# Decision-Making Framework for an Effective Sanitizer to Reduce COVID-19 under Fermatean Fuzzy Environment

Authors: Muhammad Akram; Gulfam Shahzadi; Abdullah Ali H. Ahmadini
Journal: Journal of Mathematics
 (2020)

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## Abstract

The purpose of this article is to develop some general aggregation operators (AOs) based on Einstein’s norm operations, to cumulate the Fermatean fuzzy data in decision-making environments. A Fermatean fuzzy set (FFS), possessing the more flexible structure than the intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS), is a competent tool to handle vague information in the decision-making process by the means of membership degree (MD) and nonmembership degree (NMD). Our target is to empower the AOs using the theoretical basis of Einstein norms for the FFS to establish some advantageous operators, namely, Fermatean fuzzy Einstein weighted averaging (FFEWA), Fermatean fuzzy Einstein ordered weighted averaging (FFEOWA), generalized Fermatean fuzzy Einstein weighted averaging (GFFEWA), and generalized Fermatean fuzzy Einstein ordered weighted averaging (GFFEOWA) operators. Some properties and important results of the proposed operators are highlighted. As an addition to the MADM strategies, an approach, based on the proposed operators, is presented to deal with Fermatean fuzzy data in MADM problems. Moreover, multiattribute decision-making (MADM) problem for the selection of an effective sanitizer to reduce coronavirus is presented to show the capability and proficiency of this new idea. The results are compared with the Fermatean fuzzy TOPSIS method to exhibit the potency of the proposed model.


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## Body

## 1. Introduction

In decision sciences, it is an important aspect to find the ranking order of the alternatives corresponding to different attributes according to the preferences of the decision-making experts. Therefore, selection of various attributes of the alternatives is a very complex task. These decisions cannot be interpreted by the exact data so the need of a powerful model was raised to handle the ambiguous data. For that issue, Zadeh [1] initiated the innovative idea of fuzzy set (FS) which served as the backbone of the FS theory. FS permits the experts to describe their satisfaction level (membership degree) regarding performance of a member within the unit interval. Although, the FSs provide the grounds to the uncertain assessments but they were not adequate enough to describe the NMD. To overcome the limitations of FS, Atanassov [2] introduced a more dominant model, namely, IFS which has both MD μ and NMD ν with condition μ+ν≤1. The theory of IFS was felt to be inept and insufficient to represent the inexact data as there are a lot of problems where the sum of MD and NMD is exceeded by 1. To reduce such type of complications, Yager [3] delivered the idea of PFS with condition μ2+ν2≤1. However, PFS has also some limitations if MD of an element is 0.8 and NMD is 0.7, then sum of square of these values is greater than 1. Then, Yager [4] developed the theory of q-rung orthopair fuzzy set (q-ROFS) with condition μq+νq≤1. Recently, Senapati and Yager [5] gave the concept of FFS as a generalization of IFS and PFS.The worthwhile theory of AOs is widely applied to decision-making scenarios for the sake of data aggregation and to identify the best alternative from the possible choices. Xu [6] gave the idea of intuitionistic fuzzy (IF) AOs. The concept of generalized AOs for IFS was developed by Zhao et al. [7]. Rahman et al. [8] introduced Pythagorean fuzzy (PF) AOs. Zhao and Wei [9] studied the Einstein hybrid AOs under IF environment. The idea of IF aggregation using Einstein operations was discussed by Wang and Liu [10]. The induced interval-valued IF Einstein AOs were developed by Cai and Han [11]. Garg [12] studied the generalized PF Einstein weighted arithmetic AOs. Garg [13] also proposed the generalized PF Einstein weighted geometric AOs. The Pythagorean Dombi fuzzy AOs with applications were discussed by Akram et al. [14]. Shahzadi et al. [15] proposed the decision-making approach using PF Yager AOs. Liu and Wang [16] expressed q-rung orthopair fuzzy (q-ROF) weighted AOs. Wei et al. [17] studied weighted Heronian mean AOs under q-ROF information. q-ROF power Maclaurin AOs were developed by Liu et al. [18]. Jana et al. [19] studied Dombi AOs for q-ROFS. Liu and Liu [20] proposed q-ROF Bonferroni mean operators. Joshi and Gegov [21] studied the confidence levels q-ROF aggregation operators. Akram and Shahzadi [22] developed the hybrid decision-making model under q-ROF Yager AOs. Liu et al. [23] extended the concept of prioritized weighted AOs for complex q-ROFS. Senapati and Yager [24] studied subtraction, division, and Fermatean arithmetic mean operations over FFS. The idea of Fermatean fuzzy (FF) weighted averaging/geometric operators was also given by Senapati and Yager [25]. For more information and applications, the readers can refer to [26–59].The motivations of this article are described as follows:(1)
The judgement of a perfect alternative in an FF environment is a laborious MADM problem. The prevalent model, possessing the more space than the IF model and PF model, vigorously elaborates the imprecise decisions for the selection of best alternative.(2)
As Einstein AOs are the simplest and quite creative approach for dealing with DM affairs, basically, this article directs Einstein AOs in FF surroundings to face complex issues.(3)
The outcomes based on conclusion are quite accurate under Einstein AOs when it is put on to the reality-based MADM problems in FF data.(4)
The proposed operators are keen to provide the optimal solution not only for FF environment but also to work efficiently for IF and PF environment.The contributions of this article are described as follows:(1)
The feasibility of FFNs is merged with the aggregation skills of Einstein norms to establish more powerful, multiskilled, and practical AOs which can be deployed to aggregate FF data and to get more accuracy in decision-making scenarios(2)
The dominant properties as well as the notable results of the proposed operators are highlighted(3)
An algorithm is studied to handle complex realistic problems with FF data(4)
A MADM problem for the selection of an effective sanitizer to reduce coronavirus is discussed by using proposed operators(5)
A validity test is discussed for the approval and authenticity of proposed theory(6)
At the end, the benefits and characteristics of the proposed work are discussed by comparison analysisThe remaining paper is as follows. In Section2, we recall the concept of FFS and related score functions. Section 3 provides Einstein operational laws for FFNs. In Sections 4 and 5, we study the FFEWA and FFEOWA operators, respectively, and related properties to them. In Sections 6 and 7, we present the idea of GFFEWA and GFFEOWA operators, respectively. In Section 8, we propose an algorithm for our new model and discuss a MADM problem for the selection of a good sanitizer to reduce the coronavirus. Section 9 provides the validity criteria to prove the consistency of the proposed work. Section 10 gives the comparison analysis of proposed theory with the FF TOPSIS method. In Section 11, we have concluded the results related to the proposed model.The list of acronyms in research paper is given in Table1.Table 1
List of acronyms.
AcronymsDescriptionFSFuzzy setIFSIntuitionistic fuzzy setPFSPythagorean fuzzy setq-ROFSq-rung orthopair fuzzy setFFSFermatean fuzzy setFFNFermatean fuzzy numberAOsAggregation operatorsMADMMultiattribute decision-makingFF TOPSISFermatean fuzzy TOPSIS


