A novel approach to MADM problems using Fermatean fuzzy Hamacher prioritized aggregation operators

A generalized form of union and intersection on FFS can be formulated from a generalized t-norm (TN) and t-conorm (TCN). Hamacher operations such as Hamacher product and Hamacher sum are good alternatives to produce such product and sum. The Hamacher operations can generate more flexible and more accurate results in decision-making process due to the working parameter involved in these operations. The intuitionistic fuzzy set, briefly as IFS and its extension involving Pythagorean fuzzy set (PFS) and Fermatean fuzzy set (FFS), are all effective tools to express uncertain and incomplete cognitive information with membership, nonmembership and hesitancy degrees. The Fermatean fuzzy set (FF-set) carries out uncertain and imprecise information smartly in exercising decision making than IFS and PFS. By adjusting the prioritization of attributes in FF environment, in this course of this article, we first device new operations on FF information using prioritized attributes and by employing HTN and HTCN, we discuss the basic operations. Induced by the Hamacher operations and FF-set, we propose FF Hamacher arithmetic and also geometric aggregation operators (AOs). In the first section, we introduce the concepts of an FF Hamacher prioritized AO and FF Hamacher prioritized weighted AO. In the second part, we develop FF Hamacher prioritized geometric operator (GO) and FF Hamacher prioritized weighted GO. We study essential properties and a few special cases of our newly proposed operators. Then, we make use of these proposed operators in developing tools which are key factors in solving the FF multi-attribute decision-making situations with prioritization. The university selection phenomena are considered as a direct application for analysis and to demonstrate the practicality and efficacy of our proposed model. The working parameter considered in these AOs is analyzed in different existing and proposed AOs. Further, comparison analysis is conducted for the authenticity of proposed & existing operators.


