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Fermatean Fuzzy Entropy Measure with Application in Decision Making Using COPRAS Approach

Mansi Bhatia1*, H. D. Arora1, Anjali Naithani1 and Vijay Kumar2

1Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Uttar Pradesh, Noida, India

2Department of Mathematics, Manav Rachna International Institute of Research and Studies, Faridabad, India

Abstract

Due to the influence of media and easy access to other mediums uncertainty has significantly impacted decision making process in every field and aspect of human life. Over the last few decades improved fuzzy decision making models have proved themselves as more efficient tool in handling decision making as compared to classical set theory. One such concept is the notion of Fermatean fuzzy sets (FFS) which states that the sum of cubes of membership and non-membership is restricted to unity. FFS is an extension of Pythagorean fuzzy sets (PFS) and intuitionistic fuzzy sets (IFS). FFS can also handle situations where IFS and PFS fails and hence can be easily combined with different decision making techniques to process uncertain information and hence simplify complex decisions. The purpose of this article is to propose a novel entropy measure that can be used to solve real life application using COmplex PRoportional ASsessment (COPRAS). Weights are calculated using fuzzy entropy and example is taken to numerically evaluate the rationality of the measure. Finally, practical examples are given to validate the reliability and effectiveness of the proposed measure.

Keywords: COPRAS, decision-making, entropy measure, fermatean fuzzy sets

1.1 Introduction

Multi Criteria Decision Making (MCDM) is a branch of Operation Research (OR) which deals with decision making situations involving conflicting criteria. It is used in wide varieties of fields but not limited to management, education, medical and environment. Humans have limitations when it comes to conflicting decision making situations hence MCDM techniques makes it convenient to evaluate quantitatively and makes sound decisions. There are various types of MCDM techniques like AHP, SAW, TOPSIS, COPRAS, ELECTRE, VIKOR, etc. which has been combined time to time with different fuzzy sets to answer and explore the uncertainty in the path of decision making.

In real life data is not in crisp form hence a need was felt by Zadeh [1] to extend the classical set theory of George Cantor to the fuzzy set theory that can address the uncertainty involved in practical situations. Unlike crisp set where an element either belongs to a set or it does not, fuzzy set theory also takes into account all the values that falls between them. FS has proved to be an ideal tool to handle the vagueness but a situation is possible where belongingness of an object may be accompanied with hesitancy. This problem was solved with the use of Intuitionistic Fuzzy Set (IFS) that incorporated the hesitancy by Atanassov [2, 3].

Later on, it was observed that for some situations Atanassov’s theory was not satisfied and hence Yager [4–7] formulated the theory of Pythagorean Fuzzy Sets (PFS) which states that the aggregate of square of membership and non-membership lies between 0 and 1. In 2017, Yager [8] extended the concept of IFS and PFS with the introduction of q-rung orthopair fuzzy sets. If is the measure of belongingness, is the measure of non-belongingness function and τ(θ) is the hesitancy then and . For q = 1 the inequality reduces to which is the condition of IFS, for q = 2 it becomes inequality satisfied by PFS and q=3 denotes the Fermatean Fuzzy sets i.e., . The theory of FFS given in 2020 by Senapati and Yager [9, 10] is the latest addition to FS theory. The whole theoretical concept can be explained with the help of example as follows: Let <0.6, 0.9> be a fuzzy set then 0.6+0.9 ≰ 1 and 0.62 + 0.92 = 0.36 + 0.81 > 1 but 0.63 + 0.93 = 0.216 + 0.729 <1. The example showed a situation where IFS, PFS fails but FFS satisfies the case. The evolution of Fuzzy theory can also be understood with the help of flow Figure 1.1 given below.

