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Complex q‐rung orthopair fuzzy 2‐tuple linguistic Maclaurin symmetric mean operators and its application to emergency program selection

International Journal of Intelligent Systems

September 2020
35(11)

DOI:10.1002/int.22271

Authors:

Yuan Rong

Shanghai University

Yi Liu

Neijiang Normal University

Zheng Pei

Xihua University

This essay designs an innovate approach to work out linguistic multiattribute group decision‐making (MAGDM) issues with complex q‐rung orthopair fuzzy 2‐tuple linguistic (Cq‐ROF2TL) evaluation information. To begin with, the conception of Cq‐ROF2TL set is propounded to express uncertain and fuzzy assessment information. Meanwhile, the score and accuracy function, a comparison approach, Cq‐ROF2TL weighted averaging, and Cq‐ROF2TL weighted geometric operator are put forward. Furthermore, to take into consideration the correlation among multiple input data, the Cq‐ROF2TL Maclaurin symmetric mean (MSM) operator, the Cq‐ROF2TL dual MSM operator and their weighted forms are presented. Several attractive characteristics and particular instances of the developed operators are also explored at length. Later, an innovative MAGDM methodology is designed based upon the propounded operators to settle the emergency program evaluation issue under the Cq‐ROF2TL circumstance. Consequently, the efficiency and outstanding superiority of the created approach are severally substantiated by parameter exploration and detailed comparative analysis.

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Content may be subject to copyright.

Received: 29 February 2020

Revised: 6 July 2020

Accepted: 2 August 2020

DOI: 10.1002/int.22271

RESEARCH ARTICLE

Complex q‐rung orthopair fuzzy 2‐tuple

linguistic Maclaurin symmetric mean

operators and its application to emergency

program selection

Yuan Rong

|Yi Liu

2,3

|Zheng Pei

School of Science, Xihua University,

Sichuan, China

Data Recovery Key Lab of Sichuan

Province, Neijiang Normal University,

Sichuan, China

Numerical Simulation Key Laboratory

of Sichuan Province, Neijiang Normal

University, Sichuan, China

Correspondence

Zheng Pei, School of Science, Xihua

University, Chengdu, 610039 Sichuan,

China.

Email: pqyz@263.net

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 61372187;

Scientific and Technological Project of

Sichuan Province,

Grant/Award Number: 2019YFG0100;

Sichuan Province Youth Science and

Technology Innovation Team,

Grant/Award Number: 2019JDTD0015;

Application Basic Research Plan Project

of Sichuan Province,

Grant/Award Number: 2017JY0199;

Scientific Research Project of

Department of Education of Sichuan

Province, Grant/Award Numbers:

18ZA0273, 15TD0027; Scientific

Research Project of Neijiang Normal

University, Grant/Award Number:

18TD08; Innovation Fund of

Postgraduate Xihua University,

Abstract

This essay designs an innovate approach to work

out linguistic multiattribute group decision‐making

(MAGDM) issues with complex q‐rung orthopair

fuzzy 2‐tuple linguistic (Cq‐ROF2TL) evaluation

information. To begin with, the conception of

Cq‐ROF2TL set is propounded to express uncertain

and fuzzy assessment information. Meanwhile, the

score and accuracy function, a comparison approach,

Cq‐ROF2TL weighted averaging, and Cq‐ROF2TL

weighted geometric operator are put forward. Fur-

thermore, to take into consideration the correlation

among multiple input data, the Cq‐ROF2TL Maclaurin

symmetric mean (MSM) operator, the Cq‐ROF2TL

dual MSM operator and their weighted forms are pre-

sented. Several attractive characteristics and particular

instances of the developed operators are also explored

at length. Later, an innovative MAGDM methodology

is designed based upon the propounded operators to

settle the emergency program evaluation issue under

the Cq‐ROF2TL circumstance. Consequently, the

efficiency and outstanding superiority of the created

approach are severally substantiated by parameter ex-

ploration and detailed comparative analysis.

Grant/Award Number: YCJJ2020028;

University Students Innovation and

Entrepreneurship Project of Xihua Cup,

Grant/Award Number: 2020107

KEYWORDS

complex q‐rung orthopair fuzzy 2‐tuple linguistic set, emergency

management program evaluation, Maclaurin symmetric mean

operator, multiattribute group decision‐making

1|INTRODUCTION

Decision‐making (DM) is a common task associated with intelligent and sophisticated ac-

tivities that take into consideration various kinds of vagueness and uncertainty in which

human beings face situations. During daily life and economic development, various selec-

tion and assessment issues can be deemed as a DM problem. Accordingly, the exploration of

DM approaches for modern decision science and management has received more attention

from multitude scholars.

1‐5

To effectual portray the ambiguous and undetermined assess-

ment data, the propounded fuzzy set (FS)

is an important technique during settling DM

issues in uncertain environment. Since it was reported in 1965, FS has attracted a large

number of researchers and attained investigation achievements in various aspects.

7‐11

Nonetheless, one of the deficiencies of FS is that the range of FS is limited to [0, 1], which

will lead to the situation of information loss for expressing evaluation information.

Therefore, Ramot et al.

originally introduced the complex fuzzy set (CFS) through ex-

panding the membership grade (MG) from real value to complex value within the unit disc.

In light of its advantages, CFS is omnipresent applied to fuzzy logic, decision science and

other science domains.

13‐15

Because the FS and CFS have a common defect which fail to

take into account the nonmembership grade (NMG) of an element belonging to the given

objective. Then Atanassov

advanced an expansion form of FS called intuitionistic fuzzy set

(IFS), which validly remedies the deficiency of FS through attaching a NMG. Since its

introduction, the fundament theory and application on it have gotten numerous attentions,

such as aggregation operator,

distance measure,

information entropy,

decision

approach

20,21

and so forth. Later on, Alkouri et al.

proffered the definition of complex

intuitionistic fuzzy set (CIFS) to describe the undetermined and ill‐defined judgement in-

formation in practical issues. The CIFS makes up of complex‐valued MG and complex‐

valued NMG, which are indicated through the polar coordinates. Rani and Garg

defined

the basic algorithm of CIFS and proffered several power operators to establish multiple

attribute decision making (MADM) method. Inspired the interval‐valued IFS presented by

Attanassov,

Garg and Rani

introduced the framework of the complex interval‐valued IFS

and studied its related operational rules and aggregation operators. For the information

measure theory, Garg and Rani

introduced several information measures including si-

milarity, entropies measure and so forth, and further presented a clustering algorithm on

the basis of these measures. Garg and Rani

propounded generalized Bonferroni mean

(BM) operators based upon Archimedean operations to integrate complex intuitionistic

fuzzy information. The more research results can refer to.

28‐30

However, if DMs judge their

evaluation information as (0.5, 0.7) for MG and NMG, the IFS cannot depict it because of

.5 + 0.7 = 1.2 >

. Hence, Yager

first proffered the Pythagorean fuzzy set (PFS) to por-

tray undetermined opinion for DM issues. It is obvious that the PFS is more universal than

FS and IFS because of 0.5

+0.7

= 0.74 < 1. Aiming at the PFS, Qin et al.

proposed some

ordered weighted distance measures to do with DM problems. Garg

defined novel

RONG ET AL.

logarithmic operations and propounded several logarithmic operators under Pythagorean

fuzzy environment. Liang et al.

combined the TOPSIS technique and three‐way decision

theory to construct a new approach to resolve DM problems. Ullah et al.

proposed several

distance measures of complex Pythagorean fuzzy set (CPFS) and an algorithm for addres-

sing pattern recognition issues.

Because the PFS has a prerequisite that

≤MG NMG

(

)+( )

, but when we come cross

several practical situations which the evaluation information provided by DMs in the form of

PFS cannot meet the precondition. For example, the MG and NMG are provided as (0.7, 0.8),

IFS and PFS fail to process it effectively because of

.7 + 0.8 >

and 0.7

+ 0.8

> 1. Based on

this restriction, Yager

developed the idea of q‐rung orthopair fuzzy set (q‐ROFS), which must

meet ≤MG NMG

(

)+( )

. The q‐ROFS can valid dispose of the aforementioned example

because of 0.7

+ 0.8

= 0.855 < 1. It is evident that q‐ROFS has more evaluation space than IFS

and PFS because they are particular situations of q‐ROFS for

and

, respectively.

Based upon it, Liu and Wang

introduced several q‐rung orthopair fuzzy Archimedean

BM operators. Li et al.

expanded the EDAS method to q‐rung orthopair fuzzy context to

establish DM approach. Further, Liu et al.

presents the concept of complex q‐ROFS

(Cq‐ROFS) and Cq‐ROFLS set (Cq‐ROFLS) and further advanced several Cq‐ROFL Heronian

mean (HM) operators to build up decision model.

The above fuzzy sets can only depict information from the point of quantitative and DMs

are difficult to provide the precise numerical values to express their viewpoint. So Zadeh

proposed the linguistic variable (LV) to describe the qualitative information in DM pro-

blems. Thereafter, several novel notions by combining the LV and fuzzy set have been

propounded, such as intuitionistic linguistic numbers,

single‐valued neutrosophic lin-

guistic set

and linguistic q‐rung orthopair fuzzy number.

Furthermore, Herrera and

Martłnez

proposed the notion of 2‐tuple fuzzy linguistic established by a LV and a nu-

merical to prevent information loss in the procedure DM. Later on, a lot of scholars com-

bined the 2‐tuple linguistic with other fuzzy sets and propounded intuitionistic 2‐tuple

linguistic model,

2‐tuple linguistic PFS,

and so forth. These extensions can efficient

express the ill‐defined and fuzzy information in addressing DM issues.

As discussed above‐mentioned, the Cq‐ROFS and 2‐tuple linguistic model are two sig-

nificant techniques for disposing the quantitative and qualitative assessment information.

Motivated by the thought of Pythagorean fuzzy 2‐tuple linguistic, it is meaningful to pro-

pound a novel conception called Cq‐ROF2TLS through synthesizing the Cq‐ROFS and

2‐tuple linguistic to depict the complex uncertain and vague information. Obviously, the

Cq‐ROF2TLS is more universal than existing fuzzy sets because we can attain several

particular examples by taking into some special situations. When we take the parameter

and

in the context of Cq‐ROF2TLS, it will degenerate into the complex in-

tuitionistic fuzzy 2‐tuple linguistic set and complex Pythagorean fuzzy 2‐tuple linguistic set,

respectively. Besides, if the imaginary part of Cq‐ROF2TLS is assigned as zero, then it will

be simplified to q‐rung orthopair fuzzy 2‐tuple linguistic set. From previous research of

linguistic set the Cq‐ROF2TLS is stronger because: (1) it can prevent the information dis-

tortion during the process of linguistic information disposing; (2) it can avoid information

loss through utilizing complex‐valued MG complex‐valued NMG to express assessment

information; (3) it can valid tackle the problems with two dimension information in real‐life

applications.

It is well known that aggregation operator is an indispensable technique for information

fusion field and it has achieved a multitude of researcher results on diverse aspects.

RONG ET AL.

presented several geometric operators to aggregate intuitionistic fuzzy information.

