Multiple-Attribute Decision-Making Using Fermatean Fuzzy Hamacher Interactive Geometric Operators

Fermatean fuzzy set (FFS) is a more efficient, flexible, and generalizedmodel to deal with uncertainty, as compared to intuitionistic and Pythagorean fuzzy models. +is research article presents a novel multiple-attribute decision-making (MADM) technique based on FFS. Aggregation operators (AOs), for example, Dombi, Einstein, and Hamacher, are frequently being used in the MADM process and are considered useful tools for evaluating the given alternatives. Among these, one of the most effective is the Hamacher operator. +e salient feature of this operator is that it reduces the impact of negative information and provides more accurate results. Motivated by the primary characteristics of the Hamacher operator, we apply Hamacher interactive aggregation operators based on FFSs to determine the best alternative. Using Hamacher’s norm operations, we introduce some new geometric operators, namely, Fermatean fuzzy Hamacher interactive weighted geometric (FFHIWG) operator, Fermatean fuzzy Hamacher interactive ordered weighted geometric (FFHIOWG) operator, and Fermatean fuzzy Hamacher interactive hybrid weighted geometric (FFHIHWG) operator. Some important results and properties of the proposed AOs are discussed, and to achieve the optimal alternative, the proposed MADM technique is carried out in a real-life application of the medical field. An algorithm of the proposed technique is also developed. +e significance of the proposed method is that Fermatean fuzzy Hamacher interactive geometric (FFHIG) operators deal with the relationship among belongingness degree (BD) and nonbelongingness degree (NBD) of the objects, which perform a crucial role in decision-making (DM). At last, to show the exactness and validity of the proposed work, a comparative analysis of the proposed model and the existing models is presented.


Introduction
Ambiguous or uncertain information is one of the greatest dilemmas dealing with the MADM process. e uncertain information can be captured in different ways. In the last few years, Zadeh's fuzzy set theory (FST) [1] and its extensions, i.e., intuitionistic fuzzy set theory (IFST) [2], Pythagorean fuzzy set theory (PFST) [3], hesitant fuzzy set theory (HFST) [4,5], and intervalvalued fuzzy set theory (IVFST) [6], have been proved to be efficient tools handling uncertainty in numerous applications of MADM. However, there are some restrictions involved in all these theories, for example, FST deals with belongingness degree only, whereas IFST deals with both BD and NBD but it restricts their sum to be less or equal to 1. To overcome this issue, PFST replaces the condition of the sum to "the sum of squares of BD and NBD to be less or equal to 1." Recently, a more generalized theory, namely, Fermatean fuzzy set theory (FFST), was introduced by Senapati and Yager [7]. e notion of FFS was initiated from IFSs and PFSs, where the sum of cubes of NBD and BD is less than or equal to one. erefore, FFSs are more flexible and generalized as compared to both IFSs and PFSs.
Aggregation methods based on IFSs and PFSs were widely used in MADM. Xu [8] and Zhao et al. [9] defined aggregation operators (AOs) based on IFSs. Wei and Lu [10] developed Pythagorean fuzzy (PF) power AOs and used them in DM problems. Ordered weighted averaging aggregation operators (OWAAOs) for MADM were defined by Yager [11]. Many multiple-attribute decision-making models (MADMMs) use algebraic operators and these are Dombi, Einstein, and Hamacher operators. In recent years, many theories were developed based on these operators. Dombi [12] defined Dombi triangular norm and conorm operators. Many authors contributed their work to Dombi AOs. Akram et al. [13] worked on Pythagorean Dombi fuzzy AOs (PFDAOs). Wei [14] introduced interaction AOs based on PFSs and also applied these to MADMMs.
Hamacher AOs were introduced in 1978 [15]. Wei [16] defined Hamacher AOs based on PFSs and gave a comparative analysis for MADM. Garg [17,18] presented some series of IF interactive averaging AOs by applying interactive averaging AOs on IFSs and also gave the idea of IF Hamacher AOs having entropy weight. Wu and Wei [19] presented MADMMs based on PF Hamacher AOs (PFHAOs). Wei [14] introduced the PF interaction AOs (PFIAOs) with their application to MADM. Waseem et al. [20] discussed MADM based on m-polar fuzzy Hamacher AOs (mFHAOs). Zhao and Wei [21] gave the idea of IF Einstein hybrid AOs (IFEHAOs). Wang and Liu [22] elaborated IF information AOs using Einstein operations. e idea for assessment of express service quality with entropy weight was explained by Wang et al. [23] under PF interactive Hamacher power AO.
Senapati and Yager [24] introduced Fermatean fuzzy averaging/geometric operators (FFAOS/FFFOs). ey also defined operations over Fermatean fuzzy numbers (FFNs) [25]. Garg et al. [26] presented a method for the most suitable laboratory selection for COVID-19 test under Fermatean fuzzy environment (FFE). Akram et al. [27] discussed a MADMM to show the benefits of a sanitizer in COVID-19 under FFE. Shahzadi and Akram [28] proposed the idea of Fermatean fuzzy soft AOs and applied this idea in the field of group decision-making for the selection of an antivirus mask. Recently, Aydemir and Gunduz [29] explained the Fermatean fuzzy TOPSIS (FF-TOPSIS) method consisting of Dombi AOs. Shahzadi et al. [30] introduced the idea of Hamacher interactive hybrid weighted averaging operators under FFE. Feng et al. [31] investigated membership grades of q− rung orthopair fuzzy sets geometrically. For more comprehension and understanding, the readers are referred to study [24,25,[32][33][34][35]. e motivations of this study are defined as follows: (i) Some new HIAOs such as FFHIWG, FFHIOWG, and FFHIHWG are proposed here (ii) e attractive properties alongside their special cases are discussed which reduce the loopholes in the existing operators (iii) An algorithm for MADM using the proposed operators is described and an application is presented to show the applicability of the intended method in the real world (iv) A comparison is also presented which shows the innovation and importance of the contemplated model e remaining part of the article is arranged as follows. In Section 2, some elementary notions are presented. Section 3 explains a hybrid structure of Hamacher interactive operators based on FFSs such as FFHIWG operators with a few important results and basic properties, for example, boundedness, homogeneity, idempotency, monotonicity, and shift invariance. In Section 4, the basic concept and results of the FFHIOWG operator are presented. Section 5 presents the notions related to the FFHIHWG operator. In Section 6, a MADMM under Fermatean fuzzy environment is explained through a real-life application. In Section 7, the influence of distinct values of the parameter is shown. In Section 8, a comparative analysis with existing theories is discussed which shows the efficacy and importance of the intended model. In Section 9, the presented work is summarized with concluding remarks.