Introduction
Pythagorean fuzzy sets (PFSs) Xu (2007); Xu and Yager (2006), an augmentation of intuitionistic fuzzy sets (IFSs), tremendously have attracted many potential researchers in recent times. Yager Xu and Yager (2008) was the first to develop a useful decision-making technique based on Pythagorean fuzzy information for use in MCDM scenarios involving Pythagorean fuzzy information. In Yager and Abbasov (2013), Yager and Abbasov treated the Pythagorean membership grades (PMGs) and they found it identical with PFSs. Their work also showed the relation between the PMGs and the complex numbers. PFNs were used by Reformat and Yager Xu and Yager (2008) to create a framework for dealing with collaboration-based recommendation. Gou et al. Gou et al. (2016) studied Pythagorean fuzzy mappings and investigated fundamental properties called derivability, continuity and differentiability. Zeng et al. Zeng et al. (2018) introduced an aggregation procedure involving PFS and applied its notion in solving MADM. Zhang Zhang (2016) proposed an approach to MCDM problems in terms of the idea of similarity measure for Pythagorean fuzzy sets. PFSs have been successfully introduced in different research area; in particular, Garg used PFSs in investment decision process (see Garg, Garg (2016); Peng and Yang, Peng and Yang (2015)), utilized PFSs in the candidate selection procedure for Asian Infrastructure Investment Bank ) and the service excellence of national airlines (Zhang and Xu, Zhang and Xu (2014)). Senapati and Yager Senapati and Yager (2019a, b) introduced the notion of a Fermatean fuzzy set (FFS). They offered various numerical examples of FFS to help people understand the concept. It is also important to mention that the class of this type of fuzzy set has more ability to capture the uncertainties as compared to IFSs and PFSs, and is qualified to handle higher degree of vagueness. MADM has been extensively used in many area of sciences, for example, (Xu & Xia;Xu & Chen;Xu Xu and Xia (2011); Xu and Chen (2011);Xu (2011)) introduced the intuitionistic fuzzy weighted averaging (IFWA) operator, intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and the intuitionistic fuzzy hybrid aggregation (IFHA) operator.
The fuzzy information aggregation operators are necessarily appealing and significant research topics and are given profound regard among the researchers. Various forms of generalizations of T-norms and T-conorms exist in text, such as Archimedean T-norms and T-conorms, Hamacher Tnorms and T-conorms, algebraic T-norms and T-conorms, Einstein T-norms and T-conorms, Frank T-norms and Tconorms and Dombi T-norms and T-conorms. Liu Liu (2014) used Hamacher aggregation operators in interval-valued intuitionistic fuzzy numbers (IVIFNs) and discussed MAGDM techniques. Zhao and Wei Zhou et al. (2014) initiated Einstein hybrid aggregation operators for IFNs and applied it to multiattribute decision-making method. Wei et al. Chen (2012) studied multi-attribute decision-making problems by proposing the bipolar fuzzy Hamacher arithmetic and geometric aggregation operators and discussed the basic properties of these proposed operators. Hamacher T-conorm and Tnorm, which are the generalization of algebraic and Einstein T-conorm and T-norm Beliakov et al. (2007), are more universal and adoptable. The value of research on aggregation operators based on Hamacher operation and their application to MADM problems is significant. Xiao Xiao (2014) gave induced interval-valued intuitionistic fuzzy Hamacher ordered weighted geometric (IIVIFHOWG) operator. Li Li (2014) studied interval-valued intuitionistic fuzzy sets with various operations, known as the Hamacher sum and the Hamacher product, and introduced the interval-valued intu-itionistic fuzzy Hamacher correlated averaging (IVIFHCA) operator. Tan et al. Tan et al. (2015) developed hesitant fuzzy Hamacher aggregation operators for multi-attribute decisionmaking. Senapati and Yager Chen (2014a) introduced four new types of weighted aggregation operators for FFS, namely, Fermatean fuzzy weighted average (FFWA) operator, Fermatean fuzzy weighted geometric (FFWG) operator, Fermatean fuzzy weighted power average (FFWPA) operator and Fermatean fuzzy weighted power geometric (FFWPG) operator. Recently, in a paper Aydemir and Gunduz (2020), Aydemir & Gunduz discussed TOPSIS method in terms of Dombi aggregation operators based on FF-sets and gave a complete overview of FF-sets in the framework of Dombi operations. In [43], the authors have extended FF-set into Hamacher operations and investigated the basic properties of FF-sets in Hamacher operations. Some practical examples of real-world scenario were discussed as practical example for the validation of the theory. Keeping in view the work on FF-set, we intend to extend the work of [43] and propose a series of new aggregation operators in Fermatean fuzzy environment based upon the Hamacher operations with the prioritization of attributes. The organization and novel contributions of this article are mentioned as: • The FF-set has more potential than the traditional IFS and PFS for decision makers to study the uncertain situation in the real-world problems. • The flexibility parameter involved in Hamacher operations has the ability to produce more accurate results in a decision process. • The proposed model can be applied in situations where the traditional models of IFS and PFS are failed. • The prioritization factor of attributes makes the proposed operators more advanced in modern decision process.
The remaining parts of the paper are organized in the following lines.
The second section, in brief, recalls basic knowledge of the IFS, PFSs and FFSs and the elementary operational laws of FFSs. In Sect. 3, we develop Fermatean fuzzy Hamacher prioritized average (FFHPA) operator, and Fermatean fuzzy Hamacher prioritized weighted average (FFHPWA) operator, Fermatean fuzzy Hamacher prioritized geometric (FFHPG) operator, and Fermatean fuzzy Hamacher prioritized weighted geometric (FFHPWG) operator. In Sect. 4, we make the use of these operators to develop tools that are handy in solving the Fermatean fuzzy multiattribute decision-making problems. A case study example of the university selection committee is analyzed in Sect. 5, and some comparisons of proposed operators are studied. The comparison of proposed and existing operators is stud-ied, and some future directions are given at the end of the paper.

Preliminaries and basic results
In this section, we sum up requisite knowledge associated with IFS, PFS and FFS along with corresponding operations and related properties. We will consider more familiarized ideas, which are useful in the sequential analysis.
Definition 2.1 [1,2] For a universe X , intuitionistic fuzzy set (IFS) A is an expression of the form for all x ∈ X , A turns to be a fuzzy set.
We shall notice that the Fermatean membership grades (FMGs) are greater than the Pythagorean membership grades (PMGs) and intuitionistic membership grades (IMGs), respectively.
Theorem 2.4 Senapati and Yager (2019a) The set of FMGs is larger than the set of PMGs and IMGs.
Definition 2.9 [19] Let F 1 = (μ 1 , ν 1 ) , and F 2 = (μ 2 , ν 2 ) , be any two FFEs, and let rt(F 1 ) and rt(F 2 ) be the respective ratings of F 1 and F 2 , then , be a FFE. The accuracy function of F can be defined as follows: Using the analogy of rating function (briefly as; rt) and accuracy function (briefly as; Acc), we give a complete criterion for the ranking of FFEs in the following.

