# Extending COPRAS Method with Linguistic Fermatean Fuzzy Sets and Hamy Mean Operators

Authors: Muhammad Akram; Naila Ramzan; Feng Feng
Journal: Journal of Mathematics
 (2022)

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## Abstract

The utilization of linguistic variables is of vital importance for qualitative information processing in mathematical modelling and decision making under uncertainty. The primary purpose of our study is to introduce a novel structure by the fusion of Fermatean fuzzy sets and linguistic term sets. This hybrid structure is termed as the linguistic Fermatean fuzzy set which can be employed for dealing with decision-making problems involving qualitative information. Several fundamental operations of linguistic Fermatean fuzzy sets are introduced. Furthermore, we present four classes of linguistic Fermatean fuzzy Hamy mean operators, namely, the linguistic Fermatean fuzzy Hamy mean operator, the linguistic Fermatean fuzzy dual Hamy mean operator, the linguistic Fermatean fuzzy weighted Hamy mean operator, and the linguistic Fermatean fuzzy weighted dual Hamy mean operator. Some basic properties of these linguistic Fermatean fuzzy Hamy mean operators are examined as well. In addition, we develop a linguistic Fermatean fuzzy extension of the COPRAS method. We also propose a novel approach to multi-attribute group decision making with linguistic Fermatean fuzzy information. With our proposed methods, we solve two practical problems regarding food company ranking and green supplier selection, respectively. Finally, the efficacy of our developed method is validated through a comparative analysis with existing methods.


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## Body

## 1. Introduction

Multi-attribute decision making (MADM) is a process of evaluating the superlative alternative supporting certain attributes. If a panel of decision experts is involved in this procedure, it is referred to as multi-attribute group decision making (MAGDM). The crisp set theory was only limited to exact information. Therefore, there is a need that a certain model should be assembled to tackle vague information. Zadeh [1] put forward the foundation of fuzzy sets (FSs). In FSs, the satisfaction level of an expert is asserted as degree of membership function and lies in unit interval. Fuzzy set theory (FST) is not confined to only decision making, but it is applicable in numerous fields, including medicine, business and administration, control system engineering, and facial pattern recognition.Although FSs are capable of handling the vague information, they cannot deal with dissatisfaction level of human beings. Atanassov [2] put forward the idea of intuitionistic fuzzy sets (IFSs) to make up this lack for fuzzy set theory. In IFS, the membership degree (MD) and the non-membership degree (NMD) of an element can be described with the condition that their sum is less or equal to 1. IFSs were considered inadequate to handle the vague information when sum of MD and NMD exceeds 1. Yager [3] evolved the concept of Pythagorean fuzzy sets (PFSs) as a generalization of IFSs. In PFS, the sum of square of MD and NMD cannot exceed 1. However, PFSs have some limitations when membership and non-membership degree exceed 1. PFSs are more efficient to deal with imprecise information than IFSs. But when the MD is 0.8 and NMD is 0.7, PFS theory is not enough to handle this type of information. As a generalization of IFSs and PFSs, Senapati and Yager [4] gave the concept of Fermatean fuzzy sets (FFSs) in which sum of a cube of MD and a cube of NMD is less than or equal to 1. For example, 0.7,0.2 is an intuitionistic fuzzy number (IFN) as 0.7+0.2≤1, but if the NMD is 0.6, then it is a Pythagorean fuzzy number (PFN) but not a IFN. But if the NMD is 0.8, then 0.7,0.8 is neither IFN nor PFN. So, decision makers prefer to use FFSs to handle such kind of situation. Yager [5] developed q-rung orthopair fuzzy sets in which the sum of the q-th power MD and the q-th power of NMD cannot exceed 1. q-rung orthopair fuzzy sets (q-ROFSs) are more effective to deal with imprecise information than PFSs and IFSs. Certain applications of FFSs and q-ROFSs have been discussed in [6–15].Many aspects of real life cannot be handled in quantitative way but rather in qualitative way. In decision making, there are many criteria which cannot be dealt with real numbers. Therefore, there is a need to express the imprecise information in qualitative form. For example, to describe the quality of mobile phone, the decision makers use the term good or bad. Realizing this, Zadeh [16] introduced linguistic fuzzy sets (LFSs). Herrera and Verdegay [17] defined the operational laws for linguistic fuzzy sets. Normally, linguistic fuzzy set is based on seven terms which can be defined as S=s0=verypoor,s1=poor,s2=mediumpoor,s3=medium,s4=mediumgood,s5=good,s6=verygood. The quality of company considered as good (0.7) can be represented as s6, and the membership degree is taken as 0.8. But the linguistic fuzzy sets cannot describe the qualitative vague information in proper way. If one group of decision makers takes the membership degree of the quality of mobile phone as 0.7 and the other group of decision makers considers it as 0.5, then linguistic fuzzy set cannot deal with this type of information. Therefore, Chen and Hong [18] proposed linguistic intuitionistic fuzzy sets (LIFSs) by uniting LTSs and IFSs. LIFSs describe the linguistic membership (LMD) and linguistic non-membership degree (LNMD) of element in qualitative way in more effective manner. LIFSs are restricted to that information where the sum of LMD and LNMD is less than or equal to t. If decision makers give their partiality in form of LIFN s3,s5 where ssh denotes the linguistic term in linguistic term set and h∈0,6, this preference cannot be solved under LIFN. To overcome this limitation of LIFSs, Garg [19] defined linguistic Pythagorean fuzzy sets (LPFSs) in which the sum of square of LMD and LNMD cannot exceed t2. But LPFSs cannot deal with that type of imprecise information where LMD and LNMD exceed t2. For other applications and notions, the readers can refer to [20–23].Following linguistic fuzzy set, a lot of aggregation operators were defined. Peng et al. [24] developed an LIFN-based decision-making scheme and defined the notion of Frank Heronian mean operators. Liu et al. [25] elongated it to partitioned Heronian mean operator. Liu and Qin [26] defined Maclaurin symmetric mean operator for LIFNs. Hara et al. [27] proposed the Hamy mean operator which considered the connection between different parameters by modifying their values. Qin [28] developed interval type-2 fuzzy Hamy mean operator. Deng et al. [29] defined 2-tuple linguistic fuzzy Hamy mean operator. Li et al. [30] defined the (weighted) Pythagorean fuzzy Hamy mean operator. Wei et al. [21] introduced dual hesitant Pythagorean fuzzy Hamy mean operator.Zavadaskas et al. [31] proposed a methodology to deal with decision-making problems, named as COmplex PRoportional ASsessment (COPRAS) method. Wang et al. [32] utilized the COPRAS method to determine the ranking order of the failure modes which was recognized in failure mode and effect analysis (FMEA). Yazdani et al. [33] utilized the fuzzy COPRAS method to develop the risk analysis of critical infrastructures. Hajiagha et al. [34] developed a COPRAS method for group decision making under interval-valued intuitionistic fuzzy environment. Lu et al. [35] devised a COPRAS method under picture fuzzy environment.The major motivation for the proposed model comes from the following observation:(i)
Neither LIFSs nor LPFSs can handle the situation when the sum of squares of linguistic membership and non-membership degrees exceeds 1. Hence, it is necessary to develop a new model in this case.(ii)
The Hamy mean operator was designed to describe the correlation with numerous parameters. It is necessary to broaden the concept of Hamy mean operators using LFSs.(iii)
The COPRAS method was developed to determine the superlative alternative in classical decision making. Hence, this technique should be further extended to solve MAGDM problems involving linguistic term sets.Motivated by the above facts, we introduce a novel concept termed as the linguistic Fermatean fuzzy set (LFFS). The main contributions can be summarized as follows:(i)
We devise a mathematical approach for group decision making by the fusion of linguistic fuzzy sets and Fermatean fuzzy sets.(ii)
The linguistic Fermatean fuzzy Hamy mean operators are constructed to capture interrelationships among different parameters.(iii)
The classical COPRAS method is extended to handle MAGDM problems based on linguistic Fermatean fuzzy sets.The remainder of this article is structured as follows. In Section2, we define LFFNs and related operations, linguistic score and linguistic accuracy function, linguistic Fermatean fuzzy Hamy mean operator, and linguistic Fermatean fuzzy weighted Hamy mean operator and examine their properties. In Section 3, we present linguistic Fermatean fuzzy dual Hamy mean operator and linguistic Fermatean fuzzy weighted dual Hamy mean operator and investigate their properties. In Section 4, we develop an extension of the COPRAS method on the basis of LFFNs and apply it to a problem regarding food company ranking. In Section 5, we propose another approach to linguistic Fermatean fuzzy MAGDM and apply it to a problem regarding green supplier selection. In Section 6, we check the validity of our methods by comparing with existing methods. Section 7 presents conclusions and future directions.