Liu and Wang

propounded weighted average and geometric operator for q‐ROFS to es-

tablished MAGDM approach. However, these operators fail to take into consideration the

interrelationship of discussed attributes in decision problems. In order to conquer this

restriction, the BM and HM operator are proposed to consider the relevance of ant two data.

Nevertheless, the BM and HM operator fail to catch the interconnection among multi‐input

data. Accordingly, Maclaurin

originally propounded the Maclaurin symmetric mean

(MSM) operator to conquer the aforementioned defects. Thereafter, Qin and Liu

pro-

pounded dual MSM operator to aggregation intuitionistic fuzzy information. Liu and Qin

presented several linguistic intuitionistic fuzzy MSM operators to build up MAGDM

method. Wei and Lu

extended MSM operator to Pythagorean fuzzy environment to do

with decision problems. As we have seen, the MSM operator is not generalized to complex

fuzzy setting. Consequently, it is valuable to expand MSM operator to Cq‐ROF2TLS to

propound several novel operators.

As aforementioned discussion and analysis, the motivations of this essay are illustrated

as follows: (1) The existing theories about CFS fail to depict uncertain information through

the 2‐tuple linguistic representation madel, because it has stronger capability to describe

linguistic information and it also can avoid information distortion loss in dealing with

linguistic decision issues. Thus, we first proposed the complex q‐rung orthopair fuzzy

2‐tuple linguistic set and related fundamental conceptions, which will further rich the CFS

theories and provide a valid tool for experts to express assessment information; (2) The

information fusion plays an important role in aggregating preference information of deci-

sion experts. In addition, multitude practical issues need to take into consideration the

correlation of the identified attributes. In light of the outstanding superiority of MSM

operator, several complex q‐rung orthopair fuzzy 2‐tuple linguistic MSM operators are

created to address two‐dimensional fuzzy information; (3) The emergency program eva-

luation and selection has been viewed as vital and hot topic in management science. In view

of the complexity and unpredictability of emergency events, different types of assessment

methods need to be explored to valid evaluate emergency programs.

According to the above inspiration, the goal of this essay is to develop a innovative

decision approach to evaluate the emergency management programs. To achieve the

objective, we first find a suitable information expression tool to portray complex cognitive

information. Then we need to explore how to build up the decision algorithm and to

efficiently select the optima emergency program. Eventually, we need to confirm that the

present decision algorithm is valid and advantageous from diverse aspects. Accordingly,

the concrete aims of this article can be summarized as below:

(1) To proffer a novel conception called Cq‐ROF2TLS and several fundamental operations,

then define its score function and comparative approach;

(2) To present Cq‐ROF2TL weighted averaging (Cq‐ROF2TLWA) operator and complex

Cq‐ROF2TL weighted geometric (Cq‐ROF2TLWG) operator;

(3) To propound several MSM operators including the Cq‐ROF2TL Maclaurin symmetric

mean (Cq‐ROF2TLMSM) operator, the Cq‐ROF2TL weighted Maclaurin symmetric

mean (Cq‐ROF2TLWMSM) operator, the Cq‐ROF2TL dual Maclaurin symmetric mean

(Cq‐ROF2TLDMSM) operator and the Cq‐ROF2TL weighted Maclaurin symmetric

mean (Cq‐ROF2TLWDMSM) operator, as well as the fundamental characteristics are

discussed in detail.

RONG ET AL.

(4) To design a MAGDM methodology on the basis of these operators;

(5) To expound the validation and performance of the developed approach by an empirical

example for assessing emergency project.

To accomplish the aforementioned objectives, the overall structure of the essay is allo-

cated as below. In Section 2, we succinctly retrospect several fundamental concepts and

definitions including 2‐Tuple linguistic, Cq‐ROFS and MSM operator. Section 3propounds

the notion of Cq‐ROF2TLS, basic operation rules, comparison methodology, fundamental

operators. Section 4presents the Cq‐ROF2TLMSM, Cq‐ROF2TLWMSM, Cq‐ROF2TLDMSM,

Cq‐ROF2TLDWMSM operator, and also study several worthwhile features and particular

cases of them. Section 5is concerned with the novel MAGDM methodology on the basis of

Cq‐ROF2TLWMSM and Cq‐ROF2TLWDMSM operator. In Section 6,anevaluationproblem

of emergency program is utilized to show the efficiency and a contrastive study is performed

to highlight the merits of the developed method. Several conclusion remarks are listed in

the end.

2|PRELIMINARIES

This part succinctly retrospected several fundamental knowledge, such as 2‐tuple linguistic,

Cq‐ROFS and MSM operators.

2.1 |2‐tuple linguistic set

To efficiently dispose the vague linguistic information, Herrera and Martinez

first prof-

fered the 2‐tuple linguistic model based upon the symbolic transformation and LV, which is

stated as below.

Definition 1 (Herrera and Martinez

). Suppose



ss s={ , ,…, }

l01 be a LTS. ∈

l[0, ] be

a value denoting the result of a symbolic aggregation operation, then the mapping

utilized to acquire the 2‐tuple that expresses the equivalent information to

depicted as

→lS βs

:[0, ] ×[−0.5, 0.5)Δ()=(,ϱ)

i(1)

where

∈

round ββi=(),ϱ=−,ϱ[−0.5, 0.5)

,andround denotes the usual round

operation.

Definition 2 (Herrera and Martinez

). Suppose



ss s={ , ,…, }

l01 be a LTS and

(

,ϱ)

a2‐tuple. there exists a mapping

−

, which can attain ∈

l[0, ] on the basis of 2‐tuple,

described as follows:

→Sl

si β

Δ:×[−0.5, 0.5) [0, ]

Δ(,ϱ)= +ϱ=.

−1

−1(2)

RONG ET AL.

Based upon the aforementioned notions, we can easily find that a linguistic term sican be

shifted to a 2‐tuple linguistic, that is, ss

()=(,0)

2.2 |Cq‐ROFS

Liu et al.

propounded the conception of Cq‐ROFS, which is defined as below.

Definition 3 (Liu et al.

). Given a fixed set

, the Cq‐ROFS is defined as follows:

ab〈〉∣∈

ζζ ζζ={ ,˙(),˙() Ξ}

(3)

in which



ζe

˙=()

iπ2ζ(

)

and



ζe

˙=()

iπ2

ζ()

indicate complex‐valued MG and

NMG, respectively, meeting the following characteristic ab≤≤

≤

ζζ

()+ () 1,0



≤≥q+1(1

)

() () . The indeterminacy degree is expressed as μζ()

ab ab



()

ζζe

(

1−( ( ) + ( )))

iπ21−+

qζ

1() ()

. Additionally, ab



()

ζeζe() , ()

iπiπ22

ζζ() () is

called complex q‐rung orthopair fuzzy number (Cq‐ROFN). Conveniently, we rewrite

as ab



(

)

ee,

iπiπ22

Definition 4 (Liu et al.

). Given two Cq‐ROFNs ab



()

ee=,

iπiπ

212

and



()

ee=,

iπiπ

222

with ≥

λ1

, the basic algorithms between

and

are

given as follows:

aaaa bb

aaa bb

 

⊕

()

QQ e e

(

1) = + −,

;

qqqq

iπiπ

12 12122+−12 2

qqqqq

1212

⎛

⎝

⎜⎞

⎠

⎟(4)

aa bbbb b

aa bbb

 

⊗

()

QQ e e

(

2) = , + −

;

iπqqqq

iπ

1212

212122+−

qqq qqq

1212

⎛

⎝

⎜⎞

⎠

⎟(5)



()

λQee

(

3) = 1 −1−,;

qλiπλiπ

21−1−

qqλqλ

⎛

⎝

⎜

⎞

⎠

⎟

⎟(6)



()

Qe e

(

4) = , 1 −1−

λλ

iπqλiπ

21−1−

λqqλq

⎛

⎝

⎜

⎞

⎠

⎟

(7)

Definition 5 (Liu et al.

). Given a Cq‐ROFN ab



()

ee=,

iπiπ22

, the score index and

accuracy index are defined as below:

ab ab



()

Q()=

2−+−

qq qq(8)

RONG ET AL.

ab ab



()

HQ()=

2++ + .

qq qq(9)

2.3 |MSM operator

Definition 6 (Maclaurin

). The mathematical expression of MSM operator is defined as

below:

∑∏

≤⋯ ≤

()

SM e e e

(, ,…, )=

rrnj

() 12

1<< =1

⎛

⎝

⎜

⎞

⎠

⎟

(10)

in which

is a parameter and knrrr=1,2,…, , , ,…,

12 are

integer values taken from

the collection n{1, 2, …, } of ninteger values,

stands for the binomial coefficient whose

expression is C=

kn k

!( −)!.

The MSM operator has the following properties:

(1)

SM (0,0,…,0)=

k() ;

(2)

SM e e e

(, ,…, )=

k()

,ifeet n=(=1,2,…,

)

(3)

≤

SM e e e MSM e e e(, ,…, ) (

˜,˜,…, ˜

)

() () 12 ,if ≤ee

. for all

;

(4) ≤≤eMSMeee emin { } ( , , …, ) max { }

tt kntt

() 12 .

Based upon the definition of MSM operator, the dual form of MSM operator is propounded

by Qin and Liu,

which is stated as below.

Definition 7. The mathematical expression of DMSM operator is defined as below:

∏∑

≤⋯ ≤

DMSM e e e ke(, ,…, )=

rrnj

() 12

1<< =1

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎟(11)

The DMSM operator has the same characteristics with the MSM operator.

3|COMPLEX Q‐RUNG ORTHOPAIR FUZZY 2‐TUPLE

LINGUISTIC SET

Inspired by the conception of 2‐tuple linguistic model and Cq‐ROFS, we shall first propound a

novel notion named Cq‐ROF2TLS, which can be viewed as a valid expansion of Cq‐ROFS.

Moreover, the fundamental algorithms, score function, accuracy function and basic operators

of Cq‐ROF2TLS are developed in detail.

RONG ET AL.

Definition 8. Given a fixed set

, the Cq‐ROF2TLS is expressed as follows



〈〉∣∈sζζζ={ ( ,ϱ), ˙(),˙() Ξ}

θζ() (12)

where aa a



∈∈sζe,ϱ[−0.5, 0.5), ˙=()

θζ iπ

() 2ζ(

)

and



ζe

˙=()

iπ2

ζ()

indicate complex‐

valued MG and NMG of the element

belonging to the LV s

(

,ϱ)

θζ() , respectively, meeting

the following characteristic ab≤≤ζζ

()+ ()

and ab

≤≤≥q

+1(1

)

() () . For

any ab ab



∈

()

ζμζ ζ ζeΞ,()=(1−( ( ) + ( )))

iπ21−+

qζ

1() ()

is called the refusal MG of

the element

belonging to the LV s

(

,ϱ)

θζ() Additionally, ab





(( )

)

see=( ,ϱ),

θiπiπ

() 22

is called complex q‐rung orthopair fuzzy 2‐tuple linguistic number (Cq‐ROF2TLN).

