# Hamacher Interactive Hybrid Weighted Averaging Operators under Fermatean Fuzzy Numbers

Authors: Gulfam Shahzadi; G. Muhiuddin; Muhammad Arif Butt; Ather Ashraf
Journal: Journal of Mathematics
 (2021)

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## Abstract

A Fermatean fuzzy set is a more powerful tool to deal with uncertainties in the given information as compared to intuitionistic fuzzy set and Pythagorean fuzzy set and has energetic applications in decision-making. Aggregation operators are very helpful for assessing the given alternatives in the decision-making process, and their purpose is to integrate all the given individual evaluation values into a unified form. In this research article, some new aggregation operators are proposed under the Fermatean fuzzy set environment. Some deficiencies of the existing operators are discussed, and then, new operational law, by considering the interaction between the membership degree and nonmembership degree, is discussed to reduce the drawbacks of existing theories. Based on Hamacher’s norm operations, new averaging operators, namely, Fermatean fuzzy Hamacher interactive weighted averaging, Fermatean fuzzy Hamacher interactive ordered weighted averaging, and Fermatean fuzzy Hamacher interactive hybrid weighted averaging operators, are introduced. Some interesting properties related to these operators are also presented. To get the optimal alternative, a multiattribute group decision-making method has been given under proposed operators. Furthermore, we have explicated the comparison analysis between the proposed and existing theories for the exactness and validity of the proposed work.


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## Body

## 1. Introduction

The process of multiattribute group decision-making (MAGDM) yields the best alternative when the list of all possible alternatives has been compiled according to some certain attributes. Previously, the data about alternatives corresponding to attributes and their weights were given in crisp values. However, nowadays, uncertainties play an important part in the decision-making (DM) approach. Each alternative is allotted a preference to some certain degree to deal with the complicated system. However, information regarding real-world system is indefinite and fuzzy with a lot of ambiguities. Such type of conditions is appropriately explained by fuzzy set (FS) [1] and intuitionistic fuzzy set (IFS) [2] rather than crisp values. IFS is a more efficient tool to deal with vague information because it has both the membership degree (MD) and nonmembership degree (NMD), but there are some drawbacks. The sum of MD and NMD is constrained to unit interval in IFS’s model. Pythagorean fuzzy set (PFS) was introduced by Yager [3] to tackle vague decisions more effectively. However, this model also has some restrictions; if MD of an element is 0.8 and NMD is 0.76, then 0.82+0.762>1. Therefore, Yager [4] narrated the theory of q-rung orthopair fuzzy set (q-ROFS) with condition 0≤ϱq+σq≤1. The basic notions about Fermatean fuzzy set (FFS) were studied by Senapati and Yager [5].The idea of aggregation operators (AOs) performs a crucial role in getting an optimal solution when there are a lot of choices for one given problem. The idea of aggregation of infinite sequences was presented by Mesiar and Pap [6]. Xu [7] gave the theory of intuitionistic fuzzy (IF) AOs. Zhao et al. [8] developed the theory of generalized AOs for IFS. The Einstein hybrid AOs under IF environment were studied by Zhao and Wei [9]. The concept of IF AOs using Einstein operations were discussed by Wang and Liu [10]. Garg [11] combined the theories of IFS and interactive averaging AOs. Garg et al. [12] gave the idea of Choquet integral aggregation operators for interval-valued IFS. Garg [13] introduced IF Hamacher AOs with entropy weight. Alcantud et al. [14] elaborated the idea of aggregation of infinite chains of IFS. Wu and Wei [15] gave the theory of Pythagorean fuzzy (PF) Hamacher AOs. Wei [16] proposed the PF interaction AOs. Shahzadi and Akram [17] combined the concept of PF numbers and Yager operators. The theory of novel interactive hybrid weighted AOs with PF environment was studied by Li et al. [18]. The idea of q-ROF power Maclaurin symmetric mean operators was narrated by Liu et al. [19]. q-rung orthopair fuzzy (q-ROF) weighted AOs were expressed by Liu and Wang [20]. The exponential aggregation operators for q-ROFS were defined by Peng et al. [21]. Some confidence levels about q-ROF AOs were studied by Joshi and Gegov [22]. The hybrid DM model under q-ROF Yager AOs was developed by Akram and Shahzadi [23]. Akram et al. [24] presented the Einstein geometric operators for q-ROF information. Akram et al. [25] gave the protraction of Einstein operators under q-ROF environment. Darko and Liang [26] examined q-ROF Hamacher AOs and their application in MAGDM with modified EDAS method. Senapati and Yager [27] elaborated the theory of Fermatean fuzzy (FF) averaging/geometric operators. Senapati and Yager [28] studied subtraction, division, and Fermatean arithmetic mean operations over FFS. Many new operations for FFS were defined by Senapati and Yager [28]. Garg et al. [29] developed the theory for the choice of a most suitable laboratory for COVID-19 test under FF environment. The effectiveness of a sanitizer in COVID-19 was discussed by Akram et al. [30]. For more knowledge and applications, the readers are suggested to study [31–44].


### 1.1. Motivations of Proposed Work

(i)
The proposed operators have the ability to deal with the interaction between the MD and NMD.(ii)
The proposed theory shows that the change in MD will affect the NMD.(iii)
The developed operators show that there will be nonzero NMD of the whole aggregated FF numbers (FFNs) even if at least one of them is zero. Therefore, the others grades of nonmembership function of FFNs perform a significant role in the aggregation process (AP).


### 1.2. Contributions of Proposed Work

(i)
Some novel operators such as Fermatean fuzzy Hamacher interactive weighted averaging (FFHIWA), Fermatean fuzzy Hamacher interactive ordered weighted averaging (FFHIOWA), and Fermatean fuzzy Hamacher interactive hybrid weighted averaging (FFHIHWA) operators are explored here.(ii)
Some special cases of these operators along with their attractive properties are discussed, which reduce the shortcomings of the existing operators.(iii)
Some basic steps for MAGDM under proposed operators are explained with the help of a numerical example.(iv)
The comparison analysis with other developed approaches shows the validity of proposed theory.


### 1.3. Framework and Organization of the Paper

The remaining paper is arranged as follows: Section2 recalls some elementary definitions. Section 3 defines the hybrid structure of Hamacher, interactive operators, and FFNs such as FFHIWA operator along with some fundamental properties. In Section 4, we elaborate the idea of FFHIOWA operator with some attractive properties. Section 5 presents the notion of FFHIHWA operator. Section 6 discusses an algorithm to deal with MAGDM along with a numerical example. Section 7 gives a comparison analysis with FF Einstein weighted averaging (FFEWA) operator for the validity and importance of proposed theory. In Section 8, we have summarized the results.


## 1.1. Motivations of Proposed Work

(i)
The proposed operators have the ability to deal with the interaction between the MD and NMD.(ii)
The proposed theory shows that the change in MD will affect the NMD.(iii)
The developed operators show that there will be nonzero NMD of the whole aggregated FF numbers (FFNs) even if at least one of them is zero. Therefore, the others grades of nonmembership function of FFNs perform a significant role in the aggregation process (AP).


## 1.2. Contributions of Proposed Work

(i)
Some novel operators such as Fermatean fuzzy Hamacher interactive weighted averaging (FFHIWA), Fermatean fuzzy Hamacher interactive ordered weighted averaging (FFHIOWA), and Fermatean fuzzy Hamacher interactive hybrid weighted averaging (FFHIHWA) operators are explored here.(ii)
Some special cases of these operators along with their attractive properties are discussed, which reduce the shortcomings of the existing operators.(iii)
Some basic steps for MAGDM under proposed operators are explained with the help of a numerical example.(iv)
The comparison analysis with other developed approaches shows the validity of proposed theory.


## 1.3. Framework and Organization of the Paper

The remaining paper is arranged as follows: Section2 recalls some elementary definitions. Section 3 defines the hybrid structure of Hamacher, interactive operators, and FFNs such as FFHIWA operator along with some fundamental properties. In Section 4, we elaborate the idea of FFHIOWA operator with some attractive properties. Section 5 presents the notion of FFHIHWA operator. Section 6 discusses an algorithm to deal with MAGDM along with a numerical example. Section 7 gives a comparison analysis with FF Einstein weighted averaging (FFEWA) operator for the validity and importance of proposed theory. In Section 8, we have summarized the results.