## 2. Preliminaries

In this section, we recall some basic definitions including IFS, PFS, FFS, and score functions related to FFS.Definition 1.
(see [2]). An IFS I on nonempty set V is given by(1)I=x,μIx,νIx,where μI:V⟶0,1 and νI:V⟶0,1 specify MD and NMD of an element x∈V, respectively. ϖIx=1−μIx−νIx is indeterminacy degree (InD) of an element x∈V.Definition 2.
(see [3]). A PFS P on nonempty set V is given by(2)P=x,μPx,νPx,where μP:V⟶0,1 and νP:V⟶0,1 specify MD and NMD of an element, respectively. ϖPx=1−μPx2−νPx2 is InD.Definition 3.
(see [5]). An FFS ℛ on nonempty set V is given by(3)ℛ=x,μℛx,νℛx,where μℛ:V⟶0,1, νℛ:V⟶0,1, and ϖℛx=1−μℛx3−νℛx33 specify MD, NMD, and InD, respectively. FFNs are components of the FFS.Definition 4.
(see [5]). The score function and accuracy function for FFN ℛ=μℛ,νℛ are represented by(4)Sℛ=μℛ3−νℛ3,whereSℛ∈−1,1,Aℛ=μℛ3+νℛ3,whereAℛ∈0,1.Definition 5.
(see [5]). Consider two FFNs ℛ1=μℛ1,νℛ1 and ℛ2=μℛ2,νℛ2. Then,(1)
IfSℛ1<Sℛ2, then ℛ1≺ℛ2.(2)
IfSℛ1>Sℛ2, then ℛ1≻ℛ2.(3)
IfSℛ1=Sℛ2, then(a)
IfAℛ1<Aℛ2, then ℛ1≺ℛ2.(b)
IfAℛ1>Aℛ2, then ℛ1≻ℛ2.(c)
IfAℛ1=Aℛ2, then ℛ1∼ℛ2.


## 3. Einstein Operational Law of FFNs

In this section, we present concepts of the Einsteint-norm and t-conorm operations for FFNs and some of their properties. The Einstein operations on FFNs are defined as follows.Definition 6.
Letℛ=μ,ν, ℛ1=μ1,ν1, and ℛ2=μ2,ν2 be FFNs and λ>0; then,(i)
ℛ¯=νℛ,μℛ(ii)
ℛ1∧ℛε2=minμ1,μ2,maxν1,ν2(iii)
ℛ1∨ℛε2=maxμ1,μ2,minν1,ν2(iv)
ℛ1⊕εℛ2=μ13+μ23/1+μ13⋅με233,ν1⋅νε2/1+1−ν13⋅ε1−ν233(v)
ℛ1⊕εℛ2=μ1⋅με2/1+1−μ13⋅ε1−μ233,ν13+ν23/1+ν13⋅νε233(vi)
λ⋅ℛε=1+μ3λ−1−μ3λ/1+μ3λ+1−μ3λ3,23νλ/2−ν3λ+ν3λ3(vii)
ℛλ=23μλ/2−μ3λ+μ3λ3,1+ν3λ−1−ν3λ/1+ν3λ+1−ν3λ3Theorem 1.
Letℛ=μℛ,νℛ, ℛ1=μ1,ν1, and ℛ2=μ2,ν2 be three FFNs; then, ℛ3=ℛ1⊕εℛ2 and ℛ4=λ⋅ℛε are also FFNs.Proof.
Sinceλ>0 and ℛ is an FFN, therefore, 0≤μℛx≤1, 0≤νℛx≤1, and 0≤μℛx3+νℛx3≤1; then, 1−μℛx3≥νℛx3≥0, 1−νℛx3≥μℛx3≥0, and 1−μℛx3λ≥νℛx3; then,(5)1+μℛx3λ−1−μℛx3λ1+μℛx3λ+1−μℛx3λ3≤1+μℛx3λ−νℛx3λ1+μℛx3λ+νℛx3λ3,23νℛxλ2−νℛx3λ+νℛx3λ3≤23νℛxλ1+μℛx3λ+νℛx3λ3.
Thus,(6)1+μℛx3λ−1−μℛx3λ1+μℛx3λ+1−μℛx3λ33+23νℛxλ2−νℛx3λ+νℛx3λ33≤1.
Furthermore,(7)1+μℛx3λ−1−μℛx3λ1+μℛx3λ+1−μℛx3λ33+23νℛxλ2−νℛx3λ+νℛx3λ33=0,iff μℛx=νℛx=0 and(8)1+μℛx3λ−1−μℛx3λ1+μℛx3λ+1−μℛx3λ33+23νℛxλ2−νℛx3λ+νℛx3λ33=1,iff μℛx3+νℛx3=1.
Thus,ℛ4=λ⋅ℛε is an FFN for λ>0.Theorem 2.
Letλ,λ1,λ2≥0; then,(i)
ℛ1⊕εℛ2=ℛ2⊕εℛ1(ii)
ℛ1⊗εℛ2=ℛ1⊗εℛ2(iii)
λ⋅ℛ1⊕εℛ2ε=λ.εℛ1⊕ελ⋅ℛε2(iv)
ℛ1⊗εℛ2λ=ℛ1λ⊗εℛ2λ(v)
λ1⋅ℛε⊕ελ2⋅ℛε=λ1+λ2⋅ℛε(vi)
ℛλ1⊗εℛλ2=ℛλ1+λ2Proof.
(i)(9)ℛ1⊕εℛ2=μ13+μ231+μ13⋅με233,ν1⋅νε21+1−ν13⋅ε1−ν233=μ23+μ131+μ23⋅με133,ν2⋅νε11+1−ν23⋅ε1−ν133=ℛ2⊕εℛ1.
(ii)(10)ℛ1⊕εℛ2=μ13+μ231+μ13⋅με233,ν1⋅νε21+1−ν13⋅ε1−ν233is equivalent to(11)ℛ1⊕εℛ2=1+μ13⋅1+μ23ε−1−μ13⋅1−μ23ε1+μ13⋅1+μ23ε+1−μ13⋅1−μ23ε3,23ν1⋅νε22−ν13⋅ε2−ν23+ν13⋅νε233.
Takea=1+μ13⋅1+μ23ε, b=1−μ13⋅1−μ23ε, c=ν13⋅νε23, and d=2−ν13⋅ε2−ν23; then,(12)ℛ1⊕εℛ2=a−ba+b3,2c3d+c3.
By the Einstein FF law,(13)λ⋅ℛ1⊕εℛ2ε=λ⋅a−ba+b3,2c3d+c3ε=1+a−b/a+bλ−1−a−b/a+bλ1+a−b/a+bλ+1−a−b/a+bλ3,23⋅2c3/d+c3λ2−2c/d+cλ+2c/d+cλ3=aλ−bλaλ+bλ3,2cλ3dλ+cλ3=1+μ13λ⋅1+μ23λε−1−μ13λ⋅1−μ23λε1+μ13λ⋅1+μ23λε+1−μ13λ⋅1−μ23λε3,23ν1λ⋅ν2λε2−ν13λ⋅ε2−ν23λ+ν13λ⋅ν23λε3.
On the other hand,(14)λ⋅ℛε1=1+μ13λ−1−μ13λ1+μ13λ+1−μ13λ3,23ν1λ2−ν13λ+ν13λ3=a1−b1a1+b13,2c13d1+c13,λ⋅ℛε2=1+μ23λ−1−μ23λ1+μ23λ+1−μ23λ3,23ν2λ2−ν23λ+ν23λ3=a2−b2a2+b23,2c23d2+c23,where a1=1+μ13λ, b1=1−μ13λ, c1=ν13λ, d1=2−ν13λ, a2=1+μ23λ, b2=1−μ23λ, c2=ν23λ, and d2=2−ν23λ; therefore,(15)λ⋅ℛε1⊕ελ⋅ℛε2=a1−b1a1+b13,2c13d1+c13⊕εa2−b2a2+b23,2c23d2+c23=a1−b1/a1+b1+a2−b2/a2+b21+a1−b1/a1+b1⋅a2−b2/a2+b2ε3,22/3c1⋅cε2/d1+c1⋅d2+c2ε31+1−2c1/d1+c1.1−2c2/d2+c2ε3=a1⋅aε2−b1⋅bε2a1⋅aε2+b1⋅bε23,2c1⋅cε23d1⋅dε2+c1⋅cε23=1+μ13λ⋅1+μ23λε−1−μ13λ⋅1−μ23λε1+μ13λ⋅1+μ23λε+1−μ13λ⋅1−μ23λε3,23ν1λ⋅ν2λε2−ν13λ⋅ε2−ν23λ+ν13λ⋅ν23λε3.
Hence,λ⋅ℛ1⊕εℛ2ε=λ⋅ℛε1⊕ελ⋅ℛε2.
(v) Forλ1,λ2>0,(16)λ1⋅ℛε=1+μ3λ1−1−μ3λ11+μ3λ1+1−μ3λ13,23νλ12−ν3λ1+ν3λ13=a1−b1a1+b13,2c13d1+c13,λ2⋅ℛε=1+μ3λ2−1−μ3λ21+μ3λ2+1−μ3λ23,23νλ22−ν3λ2+ν3λ23=a2−b2a2+b23,2c23d2+c23,where aj=1+μ3λj, bj=1−μ3λj, cj=ν3λj, and dj=2−ν3λj, for j=1,2.(17)λ1⋅ℛε⊕ελ2⋅ℛε=a1−b1a1+b13,2c13d1+c13⊕εa2−b2a2+b23,2c23d2+c23=a1−b1/a1+b1+a2−b2/a2+b21+a1−b1/a1+b1⋅a2−b2/a2+b2ε3,22/3c1⋅cε2/d1+c1⋅d2+c2ε31+1−2c1/d1+c1⋅1−2c2/d2+c2ε3=a1⋅aε2−b1⋅bε2a1⋅aε2+b1⋅bε23,2c1⋅cε23d1⋅dε2+c1⋅cε23=1+μ3λ1+λ2−1−μ3λ1+λ21+μ3λ1+λ2+1−μ3λ1+λ23,23νλ1+λ22−ν3λ1+λ2+ν13λ1+λ23=λ1+λ2⋅ℛε.
Hence,λ1⋅ℛε⊕ελ2⋅ℛε=λ1+λ2⋅ℛε.
Similarly, others can be verified.Theorem 3.
Letℛ1=μ1,ν1 and ℛ2=μ2,ν2 be FFNs; then,(i)
ℛ1c∧ℛε2c=ℛ1∨ℛε2c(ii)
ℛ1c∨ℛε2c=ℛ1∧ℛε2c(iii)
ℛ1c⊕εℛ2c=ℛ1⊗εℛ2c(iv)
ℛ1c⊗εℛ2c=ℛ1⊕εℛ2c(v)
ℛ1∨ℛε2⊕εℛ1∧ℛε2=ℛ1⊕εℛ2(vi)
ℛ1∨ℛε2⊕εℛ1∧ℛε2=ℛ1⊕εℛ2Proof.
It is obvious.Theorem 4.
Letℛ1=μ1,ν1, ℛ2=μ2,ν2, and ℛ3=μ3,ν3 be three FFNs; then,(i)
ℛ1∨ℛε2∧ℛε3=ℛ1∧ℛε3∨ℛ2∧ℛε3ε(ii)
ℛ1∧ℛε2∨ℛε3=ℛ1∨ℛε3∧ℛ2∨ℛε3ε(iii)
ℛ1∨ℛε2⊕ℛε3=ℛ1⊕ℛε3∨ℛ2⊕εℛ3ε(iv)
ℛ1∧ℛε2⊕ℛε3=ℛ1⊕ℛε3∧ℛ2⊕εℛ3ε(v)
ℛ1∨ℛε2⊗ℛε3=ℛ1⊗ℛε3∨ℛ2⊗εℛ3ε(vi)
ℛ1∧ℛε2⊗ℛε3=ℛ1⊗ℛε3∧ℛ2⊗εℛ3εProof.
The proof is trivial, so we omit it.


