Abstract

The purpose of aggregation methods is to convert a list of objects of a set into a single object of the same set usually by an -arry function, so-called aggregation operator. The key features of this work are the aggregation operators, because they are based on a novel set called Fermatean cubic fuzzy set (F-CFS). F-CFS has greater spatial scope and can deal with more ambiguous situations where other fuzzy set extensions fail to support them. For this purpose, the notion of F-CFS is defined. F-CFS is the transformation of intuitionistic cubic fuzzy set (I-CFS), Pythagorean cubic fuzzy set (P-CFS), interval-valued cubic fuzzy set, and basic orthopair fuzzy set and is grounded on the constraint that “the cube of the supremum of membership plus nonmembership degree is ”. We have analyzed some properties of Fermatean cubic fuzzy numbers (F-CFNs) as they are the alterationof basic properties of I-CFS and P-CFS. We also have defined the score and deviation degrees of F-CFNs. Moreover, the distance measuring function between two F-CFNs is defined which shows the space between two F-CFNs. Based on this notion, the aggregation operators namely Fermatean cubic fuzzy-weighted averaging operator (F-CFWA), Fermatean cubic fuzzy-weighted geometric operator (F-CFWG), Fermatean cubic fuzzy-ordered-weighted averaging operator (F-CFOWA), and Fermatean cubic fuzzy-ordered-weighted geometric operator (F-CFOWG) are developed. Furthermore, the notion is applied to multiattribute decision-making (MADM) problem in which we presented our objectives in the form of F-CFNs to show the effectiveness of the newly developed strategy.

1. Introduction

The multiattribute decision-making (MADM) approach is a well-known and influential procedure for selecting the best alternative for many problems in our practical life. Despite its popularity, there are two fundamental issues for MADM: (i) how decision makers suitably establish their assessment data and (ii) how the best alternative is determined. The selection may be accessible in the case of crisp data, but it becomes challenging when there are uncertainties and fuzziness in the available data. Zadeh’s fuzzy set [1] has played a significant role in dealing with such data types. The proficient concept called fuzzy set theory (FST) was established in 1965. A fuzzy set contains membership degree of an object , such that . Considering the importance of FST in real-life problems, fuzzy MADM has become a hot topic for researchers. McBratney and Odeh [2] have shown the best solid example of the application of FST in real life and showed that FST provides a rich and meaningful improvement or extension of conventional logic and explored that the Mathematics produced by FST is consistent. They exposed that the applications in soil sciences, numerical classification of soil and mapping, land evaluation, modeling and simulation of soil physical process [2], etc. are emerging roles of FST.

In a simple fuzzy set, we cannot assess complete information about an object in the universe to explain a decision-making problem more realistically. Atanassov [3] introduced an intuitionistic fuzzy set (IFS) by defining the membership grade and nonmembership grade for an element in the set , such that The concept of IFS is proficient however; Yager [4] generalized the concept of IFS by introducing the Pythagorean fuzzy set (PFS). In PFS, the square sum of membership grade and nonmembership grade is no more than one. Many authors contributed to the IFS and PFS in various directions, like decision making, medical diagnoses, and information measures. Yang and Yao presented set-theoretic operations and relationships in the three systems (i.e., fuzzy sets, Atanassov’s IFS, and shadowed sets) and the applications of the constructed shadowed sets for three-way decision making [5]. Recently, Verma worked on parametric information measures under the IFS and proposed four new order- divergence measures between two IFSs [6]. Verma and Merigó [7] developed a new and flexible method for Pythagorean fuzzy decision making using some trigonometric similarity measures. Bakioglu and Atahan [8] investigated the prioritization of risks involved with self-driving vehicles by proposing new hybrid MCDM methods based on the analytic hierarchy process (AHP), the technique for order preference by similarity to an ideal solution (TOPSIS), and Vlse Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) under Pythagorean fuzzy environment. Zhang and Xu [9] and Rani et al. [10] introduced the technique for order of preferences by similarity to ideal solution (TOPSIS) to tackle the MADM problems under PFS. Similarly, Sajjad et al. [11] presented a new extension of the TOPSIS method based on Pythagorean hesitant fuzzy sets with incomplete weight information. Peng and Yang [12] and Khan [13] worked on the fundamental properties of interval-valued Pythagorean fuzzy (IVPF) aggregation operators. They presented some emphasized aggregation operators based on interval-valued intuitionistic fuzzy numbers and their operation to group decision-making problems. Yager extended set measures to Pythagorean fuzzy sets [14]. Xu et al. presented the Pythagorean undefined decision-making method based on overall entropy [15], and Garg [16] generalized Pythagorean fuzzy information aggregation using Einstein operations. On the other hand, in the intuitionistic cubic fuzzy set (I-CFS) [17], both membership and nonmembership degrees are cubic sets. It is clear that the definition is more reliable and accurate for using it in multicriteria decision-making problems (MCDM), and numerous works have been done in this environment. Scholars evaluate MCDM problems through modified aggregation operators in terms of I-CFS [18]. Khan et al. [19] presented P-CFS, wherein membership and nonmembership degrees are cubic fuzzy numbers. The condition is that the supremum of the square of the membership degree plus the square of the nonmembership degree should be less than or equal to one, i.e., . Since P-CFNs are the modification of I-CFNs, the definition has more powerful results in MADM problems and is quite a straightforward methodology to deal with such problems. Readers can easily understand the value of the work by comparing it with the early results obtained through different aggregation operators. For example, Fahmi et al. [20, 21] defined Einstein aggregation operators for trapezoidal cubic fuzzy and cubical fuzzy sets. They also introduced geometric operators with triangular cubic linguistic hesitant fuzzy numbers [21]. Similarly, Rahman et al. in [22, 23], respectively, defined generalized intuitionistic fuzzy Einstein hybrid aggregation operators and interval-valued Pythagorean fuzzy Einstein hybrid aggregation operators and utilized them in decision-making problems.

