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

Entropy is one of many important mathematical tools for measuring uncertain/fuzzy information. As a subclass of neutrosophic sets (NSs), simplified NSs (including single-valued and interval-valued NSs) can describe incomplete, indeterminate, and inconsistent information. Based on the concept of fuzzy exponential entropy for fuzzy sets, this work proposes exponential entropy measures of simplified NSs (named simplified neutrosophic exponential entropy (SNEE) measures), including single-valued and interval-valued neutrosophic exponential entropy measures, and investigates their properties. Then, the proposed exponential entropy measures of simplified NSs are compared with existing related entropy measures of interval-valued NSs to illustrate the rationality and effectiveness of the proposed SNEE measures through a numerical example. Finally, the developed exponential entropy measures for simplified NSs are applied to a multi-attribute decision-making example in an interval-valued NS setting to demonstrate the application of the proposed SNEE measures. However, the SNEE measures not only enrich the theory of simplified neutrosophic entropy, but also provide a novel way of measuring uncertain information in a simplified NS setting.


Introduction
Entropy is an important mathematical tool for measuring the fuzziness of a fuzzy event. Zadeh [1] first presented the entropy of a fuzzy event, based on a probabilistic framework, in 1968 to measure the fuzziness of a fuzzy event. Then, De Luca and Termini [2] defined the entropy of a fuzzy set (FS) based on Shannon's function, and formulated the axioms of a fuzzy entropy measure. Pal and Pal [3] proposed fuzzy exponential entropy measures for fuzzy sets. Later, Verma and Sharma [4] presented a generalized exponential fuzzy entropy measure. Since intuitionistic FSs [5] and interval-valued intuitionistic FSs [6] are extension forms of FSs, Bustince and Burrillo [7] introduced intuitionistic fuzzy and interval-valued fuzzy entropy measures as an extension of fuzzy entropy. Szmidt and Kacprzyk [8] proposed intuitionistic fuzzy entropy based on an extension of De Luca and Termini's [2] axioms of fuzzy entropy. Subsequently, Vlachos and Sergiagis [9] presented an intuitionistic fuzzy entropy measure and indicated an intuitive and mathematical connection between the notions of fuzzy entropy and intuitionistic fuzzy entropy. Zhang and Jiang [10] put forward an entropy measure of vague sets (intuitionistic FSs) as an extension of logarithmic fuzzy entropy [2]. Furthermore, Ye [11] proposed cosine and sine entropy measures of intuitionistic FSs. As a generalization of fuzzy exponential entropy in Reference [3], Verma and Sharma [12] introduced an intuitionistic fuzzy exponential entropy measure based on the concept of fuzzy entropy. Thereafter, Verma and Sharma [13,14] introduced intuitionistic fuzzy entropy measures of order-alpha and R-norm. Ye [15] presented an

Preliminaries of Simplified Neutrosophic Sets
In 2014, Ye [21] presented the concept of simplified NSs, including single-valued and interval-valued NSs, as a subclass of NSs, in order to be conveniently applied in science and engineering fields. A simplified NS, S, is defined as S = {<x 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 ), for an interval-valued NS. Let SNS(X) denote the family of all simplified NSs in the universe set, X = {x 1 , x 2 , . . . , x n }, and set N, Then, some operations of simplified NSs can be defined as follows [21,25]: for interval-valued NSs, and x j ∈ X; 2. N = S if and only if N ⊆ S and S ⊆ N; for interval-valued NSs; for the single-valued NS, S, and δ > 0, and |x j ∈ X for the interval-valued NS, S, and δ > 0; 9.
) δ |x j ∈ X for the single-valued NS, S, and for the interval-valued NS, S, and δ > 0.

Simplified Neutrosophic Exponential Entropy
For a fuzzy set, A = {x 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: Based on the extension of fuzzy exponential entropy, we can propose SNEE for a simplified NS.
Definition 1. Let S = {<x 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: Corresponding to an axiomatic definition of an entropy measure for interval-valued NSs introduced by Ye and Du [3], the exponential entropy measure of a simplified NS can be given by the following theorem.
Theorem 1. For any S ∈ SNS(X), Y k (S) for k = 1, 2 is the SNEE measure of the simplified NS, S, which satisfies the following properties, (P1)-(P4): (P3) If the closer a simplified NS, S, is to A than P, the fuzzier S is than P, then we have the following results: and then by using Equation (2) we have the following results: (P2) Let us consider the following function: Differentiating Equation (3) with respect to z S (x j ) and equating to zero, we can obtain the following results: Further differentiating Equation (4) with respect to z S (x j ), we get Thus, we can find ∂ 2 f (z S (x j )) ∂(z S (x j )) 2 < 0 for z S (x j ) = 0.5 and x j X.
Therefore, f (z 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: (P3) Based on Equation (4), there exists the following result: For any z 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.
Obviously, the closer S is to A than P, the fuzzier S is than P, and then Y k (P) ≤ Y k (S) (k = 1, 2) for S, P ∈ SNS(X), and x j ∈ X.
This proves the theorem.
It is worth noting that a single-valued NS is a special case of an interval-valued NS if there In this case, Equation (2) is reduced to Equation (1).