## 2. Linguistic Fermatean Fuzzy Hamy Mean Operators

Definition 1.
(see [4]). Letχbe a universe of discourse. A Fermatean fuzzy set (FFS)Finχis an object of the form(1)F=<z,ξFz,ηFz>:z∈χ,where ξFz:χ⟶0,1 and ηFz:χ⟶0,1 including the condition(2)0≤ξFz3+ηFz3≤1,for all z∈χ. ξFz and ηFz denote, respectively, the MD and NMD of element z in the set F. For any FFS and z∈χ,(3)ΠFz=1−ξFz3−ηFz33is called the degree of indeterminacy of z to F. The pair ξFz,ηFz in F is defined as Fermatean fuzzy number (FFN).Definition 2.
(see [36]). Let there exist a linguistic term setS=sγ|γ=0,1,…,t, wheresγdenotes a possible linguistic term for a linguistic variable and the cardinality of the setSis odd. For instance, a linguistic term setShaving five terms can be described as follows:(4)S=s0=verypoor,s1=poor,s2=medium,s3=good,s4=verygood.
Ifsα,sβ∈S, then the linguistic term set satisfies the following properties:(i)
sα>sb, if and only ifα>β.(ii)
maxsα,sβ=sα, if and only ifα≥β.(iii)
minsα,sβ=sα, if and only ifα≤β.(iv)
Negsα=sβsuch thatβ=t−α.Next, let us propose the notion of linguistic Fermatean fuzzy sets and some basic operations of linguistic Fermatean fuzzy numbers. For other notions and applications, the readers are suggested to [37–51].Definition 3.
Suppose thatχis a universal set andS¯=sγ∣s0≤sγ≤st,γ∈0,tis a continuous linguistic term set (CLTS). A linguistic Fermatean fuzzy set (LFFS) is defined as(5)F=<z,sξz,sηz>∣z∈χ,wheresξz,sηz∈S¯stand for linguistic membership and non-membership degrees of the elementz. It is required thatξ3+η3≤t3with0≤ξ≤t,0≤η≤t. We refer toπzas the linguistic indeterminacy degree ofztoF, which is defined asπz=s3t3−ξ3−η3. The pairsξz,sηzrepresents a linguistic Fermatean fuzzy number (LFFN) inF.Definition 4.
LetP=sξ1,sη1andQ=sξ2,sη2be two LFFNs. Some relations and operations of LFFNs can be defined as follows:(i)
P=Qifsξ1=sξ2,sη1=sη2.(ii)
Pc=sη1,sξ1.(iii)
P∩Q=minsξ1,sξ2,maxsη1,sη2.(iv)
P∪Q=maxsξ1,sξ2,minsη1,sη2.(v)
P≤Qifsξ1≤sξ2,sη1≥sη2.Definition 5.
LetP=sξ1,sη1andQ=sξ2,sη2be two LFFNs. We can define the following operations of LFFNs:(i)
P⊕Q=stξ1/t3+ξ2/t3−ξ1/tξ2/t33,stη1η2/t2.(ii)
P⊗Q=stξ1ξ2/t2,stη1/t3+η2/t3−η1/tη2/t33.(iii)
λP=st1−1−ξ1/t3λ3,stη1/tλ.(iv)
Pλ=stξ1/tλ,st1−1−η1/t3λ3.Definition 6.
LetF=sξ,sηbe LFFS. The linguistic score function and the linguistic accuracy function of LFFS are represented as(6)LsF=ξ3−η32,LhF=ξ3+η3,where ls∈−t3/2,t3/2 and lh∈0,t3.Definition 7.
LetFv=sξv,sηvv=1,2,…,kbe a collection of LFFNs. We can define LFFHM operator as follows:(7)LFFHMyF1,F2,…,Fn=⊕1<u1<⋯<uy⊗yv=1FuvCyk,whereyis a parameter andv=1,2,…,y,u1,u2…uyareyintegers belonging to the set1,2,…,kofkinteger values,Cyk=k!/y!k−y!.Theorem 1.
LetFv=sξv,sηvv=1,2,…,kbe a collection of LFFNs. Then, the aggregate result of them by the above definition is still a LFFN:(8)LFFHMyF1,F2,…,FN=s1−∏1<u1<…<uy1−∏yv=1ξuvt3/y1/Cykst∏1<u1<⋯<uy1−∏yv=11−ηuvt31/y1/Cyk3t3.Proof.
From Definition5, we define(9)⊗v=1yFuv=st∏yv=1ξuvt,st1−∏yv=11−ηuvt33,⊗v=1yFuv1/y=st∏yv=1ξuvt1/y,st1−∏yv=11−ηuvt31/y3,⊕1<u1<…<uy<k⊗yv=1Fuv1/y=st1−∏1<u1<…<uy1−∏yv=1ξuvt3/y3,st∏1<u1<…<uy1−∏yv=11−ηuvt31/y3,LFFHMyF1,F2,…,Fn=⊕1<u1<…<uy<k⊗yv=1Fuv1/yCyk=st1−∏1<u1<…<uy1−∏yv=1ξuvt3/y1/Cyk3,st∏1<u1<…<uy1−∏yv=11−ηuvt31/y1y1/Cyk3.
Hence, (8) is established. We need to show that (8) is a LFFN. It satisfies the following two conditions:(1)
0≤ξ≤t,0≤η≤t.(2)
0≤ξ3+η3≤t3.
Let(10)ξ=t1−<1u1<⋯<uy∏1−∏yv=1ξuvt3/y1/Cyk3,η=t∏1<u1<…<uy1−∏yv=11−ηuvt31/y1/Cyk3.
Since0≤ξu≤t,(11)0≤∏yv=1ξuvt3/y≤1,0≤1−∏v=1xξuvt3/y≤1,0≤1−∏1<u1<⋯<uy1−∏yv=1ξuvt3/y1/Cyk3≤1,0≤t1−∏1<u1<⋯<uy1−∏yv=1ξuvt3/y1/Cyk3≤t.
This means0≤ξ≤t; similarly, 0≤η≤t. So, condition (1) is verified. Since 0≤ξu3+ηu3≤t3, we get(12)ξ3+η3=t31−∏1−∏yv=1ξuvt3/y1<u1<…<uy1/Cyk+t3∏1<u1<…<uy1−∏yv=11−ηuvt31/y1/Cyk≤t31−∏1<u1<…<uy1−∏yv=11−ηuvt31/y1/Cyk+t3∏1<u1<…<uy1−∏yv=11−ηuvt31/y1/Cyk.
This means0≤ξu3+ηu3≤t3, so condition (2) is verified.The LFFHM operator has three properties.Theorem 2.
IfFv=sξv,sηvv=1,2,…,nare equal, then(13)LFFHMyF1,F2,…,Fn=F.Proof.
(14)LFFHMyF1,F2,…,Fn=st1−∏1<u1<…<uy1−∏yv=1ξt3/y1/Cyk3,st∏1<u1<…<uy1−∏yv=11−ηt31/y1/Cyk3=st1−∏1<u1<…<uy1−ξty3/y1/Cyk3,st∏1<u1<…<uy1−1−ηt3y1/y1/Cyk3=sξ,sη.Theorem 3.
LetFav=sξav,sηavv=1,2,…,nandFbv=sξbv,sηbvv=1,2,…,nbe two sets of LFFNs. Ifsξav≤sξbvandsηav≥sηbvhold for all u and v, then(15)LFFHMyFa1,Fa2,…,Fan=LFFHMyFb1,Fb2,…,Fbn.Proof.
LetFav=sξav,sηav and Fbv=sξbv,sηbv. Suppose that sξav≤sξbv, and we obtain(16)∏v=1yξavt≤∏v=1yξbvt1−∏v=1yξavt2/y≥1−∏v=1yξbvt2/y.
Thereafter,(17)∏1−∏v=1yξavt3/y1<u1<⋯<uy<n≥∏1−∏v=1yξbvt3/y1<u1<…<uy<n.
Furthermore,(18)t1−∏1−∏v=1yξavt3/y1<u1<⋯<uy<n1/Cyk3≤t1−∏1−∏v=1yξbvt3/y1<u1<⋯<uy<n1/Cyk3.
This meanssξa≤sξb. Similarly, we get sηa≥sηb.
Ifsξa≤sξb and sηa≥sηb,(19)LFFHMyFa1,Fa2,…,Fan<LFFHMyFb1,Fb2,…,Fbn,
Ifsξa=sξb and sηa=sηb,(20)LFFHMyFa1,Fa2,…,Fan=LFFHMyFb1,Fb2,…,Fbn.Theorem 4.
LetFv=sξv,sηvv=1,2,…,nbe a collection of LFFNs. IfFv+=smaxξv,sminηvandFv−=sminξv,smaxηv, then(21)Fv−≤LFFHMyF1,F2,…,Fn≤Fv+.Proof.
From Theorem2,(22)LFFHMyF1−,F2−,…,Fn−=F−,LFFHMyF1+,F2+,…,Fn+=F+.
From Theorem3,(23)Fv−≤LFFHMyF1,F2,…,Fn≤Fv+.Definition 8.
LetFu=sξu,sηuu=1,2,…,nbe a collection of LFFNs with weight vectorw=w1,w2,…,wnT, satisfyingwv∈0,1and∑v=1nwv=1. Then, we can define LFFWHM operator as follows:(24)LFFWHMyF1,F2,…,Fn=⊕1<u1<…<uy⊗yv=1FuvwuvCyk.Theorem 5.
LetFv=sξv,sηvv=1,2,…,kbe a collection of LFFNs; then, the aggregate result of them by the above definition is still a LFFN:(25)LFFWHMyF1,F2,…,FN=st1−∏1<u1<…<uy1−∏yv=1ξuvtwuv3/y1/Cyk3,st∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y1/Cyk3.Proof.
From Definition5, we define(26)⊗v=1yFuv=st∏yv=1ξuvtwuv,st1−∏yv=11−ηuvt3wuv3,⊗v=1yFuv1/y=st∏yv=1ξuvtwuv1/y,st1−∏yv=11−ηuvt3wuv1/y3,⊕1<u1<…<uy<k⊗v=1yFuv1/y=st1−∏1<u1<…<uy1−∏yv=1ξuvtwuv3/y3,st∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y3,LFFWHMyF1,F2,…,Fn=⊕1<u1<…<uy<k⊗yv=1Fuv1/yCyk=st1−∏1<u1<…<uy1−∏yv=1ξuvtwuv3/y1/Cyk3,st∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y1/Cyk3.
Hence, (24) is established. We need to show that (24) is a LFFN. The following two conditions are satisfied:(1)
0≤ξ≤t,0≤η≤t.(2)
0≤ξ3+η3≤t3.
Let(27)ξ=t1−∏1<u1<…<uy1−∏yv=1ξuvt3/ywuv1/Cyk3,η=t∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y1/Cyk3.
Since0≤ξu≤t,(28)0≤∏yv=1ξuvtwuv3/y≤1,0≤1−∏yv=1ξuvtwuv3/y≤1,0≤1−∏1<u1<…<uy1−∏yv=1ξuvtwuv3/y1/Cyk3≤1,0≤t1−∏1<u1<…<uy1−∏yv=1ξuvtwuv3/y1/Cyk3≤t.
This means0≤ξ≤t; similarly, 0≤η≤t. So, condition (1) is proved.
(2) Since0≤ξu3+ηu3≤t3,(29)ξ3+η3=t31−∏1−∏yv=1ξuvtwuv3y1<u1<…<uy1/Cyk+t3∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y1/Cyk≤t31−∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y1/Cyk+t3∏1<u1<…<uy1−∏yv=11−ηuvt3wuv1/y1/Cyk.
This means0≤ξu3+ηu3≤t3, so condition (2) is proved.Theorem 6.
LetFav=sξav,sηavv=1,2,…,nandFbv=sξbv,sηbvv=1,2,…,nbe two sets of LFFNs. Ifsξav≤sξbvandsξav≥sξbvhold for all i and v, then(30)LFFWHMyFa1,Fa2,…,Fan=LFFWHMyFb1,Fb2,…,Fbn.Proof.
LetFav=sξav,sηav and Fbv=sξbv,sηbv; suppose that sξav≤sξbv, and we can obtain(31)∏yv=1ξavtwuv≤∏yv=1ξbvtwuv,1−∏yv=1ξavtwuv2/y≥1−∏yv=1ξbvtwuv2/y.
Thereafter,(32)∏1<u1<…<uy<n1−∏yv=1ξavtwuv3/y≥∏1<u1<…<uy<n1−∏yv=1ξbvtwuv3/y.
Furthermore,(33)t1−∏1<u1<…<uy<n1−∏yv=1ξavtwuv3/y1/Cyk3≤t1−∏1<u1<…<uy<n1−∏yv=1ξbvtwuv3/y1/Cyk3.
This meanssξa≤sξb. Similarly, sηa≥sηb.
Ifsξa≤sξb and sηa≥sηb,(34)LFFWHMyFa1,Fa2,…,Fan<LFFWHMyFb1,Fb2,…,Fbn.
Ifsξa=sξb and sηa=sηb,(35)LFFWHMyFa1,Fa2,…,Fan=LFFWHMyFb1,Fb2,…,Fbn.Theorem 7.
LetFv=sξv,sηvv=1,2,…,nbe a set of LFFNs. IfFv+=smaxξv,sminηvandFv−=sminξv,smaxηv, then(36)Fv−≤LFFWHMyF1,F2,…,Fn≤Fv+.Proof.
From Theorem5,(37)LFFWHMyF1−,F2−,…,Fn−=F−,LFFWHMyF1+,F2+,…,Fn+=F+.
From Theorem6,(38)Fv−≤LFFWHMyF1,F2,…,Fn≤Fv+.


