generalizedcomplexq-rungorthopairfuzzyeinsteinaveraging ......rani and garg [41, 42] introduced...

Complex & Intelligent Systems (2021) 7:511–538https://doi.org/10.1007/s40747-020-00197-6

ORIG INAL ART ICLE

Generalized complex q-rung orthopair fuzzy Einstein averagingaggregation operators and their application in multi-attribute decisionmaking

Peide Liu1,3 · Zeeshan Ali2 · Tahir Mahmood2

Received: 10 February 2020 / Accepted: 4 September 2020 / Published online: 3 November 2020© The Author(s) 2020

AbstractThe recently proposed q-rung orthopair fuzzy set, which is characterized by a membership degree and a non-membershipdegree, is effective for handling uncertainty and vagueness. This paper proposes the concept of complex q-rung orthopairfuzzy sets (Cq-ROFS) and their operational laws. A multi-attribute decision making (MADM) method with complex q-rungorthopair fuzzy information is investigated. To aggregate complex q-rung orthopair fuzzy numbers, we extend the Einsteinoperations to Cq-ROFSs and propose a family of complex q-rung orthopair fuzzy Einstein averaging operators, such as thecomplex q-rung orthopair fuzzy Einstein weighted averaging operator, the complex q-rung orthopair fuzzy Einstein orderedweighted averaging operator, the generalized complex q-rung orthopair fuzzy Einstein weighted averaging operator, and thegeneralized complex q-rung orthopair fuzzy Einstein ordered weighted averaging operator. Desirable properties and specialcases of the introduced operators are discussed. Further, we develop a novelMADMapproach based on the proposed operatorsin a complex q-rung orthopair fuzzy context. Numerical examples are provided to demonstrate the effectiveness and superiorityof the proposed method through a detailed comparison with existing methods.

Keywords Pythagorean fuzzy sets · Complex pythagorean fuzzy sets · q-Rung orthopair fuzzy sets · Complex q-rungorthopair fuzzy sets · Einstein aggregation operators

Introduction

The intuitionistic fuzzy set (IFS), which was pioneered byAtanassove [1], is a generalization of the fuzzy set (FS)[2]. The IFS is an important tool for coping with unreli-able and difficult information. The prominent characteristicof the IFS is that it assigns each element a membership gradeand a non-membership grade, whose sum is≤1. Therefore,the IFS has received increasing research attention since itwas proposed [3, 4]. However, in many cases, the IFS can-not work effectively, for example, when a decision maker

B Peide [email protected]

1 School of Economics and Management, Civil AviationUniversity of China, Tianjin, China

2 Department of Mathematics and Statistics, InternationalIslamic University Islamabad, Islamabad, Pakistan

3 School of Management Science and Engineering, ShandongUniversity of Finance and Economics, Jinan 250015,Shandong, China

provides information for which the aforementioned sumis>1. For example, if he/she assigns 0.52 as the member-ship grade and 0.63 as the non-membership grade, we notethat 0.52 + 0.63 �1. To deal with such problems, Yager[5] pioneered the notion of the Pythagorean FS (PFS) asa useful tool to describe uncertain and unreliable informa-tion effectively, where the sum of the square of membershipgrade and the square of the non-membership grade is lim-ited to [0,1]. Sometimes, the PFS cannot work effectively.For example, when a decision maker assigns 0.9 as themembership grade and 0.7 as the non-membership grade,0.92 + 0.72 = 0.81 + 0.47 = 1.28 > 1; thus, the PFSis not applicable. Yager [6] developed the q-rung orthopairFS (q-ROFS) for handling such situations, where the sum ofthe q-power of the membership grade and the q-power of thenon-membership grade is constrained to [0,1]. For this exam-ple, we have 0.94 + 0.74 = 0.66 + 0.24 = 0.9. Clearly, theq-ROFS is more general than the IFS and PFS (see Fig. 1).Because of its advantages, the q-ROFS is a fundamental toolfor processing troublesome fuzzy data, and it is used in thefields of aggregation operators, similarity measures (SMs),

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512 Complex & Intelligent Systems (2021) 7:511–538

Fig. 1 Geometric comparison of the q-ROFS, PFS, and IFS

hybrid aggregation operators, etc. Various studies have beenperformed on q-ROFSs, as described below.

1. Operator-based approaches: According to the aggre-gation operators, many scholars have successfullydeveloped operators for the q-ROFS environment. Forinstance, Liu and Wang [7] developed aggregation oper-ators using q-ROFSs. Garg and Chen [8] investigatedneutrality aggregation operators for q-ROFSs. Using thenew score function, Peng et al. [9] presented exponen-tial and aggregation operators based on q-ROFSs. Xinget al. [10] established point-weighted aggregation oper-ators based on q-ROFSs.

2. Measures-based approaches: The SM is useful for accu-rately examining the degree of similarity between anytwo objects. Many scholars have applied SMs in differ-ent environments. For example, Wang et al. [11] used acosine function to evaluate the SMs for q-ROFSs.Du [12]established Makowski-type distance measures based ongeneralized q-ROFSs. Peng and Liu [13] examined infor-mation measures for q-ROFSs.

3. Hybrid operator-based approaches: For investigating therelationships between objects, the hybrid aggregationoperators play an essential role in the realistic deci-sion environment. Scholars have investigated varioushybrid aggregation operators based on the q-ROFS, forexample, Archimedean Bonferroni mean operators [14],Maclaurin symmetric mean operators [15], Muirheadmean operators [16], power Maclaurin symmetric meanoperators [17],Heronianmeanoperators [18], partitionedBonferroni mean operators [19], partitioned Maclaurinsymmetric mean operators [20], and others [21–33].

Figure 1 presents the geometric representations of theIFSs, PFSs, and q-ROFSs. As shown, the q-ROFSs providethe largest amount of space for expressing the fuzzy infor-mation. Ramot et al. [34] introduced the complex FS (CFS),which extended the range of the membership grades from

real numbers to complex numbers with unit discs. The CFSis an extension of the FS for coping with uncertainty andunreliability. Additionally, Ramot et al. [35] reintroduced thenotion of complex fuzzy logic. However, the CFS is com-pletely different from the complex fuzzy number proposedbyBuckley [36]. Furthermore,Alkouri and Salleh [37] devel-oped the complex IFS (CIFS), which extended the rangeof the membership and non-membership grades from realnumbers to complex numbers with a unit disc. In contrast tothe IFS, which can process only one-dimensional informa-tion, the CIFS can cope with two-dimensional information;hence, information loss can be avoided.Kumar andBajaj [38]developed complex intuitionistic fuzzy soft setswith distancemeasures and entropies.Garg andRani [39, 40] proposed cor-relation coefficients and aggregation operators based on theCIFS. Rani and Garg [41, 42] introduced distance measuresand power aggregation operators based on the CIFS.

Recently, Ullah et al. [43] proposed the complex PFS(CPFS), which is useful for efficiently describing uncer-tain and unreliable information. A characteristic of CPFSis that the sum of the square of the real part (and imag-inary part) of complex membership grade and the squareof the real part (and imaginary part) of complex non-membership grade is limited to [0,1]. Sometimes, the CPFScannot work effectively; for example, when a decision makerprovides 0.9ei2π(0.8) for the complex membership gradeand 0.7ei2π(0.7) for the complex non-membership grade,0.92 + 0.72 = 0.81 + 0.47 = 1.28 > 1 and 0.82 + 0.62 =0.64 + 0.49 = 1.13 > 1, making the CPFS inapplicable.To solve such problems, we developed a complex q-rungorthopair FS (Cq-ROFS) characterized by complex mem-bership and non-membership grades, i.e., 0.94 + 0.74 =0.66 + 0.24 = 0.9 and 0.84 + 0.74 = 0.41 + 0.24 = 0.65.In the Cq-ROFS, the sum of the q-power of the real part (andimaginary part) of the complex membership grade and theq-power of the real part (and imaginary part) of the complexnon-membership grade is constrained to [0,1]. TheCq-ROFSis more general than the CIFS and CPFS.

Real-life situations involve numerous complex circum-stances, and data measures are useful for handling uncertaindata. Various data measures, e.g., similitude, separation, andentropy, are used in the decision-making process, for exam-ple, in design acknowledgment, clinical determination, andbunching investigation. Some existing MADM methodolo-gies for fuzzy information are based on the PFS and q-ROFS,because they have shortcomings. In the proposed Cq-ROFS,the sum of the q-power of the real part (and imaginarypart) of the complex membership grade and the q-powerof the real part (and imaginary part) of the complex non-membership grade is constrained to [0,1]. It is more generaland can express awider information scope.Next, the q-ROFSis turned into a fundamental tool to process troublesomefuzzy data. Now, we provide an example. Assume that XYZ

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Complex & Intelligent Systems (2021) 7:511–538 513

Corp. chooses to set up biometric-based participation gadgets(BBPGs) in all its workplaces throughout the nation. For this,the organization invites a specialist to provide the data withrespect to the (i) shows of BBPGs and (ii) creation dates ofBBPGs. The organization must select the optimal model ofBBPGswith its creation date at the same time. Here, the issueincludes two-dimensional information: the BBPGmodel andthe creation date of the BBPGs. Obviously, they cannot beexpressed precisely by the q-ROFS, because the q-ROFScan-not handle multiple types of one-dimensional informationsimultaneously. The sufficiency terms in the Cq-ROFS canbe utilized to determine an organization’s choice with regardto the model of the BBPGs, and the stage terms can be uti-lized to express the organization’s judgment regarding thecreation date of the BBPGs.

Keeping the advantages of the Cq-ROFS, the objectivesof this study were as follows:

1. To propose the notion of Cq-ROFS and its operationallaws, which was the basis of this study

2. To aggregate complex q-rung orthopair fuzzy numbers(Cq-ROFNs), we extend the Einstein operations (EOs)to Cq-ROFSs and propose a family of complex q-rungorthopair fuzzy Einstein averaging operators, such asthe complex q-rung orthopair fuzzy Einstein weightedaveraging (Cq-ROFEWA) operator, the complex q-rungorthopair fuzzy Einstein ordered weighted averaging(Cq-ROFEOWA) operator, the generalized complex q-rung orthopair fuzzy Einstein weighted averaging (GCq-ROFEWA) operator, and the generalized complex q-rungorthopair fuzzy Einstein ordered weighted averaging(GCq-ROFEOWA) operator. Desirable properties andspecial cases of the introduced operators are discussed.

3. To develop a novel MADM approach in the complexq-rung orthopair fuzzy context based on the proposedoperators

4. To demonstrate the effectiveness and superiority of theproposed method by comparing numerical results withthe results of existing methods

The remainder of this paper is organized as follows. Inthe next section, we briefly review existing concepts such asPYFSs, CPYFSs, q-ROFSs, and their operational laws. In thefollowing section, the concept of the Cq-ROFS and its oper-ational laws are investigated. In the next section, we extendthe EOs to Cq-ROFSs and propose a family of complexq-rung orthopair fuzzy Einstein averaging operators, suchas the Cq-ROFEWA, Cq-ROFEOWA, GCq-ROFEWA, andGCq-ROFEOWAoperators. Desirable properties and specialcases of the introduced operators are discussed. In the fol-lowing section, we develop a novel approach for MADM inthe complex q-rung orthopair fuzzy context based on the pro-posed operators. Finally, a detailed comparison is performed

between existing methods and the proposed approach. Theconclusions are presented in the last section.

Preliminaries

In this section, we briefly review existing concepts such asPYFSs, CPYFSs, q-ROFSs, and their operational laws. Inthis study, the finite universal set is denoted as X .

Definition 1 [15] A PYFS E on a finite universal set X isgiven by:

E = {(T ′E(X),N ′

E(X)

) : X ∈ X}, (1)

where T ′E

and N ′E

represent the membership and non-membership grades, respectively, under the following condi-tion: 0 ≤ T ′2

E+N ′2

E≤ 1. The hesitancy degree is defined as

HE = (1 − T ′2

E− N ′2

E

) 12 . The Pythagorean fuzzy number

(PYFN) is given as E = (T ′E, N ′

E

).

Definition 2 [43] A CPYFS E on a finite universal set X isgiven as follows:

E = {(T ′E(X),N ′

E(X)

) : X ∈ X} →, (2)

where T ′E

= TEei2πWTE and N ′E

= NEei2πWNE represent

the complex-valued membership grade and complex-valuednon-membership grade, respectively, under the followingconditions: 0 ≤ T 2

E+ N 2

E≤ 1 and 0 ≤ W2

TE + W2NE

≤ 1.Further, the complex-valued hesitancy degree is defined as

HE = (1 − T 2

E− N 2

E

) 12 e

i2π(1−W2

TE−W2

NE

) 12

. The complex

PYFN is given as E =(TEei2πWTE , NEe

i2πWNE

).