Hamacher operations
Hamacher proposed generalized form of T-norm as well as T-conorm, called Hamacher operations, which consists of Hamacher product and Hamacher sum. These are the respective blueprints of the well-known T-norm and T -conorm, mentioned in the definition below. Hamachar (1978) Assume a 1 , b 1 , a 2 , b 2 ∈ R. Then, Hamacher T-norms (HT-norms) and Hamacher Tconorms (HT-conorms) are expressed as:

FF Hamacher operators
In this section, utilizing the notion of HTN and HTCN, we explain Hamacher operations with respect to FFEs. We propose the Hamacher arithmetic AOs with FFEs. In this regard, the operation rules for FF Hamacher operation are recalled in the following definition.

Fermatean fuzzy Hamacher prioritized arithmetic aggregation operators
Let us denote by £ the set of all nonempty FFNs, i.e., ⇐⇒ ω 1 ≤ ω 2 and 1 ≥ 2 . The top and bottom elements of £ are defined by 1 £ = (1, 0) and 0 £ = (0, 1). Then £ becomes a lattice with the partial order £ . If F 1 and F 2 are two FFNs, then The concept of prioritized average (PA) was first introduced by Yager [39] in 2008 and was defined as follows: Definition 3.2 (Yager [39]) Assume that C = {C 1 , C 2 , ..., C n } is a collection of criteria with the prioritization among the criteria defined by the linear ordering as C 1 C 2 ... C n , where criteria C j have a higher priority than C k for j < k and n ∈ N. The real number C j (x) ∈ [0, 1] is the performance of any alternative x under the criteria C j and Then PA is called the prioritized average (PA) operator.
The PA operators are used in the situations where the input arguments are exact values. By combining the Hamacher operations with prioritized inputs based on FF-sets, develop prioritized arithmetic aggregation operators with Fermatean fuzzy numbers based on Hamacher operations. Let F r = (μ r , ν r ) (r = 1, ..., p) be a family of FFEs. We define Fermatean fuzzy Hamacher prioritized arithmetic aggregation operator as follows: Definition 3.3 The Fermatean fuzzy Hamacher prioritized averaging (FFHPA) operator is a mapping FFHPA: where T r = r −1 k=1 rt (F r ) (r = 2, 3, ..., p) , T 1 = 1 and rt (F r ) represent the score value of F r (r = 1, 2, ..., p) .
In the following theorem we use the mathematical induction and operational rule of FFEs to prove that the aggregation value of a family of FFEs by using F F H P A operator is again an FFE.

Theorem 3.7 A Fermatean fuzzy Hamacher prioritized weighted average (FFHPWA) operator of a dimension p returns a FF-set and
where T r = r −1 k=1 rt (F r ) (r = 2, 3, ..., p) , T 1 = 1 and rt (F r ) represent the rating value of F r (r = 1, 2, ..., p) and = 1 , 2 , ..., p T is the weight vector such that r > 0, and p r =1 r = 1. In the following we discuss two special cases of FFHPWA operator for the working parameter λ.
In the following theorem we use the mathematical induction and operational rule of FFEs to prove that the aggregation value of a family of FFEs by using F F H PG operator is again an FFE.
Definition 4.4 Let F r = (μ r , ν r ) (r = 1, 2, ..., p) be a family of FFEs. The FFHPWG is a mapping from F p to F such that 2, 3, ..., p) , T 1 = 1 and rt (F r ) represent the score value of F r and = 1 , 2 , ..., p T is the weighting vector of F r (r = 1, br eak2, ..., p) such that r > 0 and p r =1 r = 1. Using FFHPWG operator, and the operational rules of Definition 3.1, we can prove the following subsequent theorem easily.

