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 .
3. Fermatean Cubic Fuzzy Numbers (F-CFNs) and Related Properties
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
for which is a F-CFN and this completes the proof. Same procedure can be repeated for F-CFWG operator.
(2)Boundary: , for all , where and .(3)Monotonicity: let and for be the group of F-CFNs if we have , ; then,
The proofs of boundary and monotonicity are straightforward based on [19].
Lemma 30. Consider for and ; then,
for ; here, the equality holds only if .
Theorem 31. Let be the group of all F-CFNs; then, , where is the associated weight vector of such that and for .
Proof. The proof of this theorem is based on Lemma 30.
5. Application in MADM Problem
Here, we present a MADM problem and apply our modified aggregation operators to prove the applicability and smoothness of our newly introduced operators. To complete the process and find out a solution to a decision-making problem, the following is a flowchart in Figure 1.
5.1. Illustrative Example
To show the application of the proposed method to MADM, we consider a practical example of university ranking. Currently, universities pay considerable attention to their ranking because it significantly affects government investment, social donations, recruitment, etc. There are many systems to rank the universities throughout the world. Here, we give four more influential and authoritative evaluation systems. Assume that be the set of four universities or alternatives that are to be ranked upon the following set of five criteria . Namely, they are (1)Stability of the indicator system ()(2)Stability of weight setting ()(3)Correlation of historical ranking data ()(4)Consistency of data trends ()(5)Consistency of teacher students ratio ()
To evaluate the stability of the evaluation system, first we obtained expert’s opinions in the form of F-CFNs. There are a number of parameters to be considered, but the above set of criterion was established by the four experts/decision-makers This means that the decision makers will give their opinions in the form of F-CFNs according to the set of attributes. It is obvious that, in a set of criteria, the importance of criterion may not be the same, and hence, we take the weight vector that gives the weight or importance of each criterion. To evaluate the best university from the set , expert’s opinions were obtained in the form of F-CFNs and following decision matrices were constructed on the basis of their opinions.
Step 1. In the following, Tables 1–4 are the rating values arranged in matrix form that were assigned by the experts about the four alternatives, considering the set of parameters
Step 2. In this step, the data is examined for the purpose of normalization. The normalization is done by converting the cost type parameters into benefit type if there is any. Since, in our case, there is no cost type parameter, so this step is skipped.
Step 3. This is the aggregation phase and so we first use the novel F-CFWA operator (and later F-CFWG operator) in order to aggregate the data about the alternatives. Table 5 is the merged matrix or aggregated matrix by utilizing the F-CFWA operator on Tables 1–4 and is given below.
Further, the F-CFWA operator is once again used to aggregate Table 5 and the following preference values of the alternatives are obtained:
Step 3. In this step, we utilize the score function to find the score values of for ,
Step 4. All the alternatives are ranked in descending order on the basis of their score values. Therefore,
Using the score function, we find that is the best choice. Thus, is the best university upon the set of five criteria. Furthermore, can be selected as the best option in case is not available.
5.2. Using Fermatean Cubic Fuzzy-Weighted Geometric Operator (F-CFWG)
Similarly, utilizing F-CFWG on Tables 1–4, we obtained the following aggregated decision matrix (as given in Table 6).
Note that same steps are followed for the whole calculations.
The preferences values obtained through the proposed F-CFWG operator are given below:
The score function is used to find the score values of for ,
Therefore, the final ranking is:
Again, we see that the ranking order is the same as the one obtained through F-CFWA operator and that is the best alternative.
The graphical interpretation of the decision-making process to best examine the nature and behavior of the modified operators and the results obtained through these operators on the basis of associated score values are presented in Figures 2 and 3. They illustrate the graphical behavior of the results obtained through the F-CFWA and F-CFWG, respectively.