Definition 3 [24] A q-ROFS E on a finite universal set X isgiven as:

E = {(T ′E(X),N ′

E(X)

) : X ∈ X} →, (3)

where T ′E

and N ′E

represent the membership and non-membership grades, respectively, under the following con-dition: 0 ≤ T ′q

E+ N ′q

E≤ 1. The hesitancy degree is given

as HE =(1 − T ′q

E− N ′q

E

) 1q. The q-rung orthopair fuzzy

number (q-ROFN) is given as E = (T ′E, N ′

E

).

Definition 4 [24] For any two q-ROFNs E1 = (T ′1 , N ′

1

)and

E2 = (T ′2 , N ′

2

)with any δ > 0,

1. E1 ⊕ E2 =⎛

⎝(

T ′q1 +T ′q

2

1+T ′q1 T ′q

2

) 1q

,

⎛

⎝ N ′1N ′

2(1+

(1−N ′q

1

)(1−N ′q

2

)) 1q

⎞

⎠

⎞

⎠;

123


2. E1 ⊗ E2 =⎛

⎝

⎛

⎝ T ′1T

′2

(1+

(1−T ′q

1

)(1−T ′q

2

)) 1q

⎞

⎠,

(N ′q

1 +N ′q2

1+N ′q1 N ′q

2

) 1q

⎞

⎠;

3. δE1 =⎛

⎜⎝

⎛

⎝(1+T

′q1

)δ

−(1−T

′q1

)δ

(1+T

′q1

)δ+(1−T

′q1

)δ

⎞

⎠

1q

,

⎛

⎜⎝

(2)1q N ′δ

1((

2−N′q1

)δ+(N

′q1

)δ) 1

q

⎞

⎟⎠

⎞

⎟⎠; and

4. Eδ1 =

⎛

⎜⎝

⎛

⎜⎝

(2)1q T ′δ

1((

2−T ′q1

)δ+(T ′q1

)δ) 1

q

⎞

⎟⎠,

((1+N ′q

1

)δ−(1−N ′q

1

)δ

(1+N ′q

1

)δ+(1−N ′q

1

)δ

) 1q

⎞

⎟⎠.

Definition 5 [24] For any q-ROFNE1 = (T ′1 , N ′

1

), the score

function and accuracy function are defined as follows:

S(E1) = T ′q1 − N ′q

1 , (4)

H(E1) = T ′q1 + N ′q

1 , (5)

where S(E1) ∈ [−1, 1] and H(E1) ∈ [0, 1].

Cq-ROFSs and their operational laws

In this section, we propose the Cq-ROFSs and their opera-tional laws.

Definition 6 ACq-ROFSEon afinite universal set X is givenas:

E = {(T ′E(X),N ′

E(X)

) : X ∈ X}, (6)

where T ′E

= TEei2πWTE and N ′E

= NEei2πWNE represent

the degrees of the complex-valuedmembership and complex-valued non-membership, respectively, under the followingconditions:0 ≤ T q

E+ N q

E≤ 1, 0 ≤ Wq

TE + WqNE

≤ 1,and q ≥ 1. The complex-valued hesitancy degree is given

as HE = (1 − T q

E− N q

E

) 1q e

i2π(1−Wq

TE−Wq

NE

) 1q

. The Cq-

ROFN is given as E =(TEei2πWTE , NEe

i2πWNE

).

Definition 7 For any two Cq-ROFNs E1 =(T1ei2πWT1 , N1e

i2πWN1

)and E2 =

(T2ei2πWT2 , N2e

i2πWN2

)

1. Ec1 =

(N1e

i2πWN1 , T1ei2πWT1);

2. E1 ∨ E2 =(max(T1, T2)ei2π max

(WT1 ,WT2

),

min(N1, N2)ei2π min

(WN1 ,WN2

)); and

3. E1 ∧ E2 =(min(T1, T2)ei2π min

(WT1 ,WT2

),

max(N1, N2)ei2π max

(WN1 ,WN2

)).

Next, we investigate Einstein’s operational laws for Cq-ROFNs, which are defined as follows.

Definition 8 For any two Cq-ROFNs E1 =(T1ei2πWT1 , N1e

i2πWN1

)and E2 =

(T2ei2πWT2 , N2e

i2πWN2

)with any δ > 0,

1. E1 ⊕ E2 =

⎛

⎜⎜⎜⎜⎜⎜⎝

(T q1 + T q

2

1 + T q1 T q

2

) 1q

ei2π

(Wq

T1+Wq

T21+Wq

T1Wq

T2

) 1q

,

⎛

⎝ N1N2(1 + (

1 − N q1

)(1 − N q

2

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

WN1WN2

(1+

(1−Wq

N1

)(1−Wq

N2

)) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎠

;

2. E1 ⊗ E2 =

⎛

⎜⎜⎜⎜⎜⎜⎝

⎛

⎝ T1T2(1 + (

1 − T q1

)(1 − T q

2

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

WT1WT2(1+

(1−Wq

T1

)(1−Wq

T2

)) 1q

⎞

⎟⎟⎠

,

(N q

1 + N q2

1 + N q1 N

q2

) 1q

ei2π

(Wq

N1+Wq

N21+Wq

N1Wq

N2

) 1q

⎞

⎟⎟⎟⎟⎟⎟⎠

;

3. δE1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

((1 + T q

1

)δ − (1 − T q

1

)δ

(1 + T q

1

)δ + (1 − T q

1

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ−

(1−Wq

T1

)δ

(1+Wq

T1

)δ+

(1−Wq

T1

)δ

⎞

⎟⎠

1q

,

⎛

⎜⎜⎝

(2)1q N δ

1((2 − N q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

N1((

2−WqN1

)δ+

(Wq

N1

)δ) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

; and

123


4. Eδ1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

(2)1q T δ

1((2 − T q

1

)δ + (T q1

)δ) 1

q

⎞

⎟⎟⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

T1((

2−WqT1

)δ+

(Wq

T1

)δ) 1q

⎞

⎟⎟⎟⎠

,

((1 + N q

1

)δ − (1 − N q

1

)δ

(1 + N q

1

)δ + (1 − N q

1

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

N1

)δ−

(1−Wq

N1

)δ

(1+Wq

N1

)δ+

(1−Wq

N1

)δ

⎞

⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

.

Definition 9 For any q-ROFN E1 =(T1ei2πWT1 , N1e

i2πWN1

), the score function and accuracy

function are defined as follows:

S(E1) = 1

2

(T q1 + Wq

T1 − N q1 − Wq

N1

), (7)

H(E1) = 1

2

(T q1 + Wq

T1 + N q1 + Wq

N1

), (8)

where S(E1) ∈ [−1, 1] and H(E1) ∈ [0, 1].

Theorem 1 For any two Cq-ROFNs E1 =(T1ei2πWT1 , N1e

i2πWN1

)and E2 =

(T2ei2πWT2 , N2e

i2πWN2

)with any δ, δ1, δ2 > 0,

1. E1 ⊕ E2 = E2 ⊕ E1;2. E1 ⊗ E2 = E2 ⊗ E1;3. δ(E1 ⊕ E2) = δE1 ⊕ δE2;4. (E1 ⊗ E2)

δ = Eδ1 ⊗ Eδ

2;5. δ1E1 ⊕ δ2E1 = (δ1 + δ2)E1; and6. Eδ1

1 ⊗ Eδ21 = Eδ1+δ2

1 .

The proof of Theorem 1 is presented in the "Appendix".

Theorem 2 For any two Cq-ROFNs E1 =(T1ei2πWT1 , N1e

i2πWN1

)and E2 =

(T2ei2πWT2 , N2e

i2πWN2

),

1. Ec1 ∨ Ec

2 = (E1 ∧ E2)c;

2. Ec2 ∧ Ec

1 = (E1 ∨ E2)c;

3. Ec1 ⊕ Ec

2 = (E1 ⊗ E2)c;

4. Ec1 ⊗ Ec

2 = (E1 ⊕ E2)c;

5. (E1 ∨ E2) ⊕ (E1 ∧ E2) = E1 ⊕ E2; and6. (E1 ∨ E2) ⊗ (E1 ∧ E2) = E1 ⊗ E2.

Proof Similar to the case of Theorem 1, the proofs are omit-ted.

Theorem 3 For any three Cq-ROFNs

E1 =(T1ei2πWT1 , N1e

i2πWN1

), E2 =

(T2ei2πWT2 , N2e

i2πWN2

), and E3 =

(T3ei2πWT3 , N3e

i2πWN3

),

1. (E1 ∨ E2) ∧ E3 = (E1 ∧ E3) ∨ (E2 ∧ E3);2. (E1 ∧ E2) ∨ E3 = (E1 ∨ E3) ∧ (E2 ∨ E3);3. (E1 ∨ E2) ⊕ E3 = (E1 ⊕ E3) ∨ (E2 ⊕ E3);4. (E1 ∧ E2) ⊕ E3 = (E1 ⊕ E3) ∧ (E2 ⊕ E3);5. (E1 ∨ E2) ⊗ E3 = (E1 ⊗ E3) ∨ (E2 ⊗ E3); and6. (E1 ∧ E2) ⊗ E3 = (E1 ⊗ E3) ∧ (E2 ⊗ E3).

Proof Similar to the case of Theorem 1, the proofs are omit-ted.

Complex q-rung orthopair fuzzy Einsteinarithmetic aggregation operators

In this section, we propose complex q-rung orthopair fuzzyEinstein aggregation operators.

Complex q-rung orthopair fuzzy Einstein weightedaveraging operator

The complex q-rung orthopair fuzzy Einstein weightedaveraging operator is investigated, and its properties are dis-cussed.

Definition 10 Suppose thatEi =(Tiei2πWTi , Nie

i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theCq-ROFEWA operator is given as Cq-ROFEWA : ℵn → ℵ,and defined by:

Cq-ROFEWA(E1,E2, . . . ,En)

= ω1E1 ⊕ ω2E2⊕, · · · , ⊕ ωnEn = ⊕ni=1ωiEi (9)

Here, the symbol ℵ represents the family of Cq-ROFNs,and the weight vector is defined as

∑ni=1 ωi = 1, ωi ∈ [0,1].

Remark 1 By substituting ωi = 1n for all instances of i ,

the Cq-ROFEWA operator is reduced to the complex q-rungorthopair fuzzy averaging (Cq-ROFA) operator, as follows:

Cq-ROFA(E1,E2, . . . ,En) = 1

n(E1 ⊕ E2⊕, · · · , ⊕ En)

= 1

n⊕n

i=1Ei . (10)

123


According to Einstein’s operational laws, we obtain thefollowing result.

Theorem 4 Suppose that Ei =(Tiei2πWTi , Nie

i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theaggregating value from Definition 10 is a Cq-ROFN, and:

Cq-ROFEWA(E1,E2, . . . ,En) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏ni=1

(1+T q

i

)ωi −∏ni=1

(1−T q

i

)ωi∏n

i=1(1+T q

i

)ωi +∏ni=1

(1−T q

i

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏ni=1

(1+Wq

Ti

)ωi −∏ni=1

(1−Wq

Ti

)ωi

∏ni=1

(1+Wq

Ti

)ωi +∏ni=1

(1−Wq

Ti

)ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q

∏ni=1N

ωii

(∏ni=1

(2−N q

i

)ωi +∏ni=1

(N q

i

)ωi) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q ∏n

i=1WωiNi

(∏ni=1

(2−Wq

Ni

)ωi +∏ni=1

(Wq

Ni

)ωi) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (11)

The proof of Theorem 4 is presented in the Appendix.


i2πWNi

)∈

Cq-ROFN, i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, the relationship between theCq-ROFEWAandCq-ROFWA operators is expressed as follows:

Cq-ROFEWA(E1, E2, . . . , En)

≤ Cq-ROFWA(E1, E2, . . . , En),

where the weight vector is defined by∑n

i=1 ωi = 1, ωi ∈[0,1].

The proof of Theorem 5 is presented in the Appendix.

The Cq-ROFEWA operator has the following properties.

Suppose that Ei =(Tiei2πWTi , Nie

i2πWNi

)∈

Cq-ROFN, i = 1,2, . . . , n comprise the family of Cq-ROFNs and the weight vector is defined by

∑ni=1 ωi = 1,

ωi ∈ [0,1]. Then, we have the following:

1. Idea: When Ei = E0 =(T0ei2πWT0 , N0e

i2πWN0

), ∀i ,

Cq-ROFEWA(E1, E2, . . . , En) = E0.2. Boundedness: Let E−

i =(min(Ti )e

i2πmin(WTi

)

, max(Ni )ei2πmax

(WNi

)),

E+i =

(max(Ti )e

i2πmax(WTi

)

, min(Ni )ei2πmin

(WNi

));

then, E−i ≤ Cq-ROFEWA(E1, E2, . . . , En) ≤ E+

i .

