Novelty of Different Distance Approach for Multi-Criteria Decision-Making Challenges Using q-Rung Vague Sets

Abbreviations

1  Introduction

Decision-makers find it increasingly difficult to identify the optimal solution as real-world systems become increasingly complex. Selecting the best option is possible difficulty of deciding between the alternatives. Opportunities, objectives, and viewpoint constraints are challenging to create for many firms. In line with this, when decisions making (DM), individuals or groups should consider multiple objectives at the same time. A wide variety of MADM-related issues are dealt with every day. Our DM abilities need to be improved as a result. This field of study has been studied by a variety of researchers using a variety of methods. There are several uncertain theories proposed by them to deal with the uncertainties, including fuzzy set (FS) [1], intuitionistic fuzzy set (IFS) [2], interval valued FS (IVFS) [3], vague set [4], Pythagorean fuzzy set (PFS) [5], IVPFS [6], spherical FS (SFS) [7]. A membership grade (MG) indicates how well an FS fits into the specified set with ranging from 0 to one. An IFS was defined by Atanassov [2] as having a total of membership grade (MG) and non-membership grade (NMG) less than one. The sum of the MG and NMG is sometimes greater than one when a DM method is applied. Yager [5] developed PFS, which is characterized by a square sum of its MG and NMG not exceeding one. In order to generalize IFS, Yager used PFS to build a model. A new concept has been proposed by Yager [8] in light of society’s continuous complexity and theory development. The MG and NMG in the q-rung orthogonal pair FS (q-ROFS) have power q, but the sum can never exceed one. The IFSs and PFSs can all be considered special cases of q-ROFSs, therefore they are general. There are more orthopairs that meet the bounding constraint as rung q increases, and as rung q increases, the space of acceptable orthopairs increases. The use of q-ROFSs can thus express fuzzy information in a broader range. Because the parameter q can be adjusted, q-ROFSs are flexible and better suited to uncertain environments. An increase in q can be made as ambiguity in decision information increases. It is possible that some experts are influenced by both their own desires and their surroundings. Therefore, they may have an MDG of 0.95 and an NMG of 0.55 when evaluating certain decision-making things. The fuzzy information cannot be described by IFNs and PFNs, but q-ROFNs can be described if parameter q is increased. Due to this, the q-ROFS is more flexible and suitable for describing uncertain data.

A discussion of the q-Rung Orthopair fuzzy weighted Archimedean Bonferroni mean (q-ROFWABM) and q-Rung Orthopair fuzzy Archimedean Bonferroni mean (q-ROFABM) operators was given in Liu et al. [9]. Liu et al. [10] proposed a concept of q-rung orthopair fuzzy power averaging (q-ROFPA), q-rung orthopair fuzzy power weighted average (q-ROFPWA), q-ROFPMSM and q-rung orthopair fuzzy power weighted MSM (q-ROFPWMSM) operators for q-ROFNs, describing their properties. Liu et al. [11] discussed the q-rung orthopair fuzzy weighted average (q-ROFWA) and q-rung orthopair fuzzy weighted geometric (q-ROFWG) operators are introduced and their basic properties are discussed. The concept of an incomplete probabilistic linguistic preference relation (InPLPR) was introduced by Wang et al. [12]. In 2013, Liu et al. [13] presented the concept of unit cost consensus adjustment based on a group consensus decision model based on InPLPR that takes into account social trust networks, consistency, and social trust networks. In a recent study, Zhang et al. [14] discussed three types of multi-granularity q-rung orthopair fuzzy preference relations (PRS) as well as their interesting properties. With the MAGDM algorithm based on q-ROF multi-attribute rules, the MG-3WD approach can also be applied to q-ROF complex information systems. Zhang et al. [15] analyzed a UCI dataset using MGq-ROF PRSs, the MULTIMOORA method, and the TPOP method using the MAGDM method. Zhang et al. [16] discussed neutosophic fusion of RST based on basic models and soft sets models. Based on fuzzy granularity spaces with properties that correspond to fuzzy knowledge distances, Lian et al. [17] discussed fuzzy relative knowledge distances. Furthermore, it has been demonstrated that fuzzy knowledge distances contain different structure information than precise knowledge distances. The hybridization of archimedean copulas and generalized MSM operators, based on q-rung probabilistic dual hesitant fuzzy sets, was discussed by Anusha et al. [18]. Multi-attribute decision-making (MADM) [19,20] offers an efficient means of evaluating multiple alternatives based on their evaluation values. Usually, MADM problems can be solved in one of two ways. Traditional approaches, such as TOPSIS, VIKOR, ELECTRE, are examples. Information integration problems are more effectively solved by AOs than by traditional approaches. In contrast to traditional approaches, AOs provide comprehensive values of all alternatives, rather than ranking results only. In his article, Bairagi [21] used extended TOPSIS to select homogeneous groups of robotic systems.

