Similarity measures for Fermatean fuzzy sets and its applications in group decision-making

The intention of this paper is to propose some similarity measures between Fermatean fuzzy sets (FFSs). Firstly, we propose some score based similarity measures for finding similarity measures of FFSs and also propose score based cosine similarity measures between FFSs. Furthermore, we introduce three newly scored functions for effective uses of Fermatean fuzzy sets and discuss some relevant properties of cosine similarity measure. Fermatean fuzzy sets introduced by Senapati and Yager can manipulate uncertain information more easily in the process of multicriteria decision making (MCDM) and group decision making. Here, we investigate score based similarity measures of Fermatean fuzzy sets and scout the uses of FFSs in pattern recognition. Based on different types of similarity measures a pattern recognition problem viz. personnel appointment is presented to describe the use of FFSs and its similarity measure as well as scores. The counterfeit results show that the proposed method is more malleable than the existing method(s). Finally, concluding remarks and the scope of future research of the proposed approach are given.

The intention of this paper is to propose some similarity measures between Fermatean fuzzy sets (FFSs). Firstly, we propose some score based similarity measures for finding similarity measures of FFSs and also propose score based cosine similarity measures between FFSs. Furthermore, we introduce three newly scored functions for effective uses of Fermatean fuzzy sets and discuss some relevant properties of cosine similarity measure. Fermatean fuzzy sets introduced by Senapati and Yager can manipulate uncertain information more easily in the process of multicriteria decision making (MCDM) and group decision making. Here, we investigate score based similarity measures of Fermatean fuzzy sets and scout the uses of FFSs in pattern recognition. Based on different types of similarity measures a pattern recognition problem viz. personnel appointment is presented to describe the use of FFSs and its similarity measure as well as scores. The counterfeit results show that the proposed method is more malleable than the existing method(s). Finally, concluding remarks and the scope of future research of the proposed approach are given. .