Definition 9. Given two Cq‐ROF2TLNs ab





(( ))

se e t=(,ϱ), (=1,2

)

ttiπtiπ22

with ≥

λ1

, the basic algorithms between

1

and

2

are given as follows:

aaaa bb

aa bb



 

⊕

()()

()

ss e e

(1)

=ΔΔ (,ϱ)+( ,ϱ), + −,;

qqqq

iπiπ

−1112212122+−12 2

qqqqqq

1212

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟⎞

⎠

⎟

(13)

aa bbbb bb

aa bb



 

⊗

()()

()

ss e e

(2)

=ΔΔ (,ϱ)×( ,ϱ), , + −;

iπqqqq

iπ

−1112212 212122+−

qqq qqq

1212

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟⎞

⎠

⎟

(14)



()

()()

()

λλsee

(

3) = ΔΔ(,ϱ), 1−1−,;

qλiπλiπ

1−1111

21−1−

qqλqλ

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎟(15)



()( )

() ()

()

se e

(

4) = ΔΔ(,ϱ), ,1−1−

λλλiπqλiπ

1−111121

21−1−

λqqλq

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

(16)

Definition 10. Given a Cq‐ROF2TLN ab





(( )

)

see=( ,ϱ),

θiπiπ

() 22

the score index

and accuracy index are defined as below:







()

˜()=

−+−×Δ(,ϱ)

qq qqθ

−1() (17)

RONG ET AL.







()

˜()=

++ + ×Δ(,ϱ)

qq qqθ

−1() (18)

Definition 11. Given two Cq‐ROF2TLNs ab





(( ))

se e t=(,ϱ), (=1,2

)

ttiπtiπ22

tt ,

then the comparison algorithm of

1

and

2

is given as follows:

1. If 

˜()>

˜(

)

, then



>

2. If 

˜()=

˜(

)

, then

•If HH

˜()<

˜(

)

, then



<

•If HH

˜(=

˜(

)

, then



=

To enhance the comparison analysing, in the next, we shall present two basic

weighted operators, namely, complex q‐rung orthopair fuzzy 2‐tuple linguistic weighted

averaging and complex q‐rung orthopair fuzzy 2‐tuple linguistic weighted geometric

operator.

Definition 12. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family

of Cq‐ROF2TLNs. A mapping:

→Φ

called Cq‐ROF2TLWA operator is defined

  ⊕Cq ROF TLWA w−2(,,…,)=

12 =1

(19)

where

stands for the collection of Cq‐ROF2TLNs,

be the weight of



t, meeting

∈

[0, 1]

tand

∑

=1 .

Theorem 1. Let ab





〈〉

()

se e t n=(,ϱ), (=1,2,…,)

ttiπtiπ22

tt be a family of

Cq‐ROF2TLNs,

be the weight of



t, meeting ∈

[0, 1]

tand

∑

=1 . Then the

fused result from Cq‐ROF2TLWA operator is also a Cq‐ROF2TLN and

 



∏∏ ∏

∏∏

() ()

()

Cq ROF TLWA

ws e e

−2(,,…,)

=ΔΔ(,ϱ), 1−1−,.

qwiπ

wiπ

−1

21−1−

qwtq

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(20)

The proof of this theorem can be easily proved through mathematical induction, so we omit

it here. Additionally, the following properties can be easily attained.

Theorem 2. Let ab





(( ))

se e=(,ϱ),

ttiπtiπ22

and ab





(( ))

se et n′=(′,ϱ′)′,′(=1,2,…, )

ttiπttiπt

2′2′

be two families of Cq‐ROF2TLNs, then the Cq‐ROF2TLWA operator possesses following

features:

RONG ET AL.

(1) Idempotentency: If all Cq‐ROF2TLNs are equal, that is,



=

tfor all

, then

  Cq ROF TLWA−2(,,…,)=

n12

(2) Monotonicity: If aabb aa

≤≤≥≤ss

(

,ϱ)(′,ϱ′), ′,′,

′

tttttttt ttand



≥

′

. Then

    ≤Cq ROF TLWA Cq ROF TLWA−2(,,…,)−2(′,′,…, ′)

nn12 1 2

(3) Boundedness: Let



tn(=1,2,…, )

tbe a family of Cq‐ROF2TLNs, and



 =min{ }, =max{ }

tt tt

−+

. Then



  ≤≤Cq ROF TLWA−2(,,…,).

−12 +

Definition 13. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family of

Cq‐ROF2TLNs. A mapping:

→Φ

called Cq‐ROF2TLWG operator is defined by

  ⊗Cq ROF TLWG−2(,,…,)=

12 =1

(21)

where

stands for the collection of Cq‐ROF2TLNs,

is the weight of



tmeeting

∈

[0, 1]

tand

∑

=1 .

Theorem 3. Let ab





〈〉

()

se e t n=(,ϱ), (=1,2,…,)

ttiπtiπ22

tt be a family of

Cq‐ROF2TLNs,

is the weight of



tmeeting ∈

[0, 1]

tand

∑

=1 . Then the

fused result from Cq‐ROF2TLWG operator is also a Cq‐ROF2TLN and

 



∏∏ ∏

∏∏

() ()

()

Cq ROF TLWG

se e

−2(,,…,)

=ΔΔ(,ϱ), ,1−1−.

wiπ

qwiπ

−1

21−1−

ttt

wtt

qwtq

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(22)

The Cq‐ROF2TLWG operator also meets the properties of idempotentency, monotonicity,

and boundedness.

4|THE CQ‐ROF2TLMSM OPERATOR AND

CQ‐ROF2TLDMSM FOR CQ‐ROFLNS

Under this part, we generalize the MSM operator to the propounded Cq‐ROF2TL en-

vironment to build up the Cq‐ROF2TLMSM operator and Cq‐ROF2TLDMSM operator.

Moreover, we develop their weighted form named Cq‐ROF2TLWMSM operator and

Cq‐ROF2TLWDMSM operator.

RONG ET AL.

4.1 |The Cq‐ROF2TLMSM and Cq‐ROF2TLDMSM operator

Definition 14. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family of

Cq‐ROF2TLNs. A mapping:

→Φ

called Cq‐ROF2TLMSM operator is defined by

 

⊕⊗

≤⋯ ≤

()

Cq ROF TLMSM C

−2(,,…,)=

rrnt

() 12

1<< =1

⎛

⎝

⎜

⎞

⎠

⎟

⎟(23)

where

stands for the collection of Cq‐ROF2TLNs,

is a parameter and

knrrr=1,2,…, , , ,…,

12 are

integer values taken from the collection n{1, 2, …, } of n

integer values,

stands for the binomial coefficient.

Theorem 4. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family of

Cq‐ROF2TLNs, the fusion result of

 {, ,…, }

n12 by utilizing Cq‐ROF2TLMSM

operator is expressed as

 



∏∏

∑∏

∏∏

()

Cq ROF TLMSM

−2(,,…,)

=Δ

Δ(,ϱ)

1−1−,

1−1−1−1−

ξt

krr

ξt

qiπ

ξt

iπ

() 12

=1 −1

21−1−

21−1−1−1−

kqk

ξt

qCn

kqk

kkq

ξt

qCn

kkq

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(24)

where

denotes the subscript ≤⋯≤rrn

(

1<<

)

k1.

RONG ET AL.

Proof. In light of the basic algorithms of Cq‐ROF2TLNs, we have





∏∏ ∏

∏

()

() ()

()

=ΔΔ(,ϱ), ,

1−1−

iπ

qiπ

=1 =1

−1

21−1−

ttt

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

and





∑∏ ∑∏

∏∏

≤⋯ ≤

()

() ()

()

=ΔΔ(,ϱ),

1−1−,

1−1−

rrnt

ξt

qiπt

ξt

qiπ

1<< =1 =1

−1

21−−

21−1−

ξt

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

Then





∑∏ ∑∏

∏∏

≤⋯ ≤

()

() ()

()

=Δ

Δ(,ϱ)

1−1−,

1−1−

rrnt

ξt

qiπ

ξt

qiπ

1<< =1 =1

−1

21−1−

kttt

ξt

qCn

ξt

qqCn

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

Accordingly

RONG ET AL.





∏∏

∑∏

∏∏

≤⋯ ≤

()

=Δ

Δ(,ϱ)

1−1−,

1−1−1−1−

rrnt

ξt

krr

ξt

qiπ

ξt

iπ

1<< =1

=1 −1

21−1−

21−1−1−1−

kqk

ξt

qCn

kqk

kkq

ξt

qCn

kkq

111

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

□

Theorem 5. Assume ab





(( ))

se e=(,ϱ),

ttiπttiπt

and ab





(( ))

se et n′=(′,ϱ′)′,′(=1,2,…, )

ttiπttiπt

2′2′

be two families of Cq‐ROF2TLNs, then the Cq‐ROF2TLWG operator possesses following

features:

(1) Idempotentency: If all Cq‐ROF2TLNs are equal, that is,



=



(( )

)

se e

(,ϱ), ,

iπiπ22

, for all

, then

  Cq ROF TLMSM−2(,,…,)=.

n12

(2) Monotonicity: For



tand



tn′(=1,2,…,

)

t,if



≤

′

, for all

. Then

 

 ≤

Cq ROF TLMSM

−2(,,…,)

−2(′,′,…, ′)

(3) Boundedness: Let



tn(=1,2,…, )

tbe a family of Cq‐ROF2TLNs, and



 =min{ }, =max{ }

tt tt

−+

. Then



  ≤≤Cq ROF TLMSM−2(,,…,).

−12 +

RONG ET AL.

Proof.

(1) Idempotentency:

□

 Cq ROF TLMSM−2(,,…,)

k()





∏∏

∑∏

∏∏

∏

()

se e

=Δ

Δ(,ϱ)

1−1−,

1−1−1−(1−())

=Δ(Δ(,ϱ)),

1−1−,

1−1−1−(1−())

=Δ((Δ(,ϱ))),

1−(1−() ),

1−1−(1−(1−()))

=Δ(Δ(,ϱ)),(1−(1−() )) ,

(())

=((,ϱ), (,)).

1−1

ξt

iπ

ξt

iπ

ξt

iπ

ξt

iπ

qkC

iπ

qk iπ

qiπ

iπiπ

−1=1

21−1−

21−1−1−1−

−1=1

21−1−

21−1−1−(1−()

)

−1

−(1−() )

21−1−(−()

)

−1×2(1−(1−() ))

2(()

)

ξt

qCn

ξt

qCn

ξt

kqCn

ξt

kqCn

kCn

qkξ

kCn

qkqkqk

qqq

111

111×

111

11×11

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

RONG ET AL.

Proof.

(2) Monotonicity: Because



≤

′

, then we have aa aa

≤≤≤ss

(

,ϱ)(′,ϱ′), ′,

′

tttttt tt

and

≥≥′,

′

tt ttfor all

tn(=1,2,…, )

. Then, for the 2‐tuple linguistic

information

∑∏ ∑∏

≤

≤⋯ ≤ ≤⋯ ≤

()

() ()

CsCs

1Δ(,ϱ)1Δ′,ϱ′

rrnt

1<< =1

−1

1<< =1

−1

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

Further, for the real‐valued membership degrees, because

a≤

′

, one has

aa a a

∏∏∏∏∏∏

∏∏

≤⇒ ≥

⇒

≤

⇒

≤

′1−1−′

1−1−

1−1−′

1−1−

1−1−′ .

