Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

New algebraic structures such as q-rung interval-valued intuitionistic fuzzy set
applied to logarithm operator and its extension

G. Shanmugam1, M. Palanikumar2, Aiyared Iampan3,∗

1,2Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical
Sciences, Chennai-602105, India.

3Department of Mathematics, School of Science, University of Phayao, 19 Moo 2,Tambon Mae Ka, Amphur
Mueang, Phayao 56000, Thailand.

E-mails:1gsm.maths@gmail.com, 2palanimaths86@gmail.com, 3aiyared.ia@up.ac.th,
∗Corresponding author: Aiyared Iampan.

Received: 09-03-2024 Revised: 26-07-2024 Accepted: 16-09-2024.

Abstract

We introduce the interval-valued intuitionistic fuzzy set used with the q-rung logarithimic operator (q-LIVIFS).
A q-rung interval-valued intuitionistic fuzzy set may be developed by expanding the Pythagorean interval-
valued fuzzy set (PIVFS). We discuss the q-logarithimic operator applied to interval-valued intuitionistic fuzzy
weighted averaging (q-LIVIFWA), interval-valued intuitionistic fuzzy weighted geometric (q-LIVIFWG), ex-
tended q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted averaging (q-ELIVIFWA),
and extended q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted geometric (q-
ELIVIFWG). Several algebraic properties of q-LIVIFSs, such as distributivity, idempotency, and associativity,
have been identified.

Keywords: q-LIVIFWA; q-LIVIFWG; q-ELIVIFWA; q-ELIVIFWG.

1 Introduction

Several authors have contributed to this field of research using a variety of methods. Fuzzy set (FS),1 intuition-
istic FS (IFS),2 interval valued FS (IVFS),3 vague set (VS),4 Pythagorean FS (PFS),5 IVPFS,6 spherical FS
(SFS),7 neutrosophic set (NS)8 can all arise from uncertainties. There are several theories about uncertainty
that have been proposed. For example, fuzzy set (FS)1 has membership grade (MG) ranging from 0 to 1.
Atanassov2 developed an intuitionistic fuzzy set (IFS) where the two MGs of each component, positive β and
negative α, satisfy 0 ≤ β + α ≤ 1, for β, α ∈ [0, 1]. Yager5 invented Pythagorean fuzzy sets (PFS). With the
assumption that β + α ≥ 1 to β2 + α2 ≤ 1, they are distinguished by their MG and non-membership grade
(NMG). Numerous studies have been conducted on the use of PFSs and IFSs in different fields of research.
They are still not adept at explaining concepts. Because of this, the experts were still having problems inter-
preting the data in these sets and the supporting data. Cuong et al.9 developed the concept of a picture fuzzy
set as an alternative for this information.
As a result, it has been noted that the picture fuzzy set is an expanded version of IFS with more ambiguity han-
dling capabilities. As 0 ≤ β+α+ζ ≤ 1, it was noted that in the picture fuzzy set, MG β, NMG ζ, and neutral
grade α; for β, α, ζ ∈ [0, 1]. It will be feasible to ensure that expert assessments like ”yes,” ”abstain,” ”no,” and
”refusal” are expressed by following to the PFS definition. Furthermore, there will be consistency between the
outcome data and the actual decision-making environment, and no evaluative detail will be missed. Although
picture fuzzy sets have been used and studied extensively, less emphasis has been dedicated to their concept.
In their discussion of the new aggregating operator, Palanikumar et al.10-.15 Aggregation operators (AO) were
introduced in addition to extended PFS to handle issues with multiple truth membership values (TMD), false
membership values (FMD), and indeterminacy membership values (IMD) that are handled by the DM method,
as explained by Liu et al.16 The novel aggregating operators have been studied by Palanikumar et al.17-.18 The
concept of neutrosophic sets was proposed by Smarandache et al.19 The capacity for impartial thought is one
of the primary differences between FS and IFS. Neutroposophy is the study of neutral cognition. TMD, IMD,
and FD are used in this reasoning to calculate a value for every proposition. A strategy for MCDM under

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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

interval NS based on AOs was given by.20 This universe has all elements with values between 0 and 1, totaling
1. In several applications, AOs and their algebraic structures are explained by Palanikumar et al.21

Xu et al.22 proposed using various IFS averaging operators to manage IFS data. In addition, weighted, ordered
weighted, and hybrid geometric operators based on IFSs were developed by Xu et al.23 In order to construct
the IF ordered weighted distance (IFOWD) operators that are covered in25 by Zeng et al., Li et al.24 introduced
the generalized ordered weighted averaging (GOWA) operators. Peng et al.26 used AOs to study the basic
characteristics of PFS. Spherical fuzzy Dombi AOs have recently been suggested by Ashraf et al.27 This new
power-moirhead mean was defined in the SFS in 2022 by Temel et al.,28 and it also applies to MADM. New
aggregating operators have been developed recently by.29–31 Section 1 has an introduction. Section 2 for infor-
mation on FS and PNS. Section 3 contains a definition and an explanation of some of the uses for q-LIVIFNs.
The article draws two main conclusions: 1) It has been shown that the q-rungs of LIVIFS exhibit the following
algebraic properties: distributive, idempotent and associative. (2) We examine the q-LIVIFWA, q-LIVIFWG,
q-ELIVIFWA and q-ELIVIFWG.