Entropy is defined as the degree of uncertainty or randomness. It is used to measure the fuzziness within the fuzzy sets. Entropy is directly proportional to the amount of information conveyed by an object that is high value of entropy implies more information and low value indicates less information. Entropy is used to produce the parameter weights based on the decision matrix. Entropy always gets confused with probability but in case of probability we deal with events that might or might not occur but entropy deals with truth values only. Entropy has been combined with many fuzzy sets as per the requirement to get desired information. Hung and Yang [11] proposed two entropies for IFS. Verma and Sharma in 2014 [12] explored entropy of order – α for IFS. Rahimi et al. [13] suggested entropy approach for IFS based on supplier selection. Yuan and Zeng [14] proposed an entropy function based on special function for IFS with its applications in regional collaborative innovation. Pandey et al. [15] introduced novel entropy based on IFS for feature selection that tends to improve data management by enhancing machine learning. Further, entropy has been applied with PFS to enhance the decision making by many researchers. Yang and Hussain [16] in 2018 applied fuzzy entropy on PFS with its applications in MCDM. A new entropy measure proposed by Thao et al. [17] in 2019 based on PFS.IN 2021 Gandotra [18] gave us two entropy measures first with R. Kumar to assess best automative company and then with Kizielewicz [19] and others to solve various applications in MCDM. Chaurasiya and Jain [20] proposed the entropy measure on PFS and solved health care waste management problem. Mohagheghi and Mousavi [21] suggested entropy approach on Interval valued PFS for sustainable project decision. FFS are the most recent type of fuzzy sets and since they have an advantage over existing fuzzy sets it is expanding its roots quickly in decision making problems to solve the real life problems faced by humans. Use of entropy in FFS relatively new and only few researchers have explored this area like Murat Kirişci [22] has proposed new entropy and distance measure based on FFS with its applications in medical diagnosis. Chang et al. [23] have suggested a new entropy method based on FFS using risk assessment.

Figure 1.1 Fuzzy sets.

Entropy has been applied with different fuzzy sets for giving information on various decision making problems using Lagrange entropy [24] or MCDM techniques like TOPSIS [25], VIKOR [26], COPRAS [27] etc. The COmplex PRoportional ASsessment (COPRAS) method was introduced by Zavadskas [28] in 1994. Many entropy measures have been proposed in the literature like by Wei [30], Wang [31] using IFS, Xue et al. [29] on the basis of PFS etc. Many fuzzy techniques have been combined with COPRAS to solve decision making problems. In this article we have formulated entropy measure using COPAS method to provide an easy way for decision making problems.

The manuscript has been organized as follows: Section 1.2 helps you to understand the basic fundamentals used in the article. Section 1.3 is dedicated to the proposed entropy measure and its properties. Section 1.4 describes the COPRAS approach and the application of the measure. Section 1.5 shows comparative analysis with other authors. At last Section 1.6 concludes the article followed by references.

1.2 Preliminaries

1.2.1 Intuitionistic Fuzzy Sets

In a universe of discourse X, an intuitionistic fuzzy set can be stated as

Where is the degree of affiliation and is the degree of non-affiliation and the both the function satisfies the relation

Degree of hesitancy is the part of membership or non - membership or both and is specified as and it expresses the insufficient information or uncertainty.

1.2.2 Pythagorean Fuzzy Sets

Let be a PFS in X then it can be stated as

Where and is the degree of inclusion and non-inclusion and the membership.

Given θ be the element the set satisfies the relation .

The measure of uncertainty in this case is outlined as .

1.2.3 Fermatean Fuzzy Sets

is called Fermatean fuzzy set if there exists an element such that and where is called the measure of inclusion, is the measure of non-inclusion and is the degree of indeterminacy or hesitancy.

1.3 Novel Fermatean Entropy Measure

In this segment we have projected a new entropy measure based on FFS and shown its legitimacy by proving the properties for the proposed measure.

1.3.1 Entropy

Let be Fermatean fuzzy set then the entropy on is defined as

The proposed measure satisfies the following properties:

The entropy is said to be Fermatean entropy if

1. is a crisp set.
2. where and are FFS such that that...