This is insufficient for demonstrating neutrality (neither favor nor disfavor). It was developed by Cuong et al. [22] with a total grade no higher than one for three pointers such as positive, neutral, and negative. As a result, it would be appropriate for the DM method to use this set over IFS or PFS for selective applications. Liu et al. first presented the concept of an aggregation operator (AO) in generalized PFS [23]. A PIVFS algorithm for the problem of identifying truth membership grades (TMGs), indeterminacy membership grades (IMGs) and false membership grades (FMGs) with AOs [6,24–26] have the feature that the sum of the three grades (TMG, IMG, and FMG) is greater than one. It has been suggested by Ashraf et al. [7] that the SFS should contain the following graph: this diagram shows that the sum of the squares of the TMG, IMG and FMG should be not exceeds one. To analyze the idea of SFS, Fatmaa et al. [27] used the TOPSIS technique as part of their study. There have been several different concepts of q-Rung picture FS with AO for DM that have been demonstrated by Liu et al. [28]. In addition to Biswas [4], there is a concept of VSs VS is called to the two functions TMG Tv and FMG Fv as well as a set of transformations. Suppose that Tv(x) is the total likelihood estimate of x, derived from the evidence for x and Fv(x) is the total likelihood estimate for x derived from the evidence against x. It can be noted that these functions fall into the interval [0,1], where their sum is less than one. Various extensions have been made to the VS such as the IVFS and the FS [29–31]. Zhang et al. [32] was first introduced that suggested PFS can be extended to multi-criteria decision making (MCDM) using TOPSIS. The application of the bipolar fuzzy soft set (BFSS) was explored by Jana et al. [33] for the purpose of discovering how to broaden the set of bipolar fuzzy terms. Ullah et al. [34] described how pattern recognition applications can be used to estimate PFS distances using complex separation algorithms. It has been discussed that MCDM can be utilized using the neutrosophic set as well as the Dombi power AOs [35]. A number of algebraic structures and their applications have been investigated by Palanikumar et al. [36,37]. The notion of fuzzy c-number clustering procedures for fuzzy data was discussed by Yang et al. [38,39].

As an alternative to algebraic operations, a log q-rung arithmetic operation can provide a smooth estimate quality that is similar to that of a continuous algebraic operation when compared with its smoothness. Compared to the log q-rung arithmetic operations on the IFS and PFS only a limited research has been done on log q-rung arithmetic operations. Our method of VS is based on log q-rung arithmetic AOs within VSs rather than using log q-rung arithmetic operations. The use of spherical fuzzy q-rung arithmetic AOs based on entropy in DM, as well as their real-life application to the problem, were introduced by Jin et al. [40]. Ashraf et al. [41] proposed by that linear-logarithmic hybrid AOs be used for single-valued neutrosophic sets. Palanikumar et al. have examined the new type Pythagorean fuzzy set with AO [42]. Yager [5] has also presented an average and geometric AO using PFS weighted and weighted power cases. A number of basic PFS features were discussed by Peng et al. [43]. A generalized PFS under AO has been developed by Liu et al. [23]. Adak et al. dicussed the concept of spherical distance measurement method for solving MCDM problems under PFS [44]. Some picture fuzzy mean operators and their applications in DM is discussed by Hasan et al. [45]. Mishra et al. [46] discussed the new concept of Pythagorean and Fermatean fuzzy sub-group redefined in context of T-norm and S-conorm. DM analysis of minimizing the death rate due to COVID-19 by using q-rung orthopair fuzzy soft bonferroni mean operator discussed by Abbas et al. [47]. Yaman et al. [48] discussed the new approach for warehouse location decisions changed in medical sector after pandemic study. Recently, FS and its extension including q-rung orthopair fuzzy set, T-spherical fuzzy set based on decision making approach [49–55]. The log q-rung information about the VNS was obtained utilizing OAs. Section 2 explains the given information about the FS and VS components. Section 3 explains the definition of q-rung vague sets as well as the different operations involved with them. There is a discussion on ED and HD in Section 4 using the log q-rung vague normal number (log q-rung VNN). A MADM connection is established through the Section 5 using log q-rung VNNs. Section 6 contains a numerical example and a description of log q-rung VS as well as the insert algorithm and log q-rung VS application. We provide a conclusion in Section 7. An overview of the key things that were taken into account during the research process is given below:

1.   As a result of log-rung VNSs, ED and HD were introduced.

2.   The log q-rung VNVWA, log q-rung VNVWG, log Gq-rung VNVWA, and log Gq-rung VNVWG operators were suggestions.

3.   A log-rung VNS is used in order to explore the MADM technique.

4.   We evaluate log q-rung VNVWA, log q-rung VNVWG, log Gq-rung VNVWA and log Gq-rung VNVWG in order to establish optimal value parameters.

5.   An analysis of the proposed and early investigations is presented along with a comparative analysis.

6.   DM outcomes for natural numbers with a value of q.

2  Basic Concepts

We will introduce some basic literature on Pythagorean fuzzy set, Pythagorean interval-valued fuzzy set, vague set and spherical fuzzy set in the section, which will be useful in later section. Additionally, the basic operating rules and zero vague and unit vague set of these concepts are discussed.

Definition 2.1. [5] Let U be the universal, PFS M in U is M={ρ,⟨ηMT(ρ),ηMF(ρ)⟩|ρ∈U}, ηMT,ηMF:U→[0,1] denote MG and NMG of ρ∈U, respectively and 0⪯(ηMT(ρ))2+(ηMF(ρ))2⪯1. For M=⟨ηMT,ηMF⟩ is represents a Pythagorean fuzzy number (PFN).

Definition 2.2. [6] The PIVFS M in U is M={ρ,⟨ηMT~(ρ),ηMF~(ρ)⟩|ρ∈U}, where ηMT~,ηMF~:U→Int([0,1]) denote MG and NMG of ρ∈U, respectively, and 0⪯(ηMTU(ρ))2+(ηMFU(ρ))2⪯1.

For M=⟨[ηMTL,ηMTU],[ηMFL,ηMFU]⟩ is represent a PIVFN.

Definition 2.3. [4] (i) A VS M in U is a pair (MT,MF), MT,MF:U→[0,1] are mappings such that MT(ρ)+MF(ρ)⪯1,∀ρ∈U, MT and MF are called the TMG and FMG, respectively.

(ii) M(ρ)=[MT(ρ),1−MF(ρ)] is represent the vague value of ϱ in M.

Definition 2.4. [4] (i) A VS M is contained in VS M1, M⊆M1 if and only if M(ρ)⪯M1(ρ). That is, MT(ρ)⪯TM1(ρ) and 1−MF(ρ)⪯1−FM1(ρ),∀ρ∈U.

(ii) Union of M and M1, as X=M∪M1, TX=max{MT,TM1} and 1−FX=max{1−MF,1−FM1}=1−min{MF,FM1}.

(iii) Intersection of M and M1 as X=M∩M1, TX=min{MT,TM1} and 1−FX=min{1−MF,1−FM1}=1−max{MF,FM1}.

Definition 2.5. [4] A VS M of U,∀ρ∈U. Then

(i) MT(ρ)=0 and MF(ρ)=1 is represent a zero VS of U.

(ii) MT(ρ)=1 and MF(ρ)=0 is represent a unit VS of U.

Definition 2.6. [38] The fuzzy number M(x)=exp−(x−χ)2ψ2,(ψ>0) and M=(χ,ψ) is represent a normal fuzzy number (NFN), here R is a real numbers.

Definition 2.7. [39] Let L1=(χ1,ψ1)∈N and L2=(χ2,ψ2)∈N, (ψ1,ψ2>0), then their distance is Υ(L1,L2)=(χ1−χ2)2+12(ψ1−ψ2)2.

3  log q-rung NVN and Its Basic Operations

In this section, we develop some novel logarithmic operational laws for normal vague numbers and discuss their properties. The main purpose of this section is to propose novel logarithmic aggregation operators based on normal vague information. The logarithmic function offers versatility in supporting the decision experts choices during object appraisal due to its periodic and symmetric character. An important fundamental operation of the log q-rung normal number is defined.