2 Preliminaries

In this section, we will go over PFS and PIVFS concepts.

Definition 2.1. 5 Let X be the universal set. The PFS M =
{
u,
〈
µT
M (u), µF

M (u)
〉∣∣u ∈ X

}
, µT

M : X →
(0, 1) and µF

M : X → (0, 1) denotes MD and NMD of u ∈ X to M , respectively and 0 ≤ (µT
M (u))2 +

(µF
M (u))2 ≤ 1. For convenience, M =

〈
µT
M , µ

F
M

〉
is called the Pythagorean fuzzy number(PFN).

Definition 2.2. 6 The Pythagorean IVFS (PIVFS) M =
{
u,
〈
µ̃T
M (u), µ̃F

M (u)
〉∣∣∣u ∈ X

}
, where µ̃T

M : X →

Int((0, 1)) and µ̃F
M : X → Int((0, 1)) denotes MD and NMD of u ∈ X to M , respectively, and 0 ≤

(µT +
M (u))2 + (µF+

M (u))2 ≤ 1. For convenience, M =
〈(
µT −
M , µT +

M

)
,
(
µF−
M , µF+

M

)〉
is called the PIVFN.

Definition 2.3. The Pythagorean NSM =
{
u,
〈
µT
M (u), µIM (u), µF

M (u)
〉∣∣u ∈X

}
, where µT

M : X → (0, 1),

µIM : X → (0, 1) and µF
M : X → (0, 1) denotes TMD, IMD and FMD of u ∈ X to M , respectively and

0 ≤ (µT
M (u))2+(µIM (u))2+(µF

M (u))2 ≤ 2. For convenience,M =
〈
µT
M , µ

I
M , µ

F
M

〉
is called the Pythagorean

neutrosophic number.

3 q-LIVIFN and its fundamental operations

The q-LIVIFN has several intriguing basic operations connected to it.

Definition 3.1. The q-LIVIFS M =
{
u,
〈(
logT l

M (u), log(T u
M (u))

)
(
logF l

M (u), log(Fu
M (u))

)〉∣∣∣u ∈ X
}

, µ̃T
M : X → Int((0, 1)) and µ̃F

M : X → Int((0, 1)) denotes TMD
and FMD of u ∈X toM , respectively and 0 ≤ (log∏

i
T u

M (u))q+(log∏
i
Fu

M (u))q ≤ 1, where q are positive
integers and

∏
= �(T l

M ,T
u
M ), (F l

M ,F
u
M ).

For convenience, M =
〈(
logT l

M , log(T
u
M )
)
,
(
logF l

M , log(F
u
M )
)〉

is called the q-LIVIFN, where q ≥ 1.

Definition 3.2. Let M =
〈(
logT l

M , log(T
u
M )
)
,
(
logF l

M , log(F
u
M )
)〉

be the q-LIVIFN, the score function

of M is defined as S(M) = S1(M)+S2(M)
2 , −1 ≤ S(M) ≤ 1. where

S1(M) =

(
L1

2
+ 1− L2

2

)
,S2(M) =

(
L1

2
+ 1− L2

2

)
,

The accuracy function of M is A(M) = A1(M)+A2(M)
2 , where 0 ≤ A(M) ≤ 1.

A1(M) =

(
L1

2
+ 1 +

L2

2

)
,A2(M) =

(
L1

2
+ 1 +

L2

2

)
,

where L1 = (log∏
i
T l

M )2 + (log∏
i
(T u

M ))2, L2 = (log∏
i
F l

M )2 + (log∏
i
(Fu

M ))2

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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

Definition 3.3. Let M =
〈
(logT l

M , log(T
u
M )), (logF l

M , log(F
u
M ))

〉
,

B =
〈
(logT l

B , log(T
u
B )), (logF l

B , log(F
u
B))
〉

and C =
〈
(logT l

C , log(T
u
C )), (logF l

C , log(F
u
C))
〉

be any three q-LIVIFNs and∏
= �(TMi

,T u
Mi

), (FMi
,Fu

Mi
). Their following operations are defined as follows:

1. B
∨
C =

 q

√
(log∏

i
T l

B)
q + (log∏

i
T l

C)
q − (log∏

i
T l

B)
q · (log∏

i
T l

C)
q,

q

√
(log∏

i
(T u

B ))q + (log∏
i
(T u

C ))q − (log∏
i
(T u

B ))q · (log∏
i
(T u

C ))q

 ,(
log∏

i
(F l

B)
q · log∏

i
(F l

C)
q, log∏

i
(Fu

B)
q · log∏

i
(Fu

C)
q
)

