(2,1)-Fuzzy sets: properties, weighted aggregated operators and their applications to multi-criteria decision-making methods

Orthopair fuzzy sets are fuzzy sets in which every element is represented by a pair of values in the unit interval, one of which refers to membership and the other refers to non-membership. The different types of orthopair fuzzy sets given in the literature are distinguished according to the proposed constrain for membership and non-membership grades. The aim of writing this manuscript is to familiarize a new class of orthopair fuzzy sets called “(2,1)-Fuzzy sets” which are good enough to control some real-life situations. We compare (2,1)-Fuzzy sets with IFSs and some of their celebrated extensions. Then, we put forward the fundamental set of operations for (2,1)-Fuzzy sets and investigate main properties. Also, we define score and accuracy functions which we apply to rank (2,1)-Fuzzy sets. Moreover, we reformulate aggregation operators to be used with (2,1)-Fuzzy sets. Finally, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making (MCDM) problems in the environment of (2,1)-Fuzzy sets. To show the effectiveness and usability of the proposed technique in MCDM problems, an illustrative example is provided.


Introduction
In the real world, we deal with ideas that are loaded with uncertainties and imprecision in several territories such as engineering, medicinal science, economics, and natural science. To handle this scenario Zadeh [34], in 1965, familiarized the concept of fuzzy set that extensively applied in many areas of multi-criteria decision-making (MCDM). Zadeh allotted a membership degree for each element in the domain; however, there are various real-life cases, the nonmembership degree is not come from the membership degree. To overcome this shortcoming, Atanassov [5] proposed an extension of fuzzy sets called an intuitionistic fuzzy set (IFS) which was successfully applied in various areas like medical diagnosis and decision-making [1,8].
Then, for sake of enlarging the domain of membership and non-membership degrees, Yager [28] defined a Pythagorean fuzzy set (PFS) as a generalization of intuitionistic fuzzy set. It efficiently deals with the situations which the sum of B Tareq M. Al-shami tareqalshami83@gmail.com 1 Department of Mathematics, Sana'a University, Sana'a, Yemen their membership and non-membership degrees of a specified attribute is greater than one. To made a general umbrella of the generalization class of intuitionistic fuzzy set, Yager [29] presented the idea of q-rung orthopair fuzzy set (q-ROFS). In 2019, Senapati and Yager [22] discussed a Fermatean fuzzy sets (FFS) as a special case of q-rung orthopair fuzzy sets obtained by putting q = 3. Recently, Ibrahim et al. [9] have brought a new class of fuzzy sets which lies between the grade spaces of Pythagorean and Fermatean fuzzy sets called (3,2)-Fuzzy sets. They applied to establish new kinds of weighted aggregation operators and address more uncertainty situations than Pythagorean fuzzy sets. Then, Al-shami et al. [3] have investigated the concept of SR-fuzzy sets as a new extension of fuzzy sets and applied to generate new aggregated operators.
Since vagueness is a noteworthy issue in numerous territories and its complexity increases day by day, some improvements for fuzzy theory become necessary to keep up with these developments. In this regard, study fuzziness with bipolarity view was investigated in some published literature like [18,19]. Also, hybridization of fuzzy sets with some uncertainty tools such as rough and fuzzy soft was the goal of some articles such as [2,6,7,31,32]. Other classes of fuzzy sets were established and investigated in many manuscripts such as [10,11]. Besides all of these, abstract structures like topologies and their main properties were studied in fuzzy settings; see, for example, [4,20].
Decision-making, as a widely used concept of human daily life, gets more complicated with the progression of communication and technology. One of the major issues for decision-makers is how to obtain a unique result from the collective information given by different sources. To do this, different types of aggregation operators have been introduced which reduce the set of finite values in the decision-making process into a single value. Under intuitionistic fuzzy environment Xu [24] initiated a weighted averaging aggregation operator, and Xu and Yager [27] studied a weighted geometric aggregation operator. Lately, several types of aggregation operators have been explored in the environment of intuitionistic fuzzy sets in the published literature; see, [13,16,25,26,33]. Also, these operations have been investigated in the frame of Pythagorean fuzzy sets as given by Khan et al. [12], Peng and Yuan [15], Shahzadi et al. [23], Rahman et al. [17], and Yager and Abbasov [30]. Investigation of aggregation operators in the frame of Fermatean fuzzy sets was conducted in [21].
Multi-attribute decision-making (MADM) problems are constructed of a finite set of options/alternatives and a finite set of criteria/attributes. In this type of problems, it is important to evaluate the quality of the input data. But it's not only about selecting the environment (FS,IFS,PFS,FFS,etc.), it's also about how you are modeling the problem. In other words, which one of these environments frames the phenomena or problem under study? That is, it is not possible to use some types of fuzzy sets to model some actual problems because the information form (with respect to their membership and non-membership grades) in this problem does not satisfy these types of fuzzy sets (with respect to their constraints); hence, the comparison between the effectiveness or who is the best of these types of fuzzy sets is meaningless.
The motivation of doing this research is, first, to define a new generalization of intuitionistic fuzzy set, namely, (2,1)-Fuzzy sets. This generalization enlarges the space of membership and non-membership degrees more than intuitionistic fuzzy sets. As we see this class does not obtain from the class of q-rung orthopair fuzzy sets since the difference of the values q of membership and non-membership grades. Second, to establish a new kind of weighted aggregation operators which can be employed to handle some practical problems; especially, those that are evaluated with different importance of their membership and non-membership grades. Finally, to display a multi-criteria decision-making methods based on the introduced operators for choosing the optimal alternative. It worthily noting that the grades space of our class is smaller than the grades space of all types of qrung orthopair fuzzy sets; however, it provides another frame more convenient to represent the input data for some real-life issues.
The rest of this manuscript is arranged as follows: (1) In "Preliminaries", we recall some definitions to make this article self-contained. (2) We devote "(2,1)-Fuzzy sets" to introduce a new family of generalized IFSs called (2,1)-Fuzzy sets. We display a set of operations for (2,1)-Fuzzy sets and scrutinize main properties. (3) In "Aggregation of (2,1)-fuzzy sets with applications", the concepts of weighted aggregated operators via (2,1)-Fuzzy sets are investigated and characterized. (4) In "Application of (2,1)-FSs to MCDM problems", we describe an MCDM method under these operators and present a practical example to show how it carries out. (5) Ultimately, we outline the main achievements of the paper and propose some upcoming works in "Conclusions".