Definition 3.1. The log VS M in U is M={ρ,⟨[log⁡MT(ρ),log⁡(M1−F(ρ))],

[log⁡MF(ρ),log⁡(M1−T(ρ))]⟩| ρ∈U}, ηMT~:U→Int([0,1]) and ηMF~:U→Int([0,1]) denote the TMG, IMG and FMG of ρ∈U to M, respectively and 0⪯(logΩi⁡M1−F(ρ))q+(logΩi⁡M1−T(ρ))q⪯1, where Ωi=∏[MT,M1−F],[MF,M1−T]. For, M=⟨[logΩi⁡MT,logΩi⁡(M1−F)],[logΩi⁡MF,logΩi⁡(M1−T)]⟩ is called a log q-rung vague number.

Definition 3.2. Let (χ,ψ)∈N, M=⟨(χ,ψ);[log⁡MT,log⁡(M1−F)],[log⁡MF,log⁡(M1−T)]⟩ be a log q-rung normal vague number (NVN). The TMG, IMG and FMG are defined as [logΩi⁡MT,logΩi⁡(M1−F)]=[logΩi⁡MT⋅exp−(x−χ)2ψ2,logΩi⁡(M1−F)⋅exp−(x−χ)2ψ2], [logΩi⁡MF,logΩi⁡(M1−T)]=[1−(1−logΩi⁡MF)⋅exp−(x−χ)2ψ2,1−(1−logΩi⁡(M1−T))⋅exp−(x−χ)2ψ2] respectively, [logΩ⁡MT,logΩ⁡(M1−F)] and

0⪯(logΩ⁡(M1−F)(ρ))q+(logΩ⁡(M1−T)(ρ))q⪯1, where Ω=∏[MT,M1−F],[MF,M1−T], where x ∈ X is a non-empty set.

Definition 3.3. Let M=⟨(χ,ψ);[log⁡MT,log⁡(M1−F)],[log⁡MF, log⁡(M1−T)]⟩ be the log q-rung NVN.

The score function of M is S(M)=χ2(X12+1−Z12)+ψ2(X22+1−Z22)2, where −1⪯S(M)⪯1.

The accuracy function of M is A(M)=χ2(X12+1+Z12)+ψ2(X22+1+Z22)2, where 0⪯A(M)⪯1.

where X1=(log⁡MT)q,Z1=(log⁡MF)q and X2=(log⁡(M1−F))q,Z2=(log⁡(M1−T))q.

Definition 3.4. Let M=⟨(χ,ψ);[log⁡MT,log⁡(M1−F)],[log⁡MF,log⁡(M1−T)]⟩,

M1=⟨(χ1,ψ1);[log⁡M1T,log⁡(M11−F)], [log⁡M1F,log⁡(M11−T)]⟩ and

M2=⟨(χ2,ψ2);[log⁡M2T,log⁡(M21−F)],[log⁡M2F,log⁡(M21−T)]⟩ be the three log q-rung NVNs, q is a natural number and Ω=∏[TMi,Mi1−F],[FMi,Mi1−T], then

1.   M1⨁M2=[(χ1+χ2,ψ1+ψ2);[(logΩi⁡M1T)q+(logΩi⁡M2T)q−(logΩi⁡M1T)q⋅(logΩi⁡M2T)qq,(logΩi⁡(M11−F))q+(logΩi⁡(M21−F))q−(logΩi⁡(M11−F))q⋅(logΩi⁡(M21−F))qq],[logΩi⁡M1F⋅logΩi⁡M2F,logΩi⁡(M11−T)⋅logΩi⁡(M21−T)]],

2.   M1⨂M2=[(χ1⋅χ2,ψ1⋅ψ2);[logΩi⁡M1T⋅logΩi⁡M2T,logΩi⁡(M11−F)⋅logΩi⁡(M21−F)],[(logΩi⁡M1F)q+(logΩi⁡M2F)q−(logΩi⁡M1F)q⋅(logΩi⁡M2F)qq,(logΩi⁡(M11−T))q+(logΩi⁡(M21−T))q−(logΩi⁡(M11−T))q⋅(logΩi⁡(M21−T))qq]],

3.   Θ⋅M=[(Θ⋅χ,Θ⋅ψ);[1−(1−(logΩi⁡MT)q)qq,1−(1−(logΩi⁡(M1−F))q)qq],[(logΩi⁡MF)q,(logΩi⁡(M1−T))q]],

4.   MΘ=[(χΘ,ψΘ);[(logΩi⁡MT)q,(logΩi⁡(M1−F))q],[1−(1−(logΩi⁡MF)q)qq,1−(1−(logΩi⁡(M1−T))q)qq]].

4  Distance between log q-rung Normal Vague Numbers

The Euclidean distance and Hamming distance are useful technique for calculating the distance between two elements, two sets, etc. In order to define the Euclidean distance and Hamming distance, first, we will define a distance measure. Basically, a distance measure has to accomplish the following properties. This study examined the mathematical properties of log q-rung NVNs and measured the expectation of ED and HD.