## 2. Preliminaries

In this section, we recall some basic definitions.Definition 1.
(see [5]). A FFS P on nonempty set O is given by(1)P=r,ϱPr,σPr,where ϱP:O⟶0,1, σP:O⟶0,1, and ϖPr=1−ϱPr3−σPr33 indicate MD, NMD, and indeterminacy degree (InD), respectively.Definition 2.
(see [5]). For FFN P=ϱP,σP, the score function and accuracy function are given as(2)SP=ϱP3−σP3,SP∈−1,1,AP=ϱP3+σP3,AP∈0,1.Definition 3.
(see [5]). Consider two FFNs P1=ϱP1,σP1 and P2=ϱP2,σP2. Then, the following holds:(1)
IfSP1<SP2, then P1≺P2.(2)
IfSP1>SP2, then P1≻P2.(3)
IfSP1=SP2, then(a)
IfAP1<AP2, then P1≺P2.(b)
IfAP1>AP2, then P1≻P2.(c)
IfAP1=AP2, then P1∼P2.(3)Tr,s=rsδ+1−δr+s−rs,T∗r,s=r+s−rs−1−δrs1−1−δrs.(i)
Forδ=1, these operations become algebraic t-norm and t-conorm Tr,s=rs and T∗r,s=r+s−rs.(ii)
Forδ=2, these operations become Einstein t-norm and t-conorm Tr,s=rs/1+1−r1−s and T∗r,s=r+s/1+rs.


## 3. Fermatean Fuzzy Hamacher Interactive Average Operators

Definition 4.
LetP1=ϱ1,σ1, P2=ϱ2,σ2, and P=ϱ,σ be three FFNs and β>0. Then, some arithmetic operations between them by using Hamacher norms are as follows:(i)
P1⊕P2=〈∏i=121+δ−1ϱi3−∏i=121−ϱi3/∏i=121+δ−1ϱi3+δ−1∏i=121−ϱi33,δ∏i=121−ϱi3−∏i=121−ϱi3−σi3/∏i=121+δ−1ϱi3+δ−1∏i=121−ϱi33〉,(ii)
β.P=1+δ−1ϱ3β−1−ϱ3β/1+δ−1ϱ3β+δ−11−ϱ3β3,δ1−ϱ3−1−ϱ3−σ3β/1+δ−1ϱ3β+δ−11−ϱ3β3.


