The entropy of rough neutrosophic multisets

The entropy of rough neutrosophic multisets is introduced to measure the fuzziness degree of rough multisets information. The entropy is defined in two ways, which is the entropy of rough neutrosophic multisets generalize from existing entropy of single value neutrosophic set and the rough neutrosophic multisets entropy based on roughness approximation. The definition is derived from being satisfied in the following conditions required for rough neutrosophic multisets entropy. Note that the entropy will be null when the set is crisp, while maximum if the set is a completely rough neutrosophic multiset. Moreover, the rough neutrosophic multisets entropy and its complement are equal. Also, if the degree of lower and upper approximation for truth membership, indeterminacy membership, and falsity membership of each element decrease, then the sum will decrease. Therefore, this set becomes fuzzier, causing the entropy to increase.


1.
Introduction The information of independent opinion given by the experts always consists of differences due to the uncertainty condition. This may result in the agreed and disagreed discussion between them. The fuzzy set theory (FS) [1] is introduced as a solution to this discussion by representing a fuzzy system with values denoted from 0 to 1. The value for the degree of membership function for an agreed group and a disagreed group is that an absolute agree value is 1.0 while an absolute disagreed value is 0.0. Moreover, the intuitionistic fuzzy set theory (IFS) [2] as a generalization of FS also gives an opinion to this argument. Other than the degree of membership function, a non-membership function exists, such as absolute agree value for membership function is 1.0, and absolute agree for non-membership function is 0.0. This theory clarifies the expert when involving the truth (membership function) and falsity (nonmembership function) condition. Then, the neutrosophic set theory (NS) [3] covers up all the opinion by introducing a new argument between the prior, which is a neutral opinion, where the opinion either considers membership function or non-membership function between the two expert discussion.
The triple set element is introduced for NS, representing a degree of truth membership function, a degree of indeterminacy (neutral) membership function, and falsity (non-membership function). The FS, IFS, and NS theories are motivation arising from the uncertainty condition to solve the vagueness, imprecision, ambiguity and inconsistency from data and information collection. Therefore, the neutrosophic set theory takes a novel approach to solve the ambiguity between the expert information of opinion. Next, a refined neutrosophic set theory [4] was introduced to solve the multiple information regarding the neutrosophic set condition. Then, the extended operation for the refined neutrosophic set was studied, introducing the single-valued neutrosophic multiset [5].
After a decade, a hybrid theory using neutrosophic set theory and neutrosophic multisets theory is gaining more attention. The advantage of this theory is having a natural extension knowledge concept of fuzzy set theory, fuzzy multisets theory, intuitionistic fuzzy set theory, and intuitionistic fuzzy multisets theory such as the relation, distance, similarity, and entropy. Hence, a rough neutrosophic multisets theory [6] is introduced by the developing of neutrosophic set theory. The rough neutrosophic multisets is a generalization of a rough set by Pawlak's [7] and neutrosophic multisets [5]. This theory discusses the neutrosophic multisets in terms of lower and upper approximations in equivalence relation. The lower approximation gives a sign of surely belonging information, while the upper approximation gives a sign of possibly belonging information. All the properties for both approximations follow the natural concept of a rough set, while neutrosophic multisets properties represent the multiple information. Some of the properties are motivated by rough neutrosophic set [8].
Besides information evaluation, rough neutrosophic multisets also measure information similarity, aggregate the information, and formulate the fuzziness of information by entropy measure. The discussion of this information evaluation is a well-known study in the literature. Therefore, this paper discusses the entropy measure, where it is an important concept for information evaluation. Some literature for fuzzy entropies [9][10][11] and neutrosophic set entropy [12][13][14][15][16][17][18] are used for references. According to all the literature, a study for rough neutrosophic multisets entropy is not yet explored. This becomes the novelty for this paper to introduce the rough neutrosophic multisets entropy to evaluate the information fuzziness in a rough neutrosophic multisets environment.
As a generalization of the fuzzy set, if the entropy value is smaller, then the information provided is more useful. Therefore, this paper aims to quantify the number of fuzziness measures by defining the rough neutrosophic multisets entropy based on the natural extension of the fuzzy set. The next objective is to show the effect of roughness for a lower and upper approximation of multiple values of neutrosophic multisets by defining the rough neutrosophic multisets entropy based on roughness approximation. This paper is presented in four sections. The first section is an overview of the rough neutrosophic multisets environment with entropy literature. Next, section two recalls the definition of rough neutrosophic multisets with some of the operations used. Then, section three defines the two new entropy definitions for rough neutrosophic multisets based on the natural extension of fuzzy set and roughness approximation. All the entropy conditions are successfully proven in this section. Lastly, the conclusion, as in section four, concludes the novelty of this research paper.

2.
Preliminaries This section recalls the definition of rough neutrosophic multisets and some of the operations, the entropy of the neutrosophic set, and the accuracy and roughness measure of Pawlak's approximation.
The truth membership sequences (2) Definition 3 [19]. For a subset of object ⊆ , the accuracy measure is defined as: where is a non-empty set, ∈ , ( ) is the lower approximation of set , Definition 4 has a limit on the use of entropy in condition (E4). This entropy is questionable and well discussed in [17]. But this entropy has an advantage because it still follows the accurate extension of the concept of fuzzy sets and intuitionistic fuzzy set where the complement of a single valued neutrosophic set is = {( ( ), 1 − ( ), ( )| ∈ }. .
(6) Definition 4 proposed in [17] discusses the comparison of the entropy concept for a neutrosophic set.

3.
The entropy of Rough Neutrosophic Multisets In this section, the entropy of Rough Neutrosophic Multisets (RNM) is introduced to measure the fuzziness degree of RNM information. The entropy of RNM is defined in two ways, the entropy of RNM generalizes from the existing entropy of a single value neutrosophic set and the RNM entropy based on roughness approximation.
The definition is derived by satisfying the following conditions, required for RNM entropy: (i) The entropy will be null when the set is crisp, (ii) The entropy will be maximum if the set is completely RNM, (iii) The RNM entropy and its complement is equal, and (iv) Suppose the degree of lower and upper approximations for truth membership, indeterminacy membership, and falsity membership of each element decreases. In that case, the sum decreases as well. Therefore, this set becomes fuzzier, increasing the entropy.
Given the condition stated, the definition of RNM entropy is defined as follows: Now notice that in RNM, the present uncertainty is due to the factors of possibly belongingness and surely belongingness, representing lower and upper approximations of RNM. Considering these factors, two types of entropy measure ( ) and ∆ ( ) of rough neutrosophic multisets, are defined as follows.

Rough Neutrosophic Multisets entropy based on the natural extension of fuzzy set concept
The concept of complement for single value neutrosophic set (SVNS) is based on the natural extension of fuzzy set and intuitionistic fuzzy set where = {( ( ), 1 − ( ), ( )| ∈ }. Therefore, this complement concept is also derived for rough neutrosophic multisets (RNM) complement as in equation (2). The complement concept is used to derive a new definition for RNM entropy based on the natural extension of a fuzzy concept. Following Definition 4, it is derived as follows:

Rough Neutrosophic Multisets entropy based on roughness approximation
The accuracy and roughness measure of Pawlak's approximation gives a motivation to derive the rough neutrosophic multisets entropy. The roughness approximation in equation 4 for truth membership sequence, indeterminate membership sequence, and falsity membership sequence is used simultaneously in RNM entropy.