Fuzzy modeling of MADM: the case of Fermatean fuzzy information
We shall apply FF-Dombi prioritized AOs constructed in the previous sections to solve a MADM problem with FF information. Denote a discrete set of alternatives by A = { A 1 , A 2 , ..., A m }, we also denote by G = {G 1 , G 2 , ..., G r }, the set of attributes, we assume that there is a prioritization among these attributes and let the prioritization be a linear ordering G 1 G 2 ... G r , indicating that the attribute G ζ has a higher priority than G ξ if ζ < ξ. Let ∅ = {∅ 1 , ∅ 2 , . . . , ∅ r } be the weight vector for the attributes G ξ (ξ = 1, 2, 3, . . . , r ) such that ∅ ξ > 0 and r ξ =1 ∅ ξ = 1. Suppose that M = F cξ m×n = μ cξ , δ cξ m×n is the FF decision matrix, where μ cξ represents the degree of membership, that the alternative A ξ ∈ A satisfies the alternative G ξ , and δ cξ denotes the degree of nonmembership that the alternative A ξ ∈ A does not satisfy the attribute G ξ considered by the decision makers such that μ 3 cξ + δ 3 cξ ≤ 1 and μ cξ , δ cξ ⊂ [0, 1], ( c = 1, 2, . . . , m) and (ξ = 1, 2, . . . , n) that the DMs proposed for the attributes G ξ . To follow the above discussion we utilize the methods developed in previous section and design an algorithm to solve multiple attribute decision-making problem based on FF-environment.
Step 2. Apply the operator FFHPWA on the decision matrix M where or apply FFHPWG operator to get the aggregated values of F c ( c = 1, 2, ..., m) of the alternatives A c .
Step 3. Calculate the values of the score function Score F c ( c = 1, 2, ..., m) of all the aggregated FFNs F c ( c = 1, 2, ..., m) obtained in Step 2. If the value of score functions Score F c and Score F ξ is not different, then apply the accuracy function acc F c and acc F ξ for the ranking order of alternatives A c ( c = 1, 2, ..., m).

Example description
We discuss the selection process of teaching staff of our university. To promote the education system of Abdul Wali Khan University, the Department of Mathematics wants to recruit overseas outstanding educationists. After some important meetings in the department, an expert team is selected to complete the process of selection of outstanding teachers. The panel of experts consists on university vice chancellor (VC), dean of physical and numerical sciences (P&NS) and human resource development officer. This team of expert will analyze a set of five candidates A c ( c = 1, 2, 3, 4, 5) following the four attributes G1: qualification, G2: teaching ability, G3: research expertise and G4: quality research publications. University VC has absolute priority in decisionmaking, dean of P&NS comes next. Further they will be strict in their principle of combine ability and will not influence by any political integrity. The prioritization criteria are defined as G1 G2 G3 G4, where the symbol is used to represent prefer than relation. The team will use FFNs in the evaluation of candidates A c ( c = 1, 2, 3, 4, 5). The attribute weight vector in the selection process is ∅ = (0.2, 0.2, 0.3, 0.3) T , and the decision matrix for this model is F = F cξ 4×5 which is represented in Table 1, where F cξ are FFNs. In order to select the most desirable candidate A c ( c = 1, 2, 3, 4, 5), we apply FFHPWA and FFHPWG operators in the following steps of algorithm
Step 4. A 4 is selected as the best candidate for the post. From the above discussion, we observe that the overall ranking orders of all the candidates are different by utilizing the two operators, in FFHPWA operator A 1 is the most suitable candidate for the post, while by applying FFHPWG operator, the most desirable candidate is A 4 .

Comparison of proposed and existing operators
From Table 2, we observe that the ranking order of alternative by using different methods is different. But the best alternatives of the same type of aggregation operator are same, e.g., the best alternative in FFWA, FFWG, FFWPA and FFWPG operators is A 3 . The best alternatives for the methods FFHWA operator are A 5 and for FFHWG operator is A 4 . This means that the FFHWA and FFHWG operators have a The graphical views of these ranking orders are shown in Fig. 1.
From Fig. 1, we observe that the ranking orders of all the alternatives using several existing and proposed operators are different. We observe that the graphs of existing operators (FFWA (green line), FFWG (brown line), FFWPA (yellow line) and FFWPG (aqua line)) are monotonically increasing & decreasing between the alternatives A 1 to A 5 , and we cannot observe stability in these operators. On the other hand, the graphs of proposed operators (FFHWA (purple line), FFHWG (red line), FFHOWA (teal line), and FFHWG (black line)) are more stable and their graphs have very rare fluctuations. Therefore, the proposed operators seem to be more stable.  Fig. 2 Comparison of proposed operators. X-axis represents alternatives and y-axis represents score values