3. Monotonicity: If Ei ≤ E′ ′i , ∀i , Cq-ROFEWA

(E1, E2, . . . , En) ≤ Cq-ROFEWA(E′ ′1, E′ ′

2, . . . , E′ ′n

).

Proofs of these properties are presented in the Appendix.

Complex q-rung orthopair fuzzy Einstein orderedweighted averaging operator


i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theCq-ROFEOWAoperator is given by Cq-ROFEOWA : ℵn →ℵ

Cq-ROFEOWA(E1,E2, . . . ,En) = ω1EO(1)

⊕ ω2EO(2)⊕, · · · , ⊕ ωnEO(n) = ⊕ni=1ωiEO(i) (12)

where the symbol ℵ represents the family of Cq-ROFNs,and the weight vector is defined by

∑ni=1 ωi = 1,

ωi ∈ [0,1]. (O(1), O(2), . . . , O(n)) is a permutation of(i = 1,2, . . . , n) such that EO(i) ≤ EO(i−1).

Remark 2 By substituting ωi = 1n for all instances of i , the

Cq-ROFEOWA operator is reduced to the complex q-rungorthopair fuzzy ordered averaging (Cq-ROFOA) operator, asfollows:

Cq-ROFA(E1,E2, . . . ,En)

= 1

n

(EO(1) ⊕ EO(2)⊕, · · · , ⊕ EO(n)

)

= 1

n⊕n

i=1 EO(i). (13)

According to Einstein’s operational laws, we examine thefollowing results.


i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theaggregating value from Definition 11 is a Cq-ROFN, and

123


Cq-ROFEOWA(E1,E2, . . . ,En) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏ni=1

(1+T q

O(i)

)ωi −∏ni=1

(1−T q

O(i)

)ωi

∏ni=1

(1+T q

O(i)

)ωi +∏ni=1

(1−T q

O(i)

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏ni=1

(1+Wq

TO(i)

)ωi −∏ni=1

(1−Wq

TO(i)

)ωi

∏ni=1

(1+Wq

TO(i)

)ωi +∏ni=1

(1−Wq

TO(i)

)ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q

∏ni=1N

ωiO(i)

(∏ni=1

(2−N q

O(i)

)ωi +∏ni=1

(N q

O(i)

)ωi) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q ∏n

i=1WωiNO(i)

(∏ni=1

(2−Wq

NO(i)

)ωi +∏ni=1

(Wq

NO(i)

)ωi) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(14)

where the weight vector is defined by∑n

i=1 ωi = 1,ωi ∈ [0,1]. (O(1), O(2), . . . , O(n)) is a permutation of(i = 1,2, . . . , n) such that EO(i) ≤ EO(i−1). When we con-sider N q

O(i) = 1 − T qO(i), Eq. (14) is converted into the

following equation:

Cq-ROFEOWA(E1, E2, . . . , En) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏ni=1

(1+T q

O(i)

)ωi −∏ni=1

(1−T q

O(i)

)ωi

∏ni=1

(1+T q

O(i)

)ωi +∏ni=1

(1−T q

O(i)

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏ni=1

(1+Wq

TO(i)

)ωi −∏ni=1

(1−Wq

TO(i)

)ωi

∏ni=1

(1+Wq

TO(i)

)ωi +∏ni=1

(1−Wq

TO(i)

)ωi

⎞

⎟⎠

1q

,

(

1 −∏n

i=1

(1+T q

O(i)

)ωi −∏ni=1

(1−T q

O(i)

)ωi

∏ni=1

(1+T q

O(i)

)ωi +∏ni=1

(1−T q

O(i)

)ωi

) 1q

e

i2π

⎛

⎜⎝1−

∏ni=1

(1+Wq

TO(i)

)ωi −∏ni=1

(1−Wq

TO(i)

)ωi

∏ni=1

(1+Wq

TO(i)

)ωi +∏ni=1

(1−Wq

TO(i)

)ωi

⎞

⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Proof Similar to the case of Theorem 4, the proof is omitted.

Next, we discuss the properties of this operator.

Suppose that Ei =(Tiei2πWTi , Nie

i2πWNi

)∈

Cq-ROFN, i = 1,2, . . . , n comprise the family of Cq-ROFNs and the weight vector is defined by

∑ni=1 ωi = 1,

ωi ∈ [0,1]. Then, we have:

1. Idea: When Ei = E0 =(T0ei2πWT0 , N0e

i2πWN0

), ∀i ,

Cq-ROFEOWA(E1, E2, . . . , En) = E0.2. Boundedness: Let E−

i =(min(Ti )e

i2πmin(WTi

)

, max(Ni )ei2πmax

(WNi

)),

E+i =

(max(Ti )e

i2πmax(WTi

)

, min(Ni )ei2πmin

(WNi

));

then, E−i ≤ Cq-ROFEOWA(E1, E2, . . . , En) ≤ E+

i .

3. Monotonicity: If Ei ≤ E′ ′i , ∀i , Cq-ROFEOWA

(E1, E2, . . . , En) ≤ Cq-ROFEOWA(E′ ′1, E′ ′

2, . . . , E′ ′n

).

Generalized complex q-rung orthopair fuzzyEinstein weighted averaging operator


i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theGCq-ROFEWAoperator is given byGCq-ROFEWA : ℵn →ℵ

GCq-ROFEWA(E1,E2, . . . ,En)

= (ω1Eδ

1 ⊕ ω2Eδ2⊕, · · · , ⊕ ωnEδ

n

) 1δ = (⊕n

i=1ωiEδi

) 1δ →(15)

where the symbol ℵ represents the family of Cq-ROFNs, andthe weight vector is defined by

∑ni=1 ωi = 1, ωi ∈ [0,1],

δ > 0.


the GCq-ROFEWA operator is reduced to the generalizedcomplex q-rung orthopair fuzzy averaging (GCq-ROFA)operator, as follows:

GCq-ROFA(E1,E2, . . . ,En)

=(1

n

(Eδ1 ⊕ Eδ

2⊕, · · · , ⊕ Eδn

))1δ =

(1

n⊕n

i=1Eδi

) 1δ

(16)

When δ = 1, the GCq-ROFEWA operator is reduced to theCq-ROFEWA operator.

Using Einstein’s operational laws, the following result isobtained.


i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theaggregating value from Definition 12 is a Cq-ROFN, and:

123


GCq-ROFEWA(E1,E2, . . . ,En)

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(2)1q

⎛

⎜⎜⎝

∏ni=1

{(2 − T q

i

)δ + 3(T qi

)δ}ωi −

∏ni=1

{(2 − T q

i

)δ − (T qi

)δ}ωi

⎞

⎟⎟⎠

12δ

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎝∏n

i=1

{(2 − T q

i

)δ + 3(T qi

)δ}ωi +

3∏n

i=1

{(2 − T q

i

)δ − (T qi

)δ}ωi

⎞

⎠

1δ

+⎛

⎝∏n

i=1

{(2 − T q

i

)δ + 3(T qi

)δ}ωi −

∏ni=1

{(2 − T q

i

)δ − (T qi

)δ}ωi

⎞

⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1qe

i2π

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(2)1q

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

∏ni=1

{(2 − Wq

Ti

)δ + 3(Wq

Ti

)δ}ωi

−∏n

i=1

{(2 − Wq

Ti

)δ −(Wq

Ti

)δ}ωi

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

12δ

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(2 − Wq

Ti

)δ + 3(Wq

Ti

)δ}ωi

+

3∏n

i=1

{(2 − Wq

Ti

)δ −(Wq

Ti

)δ}ωi

⎞

⎟⎟⎠

1δ

+

⎛

⎜⎜⎝

∏ni=1

{(2 − Wq

Ti

)δ + 3(Wq

Ti

)δ}ωi

−∏n

i=1

{(2 − Wq

Ti

)δ −(Wq

Ti

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎝∏n

i=1

{(1 + N q

i

)δ + 3(1 − N q

i

)δ}ωi +

3∏n

i=1

{(1 + N q

i

)δ − (1 − N q

i

)δ}ωi

⎞

⎠

1δ

−⎛

⎝∏n

i=1

{(1 + N q

i

)δ + 3(1 − N q

i

)δ}ωi −

∏ni=1

{(1 + N q

i

)δ − (1 − N q

i

)δ}ωi

⎞

⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎝∏n

i=1

{(1 + N q

i

)δ + 3(1 − N q

i

)δ}ωi +

3∏n

i=1

{(1 + N q

i

)δ − (1 − N q

i

)δ}ωi

⎞

⎠

1δ

+⎛

⎝∏n

i=1

{(1 + N q

i

)δ + 3(1 − N q

i

)δ}ωi −

∏ni=1

{(1 + N q

i

)δ − (1 − N q

i

)δ}ωi

⎞

⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1q

e

i2π

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

Ni

)δ + 3(1 − Wq

Ni

)δ}ωi

+

3∏n

i=1

{(1 + Wq

Ni

)δ −(1 − Wq

Ni

)δ}ωi

⎞

⎟⎟⎠

1δ

−

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

Ni

)δ + 3(1 − Wq

Ni

)δ}ωi

−∏n

i=1

{(1 + Wq

Ni

)δ −(1 − Wq

Ni

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

Ni

)δ + 3(1 − Wq

Ni

)δ}ωi

+

3∏n

i=1

{(1 + Wq

Ni

)δ −(1 − Wq

Ni

)δ}ωi

⎞

⎟⎟⎠

1δ

+

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

Ni

)δ + 3(1 − Wq

Ni

)δ}ωi

−∏n

i=1

{(1 + Wq

Ni

)δ −(1 − Wq

Ni

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(17)


123


Generalized complex q-rung orthopair fuzzyEinstein ordered weighted averaging operator


i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theGCq-ROFEOWA operator is given by GCq-ROFEOWA :ℵn → ℵ

GCq-ROFEOWA(E1,E2, . . . ,En)

=(ω1Eδ

O(1) ⊕ ω2EδO(2)⊕, · · · , ⊕ ωnEδ

O(n)

) 1δ

=(⊕n

i=1ωiEδO(i)

) 1δ → (18)

where the symbol ℵ represents the family of Cq-ROFNs, andthe weight vector is defined by

∑ni=1 ωi = 1, ωi ∈ [0,1],

δ > 0.


the GCq-ROFEOWA operator is reduced to the generalizedcomplex q-rung orthopair fuzzy ordered averaging (GCq-ROFOA) operator, as follows:

GCq-ROFOA(E1,E2, . . . ,En)

=(1

n

(EδO(1) ⊕ Eδ

O(2)⊕, · · · , ⊕ EδO(n)

)) 1δ

=(1

n⊕n

i=1EδO(i)

) 1δ

(19)

When δ = 1, the GCq-ROFEOWA operator is reduced tothe Cq-ROFEOWA operator.

According to Einstssein’s operational laws, we obtain thefollowing result.


i2πWNi

),

i = 1,2, . . . , n comprise the family of Cq-ROFNs; then, theaggregating value from Definition 13 is a Cq-ROFN, and

123


GCq-ROFEOWA(E1,E2, . . . ,En)

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(2)1q

⎛

⎜⎜⎜⎜⎝

∏ni=1

{(2 − T q

O(i)

)δ + 3(T qO(i)

)δ}ωi

−∏n

i=1

{(2 − T q

O(i)

)δ −(T qO(i)

)δ}ωi

⎞

⎟⎟⎟⎟⎠

12δ

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(2 − T q

O(i)

)δ + 3(T qO(i)

)δ}ωi

+

3∏n

i=1

{(2 − T q

O(i)

)δ −(T qO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

+

⎛

⎜⎜⎝

∏ni=1

{(2 − T q

O(i)

)δ + 3(T qO(i)

)δ}ωi

−∏n

i=1

{(2 − T q

O(i)

)δ −(T qO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1qe

i2π

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(2)1q

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

∏ni=1

{(2 − Wq

TO(i)

)δ + 3(Wq

TO(i)

)δ}ωi

−∏n

i=1

{(2 − Wq

TO(i)

)δ −(Wq

TO(i)

)δ}ωi

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

12δ

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(2 − Wq

TO(i)

)δ + 3(Wq

TO(i)

)δ}ωi

+

3∏n

i=1

{(2 − Wq

TO(i)

)δ −(Wq

TO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

+

⎛

⎜⎜⎝

∏ni=1

{(2 − Wq

TO(i)

)δ + 3(Wq

TO(i)

)δ}ωi

−∏n

i=1

{(2 − Wq

TO(i)

)δ −(Wq

TO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(1 + N q

O(i)

)δ + 3(1 − N q

O(i)

)δ}ωi

+

3∏n

i=1

{(1 + N q

O(i)

)δ −(1 − N q

O(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

−

⎛

⎜⎜⎝

∏ni=1

{(1 + N q

O(i)

)δ + 3(1 − N q

O(i)

)δ}ωi

−∏n

i=1

{(1 + N q

O(i)

)δ −(1 − N q

O(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(1 + N q

O(i)

)δ + 3(1 − N q

O(i)

)δ}ωi

+

3∏n

i=1

{(1 + N q

O(i)

)δ −(1 − N q

O(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

+

⎛

⎜⎜⎝

∏ni=1

{(1 + N q

O(i)

)δ + 3(1 − N q

O(i)

)δ}ωi

−∏n

i=1

{(1 + N q

O(i)

)δ −(1 − N q

O(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1q

e

i2π

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

NO(i)

)δ + 3(1 − Wq

NO(i)

)δ}ωi

+

3∏n

i=1

{(1 + Wq

NO(i)

)δ −(1 − Wq

NO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

−

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

NO(i)

)δ + 3(1 − Wq

NO(i)

)δ}ωi

−∏n

i=1

{(1 + Wq

NO(i)

)δ −(1 − Wq

NO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

NO(i)

)δ + 3(1 − Wq

NO(i)

)δ}ωi

+

3∏n

i=1

{(1 + Wq

NO(i)

)δ −(1 − Wq

NO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

+

⎛

⎜⎜⎝

∏ni=1

{(1 + Wq

NO(i)

)δ + 3(1 − Wq

NO(i)

)δ}ωi

−∏n

i=1

{(1 + Wq

NO(i)

)δ −(1 − Wq

NO(i)

)δ}ωi

⎞

⎟⎟⎠

1δ

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(20)


123


Table 1 Complex q-rung orthopair fuzzy decision matrix

Symbols G1 G2 . . . . . . . . . . . . Gn

x1(T11ei2πWT11 , N11e

i2πWN11

) (T12ei2πWT12 , N12e

i2πWN12

). . . . . . . . . . . .

