# Exponential Entropy for Simplified Neutrosophic Sets and Its Application in Decision Making

^{*}

## Abstract

**:**

## 1. Introduction

^{−}0, 1

^{+}[ intervals. For convenient engineering applications of NSs, some researchers constrained the membership functions of truth, falsity, and indeterminacy in a real standard interval [0, 1], and defined single-valued NSs [19], interval-valued NSs [20], and simplified NSs (containing single-valued and interval-valued NSs) [21] as subclasses of NSs, which generalize intuitionistic FSs and interval-valued intuitionistic FSs. As a generalization of intuitionistic fuzzy entropy and similarity measures, Majumder and Samanta [22] developed some similarity measures and an entropy measure for single-valued NSs. Aydoğdu [23] then proposed entropy and similarity measures of interval-valued NSs, and indicated their relationship. Ye and Du [24] defined distances, similarity measures, and entropy for interval-valued NSs, and indicated their relationship.

## 2. Preliminaries of Simplified Neutrosophic Sets

_{j}, T

_{S}(x

_{j}), I

_{S}(x

_{j}), F

_{S}(x

_{j})>| x

_{j}∊ X} in a universal set, X = {x

_{1}, x

_{2}, …, x

_{n}}, which is described independently by T

_{S}(x

_{j}), I

_{S}(x

_{j}), F

_{S}(x

_{j}) ∊ [0, 1] for a single-valued NS and T

_{S}(x

_{j}) = $[{T}_{S}^{L}({x}_{j}),{T}_{S}^{U}({x}_{j})]$, I

_{S}(x

_{j}) = $[{I}_{S}^{L}({x}_{j}),{I}_{S}^{U}({x}_{j})]$, F

_{S}(x

_{j}) = $[{F}_{S}^{L}({x}_{j}),{F}_{S}^{U}({x}_{j})]$ ⊆ [0, 1] for an interval-valued NS, satisfying the conditions $0\le {T}_{S}^{}({x}_{j})+{I}_{S}^{}({x}_{j})+{F}_{S}^{}({x}_{j})\le 1$ for a single-valued NS and $0\le {T}_{S}^{U}({x}_{j})+{I}_{S}^{U}({x}_{j})+{F}_{S}^{U}({x}_{j})\le 1$ for an interval-valued NS.

_{1}, x

_{2}, …, x

_{n}}, and set N, S ∊ SNS(X) as N = {<x

_{j}, T

_{N}(x

_{j}), I

_{N}(x

_{j}), F

_{N}(x

_{j})>| x

_{j}∊ X} and S = {<x

_{j}, T

_{S}(x

_{j}), I

_{S}(x

_{j}), F

_{S}(x

_{j})>| x

_{j}∊ X}. Then, some operations of simplified NSs can be defined as follows [21,25]:

- N ⊆ S if and only if T
_{N}(x_{j}) ≤ T_{S}(x_{j}), I_{N}(x_{j}) ≥ I_{S}(x_{j}), and F_{N}(x_{j}) ≥ F_{S}(x_{j}) for single-valued NSs, and ${T}_{N}^{L}({x}_{j})\le {T}_{S}^{L}({x}_{j})$, ${T}_{N}^{U}({x}_{j})\le {T}_{S}^{U}({x}_{j})$, ${I}_{N}^{L}({x}_{j})\ge {I}_{S}^{L}({x}_{j})$, ${I}_{N}^{U}({x}_{j})\ge {I}_{S}^{U}({x}_{j})$, ${F}_{N}^{L}({x}_{j})\ge {F}_{S}^{L}({x}_{j})$, and ${F}_{N}^{U}({x}_{j})\ge {F}_{S}^{U}({x}_{j})$ for interval-valued NSs, and x_{j}∊ X; - N = S if and only if N ⊆ S and S ⊆ N;
- S
^{c}= {<x_{j}, F_{S}(x_{j}), 1 − I_{S}(x_{j}), T_{S}(x_{j})>| x_{j}∊ X} for the complement of the single-valued NS, S, and ${S}^{c}=\{<{x}_{j},[{F}_{S}^{L}({x}_{j}),{F}_{S}^{U}({x}_{j})],[1-{I}_{S}^{U}({x}_{j}),1-{I}_{S}^{L}({x}_{j})],[{T}_{S}^{L}({x}_{j}),{T}_{S}^{U}({x}_{j})]>|{x}_{j}\in X\}$ for the complement of the interval-valued NS, S; - $N\cup S=\{<{x}_{j},{T}_{N}^{}({x}_{j})\vee {T}_{S}^{}({x}_{j}),{I}_{N}^{}({x}_{j})\wedge {I}_{S}^{}({x}_{j}),{F}_{N}^{}({x}_{j})\wedge {F}_{S}^{}({x}_{j})>|{x}_{j}\in X\}$ for single-valued NSs, and $N\cup S=\left\{\begin{array}{l}<{x}_{j},[{T}_{N}^{L}({x}_{j})\vee {T}_{S}^{L}({x}_{j}),{T}_{N}^{U}({x}_{j})\vee {T}_{S}^{U}({x}_{j})],\\ [{I}_{N}^{L}({x}_{j})\wedge {I}_{S}^{L}({x}_{j}),{I}_{N}^{U}({x}_{j})\wedge {I}_{S}^{U}({x}_{j})],\\ [{F}_{N}^{L}({x}_{j})\wedge {F}_{S}^{L}({x}_{j}),{F}_{N}^{U}({x}_{j})\wedge {F}_{S}^{U}({x}_{j})]>|{x}_{j}\in X\end{array}\right\}$ for interval-valued NSs;
- $N\cap S=\{<{x}_{j},{T}_{N}^{}({x}_{j})\wedge {T}_{S}^{}({x}_{j}),{I}_{N}^{}({x}_{j})\vee {I}_{S}^{}({x}_{j}),{F}_{N}^{}({x}_{j})\vee {F}_{S}^{}({x}_{j})>|{x}_{j}\in X\}$ for single-valued NSs, and $N\cap S=\left\{\begin{array}{l}<{x}_{j},[{T}_{N}^{L}({x}_{j})\wedge {T}_{S}^{L}({x}_{j}),{T}_{N}^{U}({x}_{j})\wedge {T}_{S}^{U}({x}_{j})],\\ [{I}_{N}^{L}({x}_{j})\vee {I}_{S}^{L}({x}_{j}),{I}_{N}^{U}({x}_{j})\vee {I}_{S}^{U}({x}_{j})],\\ [{F}_{N}^{L}({x}_{j})\vee {F}_{S}^{L}({x}_{j}),{F}_{N}^{U}({x}_{j})\vee {F}_{S}^{U}({x}_{j})]>|{x}_{j}\in X\end{array}\right\}$ for interval-valued NSs;