 ,

2. B
∧
C =

(
log∏

i
(T l

B)
q · log∏

i
(T l

C)
q, log∏

i
(T u

B )q · log∏
i
(T u

C )q
)
, q

√
(log∏

i
F l

B)
q + (log∏

i
F l

C)
q − (log∏

i
F l

B)
q · (log∏

i
F l

C)
q,

q

√
(log∏

i
(Fu

B))
q + (log∏

i
(Fu

C))
q − (log∏

i
(Fu

B))
q · (log∏

i
(Fu

C))
q


 ,

3. ℵ ·M =

( q

√
1−

(
1− (log∏

i
T l

M )q
)ℵ
, q

√
1−

(
1− (log∏

i
(T u

M ))q
)ℵ )

,(
(log∏

i
F l

M )qℵ, (log∏
i
(Fu

M ))qℵ
)

 ,

4. Mℵ =


(
(log∏

i
T l

M )qℵ, (log∏
i
(T u

M ))qℵ
)
,(

q

√
1−

(
1− (log∏

i
F l

M )q
)ℵ
, q

√
1−

(
1− (log∏

i
(Fu

M ))q
)ℵ )

 .

4 q-LIVIFS concept

The weighed averaging operators for q-LIVIFN are given, based on the operational rules of q-LIVIFNs.

4.1 q-LIVIF weighted averaging( q-LIVIFWA) operator

Definition 4.1. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs, ω =

(ω1, ω2, ..., ωn) be the weight of Mi, ωi ≥ 0 and �n
i=1ωi = 1 and∏

= �(T l
Mi
,T u

Mi
), (FMi

,Fu
Mi

). Then q-LIVIFWA operator is q-LIVIFWA (M1,M2, ...,Mn) = �n
i=1ωiMi

for i = 1, 2, ..., n.

Theorem 4.2. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs. Then

q-LIVIFWA (M1,M2, ...,Mn) = (associativity property).
(

q

√
1−�n

i=1

(
1− (log∏

i
T l

M i)
q
)ωi

, q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mi))
q
)ωi

)
,(

�n
i=1 (log

∏
i
F l

M i)
qωi ,�n

i=1(log
∏

i
(Fu

Mi))
qωi

)
 .

Proof. If n = 2, then q-LIVIFWA (M1 = B,M2 = C) = ω1B
∨
ω2C, where

ω1B =


(

q

√
1−

(
1− (log∏

i
T l

B)
q
)ω1

, q

√
1−

(
1− (log∏

i
(T u

B ))q
)ω1

)
,(

(log∏
i
F l

B)
qω1 , (log∏

i
(Fu

B))
qω1

)


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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

and

ω2C =


(

q

√
1−

(
1− (log∏

i
T l

C)
q
)ω2

, q

√
1−

(
1− (log∏

i
(T u

C ))q
)ω2

)
,(

(log∏
i
F l

C)
qω2 , (log∏

i
(Fu

C))
qω2

)
 .

Hence,

ω1B
∨

ω2C =




q

√√√√√
(
1−

(
1− (log∏

i
T l

B)
q
)ω1
)
+
(
1−

(
1− (log∏

i
T l

C)
q
)ω2
)

−
(
1−

(
1− (log∏

i
T l

B)
q
)ω1
)
·
(
1−

(
1− (log∏

i
T l

C)
q
)ω2
)
,

q

√√√√√
(
1−

(
1− (log∏

i
(T u

B ))q
)ω1
)
+
(
1−

(
1− (log∏

i
(T u

C ))q
)ω2
)

−
(
1−

(
1− (log∏

i
(T u

B ))q
)ω1
)
·
(
1−

(
1− (log∏

i
(T u

C ))q
)ω2
)


,

(
(log∏

i
F l

B)
qω1 · (log∏

i
F l

C)
qω2 , (log∏

i
(Fu

B))
qω1 · (log∏

i
(Fu

C))
qω2

)



=




q

√
1−

(
1− (log∏

i
T l

B)
q
)ω1

·
(
1− (log∏

i
T l

C)
q
)ω2

,

q

√
1−

(
1− (log∏

i
(T u

B ))q
)ω1

·
(
1− (log∏

i
(T u

C ))q
)ω2

 ,

(
(log∏

i
F l

B)
qω1 · (log∏

i
F l

C)
qω2 , (log∏

i
(Fu

B))
qω1 · (log∏

i
(Fu

C))
qω2

)

 .

Thus, q-LIVIFWA (M1 = B,M2 = C)

=


(

q

√
1−�n

i=1

(
1− (log∏

i
T l

M i)
q
)ωi

, q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mi))
q
)ωi

)
,(

�n
i=1 (log

∏
i
F l

M i)
qωi ,�n

i=1(log
∏

i
(Fu

Mi))
qωi

)
 .