(T1nei2πWT1n , N1ne

i2πWN1n

)


i2πWN21

) (T22ei2πWT22 , N22e

i2πWN22

). . . . . . . . . . . .


i2πWN2n

)


i2πWN31

) (T32ei2πWT32 , N32e

i2πWN32

). . . . . . . . . . . .


i2πWN3n

)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xm(Tm1e

i2πWTm1 , Nm1ei2πWNm1

) (Tm2e

i2πWTm2 , Nm2ei2πWNm2

). . . . . . . . . . . .

(Tmnei2πWTmn , Nmnei2πWNmn

)

Table 2 Symbols of the alternatives

Symbols x1 x2 x3 x4 x5

Representations Computer company Furniture company Car company Chemical company Food company

Table 3 Representations of the attributes

Symbols G1 G2 G3 G4

Representations Technical ability Accepted benefits Competitive power in market Management capability

In Eq. (20), δ plays a key role in the aggregation of theCq-ROFNs. The parameter δ takes a special number; then,the GCq-ROFEWA operator can be reduced. For example,when δ = 1, Eq. (20) is converted into Eq. (14).

MAGDM approach based on proposedoperators

Description of decision-making problem

Suppose an MAGDM problem in which A ={x1, x2, . . . , xm} represents the family of alternatives,and G = {G1, G2, . . . , Gn} represents the family offinite attributes with weight vector ω = (ω1, ω2, . . . , ωn).The decision matrix A = (

Ei j)m×n is given by decision

makers through evaluation of the attribute G j under thealternativexi , which is represented by the Cq-ROFN. Thecomplex q-rung orthopair fuzzy decision matrix is presentedin Table 1.

We use the proposed operators to solve this decision-making problem, and the steps are as follows.

Step 1: Normalize the decision matrix with respect to.

ri j =(Ti jei2πWTi j ,Ni je

i2πWNi j

)=

{Eci j j is benefit types

Ei j j is cost types(21)

Step 2:Use Eq. (20) to aggregate the normalized decisionmatrix.

ri = GCq − ROFOW A(ri1, ri2, . . . , rin), i = 1,2, 3, . . . , m

Step 3: Use Eq. (7) to examine the score values of theaggregated values obtained in step 2.

Step 4: Rank all the alternatives according to the scorevalues from step 3, and select the best one.

Step 5: End.

Illustrative numerical example

An examplewas taken from the area of the investment, wherethe investor wishes to invest money in a standard company.After a careful analysis, five possible alternatives representedby A = {x1, x2, . . . , xm}were considered, as shown inTable2.

Furthermore, these alternatives were evaluated using fourattributes represented byG = {G1, G2, . . . , Gn}, which arepresented in Table 3.

The weight vector for the given attributes was given asω = {0.25,0.15,0.4,0.2}; then, the complex q-rung orthopairfuzzy decision matrix was obtained, as shown in Table 4.

The steps of the decision-making process were as follows:Step 1: Normalize the decision matrix and obtain the

results shown in Table 5.Step 2: Obtain the aggregation results using Eq. (20),

which are presented in Table 6.Step 3: Obtain the scores of the aggregated values using

Eq. (7), which are presented in Table 7.Step 4: Rank all the alternatives and select the best one.

E5 ≤ E4 ≤ E1 ≤ E3 ≤ E2

123


Table 4 Complex q-rung orthopair fuzzy decision matrix

Symbols G1 G2 G3 G4

x1(0.22ei2π(0.31), 0.4ei2π(0.45)

) (0.35ei2π(0.40), 0.33ei2π(0.3)

) (0.5ei2π(0.45), 0.23ei2π(0.11)

) (0.21ei2π(0.26), 0.19ei2π(0.33)

)

x2(0.24ei2π(0.31), 0.41ei2π(0.46)

) (0.36ei2π(0.41), 0.34ei2π(0.31)

) (0.51ei2π(0.46), 0.24ei2π(0.12)

) (0.22ei2π(0.27), 0.20ei2π(0.23)

)

x3(0.25ei2π(0.33), 0.42ei2π(0.47)

) (0.37ei2π(0.42), 0.35ei2π(0.32)

) (0.52ei2π(0.47), 0.25ei2π(0.13)

) (0.23ei2π(0.28), 0.21ei2π(0.24)

)

x4(0.27ei2π(0.34), 0.43ei2π(0.48)

) (0.38ei2π(0.43), 0.36ei2π(0.33)

) (0.53ei2π(0.48), 0.26ei2π(0.14)

) (0.24ei2π(0.29), 0.22ei2π(0.25)

)

x5(0.35ei2π(0.35), 0.44ei2π(0.49)

) (0.39ei2π(0.44), 0.37ei2π(0.34)

) (0.54ei2π(0.49), 0.27ei2π(0.15)

) (0.25ei2π(0.30), 0.23ei2π(0.26)

)

Table 5 Normalized complex q-rung orthopair fuzzy decision matrix

Symbols G1 G2 G3 G4

x1(0.4ei2π(0.45), 0.22ei2π(0.31)

) (0.35ei2π(0.40), 0.33ei2π(0.3)

) (0.5ei2π(0.45), 0.23ei2π(0.11)

) (0.21ei2π(0.26), 0.19ei2π(0.33)

)

x2(0.41ei2π(0.46), 0.24ei2π(0.31)

) (0.36ei2π(0.41), 0.34ei2π(0.31)

) (0.51ei2π(0.46), 0.24ei2π(0.12)

) (0.22ei2π(0.27), 0.20ei2π(0.23)

)

x3(0.42ei2π(0.47), 0.25ei2π(0.33)

) (0.37ei2π(0.42), 0.35ei2π(0.32)

) (0.52ei2π(0.47), 0.25ei2π(0.13)

) (0.23ei2π(0.28), 0.21ei2π(0.24)

)

x4(0.43ei2π(0.48), 0.27ei2π(0.34)

) (0.38ei2π(0.43), 0.36ei2π(0.33)

) (0.53ei2π(0.48), 0.26ei2π(0.14)

) (0.24ei2π(0.29), 0.22ei2π(0.25)

)

x5(0.44ei2π(0.49), 0.35ei2π(0.35)

) (0.39ei2π(0.44), 0.37ei2π(0.34)

) (0.54ei2π(0.49), 0.27ei2π(0.15)

) (0.25ei2π(0.30), 0.23ei2π(0.26)

)

Table 6 Aggregation values ofthe Cq-ROFNs Method Getting values

r1 = GCq - ROFOWA(r11, r12, . . . , r1n)(0.317ei2π(0.319), 0.082ei2π(0.073)

)


)


)


)


)

Table 7 Score values of the Cq-ROFNs

Score values of the alternatives

S(x1) = S(r1) = S(E1) = 0.241

S(x2) = S(r2) = S(E2) = 0.244

S(x3) = S(r3) = S(E3) = 0.242

S(x4) = S(r4) = S(E4) = 0.239

S(x5) = S(r5) = S(E5) = 0.233

Hence, the best alternative is E2, which represents thefurniture company.

Step 5: End.

Effect of parameter ı

In this subsection, we examine the effect of the parameter δ

on the ranking results. We obtained the ranking results fordifferent values of the parameter δ, as shown in Table 8 (qwas 1).

As shown in Table 8, the parameter δ affected the rankingresults. However, the best alternative did not change.

Effect of parameter q

In this subsection, we examine the effect of the parameter qon the ranking results. We obtained the ranking results fordifferent values of q, as shown in Table 9 (δ was 1).

As indicated by Table 9, the parameter q affected the rank-ing results. However, the best alternative did not change.

Advantages and comparative analysis

The proposed method was compared with existingapproaches to highlight its advantages.

We compared Garg and Rani’s method [40] based onthe power aggregation operators for the CIFS and Garg andRani’smethod [42] based on the aggregation operators for theCIFS with the proposed method for the CPFS and Cq-ROFS.The results are presented in Table 10 and Fig. 2.

Figure 2 contains five alternatives represented by Ei

(i = 1,2, 3,4, 5). To identify the best alternative amongthem, we adopt four types of operators, whose representa-tions are presented in Table 10. Two of the operators existedpreviously, and the other two are newly proposed herein.

123


Table 8 Effect of the parameter δ, for q = 1

Value of δ Score values of the methods Rankings of the methods

δ = 2 S(E1) = 0.336, S(E2) = 0.339, S(E3) = 0.331, S(E4) = 0.321, S(E5) = 0.295 E5 ≤ E4 ≤ E3 ≤ E1 ≤ E2

δ = 3 S(E1) = 0.282, S(E2) = 0.284, S(E3) = 0.181, S(E4) = −0.72, S(E5) = −0.23 E4 ≤ E5 ≤ E3 ≤ E1 ≤ E2




δ = 15 S(E1) = −0.004, S(E2) = −0.003, S(E3) = −0.004, S(E4) = −0.005, S(E5) = −0.008 E5 ≤ E4 ≤ E3 ≤ E1 ≤ E2

δ = 20 S(E1) = −0.004, S(E2) = −0.004, S(E3) = −0.005, S(E4) = −0.006, S(E5) = −0.009

E5 ≤ E4 ≤ E3 ≤ E1 ≤ E2

Table 9 Effect of the parameter q for δ = 1

Value of q Score values of the methods Rankings of the methods

q = 2 S(E1) = 0.0082, S(E2) = 0.0088, S(E3) = 0.0084, S(E4) = 0.0078, S(E5) = 0.0059 E5 ≤ E4 ≤ E1 ≤ E3 ≤ E2

q = 3 S(E1) = −0.0021, S(E2) = −0.0021, S(E3) = −0.0025, S(E4) = −0.003, S(E5) = −0.0039 E5 ≤ E4 ≤ E3 ≤ E1 ≤ E2



Table 10 Comparison of the proposed method with existing approaches

Methods Score values of the methods Rankings of the methods

Garg and Rani [40] S(E1) = 0.231, S(E2) = 0.234, S(E3) = 0.232, S(E4) = 0.229, S(E5) = 0.222 E5 ≤ E4 ≤ E1 ≤ E3 ≤ E2

Rani and Garg [42] S(E1) = 0.251, S(E2) = 0.254, S(E3) = 0.252, S(E4) = 0.249, S(E5) = 0.243 E5 ≤ E4 ≤ E1 ≤ E3 ≤ E2

Proposed Method for CPFS S(E1) = 0.0082, S(E2) = 0.0088, S(E3) = 0.0084, S(E4) = 0.0078, S(E5) = 0.0059 E5 ≤ E4 ≤ E1 ≤ E3 ≤ E2

Proposed methods for Cq-ROFS S(E1) = −0.0021, S(E2) = −0.0021, S(E3) = −0.0025, S(E4) = −0.003, S(E5) = −0.0039 E5 ≤ E4 ≤ E3 ≤ E1 ≤ E2

Fig. 2 Graphical representationof Table 10

Geometric comparison of the presented work with existing works

1 2 3 4 5Alternatives

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Score

valu

es

Garg and Rani [40] Rani and Garg [42] Proposed Method for CPFS Proposed methods for Cq-ROFS

As indicated by Table 10 and Fig. 2, we obtained the sameranking results for the four methods. This validates the pro-posed method.