- $N\oplus S=\left\{\langle {x}_{j},{T}_{N}^{}({x}_{j})+{T}_{S}^{}({x}_{j})-{T}_{N}^{}({x}_{j}){T}_{S}^{}({x}_{j}),{I}_{N}^{}({x}_{j}){I}_{S}^{}({x}_{j}),{F}_{N}^{}({x}_{j}){F}_{S}^{}({x}_{j})\rangle |{x}_{j}\in X\right\}$ for single-valued NSs, and $N\oplus S=\left\{\langle \begin{array}{l}{x}_{j},[{T}_{N}^{L}({x}_{j})+{T}_{S}^{L}({x}_{j})-{T}_{N}^{L}({x}_{j}){T}_{S}^{L}({x}_{j}),{T}_{N}^{U}({x}_{j})+{T}_{S}^{U}({x}_{j})-{T}_{N}^{U}({x}_{j}){T}_{S}^{U}({x}_{j})],\\ [{I}_{N}^{L}({x}_{j}){I}_{S}^{L}({x}_{j}),{I}_{N}^{U}({x}_{j}){I}_{S}^{U}({x}_{j})],[{F}_{N}^{L}({x}_{j}){F}_{S}^{L}({x}_{j}),{F}_{N}^{U}({x}_{j}){F}_{S}^{U}({x}_{j})]\end{array}\rangle |{x}_{j}\in X\right\}$ for interval-valued NSs;
- $N\otimes S=\left\{\langle \begin{array}{l}{x}_{j},{T}_{N}^{}({x}_{j}){T}_{S}^{}({x}_{j}),{I}_{N}^{}({x}_{j})+{I}_{S}^{}({x}_{j})-{I}_{N}^{}({x}_{j}){I}_{S}^{}({x}_{j}),\\ {F}_{N}^{}({x}_{j})+{F}_{S}^{}({x}_{j})-{F}_{N}^{}({x}_{j}){F}_{S}^{}({x}_{j}),\end{array}\rangle |{x}_{j}\in X\right\}$ for single-valued NSs, and $N\otimes S=\left\{\langle \begin{array}{l}{x}_{j},[{T}_{N}^{L}({x}_{j}){T}_{S}^{L}({x}_{j}),{T}_{N}^{U}({x}_{j}){T}_{N}^{U}({x}_{j})],[{I}_{N}^{L}({x}_{j})+{I}_{S}^{L}({x}_{j})-{I}_{N}^{L}({x}_{j}){I}_{S}^{L}({x}_{j}),\\ {I}_{N}^{U}({x}_{j})+{I}_{S}^{U}({x}_{j})-{I}_{N}^{U}({x}_{j}){I}_{S}^{U}({x}_{j})],[{F}_{N}^{L}({x}_{j})+{F}_{S}^{L}({x}_{j})-{F}_{N}^{L}({x}_{j}){F}_{S}^{L}({x}_{j}),\\ {F}_{N}^{U}({x}_{j})+{F}_{S}^{U}({x}_{j})-{F}_{N}^{U}({x}_{j}){F}_{S}^{U}({x}_{j})]\end{array}\rangle |{x}_{j}\in X\right\}$ for interval-valued NSs;
- $\delta S=\left\{\langle {x}_{j},1-{\left(1-{T}_{S}^{}({x}_{j})\right)}^{\delta},{I}_{S}^{\delta}({x}_{j}),{F}_{S}^{\delta}({x}_{j})\rangle |{x}_{j}\in X\right\}$ for the single-valued NS, S, and δ > 0, and $\delta S=\left\{\langle \begin{array}{l}{x}_{j},[1-{\left(1-{T}_{S}^{L}({x}_{j})\right)}^{\delta},1-{\left(1-{T}_{S}^{U}({x}_{j})\right)}^{\delta}],\\ [{({I}_{S}^{L}({x}_{j}))}^{\delta},{({I}_{S}^{U}({x}_{j}))}^{\delta}],[{({F}_{S}^{L}({x}_{j}))}^{\delta},{({F}_{S}^{U}({x}_{j}))}^{\delta}]\end{array}\rangle |{x}_{j}\in X\right\}$ for the interval-valued NS, S, and δ > 0;
- ${S}^{\delta}=\left\{\langle {x}_{j},{T}_{S}^{\delta}({x}_{j}),1-{(1-{I}_{S}^{}({x}_{j}))}^{\delta},1-{(1-{F}_{S}^{}({x}_{j}))}^{\delta}\rangle |{x}_{j}\in X\right\}$ for the single-valued NS, S, and δ > 0, and ${S}^{\delta}=\left\{\langle \begin{array}{l}{x}_{j},[{({T}_{S}^{L}({x}_{j}))}^{\delta},{({T}_{S}^{U}({x}_{j}))}^{\delta}],[1-{(1-{I}_{S}^{L}({x}_{j}))}^{\delta},1-{(1-{I}_{S}^{U}({x}_{j}))}^{\delta}],\\ [1-{(1-{F}_{S}^{L}({x}_{j}))}^{\delta},1-{(1-{F}_{S}^{U}({x}_{j}))}^{\delta}]\end{array}\rangle |{x}_{j}\in X\right\}$ for the interval-valued NS, S, and δ > 0.