It is valid for n = m and m ≥ 3.
Hence, q-LIVIFWA (M1,M2, ...,Mm)

=


(

q

√
1−�m

i=1

(
1− (log∏

i
T l

M i)
q
)ωi

, q

√
1−�m

i=1

(
1− (log∏

i
(T u

Mi))
q
)ωi

)
,(

�m
i=1 (log

∏
i
F l

M i)
qωi ,�m

i=1(log
∏

i
(Fu

Mi))
qωi

)
 .

If n = l + 1 and we apply, q-LIVIFWA (M1,M2, ...,Mm,Mm+1)

=




q

√√√√√ �m
i=1

(
1−

(
1− (log∏

i
T l

M i)
q
)ωi
)
+
(
1−

(
1− (log∏

i
T l

Mm+1)
q
)ωm+1

)
−�m

i=1

(
1−

(
1− (log∏

i
T l

M i)
q
)ωi
)
·
(
1−

(
1− (log∏

i
T l

Mm+1)
q
)ωm+1

)
,

q

√√√√√ �m
i=1

(
1−

(
1− (log∏

i
(T u

Mi))
q
)ωi
)
+
(
1−

(
1− (log∏

i
(T u

Mm+1))
q
)ωm+1

)
−�m

i=1

(
1−

(
1− (log∏

i
(T u

Mi))
q
)ωi
)
·
(
1−

(
1− (log∏

i
(T u

Mm+1))
q
)ωm+1

)


,

(
�m

i=1 (log
∏

i
F l

M i)
qωi · (log∏

i
F l

Mm+1)
qωm+1 ,�m

i=1(log
∏

i
(Fu

Mi))
qωi · (log∏

i
(Fu

Mm+1))
qωm+1

)



=


(

q

√
1−�m+1

i=1

(
1− (log∏

i
T l

M i)
q
)ωi

, q

√
1−�m+1

i=1

(
1− (log∏

i
(T u

Mi))
q
)ωi

)
,(

�m+1
i=1 (log∏

i
F l

M i)
qωi ,�m+1

i=1 (log∏
i
(Fu

Mi))
qωi

)
 .

Theorem 4.3. (idempotency property) If allMi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
(i = 1, 2, ..., n)

are equal and Mi =M . Then q-LIVIFWA (M1,M2, ...,Mn) =M .

Proof. Given that (logT l
M i, logT

u
Mi) = (logT l

M , log(T
u
M )) and (logF l

M i, logF
u
Mi) = (logF l

M , log(F
u
M )),

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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
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for i = 1, 2, ..., n and �n
i=1ωi = 1. Now, q-LIVIFWA (M1,M2, ...,Mn)

=


(

q

√
1−�n

i=1

(
1− (log∏

i
T l

M i)
q
)ωi

, q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mi))
q
)ωi

)
,(

�n
i=1 (log

∏
i
F l

M i)
qωi ,�n

i=1(log
∏

i
(Fu

Mi))
qωi

)


=

( q

√
1−

(
1− (log∏

i
T l

M )q
)�n

i=1ωi

, q

√
1
(
1− (log∏

i
(T u

M ))q
)�n

i=1ωi

)
,(

(log∏
i
F l

M )�
n
i=1qωi , (log∏

i
(Fu

M ))�
n
i=1qωi

)


=


(

q

√
1−

(
1− (log∏

i
T l

M )q
)
, q

√
1−

(
1− (log∏

i
(T u

M ))q
))

,(
(log∏

i
F l

M )q, (log∏
i
(Fu

M ))q
)


= M.

Theorem 4.4. Let Mi =
〈
(logT l

M ij , log(T
u
Mij)), (logF

l
M ij ,log(F

u
Mij))

〉
(i = 1, 2, ..., n); (j = 1, 2, ..., ij)

be the collection of q-LIVIFWA, where log∏
i
T l

M︸ ︷︷ ︸ = min log∏
i
T l

M ij ,
︷ ︸︸ ︷
log∏

i
T l

M = max log∏
i
T l

M ij , log
∏

i
(T u

M )︸ ︷︷ ︸ =
min log∏

i
(T u

Mij),
︷ ︸︸ ︷
log∏

i
(T u

M ) = max log∏
i
(T u

Mij), log
∏

i
F l

M︸ ︷︷ ︸ = min log∏
i
F l

M ij ,
︷ ︸︸ ︷
log∏

i
F l

M = max log∏
i
F l

M ij ,

log∏
i
(Fu

M )︸ ︷︷ ︸ = min log∏
i
(Fu

Mij),
︷ ︸︸ ︷
log∏

i
(Fu

M ) = max log∏
i
(Fu

Mij).

Then,
〈
(log∏

i
T l

M︸ ︷︷ ︸, log∏i
(T u

M )︸ ︷︷ ︸), (
︷ ︸︸ ︷
log∏

i
F l

M ,
︷ ︸︸ ︷
log∏

i
(Fu

M ))
〉

≤ new type LIV IFWA(M1,M2, ...,Mn)

≤
〈
(
︷ ︸︸ ︷
log∏

i
T l

M ,
︷ ︸︸ ︷
log∏

i
(T u

M )), (log∏
i
F l

M︸ ︷︷ ︸, log∏i
(Fu

M )︸ ︷︷ ︸)
〉
.

where 1 ≤ i ≤ n, j = 1, 2, ..., ij , (boundedness property).