Furthermore, the geometric interpretations of the rangeof Cq-ROFSs and the existing methods are shown in Fig. 3,which represents the unit disc in a complex plane.

The advantages of the proposed method are explainedbelow.

First numerical example for comparison

An example is taken from the area of the investment, wherethe investor wishes to invest money in a stranded company.After a careful analysis, we consider the five possible alter-natives represented by A = {x1, x2, . . . , x5}, as shown inTable 2. The alternatives are evaluated using four attributesrepresented by G = {G1, G2, . . . , G4}, as shown in Table

123


Fig. 3 Range for Cq-ROFS and the existing approaches

3. The weight vector for the given attributes is given asω = {0.25,0.15,0.4,0.2}, and the normalized complex q-rungorthopair fuzzy decision matrix is presented in Table 11.

Using Eqs. (20) and (7) for δ = 20, then the score valuesof the alternatives are obtained, as shown in Table 12 andFig. 4.

Then, all the alternatives are ranked as follows, and thebest one is selected.

E5 ≤ E4 ≤ E3 ≤ E2 ≤ E1


The proposed method is compared with existingapproaches in Table 13 and Fig. 5.

Figure 5 shows five alternatives represented by Ei

(i = 1,2, 3,4, 5). Because the two existing methods basedon the CIFS [40] and [42] could not solve this problem, theyare not included in Fig. 5. In this example, we select thecomplex Pythagorean fuzzy information form.

Therefore, this example demonstrates the advantages ofthe proposed method.

Table 12 Score values of the proposed methods


S(x1) = S(r1) = S(E1) = 0.099

S(x2) = S(r2) = S(E2) = 0.096

S(x3) = S(r3) = S(E3) = 0.094

S(x4) = S(r4) = S(E4) = 0.093

S(x5) = S(r5) = S(E5) = 0.089

Second numerical example for comparison

An example is taken from the area of the investment, wherethe investor wishes to invest money in a stranded company.After a careful analysis, we consider the five possible alter-natives represented by A = {x1, x2, . . . , x5}, as shownin Table 2, which are evaluated using four attributes rep-resented by G = {G1, G2, . . . , G4}, as shown in Table3. The weight vector for the given attributes is given asω = {0.25,0.15,0.4,0.2}, and the normalized complex q-rungorthopair fuzzy decision matrix is presented in Table 14.

Using Eqs. (20) and (7) for δ = 20, the score values of thealternatives are obtained, as shown in Table 15 and Fig. 6.

Then, all the alternatives are ranked as follows, and thebest one is selected.

E1 ≤ E3 ≤ E4 ≤ E2 ≤ E5


The proposed method is compared with existingapproaches in Table 16 and Fig. 7.

Figure 7 shows five alternatives represented by Ei

(i = 1,2, 3,4, 5), as the methods based on the CIFS andCPFS cannot solve this problem. In Fig. 7, only the pro-posed method is shown, indicating that the proposed methodis more general than the other methods.

Defense budget

Next, we discuss the defense budget of Pakistan. Informa-tion regarding the defense budget, national saving scheme,


Symbols G1 G2 G3 G4

x1(0.7ei2π(0.65), 0.22ei2π(0.31)

) (0.61ei2π(0.7), 0.33ei2π(0.3)

) (0.8ei2π(0.94), 0.23ei2π(0.11)

) (0.84ei2π(0.7), 0.19ei2π(0.33)

)

x2(0.71ei2π(0.66), 0.24ei2π(0.31)

) (0.62ei2π(0.71), 0.34ei2π(0.31)

) (0.81ei2π(0.93), 0.24ei2π(0.12)

) (0.83ei2π(0.72), 0.20ei2π(0.23)

)

x3(0.72ei2π(0.67), 0.25ei2π(0.33)

) (0.63ei2π(0.72), 0.35ei2π(0.32)

) (0.82ei2π(0.92), 0.25ei2π(0.13)

) (0.82ei2π(0.73), 0.21ei2π(0.24)

)

x4(0.73ei2π(0.68), 0.27ei2π(0.34)

) (0.64ei2π(0.73), 0.36ei2π(0.33)

) (0.83ei2π(0.91), 0.26ei2π(0.14)

) (0.81ei2π(0.74), 0.22ei2π(0.25)

)

x5(0.74ei2π(0.69), 0.35ei2π(0.35)

) (0.65ei2π(0.74), 0.37ei2π(0.34)

) (0.84ei2π(0.9), 0.27ei2π(0.15)

) (0.8ei2π(0.76), 0.23ei2π(0.26)

)

123


Fig. 4 Graphical representationof Table 12

The score values got by the GCq-ROFEOWA


0.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1

Sco

reva

lues

0.099

0.096

0.0940.093

0.089



Garg and Rani [40] Cannot be Calculated Cannot be Calculated

Rani and Garg [42] Cannot be Calculated Cannot be Calculated

Proposed Method for CPFS S(E1) = 0.069, S(E2) = 0.068, S(E3) = 0.066, S(E4) = 0.064, S(E5) = 0.061

E5 ≤ E4 ≤ E3 ≤ E2 ≤ E1

Proposed methods for Cq-ROFS S(E1) = 0.099, S(E2) = 0.096, S(E3) = 0.094, S(E4) = 0.093, S(E5) = 0.089

E5 ≤ E4 ≤ E3 ≤ E2 ≤ E1

0

0.02

0.04

1

0.06

Sco

re v

alue

s

0.08

2

0.1

Geometric comparison of the presented work with existing works

Alternatives

3 44 3

Methods5 2

1

Garg and Rani [41]Rani and Garg [43]Proposed Method for CPFSProposed methods for Cq-ROFS

Fig. 5 Graphical representation of Table 13

deposit and reserves, and development expenditure and rev-enue account for this country is presented below. Accordingto this information, we discuss the strength and capability ofarmies or which one is powerful militaries.

1. The total defense budget of the Pakistan army removesthe lion’s share:

Pakistan army budget: 0.476%;Pakistan air force budget: 0.21%;Pakistan navy budget: 0.11%;Pakistan’s inter-service budget: 0.20%.

2. The total national saving scheme budget of Pakistanremoves the lion’s share:

Pakistan investment deposit account budget: 0.8%;Pakistan’s other accounts budget: 0.5%;Pakistan total receipts budget: 0.17%;Pakistan net receipts budget: 0.13%.


Symbols G1 G2 G3 G4

x1(0.7ei2π(0.65), 0.61ei2π(0.64)

) (0.61ei2π(0.7), 0.51ei2π(0.69)

) (0.8ei2π(0.94), 0.79ei2π(0.9)

) (0.84ei2π(0.7), 0.19ei2π(0.33)

)

x2(0.71ei2π(0.66), 0.62ei2π(0.65)

) (0.62ei2π(0.71), 0.52ei2π(0.7)

) (0.81ei2π(0.93), 0.8ei2π(0.91)

) (0.83ei2π(0.72), 0.20ei2π(0.23)

)

x3(0.72ei2π(0.67), 0.63ei2π(0.66)

) (0.63ei2π(0.72), 0.53ei2π(0.71)

) (0.82ei2π(0.92), 0.81ei2π(0.92)

) (0.82ei2π(0.73), 0.21ei2π(0.24)

)

x4(0.73ei2π(0.68), 0.64ei2π(0.67)

) (0.64ei2π(0.73), 0.54ei2π(0.72)

) (0.83ei2π(0.91), 0.82ei2π(0.89)

) (0.81ei2π(0.74), 0.22ei2π(0.25)

)

x5(0.74ei2π(0.69), 0.65ei2π(0.68)

) (0.65ei2π(0.74), 0.55ei2π(073)

) (0.84ei2π(0.9), 0.83ei2π(0.79)

) (0.8ei2π(0.76), 0.23ei2π(0.26)

)

123


Table 15 Score values of the proposed method


S(x1) = S(r1) = S(E1) = −0.00357

S(x2) = S(r2) = S(E2) = −0.00234

S(x3) = S(r3) = S(E3) = −0.00287

S(x4) = S(r4) = S(E4) = −0.00283

S(x5) = S(r5) = S(E5) = −0.00224

3. The total deposit and reserves budget of Pakistan removesthe lion’s share:

Pakistan zakat collection account budget: 0.25%;Pakistan civil and criminal deposit amount budget:0.16%;Pakistan’s personal deposit budget: 0.11%;Pakistan’s post office welfare found budget: 0.1%.

4. The total development expenditure and revenue accountbudget of Pakistan take away the lion’s share:

Pakistan health budget: 0.27%;Pakistan culture and religion budget: 0.15%;Pakistan’s social protection budget: 0.17%;Pakistan’s education budget: 0.18%.

Using Eq. (11), we obtain the following values: the imag-inary part and the non-membership values are considered as

zero, with weight vectors (0.2,0.2,0.2,0.3). Then, we havethe following:

Defense budget of Pakistan: 0.249.National saving scheme of Pakistan: 0.4Deposit and reserves of Pakistan: 0.155.Development expenditure and revenue account for Pak-istan: 0.1925.

Sco

re v

alu

es

Methods

0

-1

-2

1

10-3

-3

2

-4

Geometric comparison of proposed method with existing methods

Alternatives

3 44 35 2

1

Garg and Rani [41]Rani and Garg [43]Proposed Method for CPFSProposed methods for Cq-ROFS

Fig. 7 Graphical representation of Table 16

Fig. 6 The graphicalrepresentation of the Table 15

The score values got by the GCq-ROFEOWA


-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Sco

re v

alues

10-3

-0.0036

-0.0023

-0.0029 -0.0028

-0.0022



Garg and Rani [40] Cannot be calculated Cannot be calculated

Rani and Garg [42] Cannot be calculated Cannot be calculated

Proposed Method for CPFS Cannot be calculated Cannot be calculated

Proposed methods for Cq-ROFS S(E1) = −0.00357, S(E2) = −0.00234, S(E3) = −0.00287, S(E4) = −0.00283, S(E5) = −0.00224

E1 ≤ E3 ≤ E4 ≤ E2 ≤ E5

123


The ranking values of the foregoing results clearly indi-cate that the best alternative is the national saving scheme ofPakistan, with a score of 0.4. Similarly, we can analyze thebudgets of any country.

The key advantages of the proposed approach are as fol-lows:

1. The complexq-rungorthopair fuzzyEinstein aggregationoperators generalize the existing Einstein aggregationoperators.

2. The complexq-rungorthopair fuzzyEinstein aggregationoperators take into account the refusal grade comparedwith complex intuitionistic and complex Pythagoreanfuzzy Einstein aggregation operators.

3. The proposed method based on the proposed operators ismore general than some existing methods, as the param-eters δandq can be changed.

The Cq-ROFS contains two complex-valued functionscalled the complex-valued membership and complex-valued non-membership grades. Because Cq-ROFN E =(TEei2πWTE , NEe

i2πWNE

)satisfies the conditions 0 ≤

T qE

+N qE

≤ 1, 0 ≤ WqTE +Wq

NE≤ 1, andq ≥ 1, the CIFSs

and CPFSs are clearly special cases of the proposed Cq-ROFS.By setting q = 1, the proposedCq-ROFS is convertedinto a CIFS, and by setting q = 2, the proposed Cq-ROFS isconverted into a CPFS. Moreover, the foregoing discussionsindicate that the proposed method is more general and accu-rate than some existing approaches. The proposed method isuseful for designing intelligent systems in real applications,such as image recognition.

Conclusion

This paper proposes the Cq-ROFS and its operational laws.An MADM method based on complex q-rung orthopairfuzzy information was investigated. To aggregate the Cq-ROFNs, we extended the EOs to Cq-ROFSs and proposeda family of complex q-rung orthopair fuzzy Einstein aver-aging operators, such as the Cq-ROFEWA operator, theCq-ROFEOWA operator, the GCq-ROFEWA operator, andthe GCq-ROFEOWA operator. Desirable properties andspecial cases of the introduced operators were discussed.Additionally, we developed a novel approach for MADMin the complex q-rung orthopair fuzzy context based on theproposed operators. Numerical examples were presented todemonstrate the effectiveness and superiority of the proposedmethod via comparison with existing methods. The maincontributions of this study are as follows:

1. The concept of the Cq-ROFS was proposed in the formof polar coordinates belonging to a unit disc in a complexplane.

2. Complex q-rung orthopair fuzzyEinstein averaging oper-ators were introduced, such as the Cq-ROFEWA opera-tor, Cq-ROFEOWA operator, GCq-ROFEWA operator,and GCq-ROFEOWA operator. Desirable properties andspecial cases of the introduced operators were discussed.

3. AnMADMmethod based on the proposed operators wasdeveloped, which is more general than some existingmethods, as the parameters δ and q can be changed.