Introduction
The theory of fuzzy set was established by Zadeh (Zadeh, 1965) in 1965 and it dealt with imprecision, vagueness in real life situations. In the year 1970, Bellman and Zadeh (1970) presented the notion of decision making problems entailing uncertainty. The concept of intuitionistic fuzzy sets (IFSs) was introduced by Atanassov ( Atanassov, 1986) in1986 by presenting the objective world from three aspects of support, opposition and neutrality, respectively and thus have been widely considered and applied by many researchers ( Fei et al., 2018;Zhang et al. 2018). Also, many researchers have given additional attention to interval valued fuzzy sets (IVFSs) (Turksen, 1986;Gorzalczany, 1987) interval valued intuitionistic fuzzy sets (IVIFSs) (Atanassov & Gargov, 1987;Atanassov, 2012), which are all the generalization of the fuzzy set proposed by Zadeh (Zadeh, 1965) and applied them in diverse decision making problems. However, the fuzzy set takes only a membership function and the degree of non-membership function which is just a complement of the degree of membership function. There may be a situation where the sum of the membership function and non-membership function is greater than one. Thus orthopair fuzzy sets have been introduced in which the membership grades of an element are pairs of values in the unit interval, ( ), ( ) x x α β , one of which indicates support of membership in the fuzzy set and other indicates support against membership in the fuzzy set. For example, Atanassov's classical intuitionistic fuzzy sets (Atanassov,1983;Atanassov, 1986;Atanassov et al., 2013) and Atanassov's second kind of intuitionistic fuzzy sets ( Atanassov, 2016). Yager (Yager, 2013;Yager, 2014) introduced another orthopair of fuzzy sets, known as Pythagorean fuzzy set (PFS), where the square sum of the support of membership and support against membership value is equal to or x X ∈ less than one. Also, many researchers have paid more attention to interval valued Pythagorean fuzzy sets (IVPFSs) (Garg, 2017;Garg, 2018), which is the generalization of the Pythagorean fuzzy sets (PFSs) set proposed by Yager and applied them in several decision making problems. PFSs and IVPFSs have attracted the attention of many researchers within a very short period of time. There are several methods in the field of PFS to solve real-life multi-criteria, decision-making problems (Ye, 2009;Zhang et al., 2014;Zhang, 2016;Gou et al., 2016;Geng et al., 2017;Jing et al., 2017;Li et al., 2018). Several researchers have also proposed real-life applications under a Pythagorean fuzzy environment. For more details one may mention the works of Li et al. , Zhou et al. (Zhou et al., 2018), Bolturk (Boltruk, 2018), Qin (Qin, 2018), Wan et al. (Wan et al., 2018), Lin et al. (Lin et al., 2018) and Chen (Chen, 2018). But, if orthopair fuzzy sets as <0.9, 0.6>, where 0.9 is the support of the membership of certain criteria of a parameter and 0.6 is the support against membership then it does not follow the condition of IFS as well as PFS. However, the cubic sum of the support of membership and support against membership degrees is equal to or less than one. And in this situation Senapati and Yager (Senapati & Yager, 2019;Senapati & Yager, 2020) very recently introduced the Fermatean Fuzzy set (FFS). They also showed that FFSs have more uncertainty than IFSs and PFSs and are capable of handling higher levels of uncertainties (Bai, 2013) and solved MCDM problems.
A similarity measure is an important concept for controlling the degree of similarity between two objects in many fields, such as pattern recognition, medical diagnosis, personnel appointment etc. various types of similarity measures have been introduced (Pappis & Karacapilidis, 1993;Chen, 1995;Li & Cheng, 2002;Liang & Shi, 2003 ;Hung & Yang, 2004;Ye, 2011;Zhou et al., 2014;. Among them, some similarity measures of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs) have been proposed. For example, Li and Cheng (2002) studied a similarity measure between IFSs and applied it to pattern recognition. Huang and Yang (2004) proposed the similarity measure between IFSs based on the Hausdorff distance and used it to calculate the degree of similarity between IFSs. Nguyen (2016) proposed a new knowledge-based similarity measure between IFSs and applied it to pattern recognition. Zhang (2016) introduced a novel approach based on similarity measures for Pythagorean fuzzy multiple criteria group decision making. Zhang et al. (2012) presents a type of score function on intuitionistic fuzzy sets with double parameters and its application to pattern recognition and medical diagnosis. Ejegwa (Ejegwa, 2020;Ejegwa, 2019) introduced distance and similarity measures of Pythagorean fuzzy sets. Ye et al. (2011) proposed a cosine similarity measure for IFSs (CIFS) and applied it to pattern recognition. Also, Ye (2013) presented the cosine similarity measure for IVIFSs (CIVIFS) and applied it to group decision-making problems. On the other hand  studied Cosine Similarity Measure between Hybrid Intuitionistic Fuzzy Sets and Its Application in Medical Diagnosis. Very recently, Liu et al. (2019) introduced Distance measure for Fermatean fuzzy linguistic term sets based on linguistic scale function. But as per our knowledge and belief, no one studied the distance based similarity measure and cosine similarity measure based on score function between FFSs. Influenced by this, in this paper, we shall propose distance based similarity measure and the cosine similarity measure of FFSs to handle uncertain information. In addition, applying the proposed similarity measures, we have solved group decision-making problems which are very fascinating in the real-world. In this paper, we have introduced three newly improved score functions for ranking of Fermatean fuzzy sets. We have applied the proposed score function to calculate the similarity measure and applied it to solve pattern recognition problem viz. personnel appointment. Finally, a numerical example is given to illustrate the effectiveness of the proposed distance based and cosine similarity measures, which are also compared with the existing similarity measures.
The contributions of the present paper are the following. In section 2, some definitions and basic concepts related to Fermatean fuzzy sets are described. Section 3 gives the ranking of FFSs based on proposed score functions. In section 4, a similarity measure of Fermatean fuzzy sets has been presented. Section 5 gives the group decision making with similarity measures between FFSs. Numerical example is given in Section 6. Section 7 contains the conclusion of the paper with the future scope of research.

Preliminaries
In this section some basic definitions about Fermatean fuzzy sets (FFSs) are discussed. After that some score functions are proposed to implement the entire paper.
Definition 2.1. (Senapati & Yager, 2019;Senapati & Yager, 2020) Let X be a Universal set. A Fermatean fuzzy set (FFS) is an object of the form Then the basic arithmetical operations of two Fermatean fuzzy sets 1 F and 2 F are defined as follows: (i) Addition: Then their set operations are defined as follows: (i) Union:

Ranking of Fermatean fuzzy sets
For the purpose of ranking, we have proposed some score functions ( ) F ψ , ( ) [0,1] F ψ ∈ which are as follows: Hence for Type 3 score function 3 ( ) [0,1] F ψ ∈ . In similar manner one can prove that 1 ( So we claim that our proposed score functions are justified as all score functions gives the same results similar to Senapati and Yager (Senapati & Yager, 2019;Senapati & Yager, 2020).
It is either greater than equal to 0.5 or less than 0.5. If we consider So score values are same although

