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A comparison of the performance of entropy measures for interval- valued intuitionistic fuzzy sets

Entropy measure is a significant tool to define unclear information. But, entropy measures for interval-valued intuitionistic fuzzy sets (IVIFSs) cannot be easily understood intuitively. So, it is highly important to compare the existing measures to select a reliable entropy measure in studies. The purpose of this study is to compare the performance of different entropy measures developed for IVIFSs. The numerical examples are presented to show whether entropy measures for IVIFSs are effective in representing the fuzziness degree. In order to understand whether a variation of fuzziness degree of one or more elements of IVIFSs change the ranking results, selected IVIFSs are modified diversely. Cite this article as: KOKOÇ M, ERSÖZ S. A comparison of the performance of entropy measures for interval-valued intuitionistic fuzzy sets. Sigma J Eng Nat Sci 2021;39(2):131–147. Sigma Journal of Engineering and Natural Sciences – Sigma Mühendislik ve Fen Bilimleri Dergisi Web page info: https://sigma.yildiz.edu.tr DOI: 10.14744/sigma.2021.00003


INTRODUCTION
In real-life problems include uncertainty and complexity. However, crisp numbers are not enough to describe uncertainty and imprecision properly. To deal with ambiguity and imprecision, Zadeh [1] proposed fuzzy set (FS) theory in 1965. Although the FSs characterized by a membership function has received attraction from decision experts in the various fields, Atanassov [2] put forward that it was insufficient to represent the state of belonging only with the membership function. So, he offered intuitionistic fuzzy sets (IFSs) theory in 1986. In this theory, elements belonging to a set are described by a membership degree and a non-membership degree in [0,1]. Three years later, Atanassov and Gargov [3] proposed the theory of interval-valued intuitionistic fuzzy sets (IVIFSs) theory in 1989, which is an extension of the IFSs theory. The IVIF sets are characterized by an interval membership degree, an interval non-membership degree, and an interval hesitancy degree. In a famous monograph, Pedrycz [4] stated that expressing membership degree and non-membership with a single value is not realistic enough and technically sufficient. Because the degree of an element belonging to a set is expressed by interval-valued numbers rather than crisp numbers, IVIF sets theory provides a powerful tool to cope with ambiguity and vagueness in real applications [5][6][7][8][9].
The fuzzy entropy presents a global measure of an average amount of intrinsic information which is lost when turning from a classical pattern to the fuzzy pattern. Owing to the similarity of the equation to the Shannon entropy form, this measurement was given the name of 'entropy' [10]. However, Shannon entropy and fuzzy entropy are different in terms of uncertainty. The Shannon entropy measures the average uncertainty in bits related to the estimation of outcomes in an experiment, but the fuzzy entropy explains the degree of fuzziness in a fuzzy set [11]. Entropy is an important concept for the fuzzy set theory proposed by Zadeh [1]. Burillo and Bustince [12] extended the fuzzy entropy for intuitionistic fuzzy sets. Then, in order that fuzziness degree of interval-valued intuitionistic fuzzy sets can be measured, Liu et al. [13] developed the interval-valued intuitionistic fuzzy entropy. In order to measure the uncertainty of IVIFSs, many researchers have investigated the definition and formulation of entropy of IVIFSs from various aspects. For instance, Liu et al. [13] first introduced an axiomatic definition of entropy for IV-IFSs. Based on this, Wei et al. [14], Gao and Wei [15] and Jin et al. [16] constructed a variety of entropy measures. Zhang and Jiang [17] and Zhang et al. [11] presented the entropy model depending on De Luca and Termini [18] model extending the definition of entropy for FSs. Zhang et al. [19] extended the definition of entropy for IVIFSs by taking inspiration from Burillo and Bustince [12]'s study for IFSs. Zhang et al. [20], Rashid et al. [21] developed distance-based entropy measures for unclear information in the interval-valued intuitionistic fuzzy set environment. Furthermore, more studies on entropy measures for IV-IFSs has been introduced in different viewpoints [22][23][24][25][26][27][28][29][30][31][32].
When the literature review, it is seen that there are few studies about entropy measures for IVIFSs. Moreover, these existing entropy measures for IVIFSs cannot be easily understood and compared heuristically. So, this paper is aimed to compare the performance of IVIF entropy measures about representing the fuzziness degree of IVIFSs. This comparative analysis is provided with ten numerical examples constructed by considering the variation of fuzziness degree of one or more elements of an IVIF set. In the studies mentioned in section 5, a comparison has been made with several entropy models to compare the performance of the developed entropy model. But this study is prominent with a comprehensive comparing of the IVIF entropy measures.
The article is organized as follows: Section 2 briefly reviews some basic concepts related to IVIF sets. Section 3 presents the properties of the IVIF entropy measure and some IVIF entropy measures to be compared. To understand whether a variation of fuzziness degree of one or more elements of IVIFSs changes the ranking results, ten different tests are conducted in Section 4. Test results are discussed and conclusions are presented, in the last section.