In the future, we will extend the present work to con-sider (1) aggregation operators for different FSs, e.g., thelinguistic neutrosophic set [44], probabilistic linguistic infor-mation [45], linguistic D number [46], interval type-2 FS[47], T-spherical FS [48, 49], and others [50], and (2)MADMmethods [51–59], which will be more flexible for futuredirects.

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indi-cate if changes were made. The images or other third party materialin this article are included in the article’s Creative Commons licence,unless indicated otherwise in a credit line to the material. If materialis not included in the article’s Creative Commons licence and yourintended use is not permitted by statutory regulation or exceeds thepermitted use, youwill need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix

Proof 1 We know that 0 ≤ T qE

+ N qE

≤ 1, 0 ≤ WqTE +

WqNE

≤ 1, then 0 ≤ T qE

≤ 1 − N qE, 0 ≤ Wq

TE ≤ 1 − WqNE

and 0 ≤ N qE

≤ 1 − T qE, 0 ≤ Wq

NE≤ 1 − Wq

TE implies that

0 ≤ (N q

E

)δ ≤ (1 − T q

E

)δ, 0 ≤

(Wq

NE

)δ ≤(1 − Wq

TE

)δ

implies that

((1 + T q

1

)δ − (1 − T q

1

)δ

(1 + T q

1

)δ + (1 − T q

1

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ−

(1−Wq

T1

)δ

(1+Wq

T1

)δ+

(1−Wq

T1

)δ

⎞

⎟⎠

1q

≤((

1 + T q1

)δ − (N q

E

)δ

(1 + T q

1

)δ + (N q

E

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ−

(Wq

NE

)δ

(1+Wq

T1

)δ+

(Wq

NE

)δ

⎞

⎟⎠

1q

and

123

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http://creativecommons.org/licenses/by/4.0/


⎛

⎜⎜⎝

(2)1q N δ

1((2 − N q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠e

i2π

⎛

⎜⎜⎝

(2)1q Wδ

N1((

2−WqN1

)δ+(Wq

N1

)δ) 1q

⎞

⎟⎟⎠

≤

⎛

⎜⎜⎝

(2)1q N δ

1((1 + T q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

N1((

1+WqTE

)δ+

(Wq

N1

)δ) 1q

⎞

⎟⎟⎟⎠

Thus,

⎛

⎝((

1 + T q1

)δ − (1 − T q

1

)δ

(1 + T q

1

)δ + (1 − T q

1

)δ

) 1q⎞

⎠

q

+

⎛

⎜⎜⎝

⎛

⎜⎜⎝

(2)1q N δ

1((2 − N q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠

⎞

⎟⎟⎠

q

≤ 1

⎛

⎜⎜⎝

⎛

⎜⎝

(1 + Wq

T1

)δ −(1 − Wq

T1

)δ

(1 + Wq

T1

)δ +(1 − Wq

T1

)δ

⎞

⎟⎠

1q⎞

⎟⎟⎠

q

+

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

(2)1q Wδ

N1((

2 − WqN1

)δ +(Wq

N1

)δ) 1

q

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠

q

≤ 1

Moreover,

⎛

⎝((

1 + T q1

)δ − (1 − T q

1

)δ

(1 + T q

1

)δ + (1 − T q

1

)δ

) 1q⎞

⎠

q

+

⎛

⎜⎜⎝

⎛

⎜⎜⎝

(2)1q N δ

1((2 − N q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠

⎞

⎟⎟⎠

q

= 0

⎛

⎜⎜⎝

⎛

⎜⎝

(1 + Wq

T1

)δ −(1 − Wq

T1

)δ

(1 + Wq

T1

)δ +(1 − Wq

T1

)δ

⎞

⎟⎠

1q⎞

⎟⎟⎠

q

+

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

(2)1q Wδ

N1((

2 − WqN1

)δ +(Wq

N1

)δ) 1

q

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠

q

= 0

If and only if T1 = N1 = WT1 = WN1 = 0.

⎛

⎝((

1 + T q1

)δ − (1 − T q

1

)δ

(1 + T q

1

)δ + (1 − T q

1

)δ

) 1q⎞

⎠

q

+

⎛

⎜⎜⎝

⎛

⎜⎜⎝

(2)1q N δ

1((2 − N q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠

⎞

⎟⎟⎠

q

= 1

⎛

⎜⎜⎝

⎛

⎜⎝

(1 + Wq

T1

)δ −(1 − Wq

T1

)δ

(1 + Wq

T1

)δ +(1 − Wq

T1

)δ

⎞

⎟⎠

1q⎞

⎟⎟⎠

q

+

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

(2)1q Wδ

N1((

2 − WqN1

)δ +(Wq

N1

)δ) 1

q

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠

q

= 1

If and only if T q1 + N q

1 = 1, WqT1 + Wq

N1= 1. Thus,

δE1 = E4 is also Cq-ROFNs.

Proof 2 We examine the following parts (1,3,5), and the oth-ers is similar with them.

1. Let us consider the part (1), we have

123


E1 ⊕ E2 =

⎛

⎜⎜⎜⎜⎜⎜⎝

(T q1 + T q

2

1 + T q1 T q

2

) 1q

ei2π

(Wq

T1+Wq

T21+Wq

T1Wq

T2

) 1q

,

⎛

⎝ N1N2(1 + (

1 − N q1

)(1 − N q

2

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

WN1WN2

(1+

(1−Wq

N1

)(1−Wq

N2

)) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎝

(T q2 + T q

1

1 + T q2 T q

1

) 1q

ei2π

(Wq

T2+Wq

T11+Wq

T2Wq

T1

) 1q

,

⎛

⎝ N2N1(1 + (

1 − N q2

)(1 − N q

1

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

WN2WN1

(1+

(1−Wq

N2

)(1−Wq

N1

)) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎠

= E2 ⊕ E1

1. The proof of part (2) is straightforward.2. Let us consider the part (3), we have

E1 ⊕ E2 =

⎛

⎜⎜⎜⎜⎜⎜⎝

(T q1 + T q

2

1 + T q1 T q

2

) 1q

ei2π

(Wq

T1+Wq

T21+Wq

T1Wq

T2

) 1q

,

⎛

⎝ N1N2(1 + (

1 − N q1

)(1 − N q

2

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

WN1WN2

(1+

(1−Wq

N1

)(1−Wq

N2

)) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎠

Is equivalent to

E1 ⊕ E2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+T q

1

)(1+T q

1

)−(1−T q

2

)(1−T q

2

)(1+T q

1

)(1+T q

1

)+(1−T q

2

)(1−T q

2

)

) 1q

e

i2π

⎛

⎝

(1+Wq

T1

)(1+Wq

T1

)−

(1−Wq

T2

)(1−Wq

T2

)

(1+Wq

T1

)(1+Wq

T1

)+

(1−Wq

T2

)(1−Wq

T2

)

⎞

⎠

1q

,

((2)

1q N1N2

((2−N q

1

)(2−N q

2

)+N q1 N

q2

) 1q

)

e

i2π

⎛

⎜⎜⎝

(2)1q WN1

WN2((

2−WqN1

)(2−Wq

N2

)+Wq

N1Wq

N2

) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

We will consider for real part a = (1 + T q

1

)(1 + T q

1

),

b = (1 − T q

2

)(1 − T q

2

), c = N q

1 Nq2 , d = (

2 − N q1

)

(2 − N q

2

)and similarly for imaginary part a′ =

(1 + Wq

T1

)

(1 + Wq

T1

), b′ =

(1 − Wq

T2

)(1 − Wq

T2

), c′ = Wq

N1Wq

N2,

d ′ =(2 − Wq

N1

)(2 − Wq

N2

), then

E1 ⊕ E2 =

⎛

⎜⎜⎜⎜⎜⎝

(a − b

a + b

) 1q

ei2π

(a′ −b

′a′ +b

′) 1

q

,(2c)

1q

(d + c)1q

e

i2π

(2c

′ ) 1q

(d′ +c

′ ) 1q

⎞

⎟⎟⎟⎟⎟⎠

123


By Einstein operational laws, we get

δ(E1 ⊕ E2) = δ

⎛

⎜⎜⎝

(a − b

a + b

) 1q

ei2π

(a′−b′a′+b′

) 1q

,(2c)

1q

(d + c)1q

ei2π (2c′)

1q

(d′+c′)1q

⎞

⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+ a−b

a+b

)δ−(1− a−b

a+b

)δ

(1+ a−b

a+b

)δ+(1− a−b

a+b

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+ a′−b′

a′+b′)δ

−(1− a′−b′

a′+b′)δ

(1+ a′−b′

a′+b′)δ+

(1− a′−b′

a′+b′)δ

⎞

⎟⎠

1q

,

(2)1q

((2c)

1q

(d+c)1q

)δ

((2− 2c

d+c

)δ+(

2cd+c

)δ) 1

qe

i2π

(2)1q

⎛

⎜⎝

(2c′)1q

(d′+c′)1q

⎞

⎟⎠

δ

((2− 2c′

d′+c′)δ+

(2c′

d′+c′)δ

) 1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎝

(aδ−bδ

aδ+bδ

) 1qei2π

(a′δ−b′δa′δ+b′δ

) 1q

,

(2cδ

) 1q

(dδ+cδ)1qe

i2π

(2c′δ

) 1q

(d′δ+c′δ

) 1q

⎞

⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+T q

1

)δ(1+T q

1

)δ−(1−T q

2

)δ(1−T q

2

)δ(1+T q

1

)δ(1+T q

1

)δ+(1−T q

2

)δ(1−T q

2

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ(1+Wq

T1

)δ−

(1−Wq

T2

)δ(1−Wq

T2

)δ

(1+Wq

T1

)δ(1+Wq

T1

)δ+

(1−Wq

T2

)δ(1−Wq

T2

)δ

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q N δ

1N δ2

((2−N q

1

)δ(2−N q

2

)δ+N q1

δN q2

δ) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

N1Wδ

N2((

2−WqN1

)δ(2−Wq

N2

)δ+Wq

N1

δWqN2

δ) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

In other hand, we examine that

δE1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1 + T q

1

)δ − (1 − T q

1

)δ

(1 + T q

1

)δ + (1 − T q

1

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ−

(1−Wq

T1

)δ

(1+Wq

T1

)δ+

(1−Wq

T1

)δ

⎞

⎟⎠

1q

,

⎛

⎜⎜⎝

(2)1q N δ

1((2 − N q

1

)δ + (N q

1

)δ) 1

q

⎞

⎟⎟⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

N1((

2−WqN1

)δ+

(Wq

N1

)δ) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎝

(a1 − b1a1 + b1

) 1q

ei2π

(a′1−b′1

a′1+b′1

) 1q

,

((2c1)

1q

(d1 + c1)1q

)

ei2π

⎛

⎝ (2c′1)1q

(d′1+c′1)

1q

⎞

⎠

⎞

⎟⎟⎠

123


and

δE2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1 + T q

2

)δ − (1 − T q

2

)δ

(1 + T q

2

)δ + (1 − T q

2

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T2

)δ−

(1−Wq

T2

)δ

(1+Wq

T2

)δ+

(1−Wq

T2

)δ

⎞

⎟⎠

1q

,

⎛

⎜⎜⎝

(2)1q N δ

2((2 − N q

2

)δ + (N q

2

)δ) 1

q

⎞

⎟⎟⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

N2((

2−WqN2

)δ+

(Wq

N2

)δ) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎝

(a2 − b2a2 + b2

) 1q

ei2π

(a′2−b′2

a′2+b′2

) 1q

,

((2c2)

1q

(d2 + c2)1q

)

ei2π

⎛

⎝ (2c′2)1q

(d′2+c′2)

1q

⎞

⎠

⎞

⎟⎟⎠

where a1 = (1 + T q

1

)δ, b1 = (

1 − T q1

)δ, c1 = (

N q1

)δ,

d1 = (2 − N q

1

)δ, a2 = (

1 + T q2

)δ, b2 = (

1 − T q2

)δ,

c2 = (N q

2

)δ, d2 = (

2 − N q2

)δ, a′

1 =(1 + Wq

T1

)δ

,

b′1 =

(1 − Wq

T1

)δ

, c′1 =

(Wq

N1

)δ

, d ′1 =

(2 − Wq

N1

)δ

,

a′2 =

(1 + Wq

T2

)δ

, b′2 =

(1 − Wq

T2

)δ

, c′2 =

(Wq

N2

)δ

,

d ′2 =

(2 − Wq

N2

)δ

.

