Generalized Distance-Based Entropy and Dimension Root Entropy for Simplified Neutrosophic Sets

In order to quantify the fuzziness in the simplified neutrosophic setting, this paper proposes a generalized distance-based entropy measure and a dimension root entropy measure of simplified neutrosophic sets (NSs) (containing interval-valued and single-valued NSs) and verifies their properties. Then, comparison with the existing relative interval-valued NS entropy measures through a numerical example is carried out to demonstrate the feasibility and rationality of the presented generalized distance-based entropy and dimension root entropy measures of simplified NSs. Lastly, a decision-making example is presented to illustrate their applicability, and then the decision results indicate that the presented entropy measures are effective and reasonable. Hence, this study enriches the simplified neutrosophic entropy theory and measure approaches.


Introduction
Since entropy is an effective measure approach in quantifying the uncertainty degree of the objects, with the development of fuzzy theory, a lot of research on fuzzy entropy has been done so far. Zadeh [1] first defined fuzzy entropy for fuzzy sets regarding the probability distribution of a fuzzy event. Then, De-Luca and Termini [2] formulated axioms of fuzzy entropy and proposed a non-probabilistic logarithm of fuzzy entropy. Exponential fuzzy entropy was presented by Pal and Pal [3]. Yager [4] put forward the metric distance-based entropy by measuring the lack of distinction between the fuzzy set and its complement. The weighted fuzzy entropy with trigonometric functions of membership degree was constructed by Parkash and Sharma [5]. Thereafter, the generalized parametric exponential fuzzy entropy of order-α was introduced by Verma and Sharma [6], which reduces to the Pal and Pal exponential entropy [3] when α = 1, and becomes the De-Luca and Termini logarithmic entropy [2] when α → 0. However, for an intuitionistic fuzzy set (IFS) extended by adding a non-membership degree to a fuzzy set (FS), Burillo and Bustince [7] first proposed IFS and interval-valued IFS entropy measures and their axiom requirements. Then, Szmidt and Kacprzyk [8] redefined De-Luca and Termini's axioms [2] in IFS setting and presented an intuitionistic non-probabilistic fuzzy entropy measure by a geometric interpretation and a ratio of distance of IFSs. Valchos and Sergiadis [9] constructed a new entropy logarithm of IFS on the basis of the De-Luca and Termini fuzzy entropy logarithm [2]. As an extension of logarithmic entropy [2], Zhang and Jiang [10] proposed vague entropy by the intersection and union of the non-membership degree and membership degree for vague sets, and defined vague cross-entropy for IFSs. Further, the cosine and sine entropy of IFS was defined by Ye [11]. An exponential entropy measure of IFS was proposed by Verma and Sharma [12], and then intuitionistic fuzzy entropy was proposed corresponding to the order-α [13] and R-norm [14]. Additionally, for an interval-valued IFS (IvIFS), Ye [15] put forward the sine and cosine entropy of

Simplified Neutrosophic Sets
Simplified NS, which contains both SvNS and IvNS, was presented by Ye [25] as a subset of NS for convenient application. Assume there is a universal set A = {a 1 , a 2 , ..., a n }, then a simplified NS B in then some operations between B and C can be given as follows [25,26]: (1) The sufficient and necessary condition of If B and C are SvNSs, then: However, if B and C are IvNSs, then:

Simplified Neutrosophic Generalized Distance-based Entropy and Dimension Root Entropy
In this section, two novel simplified neutrosophic entropy measures, containing a simplified neutrosophic generalized distance-based entropy measure and a simplified neutrosophic dimension root entropy measure, are defined below.

Simplified Neutrosophic Generalized Distance-Based Entropy
Definition 1. Assume a simplified NS H in a universal set A = {a 1 , a 2 , ..., a n } is H = {<a i , T H (a i ), U H (a i ), F H (a i )> | a i ∈ A}. Then, a new generalized distance-based entropy measure of H can be defined as: where ρ is an integer value.
According to the axiomatic definition of the IvNS entropy measure [21], the proposed generalized distance-based entropy measure of a simplified NS has the theorem below. (EAP3) If one simplified NS H is closer to the fuzziest simplified NS B than the other simplified NS L, then H is fuzzier than L with EA
At first, by removing the absolute symbol, the function f (x i ) can be expressed as: For ρ = 1, the first derivative of f (x i ) with respect to x i apart from x i = 0.5 is: It is clear that f (x i ) is monotonically increasing when x i ∈ [0, 0.5) and decreasing when x i ∈ (0.5, 1]. Thus, for the interval [0, 1], f (x i ) = 1 get the maximum value at the critical point of x i = 0.5 for ρ = 1.
Then, when ρ is not equal to 1, the first derivative of f (x i ) with respect to x i can be calculated by: Obviously, the first derivative of f (x i ) is equal to zero only at the point of x i = 0.5. Because f'(x i ) is positive for 0 ≤ x i < 0.5 and negative for 0.5 < x i ≤ 1, the maximum of f (x i ) = 1 on the closed interval [0, 1] can be obtained at the critical point x i = 0.5.
Regarding the definition of f (x i ), the entropy measure of simplified NS  (3) and (4), f (x i ) is monotonically increasing when x i ∈ [0, 0.5], and monotonically decreasing when x i ∈ [0.5, 1]. Therefore, the closer the simplified NS H is to the fuzziest set B than L, the fuzzier H is than When the complement of the IvNS Thus, the proof of the Theorem 1 is completed. [24] defined a dimension root distance of SvNSs as follows:

Simplified Neutrosophic Dimension Root Entropy
Based on the dimension root distance, we can present simplified neutrosophic dimension root entropy for a simplified NS.