### 3.1. Weighted Average Aggregation Operators

LetPi=ϱi,σii=1,2,…,y be a collection of FFNs and κ=κ1,κ2,…,κyT be its weight vector (WV) such that κi>0 and ∑i=1yκi=1, then FFHIWA:Ωy⟶Ω is defined as(4)FFHIWAP1,P2,…,Py=κ1P1⊕κ2P2⊕⋯κyPy.Theorem 1.
LetPi=ϱi,σi be a collection of FFNs, then(5)FFHIWAP1,P2,…,Py=∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi/∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi/∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3.Proof.
Fory=1,κ=κ1=1,(6)FFHIWAP1=κ1P1=P1=ϱ1,σ1=1+δ−1ϱ13−1−ϱ131+δ−1ϱ13+δ−11−ϱ133,δ1−ϱ13−1−ϱ13−σ131+δ−1ϱ13δ−11−ϱ133.
Thus, the result holds fory=1. Suppose that result holds for y=p, i.e.,(7)FFHIWAP1,P2,…,Pp=∏i=1p1+δ−1ϱi3κi−∏i=1p1−ϱi3κi∏i=1p1+δ−1ϱi3+δ−1∏i=1p1−ϱi3κi3,δ∏i=1p1−ϱi3κi−∏i=1p1−ϱi3−σi3κi∏i=1p1+δ−1ϱi3+δ−1∏i=1p1−ϱi3κi3.
Now, fory=p+1,(8)FFHIWAP1,P2,…,Pp+1=⊕i=1p+1κiPi=∏i=1p1+δ−1ϱi3κi−∏i=1p1−ϱi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3,δ∏i=1p1−ϱi3κi−∏i=1p1−ϱi3−σi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3⊕1+δ−1ϱp+13κp+1−1−ϱp+13κp+11+δ−1ϱp+13+δ−11−ϱp+13κp+13,δ1−ϱp+13κp+1−1−ϱp+13−σp+13κp+11+δ−1ϱp+13κp+1+δ−11−ϱp+13κp+13=∏i=1p1+δ−1ϱi3κi−∏i=1p1−ϱi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3,δ∏i=1p1−ϱi3κi−∏i=1p1−ϱi3−σi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3.⇒ Result holds ∀y.Remark 1.
We elaborate two cases of the FFHIWA operator:(i)
Forδ=1, FFHIWA operator becomes FF interactive weighted averaging (FFIWA) operator:(9)FFIWAP1,P2,…,Py=1−∏i=1y1−ϱi3κi3,∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi3.(ii)
Forδ=2, FFHIWA operator becomes FF Einstein interactive weighted averaging (FFEIWA) operator:(10)FFEIWAP1,P2,…,Py=∏i=1y1+ϱi3κi−∏i=1y1+ϱi3κi∏i=1y1+ϱi3κi+∏i=1y1+ϱi3κi3,2∏i=1y1+ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+ϱi3κi+∏i=1y1+ϱi3κi3.Theorem 2.
LetPi=ϱi,σi be FFNs, then the accumulated value by using FFHIWA operator is a FFN, i.e.,(11)FFHIWAP1,P2,…,Py∈FFN.Proof.
AsPi′s are FFNs, 0≤ϱi,σi≤1 and 0≤ϱi3+σi3≤1. Therefore,(12)∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi=1−δ∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≤1−∏i=1y1−ϱi3κi≤1.
Also,1+δ−1ϱi3≥1−ϱi3⇒∏i=1y1+δ−1ϱi3−∏i=1y1−ϱi3≥0. Therefore,(13)∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≥0,⇒∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3≥0.
Thus,
Moreover,(14)δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≤δ∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≤≤∏i=1y1−ϱi3κi≤1.
Also,(15)∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi≥0,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≥0,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3≥0.
Thus,Property 1.
(idempotency). IfPi=Po=ϱo,σo,∀i, then(16)FFHIWAP1,P2,…,Py=Po.Proof.
SincePi=Po=ϱo,σo∀i=1,2.…,y and ∑i=1yκi=1, by Theorem 1,(17)FFHIWAP1,P2,…,Py=∏i=1y1+δ−1ϱo3κi−∏i=1y1−ϱo3κi∏i=1y1+δ−1ϱo3κi+δ−1∏i=1y1−ϱo3κi3,δ∏i=1y1−ϱo3κi−∏i=1y1−ϱo3−σo3κi∏i=1y1+δ−1ϱo3κi+δ−1∏i=1y1−ϱo3κi3=1+δ−1ϱo3∑i=1yκi−1−ϱo3∑i=1yκi1+δ−1ϱo3∑i=1yκi+δ−11−ϱo3∑i=1yκi3,δ1−ϱo3∑i=1yκi−1−ϱo3−σo3∑i=1yκi1+δ−1ϱo3∑i=1yκi+δ−11−ϱo3∑i=1yκi3=1+δ−1ϱo3−1−ϱo31+δ−1ϱo3+δ−11−ϱo33,δ1−ϱo3−1−ϱo3−σo31+δ−1ϱo3+δ−11−ϱo33=ϱo,σo.Property 2.
(boundedness). LetP−=miniϱi,maxiσi and P+=miniϱi,maxiσi, then(18)P−≤FFHIWAP1,P2,…,Py≤P+.Proof.
Letfr=1−r/1+δ−1r,r∈0,1, then f′r=−δ/1+δ−1r2<0, so fr is a decreasing function (DF). As ϱi,min3≤ϱi3≤ϱi,max3,∀i=1,2,…,y, then fϱi,max3≤fϱi3≤fϱi,min3,∀i; that is, 1−ϱi,max3/1+δ−1ϱi,max3≤1−ϱi3/1+δ−1ϱi3≤1−ϱi,min3/1+δ−1ϱi,min3, ∀i. Let κi∈0,1 and ∑i=1yκi=1, we have(19)1−ϱi,max31+δ−1ϱi,max3κi≤1−ϱi31+δ−1ϱi3κi≤1−ϱi,min31+δ−1ϱi,min3κi∏i=1y1−ϱi,max31+δ−1ϱi,max3κi≤∏i=1y1−ϱi31+δ−1ϱi3κi≤∏i=1y1−ϱi,min31+δ−1ϱi,min3κi⇔1−ϱi,max31+δ−1ϱi,max3∑i=1yκi≤∏i=1y1−ϱi31+δ−1ϱi3κi≤1−ϱi,min31+δ−1ϱi,min3∑i=1yϖi⇔1−ϱi,max31+δ−1ϱi,max3≤∏i=1y1−ϱi31+δ−1ϱmath fraki3κi≤1−ϱi,min31+δ−1ϱi,min3⇔δ−11−ϱi,max31+δ−1ϱi,max3≤δ−1∏i=1y1−ϱi31+δ−1ϱi3ϖi≤δ−11−ϱi,min31+δ−1ϱi,min3⇔δ1+δ−1ϱi,max3≤1+δ−1∏i=1y1−ϱi31+δ−1ϱi3κi≤δ1+δ−1ϱi,min3⇔1+δ−1ϱi,min3δ≤11+δ−1∏i=1y1−ϱi3/1+δ−1ϱi3κi≤1+δ−1ϱi,max3δ⇔1+δ−1ϱi,min3≤δ1+δ−1∏i=1y1−ϱi3/1+δ−1ϱi3κi≤1+δ−1ϱi,max3⇔δ−1ϱi,min3≤δ1+δ−1∏i=1y1−ϱi3/1+δ−1ϱi3κi−1≤δ−1ϱi,max3⇔ϱi,min3≤∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3ϖi+δ−1∏i=1y1−ϱi3ϖi≤ϱi,max3.
Thus,(20)ϱi,min≤∏i=1y1+δ−1ϱi3κi−∏i=1y∏1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3ϖi3≤ϱi,max.
Considergs=δ−δ−1s/δ−1s,s∈0,1, then g′s=−δ/δ−1s2; i.e., gs is a DF on 0,1. Since 1−ϱi,max3≤1−ϱi3≤1−ϱi,min3,∀i, then g1−ϱi,max3≤g1−ϱi3≤g1−ϱi,max3,∀i, that is, δ−δ−11−ϱi,min3/δ−11−ϱi,min3≤δ−δ−11−ϱi3/δ−11−ϱi3≤δ−δ−11−ϱi,max3/δ−11−ϱi,max3. Then,(21)δ−δ−11−ϱi,min3δ−11−ϱi,min3κi≤δ−δ−11−ϱi3δ−11−ϱi3κi≤δ−δ−11−ϱi,max3δ−11−ϱi,max3κi∏i=1yδ−δ−11−ϱi,min3δ−11−ϱi,min3κi≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi≤∏i=1yδ−δ−11−ϱi,max3δ−11−ϱi,max3κi⇒δ−δ−11−ϱi,min3δ−11−ϱi,min3∑i=1yκi≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi≤δ−δ−11−ϱi,max3δ−11−ϱi,max3∑i=1yκi⇒δ−δ−11−ϱi,min3δ−11−ϱi,min3≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi≤δ−δ−11−ϱi,max3δ−11−ϱi,max3⇒δδ−11−ϱi,min3≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi+1≤δδ−11−ϱi,max3⇒δ−11−ϱi,max3δ≤1∏i=1yδ−δ−11−ϱi3/δ−11−ϱi3κi+1≤δ−11−ϱi,min3δ⇒1−ϱi,max3≤δδ−1∏i=1yδ−δ−11−ϱi3/δ−11−ϱi3κi+δ−1≤1−ϱi,min3⇒1−ϱi,max3≤δδ−1∏i=1yδ−δ−1σi3/δ−1σi3κi+δ−1≤1−ϱi,min3.
Let FFHIWAP1,P2,…,Py=P=ϱP,σP, then from inequalities (20) and (21), ϱmin≤ϱP≤ϱmax,σmax≤σP≤σmin, where ϱmin=miniϱi, ϱmax=maxiϱi, σmin=miniσi, and σmax=maxiσi. So SP=ϱP3−σP3≤ϱmax3−σmax3=SP+ and SP=ϱP3−σP3≥ϱmin3−σmin3=SP−. As SP<SP+ and SP>SP−,(22)P−≤FFHIWAP1,P2,…,Py≤P+.Property 3.
(monotonicity). WhenPi≤Ti,∀i, then(23)FFHIWAP1,P2,…,Py≤FFHIWAT1,T2,…,Ty.Proof.
It is similar to above.Property 4.
(shift invariance). IfT=ϱT,σT is another FFN, then(24)FFHIWAP1⊕T,P2⊕T,…,Py⊕T=FFHIWAP1,P2,…,Py⊕T.Proof.
AsPi,T∈ FFNs, (25)Pi⊕T=1+δ−1ϱi31+δ−1ϱT3−1−ϱi31−ϱT31+δ−1ϱi31+δ−1ϱT3+δ−11−ϱi31−ϱT33,δ1−ϱi31−ϱT3−1−ϱi3−σi31−ϱT3−σT31+δ−1ϱi31+δ−1ϱT3+δ−11−ϱi31−ϱT33
Therefore,(26)FFHIWAP1⊕T,P2⊕T,…,Py⊕T=∏i=1y1+δ−1ϱi31+δ−1ϱT3κi−∏i=1y1−ϱi31−ϱT3κi∏i=1y1+δ−1ϱi31+δ−1ϱT3κi+δ−1∏i=1y1−ϱi31−ϱT3κi3,δ∏i=1y1−ϱi31−ϱT3κi−∏i=1y1−ϱi3−σi31−ϱT3−σT3κi∏i=1y1+δ−1ϱi31+δ−1ϱT3κi+δ−1∏i=1y1−ϱi31−ϱT3κi3=∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3κi−∏i=1y1−ϱi3κi1−ϱT3κi∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3κi+δ−1∏i=1y1−ϱi3κi1−ϱT3κi3,δ∏i=1y1−ϱi3κi1−ϱT3κi−∏i=1y1−ϱi3−σi3κi1−ϱT3−σT3κi∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3κi+δ−1∏i=1y1−ϱi3κi1−ϱT3κi3=∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3−∏i=1y1−ϱi3κi1−ϱT3∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3+δ−1∏i=1y1−ϱi3κi1−ϱT33,δ∏i=1y1−ϱi3κi1−ϱT3−∏i=1y1−ϱi3−σi3κi1−ϱT3−σT3∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3+δ−1∏i=1y1−ϱi3κi1−ϱT33=FFHIWAP1,P2,…,Py⊕T.Property 5.
(homogeneity). Letβ>0, then(27)FFHIWAβP1,βP2,…,βPy=βFFHIWAP1,P2,…,Py.Proof.
SincePi=ϱi,σi are FFNs, for β>0,(28)βPi=1+δ−1ϱ3β−1−ϱ3β1+δ−1ϱ3β+δ−11−ϱ3β3,δ1−ϱ3β−1−ϱ3−σ3β1+δ−1ϱ3β+δ−11−ϱ3β3.
Therefore,(29)FFHIWAβP1,βP2,…,βPy=∏i=1y1+δ−1ϱ3βκi−∏i=1y1−ϱ3βκi∏i=1y1+δ−1ϱ3βκi+δ−1∏i=1y1−ϱ3βκi3,δ∏i=1y1−ϱ3βκi−∏i=1y1−ϱ3−σ3βκi∏i=1y1+δ−1ϱ3βκi+δ−1∏i=1y1−ϱ3βκi3=∏i=1y1+δ−1ϱ3κiβ−∏i=1y1−ϱ3κiβ∏i=1y1+δ−1ϱ3κiβ+δ−1∏i=1y1−ϱ3κiβ3,δ∏i=1y1−ϱ3κiβ−∏i=1y1+δ−1ϱ3κiβ∏i=1y1+δ−1ϱ3κiβ+δ−1∏i=1y1−ϱ3κiβ3=β∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3=βFFHIWAP1,P2,…,Py.Property 6.
LetPi=ϱPi,σPi and Ti=ϱTi,σTi be two collections of FFNs, then(30)FFHIWAP1⊕T1,P2⊕T2,…,Py⊕Ty=FFHIWAP1,P2,…,Py⊕FFHIWAT1,T2,…,Ty.Proof.
AsPi=ϱPi,σPi and Ti=ϱTi,σTi are two collections of FFNs, then(31)Pi⊕Ti=1+δ−1ϱPi31+δ−1ϱTi3−1−ϱPi31−ϱTi31+δ−1ϱPi31+δ−1ϱTi3+δ−11−ϱPi31−ϱTi33,δ1−ϱPi31−ϱTi3−1−ϱPi3−σPi31−ϱTi3−σTi31+δ−1ϱPi31+δ−1ϱTi3+δ−11−ϱPi31−ϱTi33
Therefore,(32)FFHIWAP1⊕T1,P2⊕T2,…,Py⊕Ty=∏i=1y1+δ−1ϱPi31+δ−1ϱTi3κi−∏i=1y1−ϱPi31−ϱTi3κi∏i=1y1+δ−1ϱPi31+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi31−ϱTi3κi3,δ∏i=1y1−ϱPi31−ϱTi3κi−∏i=1y1−ϱPi3−σPi31−ϱTi3−σTi3κi∏i=1y1+δ−1ϱPi31+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi31−ϱTi3κi3=∏i=1y1+δ−1ϱPi3κi∏i=1y1+δ−1ϱTi3κi−∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi∏i=1y1+δ−1ϱPi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi3,δ∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi−∏i=1y1−ϱPi3−σPi3κi∏i=1y1−ϱTi3−σTi3κi∏i=1y1+δ−1ϱPi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi3=∏i=1y1+δ−1ϱPi3κi−∏i=1y1−ϱPi3κi∏i=1y1+δ−1ϱPi3κi+δ−1∏i=1y1−ϱPi3κi3,δ∏i=1y1−ϱPi3κi−∏i=1y1−ϱPi3−σPi3κi∏i=1y1+δ−1ϱPi3κi+δ−1∏i=1y1−ϱPi3κi3⊕∏i=1y1+δ−1ϱTi3κi−∏i=1y1−ϱTi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱTi3κi3,δ∏i=1y1−ϱTi3κi−∏i=1y1−ϱTi3−σTi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱTi3κi3=FFHIWAP1,P2,…,Py⊕FFHIWAT1,T2,…,Property 7.
LetPi=ϱi,σi and T=ϱ,σ be FFNs and η>0, then(33)FFHIWAηP1⊕T,ηP2⊕T,…,ηPy⊕T=ηFFHIWAP1,P2,…,Py⊕T.Proof.
By applying the Properties1, 5, and 6, we can proof it.