Similarity measures for Fermatean fuzzy sets
The similarity measure can measure the similarity degree between two different alternatives. In this section we have proposed some score based similarity measures with Fermatean fuzzy sets using the concept of distance metric and some properties are also presented.
are three FFSs. The similarity measure S between 1 F and 2 F is a mapping : satisfies the following conditions: The condition (i) expresses that S is bounded-ness in 1 2 , F F . Condition (iii) gives the fact that S is symmetric in 1 2 , F F . The inequality (iv) is generally called the triangle inequality.
Let us assume that there are two PFSs A and P in { } 1 2 , ,..., n X x x x = .Also let ψ be the score function. Then the similarity measure between A and P as follows: Now, we have also defined score based Cosine similarity measure as follows:

Group Decision-Making with Similarity Measure between Fermatean fuzzy sets
In this section, we shall apply different similarity measure between Fermatean fuzzy sets in personnel appointments problem. Here, we need several experts to evaluate properly the candidates/applicants of a particular competitive examination or personal interview for employment. The panel expert usually provides his/her preferences for candidates' qualification as well as positions in terms of Fermatean fuzzy sets.
Let us assume that a Company has organized an aptitude test for the selection of employees.

Assume that 1
A is a set of qualifications P is a set of positions and A is a set of applicants. It is to be noted that for this   Table 3 Similarity measure of Applicants vs. Positions using ( , ) S A P ( , ) S A P  Table 5.3 decision makers have to take decision as per their requirements. They should take horizontal decision if the organization can requites all the applicants. Also decision makers should take vertical decision if the organization has limited fund/resources to pay salary. It is to be noted that vertical decision is more competitive compare to horizontal decision.

Numerical example and discussion
In this section, we have taken a numerical example to discuss the proposed approach.   Table 5 gives the stipulated standing qualifications by the expert panel, for each of the positions as below: The example is numerically same with the example solved by Ejegwa (Ejegwa, 2019) but descriptively different. Now using our proposed similarity measures we have constructed the following tables as below:

Table 10
Similarity measure of Applicants vs. Positions using (4) and Type-2 score function Applicants

Table 16
Similarity measure of Applicants vs. Positions using (6) and Type-2 score function Applicants   Table 4 to 26) have been presented in Table 27. Now, we have discussed horizontal decision. For instance, if we use (9) as a similarity measures then 1 A is appropriate for 5 P , 2 A is appropriate for 4 P , 3 A is appropriate for 2 P and 4 A is appropriate for 4 P . Here it is seen that all 2 A is more appropriate than 4 A . The same result occurs for Type 1, Type 2 and Type 3 score functions (see last three row of Table 27). Consequently, the obtained result is tallied with the result reported by Ejegwa (Ejegwa, 2019). So, we claimed that our similarity measure and score functions proposed here are justified. Similar conclusion may be drawn from other rows of table 27. All the vertical decisions (From Table 6 to 26) have been presented in Table 28.

Table 27
Horizontal Decisions taken from Table 6 to 26      A , positions 3 P and 5 P are appropriate for 1 A . The same result occurs for Type 1, Type 2 and Type 3 score functions (see last three row of Table 28). Consequently, the obtained result is tallied with the result reported by Ejegwa (Ejegwa, 2019). So, we claimed that our similarity measure and score functions proposed here are justified. Similar conclusion may be drawn from other rows of Table 28.

Table 28
Vertical Decisions taken from Table 6 to 26 1 P 2 P 3 P 4 P 5 P

Concluding remarks
The notion of Fermatean fuzzy sets is a new type of fuzzy sets to handle uncertain information more easily and effectively than the others existing fuzzy sets. Application of other fuzzy sets viz. intuitionistic fuzzy sets, Pythagorean fuzzy sets have been seen hugely in the existing literature but application of Fermatean fuzzy sets is very limited. So inspired from this, in this paper, we have solved Fermatean fuzzy multiple attribute group decision making problems where all the attributes are expressed in terms of Fermatean fuzzy sets. Firstly, we discuss the concept of Fermatean fuzzy sets. After that we propose some newly score function for ranking of Fermatean fuzzy sets. Then, first time we have proposed some score based similarity measures between two Fermatean fuzzy sets. We have also proposed score based cosine similarity measure between two Fermatean fuzzy sets. Applying all these similarity measures we have solved personnel appointments problem. Finally, a numerical example has been considered and solved for illustration purpose. The results obtained after applying our proposed score functions as well as different similarity measures coincide with the other which is available in existing literature. Therefore, it has been concluded that the proposed score functions of FFSs and different similarity measures proposed here can be used to solve pattern recognition problems, medical diagnosis problems and MCDM problems considering Fermatean fuzzy sets. In future research, one may apply the proposed similarity measures in the field of group decision making problems arising in the field of science, engineering and management science etc.