The idea of the Pythagorean fuzzy set was further extended by Senapati and Yager [24], defining Fermatean fuzzy sets (FFS). Graphically, it can be shown that the spatial scope and area of acceptability of the Fermatean fuzzy set is greater than the intuitionistic fuzzy set and Pythagorean fuzzy set. They also presented the entire sets of operations and defined the score and accuracy functions for Fermatean fuzzy sets. Moreover, Senapati and Yager [25] aggressively worked more on the Fermatean fuzzy set and developed Fermatean fuzzy-weighted averaging/geometric operators and their application in MCDM methodologies. Deng and Wang [26] devised two novel distance-measure methods for Fermatean fuzzy sets. Gao et al., [27] worked in the area of continuities, derivatives, and differentials in the workplace of Fermatean fuzzy numbers, which is a milestone in the field of fuzzifications, while Ye et al. and Wei et al. [28, 29] worked on single variable differential calculus and Maclaurin operators under -rung orthopair fuzzy numbers, the associated limit, continuities, and derivatives. Liu and Wang [30] developed a multiple attribute decision-making method based on Archimedean Bonferroni operators of -rung orthopair fuzzy numbers. Subha and Sharmila [31] introduced the concept of interval-valued Fermatean fuzzy interior bi -subsemihypergroup, interval-valued Fermatean fuzzy bi -hypersemigroup and the relation between these ideals. Gul [32] presented three well-known multiattribute evaluation methods, namely, SAW, ARAS, and VIKOR, under Fermatean fuzzy environment and showed their applications in COVID-19 testing laboratories. Garg et al. [33] established some aggregation operators based on -norm and -conorm to cumulate the Fermatean fuzzy data in decision-making problems and exhibit the application in the COVID- testing facility. Recently, Zeb et al. [34] introduced aggregation operators in the environment of a Fermatean fuzzy soft set and presented their application in the decision-making problem of selecting the most critical COVID-19 patient. Shit and Ghorai [35] used Dombi -norm and -conorm operations on Fermatean fuzzy numbers. The more interesting work done by Chinnadurai et al. [36] is the introduction of Fermatean fuzzy numbers into a complex field and showing applicability in MCDM.

