fs.unm.edufs.unm.edu/neut/GeneralizedSingleValuedNeutrosophicPower.pdf · Received January 27, 2020, accepted February 13, 2020, date of publication February 18, 2020, date of current

Received January 27, 2020, accepted February 13, 2020, date of publication February 18, 2020, date of current version February 28, 2020.

Digital Object Identifier 10.1109/ACCESS.2020.2974767

Generalized Single-Valued Neutrosophic PowerAggregation Operators Based on ArchimedeanCopula and Co-Copula and Their Applicationto Multi-Attribute Decision-MakingYUAN RONG 1, YI LIU 2,3, AND ZHENG PEI 11School of Science, Xihua University, Chengdu 610039, China2Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641000, China3Numerical Simulation Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641000, China

Corresponding author: Zheng Pei ([email protected])

This work was supported in part by the National Natural Science Foundation of China under Grant 61372187, in part by the Scientific andTechnological Project of Sichuan Province under Grant 2019YFG0100, in part by the Sichuan Province Youth Science and TechnologyInnovation Team under Grant 2019JDTD0015, in part by the Application Basic Research Plan Project of Sichuan Province underGrant 2017JY0199, in part by the Scientific Research Project of Department of Education of Sichuan Province under Grant 18ZA0273 andGrant 15TD0027, and in part by the Scientific Research Project of Neijiang Normal University under Grant 18TD08.

ABSTRACT Single-valued neutrosophic set (SVN) can valid depict the incompleteness, nondeterminacyand inconsistency of evaluation opinion, and the Power average (PA) operator can take into accountthe correlation of multiple discussed data. Meanwhile, Archimedean copula and co-copula (ACC) cansignificant generate operational laws based upon diverse copulas. In this paper, we first redefine several noveloperational laws of single-valued neutrosophic number (SVNN) based on ACC and discuss the associatedproperties of them. In view of these operational rules, we propound several novel power aggregationoperators (AOs) to fuse SVN information, i.e., SVN copula power average (SVNCPA) operator, weightedSVNCPA (WSVNCPA) operator, order WSVNCPA operator, and SVN copula power geometric (SVNCPG)operator, weighted SVNCPG (WSVNCPG) operator, order WSVNCPG operator. At the same time, severalsignificant characteristics and particular cases of these operators are examined in detail. Moreover, we extendthese 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 isvalidated by a empirical example. A comparative and sensitivity analysis are carried out to elaborate thestrength and preponderance of the propounded approach.

INDEX TERMS Multi-attribute decision-making, single-valued neutrosophic set, aggregation operator,archimedean copula and co-copula, power average.

I. INTRODUCTIONMultiple attribute decisionmaking is an important constituentof modern decision science and management, which is a pro-cevss of evaluating and selecting the optimal alternative byscientific and reasonable approaches based upon the selectedmultiple attributes and assessment information. Traditionally,managers expressing their preference information take in the

The associate editor coordinating the review of this manuscript and

approving it for publication was Hualong Yu .

form of a accurate numerical value. However, the precisenumerical value is arduous for decision makers (DMs) toprovide the incomplete and vague information in the com-plex system perfectly. In order to cope with this defect anddescribe the uncertainty and fuzziness of information moreeffectively, the fuzzy set (FS) [1] theory and its expansionsform intuitionistic fuzzy set (IFS) [2], interval-valued intu-itionistic fuzzy set (IVIFS) [3] are propounded and thesefuzzy sets are wide-ranging utilized to handle decision mak-ing (DM) problems [4]–[12]. As the foundation of these