Preliminaries
Definition 1 (see [7]). A Fermatean fuzzy set (FFS) L on D (a nonempty crisp set) is defined as where indicate belongingness degree, nonbelongingness degree, and indeterminacy degree, respectively.

Mathematical Problems in Engineering
Definition 3 (see [17]). Consider two FFSs L 1 � μ L 1 , ] L 1 and L 2 � μ L 2 , ] L 2 . en, Definition 4 (see [15]). Hamacher t-norm and t-conorm are defined by (i) For ς � 1, these operations become algebraic t-norm, T(a i , a j ) � a i a j , and algebraic t-conorm, T * (a i , a j ) � a i + a j − a i a j (ii) For ς � 2, these operations become Einstein t-norm, T(a i , a j ) � (a i a j /1 + (1 − a i )(1 − a j )), and Einstein t-conorm, T * (a i , a j ) � (a i + a j /1 + a i a j ) , and L � μ, ] be three FFSs and λ > 0; then, In this section, we introduce the Fermatean fuzzy Hamacher interactive weighted geometric operator (FFHIWGO) and describe its some important characteristics.

Fermatean Fuzzy HIOWG Operators
is section elaborates the notion of Hamacher interactive ordered weighted geometric operators (HIOWGO) under FFSs.
Proof. e proof is similar to that of eorem 1.
e proof is similar to that for FFHIWG operator. □
Proof. It is the same as that of eorem 1.

Remark 3.
e FFHIHWGO also satisfies the same results as given in Property 8.

Decision-Making Analysis under FFE
In a MADMM, it is a difficult task for decision makers to select the most feasible alternative among the given choices. Let S 1 , S 2 , . . . , S y be y distinct alternatives that can be characterized under a set of m different criteria c 1 , c 2 , . . . , c m taken by decision makers. Suppose that decision makers express their preference values in terms of FFSs α ij � (μ ij , ] ij )(i � 1, 2, . . . , y; j � 1, 2, . . . , m), where μ ij and ] ij are the BD and NBD of each alternative, respectively, corresponding to the given attribute such that 0 ≤ μ 3 ij + ] 3 ij ≤ 1. ere are different steps of MADM problem, given as follows: Step 1. Decision criteria can be grouped into two opposite categories, usually called the "benefit" and the "cost" criteria. Benefit criteria may be called "reward" criteria and cost criteria "regret" or "loss" criteria. A benefit criterion means that the higher an alternative scores in terms of it, the better the alternative is. e opposite is considered true for the cost criteria. e normalized Fermatean fuzzy decision matrix (FFDMx) is obtained by interchanging the assessment values of cost attributes (CAs) with benefit attributes (BAs) [36]; that is, Step 2. By using the DMx given in Step 1, the overall aggregated value of alternatives S i under different choices of attributes c j is obtained by using any of FFHIWG, FFHIOWG, or FFHIHWG operators.
Step 3. e score function is used to determine the score values of the alternatives.
Step 4. e alternatives S 1 , S 2 , . . . , S y are firstly arranged in descending order according to the score values and after it, the most appropriate alternative is selected.
A flowchart for the proposed MADM is shown in Figure 1.