## 3.1. Weighted Average Aggregation Operators

LetPi=ϱi,σii=1,2,…,y be a collection of FFNs and κ=κ1,κ2,…,κyT be its weight vector (WV) such that κi>0 and ∑i=1yκi=1, then FFHIWA:Ωy⟶Ω is defined as(4)FFHIWAP1,P2,…,Py=κ1P1⊕κ2P2⊕⋯κyPy.Theorem 1.
LetPi=ϱi,σi be a collection of FFNs, then(5)FFHIWAP1,P2,…,Py=∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi/∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi/∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3.Proof.
Fory=1,κ=κ1=1,(6)FFHIWAP1=κ1P1=P1=ϱ1,σ1=1+δ−1ϱ13−1−ϱ131+δ−1ϱ13+δ−11−ϱ133,δ1−ϱ13−1−ϱ13−σ131+δ−1ϱ13δ−11−ϱ133.
Thus, the result holds fory=1. Suppose that result holds for y=p, i.e.,(7)FFHIWAP1,P2,…,Pp=∏i=1p1+δ−1ϱi3κi−∏i=1p1−ϱi3κi∏i=1p1+δ−1ϱi3+δ−1∏i=1p1−ϱi3κi3,δ∏i=1p1−ϱi3κi−∏i=1p1−ϱi3−σi3κi∏i=1p1+δ−1ϱi3+δ−1∏i=1p1−ϱi3κi3.
Now, fory=p+1,(8)FFHIWAP1,P2,…,Pp+1=⊕i=1p+1κiPi=∏i=1p1+δ−1ϱi3κi−∏i=1p1−ϱi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3,δ∏i=1p1−ϱi3κi−∏i=1p1−ϱi3−σi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3⊕1+δ−1ϱp+13κp+1−1−ϱp+13κp+11+δ−1ϱp+13+δ−11−ϱp+13κp+13,δ1−ϱp+13κp+1−1−ϱp+13−σp+13κp+11+δ−1ϱp+13κp+1+δ−11−ϱp+13κp+13=∏i=1p1+δ−1ϱi3κi−∏i=1p1−ϱi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3,δ∏i=1p1−ϱi3κi−∏i=1p1−ϱi3−σi3κi∏i=1p1+δ−1ϱi3κi+δ−1∏i=1p1−ϱi3κi3.⇒ Result holds ∀y.Remark 1.
We elaborate two cases of the FFHIWA operator:(i)
Forδ=1, FFHIWA operator becomes FF interactive weighted averaging (FFIWA) operator:(9)FFIWAP1,P2,…,Py=1−∏i=1y1−ϱi3κi3,∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi3.(ii)
Forδ=2, FFHIWA operator becomes FF Einstein interactive weighted averaging (FFEIWA) operator:(10)FFEIWAP1,P2,…,Py=∏i=1y1+ϱi3κi−∏i=1y1+ϱi3κi∏i=1y1+ϱi3κi+∏i=1y1+ϱi3κi3,2∏i=1y1+ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+ϱi3κi+∏i=1y1+ϱi3κi3.Theorem 2.
LetPi=ϱi,σi be FFNs, then the accumulated value by using FFHIWA operator is a FFN, i.e.,(11)FFHIWAP1,P2,…,Py∈FFN.Proof.
AsPi′s are FFNs, 0≤ϱi,σi≤1 and 0≤ϱi3+σi3≤1. Therefore,(12)∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi=1−δ∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≤1−∏i=1y1−ϱi3κi≤1.
Also,1+δ−1ϱi3≥1−ϱi3⇒∏i=1y1+δ−1ϱi3−∏i=1y1−ϱi3≥0. Therefore,(13)∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≥0,⇒∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3≥0.
Thus,
Moreover,(14)δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≤δ∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≤≤∏i=1y1−ϱi3κi≤1.
Also,(15)∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi≥0,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi≥0,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3≥0.
Thus,Property 1.
(idempotency). IfPi=Po=ϱo,σo,∀i, then(16)FFHIWAP1,P2,…,Py=Po.Proof.
SincePi=Po=ϱo,σo∀i=1,2.…,y and ∑i=1yκi=1, by Theorem 1,(17)FFHIWAP1,P2,…,Py=∏i=1y1+δ−1ϱo3κi−∏i=1y1−ϱo3κi∏i=1y1+δ−1ϱo3κi+δ−1∏i=1y1−ϱo3κi3,δ∏i=1y1−ϱo3κi−∏i=1y1−ϱo3−σo3κi∏i=1y1+δ−1ϱo3κi+δ−1∏i=1y1−ϱo3κi3=1+δ−1ϱo3∑i=1yκi−1−ϱo3∑i=1yκi1+δ−1ϱo3∑i=1yκi+δ−11−ϱo3∑i=1yκi3,δ1−ϱo3∑i=1yκi−1−ϱo3−σo3∑i=1yκi1+δ−1ϱo3∑i=1yκi+δ−11−ϱo3∑i=1yκi3=1+δ−1ϱo3−1−ϱo31+δ−1ϱo3+δ−11−ϱo33,δ1−ϱo3−1−ϱo3−σo31+δ−1ϱo3+δ−11−ϱo33=ϱo,σo.Property 2.
(boundedness). LetP−=miniϱi,maxiσi and P+=miniϱi,maxiσi, then(18)P−≤FFHIWAP1,P2,…,Py≤P+.Proof.
Letfr=1−r/1+δ−1r,r∈0,1, then f′r=−δ/1+δ−1r2<0, so fr is a decreasing function (DF). As ϱi,min3≤ϱi3≤ϱi,max3,∀i=1,2,…,y, then fϱi,max3≤fϱi3≤fϱi,min3,∀i; that is, 1−ϱi,max3/1+δ−1ϱi,max3≤1−ϱi3/1+δ−1ϱi3≤1−ϱi,min3/1+δ−1ϱi,min3, ∀i. Let κi∈0,1 and ∑i=1yκi=1, we have(19)1−ϱi,max31+δ−1ϱi,max3κi≤1−ϱi31+δ−1ϱi3κi≤1−ϱi,min31+δ−1ϱi,min3κi∏i=1y1−ϱi,max31+δ−1ϱi,max3κi≤∏i=1y1−ϱi31+δ−1ϱi3κi≤∏i=1y1−ϱi,min31+δ−1ϱi,min3κi⇔1−ϱi,max31+δ−1ϱi,max3∑i=1yκi≤∏i=1y1−ϱi31+δ−1ϱi3κi≤1−ϱi,min31+δ−1ϱi,min3∑i=1yϖi⇔1−ϱi,max31+δ−1ϱi,max3≤∏i=1y1−ϱi31+δ−1ϱmath fraki3κi≤1−ϱi,min31+δ−1ϱi,min3⇔δ−11−ϱi,max31+δ−1ϱi,max3≤δ−1∏i=1y1−ϱi31+δ−1ϱi3ϖi≤δ−11−ϱi,min31+δ−1ϱi,min3⇔δ1+δ−1ϱi,max3≤1+δ−1∏i=1y1−ϱi31+δ−1ϱi3κi≤δ1+δ−1ϱi,min3⇔1+δ−1ϱi,min3δ≤11+δ−1∏i=1y1−ϱi3/1+δ−1ϱi3κi≤1+δ−1ϱi,max3δ⇔1+δ−1ϱi,min3≤δ1+δ−1∏i=1y1−ϱi3/1+δ−1ϱi3κi≤1+δ−1ϱi,max3⇔δ−1ϱi,min3≤δ1+δ−1∏i=1y1−ϱi3/1+δ−1ϱi3κi−1≤δ−1ϱi,max3⇔ϱi,min3≤∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3ϖi+δ−1∏i=1y1−ϱi3ϖi≤ϱi,max3.
Thus,(20)ϱi,min≤∏i=1y1+δ−1ϱi3κi−∏i=1y∏1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3ϖi3≤ϱi,max.