## 3. Simplified Neutrosophic Exponential Entropy

_{j}, T

_{A}(x

_{j})| x

_{j}∊ X}, in a universal set, X = {x

_{1}, x

_{2}, …, x

_{n}}, Pal and Pal [3] introduced fuzzy exponential entropy for A:

**Definition**

**1.**

_{j}, T

_{S}(x

_{j}), I

_{S}(x

_{j}), F

_{S}(x

_{j})>| x

_{j}∊ X} be a simplified NS in a universal set X = {x

_{1}, x

_{2},…, x

_{n}}. Then, the SNEE measure of S is defined as the following two forms:

**Theorem**

**1.**

_{k}(S) for k = 1, 2 is the SNEE measure of the simplified NS, S, which satisfies the following properties, (P1)–(P4):

- (P1)
- Y
_{k}(S) = 0 if S is a crisp set; - (P2)
- Y
_{k}(S) = 1 if and only if S = A = {<x_{j}, [0.5, 0.5], [0.5, 0.5], [0.5, 0.5]>|x_{j}∊ X} for interval-valued NSs, or S = A = {<x_{j}, 0.5, 0.5, 0.5>|x_{j}∊ X} for single-valued NSs; - (P3)
- If the closer a simplified NS, S, is to A than P, the fuzzier S is than P, then Y
_{k}(P) ≤ Y_{k}(S) for P ∊ SNS(X); - (P4)
- Y
_{k}(S) = Y_{k}(S^{c}) if S^{c}is the complement of S.

**Proof.**

_{j}, 1, 0, 0>|x

_{j}∊ X} or S = {<x

_{j}, 0, 0, 1>|x

_{j}∊ X} for a single-valued NS, S, ∊ SNS(X) and x

_{j}∊ X, and S = {<x

_{j}, [1, 1], [0, 0], [0, 0]>|x

_{j}∊ X} or S = {<x

_{j}, [0, 0], [0, 0], [1, 1]>|x

_{j}∊ X} for an interval-valued NS, S, ∊ SNS(X) and x

_{j}∊ X, by using Equation (1) for S = {<x

_{j}, 1, 0, 0>|x

_{j}∊ X} or S = {<x

_{j}, 0, 0, 1>|x

_{j}∊ X}, we have the following results:

_{j}, [1, 1], [0, 0], [0, 0]>|x

_{j}∊ X} or S = {<x

_{j}, [0, 0], [0, 0], [1, 1]>|x

_{j}∊ X}, we have the following results:

_{S}(x

_{j}) and equating to zero, we can obtain the following results:

_{S}(x

_{j}) is z

_{S}(x

_{j}) = 0.5 for x

_{j}∊ X.

_{S}(x

_{j}), we get

_{S}(x

_{j})) is a concave function and has the global maximum f(z

_{S}(x

_{j})) = 1 at z

_{S}(x

_{j}) = 0.5. Thus, the SNEE of a simplified NS, S, can be written as the following forms:

_{k}(S) = 1 (k = 1, 2) ⇔ S = A = {<x

_{j}, [0.5, 0.5], [0.5, 0.5], [0.5, 0.5]>|x

_{j}∊ X} for the interval-valued NSs, or S = A = {<x

_{j}, 0.5, 0.5, 0.5>|x

_{j}∊ X} for the single-valued NSs.

_{S}(x

_{j}) ∊ [0, 1], f(z

_{S}(x

_{j})) is increasing when z

_{S}(x

_{j}) < 0.5, while f(z

_{S}(x

_{j})) is decreasing when z

_{S}(x

_{j}) > 0.5.

_{k}(P) ≤ Y

_{k}(S) (k = 1, 2) for S, P ∊ SNS(X), and x

_{j}∊ X.

_{j}, T

_{S}(x

_{j}), I

_{S}(x

_{j}), F

_{S}(x

_{j})>| x

_{j}∊ X}, is S

^{c}= {<x

_{j}, F

_{S}(x

_{j}), 1 – I

_{S}(x

_{j}), T

_{S}(x

_{j})>| x

_{j}∊ X}, i.e., (T

_{S}(x

_{j}))

^{c}= F

_{S}(x

_{j}) and (I

_{S}(x

_{j}))