## 3. Linguistic Fermatean Fuzzy Dual Hamy Mean Operators

Definition 9.
LetFu=sξu,sηube a collection of LFFNs. The LFFDHM operator is defined as follows:(39)LFFDHMyF1,F2,…,Fn=⊗1<u1<…<uy<k⊕yv=1FuvyCyk,where y is a parameter and v=1,2,…,y,u1,u2…uy are y integers belonging to the set 1,2,…,k of k integer values, Cyk=k!/y!k−y!.Theorem 8.
LetFv=sξv,sηvv=1,2,…,kbe a collection of LFFNs. Then, the aggregate result of them by the above definition is still a LFFN:(40)LFFDHMyF1,F2,…,FN=st∏1<u1<…<uy1−∏yv=11−ξuvt31/y1/Cyk3,st1−∏1<u1<…<uy1−∏yv=1ηuvt3/y1/Cyk3.Proof.
From Definition5, we define(41)⊕v=1yFuv=st1−∏yv=11−ξuvt33,st∏yv=1ηuvt,⊕yv=1Fuvy=st1−∏yv=11−ξuvt31/y3,st∏yv=1ηuvt1/y,⊗1<u1<…<uy<k⊕yv=1Fuvy=st∏1−∏v=1y1−ξuvt31/y31<u1<…<uy,st1−∏v=1y1−∏v=1yηuvt31<u1<…<uy<k1/y3,LFFDHMyF1,F2,…,Fn=⊗1<u1<…<uy<k⊕yv=1FuvyCyk=st∏1−∏v=1y1−ξuvt31/y31<u1<…<uy1/Cyk,st1−∏v=1y1−∏v=1yηuvt31<u1<…<uy<k1/y3.
Hence, (39) is established. We need to show that (39) is a LFFN. The following two conditions are satisfied:(1)
0≤ξ≤t,0≤η≤t.(2)
0≤ξ3+η3≤t3.
Let(42)ξ=t∏1<u1<…<uy1−∏yv=11−ξuvt31/y31/Cyk,η=t1−∏1<u1<…<uy<k1−∏yv=1ηuvt3/y1/Cyk3.
Since0≤ξu≤t,(43)0≤∏yv=11−ξuvt3≤1,0≤∏yv=11−ξuvt31/y≤1,0≤∏yv=11−ξuvt31/y3≤1,0≤t∏1<u1<…<uy<k1−ξuvt31/y31/Cyk≤t.
This means0≤ξ≤t; similarly, 0≤η≤t. So, condition (1) is verified.
(2) Since0≤ξu3+ηu3≤t3,(44)ξu3+ηu3=t∏∏v=1y1−ξuvt31/y1<u1<…<uy<k1/Cyk+t1−∏v=1y1−∏v=1yηuvt31<u1<…<uy<k1/y≤t31−1−∏1−∏yv=1ηuvt31/y1<u1<…<uy<k+t31−∏1−∏yv=1ηuvt31<u1<…<uy<k1/y.
This means0≤ξu3+ηu3≤t3, so condition (2) is verified.The LFFDHM operator has three properties.Theorem 9.
IfFv=sξv,sηvv=1,2,…,nare equal, then(45)LFFDHMyF1,F2,…,Fn=F.Proof.
(46)LFFDHMyF1,F2,…,Fn=st∏1<u1<…<uy1−∏yv=11−ξuvt31/y31/Cyk,st1−∏1<u1<…<uy<k1−∏yv=1ηuvt3/y1/Cyk3.=st∏1<u1<…<uy1−∏yv=11−ξt31/y31/Cyk,st1−∏1<u1<…<uy<k1−∏yv=1ηt3/y1/Cyk3=sξ,sηTheorem 10.
LetFav=sξav,sηavv=1,2,…,nandFbv=sξbv,sηbvv=1,2,…,nbe two sets of LFFNs. Ifsξav≤sξbvandsηav≥sηbvhold for all u and v, then(47)LFFDHMyFa1,Fa2,…,Fan=LFFDHMyFb1,Fb2,…,Fbn.
Ifsξa≤sξb and sηa≥sηb,(48)LFFDHMyFa1,Fa2,…,Fan<LFFHMyFb1,Fb2,…,Fbn.
Ifsξa=sξb and sηa=sηb,(49)LFFDHMyFa1,Fa2,…,Fan=LFFHMyFb1,Fb2,…,Fbn.Theorem 11.
LetFv=sξv,sηvv=1,2,…,nbe a set of LFFNs. IfFv+=smaxξv,sminηvandFv−=sminξv,smaxηv, then(50)Fv−≤LFFDHMyF1,F2,…,Fn≤Fv+.Proof.
From Theorem9,(51)LFFDHMyF1−,F2−,…,Fn−=F−,LFFDHMyF1+,F2+,…,Fn+=F+.
From Theorem10,(52)Fv−≤LFFDHMyF1,F2,…,Fn≤Fv+.Definition 10.
LetFu=sξu,sηube a collection of LFFNs with weight vectorw=w1,w2,…,wnT, satisfyingwv∈0,1and∑v=1nwv=1. Then, we define LFFWDHM operator as follows:(53)LFFWDHMyF1,F2,…,Fn=⊗1<u1<⋯<uy⊕yv=1FuvwuvyCyk.Theorem 12.
LetFv=sξv,sηvv=1,2,…,kbe a collection of LFFNs. Then, the aggregate result of them by the above definition is still a LFFN:(54)LFFWDHMyF1,F2,…,FN=st∏1<u1<…<uy1−∏yv=11−ξuvt3wuv1/y31/Cyk,st1−∏1<u1<…<uy<k1−∏yv=1ηuvtwuv3/y1/Cyk3.Proof.
From Definition5, we define(55)wuvFuv=st1−1−ξuvt3wuv3,stηuvtwuv,⊕v=1ywuvFuv=st1−∏yv=11−ξuvt3wuv3,st∏yv=1ηuvtwuv,⊗1<u1<…<uy<k⊕v=1ywuvFuvy=st∏1<u1<…<uy<k1−∏yv=11−ξuvt3wuv1/y3,st1−∏1<u1<…<uy<k1−∏yv=1ηuvtwuv3/y3,⊗1<u1<…<uy<k⊕yv=1wuvFuvy1Cyk=st∏1<u1<…<uy<k1−∏yv=11−ξuvt3wuv1/y31Cyk,st1−∏1<u1<…<uy<k1−∏yv=1ηuvtwuv3/y31Cyk,LFFWDHMyF1,F2,…,Fn=⊗1<u1<…<uy<k⊕yv=1wuvFuvy1/Cyk=st∏1<u1<…<uy<k1−∏yv=11−ξuvt3wuv1/y31/Cyk,st1−∏1<u1<…<uy<k1−∏yv=1ηuvtwuv3/y1/Cyk3.
Hence, (53) is established. We need to show that (53) is a LFFN as it satisfies the following two conditions:(1)
0≤ξ≤t,0≤η≤t.(2)
0≤ξ3+η3≤t3.
Let(56)ξ=t∏1<u1<…<uy<k1−∏v=1y1−ξuvt3wuv1/y31/Cyk,η=t1−∏1−∏v=1yηuvtwuv3/y1<u1<…<uy<k1/Cyk3.
Since0≤ξu≤t,(57)0≤∏v=1y1−ξuvt3wuv1/y≤1,0≤1−∏v=1y1−ξuvt3wuv1/y≤1,0≤∏1<u1<…<uy1−∏v=1y1−ξuvt3wuv1/y3≤1,0≤∏1<u1<…<uy1−∏v=1y1−ξuvt3wuv1/y31/Cyk≤t.
This means0≤ξ≤t; similarly, 0≤η≤t. So, condition (1) is verified.
(2) Since0≤ξu3+ηu3≤t3, we can get(58)ξ3+η3=t3∏1<u1<…<uy1−∏v=1y1−ξuvt3wuv1/y1/Cyk+t31−∏1−∏v=1yηuvtwuv3/y1<u1<…<uy<k1/Cyk≤t31−1−∏1−∏v=1yηuvtwuv3/y1/Cyk1<u1<…<uy<k+t31−∏1−∏v=1yηuvtwuv3/y1<u1<…<uy<k1/Cyk.
This means0≤ξu3+ηu3≤t3, so condition (2) is verified.Theorem 13.
LetFav=sξav,sηavv=1,2,…,nandFbv=sξbv,sηbvv=1,2,…,nbe two sets of LFFNs. Ifsξav≤sξbvandsξav≥sξbvhold for all u and v, then(59)LFFWDHMyFa1,Fa2,…,Fan=LFFDWHMyFb1,Fb2,…,Fbn.Theorem 14.
LetFv=sξv,sηvv=1,2,…,nbe a set of LFFNs. IfFv+=smaxξv,sminηvandFv−=sminξv,smaxηv, then(60)Fv−≤LFFWDHMyF1,F2,…,Fn≤Fv+.Proof.
From Theorem12,(61)LFFWDHMyF1+,F2+,…,Fn+=F+,LFFWDHMyF1−,F2−,…,Fn−=F−.
From Theorem13,(62)Fv−≤LFFWDHMyF1,F2,…,Fn≤Fv+.It is clear that LFFWDHM operator does not satisfy the property of idempotency.