Proof. Since, log∏
i
T l

M︸ ︷︷ ︸ = min log∏
i
T l

M ij ,
︷ ︸︸ ︷
log∏

i
T l

M = max log∏
i
T l

M ij log
∏

i
(T u

M )︸ ︷︷ ︸ = min log∏
i
(T u

Mij),︷ ︸︸ ︷
log∏

i
(T u

M ) = max log∏
i
(T u

Mij) and log∏
i
T l

M︸ ︷︷ ︸ ≤ log∏i
T l

M ij ≤
︷ ︸︸ ︷
log∏

i
T l

M and log∏
i
(T u

M )︸ ︷︷ ︸ ≤ log∏i
(T u

M )ij ≤︷ ︸︸ ︷
log∏

i
(T u

M ).
Now, log∏

i
T l

M︸ ︷︷ ︸+ log∏
i
(T u

M )︸ ︷︷ ︸
= q

√
1−�n

i=1

(
1− (log∏

i
T l

M︸ ︷︷ ︸)q
)ωi

+ q

√
1−�n

i=1

(
1− (log∏

i
(T u

M )︸ ︷︷ ︸)q
)ωi

≤ q

√
1−�n

i=1

(
1− (log∏

i
T l

M ij)
q
)ωi

+ q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mij))
q
)ωi

≤ q

√
1−�n

i=1

(
1− (

︷ ︸︸ ︷
log∏

i
T l

M )q
)ωi

+ q

√
1−�n

i=1

(
1− (

︷ ︸︸ ︷
log∏

i
(T u

M ))q
)ωi

=
︷ ︸︸ ︷
log∏

i
T l

M +
︷ ︸︸ ︷
log∏

i
(T u

M ) .

Since, log∏
i
F l

M︸ ︷︷ ︸ = min log∏
i
F l

M ij ,
︷ ︸︸ ︷
log∏

i
F l

M = max log∏
i
F l

M ij log
∏

i
(Fu

M )︸ ︷︷ ︸ = min log∏
i
(Fu

Mij),︷ ︸︸ ︷
log∏

i
(Fu

M ) = max log∏
i
(Fu

Mij) and log∏
i
F l

M︸ ︷︷ ︸ ≤ log∏i
F l

M ij ≤
︷ ︸︸ ︷
log∏

i
F l

M and log∏
i
(Fu

M )︸ ︷︷ ︸ ≤ log∏i
(Fu

Mij) ≤

https://internationalpubls.com 543



Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

︷ ︸︸ ︷
log∏

i
(Fu

M ). Now,

log∏
i
F l

M︸ ︷︷ ︸+ log∏
i
(Fu

M )︸ ︷︷ ︸ = �n
i=1(log

∏
i
F l

M︸ ︷︷ ︸)qωi +�n
i=1(log

∏
i
(Fu

M )︸ ︷︷ ︸)qωi

≤ �n
i=1(log

∏
i
F l

M ij)
qωi +�n

i=1(log
∏

i
(Fu

Mij))
qωi

≤ �n
i=1(

︷ ︸︸ ︷
log∏

i
F l

M )qωi +�n
i=1(

︷ ︸︸ ︷
log∏

i
(Fu

M ))qωi

=
︷ ︸︸ ︷
log∏

i
F l

M +
︷ ︸︸ ︷
log∏

i
(Fu

M ) .

Therefore,

 q

√√√√1−�n
i=1

(
1− (log∏

i
T l

M︸ ︷︷ ︸)q
)ωi

2

+

 q

√√√√1−�n
i=1

(
1− (log∏

i
(T u

M )︸ ︷︷ ︸)q
)ωi

2

2

+1−

�n
i=1(

︷ ︸︸ ︷
log∏

i
F l

M )qωi

2

+

�n
i=1(

︷ ︸︸ ︷
log∏

i
(Fu

M ))qωi

2

2



=


(

q

√
1−�n

i=1

(
1− (log∏

i
T l

M ij)
q
)ωi

)2

+

(
q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mij))
q
)ωi

)2

2

+1− (�n
i=1(log

∏
i
F l

Mij
)qωi)

2
+(�n

i=1(log
∏

i
(Fu

Mij
))qωi)

2

2



=


 q

√√√√1−�n
i=1

(
1− (

︷ ︸︸ ︷
log∏

i
T l

M )q
)ωi

2

+

 q

√√√√1−�n
i=1

(
1− (

︷ ︸︸ ︷
log∏

i
(T u

M ))q
)ωi

2

2

+1−

�n
i=1(log

∏
i
F l

M︸ ︷︷ ︸)qωi

2

+

�n
i=1(log

∏
i
(Fu

M )︸ ︷︷ ︸)qωi

2

2

 .