δE1 ⊕ δE2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

(a1−b1a1+b1

) 1qei2π

(a′1−b

′1

a′1+b

′1

) 1q

,

((2c1)

1q

(d1+c1)1q

)e

i2π

⎛

⎜⎜⎝

(2c

′1

) 1q

(d′1+c

′1

) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

⊕

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

(a2−b2a2+b2

) 1qei2π

(a′2−b

′2

a′2+b

′2

) 1q

,

((2c2)

1q

(d2+c2)1q

)e

i2π

⎛

⎜⎜⎝

(2c

′2

) 1q

(d′2+c

′2

) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(a1−b1a1+b1

+ a2−b2a2+b2

1+(a1−b1a1+b1

)(a2−b2a2+b2

)

) 1q

e

i2π

⎛

⎜⎜⎜⎝

a′1−b

′1

a′1+b

′1

+ a′2−b

′2

a′2+b

′2

1+⎛

⎝ a′1−b

′1

a′1+b

′1

⎞

⎠

⎛

⎝ a′2−b

′2

a′2+b

′2

⎞

⎠

⎞

⎟⎟⎟⎠

1q

,

⎛

⎝2(

c1c2(d1+c1)(d2+c2)

) 1q

(1+

(1− 2c1

d1+c1

)(1− 2c2

d2+c2

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎜⎜⎝

2

⎛

⎝ c′1c

′2(

d′1+c

′1

)(d′2+c

′2

)

⎞

⎠

1q

⎛

⎝1+⎛

⎝1− 2c′1

d′1+c

′1

⎞

⎠

⎛

⎝1− 2c′2

d′2+c

′2

⎞

⎠

⎞

⎠

1q

⎞

⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

(a1a2−b1b2a1a2+b1b2

) 1qei2π

(a′1a

′2−b

′1b

′2

a′1a

′2+b

′1b

′2

) 1q

,

(2c1c2

(d1d2+c1c2)

) 1qei2π

(2c

′1c

′2(

d′1d

′2+c

′1c

′2

)

) 1q

⎞

⎟⎟⎟⎟⎟⎠

123


=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+T q

1

)δ(1+T q

1

)δ−(1−T q

2

)δ(1−T q

2

)δ(1+T q

1

)δ(1+T q

1

)δ+(1−T q

2

)δ(1−T q

2

)δ

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ(1+Wq

T1

)δ−

(1−Wq

T2

)δ(1−Wq

T2

)δ

(1+Wq

T1

)δ(1+Wq

T1

)δ+

(1−Wq

T2

)δ(1−Wq

T2

)δ

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q N δ

1N δ2

((2−N q

1

)δ(2−N q

2

)δ+N q1

δN q2

δ) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ

N1Wδ

N2((

2−WqN1

)δ(2−Wq

N2

)δ+Wq

N1

δWqN2

δ) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Hence δ(E1 ⊕ E2) = δE1 ⊕ δE2.

3. The proof of part (4) is straightforward.4. Let us consider the part (5), we have

δ1E1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

1

)δ1−(1−T q

1

)δ1(1+T q

1

)δ1+(1−T q

1

)δ1

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ1−(1−Wq

T1

)δ1

(1+Wq

T1

)δ1+(1−Wq

T1

)δ1

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q N δ1

1((2−N q

1

)δ1+(N q

1

)δ1) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ1

N1((

2−WqN1

)δ1+(Wq

N1

)δ1) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

(a1−b1a1+b1

) 1qei2π

(a′1−b′1

a′1+b′1

) 1q

,

((2c1)

1q

(d1+c1)1q

)ei2π

⎛

⎝ (2c′1)1q

(d′1+c′1)

1q

⎞

⎠

⎞

⎟⎟⎟⎟⎟⎠

δ2E1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

1

)δ2−(1−T q

1

)δ2(1+T q

1

)δ2+(1−T q

1

)δ2

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ2 −(1−Wq

T1

)δ2

(1+Wq

T1

)δ2 +(1−Wq

T1

)δ2

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q N δ2

1((2−N q

1

)δ2+(N q

1

)δ2) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ2

N1((

2−WqN1

)δ2 +(Wq

N1

)δ2) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎝

(a2−b2a2+b2

) 1qei2π

(a′2−b′2

a′2+b′2

) 1q

,

((2c2)

1q

(d2+c2)1q

)ei2π

⎛

⎝ (2c′2)1q

(d′2+c′2)

1q

⎞

⎠

⎞

⎟⎟⎟⎟⎟⎠

where a1 = (1 + T q

1

)δ1 , b1 = (1 − T q

1

)δ1 , c1 = (N q

1

)δ1 ,

d1 = (2 − N q

1

)δ1 , a2 = (1 + T q

2

)δ2 , b2 = (1 − T q

2

)δ2 ,

c2 = (N q

2

)δ2 , d2 = (2 − N q

2

)δ2 , a′1 =

(1 + Wq

T1

)δ1, b′

1 =(1 − Wq

T1

)δ1, c′

1 =(Wq

N1

)δ1, d ′

1 =(2 − Wq

N1

)δ1, a′

2 =(1 + Wq

T2

)δ2, b′

2 =(1 − Wq

T2

)δ2, c′

2 =(Wq

N2

)δ2, d ′

2 =(2 − Wq

N2

)δ2. Then

δ1E1 ⊕ δ2E1 =

⎛

⎜⎜⎜⎜⎜⎝

(a1−b1a1+b1

) 1qei2π

(a′1−b′1

a′1+b′1

) 1q

,

((2c1)

1q

(d1+c1)1q

)ei2π

⎛

⎝ (2c′1)1q

(d′1+c′1)

1q

⎞

⎠

⎞

⎟⎟⎟⎟⎟⎠

⊕

⎛

⎜⎜⎜⎜⎜⎝

(a2−b2a2+b2

) 1qei2π

(a′2−b′2

a′2+b′2

) 1q

,

((2c2)

1q

(d2+c2)1q

)ei2π

⎛

⎝ (2c′2)1q

(d′2+c′2)

1q

⎞

⎠

⎞

⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(a1−b1a1+b1

+ a2−b2a2+b2

1+(a1−b1a1+b1

)(a2−b2a2+b2

)

) 1q

e

i2π

⎛

⎜⎜⎝

a′1−b′1

a′1+b′1

+ a′2−b′2

a′2+b′2

1+(a′1−b′1

a′1+b′1

)(a′2−b′2

a′2+b′2

)

⎞

⎟⎟⎠

1q

,

⎛

⎝2(

c1c2(d1+c1)(d2+c2)

) 1q

(1+

(1− 2c1

d1+c1

)(1− 2c2

d2+c2

)) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎝

2

(c′1c′2

(d′1+c′1)(d′

2+c′2)

) 1q

(

1+(

1− 2c′1d′1+c′1

)(

1− 2c′2d′2+c′2

)) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

123


=

⎛

⎜⎜⎜⎜⎝

(a1a2−b1b2a1a2+b1b2

) 1qei2π

(a′1a

′2−b′1b′2

a′1a

′2+b′1b′2

) 1q

,

(2c1c2

(d1d2+c1c2)

) 1qei2π

(2c′1c′2

(d′1d

′2+c′1c′2)

) 1q

⎞

⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

1

)δ1+δ2−(1−T q

1

)δ1+δ2

(1+T q

1

)δ1+δ2+(1−T q

1

)δ1+δ2

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)δ1+δ2 −(1−Wq

T1

)δ1+δ2

(1+Wq

T1

)δ1+δ2 +(1−Wq

T1

)δ1+δ2

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q N δ1+δ2

1((2−N q

1

)δ1+δ2+N q1

δ1+δ2) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎝

(2)1q Wδ1+δ2

N1((

2−WqN1

)δ1+δ2 +WqN1

δ1+δ2) 1q

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

= (δ1 + δ2)E1.

Hence δ1E1 ⊕ δ2E1 = (δ1 + δ2)E1.

5. The proof of part (6) is straightforward.

Proof 4 Using the process of mathematical induction toprove the Eq. (11), if n = 2, then.

Cq-ROFEWA(E1, E2, . . . , En) = ω1E1 ⊕ ω2E2

By Theorem 1, the result of ω1E1 ⊕ ω2E2 is also Cq-ROFN, Using the Definition (8), we get

ω1E1 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

1

)ω1−(1−T q

1

)ω1(1+T q

1

)ω1+(1−T q

1

)ω1

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)ω1−(1−Wq

T1

)ω1

(1+Wq

T1

)ω1+(1−Wq

T1

)ω1

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q Nω1

1((2−N q

1

)ω1+(N q

1

)ω1) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q Wω1

N1((

2−WqN1

)ω1+(Wq

N1

)ω1) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ω2E2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

2

)ω2−(1−T q

2

)ω2(1+T q

2

)ω2+(1−T q

2

)ω2

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T2

)ω2 −(1−Wq

T2

)ω2

(1+Wq

T2

)ω2 +(1−Wq

T2

)ω2

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q Nω2

2((2−N q

2

)ω2+(N q

2

)ω2) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q Wω2

N2((

2−WqN2

)ω2 +(Wq

N2

)ω2) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

then

Cq-ROFEWA(E1, E2, . . . , En) = ω1E1 ⊕ ω2E2

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

1

)ω1−(1−T q

1

)ω1(1+T q

1

)ω1+(1−T q

1

)ω1

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)ω1−(1−Wq

T1

)ω1

(1+Wq

T1

)ω1+(1−Wq

T1

)ω1

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q Nω1

1((2−N q

1

)ω1+(N q

1

)ω1) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q Wω1

N1((

2−WqN1

)ω1+(Wq

N1

)ω1) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⊕

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( (1+T q

2

)ω2−(1−T q

2

)ω2(1+T q

2

)ω2+(1−T q

2

)ω2

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T2

)ω2 −(1−Wq

T2

)ω2

(1+Wq

T2

)ω2 +(1−Wq

T2

)ω2

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q Nω2

2((2−N q

2

)ω2+(N q

2

)ω2) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q Wω2

N2((

2−WqN2

)ω2 +(Wq

N2

)ω2) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎝

(1+T q1 )

ω1−(1−T q1 )

ω1

(1+T q1 )

ω1+(1−T q1 )

ω1 + (1+T q2 )

ω2 −(1−T q2 )

ω2

(1+T q2 )

ω2 +(1−T q2 )

ω2

1+((1+T q

1 )ω1−(1−T q

1 )ω1

(1+T q1 )

ω1+(1−T q1 )

ω1

)((1+T q

2 )ω2 −(1−T q

2 )ω2

(1+T q2 )

ω2 +(1−T q2 )

ω2

)

⎞

⎟⎠

1q

e

i2π

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

(1+Wq

T1

)ω1−(1−Wq

T1

)ω1

(1+Wq

T1

)ω1+(1−Wq

T1

)ω1 +

(1+Wq

T2

)ω2 −(1−Wq

T2

)ω2

(1+Wq

T2

)ω2 +(1−Wq

T2

)ω2

1+

⎛

⎜⎜⎝

(1+Wq

T1

)ω1−(1−Wq

T1

)ω1

(1+Wq

T1

)ω1+(1−Wq

T1

)ω1

⎞

⎟⎟⎠

⎛

⎜⎜⎝

(1+Wq

T2

)ω2 −(1−Wq

T2

)ω2

(1+Wq

T2

)ω2 +(1−Wq

T2

)ω2

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

1q

,

⎛

⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎝

(2)1q Nω1

1((2−N q

1 )ω1+(N q

1 )ω1

) 1q

⎞

⎟⎠

⎛

⎜⎝

(2)1q Nω2

2((2−N q

2 )ω2 +(N q

2 )ω2

) 1q

⎞

⎟⎠

⎛

⎜⎝1+

⎛

⎜⎝1− (2)

1q Nω1

1((2−N q

1 )ω1+(N q

1 )ω1

) 1q

⎞

⎟⎠

⎛

⎜⎝1− (2)

1q Nω2

2((2−N q

2 )ω2 +(N q

2 )ω2

) 1q

⎞

⎟⎠

⎞

⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎠

e

i2π

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

(2)1q Wω1

N1((

2−WqN1

)ω1+(Wq

N1

)ω1) 1q

⎞

⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎝

(2)1q Wω2

N2((

2−WqN2

)ω2 +(Wq

N2

)ω2) 1q

⎞

⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎝1+

⎛

⎜⎜⎜⎜⎝1−

(2)1q Wω1

N1((

2−WqN1

)ω1+(Wq

N1

)ω1) 1q

⎞

⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎝1−

(2)1q Wω2

N2((

2−WqN2

)ω2 +(Wq

N2

)ω2) 1q

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠

1q

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

123


=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+T q

1

)ω1 (1+T q

2

)ω2−(1−T q

1

)ω1 (1−T q

2

)ω2(1+T q

1

)ω1 (1+T q

2

)ω2+(1−T q

1

)ω1 (1−T q

2

)ω2

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T1

)ω1(1+Wq

T2

)ω2−(1−Wq

T1

)ω1(1−Wq

T2

)ω2

(1+Wq

T1

)ω1(1+Wq

T2

)ω2+(1−Wq

T1

)ω1(1−Wq

T2

)ω2

⎞

⎟⎠

1q

,

⎛

⎝(2)