Preliminaries
To make this study self-sufficient, we briefly present a few concepts engaged in the remaining parts of this study. We also present some interpretations for the beyond motivations to initiating the extensions of fuzzy sets.
Definition 1 [5] The intuitionistic fuzzy set (IFS) is defined over a universal set B as follows.
The indeterminacy degree of each ν ∈ B with respect to an IFS is given by Remember that if δ Ω (ν) = 1 − λ Ω (ν) for every element ν ∈ B, then an intuitionistic fuzzy set Ω becomes a fuzzy set.
The natural question that puts itself is why the nonmembership degree is not the complement of membership degree in all cases? To our best knowledge, the membership and non-membership degrees are calculated with respect to independent criteria, or sometimes they are evaluated by two independent groups of experts, one specifies the membership and the other specifies the non-membership. That is, the standards of a membership degree need not be the complement of the standards of a non-membership degree. To explain this matter, the example below is provided. Example 1 Consider B is a group of students is examined in Mathematics. They are evaluated by 50 questions. Every student has two options, answer (correctly or incorrectly) the question or does not answer the question. The followed technique of evaluating the students' performance is given as an IFS Ω = { ν, δ Ω (ν), λ Ω (ν) : ν ∈ B} such that δ Ω (ν) = c 50 and λ Ω (ν) = d 50 , where c and d denote the number of correct answers and the number of incorrect answers, respectively. Assume that Mustafa is a student of this group, and his performance in the exam is as follows, he correctly answered 30 questions, incorrectly answered 15 questions, and did not answer five questions. The corresponding IFS of his performance is Ω = Musta f a, 3 5 , 3 10 . It is clear that Definition 2 [28] The Pythagorean fuzzy set (PFS) is defined over a universal set B as follows.
The indeterminacy degree of each ν ∈ B with respect to a PFS is given by It can be seen that any Pythagorean fuzzy set is an intuitionistic fuzzy set, but the converse fails as the next example shows.
To enlarge the grades space of membership and nonmembership degrees, Senapati and Yager [22] defined the concept of Fermatean fuzzy set as follows.
Definition 3 [22] The Fermatean fuzzy set (FFS) is defined over a universal set B as follows.
The indeterminacy degree of each ν ∈ B with respect to a FFS is given by With the aid of example below, we demonstrate that some Fermatean fuzzy sets fail to be Pythagorean fuzzy sets.