ξt

=1 =1 =1 =1

kqk

111

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

Analogously, for the imaginary‐valued membership degrees, we attain



∏∏ ∏∏

≤1−1−1−1−′

ξt

=1 =1

kqk

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

Accordingly, we get

RONG ET AL.



∏∏

≤

∏∏

1−1−

1−1−′ .

ξt

qiπ

ξt

qiπ

21−1−

21−1−′

kqk

ξt

qCn

kqk

ξt

qCn

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

Similar to the testify of membership degrees, we attain



∏∏

≥

∏∏

()

1−1−1−1−

1−1−1−1−′

ξt

iπ

ξt

iπ

21−1−1−1−()

21−1−1−1−(′)

kkq

ξt

tqCn

kkq

ξt

qCn

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

In light of the Definition 5, we attain ≤

˜() ˜(′

)

,thatis,Cq ROF TLMS

−2

   ≤

(

)

Cq ROF TLMSM(, ,…, ) −2′,′,…, ′

12 12

. That proves the monotonicity

of Cq‐ROF2TLMSM operator.

□

Proof.

(3) Boundedness: Due to the idempotentency and monotonicity of the propounded

Cq‐ROF2TLMSM operator, the following results can be attained:

For



≥min={}

ttt

−

, one has

     ≥Cq ROF TLMSM Cq ROF TLMSM−2(,,…,)−2(,,…,)=

n12 −− − −

For



≤max={}

ttt

, one has

     ≤Cq ROF TLMSM Cq ROF TLMSM−2(,,…,)−2(,,…,)=

n12 ++ + +

Accordingly,



  ≤≤Cq ROF TLMSM−2(,,…,)

−12 +.

RONG ET AL.

□

In the next, several novel operators shall be attained through assigning diverse values

(1) If k=

, the Cq‐ROF2TLMSM operator is yielded to Cq‐ROF2TL arithmetic averaging

(Cq‐ROF2TLAA) operator, displayed as below:

   



∑∏

∏

⊕⊕

∏

≤≤

()

() ()

()

Cq ROF TLMSM nn

−2,,…,=

1=1(let = )

=Δ1Δ(,ϱ),

1−1−,

nrn rr

rrr

iπ

(1) 12 1=1

−1=1

21−1−

⎛

⎝

⎜⎞

⎠

⎟⎛

⎝

⎜⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(25)

(2) If k=

, the Cq‐ROF2TLMSM operator is yielded to Cq‐ROF2TL Bonferroni mean

(Cq‐ROF2TLBM) operator, displayed as below:

 





⊕⊗ ⊕⊗

≤≤

≠

() ()

Cq ROF TLMSM Cnn

−2(,,…,)= =

(−1)

rr n t r

nrr

(2) 12

1<< =1

2,=1

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

 



∑

∏

≠

()()

()

()()

()

()()

()

()()

nn ss

Cq ROF TLBM

Δ1

(−1) Δ,ϱΔ,ϱ,

1−1−,

1−1−1−1−1−

=−2(,,…,).

rrrr

iπ

,=1

−1−1

,=1

21−(1−)

,=1

21−1−1−1−1−

(1,1) 12

nn q

1122

(−1)

1,2=1

(−1)

1,2=1

(−1)

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(26)

RONG ET AL.

(3) If

k=3

, the Cq‐ROF2TLMSM operator is yielded to Cq‐ROF2TL generalized Bonferroni

mean (Cq‐ROF2TLGBM) operator, displayed as below:

 





 

⊕⊗

⊕⊗⊗

≤≤

≠≠

()

() ()

Cq ROF TLMSM C

nn n

Cq ROF TLBM

−2(,,…,)=

(−1) −2

=−2(,,…,).

rrrn t r

rrr

rr r

rrr

(3) 12

1<< =1

,,=1

111

(1,1,1) 12

ijp

123

123 123

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(27)

(4) If kn=, the Cq‐ROF2TLMSM operator is yielded to Cq‐ROF2TL geometric mean

(Cq‐ROF2TLGM) operator, displayed as below:

 





∏

⊗

∏

()

() ( )

()

Cq ROF TLMSM

−2,,…,

==ΔΔ(,ϱ),

1−1−

iπ

() 12

−1

21−1−

qnq

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

(28)

In what follows, the dual operator of Cq‐ROF2TLMSM operator named Cq‐ROF2TLDMSM

operator is presented on the basis of combining the Cq‐ROF2TLS and DMSM operator named

Cq‐ROF2TLDMSM operator.

Definition 15. Let ab





()

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family of

Cq‐ROF2TLNs. A mapping:

→Φ

called Cq‐ROF2TLDMSM operator is defined by

  ⊗⊕

≤⋯ ≤

Cq ROF TLDMSM k

−2(,,…,)=

knrrn

() 12 1<< =1

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎟(29)

where

stands for the collection of Cq‐ROF2TLNs,

is a parameter,

rr r,,…,

are

integer

values taken from the collection n{1, 2, …, } of ninteger values,

stands for the binomial

coefficient.

Theorem 6. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

tt be a family of

Cq‐ROF2TLNs, the fusion result of

 {, ,…, }

tn2by using Cq‐ROF2TLMSM

operator is expressed as follows:

RONG ET AL.

 



∏∑

∏∏

≤⋯ ≤

()

Cq ROF TLDMSM

−2(,,…,)

=Δ1Δ(,ϱ),

1−1−1−1−,

1−1−

rrn

ξt

iπ

ξt

qiπ

() 12

1<< =1

−1

21−1−1−1−

21−1−

kkq

ξt

qCn

kkq

kqk

ξt

qCn

kqk

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(30)

where

denotes the subscript ≤⋯≤rrn

(

1<<

)

k1.

The proof is omitted because of the similarity with Theorem 4.

Theorem 7. Assume ab





(( ))

se e=(,ϱ),

ttiπtiπ22

and ab





se e

′=(( ′,ϱ′)( ′,′))

ttiπtiπ22

′′

be two families of Cq‐ROF2TLNs, then the Cq‐ROF2TDMSM operator possesses following

features:

(1) Idempotentency: If all Cq‐ROF2TLNs are equal, that is,



==



(

()

)

se e(,ϱ), ,

iπiπ22

, for all

, then

  Cq ROF TLDMSM−2(,,…,)=

n12

(2) Monotonicity: If

aabb

≤≤≥≤

()

(

,ϱ)′,ϱ′,′,′,

tttttttt′

and . Then

 

 ≥

()

Cq ROF TLDMSM

−2(,,…,)

−2′,′,…, ′

(3) Boundedness: Let



tn(=1,2,…, )

tbe a family of Cq‐ROF2TLNs, and



 =min{ }, =max{ }

tt tt

−+

. Then



  ≤≤Cq ROF TLDMSM−2(,,…,).

−12 +

RONG ET AL.

The proof is omitted because of the similarity with Theorem 6.

Similar to Cq‐ROF2TLMSM operator, a few novel operators shall be attained through

assigning diverse parameter values of

(1) If k=

, the Cq‐ROF2TLDMSM operator is yielded to Cq‐ROF2TLGM operator, displayed

as below:

  



∏

⊗

∏

()

Cq ROF TLDMSM

−2(,,…,)=

=ΔΔ(,ϱ),

1−1−

iπ

(1) 12 =1

−1

21−1−

qnq

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

(31)

(2) If k=

, the Cq‐ROF2TLDMSM operator is yielded to Cq‐ROF2TL geometric Bonferroni

mean (Cq‐ROF2TLGBM) operator, displayed as below:

    

 



∑

∏

⊗⊕ ⊗⊕

∏

≤⋯≤

≠

()()

()

() ( )

()()

()

Cq ROF TLDMSM k

Cq ROF TLDBM

−2(,,…,)=

1=1

Δ1

2Δ,ϱ+Δ,ϱ,

1−1−1−1−1−

1−1−

=−2(,,…,).

nrrn

rrr

krr

rrrr

iπ

(2) 12 1<< =1 ,=1

,=1

−1−1

,=1

21−1−1−1−1−

,=1

21−1−

(1,1) 12

knn

nn qrr

(−1)

1122

(−1)

1,2=1

(−1)

1,2=1

(−1)

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(32)

RONG ET AL.

(3) If

k=3

, the Cq‐ROF2TLDMSM operator is yielded to Cq‐ROF2TL generalized geometric

Bonferroni mean (Cq‐ROF2TLGGBM) operator, displayed as below:

  



 

⊗⊕

⊗⊕⊕

≤⋯ ≤

≠≠

()

Cq ROF TLDMSM k

Cq ROF TLBM

−2(,,…,)=

=−2(,,…,).

()

nrrn

rrr

krrr

(3) 12 1<< =1

,,=1

(1,1,1) 12

nn n

122

123

(−1) −2

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(33)

(4) If kn=, the Cq‐ROF2TLDMSM operator is yielded to Cq‐ROF2TLAM operator, displayed

as below:

   



∑∏

∏

⊕⊕

∏

≤≤

() () ()

Cq ROF TLDMSM nn

−2(,,…,)=

1=1(set = )

=Δ1Δ(,ϱ),

1−1−,

nnrn rr

rrr

iπ

() 12 1=1

−1=1

21−1−

⎛

⎝

⎜⎞

⎠

⎟⎛

⎝

⎜⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

(34)

4.2 |The Cq‐ROF2TLWMSM and Cq‐ROF2TLWDMSM operator

The weight of attribute is an indispensable indicator in settling practical decision issues. In

light of the situation in which Cq‐ROF2TLMSM operator fails to take into account the im-

portance of the fused data, we propound the Cq‐ROF2TLWMSM operator and Cq‐

ROF2TLWDMSM operator to make up the faultiness of Cq‐ROF2TLMSM operator as below.

Definition 16. Assume ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a set of

Cq‐ROF2TLNs,

is the weight of



tmeeting ∈

[0, 1]

tand

∑

=1 . The

Cq‐ROF2TLWMSM operator is expressed as below:

 

⊕⊗

≤⋯ ≤

()

Cq ROF TLWMSM C

−2(,,…,)=

()

rrnt

krw

() 12

1<< =1

ktrtk

⎛

⎝

⎜

⎞

⎠

⎟

⎟(35)

RONG ET AL.

where

stands for the collection of Cq‐ROF2TLNs,

is a parameter and

knrrr=1,2,…, , , ,…,

12 are

integer values taken from the collection n{1, 2, …, } of n

integer values,

stands for the binomial coefficient.

Theorem 8. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a collection of

Cq‐ROF2TLNs, the fusion result of

 {, ,…, }

tn2by Cq‐ROF2TLMSM operator is

expressed as follows:

 



∏∏

∑∏

∏∏

()

Cq ROF TLWMSM

−2(,,…,)

=Δ

Δ(,ϱ)

1−1−,

1−1−1−1−

ξt

krr

ξt

qiπ

ξt

iπ

() 12

=1 −1

21−1−

21−1−1−1−

rtk

kqk

ξt

wrt

qCn

kqk

rtCn

kkq

ξt

qwrtCn

kkq

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(36)

where

denotes the subscript ≤⋯≤rrn

(

1<<

)

k1.