## 4. Fermatean Fuzzy Einstein Weighted Averaging Operators

The Einstein weighted averaging operators under FF environment are defined here.Definition 7.
Letℛj=μj,νjj=1,2,…,s be a collection of FFNs and wj be the weight vector (WV) of ℛj with wj>0 and ∑j=1swj=1; then, FFEWA operator is a mapping Qs⟶Q such that(18)FFEWAℛ1,ℛ2,…,ℛs=w1⋅ℛε1⊕εw2⋅ℛε2⊕ε⋯⊕εws⋅ℛεs.
Ifwj=1/s, ∀j, then FFEWA operator becomes FFWA operator:(19)FFAℛ1,ℛ2,…,ℛs=1sℛ1⊕εℛ2⊕ε⋯⊕εℛs.Theorem 5.
Letℛj=μj,νj be FFNs; then, the aggregated value by using equation (18) is(20)FFEWAℛ1,ℛ2,…,ℛs=∏j=1s1+μj3wj−∏j=1s1+μj3wj∏j=1s1+μj3wj+∏j=1s1+μj3wj3,23∏j=1sνjwj∏j=1s2−νj3wj+∏j=1sνj3wj3.Proof.
Use the mathematical induction to prove equation (20).
Whens=2,(21)FFEWAℛ1,ℛ2=w1⋅ℛε1⊕εw2⋅ℛε2.
By Theorem1, both w1⋅ℛε1 and w2⋅ℛε2 are FFNs and value of w1⋅ℛε1⊕εw2⋅ℛε2 is an FFN. By using (vi) in Definition 6,(22)w1⋅ℛε1=1+μ13w1−1−μ13w11+μ13w1+1−μ13w13,2ν1w132−ν13w1+ν13w13,w2⋅ℛε2=1+μ23w2−1−μ23w21+μ23w2+1−μ23w23,23ν2w22−ν23w2+ν23w23.
Then,(23)FFEWAℛ1,ℛ2=w1⋅ℛε1⊕εw2⋅ℛε2=1+μ13w1−1−μ13w1/1+μ13w1+1−μ13w1+1+μ23w2−1−μ23w2/1+μ23w2+1−μ23w21+1+μ13w1−1−μ13w1/1+μ13w1+1−μ13w1⋅1+μ23w2−1−μ23w2/1+μ23w2+1−μ23w2ε3,23ν1w1/2−ν13w1+ν13w13⋅23ν1w2/2−ν23w2+ν23w23ε1+1−2ν13w1/2−ν13w1+ν13w1⋅1−2ν23w2/2−ν23w2+ν23w2ε3=1+μ13w1⋅ε1+μ23w2−1−μ13w1⋅ε1−μ23w21+μ13w1⋅ε1+μ23w2+1−μ13w1⋅ε1−μ23w23,23ν1w1⋅νε2w22−ν13w1⋅ε2−ν23w2+ν13w1⋅ν23εw23.
Thus, equation (20) is true when s=2.
Suppose result is true fors=k:(24)FFEWAℛ1,ℛ2,…,ℛk=∏j=1k1+μj3wj−∏j=1k1−μj3wj∏j=1k1+μj3wj+∏j=1k1−μj3wj3,23∏j=1kνjwj∏j=1k2−νj3wj+∏j=1kνj3wj3.
Now, fors=k+1,(25)FFEWAℛ1,ℛ2,…,ℛk+1=∏j=1k1+μj3wj−∏j=1k1−μj3wj∏j=1k1+μj3wj+∏j=1k1−μj3wj3,23∏j=1kνjwj∏j=1k2−νj3wj+∏j=1kνj3wj3⊕ε1+μk+13wk+1−1−μk+13wk+11+μk+13wk+1+1−μk+13wk+13,23νk+1wk+12−νk+13wk+1+νk+13wk+13=∏j=1k+11+μj3wj−∏j=1k+11−μj3wj∏j=1k+11+μj3wj+∏j=1k+11−μj3wj3,23∏j=1k+1νjwj∏j=1k+12−νj3wj+∏j=1k+1νj3wj3.
Thus, the result is true fors=k+1. Hence, equation (20) holds, ∀s.Lemma 1.
Letℛj=μj,νj,wj>0, and ∑j=1swj=1; then,(26)∏j=1sRjwj≤∑j=1swjRj,where equality holds iff R1=R2=⋯=Rs.Theorem 6.
Ifℛj=μj,νj are FFNs, then FFEWA ℛ1,ℛ2,…,ℛs is also an FFN.Proof.
Sinceℛj=μj,νj are FFNs, so 0≤μj,νj≤1 and 0≤μj3+νj3≤1. Therefore,(27)∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj=1−2∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj≤1−∏j=1s1−μj3wj≤1.
Also,1+μj3≥1−μj3⇒∏j=1s1+μj3−∏j=1s1−μj3≥0. Therefore,(28)∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj≥0.
Thus,0≤μFFEWA≤1.
Moreover,(29)2∏j=1sνj3wj∏j=1s2−νj3wj+∏j=1sνj3wj≤2∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj≤∏j=1s1−μj3wj≤1.
Also,(30)∏j=1sνj3wj≥0⟺2∏j=1sνj3wj∏j=1s2−νj3wj+∏j=1sνj3wj≥0.
Thus,0≤νFFEWA≤1. Moreover,(31)μFFEWA3+νFFEWA3=∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj+2∏j=1sνj3wj∏j=1s2−νj3wj+∏j=1sνj3wj≤1−2∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj+2∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj=1.
Hence,FFEWA∈0,1. Therefore, FFEWA ℛ1,ℛ2,…,ℛs∈FFN.Corollary 1.
The FFEWA and FFWA operators have the relationship:(32)FFEWAℛ1,ℛ2,…,ℛs≤FFWAℛ1,ℛ2,…,ℛs.Proof.
Let FFEWAℛ1,ℛ2,…,ℛs=μℛβ,νℛβ=ℛβ and FFWA ℛ1,ℛ2,…,ℛs=μℛ,νℛ=ℛ. Since ∏j=1s1+μj3wj+∏j=1s1−μj3wj≤∑j=1s1+μj3wj+∑j=1s1−μj3wj=2, then from equation (27), we obtain(33)∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj3≤1−∏j=1s1−μj3wj3⟺μℛβ≤μℛ,equality holds iff μ1=μ2=⋯=μs.