## 4. A Linguistic Fermatean Fuzzy Extension of COPRAS Method

### 4.1. Linguistic Fermatean Fuzzy COPRAS Method

Assume that the set of alternatives isZ=z1,z2,…,zp and the set of attributes is A=a1,a2,…,aq; the weighting vector for the set of criteria is W=w1,w2,…,wqT which satisfies the condition that 0≤wv≤1v=1,2,…,q and ∑v=1qwv=1. Let Y=y1,y2,…,yr be the set of decision experts. These experts are invited to evaluate the alternatives under different attributes. Their weights are given by the vector ω=ω1,ω2,…,ωrT which satisfy the condition that 0≤ωb≤1b=1,2,…,r and ∑b=1rωb=1. The decision makers determine the value of every attribute av of zu in terms of the LFFN γuvb=sξuvb,sηuvbu=1,2,…,p,v=1,2,…,q. The decision matrix is formed as Rb=γuvbp×qb=1,2,…,r accordingly.The following steps are involved.Step 1.
We utilize either LFFWHM operator given in equation (25) or LFFWDHM operator given in equation (40) to determine the collective decision matrix and find out the evaluation values under various criteria of every alternative.(63)γuv=LFFWHMγuv1,γuv2,…,γuvr,or(64)γuv=LFFWDHMγuv1,γuv2,…,γuvr.Step 2.
For positive type of attributes, each alternative is evaluated by using either LFFWHM operator in equation (25) or LFFWDHM operator given in equation (40):(65)P+u=LFFWHMγu1,γu2,…,γug,or(66)P+u=LFFWDHMγu1,γu2,…,γug.
For negative type of attributes, each alternative is evaluated by using either LFFWHM operator given in (24) or LFFWDHM operator given in (53):(67)P−u=LFFWHMγug+1,γug+2,…,γuq,or(68)P−u=LFFWDHMγug+1,γug+2,…,γuq,where g represents the totality of positive attributes and g−k is the totality of negative attributes.Step 3.
Find the linguistic score ofP+u, P−u.Step 4.
Calculate relative priority of each alternative by using the following formula:(69)Qzu=lsP+u+∑u=1plsP−ulsP−u×∑u=1p1/lsP−u.Step 5.
We determine the ranking of alternatives based on relative priority.
The flowchart of the presented algorithm is given in Figure1.Figure 1
Flowchart of the linguistic Fermatean fuzzy COPRAS method.


### 4.2. Numerical Example I: Food Company Ranking

We apply our proposed COPRAS method to find out the ranking of food companies based on different criteria. The food authority wants to select best food company in the world and there are four food companies as candidatesA=Unilever,Nestle,Danone,Ocp. The food companies are selected on the basis of six aspects which are B=governance,environment,nutrition,labourcost,productioncost,price of product. The vector W=0.2,0.2,0.3,0.05,0.1,0.15T specifies the attribute weights. Three decision experts ybb=1,2,3 are invited to rank the food companies, and each decision maker evaluates each alternative on the basis of linguistic term set (LTS) S=s0=extremelypoor,s1=verypoor,s2=poor,s3=mediumpoor,s4=medium,s5=mediumgood,s6=good,s7=verygood,s8=best. The weight vector for decision makers is ϖ=0.5,0.25,0.25; then, the decision matrices are shown in Tables 1–4. Here benefit attributes are(i)
Governance(ii)
Environment(iii)
NutritionTable 1
Decision matrix by first expert.
a1a2a3a4a5a6uns7,s1s6,s2s5,s3s1,s2s1,s1s2,s2nss6,s2s6,s2s5,s3s2,s1s1,s1s3,s2dns7,s1s6,s2s4,s4s3,s1s2,s1s2,s2ops6,s2s5,s3s4,s4s2,s2s3,s2s4,s1Table 2
Decision matrix by second expert.
a1a2a3a4a5a6uns7,s1s5,s1s4,s2s1,s2s1,s1s2,s2nss6,s2s5,s1s4,s2s2,s1s1,s1s3,s1dns7,s1s5,s1s3,s3s3,s1s2,s1s2,s2ops6,s2s4,s2s3,s3s2,s2s3,s2s4,s1Table 3
Decision matrix by third expert.
a1a2a3a4a5a6uns7,s1s6,s2s5,s3s1,s2s1,s1s1,s1nss6,s2s6,s2s5,s3s2,s1s1,s1s2,s1dns7,s1s6,s2s4,s4s3,s1s2,s1s1,s1ops6,s2s5,s3s4,s4s2,s2s3,s2s3,s2Table 4
Collective decision matrix with LFFWHM operator.
a1a2a3a4a5a6uns7.66,s0.69s7.17,s1.25s6.73,s1.92s4.11,s1.38s4.11,s0.69s4.8,s1.25nss7.28,s1.38s7.17,s1.25s6.73,s1.92s5.11,s0.69s4.11,s0.69s5.61,s1.01dns7.66,s0.69s7.17,s1.25s6.22,s2.63s5.82,s0.69s5.11,s0.69s4.8,s1.25ops7.28,s1.38s6.73,s1.92s6.22,s2.63s5.11,s1.38s5.82,s1.38s6.22,s0.93Cost attributes are as follows:(i)
Labour cost(ii)
Production cost(iii)
Price of productThe food companies are represented asZ=un,ns,dn,op, respectively, and the attributes are represented as A=a1,a2,a3,a4,a5,a6, respectively. The decision matrices in terms of LFFNs are shown in Tables 1–3.