Hence,
〈
(logT l

M︸ ︷︷ ︸, log(T u
M )︸ ︷︷ ︸), (

︷ ︸︸ ︷
logF l

M ,
︷ ︸︸ ︷
log(Fu

M ))
〉

≤ q − LIV IFWA(M1,M2, ...,Mn)

≤
〈
(
︷ ︸︸ ︷
logT l

M ,
︷ ︸︸ ︷
log(T u

M )), (logF l
M︸ ︷︷ ︸, log(Fu

M )︸ ︷︷ ︸)〉.
Theorem 4.5. (monotonicity property) Let Mi =

〈
(logT l

M tij
, log(T u

Mtij)
), (logF l

M tij
, log(Fu

Mtij
))
〉

and

ωi =
〈
(logT l

Mhij
, log(T u

Mhij
)), (logF l

Mhij
, log(Fu

Mhij
))
〉

(i = 1, 2, ..., n); (j = 1, 2, ..., ij) be the families of q-LIVIFWAs. For any i, if there is
(
log∏

i
T l

M tij

)2
+(

log∏
i
(T u

Mtij
)
)2
≤
(
log∏

i
T l

Mhij

)2
+
(
log∏

i
(T u

Mhij
)
)2

and
(
log∏

i
F l

M tij

)2
+
(
log∏

i
(Fu

M )tij

)2
≥(

log∏
i
F l

Mhij

)2
+
(
log∏

i
(Fu

Mhij)

)2
or Mi ≤ Wi. Then q-LIVIFWA (M1,M2, ...,Mn) ≤ q-LIVIFWA

(W1,W2, ...,Wn).

Proof. For any i,
(
log∏

i
T l

M tij

)q
+
(
log∏

i
(T u

Mtij
)
)q
≤
(
log∏

i
T l

Mhij

)q
+
(
log∏

i
(T u

Mhij
)
)q

.

Therefore, 1−
(
log∏

i
T l

M ti

)q
+ 1−

(
log∏

i
(T u

Mti
)
)q
≥ 1−

(
log∏

i
T l

Mhi

)q
+ 1−

(
log∏

i
(T u

Mhi
)
)q

.

Hence, �n
i=1

(
1−

(
log∏

i
T l

M ti

)q)ωi

+�n
i=1

(
1−

(
log∏

i
(T u

Mti
)
)q)ωi

≥

�n
i=1

(
1−

(
log∏

i
T l

Mhi

)q)ωi

+�n
i=1

(
1−

(
log∏

i
(T u

Mhi
)
)q)ωi

and q

√
1−�n

i=1

(
1−

(
log∏

i
T l

M ti

)q)ωi

+ q

√
1−�n

i=1

(
1−

(
log∏

i
(T u

Mti)

)q)ωi

≤ q

√
1−�n

i=1

(
1−

(
log∏

i
T l

Mhi

)q)ωi

+ q

√
1−�n

i=1

(
1−

(
log∏

i
(T u

Mhi)

)q)ωi

.

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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

For any i,
(
log∏

i
F l

M tij

)q
+
(
log∏

i
(Fu

Mtij)

)q
≥
(
log∏

i
F l

Mhij

)q
+
(
log∏

i
(Fu

Mhij)

)q
.

Therefore, 1−
(
�n

i=1log
∏

i
F l

Mtij

)q
+
(
�n

i=1log
∏

i
(Fu

Mtij)

)q

2 ≤ 1−
(
�n

i=1log
∏

i
F l

Mhij

)q
+
(
�n

i=1log
∏

i
(Fu

Mhij)

)q

2 .

=


(

q

√
1−�n

i=1

(
1− (log∏

i
T l

M ti)
q
)ωi

)2

+

(
q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mti))
q
)ωi

)2

2

+1− (�n
i=1(log

∏
i
F l

Mtij
))

2
+(�n

i=1(log
∏

i
(Fu

Mtij
)))

2

2



=


(

q

√
1−�n

i=1

(
1− (log∏

i
T l

Mhi)
q
)ωi

)2

+

(
q

√
1−�n

i=1

(
1− (log∏

i
(T u

Mhi))
q
)ωi

)2

2

+1− (�n
i=1(log

∏
i
F l

Mhij
))

2
+(�n

i=1(log
∏

i
(Fu

Mhij
)))

2

2

 .

Hence, q-LIVIFWA (M1,M2, ...,Mn) ≤ q-LIVIFWA (W1,W2, ...,Wn).

4.2 q-LIVIF weighted geometric( q-LIVIFWG) operator

Definition 4.6. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs. Then

q-LIVIFWG operator is q-LIVIFWG (M1,M2, ...,Mn) = �n
i=1M

ωi
i (i = 1, 2, ..., n).

Theorem 4.7. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs. Then

q-LIVIFWG (M1,M2, ...,Mn)

=

 (
�n

i=1(log
∏

i
T l

M i)
qωi ,�n

i=1(log
∏

i
(T u

Mi))
qωi
)
,(

q

√
1−�n

i=1

(
1− (log∏

i
F l

M i)
q
)ωi

, q

√
1−�n

i=1

(
1− (log∏

i
(Fu

Mi))
q
)ωi

) .