Also,(34)2∏j=1sνj3wj∏j=1s2−νj3wj+∏j=1sνj3wj≥2∏j=1sνj3wj∑j=1swj2−νj3+∑j=1swjνj3≥∏j=1sνj3wj⇒2∏j=1sνj3wj∏j=1s2−νj3wj+∏j=1sνj3wj3≥∏j=1sνjwj⇒νℛβ≤νℛ,equality holds iff ν1=ν2=⋯=νs.
Thus,(35)Sℛβ=μℛβ3−νℛβ3≤μℛ3−νℛ3=Sℛ.
IfSℛβ<Sℛ, then(36)FFEWAℛ1,ℛ2,…,ℛs<FFWAℛ1,ℛ2,…,ℛs.
IfSℛβ=Sℛ, that is, μℛβ3−νℛβ3=μℛ3−νℛ3, then by condition μℛβ≤μℛ and νℛβ≥νℛ; thus, the accuracy function Aℛβ=μℛβ3−νℛβ3=μℛ3−νℛ3=Aℛ. Thus,(37)FFEWAℛ1,ℛ2,…,ℛs=FFWAℛ1,ℛ2,…,ℛs.
Hence,(38)FFEWAℛ1,ℛ2,…,ℛs≤FFWAℛ1,ℛ2,…,ℛs,equality holds iff ℛ1=ℛ2=⋯=ℛs.Example 1.
Letℛ1=0.8,0.5, ℛ2=0.9,0.4, ℛ3=0.6,0.7, and ℛ4=0.8,0.7 be four FFNs and w=0.4,0.2,0.2,0.2T; then,(39)FFEWAℛ1,ℛ2,ℛ3,ℛ4=∏j=141+μj3wj−∏j=141−μj3wj∏j=141+μj3wj+∏j=141−μj3wj3,23∏j=14νjwj∏j=142−νj3wj+∏j=14νj3wj3=1.49−0.481.49+0.483,23×0.551.80+0.163=0.80,0.55.
Now,(40)FFWAℛ1,ℛ2,ℛ3,ℛ4=1−∏j=141−μj3wj3,∏j=14νjwj=0.80,0.55,⇒FFEWAℛ1,ℛ2,ℛ3,ℛ4≤FFWAℛ1,ℛ2,ℛ3,ℛ4.Proposition 1.
Letℛj=μj,νj be FFNs and wj be the WV of ℛj, such that wj∈0,1 and ∑j=1swj=1.(i)
Idempotency: ifℛj=ℛo=μo,νo for all j, then(41)FFEWAℛ1,ℛ2,…,ℛs=ℛo.(ii)
Boundedness: letℛ−=minjμj,maxjνj and ℛ+=maxjμj,minjνj; then,(42)ℛ−≤FFEWAℛ1,ℛ2,…,ℛs≤ℛ+.(iii)
Monotonicity: whenℛj≤Pj,∀j, then(43)FFEWAℛ1,ℛ2,…,ℛs≤FFEWAP1,P2,…,Ps.Proof.
(i) Asℛj=μo,νo are FFNs, ∀j, then(44)FFEWAℛ1,ℛ2,…,ℛs=∏j=1s1+μo3wj−∏j=1s1−μo3wj∏j=1s1+μo3wj+∏j=1s1−μo3wj3,23∏j=1sνowj∏j=1s2−νo3wj+∏j=1sνo3wj3=1+μo3∑j=1swj−1−μo3∑j=1swj1+μo3∑j=1swj+1−μo3∑j=1swj3,23νo∑j=1swj2−νo3∑j=1swj+νo3∑j=1swj3=μo,νo.
(ii) Considerfx=1−x/1+x,x∈0,1, then f′x=−2/1+x2<0, so fx is a decreasing function (DF). As μj,min3≤μj3≤μj,max3,∀j=1,2,…,s, then fμj,max3≤fμj3≤fμj,min3,∀j, that is, 1−μj,max3/1+μj,max3≤1−μj3/1+μj3≤1−μj,min3/1+μj,min3, for all j. Let wj∈0,1 and ∑j=1swj=1, and we have(45)1−μj,max31+μj,max3wj≤1−μj31+μj3wj≤1−μj,min31+μj,min3wj,∏j=1s1−μj,max31+μj,max3wj≤∏j=1s1−μj31+μj3wj≤∏j=1s1−μj,min31+μj,min3wj⟺1−μj,max31+μj,max3∑j=1swj≤∏j=1s1−μj31+μj3wj≤1−μj,min31+μj,min3∑j=1swj⟺1−μj,max31+μj,max3≤∏j=1s1−μj31+μj3wj≤1−μj,min31+μj,min3⟺21+μj,max3≤1+∏j=1s1−μj31+μj3wj≤21+μj,min3⟺1+μj,min32≤11+∏j=1s1−μj3/1+μj3wj≤1+μj,max32⟺1+μj,min3≤21+∏j=1s1−μj3/1+μj3wj≤1+μj,max3⟺μj,min3≤21+∏j=1s1−μj3/1+μj3wj−1≤μj,max3⟺μj,min3≤∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj≤μj,max3.
Thus,(46)μj,min≤∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj3≤μj,max.
Considergy=2−y/y,y∈0,1, then g′y=−2/y2, i.e., gy is a DF on 0,1. Since νj,min3≤νj3≤νj,max3,∀j, then gνj,max3≤gνj3≤gνj,min3,∀j, that is, 2−νj,max3/νj,max3≤2−νj3/νj3≤2−νj,min3/νj,min3. Then,(47)2−νj,max3νj,max3wj≤2−νj3νj3wj≤2−νj,min3νj,min3wj,∏j=1s2−νj,max3νj,max3wj≤∏j=1s2−νj3νj3wj≤∏j=1s2−νj,min3νj,min3wj⇒2−νj,max3νj,max3∑j=1swj≤∏j=1s2−νj3νj3wj≤2−νj,min3νj,min3∑j=1swj⇒2−νj,max3νj,max3≤∏j=1s2−νj3νj3wj≤2−νj,min3νj,min3⇒2νj,max3≤1+∏j=1s2−νj3νj3wj≤2νj,min3⇒νj,min32≤11+∏j=1s2−νj3/νj3wj≤νj,max32⇒νj,min3≤21+∏j=1s2−νj3/νj3wj≤νj,max3⇒νj,min3≤21+∏j=1s2−νj3/νj3wj≤νj,max3⇒νj,min3≤2∏j=1sνj3wj∏j=1s2−νj3wj+∏j=1sνj3wj≤νj,max3⇒νj,min≤23∏j=1sνjwj∏j=1s2−νj3wj+∏j=1sνj3wj3≤νj,max.
Let FFEWAℛ1,ℛ2,…,ℛs=ℛ=μℛ,νℛ; then, from equations (46) and (47),(48)μmin≤μℛ≤μmax,νmin≤νℛ≤νmax,where μmin=minjμj, μmax=maxjμj, νmin=minjνj, and νmax=maxjνj. So, Sℛ=μℛ3−νℛ3≤μmax3−νmin3=Sℛ+ and Sℛ=μℛ3−νℛ3≥μmin3−νmax3=Sℛ−. As Sℛ<Sℛ+ and Sℛ>Sℛ−, so(49)ℛ−≤FFEWAℛ1,ℛ2,…,ℛs≤ℛ+.
(iii) It is similar to (ii), so we omit it.