### 4.3. Calculation and Decision Making of Numerical Example I

Step 1.
We utilize LFFWHM operator and LFFWDHM operator to find out the evaluation values of different schemes under each value of alternative, and a collective decision matrix with LFFWHM operator is formed fory=2 using equation (63) and is shown in Table 4, while using LFFWDHM operator, the collective decision matrix is formed using equation (64) and is shown in Table 5.Table 5
Collective decision matrix with LFFWDHM operator.
a1a2a3a4a5a6uns5.38,s4.12s4.18,s4.8s3.37,s5.61s0.69,s5.11s0.69,s4.11s1.25,s4.8nss4.38,s5.11s4.18,s4.8s3.37,s5.61s1.38,s4.11s0.69,s4.11s1.92,s4.5dns5.38,s4.11s4.18,s4.8s2.63,s6.22s2.08,s4.11s1.38,s4.11s1.24,s4.8ops4.38,s5.11s3.37,s5.61s2.63,s6.22s1.38,s5.11s2.08,s5.11s2.63,s4.42Step 2.
The maximizing and minimizing index values for positive and negative types of attributes with LFFWHM operator are calculated using equations (65) and (67) for y=2, while with LFFWDHM operator, the maximizing and minimizing values are calculated using equations (67) and (69). These values are shown in Table 6.Table 6
Maximizing and minimizing index values.
LFFWHMLFFWDHMzuP+uP−uP+uP−uuns7.79,s0.88s7.55,s0.52s2.73,s7.13s0.44,s7.57nss7.76,s0.97s7.64,s0.39s2.47,s7.23s0.67,s7.53dns7.75,s1.07s7.66,s0.44s2.61,s7.21s0.69,s7.55ops7.67,s1.3s7.76,s0.55s2.16,s7.38s1.04,s7.61Step 3.
The linguistic score of alternatives under positive type of attributes and negative types of attributes, that is,lsP+u and lsP−u, is found out and is shown in Table 7.Table 7
Linguistic scores.
AlternativeLFFWHMLFFWDHMlsP+ulsP−ulsP+ulsP−uun236.24214.98−171−217.7ns233.276222.64−181.49−213.01dn231.88224.366−179.32−214.89op224.7233.54−196.49−219.86Step 4.
The relative priority of each alternative is calculated by using equation (69) and is shown in Table 8.Table 8
Relative priority.
LFFWHMLFFWDHMzuQzuQzuun469.19−386.05ns458.21−396.13dn455.089−401.239op439.15−409.39Step 5.
Rank the alternatives based on relative priority. The best alternative has greatest relative priority.
The ranking with LFFWHM operator fory=2 is given as(70)un>ns>dn>op,whereas the ranking with LFFWDHM operator for y=2 is given as(71)un>dn>ns>op.
As a result of evaluation of COPRAS method, it is clear that alternativeunilever is the best food company and ocp is the worst food company.


## 4.1. Linguistic Fermatean Fuzzy COPRAS Method

Assume that the set of alternatives isZ=z1,z2,…,zp and the set of attributes is A=a1,a2,…,aq; the weighting vector for the set of criteria is W=w1,w2,…,wqT which satisfies the condition that 0≤wv≤1v=1,2,…,q and ∑v=1qwv=1. Let Y=y1,y2,…,yr be the set of decision experts. These experts are invited to evaluate the alternatives under different attributes. Their weights are given by the vector ω=ω1,ω2,…,ωrT which satisfy the condition that 0≤ωb≤1b=1,2,…,r and ∑b=1rωb=1. The decision makers determine the value of every attribute av of zu in terms of the LFFN γuvb=sξuvb,sηuvbu=1,2,…,p,v=1,2,…,q. The decision matrix is formed as Rb=γuvbp×qb=1,2,…,r accordingly.The following steps are involved.Step 1.
We utilize either LFFWHM operator given in equation (25) or LFFWDHM operator given in equation (40) to determine the collective decision matrix and find out the evaluation values under various criteria of every alternative.(63)γuv=LFFWHMγuv1,γuv2,…,γuvr,or(64)γuv=LFFWDHMγuv1,γuv2,…,γuvr.Step 2.
For positive type of attributes, each alternative is evaluated by using either LFFWHM operator in equation (25) or LFFWDHM operator given in equation (40):(65)P+u=LFFWHMγu1,γu2,…,γug,or(66)P+u=LFFWDHMγu1,γu2,…,γug.
For negative type of attributes, each alternative is evaluated by using either LFFWHM operator given in (24) or LFFWDHM operator given in (53):(67)P−u=LFFWHMγug+1,γug+2,…,γuq,or(68)P−u=LFFWDHMγug+1,γug+2,…,γuq,where g represents the totality of positive attributes and g−k is the totality of negative attributes.Step 3.
Find the linguistic score ofP+u, P−u.Step 4.
Calculate relative priority of each alternative by using the following formula:(69)Qzu=lsP+u+∑u=1plsP−ulsP−u×∑u=1p1/lsP−u.Step 5.
We determine the ranking of alternatives based on relative priority.
The flowchart of the presented algorithm is given in Figure1.Figure 1
Flowchart of the linguistic Fermatean fuzzy COPRAS method.


## 4.2. Numerical Example I: Food Company Ranking

We apply our proposed COPRAS method to find out the ranking of food companies based on different criteria. The food authority wants to select best food company in the world and there are four food companies as candidatesA=Unilever,Nestle,Danone,Ocp. The food companies are selected on the basis of six aspects which are B=governance,environment,nutrition,labourcost,productioncost,price of product. The vector W=0.2,0.2,0.3,0.05,0.1,0.15T specifies the attribute weights. Three decision experts ybb=1,2,3 are invited to rank the food companies, and each decision maker evaluates each alternative on the basis of linguistic term set (LTS) S=s0=extremelypoor,s1=verypoor,s2=poor,s3=mediumpoor,s4=medium,s5=mediumgood,s6=good,s7=verygood,s8=best. The weight vector for decision makers is ϖ=0.5,0.25,0.25; then, the decision matrices are shown in Tables 1–4. Here benefit attributes are(i)
Governance(ii)
Environment(iii)
NutritionTable 1
Decision matrix by first expert.
a1a2a3a4a5a6uns7,s1s6,s2s5,s3s1,s2s1,s1s2,s2nss6,s2s6,s2s5,s3s2,s1s1,s1s3,s2dns7,s1s6,s2s4,s4s3,s1s2,s1s2,s2ops6,s2s5,s3s4,s4s2,s2s3,s2s4,s1Table 2
Decision matrix by second expert.
a1a2a3a4a5a6uns7,s1s5,s1s4,s2s1,s2s1,s1s2,s2nss6,s2s5,s1s4,s2s2,s1s1,s1s3,s1dns7,s1s5,s1s3,s3s3,s1s2,s1s2,s2ops6,s2s4,s2s3,s3s2,s2s3,s2s4,s1Table 3
Decision matrix by third expert.
a1a2a3a4a5a6uns7,s1s6,s2s5,s3s1,s2s1,s1s1,s1nss6,s2s6,s2s5,s3s2,s1s1,s1s2,s1dns7,s1s6,s2s4,s4s3,s1s2,s1s1,s1ops6,s2s5,s3s4,s4s2,s2s3,s2s3,s2Table 4
Collective decision matrix with LFFWHM operator.
a1a2a3a4a5a6uns7.66,s0.69s7.17,s1.25s6.73,s1.92s4.11,s1.38s4.11,s0.69s4.8,s1.25nss7.28,s1.38s7.17,s1.25s6.73,s1.92s5.11,s0.69s4.11,s0.69s5.61,s1.01dns7.66,s0.69s7.17,s1.25s6.22,s2.63s5.82,s0.69s5.11,s0.69s4.8,s1.25ops7.28,s1.38s6.73,s1.92s6.22,s2.63s5.11,s1.38s5.82,s1.38s6.22,s0.93Cost attributes are as follows:(i)
Labour cost(ii)
Production cost(iii)
Price of productThe food companies are represented asZ=un,ns,dn,op, respectively, and the attributes are represented as A=a1,a2,a3,a4,a5,a6, respectively. The decision matrices in terms of LFFNs are shown in Tables 1–3.