Considergs=δ−δ−1s/δ−1s,s∈0,1, then g′s=−δ/δ−1s2; i.e., gs is a DF on 0,1. Since 1−ϱi,max3≤1−ϱi3≤1−ϱi,min3,∀i, then g1−ϱi,max3≤g1−ϱi3≤g1−ϱi,max3,∀i, that is, δ−δ−11−ϱi,min3/δ−11−ϱi,min3≤δ−δ−11−ϱi3/δ−11−ϱi3≤δ−δ−11−ϱi,max3/δ−11−ϱi,max3. Then,(21)δ−δ−11−ϱi,min3δ−11−ϱi,min3κi≤δ−δ−11−ϱi3δ−11−ϱi3κi≤δ−δ−11−ϱi,max3δ−11−ϱi,max3κi∏i=1yδ−δ−11−ϱi,min3δ−11−ϱi,min3κi≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi≤∏i=1yδ−δ−11−ϱi,max3δ−11−ϱi,max3κi⇒δ−δ−11−ϱi,min3δ−11−ϱi,min3∑i=1yκi≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi≤δ−δ−11−ϱi,max3δ−11−ϱi,max3∑i=1yκi⇒δ−δ−11−ϱi,min3δ−11−ϱi,min3≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi≤δ−δ−11−ϱi,max3δ−11−ϱi,max3⇒δδ−11−ϱi,min3≤∏i=1yδ−δ−11−ϱi3δ−11−ϱi3κi+1≤δδ−11−ϱi,max3⇒δ−11−ϱi,max3δ≤1∏i=1yδ−δ−11−ϱi3/δ−11−ϱi3κi+1≤δ−11−ϱi,min3δ⇒1−ϱi,max3≤δδ−1∏i=1yδ−δ−11−ϱi3/δ−11−ϱi3κi+δ−1≤1−ϱi,min3⇒1−ϱi,max3≤δδ−1∏i=1yδ−δ−1σi3/δ−1σi3κi+δ−1≤1−ϱi,min3.
Let FFHIWAP1,P2,…,Py=P=ϱP,σP, then from inequalities (20) and (21), ϱmin≤ϱP≤ϱmax,σmax≤σP≤σmin, where ϱmin=miniϱi, ϱmax=maxiϱi, σmin=miniσi, and σmax=maxiσi. So SP=ϱP3−σP3≤ϱmax3−σmax3=SP+ and SP=ϱP3−σP3≥ϱmin3−σmin3=SP−. As SP<SP+ and SP>SP−,(22)P−≤FFHIWAP1,P2,…,Py≤P+.Property 3.
(monotonicity). WhenPi≤Ti,∀i, then(23)FFHIWAP1,P2,…,Py≤FFHIWAT1,T2,…,Ty.Proof.
It is similar to above.Property 4.
(shift invariance). IfT=ϱT,σT is another FFN, then(24)FFHIWAP1⊕T,P2⊕T,…,Py⊕T=FFHIWAP1,P2,…,Py⊕T.Proof.
AsPi,T∈ FFNs, (25)Pi⊕T=1+δ−1ϱi31+δ−1ϱT3−1−ϱi31−ϱT31+δ−1ϱi31+δ−1ϱT3+δ−11−ϱi31−ϱT33,δ1−ϱi31−ϱT3−1−ϱi3−σi31−ϱT3−σT31+δ−1ϱi31+δ−1ϱT3+δ−11−ϱi31−ϱT33
Therefore,(26)FFHIWAP1⊕T,P2⊕T,…,Py⊕T=∏i=1y1+δ−1ϱi31+δ−1ϱT3κi−∏i=1y1−ϱi31−ϱT3κi∏i=1y1+δ−1ϱi31+δ−1ϱT3κi+δ−1∏i=1y1−ϱi31−ϱT3κi3,δ∏i=1y1−ϱi31−ϱT3κi−∏i=1y1−ϱi3−σi31−ϱT3−σT3κi∏i=1y1+δ−1ϱi31+δ−1ϱT3κi+δ−1∏i=1y1−ϱi31−ϱT3κi3=∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3κi−∏i=1y1−ϱi3κi1−ϱT3κi∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3κi+δ−1∏i=1y1−ϱi3κi1−ϱT3κi3,δ∏i=1y1−ϱi3κi1−ϱT3κi−∏i=1y1−ϱi3−σi3κi1−ϱT3−σT3κi∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3κi+δ−1∏i=1y1−ϱi3κi1−ϱT3κi3=∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3−∏i=1y1−ϱi3κi1−ϱT3∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3+δ−1∏i=1y1−ϱi3κi1−ϱT33,δ∏i=1y1−ϱi3κi1−ϱT3−∏i=1y1−ϱi3−σi3κi1−ϱT3−σT3∏i=1y1+δ−1ϱi3κi1+δ−1ϱT3+δ−1∏i=1y1−ϱi3κi1−ϱT33=FFHIWAP1,P2,…,Py⊕T.Property 5.
(homogeneity). Letβ>0, then(27)FFHIWAβP1,βP2,…,βPy=βFFHIWAP1,P2,…,Py.Proof.
SincePi=ϱi,σi are FFNs, for β>0,(28)βPi=1+δ−1ϱ3β−1−ϱ3β1+δ−1ϱ3β+δ−11−ϱ3β3,δ1−ϱ3β−1−ϱ3−σ3β1+δ−1ϱ3β+δ−11−ϱ3β3.
Therefore,(29)FFHIWAβP1,βP2,…,βPy=∏i=1y1+δ−1ϱ3βκi−∏i=1y1−ϱ3βκi∏i=1y1+δ−1ϱ3βκi+δ−1∏i=1y1−ϱ3βκi3,δ∏i=1y1−ϱ3βκi−∏i=1y1−ϱ3−σ3βκi∏i=1y1+δ−1ϱ3βκi+δ−1∏i=1y1−ϱ3βκi3=∏i=1y1+δ−1ϱ3κiβ−∏i=1y1−ϱ3κiβ∏i=1y1+δ−1ϱ3κiβ+δ−1∏i=1y1−ϱ3κiβ3,δ∏i=1y1−ϱ3κiβ−∏i=1y1+δ−1ϱ3κiβ∏i=1y1+δ−1ϱ3κiβ+δ−1∏i=1y1−ϱ3κiβ3=β∏i=1y1+δ−1ϱi3κi−∏i=1y1−ϱi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3,δ∏i=1y1−ϱi3κi−∏i=1y1−ϱi3−σi3κi∏i=1y1+δ−1ϱi3κi+δ−1∏i=1y1−ϱi3κi3=βFFHIWAP1,P2,…,Py.Property 6.
LetPi=ϱPi,σPi and Ti=ϱTi,σTi be two collections of FFNs, then(30)FFHIWAP1⊕T1,P2⊕T2,…,Py⊕Ty=FFHIWAP1,P2,…,Py⊕FFHIWAT1,T2,…,Ty.Proof.
AsPi=ϱPi,σPi and Ti=ϱTi,σTi are two collections of FFNs, then(31)Pi⊕Ti=1+δ−1ϱPi31+δ−1ϱTi3−1−ϱPi31−ϱTi31+δ−1ϱPi31+δ−1ϱTi3+δ−11−ϱPi31−ϱTi33,δ1−ϱPi31−ϱTi3−1−ϱPi3−σPi31−ϱTi3−σTi31+δ−1ϱPi31+δ−1ϱTi3+δ−11−ϱPi31−ϱTi33
Therefore,(32)FFHIWAP1⊕T1,P2⊕T2,…,Py⊕Ty=∏i=1y1+δ−1ϱPi31+δ−1ϱTi3κi−∏i=1y1−ϱPi31−ϱTi3κi∏i=1y1+δ−1ϱPi31+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi31−ϱTi3κi3,δ∏i=1y1−ϱPi31−ϱTi3κi−∏i=1y1−ϱPi3−σPi31−ϱTi3−σTi3κi∏i=1y1+δ−1ϱPi31+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi31−ϱTi3κi3=∏i=1y1+δ−1ϱPi3κi∏i=1y1+δ−1ϱTi3κi−∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi∏i=1y1+δ−1ϱPi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi3,δ∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi−∏i=1y1−ϱPi3−σPi3κi∏i=1y1−ϱTi3−σTi3κi∏i=1y1+δ−1ϱPi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱPi3κi∏i=1y1−ϱTi3κi3=∏i=1y1+δ−1ϱPi3κi−∏i=1y1−ϱPi3κi∏i=1y1+δ−1ϱPi3κi+δ−1∏i=1y1−ϱPi3κi3,δ∏i=1y1−ϱPi3κi−∏i=1y1−ϱPi3−σPi3κi∏i=1y1+δ−1ϱPi3κi+δ−1∏i=1y1−ϱPi3κi3⊕∏i=1y1+δ−1ϱTi3κi−∏i=1y1−ϱTi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱTi3κi3,δ∏i=1y1−ϱTi3κi−∏i=1y1−ϱTi3−σTi3κi∏i=1y1+δ−1ϱTi3κi+δ−1∏i=1y1−ϱTi3κi3=FFHIWAP1,P2,…,Py⊕FFHIWAT1,T2,…,Property 7.
LetPi=ϱi,σi and T=ϱ,σ be FFNs and η>0, then(33)FFHIWAηP1⊕T,ηP2⊕T,…,ηPy⊕T=ηFFHIWAP1,P2,…,Py⊕T.Proof.
By applying the Properties1, 5, and 6, we can proof it.