## 5. Fermatean Fuzzy Einstein Ordered Weighted Averaging Operators

Definition 8.
Letℛj=μj,νj be a family of FFNs and wj be the WV of ℛj with wj>0 and ∑j=1swj=1; then, FFEOWA operator is a mapping Qs⟶Q such that(50)FFEOWAℛ1,ℛ2,…,ℛs=w1⋅ℛεϱ1⊕εw2⋅ℛεϱ2⊕ε⋯⊕εws⋅ℛεϱs,where ϱ1,ϱ2,…,ϱs is the permutation of j=1,2,…,s such that ℛϱj−1≥ℛϱj,∀j=1,2,…,s.Theorem 7.
Letℛj=μj,νj be FFNs; then, the aggregated value by using FFEOWA is an FFN and(51)FFEOWAℛ1,ℛ2,…,ℛs=∏j=1s1+μϱj3wj−∏j=1s1−μϱj3wj∏j=1s1+μϱj3wj+∏j=1s1−μϱj3wj3,23∏j=1sνϱjwj∏j=1s2−νϱj3wj+∏j=1sνϱj3wj3.Proof.
It is similar to Theorem4.
We give some properties without their proofs.Corollary 2.
The FFEOWA and FFOWA operators have the relation:(52)FFEOWAℛ1,ℛ2,…,ℛs≤FFOWAℛ1,ℛ2,…,ℛs.Example 2.
Letℛ1=0.6,0.7, ℛ2=0.8,0.7, ℛ3=0.6,0.9, and ℛ4=0.9,0.4 be four FFNs and w=0.3,0.3,0.2,0.2T as(53)Sℛ1=0.63−0.73=−0.13,Sℛ2=0.83−0.73=0.17,Sℛ3=0.63−0.93=−0.51,Sℛ4=0.93−0.43=0.67.
SinceSℛ4>Sℛ2>Sℛ1>Sℛ3, therefore(54)ℛϱ1=ℛ4=0.9,0.4,ℛϱ2=ℛ2=0.8,0.7,ℛϱ3=ℛ1=0.6,0.7,ℛϱ4=ℛ3=0.6,0.9.
Thus, by applying the FFEOWA operator, we obtain(55)FFEOWAℛ1,ℛ2,ℛ3,ℛ4=∏j=141+μϱj3wj−∏j=141−μϱj3wj∏j=141+μϱj3wj+∏j=141−μϱj3wj3,23∏j=14νϱjwj∏j=142−νϱj3wj+∏j=14νϱj3wj3=1.44−0.491.44+0.493,23×0.661.62+0.663=0.79,0.55.
Now,(56)FFOWAℛ1,ℛ2,ℛ3,ℛ4=1−∏j=141−μϱj3wj3,∏j=14νϱjwj=0.80,0.66,⇒FFEOWAℛ1,ℛ2,ℛ3,ℛ4<FFOWAℛ1,ℛ2,ℛ3,ℛ4.Proposition 1.
Letℛj=μj,νj be FFNs and wj be the WV of ℛj, such that wj∈0,1 and ∑j=1swj=1.(i)
Idempotency: ifℛj=ℛo=μo,νo, ∀j, then(57)FFEOWAℛ1,ℛ2,…,ℛs=ℛo.(ii)
Boundedness: letℛ−=minjμj,maxjνj and ℛ+=maxjμj,minjνj; then,(58)ℛ−≤FFEOWAℛ1,ℛ2,…,ℛs≤ℛ+.(iii)
Monotonicity: whenℛj≤Pj,∀j, then(59)FFEOWAℛ1,ℛ2,…,ℛs≤FFEOWAP1,P2,…,Ps.