(2,1)-Fuzzy Sets
The core concept of this manuscript called "(2,1)-Fuzzy Sets" is introduced herein. The aim of presenting this concept are to extend the grade space of intuitionistic fuzzy sets and create a suitable environment to model some real-life issues. We elucidate that this concept lies between the classes of intuitionistic fuzzy sets and Pythagorean fuzzy sets. Then, We define the main set of operations for (2,1)-Fuzzy sets and find out their master features.
In what follows, we compare (2,1)-FS with IFS and PFS.
Hence, the proof is completed.
The converses of the assertions furnished in Proposition 1 fail as the next example illustrates.
The operators of ∪ and ∩, given in Definition 5, are generalized for arbitrary numbers of (2,1)-FSs as follows.
We close this section by defining the score and accuracy functions of (2,1)-FSs which will be helpful later to rank (2,1)-FSs.

Definition 7
The score function score :
To efficiently make a comparison of (2,1)-FSs, we introduce the concept of accuracy function for (2,1)-FSs as follows.

Definition 9
The accuracy function acc : We make use of the score and accuracy functions to compare between (2,1)-FSs.

Aggregation of (2,1)-fuzzy sets with applications
In this section, we first introduce some new operations on (2,1)-Fuzzy sets and explore their main properties. Then, we initiate novel types of aggregation operators with respect to (2,1)-Fuzzy sets and scrutinize the interrelations between them. We display some elucidative examples.

Some operations on (2,1)-FSs
Herein, we define some operations over the family of (2,1)-Fuzzy sets, and explore the interrelations between them.
Then, we have This implies that Following similar arguments, we obtain Following similar arguments, we obtain Hence, Ω 1 ⊗ Ω 2 is a (2,1)-FS.
Proof From Definition 12, we obtain: And

Aggregation of (2,1)-fuzzy sets
Herein, we generalize some aggregation operators to the environment of (2,1)-Fuzzy sets, and display some formulas which show the relationships between them.

Application of (2,1)-FSs to MCDM problems
We dedicated this section to investigating a MCDM problem using the four types of aggregations operators given in the foregoing section. We propose some algorithms that show how this type of problem is handled, and provide an illustrative example.

Representation of MCDM problems and their algorithms under the environment of (2,1)-FSs
MCDM problems are one of the challenging and fast techniques for all decision makers for getting the best alternative(s) among the set of possible ones according to multiple criteria. To illustrate that, assume B = {b i : i = 1, 2, . . . , n} as a set of n different alternatives that have been evaluated (by the decision maker) under a set of m different criteria C = {c j : j = 1, 2, . . . , m}. Presume that the decision maker estimates the preferences in terms of (2,1)-FNs: θ i j = δ i j , λ i j i× j , where 0 ≤ δ 2 i j + λ i j ≤ 1 and δ i j , λ i j ∈ [0, 1] for all i = 1, 2, . . . , n and j = 1, 2, . . . , m such that δ i j and λ i j respectively represent the degree that the alternative b i fulfills and doesn't fulfill the attribute c j provided by the decision maker. Thus, MCDM problems can be concisely expressed in a (2,1)-Fuzzy decision matrix In what follows , we explain the steps used in the proposed methodology for MCDM: Step 1 : formulate the (2,1)-Fuzzy decision matrix θ = (θ i j ) n×m for a MCDM problem under study.
Step 4 : Compute the scores and accuracy functions for each (2,1)-FNs provided in Step 3. According Remark 3 the ordered values obtained from these operators need not be a (2,1)-FS; however, we extend the formulas of scores and accuracy functions given in Definition 10 for those ordered values.
Step 5 : Compare the given alternatives based on the scores and accuracy.
Input : The set of alternatives B and the set of multi criteria C. Output: select the most desirable alternative(s).