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Y. Rong et al.: Generalized SVN Power Aggregation Operators Based on ACC

theories, numerous investigators have been favored on theoperation laws of fuzzy numbers in the last two decades.Xu and Yager [13] pioneered several basic operational rulesbased on Algebraic t-norm and s-norm (ATS) and pro-pounded some geometric operators of intuitionistic fuzzynumber (IFN).Wang et al. [14] further developed the Einsteinoperatorts of IFN to aggregation intuitionistic fuzzy informa-tion. Xia et al. [15] defined the generalized operational lawsof IFN on the basis of ATS and developed several generalizedAOs. Lei and Xu [16] introduced the novel operational of IFNnamed derivative and differential operational laws. Besides,the neutral operational [17], exponential operational [18] andlogarithmic operational [19] of IFN are put forward one afteranother. With the aid of these operations, a multitude ofAOs have been brought forward to apply to the procedure ofinformation aggregation. Luo et al. [20] propounded severalexponential AOs based on ATS for settling MADM issues.Garg [21] developed some generalized AOs based on expo-nential and logarithmic operational laws under intuitionisticfuzzy circumstance. Garg [22] proposed a series of general-ized intuitionistic fuzzy soft power AOs based upon ATS.Except for these, other researches also developed a sea ofAOs based on different extension forms of IFS, which aresummarized as [23]–[29]. However, these above approacheshave a limitation that they disable to deal with uncertainand inconsistent evaluation information effectively in severalspecial situations.

In order to overwhelm the aforementioned deficiencyvalidly, Smarandache [30] originally pioneered a novel notionnamed neutrosophic set(NS). NS theory adds a independentindeterminacy membership degree on the basis of IFS, whichis a novel generalization of FS and IFS. However, the NStheory is presented from the philosophical perspective inthe early days, it is arduous to put it into particular issuesbecause the membership functions of NS are bounded byin ]0−, 1+[. For surmount it efficiently, Wang et al. [31]firstly developed the conception of single-valued neutro-sophic and explored related operational laws and proper-ties. Smarandache [32] defined the subtraction and divisionoperational rules of SVNN. Thereafter, a lot of researchachievements have been achieved with respect to the oper-ations and AOs of SVN. Liu et al. [33] proposed a gen-eral operations of SVNNs based on ATT and several AOs.Lu et al. [34] developed the exponential operations of SVNNand corresponding AOs to settle MADM problems. Garg[35] defined the logarithmic operational laws for computingSVNN and propounded the logarithmic SVN weight aver-aging (L-SVNWA) operator and logarithmic SVN weightgeometric (L-SVNWG) operator to integrate SVN infor-mation. However, the above-mention AOs assume that allattributes are independent of each other in the process ofinformation aggregation. This will affects the final decisionresults. Based on this, several novel AOs are extended to SVNenvironment to overcome the above deficiencies. Ji et al. [36]defined the Frank operations of SVNN and propounded SVNFrank prioritized Bonferroni mean operator, which can take

into account the interrelationship between any two discussedparameters. Liu et al. [37] developed several power Muir-head mean operators under SVN context. Zhao et al. [38]introduced some SVN power Heronian mean AOs, whichconsider the importance and correlation between discussedarguments simultaneously. Peng et al. [39] presented powerShapley Choquet average operators for SVNNs to do withMADM problems. Liu et al. [40] defined the new opera-tions based on Schweizer-Sklar norm and presented severalSVN Schweizer-Sklar prioritized AOs. Based upon decisionapproaches, Sun et al. [41] presented the extended TODIMand extended ELECTRE III approaches to SVN environmenton the basis of a novel distance measure to solve the problemof doctors assessment problems. Peng et al. [42] developedthree MADM algorithms based on new similarity measure,TOPSIS and MABACmethods. Tian et al. [43] developed anextended QUALIFLEX approach on the basis of an improvedSVN projection measure to resolve the problem of greensupplier selection with weight information fully known. Themore research results about SVN have been propoundedin [44]–[52].