Proof. In light of the basic algorithms of Cq‐ROF2TLNs, we have





∏∏ ∏

∏

()

()()

() ()

()

()=ΔΔ(,ϱ), ,

1−1−

wiπ

qwiπ

=1 =1

−1

21−1−

trttt

rtt

wrt

qwrtq

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

RONG ET AL.

and





∑∏ ∑∏

∏∏

≤⋯≤ ≤⋯≤

()

() =ΔΔ(,ϱ),

1−1−,

1−1−

rrnt

ξt

qiπ

ξt

qwiπ

<< =1 1 << =1

−1

21−1−

trt

ξt

wrt

ξt

qwrtq

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

Then





∏∏

∑∏ ∑∏

∏∏

≤⋯ ≤

()

=Δ

Δ(,ϱ)

1−1−,

1−1−

rrnt

krw

ξt

krr

ξt

qiπ

ξt

qwiπ

1<< =1 =1 −1

21−1−

ktrttt

ξt

wrt

qCn

ξt

qwrtqCn

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

RONG ET AL.

Accordingly,





∏∏

∑∏ ∑∏

∏∏

≤⋯ ≤

()

() ()

()

=Δ

Δ(,ϱ)

1−1−,

1−1−1−1−

rrnt

krw

ξt

krr

ξt

qiπ

ξt

iπ

1<< =1 =1 −1

21−1−

21−1−1−1−

ktrtk

rtk

kqk

ξt

wrt

qCn

kqk

rtCn

kkq

ξt

qwrtCn

kkq

111

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

□

In the next, several novel operators shall be attained through assigning diverse values of

(1) If k=

, the Cq‐ROF2TLWMSM operator is yielded to Cq‐ROF2TL weighted averaging

operator, displayed as below:

   



∑

∏

⊕⊕

∏

≤≤

()

() ()

()

Cq ROF TLMSM nn

−2(,,…,)=

1=1(=)

=Δ1Δ(,ϱ),

1−1−,

1−1−

nrn

qw iπ

(1) 12 1=1

−1

21−1−

qwrnq

rqn

qwrqn

⎛

⎝

⎜⎞

⎠

⎟⎛

⎝

⎜⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(37)

RONG ET AL.

(2) If k=

, the Cq‐ROF2TLWMSM operator is yielded to Cq‐ROF2TL weighted BM operator,

displayed as below:

 





⊕⊗

≤≤

≠

()

Cq ROF TLWMSM C

−2(,,…,)=

(−1)

rr n t r

(2) 12

1<< =1

,=1

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

 



∑

∏

≠

()

()()

()

() ()

()()

nn ss e

Cq ROF TLWBM

Δ1

(−1) Δ,ϱΔ,ϱ,

1−1−

1−1−1−1−1−

−2(,,…,).

qwr

−1−1

11−2

iπ

,=1

−1−1

,=1

21−(1−(( ) ( ) ) )

,=1

21−1−

(1,1) 12

nn q

rwrrwrq

(−1)

1,2=1

1122

(−1)

1,2=1

(−1)

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(3) If

k=3

, the Cq‐ROF2TLWMSM operator is yielded to Cq‐ROF2TL generalized BM

(Cq‐ROF2TLGBM) operator, displayed as below:

 





 

⊕⊗

⊕⊗⊗

≤≤

≠≠

()

() ()

Cq ROF TLWMSM C

nn n

Cq ROF TLWBM

−2(,,…,)=

(−1) −2

=−2(,,…,).

rr n t r

rrr

rr r

(3) 12

1<< =1

,,=1

(1,1,1) 12

ijp

rrr

123

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

(38)

RONG ET AL.

(4) If kn=, the Cq‐ROF2TLMSM operator is yielded to Cq‐ROF2TL geometric mean operator,

displayed as below:

  



∏

⊗

∏

()

Cq ROF TLWMSM

−2(,,…,)=

=ΔΔ(,ϱ),

() ,

1−1−

nnt

twiπ

qwiπ

() 12 =1

−1

21−1−

wtn

qwtnq

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(39)

In what follows, the dual operator of Cq‐ROF2TLWMSM operator named

Cq‐ROF2TLWDMSM operator is presented on the basis of combining the Cq‐ROF2TLS and

DMSM operator named Cq‐ROF2TLDMSM operator.

Definition 17. Assume ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family of

Cq‐ROF2TLNs,

is the weight of



tmeeting ∈

[0, 1]

tand

∑

=1 . The

Cq‐ROF2TLWDMSM operator is expressed as below:

  ⊗⊕⊗

≤⋯ ≤

Cq ROF TLWDMSM kw−2(,,…,)=

nrrn

rr12 1<< =1

ktt

⎛

⎝

⎜

⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

⎟(40)

In view of the fundamental operations of Cq‐ROF2TLNs, we can attain the following

Theorems.

Theorem 9. Let ab





(( ))

se e t n=(,ϱ), (=1,2,…,

)

ttiπtiπ22

be a family of

Cq‐ROF2TLNs, the aggregation value of

 {, ,…, }

n12 produced by Cq‐ROF2TLWDMSM

operator is still a Cq‐ROF2TLN and

RONG ET AL.

 



∏∑

∏∏

≤⋯≤

()

Cq ROF TLWDMSM

kws

−2(,,…,)

=Δ1Δ(,ϱ),

1−1−1−1−,

1−1−

rrn

ξt

iπ

ξt

qiπ

() 12

1<< =1

−1

21−1−1−1−

21−1−

ttt

rtCn

kkq

ξt

qwrtCn

kkq

rtCn

kqk

ξt

wrt

qCn

kqk

111

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

⎞

⎠

⎟

(41)

where

denotes the subscript ≤⋯≤rrn

(

1<<

)

k1.

The proof is omitted because of the similarity with Theorem 9.

5|THE PROPOUNDED MAGDM APPROACH

This part gives the depiction of traditional MAGDM issue and designs an innovative MAGDM

algorithm based upon the propounded Cq‐ROF2TLWMSM operators to cope with actual de-

cision issues.

This section shall establish a MAGDM approach on the basis of the propounded

Cq‐ROF2TLMSM operator under the Cq‐ROF2TL setting. For a classical MAGDM issues: suppose

ϒ={ϒ,ϒ,…,ϒ}

m12 be a family of alternatives,

 ={ , ,…, }

n12

be a family of attributes,

and the associated weights of attributes are

ww w={ , ,…, }

with

∑

=1 .Supposethe

set of evaluators is



 ={ , ,…, }

p12 , and the weights of evaluators are

ωω ω={ , ,…, }

with

∑

ω=

pς

=1 . The evaluator



ςp(=1,2,…,)

ςprovides his (her) assessment information for

alternative jmϒ(=1,2,…, )

jwith respect to the attribute tn(=1,2,…, )

tby the form of

Cq‐ROF2TLN, which is denoted as ab





()

see=,ϱ,,

ςjt

ςiπjt

ςiπ

ς. The decision

matrices are indicated as



()

ςjt

mn×. In order to attain the satisfies alternative, we build up a

novel MAGDM method to sort the alternatives based upon the presented Cq‐ROF2TLMSM

operator. The detailed procedure includes the following steps:

RONG ET AL.

Step 1: Standardize the DM matrices by the following transformation manner;







()

see

,ϱ, , , is benefit attribute ;

,ϱ, , , , is cost attribute .

ςjt

ςiπjt

ςiπt

ςjt

ςiπjt

ςiπt

⎧

⎨

⎪

⎩

⎪

(42)

Step 2: Utilize the Cq‐ROF2TLWMSM operator or Cq‐ROF2TLWDMSM to integrate all

individual DM matrices 

()

ςp=(=1,2,…,)

ςjt

mn×into a single DM matrix 

=( )

jt m n×:



 

(

)

Cq ROF TLWMSM=−2,,…,

jt jt jt jt

12 (43)



 

()

Cq ROF TLWDMSM=−2,,…,

jt jt jt jt

(44)

Step 3: Utilize the Cq‐ROF2TLWMSM operator or Cq‐ROF2TLWDMSM to fuse linguistic

evaluation information



tn(=1,2,…, )

jt into the comprehensive assessment value of alter-

natives jmϒ(=1,2,…, )



 Cq ROF TLWMSM=−2(,,…,

)

jjtjtjt

(45)



 Cq ROF TLWDMSM=−2(,,…,)

jjtjtjt

(46)

Step 4: Compute the score function 

˜(

)

jof each assessment value jmϒ(=1,2,…, )

through Definition 11, if the score values of diverse alternatives are same, we further calculate

the accuracy value of them;

Step 5: Sort jmϒ(=1,2,…, )

jin light of the comparison approach in Definition 12 and

choice the optimal alternative(s).

Step 6: End.

6|NUMERICAL EXAMPLE

In this section, a practical example is provided to verify the applicability of the designed

approach. Then, the sensitiveness of the

and

is studied. Also, the comparison analysing is

implemented to show the merits of the developed method.

Example 1. Emergency management is a proper term for dealing with the risk of

major accidents and disasters, which refers to the government and other public

institutions in the process of prevention, response, disposal and recovery of emergencies,

through the establishment of necessary response mechanism, take a series of necessary

measures, the use of science, technology, planning and management means, to ensure

the safety of emergencies related activities public life, health and property, and

promoting the harmonious and healthy development of society. In recent years, the

RONG ET AL.

frequent occurrence of natural disasters has caused great harm and loss to human life

and global economy. In order to effectively reduce the losses caused by major accidents

and disasters, the emergency management center will formulate different emergency

alternatives according to the types of accidents and invite experts in various fields to evaluate

the alternative emergency plans. The evaluation of emergency alternative is an important

process in emergency management. Its core is the classic DM problem, which has received a

multitude of attention from a large number of scholars. Therefore, in this part, we will apply

the propound approach to handle the evaluation problems of selecting the optimal

emergency alternative for the emergency management centre. After a series of screening,

four better alternatives will conduct further assessment. The four alternatives

{ϒ,ϒ,ϒ,ϒ}

123 4

will be assessed through three evaluators

{, , }

123

utilizing the linguistic term



sssss= { = very bad, = bad, = fair, = good, =very good }

01234

.Thefollowingfour

attributes are taken into consideration for valid modeling the characteristic of alternatives

by discussing from experts, which are displayed as: 

1preparation ability, :

2rescuing

ability, 

3restore ability, and :

4reaction capacity. Suppose the weight vector of

 {, , , }

123 4

= {0.22, 0.30, 0.20, 0.28}

Tand the weight vector of

{, , }

123

= {0.40, 0.35, 0.25}

. The individual decision matrices are provided by

{, , }

123

accordance to their knowledge level and viewpoint, which are displayed as Tables 1,2,and3.

6.1 |DM procedure

Step 1: The standardization process is omitted because all attributes are the benefit type.

Step 2: Utilize the Cq‐ROF2TLWMSM operator or Cq‐ROF2TLWDMSM to integrate all

individual DM matrices into a single DM matrix, which are displayed in Tables 4and 5

(set k=

and

Step 3: Utilize the Cq‐ROF2TLWMSM operator or Cq‐ROF2TLWDMSM to fuse linguistic

evaluation information



tn(=1,2,…, )

jt in Tables 4or 5and attain the comprehensive as-

sessment value of alternatives jmϒ(=1,2,…, )

j, which is displayed in Table 6(set k=

and

Step 4: Calculate score values 

˜(

)

jof each alternative jmϒ(=1,2,…, )

jthrough Definition

11, which are listed in Table 7.