Theorem 4.8. If all Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
are equal and Mi = M , for i =

1, 2, ..., n. Then q-LIVIFWG (M1,M2, ...,Mn) =M .

Corollary 4.9. The q-LIVIFWG operator is used to satisfy the boundedness and monotonicity properties.

4.3 Extended q-LIVIFWA ( q-ELIVIFWA) operator

Definition 4.10. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFN. Then

q-ELIVIFWA (M1,M2, ...,Mn) =
(
�n

i=1 ωiM
ℵ
i

)1/ℵ
is called the q-ELIVIFWA operator.

Theorem 4.11. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs. Then

q-ELIVIFWA (M1,M2, ...,Mn)

=



( q

√√√√ 1−�n
i=1

(
1−

(
(log∏

i
T l

M i)
q
)q)ωi

)1/ℵ

,

(
q

√√√√ 1−�n
i=1

(
1−

(
(log∏

i
(T u

Mi))
q
)q)ωi

)1/ℵ
 ,

q

√√√√1−

(
1−

(
�n

i=1

(
q

√
1−

(
1− (log∏

i
F l

M i
)q
)q)ωi

)q)1/ℵ

,

q

√√√√1−

(
1−

(
�n

i=1

(
q

√
1−

(
1− (log∏

i
(Fu

Mi
))q
)q)ωi

)q)1/ℵ




.

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ISSN: 1074-133X
Vol 32 No. 3 (2025)

Proof. We have, �n
i=1ωiM

ℵ
i

=



 q

√√√√ 1−�n
i=1

(
1−

(
(log∏

i
T l

M i)
q
)q)ωi

, q

√√√√ 1−�n
i=1

(
1−

(
(log∏

i
(T u

Mi))
q
)q)ωi

 ,(
�n

i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

, �n
i=1

(
q

√
1−

(
1− (log∏

i
(Fu

Mi))
q
)q)ωi

)
 .

If n = 2, then ω1B
∨

ω2C

=





q

√√√√√√√√√√

(
q

√
1−

(
1−

(
(log∏

i
T l

B)
q
)q)ω1

)q

+

(
q

√
1−

(
1−

(
(log∏

i
T l

C)
q
)q)ω1

)q

,

−

(
q

√
1−

(
1−

(
(log∏

i
T l

B)
q
)q)ω1

)q

·

(
q

√
1−

(
1−

(
(log∏

i
T l

C)
q
)q)ω1

)q

q

√√√√√√√√√√

(
q

√
1−

(
1−

(
(log∏

i
(T u

B ))q
)q)ω1

)q

+

(
q

√
1−

(
1−

(
(log∏

i
(T u

C ))q
)q)ω1

)q

−

(
q

√
1−

(
1−

(
(log∏

i
(T u

B ))q
)q)ω1

)q

·

(
q

√
1−

(
1−

(
(log∏

i
(T u

C ))q
)q)ω1

)q



,


(

q

√
1−

(
1− (log∏

i
F l

B)
q
)q)ω1

·

(
q

√
1−

(
1− (log∏

i
F l

C)
q
)q)ω1

,(
q

√
1−

(
1− (log∏

i
(Fu

B))
q
)q)ω1

·

(
q

√
1−

(
1− (log∏

i
(Fu

C))
q
)q)ω1





=


(

q

√
1−�q

i=1

(
1−

(
(log∏

i
T l

B)
q
)q)ωi

, q

√
1−�q

i=1

(
1−

(
(log∏

i
(T u

B ))q
)q)ωi

)
,(

�q
i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

,�q
i=1

(
q

√
1−

(
1− (log∏

i
(Fu

Mi))
q
)q)ωi

)
 .

It is valid for n = m and m ≥ 3.
Hence, �m

i=1ωiM
ℵ
i =

(
q

√
1−�m

i=1

(
1−

(
(log∏

i
T l

B)
q
)q)ωi

, q

√
1−�m

i=1

(
1−

(
(log∏

i
(T u

B ))q
)q)ωi

)
,(

�m
i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

,�m
i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

)
.

If n = m+ 1 and we apply, then �m
i=1ωiM

ℵ
i + ωm+1M

ℵ
m+1 = �m+1

i=1 ωiM
ℵ
i .