Illustrative examples
In this subsection, we explain the above-mentioned approaches by the following example which investigated a multiple criteria decision-making problem.

Example 10
Assume that a certain university wants to assign a permanent faculty member from the set of candidates U = {Redhwan, Al-Harith, Mustafa, Bushra, Sarah}. For this, the university authorities consider the following five criteria C = {c i : i = 1, 2, 3, 4, 5}, where: c 1 represents the number of research publications, c 2 represents the teaching experience, c 3 represents the regularity and punctuality, c 4 represents the number of conferences participated, and c 5 represents the behavior with students through the class.
After a deep discussion, a committee (forms by the university authorities) proposed a weight vector corresponding to every criteria ω = (0.25, 0.35, 0.1, 0.1, 0.2) T . A committee assesses the performance of these candidates under the (2,1)-FSs environment as given in Table 1. Every ordered pair (δ, λ) given in Table 1 represents the membership and non-membership degrees of a candidate to fulfill and dissatisfy the corresponding criteria (or attribute) such that 0 ≤ (δ) 2 + λ ≤ 1 and δ, λ lie in [0, 1].
Assume that the proposed approach for accessing the best candidate with appreciation to every criterion provided using the committee is furnished according to the different types of (2,1)-FS operators introduced in Definition 13. Then, we compute the score function for each candidate. If there are some candidates who have the same score function, then we compute their accuracy function to decide who is the optimal candidate(s); see, Table 2.
According to the computations induced from the four operators of aggregation, we find that the optimal ranking order of the five candidates induced from a (2, 1)-F W A operator is Sarah. It should be noted that the candidates Redhwan and Sarah are equal with respect to the score function; so that, we complete comparison by computing their accuracy functions which show that Sarah is the best candidates to get this job. The rank of the candidates induced from a (2, 1)-F W A operator is Sarah Redhwan Bushra Mustafa Al-Harith.
On the other hand, note that the values of score functions induced from the other aggregation operators are distinct for all candidates, so there is no need to compute the accuracy function. Thus, the rank of the five candidates respectively induced from (2, 1)-F W G, (2, 1)-F W P A and It can be noted from the above discussion that the selection of the optimal permanent faculty member is based on two factors, first one is the type of generalizations of IFSs, which is herein a (2,1)-FS. The second one is the aggregation operator provided by the committee to evaluate the performance of the candidates.

Conclusions
In this paper, we have established a new class of orthopair fuzzy sets, namely (2,1)-Fuzzy sets. Two of the merits of this class are to, first, enlarge the space of membership and non-membership more than IFSs, which means overcoming some limitations of IFS in handling some situations that have the sum of membership and nonmembership grades exceed one. Second, to offer a convenient frame to model some reallife problems that are evaluated with different importance of their membership and non-membership grades. On the other hand, the limitation of the proposed class is that its grades space is smaller than the grades space of q-rung orthopair fuzzy sets.
Our contributions through the manuscript are as follows. We have defined some operations for (2,1)-Fuzzy sets and presented main characterizations. In addition, we have introduced four types of aggregation operators in the environment of (2,1)-Fuzzy sets and reveled the relationships among them. Ultimately, we have exploited the proposed aggregation operators to address the decision-making issues and provided the algorithms used in the evaluation with a flow chart. A numerical example has been given to show how the followed method assisted us with being effective in decision problems.
In future works, we intend to display a novel class of orthopair fuzzy sets that forms an umbral for all the generalizations of IFSs. Theoretically, we shall benefit from (2,1)-Fuzzy sets to construct a new type of fuzzy topologies.