TABLE 1 The individual decision making matrix

provided by



12



1se

(,0),0.30 ,

0.20

iπ

2 (0.50)

2 (0.40)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.25 ,

0.35

iπ

2 (0.56)

2 (0.40)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.20 ,

0.45

iπ

2 (0.70)

2 (0.28)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.25 ,

0.15

iπ

2 (0.65)

2 (0.20)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.45 ,

0.28

iπ

2 (0.55)

2 (0.42)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.26 ,

0.32

iπ

2 (0.46)

2 (0.36)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.31 ,

0.43

iπ

2 (0.31)

2 (0.48)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.30 ,

0.18

iπ

2 (0.58)

2 (0.32)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.40 ,

0.28

iπ

2 (0.29)

2 (0.22)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.29 ,

0.35

iπ

2 (0.64)

2 (0.29)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.35 ,

0.51

iπ

2 (0.41)

2 (0.28)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.37 ,

0.19

iπ

2 (0.39)

2 (0.42)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.43 ,

0.20

iπ

2 (0.65)

2 (0.12)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.29 ,

0.30

iπ

2 (0.56)

2 (0.29)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.38 ,

0.44

iπ

2 (0.61)

2 (0.33)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.33 ,

0.10

iπ

2 (0.48)

2 (0.39)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

RONG ET AL.

Step 5: Sort the alternatives jmϒ(=1,2,…, )

jand obtain the optimal alternative, which is

displayed in Table 8.

6.2 |Sensitivity analysis

In the above‐mentioned computations, the parameters have a pivotal role for the decision

procedure and strategy affects the final decision result. Therefore, we shall conduct a parameter

analysing for the parameter

and

in this subsection. For convenience, we utilize

Cq‐ROF2TLWMSM operator to resolve the mentioned example to achieve the analysis of this

subsection. To reveal the sensitivity of parameters

and

for the ultimate decision outcomes,

we utilize the propounded Cq‐ROF2TLWMSM operator with dissimilar parameter values to

cope with the aforementioned actual example and attain the ranking results, which are re-

spectively listed in Table 9(when

) and Table 10 (when k=

From Table 9, we can discover that the sorting outcomes of alternatives are slightly diverse

when the parameter

is taken dissimilar values through the risk preference of decision makers.

TABLE 2 The individual decision making matrix

provided by

2

12



1se

(,0),0.20 ,

0.26

iπ

2 (0.80)

2 (0.17)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.15 ,

0.42

iπ

2 (0.55)

2 (0.35)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.20 ,

0.69

iπ

2 (0.37)

2 (0.48)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.22 ,

0.24

iπ

2 (0.56)

2 (0.39)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.37 ,

0.39

iπ

2 (0.34)

2 (0.28)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.26 ,

0.24

iπ

2 (0.38)

2 (0.19)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.43 ,

0.32

iπ

2 (0.19)

2 (0.47)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.20 ,

0.17

iπ

2 (0.47)

2 (0.28)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.10 ,

0.37

iπ

2 (0.27)

2 (0.38)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.36 ,

0.46

iπ

2 (0.28)

2 (0.55)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.41 ,

0.37

iπ

2 (0.46)

2 (0.29)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.28 ,

0.36

iπ

2 (0.59)

2 (0.37)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.39 ,

0.18

iπ

2 (0.64)

2 (0.25)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.17 ,

0.27

iπ

2 (0.60)

2 (0.34)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.18 ,

0.55

iπ

2 (0.49)

2 (0.38)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.24 ,

0.57

iπ

2 (0.45)

2 (0.41)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

TABLE 3 The individual decision making matrix

3provided by



12



1se

(,0),0.61 ,

0.33

iπ

2 (0.40)

2 (0.38)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.37 ,

0.24

iπ

2 (0.65)

2 (0.22)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.29 ,

0.59

iπ

2 (0.57)

2 (0.32)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.22 ,

0.21

iπ

2 (0.42)

2 (0.39)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.47 ,

0.36

iπ

2 (0.36)

2 (0.27)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.32 ,

0.23

iπ

2 (0.38)

2 (0.40)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.17 ,

0.30

iπ

2 (0.58)

2 (0.44)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.29 ,

0.15

iπ

2 (0.49)

2 (0.46)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.37 ,

0.30

iπ

2 (0.36)

2 (0.39)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.28 ,

0.45

iπ

2 (0.29)

2 (0.45)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.21 ,

0.43

iπ

2 (0.57)

2 (0.31)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.19 ,

0.27

iπ

2 (0.11)

2 (0.74)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.22 ,

0.10

iπ

2 (0.64)

2 (0.25)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.36 ,

0.29

iπ

2 (0.19)

2 (0.56)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.29 ,

0.56

iπ

2 (0.53)

2 (0.83)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.38 ,

0.57

iπ

2 (0.56)

2 (0.26)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

RONG ET AL.

TABLE 4 Collective decision making matrix by utilizing Cq‐ROF2TLWMSM operator

123



( , 1.3292), 0.6949 ,

0.1822

iπ

2 (0.8289)

2 (0.2353)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.4964), 0.6626 ,

0.5473

iπ

2 (0.8354)

2 (0.2404)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.2605), 0.6119 ,

0.4156

iπ

2 (0.8194)

2 (0.2584)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.4446), 0.6129 ,

0.1399

iπ

2 (0.8399)

2 (0.2333)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.2605), 0.7548 ,

0.2393

iπ

2 (0.7505)

2 (0.2345)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.3990), 0.6551 ,

0.1879

iπ

2 (0.7441)

2 (0.2297)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.1696), 0.6718 ,

0.2515

iπ

2 (0.6909)

2 (0.3266)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.4096), 0.6728 ,

0.1171

iπ

2 (0.7907)

2 (0.2464)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.4964), 0.6544 ,

0.2220

iπ

2 (0.6739)

2 (0.2329)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.1297), 0.6802 ,

0.2929

iπ

2 (0.7403)

2 (0.3092)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.4711), 0.6891 ,

0.3107

iπ

2 (0.7795)

2 (0.2026)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.3234), 0.6334 ,

0.1947

iπ

2 (0.6821)

2 (0.3706)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.4499), 0.7060 ,

0.1201

iπ

2 (0.8643)

2 (0.1478)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.4440), 0.6440 ,

0.1993

iπ

2 (0.7672)

2 (0.2795)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.3352), 0.6575 ,

0.3626

iπ

2 (0.8183)

2 (0.3868)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.3341), 0.7059 ,

0.3353

iπ

2 (0.8138)

2 (0.2583)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

RONG ET AL.

TABLE 5 Collective decision making matrix by utilizing Cq‐ROF2TLWDMSM operator

123



1se

(,−0.1907), 0.2726 ,

0.6397

iπ

2 (0.4379)

2 (0.6782)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.1190), 0.1849 ,

0.6989

iπ

2 (0.4141)

2 (0.6902)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.3362), 0.1585 ,

0.8045

iπ

2 (0.4076)

2 (0.7215)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.2374), 0.1608 ,

0.5886

iπ

2 (0.4031)

2 (0.6868)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.3362), 0.3009 ,

0.7014

iπ

2 (0.3046)

2 (0.6901)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,−0.0951), 0.1928 ,

0.6450

iπ

2 (0.2880)

2 (0.6773)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,−0.4476), 0.2330 ,

0.7141

iπ

2 (0.2655)

2 (0.7608)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.3276), 0.1856 ,

0.5556

iπ

2 (0.3663)

2 (0.7017)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.1190), 0.2289 ,

0.6845

iπ

2 (0.2117)

2 (0.6895)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.4383), 0.2183 ,

0.7489

iπ

2 (0.3099)

2 (0.7539)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.1972), 0.2417 ,

0.7513

iπ

2 (0.3347)

2 (0.6757)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.2247), 0.2087 ,

0.6509

iπ

2 (0.3077)

2 (0.7841)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.0591), 0.2614 ,

0.5456

iπ

2 (0.4608)

2 (0.5925)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,−0.0043), 0.1960 ,

0.6623

iπ

2 (0.3693)

2 (0.7244)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,−0.1427), 0.2076 ,

0.7520

iπ

2 (0.3883)

2 (0.7315)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,−0.2146), 0.2220 ,

0.7382

iπ

2 (0.3471)

2 (0.7109)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

RONG ET AL.

The dissimilar parameter values

reflect the interrelationship of different attributes during the

decision making process. For instance, when we take k=

in Cq‐ROF2TLWMSM operator, the

order relation of alternatives is ϒ>ϒ>ϒ>ϒ

4123

, which is different from other situations.

Because the Cq‐ROF2TLWMSM operator shall transform into Cq‐ROF2TLWA operator when

the parameter is assigned k=

, the correlation of discussed attributes fail to consider in the

course of dealing decision issues. When decision issues need to take consideration the

TABLE 9 The score value and order relation of alternatives based upon diverse

values

Score value of alternative

Parameter value

ϒ4

Sorting Optimal selection

0.6021 0.5471 0.4973 0.6111 ϒ>ϒ>ϒ>ϒ

412

0.8752 0.8474 0.8268 0.8720 ϒ>ϒ>ϒ>ϒ

142

1.0839 1.0668 1.0465 1.0727 ϒ>ϒ>ϒ>ϒ

142

TABLE 6 The integrated assessment information by utilizing Cq‐ROF2TLWMSM and Cq‐ROF2TLWDMSM

operator

Alternative Cq‐ROF2TLWMSM operator Cq‐ROF2TLWDMSM operator

( , 0.1098), 0.8935 ,

0.1606

iπ

2 (0.9552)

2 (0.1521)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.2419), 0.9082 ,

0.1241

iπ

2 (0.9130)

2 (0.2637)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.0908), 0.9109 ,

0.1275

iπ

2 (0.9300)

2 (0.1632)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.2164), 0.8985 ,

0.1439

iπ

2 (0.9169)

2 (0.1958)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.0972), 0.9031 ,

0.1628

iπ

2 (0.9209)

2 (0.1839)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.2171), 0.9179 ,

0.1408

iπ

2 (0.9248)

2 (0.1872)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

( , 0.0927), 0.9080 ,

0.1701

iπ

2 (0.9502)

2 (0.1767)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

( , 0.2293), 0.9070 ,

0.1398

iπ

2 (0.9124)

2 (0.2479)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

TABLE 7 The score values of alternatives

Operator

ϒ4

Cq‐ROF2TLWMSM operator 

˜( ) = 0.875

1

˜( ) = 0.847

2

˜( ) = 0.826

3

˜( ) = 0.8720

Cq‐ROF2TLWDMSM operator 

˜( ) = 0.180

1

˜( ) = 0.160

2

˜( ) = 0.168

3

˜( ) = 0.170

TABLE 8 The order relation of alternatives

Operator Ranking relation of j

( = 1, 2, 3, 4)

jOptimal selection

Cq‐ROF2TLWMSM operator ϒ>ϒ>ϒ>ϒ

142

Cq‐ROF2TLWMSM operator ϒ>ϒ>ϒ>ϒ

143

RONG ET AL.

interconnection among any input data in assessment process, evaluators can take k=

k=3

in Cq‐ROF2TLWMSM operator to aggregation assessment information. For different

parameter values of

, DMs can choose an appropriate parameter value through their pre-

ference attitude and further attain a reasonable and satisfied decision result.