^{c}= 1 – I

_{S}(x

_{j}) for x

_{j}∊ X and j = 1, 2,…, n, and the complement of the interval-valued NS $S=\{<{x}_{j},[{T}_{S}^{L}({x}_{j}),{T}_{S}^{U}({x}_{j})],[{I}_{S}^{L}({x}_{j}),{I}_{S}^{U}({x}_{j})],[{F}_{S}^{L}({x}_{j}),{F}_{S}^{U}({x}_{j})]>|{x}_{j}\in X\}$ is ${S}^{c}=\{<{x}_{j},[{F}_{S}^{L}({x}_{j}),{F}_{S}^{U}({x}_{j})],[1-{I}_{S}^{U}({x}_{j}),1-{I}_{S}^{L}({x}_{j})],[{T}_{S}^{L}({x}_{j}),{T}_{S}^{U}({x}_{j})]>|{x}_{j}\in X\}$ (i.e., ${[{T}_{S}^{L}({x}_{j}),{T}_{S}^{U}({x}_{j})]}^{c}=[{F}_{S}^{L}({x}_{j}),{F}_{S}^{U}({x}_{j})]$ and ${[{I}_{S}^{L}({x}_{j}),{I}_{S}^{U}({x}_{j})]}^{c}=[1-{I}_{S}^{U}({x}_{j}),1-{I}_{S}^{L}({x}_{j})]$ for x

_{j}∊ X and j = 1, 2,…, n), then, we have Y

_{k}(S

^{c}) = Y

_{k}(S) (k = 1, 2), by using Equations (1) and (2).

## 4. Comparison with Other Entropy Measures for Interval-Valued NSs

**Example**

**1.**

_{1}, x

_{2}, …, x

_{n}}. Then, we define the interval-valued NS, S

^{n}, for any positive real number, n, as follows:

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}} = {6, 7, 8, 9, 10} as S = {<6, [0.1, 0.2], [0.5, 0.5], [0.6, 0.7]>, <7, [0.3, 0.5], [0.5, 0.5], [0.4, 0.5]>, <8, [0.6, 0.7], [0.5, 0.5], [0.1, 0.2]>, <9, [0.8, 0.9], [0.5, 0.5], [0, 0.1]>, <10, [1, 1], [0.5, 0.5], [0, 0]>}, S may be viewed as “large” in X, based on the characterization of linguistic variables corresponding to these operations [24]: (1) S

^{2}may be considered as “very large”; (2) S

^{3}may be considered as “quite very large”; and (3) S

^{4}may be considered as “very very large”. Thus, these operational results can be given by the interval-valued NSs, which are shown in Table 1.

_{k}(S) > Y

_{k}(S

^{2}) > Y

_{k}(S

^{3}) > Y

_{k}(S

^{4}) (k = 2–8).

_{2}, Y

_{3}, Y

_{4}, Y

_{5}, Y

_{6}, Y

_{8}satisfy the required ranking order above, but the ranking order of Y

_{7}may be unreasonable, since Y

_{7}(S) = 1. Obviously, the proposed exponential entropy of simplified NSs can also conform to the above required ranking order, demonstrating its rationality and effectiveness.

## 5. Decision-Making Example Based on Entropy Measures of Interval-Valued NSs and Comparison

_{1}), a food company (B

_{2}), a computer company (B

_{3}), and an arms company (B

_{4}). According to the set of required attributes, C = {C

_{1}, C

_{2}, C

_{3}}, where C

_{1}, C

_{2}, and C

_{3}denote the risk, the growth, and the environmental impact, respectively, the investment company made a decision from the set of four alternatives, B = {B

_{1}, B

_{2}, B

_{3}, B

_{4}}. The four alternatives (projects), according to the three attributes, were only evaluated by the form of interval-valued NSs, and then all evaluation values could be constructed as the following interval-valued NS decision matrix [24]:

_{2}, was B

_{4}⥼ B

_{2}⥼ B

_{1}⥼ B

_{3}, corresponding to an ascending order of entropy values of Y

_{2}, where the symbol “⥼” means “superior to”. Obviously, the alternative, B

_{4}, was the best project, since the alternative with the lowest entropy value is considered as the best one.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Zadeh, L.A. Probability measure of fuzzy events. J. Math. Anal. Appl.
**1968**, 23, 421–427. [Google Scholar] [CrossRef] - De Luca, A.S.; Termini, S. A definition of nonprobabilistic entropy in the setting of fuzzy set theory. Inf. Control
**1972**, 20, 301–312. [Google Scholar] [CrossRef] - Pal, N.R.; Pal, S.K. Object background segmentation using new definitions of entropy. IEEE Proc.
**1989**, 366, 284–295. [Google Scholar] [CrossRef] - Verma, R.; Sharma, B.D. On generalized exponential fuzzy entropy. Int. J. Math. Comput. Sci.
**2011**, 5, 1895–1898. [Google Scholar] - Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K.; Gargov, G. Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Bustince, H.; Burillo, P. Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst.
**1996**, 78, 305–316. [Google Scholar] - Szmidt, E.; Kacprzyk, J. Entropy on intuitionistic fuzzy sets. Fuzzy Sets Syst.
**2001**, 118, 467–477. [Google Scholar] [CrossRef] - Valchos, I.K.; Sergiadis, G.D. Intuitionistic fuzzy information—A pattern recognition. Pattern Recognit. Lett.
**2007**, 28, 197–206. [Google Scholar] [CrossRef] - Zhang, Q.S.; Jiang, S.Y. A note on information entropy measure for vague sets. Inf. Sci.
**2008**, 178, 4184–4191. [Google Scholar] [CrossRef] - Ye, J. Two effective measures of intuitionistic fuzzy entropy. Computing
**2010**, 87, 55–62. [Google Scholar] [CrossRef] - Verma, R.; Sharma, B.D. Exponential entropy on intuitionistic fuzzy sets. Kybernetika
**2013**, 49, 114–127. [Google Scholar] - Verma, R.; Sharma, B.D. On intuitionistic fuzzy entropy of order-alpha. Adv. Fuzzy Syst.
**2014**, 14, 1–8. [Google Scholar] - Verma, R.; Sharma, B.D. R-norm entropy on intuitionistic fuzzy sets. J. Intell. Fuzzy Syst.
**2015**, 28, 327–335. [Google Scholar] - Ye, J. Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets. Appl. Math. Model.
**2010**, 34, 3864–3870. [Google Scholar] [CrossRef] - Wei, C.P.; Wang, P.; Zhang, Y.Z. Entropy, similarity measure of interval valued intuitionistic sets and their applications. Inf. Sci.
**2011**, 181, 4273–4286. [Google Scholar] [CrossRef] - Zhang, Q.S.; Xing, H.Y.; Liu, F.C.; Ye, J.; Tang, P. Some new entropy measures for interval-valued intuitionistic fuzzy sets based on distances and their relationships with similarity and inclusion measures. Inf. Sci.
**2014**, 283, 55–69. [Google Scholar] [CrossRef] - Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic; American Research Press: Rehoboth, DE, USA, 1998. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. [Google Scholar]
- Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct.
**2010**, 4, 410–413. [Google Scholar] - Ye, J. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst.
**2014**, 26, 2459–2466. [Google Scholar] - Majumder, P.; Samanta, S.K. On similarity and entropy of neutrosophic sets. J. Intell. Fuzzy Syst.
**2014**, 26, 1245–1252. [Google Scholar] - Aydoğdu, A. On entropy and similarity measure of interval valued neutrosophic sets. Neutrosophic Sets Syst.
**2015**, 9, 47–49. [Google Scholar] - Ye, J.; Du, S.G. Some distances, similarity and entropy measures for interval-valued neutrosophic sets and their relationship. Int. J. Mach. Learn. Cybern.
**2017**, 1–9. [Google Scholar] [CrossRef] - Peng, J.J.; Wang, J.Q.; Wang, J.; Zhang, H.Y.; Chen, X.H. Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int. J. Syst. Sci.
**2016**, 47, 2342–2358. [Google Scholar] [CrossRef]

**Table 1.**Operational results of the interval-valued neutrosophic set (NS), S

^{n}, for n = 1, 2, 3, 4.