Comparison with Other Entropy Measures for Interval-Valued NSs
Since a single-valued NS is a special case of an interval-valued NS corresponding to the equality of the two endpoints of the truth, indeterminacy, and falsity intervals in an interval-valued NS, this section only adopts an example from the literature [24] in order to compare our proposed exponential entropy measures of simplified NSs with the entropy measures of interval-valued NSs introduced by Ye and Du [24].
For convenience in our comparisons, we introduce various entropy measures of the interval-valued NS, S, given in previous articles [22][23][24] as follows: ] |x j ∈ X be an interval-valued NS in the finite universe set X = {x 1 , x 2 , . . . , x n }. Then, we define the interval-valued NS, S n , for any positive real number, n, as follows: When we consider the interval-valued NS, S, in the finite universe set, X = {x 1 , x 2 , x 3 , x 4 [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. Table 1. Operational results of the interval-valued neutrosophic set (NS), S n , for n = 1, 2, 3, 4. From logical consideration and human intuition, the entropy measures of these interval-valued NSs are required to satisfy the ranking order Y k (S) > Y k (S 2 ) > Y k (S 3 ) > Y k (S 4 ) (k = 2-8).
Hence, these entropy measures are calculated by Equations (2) and (9)- (14) for these cases, and all the entropy measure values are given in Table 2. In Table 2, all entropy measure values of Y 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.

Decision-Making Example Based on Entropy Measures of Interval-Valued NSs and Comparison
In this section, the proposed exponential entropy measure of simplified NSs is applied to multiple-attribute decision-making problems. In the decision-making process, the decision maker can imply his/her provision of more useful information from the alternative if the exponential entropy value of some alternative across all attributes is smaller. Since the alternative, along with the smallest entropy value, can be considered as a significant preference and priority, we can determine the priority ranking of all alternatives, and the best one(s) corresponding to the exponential entropy values of simplified NSs (single-valued and interval-valued NSs).
For convenience in our comparisons, a decision-making example was adopted from Reference [24], where an investment company wanted to invest a sum of money into the best project. Then, the four potential alternatives were considered for the investment projects of a car company (B 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]: Then, the developed exponential entropy measure of simplified NSs was applied to the decision making problem, with information from an interval-valued NS.
Using Equation (2), we computed the exponential entropy values of interval-valued NSs for the four alternatives, which are shown in Table 3, and then we also calculated the entropy values of interval-valued NSs for the four alternatives through the entropy Equations (9)- (14) given in previous articles [22][23][24] for comparative convenience, as shown in Table 3. In Table 3, the priority ranking order of the four alternatives based on the developed SNEE measure, Y 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.
All priority ranking orders of the four alternatives were identical, after comparing ranking results between the developed SNEE measure and various entropy measures of interval-valued NSs given in previous articles [22][23][24] as shown in Table 3. It is clear that the decision results indicate the effectiveness of the proposed SNEE measure in the decision-making application with information from an interval-valued NS. However, this study presents the SNEE measures, including single-valued and interval-valued neutrosophic exponential entropy measures, while previous articles only gave either single-valued neutrosophic entropy measures [22] or interval-valued neutrosophic entropy measures [23,24]. Obviously, this study not only enriches the theory of simplified neutrosophic entropy, but also provides a novel way of evaluating the entropy measure of simplified neutrosophic information.

Conclusions
This paper firstly proposed the exponential entropy measures of simplified NSs, including single-valued and interval-valued neutrosophic exponential entropy measures, based on the concept of fuzzy exponential entropy. Subsequently, the properties of the exponential entropy measures for simplified NSs were discussed based on an axiomatic definition of an entropy measure taken from Reference [24], and then the proposed SNEE measure was compared with existing related entropy measures of interval-valued NSs through a numerical example, showing its effectiveness. Finally, the proposed SNEE entropy measure was applied to an actual multiple-attribute decision-making example, and then the feasibility and effectiveness of decision making in an interval-valued NS setting was demonstrated through the comparative analysis of the actual decision-making example. However, the proposed SNEE measure not only enriches the theory of simplified neutrosophic entropy, but also provides a novel way of measuring uncertain information in a simplified NS setting. In the future, we shall extend the proposed SNEE measure to propose a neutrosophic cubic exponential entropy measure, and determine its applications.
Author Contributions: J.Y. proposed the exponential entropy measures of simplified NSs and their proof; W.C. provided the actual decision-making example and comparative analysis; all authors wrote this paper together.