From Table 10, it is obvious that the orders of alternatives are the same for diverse para-

meter

by utilizing Cq‐ROF2TLWMSM operator, which testify the decision procedure is stable.

The values of

stand for the space of the assessment information for DMs. As the parameter

increases, DMs can provide more evaluation information according to their preference.

Moreover, we can easily acquire that the score value of alternatives on the basis of

Cq‐ROF2TLWMSM operator gets smaller and smaller with the value of

increases.

6.3 |Comparison analysis

6.3.1 |Verification of validity

To elaborate the validity and practicability of the created method in this essay, we conduct a

collection of comparative analysing with other previous decision methodologies including the

method based upon complex intuitionistic fuzzy weighted averaging (CIFWA) operator pro-

posed by Garg,

the method based upon complex intuitionistic BM (CIFBM) operator proposed

by Garg

and the method based upon complex q‐rung orthopair fyzzy linguistic Heronian

mean (Cq‐ROFLHM) operator proposed by Liu et al.

We utilize these methods to cope with

the Example 1in this paper, the score values and ranking of alternatives are displayed in

Table 11. From it, we can attain the same sorting results of alternatives based on the previous

methods and the designed method in this article, which can verify the availability of the

propounded method.

6.3.2 |Generalization analysis

In the next, a numerical case is utilized to expound the generalization of the presented method.

The evaluation information provided by DMs in respect of complex q‐rung orthopair linguistic

numbers, which are displayed in Table 12, and the weights of attributes are the same as with

Example 1. Then the case is settled through our method in this article and the decision results

are displayed in Table 13.

TABLE 10 The score value and order relation of alternatives based upon diverse

values

Score value of alternative

Parameter value

ϒ4

Sorting Optimal selection

0.8764 0.8483 0.8275 0.8750 ϒ>ϒ>ϒ>ϒ

142

0.8762 0.8481 0.8269 0.87430 ϒ>ϒ>ϒ>ϒ

142

0.8736 0.8474 0.8268 0.8720 ϒ>ϒ>ϒ>ϒ

142

0.8752 0.8464 0.8248 0.8661 ϒ>ϒ>ϒ>ϒ

142

0.8704 0.8440 0.8208 0.8588 ϒ>ϒ>ϒ>ϒ

142

RONG ET AL.

TABLE 11 The score value and order relation of alternatives based upon diverse

values

Score value of alternative

Approaches

ϒ4

Sorting

CIFWA operator presented in Garg and Rani

0.7027 0.6900 0.6663 0.7023 ϒ>ϒ>ϒ>ϒ

1423

CIFBM operator presented in Garg and Rani

(pq==

) 0.8752 0.8474 0.8268 0.8720 ϒ>ϒ>ϒ>ϒ

1423

Cq‐ROFLHM operator presented in Liu et al.

(

st q==1, =3

) 0.3848 0.3509 0.3280 0.3577 ϒ>ϒ>ϒ>ϒ

1423

Cq‐ROF2TLMSM operator presented in this essay (

) 0.8764 0.8483 0.8275 0.8750 ϒ>ϒ>ϒ>ϒ

1423

Cq‐ROF2TLMSM operator presented in this essay (

) 0.8736 0.8474 0.8268 0.8720 ϒ>ϒ>ϒ>ϒ

1423

Cq‐ROF2TLMSM operator presented in this essay (

) 0.8704 0.8440 0.8208 0.8588 ϒ>ϒ>ϒ>ϒ

1423

RONG ET AL.

From Table 13, we can find that the previous approaches like CIFWA operator,

CIFBM operator,

distance measure

can not address the Example 2, but the present

method can effectively address it. That proves the presented method is more universal

than previous approaches because the CIFS and CPFS are the special examples of our

propounded method.

6.3.3 |Further contrastive analysing

In the next, we will administer a detailed contrast between the previous works with the

designed method, the significant supremacies of our propounded method are summarized as

below.

✓Compare with the approach based upon CIFWA operator propounded by Garg and

Rani.

The CIFWA operator as a fundamental aggregation technique to integrate com-

plex intuitionistic fuzzy information assumes that the considered attributes in real‐life

problems are unrelated, that is, it deems the relevance of attributes, which the decision

result ambiguous and unreasonable. The Cq‐ROF2TLMSM operator can valid conquer

the aforementioned defect and take into consideration the relationship of attributes.

TABLE 13 The score value and order relation of alternatives based upon diverse

values

Approaches Score value of alternative Sorting

CIFWA operator presented in Garg &

Rani

Cannot be computed Cannot be computed

CIFBM operator presented in Garg &

Rani

(pq==

)

Cannot be computed Cannot be computed

Distance measure on CPFS presented in

Ullah et al.

Cannot be computed Cannot be computed

Cq‐ROF2TLMSM operator presented in

this essay (

)

˜(ϒ) = 0.8924, ˜(ϒ) = 0.7290

˜(ϒ) = 0.7036, ˜(ϒ) = 0.622

ϒ>ϒ>ϒ>ϒ

123

TABLE 12 The decision making matrix from Example 2

12



1se

(,0),0.70 ,

0.36

iπ

2 (0.80)

2 (0.95)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.55 ,

0.62

iπ

2 (0.55)

2 (0.35)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.89 ,

0.69

iπ

2 (0.37)

2 (0.48)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.78 ,

0.84

iπ

2 (0.56)

2 (0.39)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.67 ,

0.69

iπ

2 (0.34)

2 (0.28)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.56 ,

0.59

iπ

2 (0.38)

2 (0.85)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.69 ,

0.78

iπ

2 (0.59)

2 (0.47)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.87 ,

0.97

iπ

2 (0.47)

2 (0.73)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.90 ,

0.96

iπ

2 (0.27)

2 (0.67)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.68 ,

0.85

iπ

2 (0.28)

2 (0.79)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.65 ,

0.76

iπ

2 (0.46)

2 (0.488)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.81 ,

0.77

iπ

2 (0.59)

2 (0.37)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.67 ,

0.78

iπ

2 (0.64)

2 (0.25)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.57 ,

0.48

iπ

2 (0.60)

2 (0.88)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟se

(,0),0.33 ,

0.66

iπ

2 (0.49)

2 (0.38)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

(,0),0.24 ,

0.87

iπ

2 (0.45)

2 (0.81)

⎛

⎝

⎜⎛

⎝

⎜⎞

⎠

⎟

⎞

⎠

⎟

RONG ET AL.

Additionally, it can reflect DM's individual favourites and show the dynamic trend of

the order relation of alternatives through the adjustable parameter. Accordingly, the

Cq‐ROF2TLMSM operator are more efficient and general to process decision analysis

problems.

✓Compare with the approach based on CIFBM operator proffered via Garg and Rani.

Although CIFBM operator can be utilized to aggregation complex intuitionistic fuzzy in-

formation, it only takes into consideration the correlation between any two attributes. The

Cq‐ROF2TLMSM operator proffered in this essay not only catches the correlation among

multiple attributes, but also reduces the computational complexity during information

aggregation process. Furthermore, our propounded method can address actual issues from a

qualitative point of view and it settles several problems that CIFBM operator can not

resolve. Hence, the presented methodology is more universal and realistic to attain a ra-

tional decision result.

✓Compared with the approach on the basis of Cq‐ROFLHM operator proposed by

Liu et al.,

the following three aspects are given highlight the difference between

Cq‐ROFLHM and Cq‐ROF2TLMSM operator. (1) For linguistic representation: the

integration information of LV may not be matched by appropriate linguistic terms. For

instance, as= ( , (0.6, 0.8)

)

1 1.432 . Aiming at this situation, the virtual linguistic term s1.43

only utilized to comparative and computation process, but does not have any semantics to

match with it. That will lead to the information loss in information fusion process.

Nevertheless, the 2‐tuple linguistic representation model can valid prevent information

loss because the linguistic terms in 2‐tuple linguistic are continuous. (2) For the corre-

lation of attributes: the HM operator can only take into the interrelationship between any

two attributes, which will produce a irrational decision result. But the MSM operator can

consider the correlation among multiple input attributes in the procedure of information

integration. (3) For the computational‐complexity: the HM operator has two parameters,

which make the computational process become more difficult, and it is arduous for DMs

to determinate two satisfied parameter values. However, the MSM operator only has one

parameter, which is more convenient for DMs to allocate appropriate parameter

value according to actual needs and their favourites. To sum up, the designed approach

based upon Cq‐ROF2TLMSM operator is more universal and flexible than the

Cq‐ROFLHM operator.

From the above comprehensive comparative analysis, we summarize the marked char-

acteristics between the propounded method with other existing approaches, which are dis-

played in Table 14. From it, we can attain that the current methods like CIFs, CPFS and

Cq‐ROFLS are the particular cases OF Cq‐ROF2TLS. The propounded approach on the basis of

Cq‐ROF2TLMSM operators is more powerful than other previous methods to fuse fuzzy in-

formation. Consequently, the merits of the developed operators are outlined as below: (a) the

proposed operators can efficiently seize the interrelationship of multiple input arguments,

which will reduce information loss during the real‐life decision procedure and attain more

rational decision outcome; (b) the presented operators can ponder the correlation between

different quantitative parameters through assigning diverse parameter, which greatly increases

the flexibility of decision process; (c) the developed can degenerate into several extant operators

via taking dissimilar values of parameter, which signifies that the propounded operators are

more generalized and suitable to dispose actual decision issues.

RONG ET AL.

TABLE 14 Characteristic comparison with existing approaches

Approaches

Capability

to fuse

information

Capture correlation

between two

attributes

Capture correlation

among multiple

attributes

Capability to tackle

two‐dimensional

information

Capability to express

information by

complex numbers

Flexibility of

decision

procedure

The method presented in

Rani and Garg

✓✓

The method presented in

Garg and Rani

✓

××

✓✓

The method presented

in Garg and Rani

✓✓

✓✓ ✓

The method presented in

Ullah et al.

××

✓✓

The method presented in

Liu

✓

××

The method presented in

Liu et al.

✓✓

✓✓ ✓

The method presented in

Li et al.

✓✓

✓

The presented

method (

)

✓✓ ✓ ✓ ✓ ✓

The presented

method (

)

✓✓ ✓ ✓ ✓ ✓

The presented

method (

)

✓✓ ✓ ✓ ✓ ✓

RONG ET AL.