We can easily exhibit the conclusion that in Figures 2 and 3, the alternative has a peak output; thus, is the best choice with respect to averaging and geometric operators. For further rectification and demolishing the novel operator, we now compare our results with some existing operators that are developed in the environment of cubic fuzzy set.
6. Comparison with Aggregation Operators under P-CFNs
In this section, we compare our results with aggregation operators of Pythagorean cubic fuzzy numbers such as Pythagorean cubic fuzzy Hamacher-weighted averaging (PCFHWA), Pythagorean cubic fuzzy Hamacher order-weighted averaging (PCFHOWA), Pythagorean Cubic fuzzy Hamacher hybrid averaging (PCFHHA), Pythagorean Cubic fuzzy Hamacher-weighted Geometric (PCFHWG), Pythagorean Cubic fuzzy Hamacher order-weighted Geometric (PCFHOWG), and Pythagorean Cubic fuzzy Hamacher hybrid geometric (PCFHHA) operators, Abdullah et al. in [38]. The performance index values of the alternatives using aggregation operators as abovementioned, each alternative’s output index value, and their respective ranking are calculated regarding all criteria. It indicates that in Table 7, with the highest performance index of 0.8566, alternative has the highest performance value.
Next, we have utilized the newly developed operator (F-CFWA) for the data given in [38] (from Tables 1–3). The aggregated results using the novel operator are given in Table 8 and their corresponding preference values are tabulated in Table 9.
Finally, the ranking order on the basis of the score values are given in Table 10. The ranking orders of the F-CFWA operator are identical with the ranking orders of the alternatives in Table 7. Overall comparison revealed to us that the best choice is under criteria () with weighted vector .
This shows the stability of our proposed operators in Fermatean cubic fuzzy environment, and hence, the presented work can be utilized instead of other methods. The developed operators are based on novel concept of F-CFS having greater capability of supporting ambiguous situations as the spatial scope of the F-CFS is lager compare to other cubic fuzzy sets. Hence, the advantage of the presented work is that the F-CFWA and F-CFWG operators yield better approximation than other existing operators.
Figures 4 and 5, respectively, elaborates that the ranking orders of the Hamacher aggregation operators defined for Pythagorean cubic fuzzy numbers [38] are identical and hence support the stability of the presented work.
7. Conclusion and Future Work
Fermatean cubic fuzzy set has prominent applications in MADM and is a highly efficient tool to deal with data involving fuzziness as compared to I-CFS and P-CFS. The aggregation operators are very effective for evaluating the given alternatives in decision-making process as they combine the different values about individuals into a unified form. In this article, we have designed Fermatean cubic fuzzy aggregation operators such as F-CFWA and F-CFWG. Some functional characteristics such as idempotency, boundary, and monotonicity of these novel operators have been discussed. We have defined Fermatean cubic fuzzy distance operator between two Fermatean cubic fuzzy numbers and illustrated it through an example showing better result as compare to I-CFS and P-CFS. Since the power of decimals in F-CFS is higher than I-CFS and P-CFS, so by expansion, their results lead to a more precised value. On the basis of F-CFWA and F-CFWG operators, we have proposed a MADM approach to show the effectiveness of the developed operators. A numerical example is also provided in order to show that the proposed operators give us a more precise way to resolve the MADM problem. Also, the stability of the proposed work is provided by comparing it with some existing work. We concluded that the I-CFS and P-CFS are subsequent to F-CFS. Furthermore, we developed the operational laws associated to it such as associativity, commutativity with respect to algebraic sum, and algebraic product. In the future, we plan to develop more generalized aggregation operators like Fermatean cubic fuzzy hybrid aggregation operators, Fermatean cubic fuzzy Einstein aggregation operators, and Fermatean cubic fuzzy Hamacher aggregation operators. We also plan to generalize our presented work to introduce -rung orthopair cubic fuzzy set and its application in MADM problem of medical diagnosis.
Data Availability
The data sets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University Makkah for supporting this work under grant code: 22UQU4310396DSR31.