Now,�m
i=1ωiM

ℵ
i + ωm+1M

ℵ
m+1 = ω1M

ℵ
1

∨
ω2M

ℵ
2

∨
...
∨

ωmMℵ
m

∨
ωm+1M

ℵ
m+1

=





q

√√√√√√√√√√

(
q

√
1−�m

i=1

(
1−

(
(log∏

i
T l

M i)
q
)q)ωi

)q

+

(
q

√
1−

(
1−

(
(log∏

i
T l

Mm+1)
q
)q)ω1

)q

,

−

(
q

√
1−�m

i=1

(
1−

(
(log∏

i
T l

M i)
q
)q)ωi

)q

·

(
q

√
1−

(
1−

(
(log∏

i
T l

Mm+1)
q
)q)ω1

)q

q

√√√√√√√√√√

(
q

√
1−�m

i=1

(
1−

(
(log∏

i
(T u

Mi))
q
)q)ωi

)q

+

(
q

√
1−

(
1−

(
(log∏

i
(T u

Mm+1))
q
)q)ω1

)q

−

(
q

√
1−�m

i=1

(
1−

(
(log∏

i
(T u

Mi))
q
)q)ωi

)q

·

(
q

√
1−

(
1−

(
(log∏

i
(T u

Mm+1))
q
)q)ω1

)q



,


�m

i=1

(
q

√
1−

(
1− (log∏

i
F l

M i
)q
)q)ωi

·

(
q

√
1−

(
1− (log∏

i
F l

Mm+1
)q
)q)ω1

,

�m
i=1

(
q

√
1−

(
1− (log∏

i
(Fu

Mi
))q
)q)ωi

·

(
q

√
1−

(
1− (log∏

i
(Fu

Mm+1
))q
)q)ω1





.

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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

Thus,

�m+1
i=1 ωiM

ℵ
i =


(

q

√
1−�m+1

i=1

(
1−

(
(log∏

i
T l

Mi)
q
)q)ωi

, q

√
1−�m+1

i=1

(
1−

(
(log∏

i
(T u

Mi))
q
)q)ωi

)
,

�m+1
i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

,�m+1
i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

 .

Hence,(
�m+1

i=1 ωiM
ℵ
i

)1/ℵ

=



( q

√√√√ 1−�m+1
i=1

(
1−

(
(log∏

i
T l

M i)
q
)q)ωi

)1/ℵ

,

(
q

√√√√ 1−�m+1
i=1

(
1−

(
(log∏

i
(T u

Mi))
q
)q)ωi

)1/ℵ
 ,

q

√√√√1−

(
1−

(
�m+1

i=1

(
q

√
1−

(
1− (log∏

i
F l

M i)
q
)q)ωi

)q)1/ℵ

,

q

√√√√1−

(
1−

(
�m+1

i=1

(
q

√
1−

(
1− (log∏

i
(Fu

Mi))
q
)q)ωi

)q)1/ℵ




.

It is valid for m ≥ 1.

Remark 4.12. If ωi = 1, then q-ELIVIFWA operator is modified to the q-LIVIFWA operator.

Theorem 4.13. If all Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
are equal and Mi = M . Then q-

ELIVIFWA (M1,M2, ...,Mn) =M .

Remark 4.14. We use the q-ELIVIFWA operator to satisfy boundedness and monotonicity conditions.

4.4 Extended q-LIVIFWG ( q-ELIVIFWG) operator

Definition 4.15. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs. Then

q-ELIVIFWG (M1,M2, ...,Mn) =
1
ℵ

(
�n

i=1 (ℵMi)
ωi

)
is called the q-ELIVIFWG operator.

Theorem 4.16. Let Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
be the collection of q-LIVIFNs. Then

q-ELIVIFWG(M1,M2, ...,Mn)

=




q

√√√√1−

(
1−

(
�n

i=1

(
q

√
1−

(
1− (log∏

i
T l

M i)
q
)q)ωi

)q)1/ℵ

,

q

√√√√1−

(
1−

(
�n

i=1

(
q

√
1−

(
1− (log∏

i
(T u

Mi))
q
)q)ωi

)q)1/ℵ

 ,

( q

√√√√ 1−�n
i=1

(
1−

(
(log∏

i
F l

M i)
q
)q)ωi

)1/ℵ

,

(
q

√√√√ 1−�n
i=1

(
1−

(
(log∏

i
(Fu

Mi))
q
)q)ωi

)1/ℵ



.

Remark 4.17. If ωi = 1, then q-ELIVIFWG operator is converted to the q-LIVIFWG operator.

Remark 4.18. q-ELIVIFWG operators satisfy boundedness and monotonicity properties.

Corollary 4.19. If all Mi =
〈
(logT l

M i, logT
u
Mi), (logF

l
M i, logF

u
Mi)

〉
are equal and Mi = M , for i =

1, 2, ..., n. Then q-ELIVIFWG(M1,M2, ...,Mn) =M .

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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 32 No. 3 (2025)

Acknowledgment. This research was supported by the University of Phayao and the Thailand Science Re-
search and Innovation Fund (Fundamental Fund 2025).

Conflicts of Interest The author(s) declare that there are no conflicts of interest regarding the publication of
this paper.

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	1 Introduction
	2 Preliminaries
	3 q-LIVIFN and its fundamental operations
	4 q-LIVIFS concept
	4.1  q-LIVIF weighted averaging( q-LIVIFWA) operator
	4.2  q-LIVIF weighted geometric( q-LIVIFWG) operator
	4.3 Extended q-LIVIFWA ( q-ELIVIFWA) operator
	4.4 Extended q-LIVIFWG ( q-ELIVIFWG) operator