_{1}(Y_{3}, Y*) = 0. 9397, and W_{1}(Y_{4}, Y*) = 0.9989. **Step****4.**- The four alternatives are ranked by Y
_{4}> Y_{2}> Y_{1}> Y_{3}since the SM values are W_{1}(Y_{4}, Y*) > W_{1}(S_{2}, S*) > W_{1}(S_{1}, S*) > W_{1}(Y_{3}, Y*). It is obvious that Y_{4}is the best scheme.

**Step****1.**- By ${y}_{j}^{*}=<{\alpha}_{j}^{*},{\beta}_{j}^{*},{\gamma}_{j}^{*}>=<[\underset{i}{\mathrm{max}}({\alpha}_{ij}^{-}),\underset{i}{\mathrm{max}}({\alpha}_{ij}^{+})],[\underset{i}{\mathrm{min}}({\beta}_{ij}^{-}),\underset{i}{\mathrm{min}}({\beta}_{ij}^{+})],[\underset{i}{\mathrm{min}}({\gamma}_{ij}^{-}),\underset{i}{\mathrm{min}}({\gamma}_{ij}^{+})>$ for j = 1, 2, 3 and i = 1, 2, 3, 4, we can give the ideal solution of INSs (ideal alternative):$${Y}^{*}=\{{y}_{1}^{*},{y}_{2}^{*},\dots ,{y}_{n}^{*}\}=\{<[0.8,0.9],[0.1,0.2],[0.1,0.2]>,<[0.7,0.8],[0.1,0.2],[0.1,0.2]>,<[0.7,0.8],[0,0.2],[0.1,0.3]>\}$$
**Step****2.**- By Equation (18), the criteria weight vector is obtained as follows:W = (w
_{1}, w_{2}, w_{3}) = (0.4976, 0.3557, 0.1467). **Step****3.**- By Equation (20), the improved weighted SM values between Y
_{i}(i = 1, 2, 3, 4) and Y* can be yielded as the following results:W_{2}(Y_{1}, Y*) = 0.9521, W_{2}(Y_{2}, Y*) = 0.9848, W_{2}(Y_{3}, Y*) = 0. 9145, and W_{2}(Y_{4}, Y*) = 0.9933. **Step****4.**- Since the SM values are W
_{2}(Y_{4}, Y*) > W_{2}(Y_{2}, Y*) > W_{2}(Y_{1}, Y*) > W_{2}(Y_{3}, Y*), the four alternatives are ranked by Y_{4}> Y_{2}> Y_{1}> Y_{3}. Hence, the alternative Y_{4}is the best one.

#### 6.2. Comparative Analysis

- (1)
- The improved SMs of SNSs not only indicate simpler algorithms than the existing SMs of SNSs [32], but also can overcome the insufficiency of the existing SMs of SNSs.
- (2)
- The improved MCDM method based on the sine entropy weight model can handle MCDM problems with unknown criteria weights, while the existing MCDM method [32] can only handle MCDM problems with known criteria weights. Hence, the former is superior to the latter in the MCDM problems.
- (3)
- The objective criteria weights obtained by the sine entropy weight model are more reasonable and more practicable than the known criteria weights/subjective criteria weights given by decision-makers’ preference.
- (4)
- The improved MCDM method based on the sine entropy weight model is simple and effective in simplified neutrosophic MCDM problems with unknown criteria weights.

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Table 1.**Decision results of existing multi-criteria decision-making (MCDM) method [32] and the improved MCDM method.

MCDM Method | SM Value between Y_{i} and Y* in SVNS Setting | SM Value between Y_{i} and Y* in INS Setting | Ranking Order in SVNS Setting | Ranking Order in INS Setting |
---|---|---|---|---|

Existing MCDM with the given weights [32] | 0.8945, 0.9964, 0.9717, 0.9730 | 0.9053, 0.9423, 0.9401, 0.9762 | Y_{2} > Y_{4} > Y_{3} > Y_{1} | Y_{4} > Y_{2} > Y_{3} > Y_{1} |

Improved MCDM method with the given weights | 0.9394, 0.9981, 0.9853, 0.9859 | 0.9472, 0.9691, 0.9675, 0.9877 | Y_{2} > Y_{4} > Y_{3} > Y_{1} | Y_{4} > Y_{2} > Y_{3} > Y_{1} |

Improved MCDM method with the sine entropy weights | 0.9757, 0.9977, 0.9397, 0.9989 | 0.9521, 0.9848, 0.9145, 0.9933 | Y_{4} > Y_{2} > Y_{1} > Y_{3} | Y_{4} > Y_{2} > Y_{1} > Y_{3} |

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## Share and Cite

**MDPI and ACS Style**

Cui, W.; Ye, J.
Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model. *Symmetry* **2018**, *10*, 225.
https://doi.org/10.3390/sym10060225

**AMA Style**

Cui W, Ye J.
Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model. *Symmetry*. 2018; 10(6):225.
https://doi.org/10.3390/sym10060225

**Chicago/Turabian Style**

Cui, Wenhua, and Jun Ye.
2018. "Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model" *Symmetry* 10, no. 6: 225.
https://doi.org/10.3390/sym10060225