Effect of prioritizations of attributes
In Table 3, the score values and their ranking orders of alternatives in FFHPA, FFHPG, FFHPWA and FFHPWG operators are given. From Table 3, we observe that the ranking order of alternatives in FFHPA and FFHPWA operators has very rare fluctuations. Similarly, the ranking orders in FFHPG and FFHPWG operator have very small changes. It is concluded that the weights of alternatives in prioritized aggregation operators of FF-sets have a very low effect on the order of alternatives based on Hamacher operations.
From Fig. 2, it is clear that the ranking order of alternatives in (FFHPA) (green line) and (FFHPWA) (dotted line) is approximately parallel, while the ranking order in (FFHPG) (yellow line) and (FFHPWG) (dotted line) is apparently parallel. This means that the weights in Fermatean fuzzy Hamacher (prioritized) weighted averaging and (prior-itized) weighted geometric operators have very small effect on the ranking order of alternatives. It is also observed that the ranking order is monotonically increasing in FFHPA and FFHPWA operators while monotonically decreasing in FFHPG and FFHPWG operators.
To compare our proposed operators with the results obtained by applying Pythagorean fuzzy aggregation operators with and without prioritization of weights of attributes, we consider the methods proposed in [19] and [29].
In [19], Pythagorean fuzzy Hamacher weighted averaging and geometric aggregation operators have been applied; if we consider the attribute weight of alternatives as = (0.1, 0.2, 0.3, 0.4) T , then the score values of alternative are given in following Table 4.
On the other hand, if we consider the prioritization of attributes and use PFHPWA and PFHPWG operators, then the score values and their corresponding ranking orders are shown in Table 5.
Ranking order of alternatives using PFHPWA and PFH-PWG operators and proposed operators with weighted prioritization is shown in Tables 6 and 7.
In the following figures, we compare our proposed operators with the existing operators of Pythagorean fuzzy Hamacher averaging and geometric operators with prioritization of attributes. From Fig. 3, the ranking order of alternatives using PFH-PWA operator (green line) is smoothly decreasing, while the ranking of alternatives using FFHPWA operator (brown line) is strictly decreasing.
In Fig. 4, the ranking order of alternatives in PFHPWG operator (green line) is smoothly decreasing, while the ranking order of alternatives in FFHPWG operator (brown line) is smoothly increasing.

Concluding remarks
In this paper, an attempt is made to present several types of aggregation operators based on Hamacher operations for use in FFS decision-making process. Previously, the FFSs environment described different aggregation operators of Hamacher operations without prioritization of attributes. We presented here arithmetic and geometric operations to initiate some Fermatean fuzzy Hamacher prioritized aggregation operators from the rationale of Hamacher operations as Fermatean fuzzy Hamacher prioritized average (FFHPA) operator, Fermatean fuzzy Hamacher prioritized weighted average (FFHPWA) operator, Fermatean fuzzy Hamacher prioritized geometric (FFHPG) operator and Fermatean fuzzy Hamacher prioritized weighted geometric (FFHPWG) operator. Several new aspects of these recommended operators are considered. As a fact check, we have applied these operators to look into strategies remedying MADM situations. Eventually, a genuine example for the selection process of teaching staff of our university is considered to develop a  strategy and usefulness about the presented method. The new operators are compared with Pythagorean fuzzy Hamacher aggregation operators which gave the reliability of these operators. In future, we will consider risk theory and other areas under uncertain conditions for the proposed Fermatean fuzzy sets with prioritized attributes structure.
Author Contributions Khan and Khan discussed and formulated the measures. Jan and Afridi wrote the paper together.

Conflict of interest
The authors declare no conflict of interest.
Ethical approval This article does not contain any studies with human participants or animal performed by any of the authors.