S^{n} | x_{1} = 6 | x_{2} = 7 | x_{3} = 8 | x_{4} = 9 | x_{5} = 10 |
---|---|---|---|---|---|

S | <x_{1}, [0.1, 0.2], [0.5, 0.5], [0.6, 0.7]> | {x_{2}, [0.3, 0.5], [0.5, 0.5], [0.4, 0.5]> | <x_{3}, [0.6, 0.7], [0.5, 0.5], [0.1, 0.2]> | <x_{4}, [0.8, 0.9], [0.5, 0.5], [0, 0.1]> | <x_{5}, [1, 1], [0.5, 0.5], [0, 0]> |

S^{2} | <6, [0.01, 0.04], [0.75, 0.75], [0.84, 0.91]> | <7, [0.09, 0.25], [0.75, 0.75], [0.64, 0.75]> | <8, [0.36, 0.49], [0.75, 0.75], [0.19, 0.36]> | <9, [0.64, 0.81], [0.75, 0.75], [0, 0.19]> | <10, [1, 1], [0.75, 0.75], [0, 0]> |

S^{3} | <6, [0.001, 0.008], [0.875, 0.875], [0.936, 0.973]> | <7, [0.027, 0.125], [0.875, 0.875], [0.784, 0.875]> | <8, [0.216, 0.343], [0.875, 0.875], [0.271, 0.488]> | <9, [0.512, 0.729], [0.875, 0.875], [0, 0.271]> | <10, [1, 1], [0.875, 0.875], [0, 0]> |

S^{4} | <6, [0.0001, 0.0016], [0.9375, 0.9375], [0.9744, 0.9919]> | <7, [0.0081, 0.0625], [0.9375, 0.9375], [0.8704, 0.9375]> | <8, [0.1296, 0.2401], [0.9375, 0.9375], [0.3439, 0.5904]> | <9, [0.4096, 0.6561], [0.9375, 0.9375], [0, 0.3439]> | <10, [1, 1], [0.9375, 0.9375], [0, 0]> |

Y_{k} | S | S^{2} | S^{3} | S^{4} |
---|---|---|---|---|

Y_{2} | 0.6954 | 0.5704 | 0.4189 | 0.3139 |

Y_{3} [24] | 0.6067 | 0.3927 | 0.2794 | 0.2096 |

Y_{4} [24] | 0.4450 | 0.3397 | 0.2332 | 0.1685 |

Y_{5} [24] | 0.5467 | 0.3240 | 0.2084 | 0.1612 |

Y_{6} [24] | 0.3000 | 0.2160 | 0.1322 | 0.0645 |

Y_{7} [22,24] | 1.0000 | 0.5715 | 0.3581 | 0.2510 |

Y_{8} [23] | 0.3365 | 0.2717 | 0.2662 | 0.2217 |

**Table 3.**Results and ranking orders of both the developed simplified neutrosophic exponential entropy (SNEE) measure and the existing various entropy measures of interval-valued NSs.

B_{1} | B_{2} | B_{3} | B_{4} | Ranking Order | |
---|---|---|---|---|---|

Y_{2} | 0.8165 | 0.7406 | 0.8429 | 0.3818 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

Y_{3} [24] | 0.6333 | 0.5333 | 0.6556 | 0.4444 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

Y_{4} [24] | 0.5654 | 0.4836 | 0.5972 | 0.3963 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

Y_{5} [24] | 0.5111 | 0.4222 | 0.5333 | 0.3333 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

Y_{6} [24] | 0.4333 | 0.3000 | 0.4667 | 0.2333 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

Y_{7} [22,24] | 0.4983 | 0.4933 | 0.5500 | 0.3817 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

Y_{8} [23] | 0.5687 | 0.3640 | 0.5728 | 0.3818 | B_{4} ⥼ B_{2} ⥼ B_{1} ⥼ B_{3} |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ye, J.; Cui, W. Exponential Entropy for Simplified Neutrosophic Sets and Its Application in Decision Making. *Entropy* **2018**, *20*, 357.
https://doi.org/10.3390/e20050357

**AMA Style**

Ye J, Cui W. Exponential Entropy for Simplified Neutrosophic Sets and Its Application in Decision Making. *Entropy*. 2018; 20(5):357.
https://doi.org/10.3390/e20050357

**Chicago/Turabian Style**

Ye, Jun, and Wenhua Cui. 2018. "Exponential Entropy for Simplified Neutrosophic Sets and Its Application in Decision Making" *Entropy* 20, no. 5: 357.
https://doi.org/10.3390/e20050357