Definition 4.1. Let M1=⟨(χ1,ψ1);[log⁡M1T,log⁡(M11−F)],[log⁡M1F, log⁡(M11−T)]⟩ and

M2=⟨(χ2,ψ2);[log⁡M2T,log⁡(M21−F)],[log⁡M2F, log⁡(M21−T)]⟩ be any two log q-rung NVNs. Then ED between M1 and M2 is

ΥE(M1,M2)=12[(logΩi⁡M1T)2+1−(logΩi⁡M1F)2+(logΩi⁡(M11−F))2+1−(logΩi⁡(M11−T))22χ1−(logΩi⁡M2T)2+1−(logΩi⁡M2F)2+(logΩi⁡(M21−F))2+1−(logΩi⁡(M21−T))22χ2]2+12[(logΩi⁡M1T)2+1−(logΩi⁡M1F)2+(logΩi⁡(M11−F))2+1−(logΩi⁡(M11−T))22ψ1−(logΩi⁡M2T)2+1−(logΩi⁡M2F)2+(logΩi⁡(M21−F))2+1−(logΩi⁡(M21−T))22ψ2]2

and HD between M1 and M2 is defined as

ΥH(M1,M2)=

12[|(logΩi⁡M1T)2+1−(logΩi⁡M1F)2+(logΩi⁡(M11−F))2+1−(logΩi⁡(M11−T))22χ1−(logΩi⁡M2T)2+1−(logΩi⁡M2F)2+(logΩi⁡(M21−F))2+1−(logΩi⁡(M21−T))22χ2|+12|(logΩi⁡M1T)2+1−(logΩi⁡M1F)2+(logΩi⁡(M11−F))2+1−(logΩi⁡(M11−T))22ψ1−(logΩi⁡M2T)2+1−(logΩi⁡M2F)2+(logΩi⁡(M21−F))2+1−(logΩi⁡(M21−T))22ψ2|]

Theorem 4.1. Let M1=⟨(χ1,ψ1);[log⁡M1T,log⁡(M11−F)],[log⁡M1F, log⁡(M11−T)]⟩,

M2=⟨(χ2,ψ2);[log⁡M2T,log⁡(M21−F)],[log⁡M2F,log⁡(M21−T)]⟩ and

M3=⟨(χ3,ψ3);[log⁡M3T,log⁡(M31−F)],[log⁡M3F, log⁡(MT1−F)]⟩ be any three log q-rung NVNs, then

1.   ΥE(M1,M2)=0, if and only if M1=M2.

2.   ΥE(M1,M2)=ΥE(M2,M1).

3.   ΥE(M1,M3)⪯ΥE(M1,M2)+ΥE(M2,M3).

Corollary 4.1. Let M1=⟨(χ1,ψ1);[log⁡M1T,log⁡(M11−F)],[log⁡M1F, log⁡(M11−T)]⟩,

M2=⟨(χ2,ψ2);[log⁡M2T,log⁡(M21−F)],[log⁡M2F,log⁡(M21−T)]⟩ and

M3=⟨(χ3,ψ3);[log⁡M3T,log⁡(M31−F)],[log⁡M3F, log⁡(MT1−F)]⟩ be any three log q-rung NVNs. Then

1.   ΥH(M1,M2)=0 if and only if M1=M2.

2.   ΥH(M1,M2) = ΥH(M2,M1).

3.   ΥH(M1,M3)⪯ΥH(M1,M2)+ΥH(M2,M3).

5  log q-rung Normal Vague Number Using AOs

We will introduce the novel concepts of log q-rung NVWA, log q-rung NVWG, log G q-rung NVWA, and log G q-rung NVWG operators utilizing log q-rung NVN.

5.1 log q-rung NVWA Operator

Definition 5.1. Let Mi=⟨(χi,ψi);[log⁡MiT,log⁡Mi1−F)],[log⁡MiF, log⁡Mi1−T)]⟩ be the family of log q-rung NVNs, Ψ=(Ψ1,Ψ2,...,Ψn) be the weight of Mi, Ψi⪰0 and ⊎i=1nΨi=1 and

Ω=∏[TMi,Mi1−F],[FMi,Mi1−T], then log q-rung NVWA operator is log q-rung NVWA (M1,M2,...,Mn)=⊎i=1nΨiMi.