## 6. Generalized Fermatean Fuzzy Einstein Weighted Averaging Operators

Definition 9.
Letℛj=μj,νj be a collection of FFNs and wj be the WV of ℛj with wj>0 and ∑j=1swj=1; then, GFFEWA operator is a mapping Qs⟶Q such that(60)GFFEWAℛ1,ℛ2,…,ℛs=⊕j=1swj⋅ℛεjλ1/λ,where λ>0.
Particularly,(i)
Ifλ=1, then GFFEWA becomes FFEWA(ii)
Ifw=1/s,1/s,…,1/sT, then GFFEWA ℛ1,ℛ2,…,ℛs=1/s⋅⊕εj=1sℛjλ1/λTheorem 8.
Letℛj=μj,νj be FFNs and wj be the WV of ℛj with wj>0 and ∑j=1swj=1; then, the aggregated value by applying the GFFEWA operator is an FFN and(61)GFFEWAℛ1,ℛ2,…,ℛs=23∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ+μj3λwj1/3λ∏j=1s2−μj3λ+3μj3λwj+3∏j=1s2−μj3λ−μj3λwj1/λ+∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj1/λ3,∏j=1s1+νj3λ+31−νj3λwj+3∏j=1s1+νj3λ−1−νj3λwj1/λ−∏j=1s1+νj3λ+31−νj3λwj−∏j=1s1+νj3λ−1−νj3λwj1/λ∏j=1s1+νj3λ+31−νj3λwj+3∏j=1s1+νj3λ−1−νj3λwj1/λ+∏j=1s1+νj3λ+31−νj3λwj−∏j=1s1+νj3λ−1−νj3λwj1/λ3.Proof.
Since(62)ℛjλ=23μjλ2−μj3λ+μj3λ3,1+νj3λ−1−νj3λ1+νj3λ+1−νj3λ3,⇒⊕j=1swj⋅ℛεjλ=∏j=1s1+2μj3λ/2−μj3λ+μj3λwj−∏j=1s1−2μj3λ/2−μj3λ+μj3λwj∏j=1s1+2μj3λ/2−μj3λ+μj3λwj+∏j=1s1−2μj3λ/2−μj3λ+μj3λwj3,23∏j=1s1+νj3λ−1−νj3λ/1+νj3λ+1−νj3λ3wj∏j=1s2−1+νj3λ−1−νj3λ/1+νj3λ+1−νj3λwj+∏j=1s1+νj3λ−1−νj3λ/1+νj3λ+1−νj3λwj3=∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj∏j=1s2−μj3λ+3μj3λwj+∏j=1s2−μj3λ−μj3λwj3,23∏j=1s1+νj3λ−1−νj3λ3wj∏j=1s1+νj3λ+31−νj3λwj+∏j=1s1+νj3λ−1−νj3λwj3.
Therefore,(63)⊕j=1swj⋅ℛεjλ1/λ=2∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj/∏j=1s2−μj3λ+3μj3λwj+∏j=1s2−μj3λ−μj3λwj1/λ32−∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj/∏j=1s2−μj3λ+3μj3λwj+∏j=1s2−μj3λ−μj3λwj1/λ+∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj/∏j=1s2−μj3λ+3μj3λwj+∏j=1s2−μj3λ−μj3λwj1/λ3,1+2∏js1+νj3λ−1−νj3λwj/∏js1+νj3λ+31−νj3λwj+∏js1+νj3λ−1−νj3λwj1/λ−1−2∏j=1s1+νj3λ−1−νj3λwj/∏j=1s1+νj3λ+31−νj3λwj+∏j=1s1+νj3λ−1−νj3λwj1/λ1+2∏j=1s1+νj3λ−1−νj3λwj/∏j=1s1+νj3λ+31−νj3λwj+∏j=1s1+νj3λ−1−νj3λwj1/λ+1−2∏j=1s1+νj3λ−1−νj3λwj/∏j=1s1+νj3λ+31−νj3λwj+∏j=1s1+νj3λ−1−νj3λwj1/λ3=23∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj1/3λ∏j=1s2−μj3λ+3μj3λwj+3∏j=1s2−μj3λ−μj3λwj1/λ+∏j=1s2−μj3λ+3μj3λwj−∏j=1s2−μj3λ−μj3λwj1/λ3,∏j=1s1+νj3λ+31−νj3λwj+3∏j=1s1+νj3λ−1−νj3λwj1/λ−∏j=1s1+νj3λ+31−νj3λwj−∏j=1s1+νj3λ−1−νj3λwj1/λ∏j=1s1+νj3λ+31−νj3λwj+3∏j=1s1+νj3λ−1−νj3λwj1/λ+∏j=1s1+νj3λ+31−νj3λwj−∏j=1s1+νj3λ−1−νj3λwj1/λ3.
Whenλ=1, then(64)GFFEWAℛ1,ℛ2,…,ℛs=∏j=1s1+μj3wj−∏j=1s1−μj3wj∏j=1s1−μj3wj∏j=1s1+μj3wj+∏j=1s1−μj3wj3,23∏j=1sνjwj/∏j=1s2−νj3wj+∏j=1sνj3wj3.


## 7. Generalized Fermatean Fuzzy Einstein Ordered Weighted Averaging Operators

Definition 10.
Letℛj=μj,νj be a collection of FFNs and wj be the WV of ℛj with wj>0 and ∑j=1swj=1; then, the GFFEOWA operator is a mapping Qs⟶Q such that(65)GFFEOWAℛ1,ℛ2,…,ℛs=⊕j=1swj⋅ℛεϱjλ1/λ,where λ>0.Theorem 9.
Letℛj=μj,νj be FFNs and wj be the WV of ℛj with wj>0 and ∑j=1swj=1; then, the aggregated value by applying the GFFEOWA operator is an FFN and(66)GFFEOWAℛ1,ℛ2,…,ℛs=23∏j=1s2−μϱj3λ+3μϱj3λwj−∏j=1s2−μϱj3λ−μϱj3λwj1/3λ∏j=1s2−μϱj3λ+3μϱj3λwj+3∏j=1s2−μϱj3λ−μϱj3λwj1/λ+∏j=1s2−μϱj3λ+3μϱj3λwj−∏j=1s2−μϱj3λ−μϱj3λwj1/λ3,∏j=1s1+νϱj3λ+31−νϱj3λwj+3∏j=1s1+νϱj3λ−1−νϱj3λwj1/λ−∏j=1s1+νϱj3λ+31−νϱj3λwj−∏j=1s1+νϱj3λ−1−νϱj3λwj1/λ∏j=1s1+νϱj3λ+31−νϱj3λwj+3∏j=1s1+νϱj3λ−1−νϱj3λwj1/λ+∏j=1s1+νϱj3λ+31−νϱj3λwj−∏j=1s1+νϱj3λ−1−νϱj3λwj1/λ3.Proof.
It is similar to Theorem6, and we can prove it.Whenλ=1, then(67)GFFEOWAℛ1,ℛ2,…,ℛs=∏j=1s1+μϱj3wj−∏j=1s1−μϱj3wj∏j=1s1+μϱj3wj+∏j=1s1−μϱj3wj3,23∏j=1sνϱjwj∏j=1s2−νϱj3wj+∏j=1sνϱj3wj3.