## 4. Ordered Weighted Averaging Operator

Definition 5.
LetPi=ϱi,σi be a collection of FFNs and κ=κ1,κ2,…,κyT be its WV such that κi>0 and ∑i=1yκi=1, then FFHIOWA:Ωy⟶Ω is defined as(34)FFHIWAP1,P2,…,Py=κ1Pσ1⊕κ2Pσ2⊕⋯κyPσy,where σ1,σ2,…,σy is a permutation of 1,2,…,y such that σi−1≥σi for any i.Theorem 3.
LetPi=ϱi,σi be a collection of FFNs, then(35)FFHIOWAP1,P2,…,Py=∏i=1y1+δ−1ϱσi3κi−∏i=1y1−ϱσi3κi∏i=1y1+δ−1ϱσi3κi+δ−1∏i=1y1−ϱσi3κi,3δ∏i=1y1−ϱσi3κi−∏i=1y1−ϱσi3−σσi3κi∏i=1y∏i=1y1+δ−1ϱσi3κi+δ−1∏i=1y1−ϱσi3κi3Proof.
It is similar to Theorem1.Remark 2.
We elaborate two cases of the FFHIOWA operator.(i)
Forδ=1, FFHIOWA operator becomes FF interactive ordered weighted averaging (FFIOWA) operator:(36)FFIOWAP1,P2,…,Py=1−∏i=1y1−ϱσi3κi3,∏i=1y1−ϱσi3κi−∏i=1y1−ϱσi3−σσi3κi3.(ii)
Forδ=2, FFHIOWA operator becomes FF Einstein interactive ordered weighted averaging (FFEIOWA) operator:(37)FFEIOWAP1,P2,…,Py=∏i=1y1+ϱσi3κi−∏i=1y1−ϱσi3κi∏i=1y1+ϱσi3κi+∏i=1y1−ϱσi3κi3,2∏i=1y1−ϱσi3κi−∏i=1y1−ϱσi3−σσi3κi∏i=1y1+ϱσi3κi+∏i=1y1−ϱσi3κi3.Property 8.
LetPi=ϱi,σi be a collection of FFNs and κ=κ1,κ2,…,κyT be its WV such that κi>0 and ∑i=1yκi=1.(i)
Idempotency: ifPi=Po=ϱo,σo,∀i, then(38)FFHIOWAP1,P2,…,Py=Po.(ii)
Boundedness: letP−=miniϱi,maxiσi and P+=maxiϱi,miniσi, then(39)P−≤FFHIOWAP1,P2,…,Py≤P+.(iii)
Monotonicity: whenPi≤Ti,∀i, then(40)FFHIOWAP1,P2,…,Py≤FFHIOWAT1,T2,…,Ty.(iv)
Shift invariance: ifT=ϱT,σT is another FFN, then(41)FFHIOWAP1⊕T,P2⊕T,…,Py⊕T=FFHIOWAP1,P2,…,Py⊕T.(v)
Homogeneity: letβ>0, then(42)FFHIOWAβP1,βP2,…,βPy=βFFHIOWAP1,P2,…,Py.Proof.
It is similar to the FFHIWA properties.


## 5. Hybrid Weighted Averaging Operator

Definition 6.
LetPi=ϱi,σi be a collection of FFNs, then FFHIHWA:Ωy⟶Ω is defined as(43)FFHIHWAP1,P2,…,Py=κ1P.1⊕κ2P.2⊕⋯κyP.y,where κ=κ1,κ2,…,κyT is the WV associated with FFHIHWA operator and ϕ=ϕ1,ϕ2,…,ϕyT is the WV of Pi such that ϕi∈0,1 and ∑i=1yϕi=1. Let P. is the ith largest of the weighted FFNs (P.=yϕiPi and σ1,σ2,…,σy is a permutation of 1,2,…,y such that σi−1≥σi for any i.Theorem 4.
LetPi=ϱi,σi be a collection of FFNs, then(44)FFHIHWAP1,P2,…,Py=∏i=1y1+δ−1ϱ˙σi3κi−∏i=1y1−ϱ˙σi3κi∏i=1y1+δ−1ϱ˙σi3κi+δ−1∏i=1y1−ϱ˙σi3κi3,δ∏i=1y1−ϱ˙σi3κi−∏i=1y1−ϱ˙σi3−σ˙σi3κi∏i=1y1+δ−1ϱ˙σi3κi+δ−1∏i=1y1−ϱ˙σi3κi3.Proof.
It is similar to Theorem1.Remark 3.
FFHIHWA operator also satisfies the same properties as given in Property8.


## 6. MAGDM under Fermatean Fuzzy Environment

In MAGDM problem, it is a biggest challenge for decision makers (DMrs) to choose the best alternative among the list of possible alternatives. LetS1,S2,…,Sy be y distinct alternatives which can be classified under the set of m different attributes c1,c2,…,cm by the DMrs. Suppose that DMrs give their preferences in terms of FFNs αij=ϱij,σiji=1,2,…,y;j=1,2,…,m, where ϱij and σij are the satisfaction and dissatisfaction degrees, respectively, of the alternative corresponding to given parameter given by the DMrs such that 0≤ϱij3+σij3≤1. The different steps for MAGDM problem are given as follows:Step 1. Attain the normalize FF decision matrix by exchanging the assessment value of cost parameter (CP) into benefit parameter (BP) [40], i.e.,(45)Pij=αijc;forCP,αij;for BP.Step 2. By using the decision matrix of step 1, the overall aggregated value of alternativeSi under the distinct choices of attributes cj is obtained by using FFHIWA or FFHIOWA or FFHIHWA operator and get the overall value of them.Step 3. By using the score function, calculate the score values of all alternatives.Step 4. Rank the alternativesS1,S2,…,Sy in the descending order of score values and then select the most suitable alternative.