1q

(N ω1

1 N ω22

)

((2−N q

1

)ω1 (2−N q

2

)ω2+(N q

1

)ω1 (N q

2

)ω2) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q

(Wω1

N1Wω2

N2

)

((2−Wq

N1

)ω1(2−Wq

N2

)ω2+(Wq

N1

)ω1(Wq

N2

)ω2) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

For n = 2, the result is kept. We suppose that it is true forn = k, then the Eq. (11), we have

Cq-ROFEWA(E1, E2, . . . , Ek) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏ki=1

(1+T q

i

)ωi −∏ki=1

(1−T q

i

)ωi∏k

i=1(1+T q

i

)ωi +∏ki=1

(1−T q

i

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏ki=1

(1+Wq

Ti

)ωi −∏ki=1

(1−Wq

Ti

)ωi

∏ki=1

(1+Wq

Ti

)ωi +∏ki=1

(1−Wq

Ti

)ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q

∏ki=1N

ωii

(∏ki=1

(2−N q

i

)ωi +∏ki=1

(N q

i

)ωi) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q ∏k

i=1WωiNi

(∏ki=1

(2−Wq

Ni

)ωi +∏ki=1

(Wq

Ni

)ωi) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

We will prove for n = k + 1, such that

Cq − ROFEW A(E1, E2, . . . , Ek+1) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏ki=1

(1+T q

i

)ωi −∏ki=1

(1−T q

i

)ωi∏k

i=1(1+T q

i

)ωi +∏ki=1

(1−T q

i

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏ki=1

(1+Wq

Ti

)ωi −∏ki=1

(1−Wq

Ti

)ωi

∏ki=1

(1+Wq

Ti

)ωi +∏ki=1

(1−Wq

Ti

)ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q

∏ki=1N

ωii

(∏ki=1

(2−N q

i

)ωi +∏ki=1

(N q

i

)ωi) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q ∏k

i=1WωiNi

(∏ki=1

(2−Wq

Ni

)ωi +∏ki=1

(Wq

Ni

)ωi) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⊕

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+T q

k+1

)ωk+1−(1−T q

k+1

)ωk+1(1+T q

k+1

)ωk+1+(1−T q

k+1

)ωk+1

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

Tk+1

)ωk+1−(1−Wq

Tk+1

)ωk+1

(1+Wq

Tk+1

)ωk+1+(1−Wq

Tk+1

)ωk+1

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q Nωk+1

k+1((2−N q

k+1

)ωk+1+(N q

k+1

)ωk+1) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q W

ωk+1Nk+1

((2−Wq

Nk+1

)ωk+1+(Wq

Nk+1

)ωk+1) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏k+1i=1

(1+T q

i

)ωi −∏k+1i=1

(1−T q

i

)ωi∏k+1

i=1

(1+T q

i

)ωi +∏k+1i=1

(1−T q

i

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏k+1i=1

(1+Wq

Ti

)ωi −∏k+1i=1

(1−Wq

Ti

)ωi

∏k+1i=1

(1+Wq

Ti

)ωi +∏k+1i=1

(1−Wq

Ti

)ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q

∏k+1i=1N

ωii

(∏k+1i=1

(2−N q

i

)ωi +∏k+1i=1

(N q

i

)ωi) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q ∏k+1

i=1WωiNi

(∏k+1i=1

(2−Wq

Ni

)ωi +∏k+1i=1

(Wq

Ni

)ωi) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

123


For n = k + 1, the result is also kept. Hence the Eq. (11),is kept for all n.

Proof 5 Let Cq-ROFEWA(E1, E2, . . . , En) =(T qi e

i2πWqTi , N q

i ei2πWq

Ni

)= Eq

i and Cq-ROFWA

(E1, E2, . . . , En) =(Tiei2πWTi , Nie

i2πWNi

)= Ei .

Firstly, we check for real part of complex-valued member-ship grades, such that

∏n

i=1

(1 + T q

i

)ωi +∏n

i=1

(1 − T q

i

)ωi

≤n∑

i=1

ωi(1 + T q

i

) +n∑

i=1

ωi(1 − T q

i

) = 2

then,

(∏ni=1

(1 + T q

i

)ωi − ∏ni=1

(1 − T q

i

)ωi

∏ni=1

(1 + T q

i

)ωi + ∏ni=1

(1 − T q

i

)ωi

) 1q

≤(1 −

∏n

i=1

(1 − T q

i

)ωi) 1

q ⇐⇒ T pi = Ti ⇐⇒ T1

= T2 = . . . = Tn .

At the same time, it is also kept for imaginary part ofcomplex-valued membership grade. Next, we check for realpart for non-membership grade, we get

2(∏n

i=1Nqi

)ωi

∏ni=1

(2 − N q

i

)ωi + (∏ni=1N

qi

)ωi

≥ 2∏n

i=1

(N q

i

)ωi

∑ni=1 ωi

(2 − N q

i

) + ∑ni=1 ωiN q

i

≥∏n

i=1

(N q

i

)ωi

⇒(

2(∏n

i=1Nqi

)ωi

∏ni=1

(2 − N q

i

)ωi + (∏ni=1N

qi

)ωi

) 1q

≥∏n

i=1(Ni )

ωi

⇒ N pi ≥ Ni ⇐⇒ N1 = N2 = . . . = Nn

At the same time, it is also kept for imaginary partof complex-valued non-membership grade. Then the scorevalue of the above, such that

S(Epi

) = 1

2

(T pi

q + W pTi

q − Nipq − W p

Ni

q)

≤ 1

2

(T qi + Wq

Ti − Niq − Wq

Ni

)= S(Ei )

When S(Epi

)< S(Ei ), then by definition of score func-

tion, we have

Cq-ROFEWA(E1, E2, . . . , En) < Cq-ROFWA(E1, E2, . . . , En)

When S(Epi

) = S(Ei ), then by definition of accuracyfunction, we have

H(Epi

) = 1

2

(T p1

q + W pT1

q − N1pq − W p

N1

q)

= 1

2

(T qi + Wq

Ti + Niq + Wq

Ni

)= H(Ei )

Cq-ROFEWA(E1, E2, . . . , En) = Cq-ROFWA(E1, E2, . . . , En)

therefore

Cq-ROFEWA(E1, E2, . . . , En) ≤ Cq-ROFWA(E1, E2, . . . , En)

Proof (Property 1: Idempotency)

1. We know that E0 =(T0ei2πWT0 , N0e

i2πWN0

)∈

Cq-ROFN, i = 1,2, . . . , n, then

Cq-ROFEWA(E1, E2, . . . , En) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(∏ni=1

(1+T q

0

)ωi −∏ni=1

(1−T q

0

)ωi∏n

i=1(1+T q

0

)ωi +∏ni=1

(1−T q

0

)ωi

) 1q

e

i2π

⎛

⎜⎝

∏ni=1

(1+Wq

T0

)ωi −∏ni=1

(1−Wq

T0

)ωi

∏ni=1

(1+Wq

T0

)ωi +∏ni=1

(1−Wq

T0

)ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q

∏ni=1N

ωi0

(∏ni=1

(2−N q

0

)ωi +∏ni=1

(N q

0

)ωi) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎝

(2)1q ∏n

i=1WωiN0

(∏ni=1

(2−Wq

N0

)ωi +∏ni=1

(Wq

N0

)ωi) 1q

⎞

⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

123


=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

((1+T q

0

)∑ni=1 ωi −(

1−T q0

)∑ni=1 ωi

(1+T q

0

)∑ni=1 ωi +(

1−T q0

)∑ni=1 ωi

) 1q

e

i2π

⎛

⎜⎝

(1+Wq

T0

)∑ni=1 ωi −

(1−Wq

T0

)∑ni=1 ωi

(1+Wq

T0

)∑ni=1 ωi +

(1−Wq

T0

)∑ni=1 ωi

⎞

⎟⎠

1q

,

⎛

⎝ (2)1q N

∑ni=1 ωi

0((2−N q

0

)∑ni=1 ωi +(

N q0

)∑ni=1 ωi

) 1q

⎞

⎠e

i2π

⎛

⎜⎜⎜⎜⎝

(2)1q W

∑ni=1 ωi

N0((

2−WqN0

)∑ni=1 ωi +

(Wq

N0

)∑ni=1 ωi

) 1q

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=(T0ei2πWT0 , N0e

i2πWN0

)= E0

2. Proof (Property 2: Boundedness)We know that f (x) = 1−x

1+x , x ∈ [0, 1], then f ′(x) =−2

(1+x)2< 0, thus f (x) is decreasing function. We check for

real part of membership grade, such that T qi .min ≤ T q

i ≤T qi .max∀i = 1, 2, . . . , n, then f

(T qi .min

) ≤ f(T qi

) ≤ f(T qi .max

), ∀i implies that

1−T qi .max

1+T qi .max

≤ 1−T qi

1+T qi

≤ 1−T qi .min

1+T qi .min

, ∀iand the weight vector is discussed above, we have(1 − T q

i .max

1 + T qi .max

)ωi

≤(1 − T q

i

1 + T qi

)ωi

≤(1 − T q

i .min

1 + T qi .min

)ωi

⇒n∏

i=1

(1 − T q

i .max

1 + T qi .max

)ωi

≤n∏

i=1

(1 − T q

i

1 + T qi

)ωi

≤n∏

i=1

(1 − T q

i .min

1 + T qi .min

)ωi

⇔(1 − T q

i .max

1 + T qi .max

) n∑

i=1ωi

≤n∏

i=1

(1 − T q

i

1 + T qi

)ωi

≤(1 − T q

i .min

1 + T qi .min

) n∑

i=1ωi

⇔(

2

1 + T qi .max

)

≤ 1 +n∏

i=1

(1 − T q

i

1 + T qi

)ωi

≤(

2

1 + T qi .min

)

⇔(1 + T q

i .min

2

)

≤ 1

1 + ∏ni=1

(1−T q

i1+T q

i

)ωi≤

(1 + T q

i .max

2

)

⇔ (1 + T q

i .min

) ≤ 2

1 + ∏ni=1

(1−T q

i1+T q

i

)ωi≤ (

1 + T qi .max

)

⇔ (T qi .min

) ≤ 2

1 + ∏ni=1

(1−T q

i1+T q

i

)ωi− 1 ≤ (

T qi .max

)

⇔ (T qi .min

) ≤∏n

i=1

(1 + T q

i

)ωi − ∏ni=1

(1 − T q

i

)ωi

∏ni=1

(1 + T q

i

)ωi + ∏ni=1

(1 − T q

i

)ωi≤ (

T qi .max

)

⇔ Ti .min ≤(∏n

i=1

(1 + T q

i

)ωi − ∏ni=1

(1 − T q

i

)ωi

∏ni=1

(1 + T q

i

)ωi + ∏ni=1

(1 − T q

i

)ωi

) 1q

≤ Ti .max

It is also kept for the imaginary part of membership grade,such that

WTi .min ≤⎛

⎜⎝

∏ni=1

(1 + Wq

Ti

)ωi − ∏ni=1

(1 − Wq

Ti

)ωi

∏ni=1

(1 + Wq

Ti

)ωi + ∏ni=1

(1 − Wq

Ti

)ωi

⎞

⎟⎠

1q

≤ WTi .max

Similarly, when we consider g(y) = 2−yy , y ∈ [0, 1],

then g′(y) = −2y2

< 0, then the function is decreasing, so thefollowings are kept.

Ni .min ≤ (2)1q

∏ni=1N

ωii

(∏ni=1

(2 − N q

i

)ωi + ∏ni=1

(N q

i

)ωi) 1q

≤ Ni .max

It is also kept for the imaginary part of non-membershipgrade, such that

WNi .min ≤ (2)1q

∏ni=1 W

ωiNi

(∏ni=1

(2 − Wq

Ni

)ωi + ∏ni=1

(Wq

Ni

)ωi) 1

q

≤ WNi .max

From the above discussion, we get

Ti .min ≤ Ti ≤ Ti .max

WTi .min ≤ WTi ≤ WTi .max

Ni .min ≤ Ni ≤ Ni .max

WNi .min ≤ WNi ≤ WNi .max

So, S(Ei ) = 12

(T qi + Wq

Ti − N qi − Wq

Ni

)≤ 1

2(T qi .max + Wq

Ti .max− N q

i .min − WqNi .min

)= S

(E+i

)

and S(Ei ) = 12

(T qi + Wq

Ti − N qi − Wq

Ni

)≥ 1

2(T qi .min + Wq

Ti .min− N q

i .max − WqNi .max

)= S

(E−i

), then we

get

E−i ≤ Cq-ROFEWA(E1, E2, . . . , En) ≤ E+

i .

3. The proof of property 3 is similar to property 2.

123


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