Theorem 5.1. Let Mi=⟨(χi,ψi);[log⁡MiT,log⁡Mi1−F)],[log⁡MiF,log⁡Mi1−T)]⟩ be the family of log q-rung NVNs, then log q-rung NVWA (M1,M2,...,Mn)=

[(⊎i=1nΨiχi,⊎i=1nΨiψi);[1−◯i=1n(1−(logΩi⁡MiT)q)Ψiq,1−◯i=1n(1−(logΩi⁡Mi1−F))q)Ψiq],[◯i=1n(logΩi⁡MiF)Ψi,◯i=1n(logΩi⁡Mi1−T))Ψi]].

Theorem 5.2. (idempotency property) If all Mi=⟨(χi,ψi);[log⁡MiT,log⁡Mi1−F)],

[log⁡MiF, log⁡Mi1−T)]⟩=M, then log q-rung NVWA (M1,M2,...,Mn)=M.

Theorem 5.3. (boundedness property) Let Mi=⟨(χij,ψij);[log⁡MijT,log⁡(Mij1−F)],

[log⁡MijF, log⁡(Mij1−T)]⟩ be the collection of log q-rung NVWA, where χ⏟=minχij, χ⏞=maxχij, ψ⏟=maxψij, ψ⏞=minψij,

logΩi⁡MT⏟=minlogΩi⁡MijT, logΩi⁡MT⏞=maxlogΩi⁡MijT, logΩi⁡(M1−F)⏟=minlogΩi⁡(Mij1−F),

logΩi⁡(M1−F)⏞=maxlogΩi⁡(Mij1−F), logΩi⁡MF⏟=minlogΩi⁡MijF, logΩi⁡MF⏞=maxlogΩi⁡MijF,

logΩi⁡(M1−T)⏟=minlogΩi⁡(Mij1−T), logΩi⁡(M1−T)⏞=maxlogΩi⁡(Mij1−T).

Then, ⟨(χ⏟,ψ⏟);[logΩi⁡MT⏟,logΩi⁡(M1−F)⏟],[logΩi⁡MF⏞,logΩi⁡(M1−T)⏞]⟩

⪯log <italic>q</italic>−rung NVWA(M1,M2,…,Mn)⪯〈(χ︷,ψ︷);[logΩiMT︷,logΩi(M1−F)︷],[logΩiMF︸,logΩi(M1−T)︸]〉.

where 1⪯i⪯n and j=1,2,...,ij.

Theorem 5.4. (monotonicity property) Let Mi=⟨(χtij,ψtij);[log⁡MtijT,log⁡(Mtij)1−F],

[log⁡MtijF, log⁡(Mtij1−T)]⟩ and Ψi=⟨(χhij,ψhij); [log⁡MhijT, log⁡(Mhij1−F)],

[log⁡MhijF,log⁡(Mhij1−T)]⟩ be the families of log q-rung NVWAs. For any i, if there is χtij⪯ψhij,

(logΩi⁡MtijT)2+(logΩi⁡Mtij1−F)2⪯(logΩi⁡MhijT)2+(logΩi⁡Mhij1−F)2 (logΩi⁡MtijF)2+(logΩi⁡Mtij1−T)2⪰(logΩi⁡MhijF)2+(logΩi⁡Mhij1−T)2 or Mi⪯𝒲i, then log q-rung NVWA (M1,M2,...,Mn)⪯q-Rung log q-rung NVWA (𝒲1,𝒲2,...,𝒲n).

5.2 log q-rung NVWG Operator

Definition 5.2. Let Mi=⟨(χi,ψi);[log⁡MiT,log⁡Mi1−F],[log⁡MiF, log⁡Mi1−T]⟩ be the family of log q-rung NVNs. Then log q-rung NVWG (M1,M2,...,Mn)=◯i=1nMiΨi.

Theorem 5.5. Let Mi=⟨(χi,ψi);[log⁡MiT,log⁡Mi1−F],[log⁡MiF,log⁡Mi1−T]⟩ be the family of log q-rung NVNs. Then log q-rung NVWG (M1,M2,...,Mn)=

[(◯i=1nχiΨi,◯i=1nψiΨi);[◯i=1n(logΩi⁡MiT)Ψi,◯i=1n(logΩi⁡Mi1−F))Ψi],[1−◯i=1n(1−(logΩi⁡MiF)q)Ψiq,1−◯i=1n(1−(logΩi⁡Mi1−T))q)Ψiq]].