### 6.1. Numerical Example

To classify the air quality (AQ) of Guangzhou for the 16th Asian Olympic Games [41] held during November 12-27, 2010, the AQ data in Guangzhou for November 2006, November 2007, November 2008, and November 2009 are collected to find out the trends in the AQ. Suppose that there are three AQ monitoring stations E1, E2, and E3, which are considered as DMrs and suppose that the 0.314, 0.355, and 0.331 are weights of E1, E2, and E3, respectively. There are three measured indexes, namely, SO2c1,NO2c2, and PM10c3, and their weight is κ=0.40,0.20,0.40T. Let S1, S2, S3, and S4 be alternatives, where S1 = November 2006, S2 = November 2007, S3 = November 2008, and S4 = November 2009. Suppose that the measured values obtained from the AQ monitoring stations E1, E2, and E3 under the measured indexes SO2 (i.e., the attribute c1), NO2 (i.e., the attribute c2), and PM10 (i.e., the attribute c3) in the form of FFNs are shown in Tables 1–3, respectively. We rank the AQ from 2006–2009 by using the proposed method.Table 1
Air quality data by expertE1.
E1c1c2c3S1(0.5, 0.7)(0.9, 0.6)(0.8, 0.7)S2(0.3, 0.7)(0.1, 0.9)(0.66, 0.8)S3(0.8, 0.3)(0.7, 0.2)(0.8, 0.4)S4(0.9, 0.1)(0.8, 0.1)(0.7, 0.3)Table 2
Air quality data by expertE2.
E1c1c2c3S1(0.6, 0.88)(0.3, 0.57)(0.46, 0.76)S2(0.3, 0.7)(0.1, 0.99)(0.4, 0.77)S3(0.96, 0.3)(0.7, 0.22)(0.6, 0.01)S4(0.88, 0.2)(0.86, 0.01)(0.96, 0.33)Table 3
Air quality data by expertE3.
E3c1c2c3S1(0.1, 0.93)(0.7, 0.3)(0.5, 0.3)S2(0.9, 0.4)(0.3, 0.56)(0.47, 0.66)S3(0.93, 0.3)(0.76, 0.2)(0.76, 0.1)S4(0.97, 0.4)(0.88, 0.3)(0.89, 0.5)By FFHIWA operator, the steps are as follows:Step 1: as all criteria are of same type, decision matrix cannot be normalized. The aggregated decision matrix by using FF weighted averaging operator with WVλ=0.314,0.355,0.331T is shown in Table 4.Step 2: to find the overall assessment of each alternative, we apply the FFHIWA operator forδ=1 as follows.ForP1,(46)=FFHIWAS11,S12,S13=1−1−0.403130.401−0.620830.201−0.580030.403,1−0.403130.401−0.620830.201−0.580030.40−1−0.40313−0.840030.401−0.62083−0.490130.201−0.58003−0.588930.403=0.5374,0.7106.ForP2,(47)=FFHIWAS11,S12,S13=1−1−0.498630.401−0.166230.201−0.504830.403,1−0.0.498630.401−0.166230.201−0.504830.40−1−0.49863−0.600730.401−0.16623−0.725230.201−0.50483−0.743030.403=0.4691,0.6934.ForP3,(48)=FFHIWAS11,S12,S13=1−1−0.899830.401−0.719930.201−0.715830.403,1−0.899830.401−0.719930.201−0.715830.40−1−0.89983−0.300030.401−0.71993−0.207130.201−0.71583−0.162330.403=0.8191,0.2754.ForP4,(49)=FFHIWAS11,S12,S13=1−1−0.916130.401−0.847830.201−0.855230.403,1−0.916130.401−0.847830.201−0.855230.40−1−0.91613−0.234830.401−0.84783−0.134330.201−0.85523−0.376930.403=0.8831,0.2951.Step 3: the score values for alternatives are(50)SS1=−0.2036,SS2=−0.2302,SS3=0.5287,SS4=0.6630.Step 4: asS4≻S3≻S1≻S2, the best AQ in Guangzhou is November of 2009.Table 4
Aggregated FF decision matrix.
c1c2c3S1(0.4031, 0.8400)(0.6208, 0.4901)(0.5800, 0.5889)S2(0.4986, 0.6007)(0.1662, 0.7252)(0.5048, 0.7430)S3(0.8998, 0.3000)(0.7199, 0.2071)(0.7158, 0.1623)S4(0.9161, 0.2348)(0.8478, 0.1343)(0.8552, 0.3769)The whole method which we have adopted in this application is given in Figure1.Figure 1
Flow chart for the classification of the air quality of Guangzhou.


## 6.1. Numerical Example

To classify the air quality (AQ) of Guangzhou for the 16th Asian Olympic Games [41] held during November 12-27, 2010, the AQ data in Guangzhou for November 2006, November 2007, November 2008, and November 2009 are collected to find out the trends in the AQ. Suppose that there are three AQ monitoring stations E1, E2, and E3, which are considered as DMrs and suppose that the 0.314, 0.355, and 0.331 are weights of E1, E2, and E3, respectively. There are three measured indexes, namely, SO2c1,NO2c2, and PM10c3, and their weight is κ=0.40,0.20,0.40T. Let S1, S2, S3, and S4 be alternatives, where S1 = November 2006, S2 = November 2007, S3 = November 2008, and S4 = November 2009. Suppose that the measured values obtained from the AQ monitoring stations E1, E2, and E3 under the measured indexes SO2 (i.e., the attribute c1), NO2 (i.e., the attribute c2), and PM10 (i.e., the attribute c3) in the form of FFNs are shown in Tables 1–3, respectively. We rank the AQ from 2006–2009 by using the proposed method.Table 1
Air quality data by expertE1.
E1c1c2c3S1(0.5, 0.7)(0.9, 0.6)(0.8, 0.7)S2(0.3, 0.7)(0.1, 0.9)(0.66, 0.8)S3(0.8, 0.3)(0.7, 0.2)(0.8, 0.4)S4(0.9, 0.1)(0.8, 0.1)(0.7, 0.3)Table 2
Air quality data by expertE2.
E1c1c2c3S1(0.6, 0.88)(0.3, 0.57)(0.46, 0.76)S2(0.3, 0.7)(0.1, 0.99)(0.4, 0.77)S3(0.96, 0.3)(0.7, 0.22)(0.6, 0.01)S4(0.88, 0.2)(0.86, 0.01)(0.96, 0.33)Table 3
Air quality data by expertE3.
E3c1c2c3S1(0.1, 0.93)(0.7, 0.3)(0.5, 0.3)S2(0.9, 0.4)(0.3, 0.56)(0.47, 0.66)S3(0.93, 0.3)(0.76, 0.2)(0.76, 0.1)S4(0.97, 0.4)(0.88, 0.3)(0.89, 0.5)By FFHIWA operator, the steps are as follows:Step 1: as all criteria are of same type, decision matrix cannot be normalized. The aggregated decision matrix by using FF weighted averaging operator with WVλ=0.314,0.355,0.331T is shown in Table 4.Step 2: to find the overall assessment of each alternative, we apply the FFHIWA operator forδ=1 as follows.ForP1,(46)=FFHIWAS11,S12,S13=1−1−0.403130.401−0.620830.201−0.580030.403,1−0.403130.401−0.620830.201−0.580030.40−1−0.40313−0.840030.401−0.62083−0.490130.201−0.58003−0.588930.403=0.5374,0.7106.ForP2,(47)=FFHIWAS11,S12,S13=1−1−0.498630.401−0.166230.201−0.504830.403,1−0.0.498630.401−0.166230.201−0.504830.40−1−0.49863−0.600730.401−0.16623−0.725230.201−0.50483−0.743030.403=0.4691,0.6934.ForP3,(48)=FFHIWAS11,S12,S13=1−1−0.899830.401−0.719930.201−0.715830.403,1−0.899830.401−0.719930.201−0.715830.40−1−0.89983−0.300030.401−0.71993−0.207130.201−0.71583−0.162330.403=0.8191,0.2754.ForP4,(49)=FFHIWAS11,S12,S13=1−1−0.916130.401−0.847830.201−0.855230.403,1−0.916130.401−0.847830.201−0.855230.40−1−0.91613−0.234830.401−0.84783−0.134330.201−0.85523−0.376930.403=0.8831,0.2951.Step 3: the score values for alternatives are(50)SS1=−0.2036,SS2=−0.2302,SS3=0.5287,SS4=0.6630.Step 4: asS4≻S3≻S1≻S2, the best AQ in Guangzhou is November of 2009.Table 4
Aggregated FF decision matrix.
c1c2c3S1(0.4031, 0.8400)(0.6208, 0.4901)(0.5800, 0.5889)S2(0.4986, 0.6007)(0.1662, 0.7252)(0.5048, 0.7430)S3(0.8998, 0.3000)(0.7199, 0.2071)(0.7158, 0.1623)S4(0.9161, 0.2348)(0.8478, 0.1343)(0.8552, 0.3769)The whole method which we have adopted in this application is given in Figure1.Figure 1
Flow chart for the classification of the air quality of Guangzhou.


