Generalised Fuzzy Models Applied to Logical Algebras and Intelligent Systems in EngineeringView this Special Issue
Multiple-Attribute Decision-Making Using Fermatean Fuzzy Hamacher Interactive Geometric Operators
Fermatean fuzzy set (FFS) is a more efficient, flexible, and generalized model to deal with uncertainty, as compared to intuitionistic and Pythagorean fuzzy models. This 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. The 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. The 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.
Ambiguous or uncertain information is one of the greatest dilemmas dealing with the MADM process. The uncertain information can be captured in different ways. In the last few years, Zadeh’s fuzzy set theory (FST)  and its extensions, i.e., intuitionistic fuzzy set theory (IFST) , Pythagorean fuzzy set theory (PFST) , hesitant fuzzy set theory (HFST) [4, 5], and interval-valued fuzzy set theory (IVFST) , 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 . The 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. Therefore, 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  and Zhao et al.  defined aggregation operators (AOs) based on IFSs. Wei and Lu  developed Pythagorean fuzzy (PF) power AOs and used them in DM problems. Ordered weighted averaging aggregation operators (OWAAOs) for MADM were defined by Yager . 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  defined Dombi triangular norm and conorm operators. Many authors contributed their work to Dombi AOs. Akram et al.  worked on Pythagorean Dombi fuzzy AOs (PFDAOs). Wei  introduced interaction AOs based on PFSs and also applied these to MADMMs.
Hamacher AOs were introduced in 1978 . Wei  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  presented MADMMs based on PF Hamacher AOs (PFHAOs). Wei  introduced the PF interaction AOs (PFIAOs) with their application to MADM. Waseem et al.  discussed MADM based on -polar fuzzy Hamacher AOs (mFHAOs). Zhao and Wei  gave the idea of IF Einstein hybrid AOs (IFEHAOs). Wang and Liu  elaborated IF information AOs using Einstein operations. The idea for assessment of express service quality with entropy weight was explained by Wang et al.  under PF interactive Hamacher power AO.
Senapati and Yager  introduced Fermatean fuzzy averaging/geometric operators (FFAOS/FFFOs). They also defined operations over Fermatean fuzzy numbers (FFNs) . Garg et al.  presented a method for the most suitable laboratory selection for COVID-19 test under Fermatean fuzzy environment (FFE). Akram et al.  discussed a MADMM to show the benefits of a sanitizer in COVID-19 under FFE. Shahzadi and Akram  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  explained the Fermatean fuzzy TOPSIS (FF-TOPSIS) method consisting of Dombi AOs. Shahzadi et al.  introduced the idea of Hamacher interactive hybrid weighted averaging operators under FFE. Feng et al.  investigated membership grades of rung orthopair fuzzy sets geometrically. For more comprehension and understanding, the readers are referred to study [24, 25, 32–35].
The motivations of this study are defined as follows:(i)The proposed Hamacher interactive AOs (HIAOs) deal with the relationship between the BD and NBD of an object(ii)The MADMM based on FFSs shows that the change in NBDs will definitely affect the BDs of the objects(iii)The proposed Fermatean fuzzy AOs generalize the BDs and NBDs of the objects; i.e., greater values of belongingness and nonbelongingness degrees can be taken as compared to IFST and PFST(iv)The HIAOs are a much convenient approach to cope with the issues in the DM process; this article aims to define HIAOs based on FFSs to handle uncertainty associated with the choice of alternatives in MADMMs(v)Hamacher interactive AOs give more precise and exact choice values in decision results when applied to MADMMs
The contributions of this article are outlined as follows:(i)Some new HIAOs such as FFHIWG, FFHIOWG, and FFHIHWG are proposed here(ii)The 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
The 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.
Definition 1. (see ). A Fermatean fuzzy set (FFS) on (a nonempty crisp set) is defined aswhere , and indicate belongingness degree, nonbelongingness degree, and indeterminacy degree, respectively.
Definition 2. (see ). The score function (SF) and accuracy function (AF) for a FFS are given by
Definition 3. (see ). Consider two FFSs and . Then,(1)If , then .(2)If , then .(3)If , then(a)If , then .(b)If , then .(c)If , then .
Definition 4. (see ). Hamacher t-norm and t-conorm are defined by(i)For , these operations become algebraic t-norm, , and algebraic t-conorm, (ii)For , these operations become Einstein t-norm, , and Einstein t-conorm,
Definition 5. Let , , and be three FFSs and ; then,(i)(ii)(iii)(iv)
3. Fermatean Fuzzy Hamacher Interactive Weighted Geometric Operators
In this section, we introduce the Fermatean fuzzy Hamacher interactive weighted geometric operator (FFHIWGO) and describe its some important characteristics.
Definition 6. Let be a family of FFSs and be its weight vector (WV) such that and , then is defined as
Theorem 1. Let be a collection of FFSs; then,
Proof. For ,Thus, the result is true for . Suppose that result holds for , i.e.,Now, for ,The result holds, .
Remark 1. Here are cases of the FFHIWGO.(i)For , FFHIWGO becomes Fermatean fuzzy interactive weighted geometric operator (FFIWGO):(ii)For , FFHIWG operator becomes Fermatean fuzzy Einstein interactive weighted geometric operator (FFEIWGO):
Theorem 2. The clumped value of FFSs , by using FFHIWGO, is a FFS, i.e.,
Proof. As are FFSs, , and . Therefore,Also, . Therefore,Thus, .
Property 1. (idempotency). If , then
Proof. Since and , by Theorem 1,
Property 2. (boundedness). Let and ; then,
Proof. Let ; then , so is a decreasing function (DF). As , then ; that is, . Let and ; we haveThus,Consider , then ; i.e., is a DF on . Since , then ; that is, . Then,Also,Let FFHIWG; then, from inequalities (20) and (22), where , , , and . So, and . As and ,
Property 3. (monotonicity). If , then
Proof. It can be proved on similar lines to the above.
Property 4. (shift iInvariance). If is another FFS, then
Proof. As FFSs, soTherefore,
Property 5. (homogeneity). Let , then
Proof. Since are FFSs and , thereforeTherefore,
Property 6. Let and be two collections of FFSs; then
Proof. As and are two collections of FFSs, thenTherefore,