PRELIMINARIES
In this section, some basic concepts related to interval-valued intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy numbers are reviewed in order to facilitate further discussions. Definition 1. [3] Let D 0 1 ,     be the set of all closed sub-intervals of the interval [0,1] and X be a uni-verse of discourse. An IVIFS  A in X is an object having the form The closed intervals ( ) and v x A  ( ) denote the degrees of membership and nonmembership of the x to  A , respectively. Lower and upper end points of these intervals are denoted by Thus an interval-valued intuitionistic fuzzy set can be denoted as below: For convenience, Xu [33] called then the given interval-valued intuitionistic fuzzy set  A is reduced to an ordinary intuitionistic fuzzy set.

Definition 2.[3] Hesitation degree of each element x in interval-valued intuitionistic fuzzy set 
A is given as: ∈ ( ) then Γ( ) A A = and by adding the modulus into the formula, the value of IVIFN will appear such that; Let . and  x = 0 5 . as the fuzzification, then: Definition 3. The operational relations are defined in [36][37][38] for   A A IVIFS X 1 2 ,

COMPARISON OF ENTROPY MEASURES
In this section, to understand whether a variation of fuzziness degree of one or more elements of IVIFSs changes the ranking results, ten different tests are conducted.
In order to state the linguistic variables' characterization, A as "High" in X . By using the exponentiation operator presented in Eq. (8), the following IVIFS pertaining to A can be calculated. In terms of mathematical operations, the entropy of these IVIFSs should have the below ranking:

.
A 1 5 . can be considered as "Medium High"; A 2 can be considered as "Very High"; A 2 5 . can be considered as "Quite Very High"; A 3 can be considered as "Absolutely High". , . , . , . , .
In order to understand whether a decrease of membership degrees of the more than one element of IVIFS change the ranking results, membership degrees of the elements '6' , '7' , '8' , '9' and '10' are decreased to a lower level. The modified IVIFS     , . , In order to understand whether a decrease of hesitation degrees of the more than one element of IVIFS change the ranking results, hesitation degrees of the elements '6' , '7' , '8' , and '9' are decreased to a lower level. The modified IVIFS

DISCUSSION AND CONCLUSION
When studies comparing entropies [20,21,23,47] are examined, it is seen that a comparison is not as comprehensive as in this study. Zhang et al. [20]  In this study, the results of 10 tests performed for comparing 27 entropies can be seen in Table 1, Table 2,  Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9 and Table 10. Performance rates of entropy measures on presenting the expected ranking are presented in Figure 1. From the entropy values in the above-mentioned tables, it is deduced that the closer the membership degree and the non-membership degree, or the higher the hesitation degree, the greater the entropy. From Figure 1, it can be easily  The authors have tried their best to contain IVIF entropy measures to get a comprehensive analysis, but there is a minor possibility of some publications being left out. In future work, it is suggested that the entropy measures extended for fuzzy sets such as hesitant, spherical, Pythagorean in addition to IVIF sets can be examined.

AUTHORSHIP CONTRIBUTIONS
M.K and S.E. designed the study. M.K. developed the theoretical framework, processed the experimental data, performed the analysis. S.E. supervised the findings. M.K. drafted the manuscript and designed the figures.

DATA AVAILABILITY STATEMENT
The published publication includes all graphics and data collected or developed during the study.

CONFLICT OF INTEREST
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ETHICS
There are no ethical issues with the publication of this manuscript.