## 4.3. Calculation and Decision Making of Numerical Example I

Step 1.
We utilize LFFWHM operator and LFFWDHM operator to find out the evaluation values of different schemes under each value of alternative, and a collective decision matrix with LFFWHM operator is formed fory=2 using equation (63) and is shown in Table 4, while using LFFWDHM operator, the collective decision matrix is formed using equation (64) and is shown in Table 5.Table 5
Collective decision matrix with LFFWDHM operator.
a1a2a3a4a5a6uns5.38,s4.12s4.18,s4.8s3.37,s5.61s0.69,s5.11s0.69,s4.11s1.25,s4.8nss4.38,s5.11s4.18,s4.8s3.37,s5.61s1.38,s4.11s0.69,s4.11s1.92,s4.5dns5.38,s4.11s4.18,s4.8s2.63,s6.22s2.08,s4.11s1.38,s4.11s1.24,s4.8ops4.38,s5.11s3.37,s5.61s2.63,s6.22s1.38,s5.11s2.08,s5.11s2.63,s4.42Step 2.
The maximizing and minimizing index values for positive and negative types of attributes with LFFWHM operator are calculated using equations (65) and (67) for y=2, while with LFFWDHM operator, the maximizing and minimizing values are calculated using equations (67) and (69). These values are shown in Table 6.Table 6
Maximizing and minimizing index values.
LFFWHMLFFWDHMzuP+uP−uP+uP−uuns7.79,s0.88s7.55,s0.52s2.73,s7.13s0.44,s7.57nss7.76,s0.97s7.64,s0.39s2.47,s7.23s0.67,s7.53dns7.75,s1.07s7.66,s0.44s2.61,s7.21s0.69,s7.55ops7.67,s1.3s7.76,s0.55s2.16,s7.38s1.04,s7.61Step 3.
The linguistic score of alternatives under positive type of attributes and negative types of attributes, that is,lsP+u and lsP−u, is found out and is shown in Table 7.Table 7
Linguistic scores.
AlternativeLFFWHMLFFWDHMlsP+ulsP−ulsP+ulsP−uun236.24214.98−171−217.7ns233.276222.64−181.49−213.01dn231.88224.366−179.32−214.89op224.7233.54−196.49−219.86Step 4.
The relative priority of each alternative is calculated by using equation (69) and is shown in Table 8.Table 8
Relative priority.
LFFWHMLFFWDHMzuQzuQzuun469.19−386.05ns458.21−396.13dn455.089−401.239op439.15−409.39Step 5.
Rank the alternatives based on relative priority. The best alternative has greatest relative priority.
The ranking with LFFWHM operator fory=2 is given as(70)un>ns>dn>op,whereas the ranking with LFFWDHM operator for y=2 is given as(71)un>dn>ns>op.
As a result of evaluation of COPRAS method, it is clear that alternativeunilever is the best food company and ocp is the worst food company.


## 5. Another Approach to Linguistic Fermatean Fuzzy MAGDM

### 5.1. A Linguistic Fermatean Fuzzy MAGDM Approach

Consider thatZ=z1,z2,…,zp is the set of alternatives and A=a1,a2,…,aq is the set of attributes. The weighting vector for the set of attributes is W=w1,w2,…,wqT which satisfies the condition that 0≤wv≤1v=1,2,…,q and ∑v=1qwv=1. Let Y=y1,y2,…,yr be the set of decision experts. These experts are invited to evaluate the alternatives under different attributes. Their weights are given by the vector ω=ω1,ω2,…,ωrT which satisfy the condition that 0≤ωb≤1b=1,2,…,r and ∑b=1rωb=1. The decision makers evaluate each attribute av of each alternative zu in the form of linguistic Fermatean fuzzy number γuvb=sξuvb,sηuvbu=1,2,…,pv=1,2,…,q; then, the decision matrix is formed as Rb=γuvbp×qb=1,2,…,r.The following steps are involved.Step 1.
Normalize each linguistic Fermatean fuzzy decision matrix provided by decision maker. The non-beneficial type of attributes is converted in to benefit type of attributes. The linguistic Fermatean fuzzy decision matrix is normalized as(72)γuvb=sξuvb,sηuvbfor benefit criterion,sηuvb,sξuvbfor cost criterion.Step 2.
Using linguistic Fermatean fuzzy weighted Hamy mean (LFFWHM) given in equation (25), construct the matrix of fused attributes given as(73)γ11γ12⋯γ1rγ21γ22⋯γ2r⋮⋮⋯⋮γp1γp2⋯γpr,whose entry γub is evaluated as follows:(74)γub=LFFWHMγu1b,γu2b,…,γuqb.Step 3.
Use the linguistic Fermatean fuzzy weighted Hamy mean (LFFWHM) given in Equation (25) to accommodate the fused attribute values of each alternative.(75)γu=LFFWHMγu1,γu2,…,γur.Step 4.
Find the score of each alternative [37].Step 5.
Rank the alternatives based on their score function.
The flowchart is given in Figure2.Figure 2
Flowchart of the proposed linguistic Fermatean fuzzy MAGDM method.


### 5.2. Numerical Example II: Green Supplier Selection

A construction company intends to select a supplier from four candidatesX=z1,z2,z3,z4. The supplier is selected on the basis of five aspects C=a1,a2,a3,a4,a5 which are production cost, production quality, supplier service performance, profile of supplier, and risk factor [38]. Attribute weights are given by the vector w=0.25,0.2,0.15,0.18,0.22T. Four experts ybb=1,2,3,4 evaluate every alternative on the basis of LTS S=s0=extremelypoor,s1=verypoor,s2=poor,s3=slightlypoor,s4=fair,s5=slightlygood,s6=good,s7=verygood,s8=extremlygood. The weight vector of the experts is ω=0.25,0.3,0.2,0.25T. The decision experts provide information in the form of LFFNs where sξ denotes the linguistic membership degree in LFFN which can be regarded as preference of decision maker and sη denotes the linguistic non-membership degree in LFFN which can be regarded as non-preference of decision maker. The normalized decision matrices are shown in Tables 9–12.Table 9
Decision matrixR1.
a1a2a3a4a5z1s7,s1s6,s2s4,s3s7,s1s5,s2z2s6,s2s5,s2s6,s1s6,s2s7,s1z3s6,s1s5,s3s7,s1s5,s1s3,s4z4s5,s2s7,s1s4,s3s6,s1s4,s4Table 10
Decision matrixR2.
a1a2a3a4a5z1s7,s1s4,s4s6,s2s5,s2s3,s5z2s7,s1s5,s1s6,s1s5,s2s4,s3z3s5,s2s6,s1s7,s1s5,s3s4,s4z4s6,s2s4,s3s5,s2s7,s1s5,s3Table 11
Decision matrixR3.
a1a2a3a4a5z1s6,s1s5,s2s3,s4s7,s1s5,s2z2s7,s1s6,s2s7,s1s6,s2s5,s1z3s5,s3s5,s2s6,s1s4,s3s3,s1z4s6,s2s7,s1s5,s1s5,s2s4,s4Table 12
Decision matrixR4.
a1a2a3a4a5z1s5,s3s4,s4s7,s1s5,s1s4,s2z2s6,s1s7,s1s6,s1s5,s2s6,s1z3s5,s2s3,s4s6,s2s3,s3s5,s2z4s4,s3s5,s1s4,s2s6,s2s5,s2


### 5.3. Calculation and Decision Making of Numerical Example II

Step 1.
Normalize the decision matrices provided by decision makers. In this way, cost attributes are converted into benefit type attributes. The decision matrices are shown in Tables13–16.Table 13
Normalized decision matrixR1.
a1a2a3a4a5z1s1,s7s6,s2s4,s3s7,s1s2,s5z2s2,s6s5,s2s6,s1s6,s2s1,s7z3s1,s6s5,s3s7,s1s5,s1s1,s3z4s1,s5s7,s1s4,s3s6,s1s4,s4Table 14
Normalized decision matrixR2.
a1a2a3a4a5z1s1,s7s4,s4s6,s2s5,s2s5,s3z2s1,s7s5,s1s6,s1s5,s2s3,s4z3s2,s5s6,s1s7,s1s5,s3s4,s4z4s2,s6s4,s3s5,s2s7,s1s3,s5Table 15
Normalized decision matrixR3.
a1a2a3a4a5z1s1,s6s5,s2s3,s4s7,s1s2,s5z2s1,s7s6,s2s7,s1s6,s2s1,s5z3s3,s5s5,s2s6,s1s4,s3s1,s3z4s2,s6s7,s1s5,s1s5,s2s4,s4Table 16
Normalized decision matrixR4.
a1a2a3a4a5z1s3,s5s4,s4s7,s1s5,s1s2,s4z2s1,s6s7,s1s6,s1s5,s2s1,s6z3s2,s5s3,s4s6,s2s3,s3s2,s5z4s3,s4s5,s1s4,s2s6,s2s2,s5Step 2.
Using LFFWHM operator given in equation (25), the alternatives are fused corresponding to each attribute provided by decision maker. The matrix R of fused values is shown as(76)R=s6.87,s2.47s6.95,s2.39s7.01,s1.86s6.95,s2.01s6.85,s2.49s6.87,s2s6.9,s2.33s6.83,s2.14s7.01,s1.86s7.18,s1.85s6.79,s1.82s2.56,s2.43s7.14,s1.83s6.97,s2.29s7.13,s1.99s6.92,s1.81.Step 3.
Use LFFWHM operator given inequation (25) to aggregate the fused values of each alternative provided by decision maker. The aggregated values are shown as follows: z1 =  s7.71,s1.47 , z2 =  s7.7,s1.41z3 =  s7.71,s1.26, z4 =  s7.75,s1.26.Step 4.
The score function of alternatives is calculated and is shown in Table17.Step 5.
Rank the alternatives based on score function. The ranking is given as(77)z4>z3>z1>z2.
It is determined that alternativez4 is the best candidate and alternative z2 is the worst candidate.Table 17
Score function.
AlternativeSzuz1s7.846z2s7.845z3s7.85z4s7.87