With the increasing complicacy of DM setting and thediversity of DMs’ knowledge level and evaluation experi-ence, the evaluation information provided by DMs may havesubjectivity. For instance, a manager provides his assessmentfor safety and cost in a programme, he shall distribute ahigher preference to safety than cost. For overwhelming thissituation and improving the accuracy of decision results,Yager [53] propounded the power average operator whichsticks out the support degree of data during informationaggregating procedure. Besides, PA operator can acquire theweight from fused data, and cut down the the impact oftoo high and too low data. Since its appearance, the theoryand application of it have received an increasingly numberextensive attention in information fusion and data mining.Zhang [54] studied the generalized power geometric operatorunder intuitionistic fuzzy context to settle MAGDM issues.Liu et al. [55] combined PA operator and Maclaurin Sym-metric mean (MSM) operator to developed sever powerMSMAOs to fuse q-rung orthopair fuzzy information. For demon-strating the flexibility of the aggregation process, Zhou et al.[56] revealed the generalized power AOs to handle group DMproblems. Ju et al. [57] extended it to q-ROF environment.Liu et al. [58] presented a novel MADM methodology basedon generalized power operators under hesitant fuzzy linguis-tic context. Therefor, it is significant to study onAOs utilizinggeneralized PA operator to propound novel operators.

As a classical instance of traditional t-norm and s-norm,copulas and co-copulas [59] have received pervasive appli-cation in various flied. Among these utilizations, the oper-ation of copulas is one of the essential core of researchhot [60]–[66]. From these research achievements, the advan-tages of copulas are summarized in the following three cat-egories: (i) it can generate operations under diverse fuzzysettings and the acquired operations based upon copulas areclosed; (ii) it can effectively prevent the loss of information

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in the process of information aggregation because it canseize the interrelationship among diverse attributes; (iii) itprovides different copulas for managers to choose throughtheir preference, and each copula has a parameter whichmakes the decision procedure more flexible. In allusion theabove-mentioned merits, Tao et al. [67] firstly developedthe copula operational on intuitionistic fuzzy environmentand proposed a weighted aggregation operator to aggregateintuitionistic fuzzy information. Tao et al. [68] constructed anovel computation model based on the ACC, which providesa valid technique to fuse unbalanced linguistic information.Chen et al. [69] introduced several new AOs of linguisticneutrosophic on the basis of ACC. Based on these foregoinganalysis, it is evident know that copulas and PA have theirdistinctive preponderance and acquire growing attention fromscholars during the application in MADM issues but theyare not generalized to SVN domain. Han et al. [70] pro-pounded the conception of probabilistic unbalanced linguisticterm set and several AOs bases upon Archimedean copula.In light of above discussion and motivations, the objectiveof this study are to combine copulas, PA operator and SVNto present several AOs and decision approach. Accordingly,the mainly intentions and contributions of this research areoutline following:

1) To extend ACC to SVN and propound several novelgeneral operational rules of SVNN;

2) To combine PA operator and the new operationto develop a series of SVN power AOs includ-ing SVN copula power average(SVNCPA) operator,weighted SVN copula power average (WSVNCPA)operator, order weighted SVN copula power aver-age (OWSVNCPA) operator and correspondinggeometric operators;

3) To extend the proposed AOs to their generalized ver-sion namely GWSVNCPA operator, GOWSVNCPAoperator,GWSVNCPG operator and GOWSVNCPGoperator;

4) To set up an MADM approach with the weight infor-mation unknown based on the propounded AOs;

5) To confirm the validity and strength of propoundedmethod through empirical application and comparativestudy.

For achieving the objective, the remainder of this inves-tigation is allocated as follows: Section 2 succinctly looksback several fundamental notions of SVNS and ACC.Section 3 develops several novel operations for SVNNbased on ACC. Section 4 presents several aggregation oper-ators based on the newly operations and discusses someproperties and particular examples. Section 5 extends thepropounded operators in Section 4 to their generalizedform. Section 6 develops an approach based on thosepresented operators for tackling decision making issues.Section 7 exhibits an empirical example to demonstrate thevalidity and legitimacy of the presented strategy. The articleends with several conclusions in Section 8.