## 7. Comparison Analysis

For the validity and importance of proposed operators, we aggregate the same information using different operator, namely, FFEWA or FFEOWA operator [30].Definition 7.
(see [30]). The FFEWA operator is as follows:(51)FFEWAP1,P2,…,Py=∏i=1y1+ϱi3κi−∏i=1y1+ϱi3κi∏i=1y1+ϱi3κi+∏i=1y1+ϱi3κi3,23∏i=1yσiκi∏i=1y2−σi3κi+∏i=1yσi3κi3.
The FF Einstein ordered weighted averaging (FFEOWA) operator is(52)FFEOWAP1,P2,…,Py=∏i=1y1+ϱσi3κi−∏i=1y1−ϱσi3κi∏i=1y1+ϱσi3κi+∏i=1y1−ϱσi3κi3,23∏i=1yσσiκi∏i=1y2−σσi3κi+∏i=1yσσi3κi3.
By FFEWA operator, the steps are as follows:Step 1: same as above.Step 2: to find the overall assessment of each alternative, we apply the FFEWA operator as(53)P1=FFEWAS11,S12,S13=1+0.403130.401+0.620830.201+0.580030.40−1−0.403130.401−0.620830.201−0.580030.401+0.403130.401+0.620830.201+0.580030.40+1−0.403130.401−0.620830.201−0.580030.403,230.84000.400.49010.200.58890.402−0.840030.402−0.490130.202−0.588930.40+0.840030.400.490130.200.588930.40=0.5347,0.6628.P2=FFEWAS11,S12,S13=1+0.498630.401+0.166230.201+0.504830.40−1−0.498630.401−0.166230.201−0.504830.401+0.498630.401+0.166230.201+0.504830.40+1−0.498630.401−0.166230.201−0.504830.403,230.60070.400.72520.200.74300.402−0.600730.402−0.725230.202−0.743030.40+0.600730.400.725230.200.743030.40=0.4674,0.6810.P3=FFEWAS11,S12,S13=1+0.899830.401+0.719930.201+0.715830.40−1−0.899830.401−0.719930.201−0.715830.401+0.899830.401+0.719930.201+0.715830.40+1−0.899830.401−0.719930.201−0.715830.403,230.30000.400.20710.200.16230.402−0.300030.402−0.207130.202−0.162330.40+0.300030.400.207130.200.162330.40=0.8137,0.2180.P4=FFEWAS11,S12,S13=1+0.916130.401+0.847830.201+0.855230.40−1−0.916130.401−0.847830.201−0.855230.401+0.916130.401+0.847830.201+0.855230.40+1−0.916130.401−0.847830.201−0.855230.403,230.23480.400.13430.200.37690.402−0.234830.402−0.134330.202−0.376930.40+0.234830.400.134330.200.376930.40=0.8831,0.2951.Step 3: the score values for alternatives are(54)SS1=−0.1383,SS2=−0.2137,SS3=0.5284,SS4=0.6704.Step 4: asS4≻S3≻S1≻S2, the best AQ in Guangzhou is November of 2009.
The results obtained from these operators are shown in Table5 and Figure 2. It is clear that the most suitable alternative obtained by using FFHIWA and FFEWA operators is the same. This implies that our proposed methods are accurate and can be utilized in DM problems.
Advantages of proposed operators: the main reason behind proposed approach is that(i)
We can see the effect of other grades of nonmembership in the aggregated value even if nonmembership of any one alternative is zero.(ii)
We can see that there is a proper interaction between the MD and NMD.
The operators defined in [30] are very concise and have been extensively used, but these operators have certain drawbacks. Few of them have been highlighted as follows:(1
)LetP1=0.9,0, P2=0.77,0.45, P3=0.80,0.63, and P4=0.58,0.67 be four FFNs and κ=0.3,03,0.2,0.2T is the WV corresponding to FFNs. By applying the FFEWA operator, we get FFEWAP1,P2,P3,P4=0.88,0. This shows that NMD of a FFN is independent of the NMD of others FFNs (which are nonzero in Pi′s) and hence does not play a significant role during the AP. The aggregated FFNs as FFHIWAP1,P2,P3,P4=0.81,0.49 for δ=1 and for δ=2FFHIWAP1,P2,P3,P4=0.88,0.43. It can be seen that NMD is nonzero of the whole aggregated FFNs even if at least one of the NMD of FFNs is zero. Thus, the others nonmembership values of FFNs play a predominant role during the AP in the proposed operator.(2)
LetP1=0.53,0.42, P2=0.98,0.34, P3=0.61,0.54, and P4=0.71,0.46 be four FFNs and κ=0.4,0.2,0.3,0.1T is the WV corresponding to FFNs. By applying the FFEWA operator, we get FFEWAP1,P2,P3,P4=0.78,0.44. If we replace FFNs P2 and P3 with T2=0.78,0.34 and T3=0.67,0.54, then their corresponding aggregated FFN become 0.66,0.44. Hence, the NMD part of aggregated FFN becomes independent of the change in MD’s values. That is why it is incompatible and does not produce an accurate information to the decision maker. The aggregated FFNs as FFHIWAP1,P2,P3,P4=0.80,0.53 for δ=1 and FFHIWAP1,P2,P3,P4=0.78,0.55 for δ=2, and if we consider modified FFNs, then FFHIWAP1,T2,T3,P4=0.67,0.45 for δ=1 and FFHIWAP1,T2,T3,P4=0.66,0.45 for δ=2. It can be seen that the modification in membership function will affect aggregated value of nonmembership function and is nonzero. That is why, there is a proper interaction between the MD and NMD, and hence, the results are unchangeable and more realistic than the existing operators results.Table 5
Comparison analysis with FFEWA operator.
MethodsSS1SS2SS3SS4Ranking orderFFEWA operator−0.1383−0.21370.52840.6704S4≻S3≻S1≻S2FFHIWA operator (proposed)−0.2036,−0.23020.52870.6630S4≻S3≻S1≻S2Figure 2
Comparison with FFEWA operator.


## 8. Conclusions

FFS is a generalized structure of IFS and PFS. It is more powerful tool to solve DM problems involving uncertainty and satisfies the condition0≤ϱ3+σ3≤1. The structure of Hamacher’s t-norm and t-conorm is more generalized that effectively integrates the complex information. The shortcomings of the existing methods and beneficial characteristics of Hamacher AOs motivate us to endeavor for the development of a fruitful fusion with FFNs. In this research article, we have developed a group of novel FF Hamacher interactive averaging AOs, such as FFHIWA, FFHIOWA, and FFHIHWA operators. These proposed operators have the characteristic of idempotency, boundedness, monotonicity, homogeneity, and shift invariance. These operators reduce the shortcomings of FFEWA operators. We have also discussed some particular cases of proposed operators. Moreover, the developed operators study the interaction between membership and nonmembership grades. We have presented an algorithm to deal with MAGDM problems. For the validity and flexibility of proposed work, we have given the comparison analysis. In short, this work focuses on role of Hamacher interactive AOs as well as the propitious characteristics of FFNs. It is concluded that the new model of uncertain data is flexible which aptly depicts imprecise and inexact information in complicated scenarios. Thus, the operators serve as a powerful tool with further applications due to their highly adaptable nature. In future, we will work on the following topics:(1)
Neutrality aggregation operators for Fermatean fuzzy sets.(2)
Fermatean fuzzy power aggregation operators.(3)
Fermatean fuzzy Hamy mean aggregation operators and their application in multiattribute decision making.(4)
Fermatean fuzzy soft Dombi aggregation operators.



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*Source: 5556017-2021-03-24.xml*