### 5.4. Sensitivity Analysis of Numerical Example I, II

Using the COPRAS method with LFFWHM and LFFWDHM operators, the ranking results for different values of parameters are calculated. The parametery plays a significant role. The ranking results for different values of parameter with LFFWHM operator are the same. Here z1 is still the best food company for different values of parameter and z4 is the worst food company, while with LFFWDHM operator, the ranking results are still the same. z1 is best food company, and z4 is the worst company. The ranking results while using COPRAS method with LFFWHM operator and with LFFWDHM operator are calculated in Tables 18 and 19, and results are shown in Figures 3 and 4, respectively.Table 18
Order for different parameters with LFFWHM operator using the COPRAS method.
Parameter valuesRelative priorityOrderQz1Qz2Qz3Qz4y=1477.174464.88464.26450.81z1>z2>z3>z4y=2469.19458.21455.09439.15z1>z2>z3>z4y=3224.67185.29168.05137.23z1>z2>z3>z4Table 19
Order for different parameters with LFFWDHM operator using the COPRAS method.
Parameter valuesRelative priorityOrderQz1Qz2Qz3Qz4y=1−397.45−412.47−410.29−417.88z1>z3>z2>z4y=2−386.05−396.13−401.24−409.39z1>z3>z2>z4y=3−48.646−67.29−58.816−70.421z1>z3>z2>z4Figure 3
Ranking results using an extended COPRAS method with LFFWDHM.Figure 4
Ranking results using the COPRAS method with LFFWHM.In MAGDM approach with LFFWHM operator, the parametery plays a significant role. For different values of y, we can see that the ranking values are the same. For different parameters, the superlative supplier is z4 and the worst supplier is still z2. The ranking values for different values of y are shown in Table 20, and the results are shown in Figure 5.Table 20
Order for different parameters with LFFWHM operator.
Parameter valuesScore functionOrderSz1Sz2Sz3Sz4y=1s7.893s7.892s7.902s7.904z4>z3>z1>z2y=2s7.846s7.845s7.85s7.87z4>z3>z1>z2y=3s7.82s7.81s7.83s7.84z4>z3>z1>z2y=4s7.81s7.79s7.82s7.85z4>z3>z1>z2Figure 5
Effect of parameters with LFFWHM.


## 5.1. A Linguistic Fermatean Fuzzy MAGDM Approach

Consider thatZ=z1,z2,…,zp is the set of alternatives and A=a1,a2,…,aq is the set of attributes. The weighting vector for the set of attributes is W=w1,w2,…,wqT which satisfies the condition that 0≤wv≤1v=1,2,…,q and ∑v=1qwv=1. Let Y=y1,y2,…,yr be the set of decision experts. These experts are invited to evaluate the alternatives under different attributes. Their weights are given by the vector ω=ω1,ω2,…,ωrT which satisfy the condition that 0≤ωb≤1b=1,2,…,r and ∑b=1rωb=1. The decision makers evaluate each attribute av of each alternative zu in the form of linguistic Fermatean fuzzy number γuvb=sξuvb,sηuvbu=1,2,…,pv=1,2,…,q; then, the decision matrix is formed as Rb=γuvbp×qb=1,2,…,r.The following steps are involved.Step 1.
Normalize each linguistic Fermatean fuzzy decision matrix provided by decision maker. The non-beneficial type of attributes is converted in to benefit type of attributes. The linguistic Fermatean fuzzy decision matrix is normalized as(72)γuvb=sξuvb,sηuvbfor benefit criterion,sηuvb,sξuvbfor cost criterion.Step 2.
Using linguistic Fermatean fuzzy weighted Hamy mean (LFFWHM) given in equation (25), construct the matrix of fused attributes given as(73)γ11γ12⋯γ1rγ21γ22⋯γ2r⋮⋮⋯⋮γp1γp2⋯γpr,whose entry γub is evaluated as follows:(74)γub=LFFWHMγu1b,γu2b,…,γuqb.Step 3.
Use the linguistic Fermatean fuzzy weighted Hamy mean (LFFWHM) given in Equation (25) to accommodate the fused attribute values of each alternative.(75)γu=LFFWHMγu1,γu2,…,γur.Step 4.
Find the score of each alternative [37].Step 5.
Rank the alternatives based on their score function.
The flowchart is given in Figure2.Figure 2
Flowchart of the proposed linguistic Fermatean fuzzy MAGDM method.


## 5.2. Numerical Example II: Green Supplier Selection

A construction company intends to select a supplier from four candidatesX=z1,z2,z3,z4. The supplier is selected on the basis of five aspects C=a1,a2,a3,a4,a5 which are production cost, production quality, supplier service performance, profile of supplier, and risk factor [38]. Attribute weights are given by the vector w=0.25,0.2,0.15,0.18,0.22T. Four experts ybb=1,2,3,4 evaluate every alternative on the basis of LTS S=s0=extremelypoor,s1=verypoor,s2=poor,s3=slightlypoor,s4=fair,s5=slightlygood,s6=good,s7=verygood,s8=extremlygood. The weight vector of the experts is ω=0.25,0.3,0.2,0.25T. The decision experts provide information in the form of LFFNs where sξ denotes the linguistic membership degree in LFFN which can be regarded as preference of decision maker and sη denotes the linguistic non-membership degree in LFFN which can be regarded as non-preference of decision maker. The normalized decision matrices are shown in Tables 9–12.Table 9
Decision matrixR1.
a1a2a3a4a5z1s7,s1s6,s2s4,s3s7,s1s5,s2z2s6,s2s5,s2s6,s1s6,s2s7,s1z3s6,s1s5,s3s7,s1s5,s1s3,s4z4s5,s2s7,s1s4,s3s6,s1s4,s4Table 10
Decision matrixR2.
a1a2a3a4a5z1s7,s1s4,s4s6,s2s5,s2s3,s5z2s7,s1s5,s1s6,s1s5,s2s4,s3z3s5,s2s6,s1s7,s1s5,s3s4,s4z4s6,s2s4,s3s5,s2s7,s1s5,s3Table 11
Decision matrixR3.
a1a2a3a4a5z1s6,s1s5,s2s3,s4s7,s1s5,s2z2s7,s1s6,s2s7,s1s6,s2s5,s1z3s5,s3s5,s2s6,s1s4,s3s3,s1z4s6,s2s7,s1s5,s1s5,s2s4,s4Table 12
Decision matrixR4.
a1a2a3a4a5z1s5,s3s4,s4s7,s1s5,s1s4,s2z2s6,s1s7,s1s6,s1s5,s2s6,s1z3s5,s2s3,s4s6,s2s3,s3s5,s2z4s4,s3s5,s1s4,s2s6,s2s5,s2


## 5.3. Calculation and Decision Making of Numerical Example II

Step 1.
Normalize the decision matrices provided by decision makers. In this way, cost attributes are converted into benefit type attributes. The decision matrices are shown in Tables13–16.Table 13
Normalized decision matrixR1.
a1a2a3a4a5z1s1,s7s6,s2s4,s3s7,s1s2,s5z2s2,s6s5,s2s6,s1s6,s2s1,s7z3s1,s6s5,s3s7,s1s5,s1s1,s3z4s1,s5s7,s1s4,s3s6,s1s4,s4Table 14
Normalized decision matrixR2.
a1a2a3a4a5z1s1,s7s4,s4s6,s2s5,s2s5,s3z2s1,s7s5,s1s6,s1s5,s2s3,s4z3s2,s5s6,s1s7,s1s5,s3s4,s4z4s2,s6s4,s3s5,s2s7,s1s3,s5Table 15
Normalized decision matrixR3.
a1a2a3a4a5z1s1,s6s5,s2s3,s4s7,s1s2,s5z2s1,s7s6,s2s7,s1s6,s2s1,s5z3s3,s5s5,s2s6,s1s4,s3s1,s3z4s2,s6s7,s1s5,s1s5,s2s4,s4Table 16
Normalized decision matrixR4.
a1a2a3a4a5z1s3,s5s4,s4s7,s1s5,s1s2,s4z2s1,s6s7,s1s6,s1s5,s2s1,s6z3s2,s5s3,s4s6,s2s3,s3s2,s5z4s3,s4s5,s1s4,s2s6,s2s2,s5Step 2.
Using LFFWHM operator given in equation (25), the alternatives are fused corresponding to each attribute provided by decision maker. The matrix R of fused values is shown as(76)R=s6.87,s2.47s6.95,s2.39s7.01,s1.86s6.95,s2.01s6.85,s2.49s6.87,s2s6.9,s2.33s6.83,s2.14s7.01,s1.86s7.18,s1.85s6.79,s1.82s2.56,s2.43s7.14,s1.83s6.97,s2.29s7.13,s1.99s6.92,s1.81.Step 3.
Use LFFWHM operator given inequation (25) to aggregate the fused values of each alternative provided by decision maker. The aggregated values are shown as follows: z1 =  s7.71,s1.47 , z2 =  s7.7,s1.41z3 =  s7.71,s1.26, z4 =  s7.75,s1.26.Step 4.
The score function of alternatives is calculated and is shown in Table17.Step 5.
Rank the alternatives based on score function. The ranking is given as(77)z4>z3>z1>z2.
It is determined that alternativez4 is the best candidate and alternative z2 is the worst candidate.Table 17
Score function.
AlternativeSzuz1s7.846z2s7.845z3s7.85z4s7.87