Numerical Example.
is section presents an application of the proposed MADM model in the field of medical science. Suppose that a doctor wants to select a patient who is more affected by cancer. Let A 1 , A 2 , A 3 , A 4   By FFHIWG operator, see the following Step 1: since the criteria set is of the same type, DMx cannot be normalized.
Step 2: to determine the combined assessment of each alternative, we apply the FFHIWGO for ς � 2 as follows.
For   (3) e influence of attributes ς on alternatives' ranking by applying FFHIHWGO From Table 1 and Figure 2, we can see that alternatives may have the same or different score values but the ranking order remains the same for all the attributes.

A Comparative Analysis and Discussion
To view the importance and validity of proposed AOs, we give a comparative analysis of the proposed model with the different existing models. From Table 2, it is clear that there are some values for which μ + ] > 1 and μ 2 + ] 2 > 1. erefore, the methods presented in [18,23] failed to handle such problems.

Comparison with Fermatean Fuzzy TOPSIS Method.
is section presents the comparison of the proposed work with the FF-TOPSIS method [7]. e steps are given as follows: (1) Table 2 provides the FFDM in which each entry is a FFS.
(3) e distance of alternatives A i from FFPIS H + and FFNIS H − is given in Table 3. (4) e revised closeness degree of each alternative is given: (56) (6) us, A 4 is the best alternative.
It is clear that ranking lists is slightly different from both methods but the most suitable alternative from both methods is A 4 .
ough, FF-TOPSIS method is a useful technique to solve MADMMs, but there are also some issues which cannot be resolved by adopting this method. erefore, such issues can be tackled through the proposed model and the results obtained from it are closer to the original results.
Advantages and limitations of intended AOs: the benefit and motivation behind this intended approach are as follows: (i) e other membership grades in the aggregated value affect the other grades even if BD of any alternative is zero. (ii) ere is a relationship between the BD and NBD of an alternative. (iii) However, there are some limitations of this model. It cannot be applied in situations where we take the parameters for the evaluation of anything. It means this theory has a lack of parametrization property.

Conclusions
A FFS having emerging applications in MADM, is a more efficient technique to cope with ambiguities involved in the given data, as compared to IFSs and PFSs. AOs are very useful for evaluating the given alternatives in DM process because they integrate the evaluation values of all the given individuals into a unified form. By comparing other AOs, the structure of Hamacher's norms is a more general framework, which effectively integrates complex information. In this article, we have intended some FF-Hamacher interactive geometric AOs such as FFHIWGO, FFHIOWGO, and FFHIHWGO. ese intended operators have some useful characteristics, for example, boundedness, homogeneity, idempotency, monotonicity, and shift invariance. Some special cases of the proposed AOs have been also explained. e interaction between belongingness and nonbelongingness degrees has been discussed in the proposed HIAOs. We have explained a MADM algorithm to cope with uncertainties present in decision problems. To show the efficacy and applicability of the intended MADM, we have applied this method in a real-life system. We have presented a comparative analysis of our model with some of the existing models and concluded that our model is more flexible and it depicts ambiguous and inexact information in complex structures.
us, the AOs due to their highly adaptable nature are very crucial. erefore, taking into account this direction further we will explore some more properties and types of these operators in Fermatean fuzzy soft sets and q-rung orthopair fuzzy soft sets. Moreover, we will extend our work on the following decision-making problems: (1) An in-depth study of the Hamacher AOs for Fermatean fuzzy information such as induced Fermatean fuzzy Hamacher interactive AOs and q-rung picture fuzzy Hamacher interactive AOs will be a hot topic in the future (2) A MADM problem in medical diagnosis under Fermatean fuzzy soft data using Hamacher interactive AOs will be discussed (3) A MADM problem for the selection of a smartphone under Fermatean fuzzy soft data using Hamacher interactive AOs will be discussed

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest
e authors declare no conflicts of interest.