## 8. MADM Problem Using FF Information

To handle an MADM problem under FF environment, letK=K1,K2,…,Km be a set of possible alternatives and J=J1,J2,…,Js be a set of possible attributes chosen by the decision maker. Let w=w1,w2,…,wsT be the WV with wj>0 and ∑j=1swj=1. Suppose that ℰ˜=μlj,νljm×s is the FF decision matrix (FFDM), where μlj and νlj are the MD and NMD of the alternative Kl for the attribute Jj, respectively, where 0≤μlj3+νlj3≤1.The following Algorithm1 is used to solve the MADM problem with FFN based on using the GFFEWA operator.Algorithm 1:
(1)
Input: selection of suitable alternatives and attributes.(2)
Use the FFDM and GFFEWA operator:(68)ℬl=GFFEWAKl1,Kl2,…,Kls=23∏j=1s2−μlj3λ+3μlj3λwj−∏j=1s2−μlj3λ−μlj3λwj1/3λ∏j=1s2−μlj3λ+3μlj3λwj+3∏j=1s2−μlj3λ−μlj3λwj1/λ+∏j=1s2−μlj3λ+3μlj3λwj−∏j=1s2−μlj3λ−μlj3λwj1/λ3,∏j=1s1+νlj3λ+31−νlj3λwj+3∏j=1s1+νlj3λ−1−νlj3λwj1/λ−∏j=1s1+νlj3λ+31−νlj3λwj−∏j=1s1+νlj3λ−1−νlj3λwj1/λ∏j=1s1+νlj3λ+31−νlj3λwj+3∏j=1s1+νlj3λ−1−νlj3λwj1/λ+∏j=1s1+νlj3λ+31−νlj3λwj−∏j=1s1+νlj3λ−1−νlj3λwj1/λ3,for overall preference valuesℬll=1,2,…,m of the alternatives Kl.(3)
Use the score functionSℬll=1,2,…,m for the ranking of alternatives. If score values are equal, then compute the accuracy functions Aℬl and rank according to these values.Output: the alternative containing maximum score value will be the decision.


### 8.1. Selection of an Effective Sanitizer to Reduce Coronavirus

Hand sanitizer is a liquid or gel mostly used to reduce infectious agents on the hands. Alcohol-based hand sanitizers are preferred for hand washing in most healthcare settings. The Centers for Disease Control and Prevention (CDC) advise the people to wash hands with soap and water to restrain the spread of infections and decrease the endanger of getting sick. In shortage of soap and water, CDC suggests people to use an alcohol-based (at least 60 percent) hand sanitizer. According to the World Health Organization (WHO), in this pandemic situation of coronavirus, good hygiene and physical distancing are the best ways to protect ourself and everyone around us from coronavirus. This virus spreads by a person who has the disease and also spread by touching a sick person. We cannot isolate ourselves entirely to prudent from coronavirus. So, good hand hygiene can be the final barrier between us and the disease. WHO recommends alcohol-based hand sanitizers to remove the novel coronavirus. Alcohol-based hand sanitizer works to prevent the proteins of microbes—including bacteria and some viruses—from functioning normally. Hand sanitizers must contain ethanol, isopropanol, n-propanol, or a combination of these alcohols. All are effective against viruses such as the novel coronavirus.Demand of a hand sanitizer is increased in such critical situation of COVID-19. Due to increasing demand, it is difficult to get good and effective hand sanitizers in local markets. Increasing demand has also led to low quality hand sanitizers entering the market. The main motive of this application is to select an effective sanitizer to mitigate transmission of coronavirus by applying the GFFYWA operator. LetK=K1,K2,K3,K4 be a set of sanitizers. Let J=J1,J2,J3 be a set of three attributes for the evaluation of an effective sanitizer, where(69)J1:represents quantity of ethanol,J2:represents quantity of glycerol,J3:represents quantity of hydrogenperoxide.(1)
The FFDM is shown in Table2.(2)
The weights assigned by the decision maker are(70)w1=0.60,w2=0.25,w3=0.15,∑j=13wj=1.We use the GFFEWA operator for the selection of an effective sanitizer.Table 2
FFDM.
ℰ˜J1J2J3K1(0.6, 0.7)(0.6, 0.2)(0.8, 0.4)K2(0.4, 0.2)(0.6, 0.4)(0.3, 0.6)K3(0.8, 0.3)(0.5, 0.3)(0.6, 0.4)K4(0.7, 0.5)(0.8, 0.1)(0.9, 0.6)Step 1.
For performance valuesℬl of sanitizers, use the GFFEWA operator for λ=1:(71)ℬ1=0.64,0.48,ℬ2=0.46,0.28,ℬ3=0.72,0.31,ℬ4=0.77,0.34.Step 2.
Calculate the scoresSℬl of FFNs ℬl and rank the sanitizers:(72)Sℬ1=0.15,Sℬ2=0.08,Sℬ3=0.34,Sℬ4=0.42.
The ranking of sanitizers is(73)K4>K3>K1>K2.Step 3.
Therefore,K4 is the best sanitizer.


## 8.1. Selection of an Effective Sanitizer to Reduce Coronavirus

Hand sanitizer is a liquid or gel mostly used to reduce infectious agents on the hands. Alcohol-based hand sanitizers are preferred for hand washing in most healthcare settings. The Centers for Disease Control and Prevention (CDC) advise the people to wash hands with soap and water to restrain the spread of infections and decrease the endanger of getting sick. In shortage of soap and water, CDC suggests people to use an alcohol-based (at least 60 percent) hand sanitizer. According to the World Health Organization (WHO), in this pandemic situation of coronavirus, good hygiene and physical distancing are the best ways to protect ourself and everyone around us from coronavirus. This virus spreads by a person who has the disease and also spread by touching a sick person. We cannot isolate ourselves entirely to prudent from coronavirus. So, good hand hygiene can be the final barrier between us and the disease. WHO recommends alcohol-based hand sanitizers to remove the novel coronavirus. Alcohol-based hand sanitizer works to prevent the proteins of microbes—including bacteria and some viruses—from functioning normally. Hand sanitizers must contain ethanol, isopropanol, n-propanol, or a combination of these alcohols. All are effective against viruses such as the novel coronavirus.Demand of a hand sanitizer is increased in such critical situation of COVID-19. Due to increasing demand, it is difficult to get good and effective hand sanitizers in local markets. Increasing demand has also led to low quality hand sanitizers entering the market. The main motive of this application is to select an effective sanitizer to mitigate transmission of coronavirus by applying the GFFYWA operator. LetK=K1,K2,K3,K4 be a set of sanitizers. Let J=J1,J2,J3 be a set of three attributes for the evaluation of an effective sanitizer, where(69)J1:represents quantity of ethanol,J2:represents quantity of glycerol,J3:represents quantity of hydrogenperoxide.(1)
The FFDM is shown in Table2.(2)
The weights assigned by the decision maker are(70)w1=0.60,w2=0.25,w3=0.15,∑j=13wj=1.We use the GFFEWA operator for the selection of an effective sanitizer.Table 2
FFDM.
ℰ˜J1J2J3K1(0.6, 0.7)(0.6, 0.2)(0.8, 0.4)K2(0.4, 0.2)(0.6, 0.4)(0.3, 0.6)K3(0.8, 0.3)(0.5, 0.3)(0.6, 0.4)K4(0.7, 0.5)(0.8, 0.1)(0.9, 0.6)Step 1.
For performance valuesℬl of sanitizers, use the GFFEWA operator for λ=1:(71)ℬ1=0.64,0.48,ℬ2=0.46,0.28,ℬ3=0.72,0.31,ℬ4=0.77,0.34.Step 2.
Calculate the scoresSℬl of FFNs ℬl and rank the sanitizers:(72)Sℬ1=0.15,Sℬ2=0.08,Sℬ3=0.34,Sℬ4=0.42.
The ranking of sanitizers is(73)K4>K3>K1>K2.Step 3.
Therefore,K4 is the best sanitizer.