Discussing the novelty of our proposed work, we are compelled to say that there are still many problems where the subjected conditions over I-CFS and P-FCS may fail. For example, a panel of experts was invited to give their opinions about the feasibility of an investment plan, and they were divided into two independent groups to make a decision. One group considered the feasibility of the investment plan 0.8, while the other group considered the nonmembership degree 0.78. It was seen that , ; hence, the situation can not be described by IFS and PFS. If the above problem is taken as , the situation can be handled if we use the Fermatean fuzzy set. Now without any doubt, the same situation will occur in the fuzzification of I-CFS and P-CFS. This problem motivated us to define a more reliable extension of I-CFS and P-CFS called the F-CFS. In F-CFS, the membership and nonmembership are cubic fuzzy numbers of degree 3 with the condition that the supremum of membership plus nonmembership degree is . Henceforth, F-CFS is an alternative and more efficient tool compared to Pythagorean and intutionistic cubic fuzzy numbers in solving MADM problems. Because the space of acceptability or the spatial scope of F-CFS is greater than I-CFS and P-CFS and thus can support more ambiguous situations. The aggregation operators in this environment are more powerful and flexible in dealing with MADM problems. The rest of the paper is arranged as follows:

Section 2 is devoted to basic concepts related to F-CFS. Section 3 briefly discusses the basic operations and related properties of F-CFNs. In Section 4, we have introduced the novel aggregation operators in the environment of F-CFS. The decision-making approach and its practical application through a case study of selecting the best university based on some essential indicators for their ranking have been discussed in Section 5. The comparative analysis is provided in Section 6, and the summary is concluded in Section 7.

2. Preliminaries and Basic Concepts

In this section, we present some basic definitions and consequential properties that will help to understand the concept of F-CFS.

Definition 1 (see [1]). Let be a fixed set; then, a fuzzy set (FS) is an entity having the given formulations: where is a function from to and is called membership degree of in .

Definition 2 (see [3]). Let be a fixed set; then, IFS set is an entity having the following texture: where and are functions from to . Also the conditions and are imposed for all in and represent membership and nonmembership degrees of element in to the set

Definition 3 (see [37]). Let be a fixed set; then, a cubic fuzzy set (CFS) is a set of the form where is interval valued fuzzy set and is a single fuzzy set in .

Definition 4 (see [4]). Let be a universal set; then, a PFS set is an object of the form having the following structure: where are functions from to , such that , for all in and , for all in , and denote the membership and nonmembership degrees of element in to set .

Equation (5) is called the Pythagorean fuzzy index of in to the set . Also for each in . We denote a Pythagorean fuzzy number by .

Definition 5 (see [4]). Let , , and be the PFNs; then, the following are some basic operations defined for Pythagorean fuzzy numbers: (1)(2);(3);(4);(5);(6).

Definition 6 (see [12]). Let be a universe of entities; the interval-valued Pythagorean fuzzy set (IVPFS) denoted by can be defined as where and are the intervals and and similarly and for each in . Also, .

Definition 7. Let for all ; then, IVPF index value of to is given by

For above equations, the following conditions must be followed: (1)If and , then an IVPFS set reduces to PFS.(2)If , then an IVPFS set reduces to an IVIFS.

Next, we define the score and accuracy function for (I-VPFSs).

Definition 8 (see [12]). Let is an IVPFN; we can define the score function S in the following way: Clearly,

Definition 9 (see [12]). Let is an IVPFN; we can define the accuracy function of as where For ranking purposes, we use the following conditions: (1)If , then ;(2)If , then (a)If , then ;(b)If , then .

Definition 10 (see [12]). Let for be the collection of IVPFNs and ; then, the following operational laws are satisfied:

Definition 11 (see [12]). Let for be the collection of IVPFNs; then, interval-valued Pythagorean fuzzy-weighted averaging IVPFWA operator is defined as where be the weight vector of for for and and

Definition 12 (see [12]). Let for be the collection of IVPFNs; then, interval-valued Pythagorean fuzzy-weighted geometric IVPFWG operator is defined as where be the weight vector of for for and and also

Definition 13 (see [19]). Let be a fixed set; then, a P-CFS can be defined as where , , and . Also, The degree of indeterminacy for P-CFS is defined as Now for simplicity we call a Pythagorean fuzzy number denoted by .

Definition 14 (see [19]). Let and be three P-CFNs and where , , , , , and ; then, the operation laws are also satisfied.

Theorem 15. Let , and be three P-CFNs and where , , , , , and ; then, the following hold in P-CFNs:

Proof. Straight forward.

Definition 16 (see [19]). Let be a Pythagorean Fuzzy number; the score value of a PFN can be obtained through the following score function: where , such that and are membership and nonmembership values of .