7|CONCLUSION

The Cq‐ROFS and 2‐tuple LV are two efficient models that can not only portray the complex

vague assessment information but also reduce information loss in MAGDM issues. Inspired by

the mentioned notion, we first propounded a novel conception called Cq‐ROF2TLS to express

ill‐defined and uncertain evaluation information in actual problems, which synthetically

consider the merits of Cq‐ROFS and 2‐tuple LV. Moreover, we present several aggregation

operators including Cq‐ROF2TLWA, Cq‐ROF2TLWG, Cq‐ROF2TLMSM, Cq‐ROF2TLWMSM,

Cq‐ROF2TLDMSM and Cq‐ROF2TWDLMSM operator to integrate Cq‐ROF2TL information

and explore several characteristics of them at length. Furthermore, we designed a new

MAGDM method based upon the Cq‐ROF2TLWMSM and Cq‐ROF2TLWDMSM operator, as

well as a numerical case that is used to show the efficiency and feasibility of the novel

approach. In the end, a comparative analysis between the extant methods and our approach is

administered to highlight the superiorities of the designed approach. Although this research

originally develops a novel conception to describe ambiguous and ill‐defined information and

expands its theory and application in real‐life, the following two defects of this research need to

further explore. It's easy to see that the different parameter can cause diverse rankings, thus the

first problem is how to determine the most suitable value of parameter for the presented

algorithm. The other is to use appropriate methods for attaining the weight of attribute

and expert. The presented approach assumes that the weights of attributes and experts

are provided by decision specialists in advance. For the expert weight, it only considers the

subjectively and ignores the weight information produced from the decision matrix. Thus, the

associative weight method to ascertain weight of attribute should be explored for handling

actual decision problems.

In future research, we shall apply the presented method to deal with several real‐life

problems, for instance, site selection for wind power plants,

assessment of the innovative

ability of universities

and so forth. Meanwhile, we will continue to study other theories of

CFS such ac fuzzy logic, basic operational, decision analysis and so forth. In addition, the

spherical fuzzy set,

56‐58

as a generalization of picture fuzzy set is originated to process un-

certainty and fuzziness recently. Hence, the research on spherical fuzzy set will become a

momentous key‐point in the following stage.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant

61372187, the Scientific and Technological Project of Sichuan Province under Grant

2019YFG0100. the Sichuan Province Youth Science and Technology Innovation Team under

Grant 2019JDTD0015, the Application Basic Research Plan Project of Sichuan Province under

Grant 2017JY0199, the Scientific Research Project of Department of Education of Sichuan

Province under Grant 18ZA0273 and Grant 15TD0027, the Scientific Research Project of Nei-

jiang Normal University under Grant 18TD08, the Innovation Fund of Postgraduate Xihua

University under Grant YCJJ2020028, the University Students Innovation and Entrepreneur-

ship Project of Xihua Cup under Grant 2020107. The author would like to thank the editors and

anonymous reviewers for their constructive comments and suggestions, which will help us to

better improve this paper. The author (Yuan Rong) would like to special thank the radio

management technology research center of Xihua University for its great support during the

preparation of the paper.

RONG ET AL.

CONFLICT OF INTERESTS

The authors declare that there are no conflict of interests.

AUTHOR CONTRIBUTIONS

This paper is a result of the common work of the authors in all aspects. All authors read and

agreed to the published version of the manuscript.

DATA AVAILABILITY STATEMENT

The data to sustain the application of this investigation are included within the essay.

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Feb 2020

Single-valued neutrosophic set (SVN) can valid depict the incompleteness, nondeterminacy and inconsistency of evaluation opinion, and the Power average (PA) operator can take into account the correlation of multiple discussed data. Meanwhile, Archimedean copula and co-copula (ACC) can significant generate operational laws based upon diverse copulas. In this paper, we first redefine several novel operational laws of single-valued neutrosophic number (SVNN) based on ACC and discuss the associated properties of them. In view of these operational rules, we propound several novel power aggregation operators (AOs) to fuse SVN information, i.e., SVN copula power average (SVNCPA) operator, weighted SVNCPA (WSVNCPA) operator, order WSVNCPA operator, and SVN copula power geometric (SVNCPG) operator, weighted SVNCPG (WSVNCPG) operator, order WSVNCPG operator. At the same time, several significant characteristics and particular cases of these operators are examined in detail. Moreover, we extend these operators to their generalized form named generalized SVNCPA and SVNCPG operator. In addition, a methodology is designed based on these operators to cope with multi-attribute decision-making (MADM) problems with SVN information. Consequently, the effectiveness and utility of the designed approach is validated by a empirical example. A comparative and sensitivity analysis are carried out to elaborate the strength and preponderance of the propounded approach.

Pythagorean fuzzy linguistic Muirhead mean operators and their applications to multiattribute decision-making

Article

Full-text available

Dec 2019

Pythagorean fuzzy sets, as an extension of intuitionistic fuzzy sets to deal with uncertainty, have attracted much attention since their introduction, in both theory and application aspects. In this paper, we investigate multiple attribute decision‐making (MADM) problems with Pythagorean linguistic information based on some new aggregation operators. To begin with, we present some new Pythagorean fuzzy linguistic Muirhead mean (PFLMM) operators to deal with MADM problems with Pythagorean fuzzy linguistic information, including the PFLMM operator, the Pythagorean fuzzy linguistic‐weighted Muirhead mean operator, the Pythagorean fuzzy linguistic dual Muirhead mean operator and the Pythagorean fuzzy linguistic dual‐weighted Muirhead mean operator. The main advantages of these aggregation operators are that they can capture the interrelationships of multiple attributes among any number of attributes by a parameter vector P and make the information aggregation process more flexible by the parameter vector P. In addition, some of the properties of these new aggregation operators are proved and some special cases are discussed where the parameter vector takes some different values. Moreover, we present two new methods to solve MADM problems with Pythagorean fuzzy linguistic information. Finally, an illustrative example is provided to show the feasibility and validity of the new methods, to investigate the influences of parameter vector P on decision‐making results, and also to analyze the advantages of the proposed methods by comparing them with the other existing methods.

On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition

Article

Full-text available

May 2019

The concept of complex fuzzy set (CFS) and complex intuitionistic fuzzy set (CIFS) is two recent developments in the field of fuzzy set (FS) theory. The significance of these concepts lies in the fact that these concepts assigned membership grades from unit circle in plane, i.e., in the form of a complex number instead from [0, 1] interval. CFS cannot deal with information of yes and no type, while CIFS works only for a limited range of values. To deal with these kinds of problems, in this article, the concept of complex Pythagorean fuzzy set (CPFS) is developed. The novelty of CPFS lies in its larger range comparative to CFS and CIFS which is demonstrated numerically. It is discussed how a CFS and CIFS could be CPFS but not conversely. We investigated the very basic concepts of CPFSs and studied their properties. Furthermore, some distance measures for CPFSs are developed and their characteristics are studied. The viability of the proposed new distance measures in a building material recognition problem is also discussed. Finally, a comparative study of the proposed new work is established with pre-existing study and some advantages of CPFS are discussed over CFS and CIFS.

EDAS method for multiple attribute group decision making under q-rung orthopair fuzzy environment

Article

Full-text available

Nov 2019

Extended q-rung orthopair fuzzy sets (q-ROFSs) is an excellent tool to depict the qualitative assessing information in multiple attribute group decision making (MAGDM) environments. The EDAS method is very effective especially when the conflicting attributes exist in the MAGDM issues in which the optimal alternative should have the biggest value of PDAS and the smallest value of NDAS. In this paper, we put forward the EDAS method for MAGDM issues under q-ROFSs, which makes use of average solution (AS) for assessing the chosen alternatives. The positive distance from AS (PDAS) and negative distance from AS (NDAS) is derived through the score of q-ROFSs. Then, the sorting order or the optimal alternative can be acquired by computing integrative appraisal score. Finally, a numerical example for buying a refrigerator is given to testify our developed EDAS method and some comparative analysis are also raised to further show the precious merits of this method. First published online 27 November 2019

Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making

Article

Dec 2018

The objective of this manuscript is to present some aggregation operators for aggregating the different complex intuitionistic fuzzy (CIF) sets by considering the dependency between the pairs of its membership degrees. In the existing studies of fuzzy and its extensions, the uncertainties present in the data are handled with the help of degrees of membership that are the subset of real numbers, which may lose some useful information and hence consequently affect the decision results. A modification to these, complex intuitionistic fuzzy set handles the uncertainties with the degrees whose ranges are extended from real subset to the complex subset with unit disc and hence handle the two‐dimensional information in a single set. Thus, motivated by this, we developed some new power aggregation operators, namely, CIF power averaging, CIF weighted power averaging, CIF ordered weighted power averaging, CIF power geometric, CIF weighted power geometric, and CIF ordered weighted power geometric. Also, some of the desirable properties of these are investigated. Further, based on these operators, a multicriteria decision‐making approach is presented under the CIF set environment. An illustrative example related to the selection of the best alternative(s) is considered to demonstrate the efficiency of the proposed approach and is validated it by comparing their results with the several existing approaches results.

A note on the MacLaurin Proof of the GA-Inequality

Preprint

Nov 2020

Joachim Jaeger

300 years ago, C. MacLaurin proved an equivalent form of the famous inequality between the arithmetic and geometric mean: Under all sets of n positive real numbers with the same sum the largest product is attained if all these numbers are equal. But MacLaurin showed only: If not all n numbers are equal one could increase their product by replacing the smallest and the largest element by their arithmetic mean. This equalizing step leaves the sum unchanged. MacLaurin supposed the existence of a maximal product. Hardy, Littlewood and Plya, and later Bullen, mention how a constructive proof using the equalizing step of MacLaurin could be given, however without actually carrying it out. This is what we do.

Some results on information measures for complex intuitionistic fuzzy sets

Article

Oct 2019

Complex intuitionistic fuzzy sets (CIFSs), modeled by complex‐valued membership and nonmembership functions with codomain the unit disc in a complex plane, handle two‐dimensional information in a single set. Under this environment, the primary objective of the present study is to introduce some novel formulae of information measures (similarity measures, distance measures, entropies, and inclusion measures) and discuss the transformation relationships among them. To demonstrate the efficiency of the proposed similarity measures, we apply it to pattern recognition problem and a detailed comparative analysis is conducted with some of the existing measures. Further, algorithms based on proposed measures are developed for handing multicriteria decision‐making problems and their working is illustrated with the help of an example. Besides this, the practicality of the proposed similarity measure is demonstrated by developing a clustering algorithm under CIFS environment.

A priority-based intuitionistic multiplicative UTASTAR method and its application in low-carbon tourism destination selection

Article

Mar 2020
APPL SOFT COMPUT

The intuitionistic multiplicative preference relation extends Saaty’s multiplicative preference relation by membership and non-membership functions to describe decision-makers’ preferences on objects (alternatives or criteria), and the product of pairwise membership and non-membership degrees is less than or equal to one. In recent years, the intuitionistic multiplicative preference relation has attracted increasing attention and many methods have been proposed to handle multiple criteria decision making (MCDM) problems. However, few of the existing methods can be used to tackle the MCDM problems with both quantitative and qualitative criteria. To overcome this challenge, in this paper, a novel MCDM method called the priority-based intuitionistic multiplicative UTASTAR method is proposed, which combines a new intuitionistic multiplicative prioritization method and the UTASTAR method within the context of intuitionistic multiplicative sets. The consistency checking and improving processes are considered in the weight-determining method. A case study regarding the low-carbon tourism destination selection is adopted to illustrate the applicability of the proposed method. The effectiveness and superiority of the proposed method are further verified by comparative analyses and discussions.

Complex q‐rung orthopair fuzzy 2‐tuple linguistic Maclaurin symmetric mean operators and its application to emergency program selection

Abstract

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