## 9. Validity Test

For the validity and authenticity of MADM methods, Wang and Triantaphyllou [53] developed testing criteria, given as follows:(i)
Criterion 1: a MADM technique is valid if the most desirable alternative remains same on changes a nonoptimal alternative with some other poor or weak alternative, without changing the respective decision criteria(ii)
Criterion 2: the transitive property should be followed by a valid MADM technique(iii)
Criterion 3: the ranking result of alternatives should not change on splitting the problem into the smaller subproblems and by applying the same MADM technique on subproblemsTable 3
Reconstructed FFDM.
ℰ˜J1J2J3K1(0.6, 0.7)(0.6, 0.2)(0.8, 0.4)K˜2(0.3, 0.4)(0.5, 0.4)(0.2, 0.8)K3(0.8, 0.3)(0.5, 0.3)(0.6, 0.4)K4(0.7, 0.5)(0.8, 0.1)(0.9, 0.6)Now, we discuss the validity of our proposed MADM technique by testing the above criteria.(1)
Validity test by criterion 1: if we replace the decision values of a nonoptimal alternativeK2 by K˜2, then the new DM is given in Table 3.By applying the GFFEWA operator forλ=1 and score function, the score values of alternatives are(74)Sℬ1=0.15,Sℬ˜2=−0.03,Sℬ3=0.34,Sℬ4=0.42.The ranking of sanitizers isK4>K3>K1>K˜2, which is the same as the original ranking order, and the best sanitizer is K4. Thus, our presented MADM model fulfills the test criterion 1.(2)
Validity test by criteria 2 and 3: for the validity of proposed algorithm, using criteria 2 and 3, we split the problem into the smaller subproblemsK1,K2,K4,K1,K3,K4,andK2,K3,K4. By utilizing the proposed technique, the ranking orders of alternatives in these subproblems are K4>K1>K2,K4>K3>K1, and K4>K3>K2, respectively. The combined ranking of alternatives is K4>K3>K1>K2, which is the same as that of the original ranking. Hence, the proposed MADM technique is authentic and proficient under criteria 2 and 3.


## 10. Comparison Analysis

Here, we discuss the comparison of proposed theory with the FF TOPSIS method [5]. The steps to find out the best alternative by the FF TOPSIS method are(1)
Table2 represents the FF decision matrix in which each entry corresponds to an FFN.(2)
The FF positive ideal solution (FFPIS)S+ and FF negative ideal solution (FFNIS) S− are(75)S+=0.8,0.3,0.8,0.1,0.9,0.6,S−=0.6,0.7,0.5,0.3,0.3,0.6.(3)
The distance between the alternativeKl and FFPIS S+ together with the FFNIS S− are given in Table 4.(4)
The revised closeness degree of each alternative is given as(76)ξK1=−3.05,ξK2=−4.11,ξK3=−1.28,ξK4=0.10.(5)
We get the following ranking list by arranging the alternatives in the decreasing order with respect toξKl:(77)K4>K3>K1>K2.(6)
K4 is the best alternative.From the outcomes of proposed operators and FF TOPSIS method, as shown in Figure1, we conclude that ranking lists obtained from both compared methods are the same and the best alternative from both approaches is K4. The FF TOPSIS method is a good approach to solve DM problems but there are many hindrances which can be solved by using our proposed theory. The FF Einstein AOs are more flexible and easy approach. A best alternative can be obtained by a short process. The results from proposed theory are more accurate and closest to original results.Figure 1
Comparison of the GFFEWA operator and FF TOPSIS method.The steps to solve any MADM problem by FF Einstein AOs and FF TOPSIS method are shown in Figure2.Figure 2
Flow chart for MADM problem using FF Einstein AOs and FF TOPSIS method.Table 4
Distance of alternatives from FFPIS and FFNIS.
DKl,S+DKl,S−0.300.080.460.290.180.210.090.32


### 10.1. Advantages and Limitations of Proposed Model

The proposed model is superior than the IF and PF models because it contains the space of IF and PF models. The cubic sum of membership and nonmembership degrees is bounded by 1 in the proposed model. The MADM approaches discussed in [10, 12, 13, 15] failed to handle the proposed application because 0.9+0.6>1 and 0.92+0.62>1 but proposed approach covers all such situations. The results are more precise and accurate by using the proposed model. However, there are some limitations of this model. It cannot be applied in the situations where we take the parameters for the evaluation of anything. It means this theory lacks parametrization property.


## 10.1. Advantages and Limitations of Proposed Model

The proposed model is superior than the IF and PF models because it contains the space of IF and PF models. The cubic sum of membership and nonmembership degrees is bounded by 1 in the proposed model. The MADM approaches discussed in [10, 12, 13, 15] failed to handle the proposed application because 0.9+0.6>1 and 0.92+0.62>1 but proposed approach covers all such situations. The results are more precise and accurate by using the proposed model. However, there are some limitations of this model. It cannot be applied in the situations where we take the parameters for the evaluation of anything. It means this theory lacks parametrization property.


## 11. Conclusions

An FFS is an extension of IFS and PFS which has more flexible structure to solve decision-making problems owing to the conditionμ3+ν3≤1. Moreover, Einstein’s t-norm and t-conorm have more generalized structure that operates efficiently to integrate the intricate information. The limitations of existing operators and beneficial characteristics of Einstein AOs motivated us to endeavor for the development of a fruitful combination of Einstein AOs with FFNs.A major contribution of the study is the development new tremendous AOs, called, FFEWA, FFEOWA, GFFEWA, and GFFEOWA operators. Some captivating properties of these operators have been discussed. Another achievement of this study is the establishment of a MADM technique on the basis of the proposed operators to manifest the application of the proposed operators. A MADM problem for the selection of an effective sanitizer to reduce the coronavirus has been presented to demonstrate the potency of the proposed strategy.The validity test has been discussed to unfold the consistency of proposed work. A comparison analysis of our proposed theory with the FF TOPSIS method has been presented to exhibit the dominance of our proposed operators over the FF TOPSIS method. In short, this article builds up a tool that has the rich properties of Einstein AOs and flexibility of the FF model. In future, our aim is to develop some worthwhile AOs using the theoretical foundations of Einstein norms for the Fermatean fuzzy soft set.



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*Source: 3263407-2020-10-30.xml*