In this section, we state the analogy of new and more approachable concept in fuzzy system called “Fermatean cubic fuzzy numbers (F-CFNs)” and will also discuss their important properties by using Pythagorean cubic fuzzy numbers and their properties. It must also be noted that in the rest of the paper we will denote a Fermatean cubic fuzzy number by

Definition 17. Let be a fixed set; then, a F-CFS denoted by can be defined as where and , i.e., and both are cubic fuzzy numbers and also denote the membership and nonmembership degrees of F-CFNs with the following conditions: The overhead restriction can also be scripted as The degree of indeterminacy for F-CFNs can be defined as

It is clear that if the power on the membership and nonmembership degrees is raised to , the above definitions reduce to I-CFS and indeterminacy function for I-CFS, while if it is raised to “2”, then they works as P-CFS and indeterminacy for P-CFS.

Example 18. Consider a fixed set and suppose we have a formulation in Z by the following texture:

Equation (23) shows that each term in is graded on the basis of Definition 17 F-CFS criteria. Each membership and nonmembership grade of , , and is restricted to the condition of F-CFNs. If Equation (23) is compared with Equation (19), then for we have

The same is also for and .

3.1. Basic Operations on F-CFNs

Definition 19. Let , , and be three F-CFNs and also , , , and then, the following are the operational laws regarding Fermatean cubic fuzzy numbers:

Theorem 20. Let , , and be three F-CFNs and for , , and with , , , and ; then, the following properties hold in Fermatean cubic fuzzy numbers: (1),(2),(3),(4),(5),(6).

Following the proofs in [19] these properties can easily be proved for F-CFNs. The score function is an essential tool to demonstrate the space analogy and to compare two or more fuzzy numbers. We define a score function as well as accuracy function for F-CFNs.

Definition 21. Let where and for be the family of F-CFNs; then, the score function for F-CFNs Fermatean cubic fuzzy numbers is scripted below: where ; the following situations should be kept in mind while finding score function for two or more than two F-CFNs: (1)If , then ,(2)If , then ,(3)If , then .

Definition 22. Suppose where and , the accuracy function for is and is defined below: where and clearly for it is the accuracy degree for I-CFNs, and for , it is an accuracy degree for P-CFNs. Furthermore, an important factor in fuzzy system is the distance measuring function which is our focal point too.

Definition 23. We define the distance measuring function between two F-CFNs by the following relation:

Example 24. Let us consider we have two F-CFNs such that and and then using Equation (28) to find the distance measure as given below.

The above value is the required distance between and obtained by using Fermatean cubic fuzzy distance measuring operator.

4. Aggregation Operators under Fermatean Cubic Fuzzy Environment

In this section, we present some aggregation operators under Fermatean cubic fuzzy environment such as F-CFWA, F-CFWG, and F-CFOWG in F-CFN environment.

Definition 25. Let for be the collection of F-CFNs. Suppose be the weight vector of for with and . Also, ; then, Fermatean cubic fuzzy-weighted averaging F-CFWA operator is defined as

The above results can be easily proved by studying the basics of fuzzy algebra.

Theorem 26. Let be a group of F-CFNs for and be the weight vector associated with for , , and . Then, the aggregation result according to our definition is also a F-CFN that is given below.

Proof. By mathematical induction, for we have Thus, From the above equation it is clear that the result is true. We suppose it is true for where is any positive integer: Let the result is true for ; then, This clearly exhibits that the result holds true for F-CFWA operator for , where is any positive integer and is index set.

Definition 27. Let for be the collection of F-CFNs and be the weight vector of for with and ; then, Fermatean cubic fuzzy-weighted geometric (F-CFWG) operator is defined as

Theorem 28. Let be a group of F-CFNs for and be the weight vector of with , , and ; then, the aggregated result according to our definition is also a F-CFN given by

Proof. By Mathematical induction, first we check it for , We suppose the result is true for , where is any positve integer. Next, we prove the result for , Hence, the result is true for all integers.

Theorem 29. Let for be the collection of F-CFNs with be the weight vector of for with and . Also, ; then, the following properties are satisfied for the operators defined under Fermatean cubic fuzzy environment. (1)Idempotency: for when for all ; then,

Proof. Given that Or If for , i.e., , , and as well as , , and .
Then, the above equation becomes <