## 5.4. Sensitivity Analysis of Numerical Example I, II

Using the COPRAS method with LFFWHM and LFFWDHM operators, the ranking results for different values of parameters are calculated. The parametery plays a significant role. The ranking results for different values of parameter with LFFWHM operator are the same. Here z1 is still the best food company for different values of parameter and z4 is the worst food company, while with LFFWDHM operator, the ranking results are still the same. z1 is best food company, and z4 is the worst company. The ranking results while using COPRAS method with LFFWHM operator and with LFFWDHM operator are calculated in Tables 18 and 19, and results are shown in Figures 3 and 4, respectively.Table 18
Order for different parameters with LFFWHM operator using the COPRAS method.
Parameter valuesRelative priorityOrderQz1Qz2Qz3Qz4y=1477.174464.88464.26450.81z1>z2>z3>z4y=2469.19458.21455.09439.15z1>z2>z3>z4y=3224.67185.29168.05137.23z1>z2>z3>z4Table 19
Order for different parameters with LFFWDHM operator using the COPRAS method.
Parameter valuesRelative priorityOrderQz1Qz2Qz3Qz4y=1−397.45−412.47−410.29−417.88z1>z3>z2>z4y=2−386.05−396.13−401.24−409.39z1>z3>z2>z4y=3−48.646−67.29−58.816−70.421z1>z3>z2>z4Figure 3
Ranking results using an extended COPRAS method with LFFWDHM.Figure 4
Ranking results using the COPRAS method with LFFWHM.In MAGDM approach with LFFWHM operator, the parametery plays a significant role. For different values of y, we can see that the ranking values are the same. For different parameters, the superlative supplier is z4 and the worst supplier is still z2. The ranking values for different values of y are shown in Table 20, and the results are shown in Figure 5.Table 20
Order for different parameters with LFFWHM operator.
Parameter valuesScore functionOrderSz1Sz2Sz3Sz4y=1s7.893s7.892s7.902s7.904z4>z3>z1>z2y=2s7.846s7.845s7.85s7.87z4>z3>z1>z2y=3s7.82s7.81s7.83s7.84z4>z3>z1>z2y=4s7.81s7.79s7.82s7.85z4>z3>z1>z2Figure 5
Effect of parameters with LFFWHM.


## 6. Comparative Analysis

In this section, we conduct a comparative analysis of our proposed extended COPRAS method and proposed MAGDM approach based on LFFWHM operator with Lin et al.’s MAGDM approach [37] based on LqROFOWA, LqROFOWG, and LqROFIWPGHM operators q=3,e=1,f=1 and Liu et al.’s MAGDM approach [12] based on aggregation operators, including LqROFPWAG and LqROFPWGG operators q=3,ε=0.4,ζ=0.5, to clarify the validity of our proposed approaches.(i)
It is clear from the first application that if an extended version of the COPRAS method based on the aggregation operators, namely, LFFWHM and LFFWDHM operators, is being used, the best food company isz1 and the worst food company is z4. It is clear from Table 21 that the ranking values of alternatives with the LFFWHM, LqROFOWG, and LqROFPWGG operators are z1>z2>z3>z4 while those with the LFFWDHM, LqROFOWA, and LqROFPIWHM operators are z1>z3>z2>z4. In all cases, the finest alternative is z1 and the unexceptional alternative is z4. This method is effective as the superlative alternative with all approaches is x1.(ii)
In the second application, the ranking result produced by the proposed MAGDM approach based on LFFWHM operator isz4>z3>z1>z2. We compare the proposed MAGDM approach with Lin et al.’s approach [37] based on the LqROFOWA, LqROFOWG, and LqROFPIWHM operators, and we use q=3 which degrades the LqROFOWA, LqROFOWG, and LqROFPIWHM operators into LFFOWA, LFFOWG, and LFFPIWHM operators. It is clear from Table 22 that best green supplier is x4, while compared with Liu et al.’s MAGDM approach [12], LqROFPWAG and LqROFPWGG operators are diversified into LFFPWAG and LFFPWGG operators, and the most estimated green supplier is z4. The proposed MAGDM method is flexible as the superlative alternative with our proposed method and all existing approaches is z4. The ranking results are described in Figure 6.Table 21
Ranking results by different methods with the COPRAS method.
MethodsRankingLFFWHMz1>z2>z3>z4LFFDWHMz1>z3>z2>z4LFFOWA [37]z1>z3>z2>z4LFFOWG [37]z1>z2>z3>z4LFFIWPGHM [37]z1>z3>z2>z4LFFPWAG [12]z1>z3>z2>z4LFFPWGG [12]z1>z2>z3>z4Table 22
Ranking results by different methods with LFFHM operator.
MethodsScore functionOrderSz1Sz2Sz3Sz4LFFWHMs7.846s7.845s7.85s7.87z4>z3>z1>z2LFFWA [37]s5.97s5.78s6.25s6.29z4>z3>z1>z2LFFWG [37]s5.98s5.83s6.25s6.3z4>z3>z1>z2LFFIWPGHM [37]s6.3491s6.3494s6.350s6.351z4>z3>z1>z2LFFPWAG [12]s6.398s6.402s6.392s6.406z4>z2>z1>z3LFFPWGG [12]s6.298s6.284s6.32s6.33z4>z3>z1>z2Figure 6
Ranking results with LFFWHM for different methods.


### 6.1. Discussion

(i)
The linguistic Fermatean fuzzy Hamy mean (LFFHM) operators model the connection between all parameters excellently. The LFFHM operators are applied more flexibly to fuse the fuzzy information.(ii)
The LFFHM operators deal with only one parameter which is easy to use and gives same ranking results for different parameters, whereas other operators such as Lq-ROFOWA, L q-ROFOWG, and L q-ROFIWPGHM operators [37] involve more than one parameter. The ranking results obtained with LFFWHM operators are the same with different parameters while by changing the value of parameters in L q-ROFOWA, L q-ROFOWG, and L q-ROFIWPGHM operators, the ranking results are altered.(iii)
The proposed MAGDM approach involves less calculations as compared to other approaches. The ranking results obtained by the proposed method are powerful and more flexible to use.


## 6.1. Discussion

(i)
The linguistic Fermatean fuzzy Hamy mean (LFFHM) operators model the connection between all parameters excellently. The LFFHM operators are applied more flexibly to fuse the fuzzy information.(ii)
The LFFHM operators deal with only one parameter which is easy to use and gives same ranking results for different parameters, whereas other operators such as Lq-ROFOWA, L q-ROFOWG, and L q-ROFIWPGHM operators [37] involve more than one parameter. The ranking results obtained with LFFWHM operators are the same with different parameters while by changing the value of parameters in L q-ROFOWA, L q-ROFOWG, and L q-ROFIWPGHM operators, the ranking results are altered.(iii)
The proposed MAGDM approach involves less calculations as compared to other approaches. The ranking results obtained by the proposed method are powerful and more flexible to use.


## 7. Conclusion

The theory of linguistic term sets performs a significant role in linguistic decision making, when decision experts express their preferences in qualitative form. In this article, we have proposed the LFFSs by the amalgamation of FFSs with LFSs. Firstly, we have presented a formal definition of LFFSs along with related operations of LFFNs. We have devised several Hamy mean operators to fuse LFFNs, which are called the LFFHM, LFFDHM, LFFWHM, and LFFWDHM operators. We have discussed some fundamental characteristics of these operators. We have extended the conventional COPRAS method to handle MAGDM problems based on LFFSs. Furthermore, we have developed another approach for solving MAGDM problems with linguistic Fermatean fuzzy information. Two examples regarding the ranking of food companies and the selection of green suppliers were presented to demonstrate the proposed methods. Finally, we compared our proposed methods with the existing approaches to exhibit the supremacy and potency of our methods. In the future, we plan to extend our research work by developing more MAGDM strategies: (i) linguistic Fermatean fuzzy AHP method, (ii) linguistic Fermatean fuzzy TOPSIS method, (iii) linguistic Fermatean fuzzy PROMETHEE method, (iv) linguistic Fermatean fuzzy ELECTRE I, (v) linguistic Fermatean fuzzy ELECTRE II method, (vi) linguistic Fermatean fuzzy ELECTRE III method, (vii) linguistic Fermatean fuzzy ELECTRE IV method, and (viii) linguistic Fermatean fuzzy MULTIMOORA method.



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