Definition 2.
Assume H = {<a i , T B (a i ), U B (a i ), F B (a i )> | a i ∈ A} is a simplified NS in a universal set A = {a 1 , a 2 , ..., a n }. Then, we can define the following dimension root entropy measure for the simplified NS H: for the IvNS H.
Similar to the proposed simplified neutrosophic distance-based entropy, the dimension root entropy of simplified NSs also has the following theorem.
(EBP4) Since the complement of the SvNS Thus, the proof of the theorem is completed.

Comparative Analysis of Entropy Measures for IvNSs
The comparative analysis between the presented simplified neutrosophic entropy measures and the existing entropy measures of simplified NSs are shown in this section. Since SvNS is a special case of IvNS when the two bounded values of its each interval are the same, the example adopted from [21] was illustrated only in IvNS setting. Then the existing entropy measures [19][20][21][22][23] of the IvNS H used for the comparison are introduced as follows:  [21]: (1) H can be regarded as "large" in A; (2) H 2 can be regarded as "very large"; (3) H 3 can be regarded as "quite very large"; (4) H 4 can be regarded as "very very large". Then the operational results are shown in Table 1.

Decision-Making Example Using Simplified Neutrosophic Entropy in IvNS Setting
In this section, the proposed entropy measures are applied in a decision-making problem, and then compared with the existing entropy measures. For convenience, an investment decision-making example adopted from the reference [21] was used for the application. In the decision-making problem, the decision makers are requested to assess four investment projects (alternatives), including a clothing company (g1), a food company (g2), a computer company (g3), and a house-building company (g4), over three attributes, like growth (a1), risk (a2), and environmental impact (a3) respectively, and then select the best alternative for the investment company. The evaluation information of the alternative set G = {g1, g2, g3, g4} over the attribute set A = {a1, a2, a3} is given by the form of IvNSs as the following matrix: By applying the proposed entropy measures of Equations (2) and (6) and the existing entropy measures of Equations (7)- (14) to the above decision-making problem, the relative entropy measure results and the ranking orders are listed in Table 3 Then, the available entropy measures of IvNSs should satisfy the ranking order EA Table 2, except for both EA 100 2 (H n ) with the ranking order EA 100 2 (H) = EA 100 2 (H 2 ) > EA 100 2 (H 3 ) > EA 100 2 (H 4 ) and R 6 (H n ) with the ranking order R 6 (H) > R 6 (H 4 ) > R 6 (H 3 ) > R 6 (H 2 ), the entropy measure values of EA ρ 2 (H n ) for ρ ∈ [1, 100) and EB 2 (H n ) and the existing R 1 (H n )-R 5 (H n ), R 7 (H n ), R 8 (H n ) satisfy the above ranking demand of the available entropy measures. Furthermore, from Figure 1, the ranking order based on the entropy measure values of EA ρ 2 (H n ) indicates some robustness regarding ρ from 1 to 100. However, with the parameter ρ increasing, especially when ρ > 30, the entropy measure values of EA ρ 2 (H) and EA ρ 2 (H 2 ) tend to the same value 0.8.

Decision-Making Example Using Simplified Neutrosophic Entropy in IvNS Setting
In this section, the proposed entropy measures are applied in a decision-making problem, and then compared with the existing entropy measures. For convenience, an investment decision-making example adopted from the reference [21] was used for the application. In the decision-making problem, the decision makers are requested to assess four investment projects (alternatives), including a clothing company (g 1 ), a food company (g 2 ), a computer company (g 3 ), and a house-building company (g 4 ), over three attributes, like growth (a 1 ), risk (a 2 ), and environmental impact (a 3 ) respectively, and then select the best alternative for the investment company. The evaluation information of the alternative set G = {g 1 , g 2 , g 3 , g 4 } over the attribute set A = {a 1 , a 2 , a 3 } is given by the form of IvNSs as the following matrix: By applying the proposed entropy measures of Equations (2) and (6) and the existing entropy measures of Equations (7)- (14) to the above decision-making problem, the relative entropy measure results and the ranking orders are listed in Table 3.

Conclusions
This study originally presented the generalized distance-based entropy measure and the dimension root entropy measure of simplified NSs, containing both the SvSN and IvSN generalized distance-based entropy measures and the SvSN and IvSN neutrosophic dimension root entropy measures. Then, their properties were discussed based on the axioms of an entropy measure of IvNSs defined in [21]. After that, a comparison between the proposed entropy and existing relative entropy measures by a numerical example in IvNS setting showed that the proposed entropy measures are effective and rational. An application of the proposed two entropy measures in an actual decision-making problem illustrated the feasibility and rationality by comparison with the existing ones, especially with the relatively small values of the parameter ρ, such as ρ < 20. The proposed simplified NS entropy not only is a complement of the entropy theory of simplified NSs, but also presents a new effective way of the uncertain measure under the simplified NS setting. Our future work will focus on research to extend the proposed entropy measures to applications in diverse engineering fields.