II. PRELIMINARIESIn this part, several fundamental concepts of SVN and ACCare succinctly reviewed as the preparation of the followingresearch.

A. SVNNDefinition 1 [30]: Let X is a finite universe of discourse.

A single-valued neutrosophic set A in X is defined as below:A = {〈x, (TA(x), IA(x),FA(x))〉 | x ∈ X }, (1)

in which TA(x) : X → [0, 1], TA(x) : X → [0, 1]and TA(x) : X → [0, 1] with the condition 0 ≤

TA(x)+IA(x)+FA(x) ≤ 3, TA(x), IA(x),FA(x) are calledthe truth-membership function, indeterminacy-membershipfunction, falsity-membership function, respectively. For sim-plicity, α = (T , I,F) is called a single-valued neutrosophicnumber and αc = (F , 1 − I, T ), 8 is a collection of allSVNNs.

B. POWER OPERATORDefinition 2 [53]: For a family of real numbers

a(1, 2, · · · , n), the power averaging operator is stated asbelow:

PA(a1, a2, · · · , an) =

∑ni=1(1+ T (ai))ai∑ni=1(1+ T (ai))

(2)

where T (εi) =∑n

j=1,j 6=i Sup(εi, εj

), and Sup

(εi, εj

)indicates

the support degree of ai to aj. The following properties aresatisfied:

(1) Sup(ai, aj

)∈ [0, 1];

(2) Sup(ai, aj

)= Sup

(aj, ai

);

(3) Sup(ai, aj

)≥ Sup(am, an) iff | ai − aj |<| am − an |.

C. ARCHIMEDEAN COPULADefinition 3 [59]: A two-dimensional function C :

[0, 1] × [0, 1] → [0, 1] is called a copula if it satisfies thefollowing conditions:

(1)C(u, 1) = C(1, u) = u;

(2)C(u, 0) = C(0, u) = 0;

(3)C(u1, v1)+ C(u2, v2) ≥ C(u1, v2)+ C(u2, v1).

where u, ui, vi ∈ [0, 1], i = 1, 2 with u1 ≤ u2, v1 ≤ v2.Definition 4 [61]: A copula C : [0, 1] × [0, 1] → [0, 1]

is said to be a Archimedean copula if there exist a continuousand strictly decreasing function ρ from [0, 1] to [0,∞] withρ(1) = 0, and function % from [0,∞) to [0, 1] with ρ(1) = 0defined by

%(m) =

{ρ−1(m), m ∈ [0, ρ(0)];0, m ∈ [ρ(0),+∞).

such that, for any (b1, b2) ∈ [0, 1]× [0, 1],

C(b1, b2) = %(ρ(b1)+ ρ(b2)). (3)

In light of the setting that if C is a strictly increasingon [0, 1] × [0, 1], then we can obtain ρ(0) = +∞ and %

35498 VOLUME 8, 2020


TABLE 1. Archimedean copula.

agrees with ρ−1 on [0,+∞]. Genest et al. [63] originallydeveloped the notion of Archimedean copula, then the Eq (3)is redefined as

C(b1, b2) = ρ−1(ρ(b1)+ ρ(b2)), (4)

in which the function ρ is called as a strict generator and Cis said to be a strict Archimedean copula, several commonArchimedean copulas are displayed in Table 1.

What’s more, Cherubini et al. [60] revealed the conceptionof co-copula.Definition 5: Let C: [0, 1] × [0, 1] → [0, 1] be a copula,

then the co-copula is defined as

C∗(b1, b2) = 1− C(1− b1, 1− b2). (5)

III. OPERATIONS OF SVNN BASED ON COPULAAND CO-COPULAIn this section, we succinctly introduce several novel opera-tions for SVNNs based onACC and explore related propertiesand particular instances.

Inspired by the investigation about generalized intersectionand generalized union of SVNNs completed by Smarandache[30]. We will redefine the definition of generalized intersec-tion and generalized union of SVNNs based on the ACC.Definition 6: Suppose αi = (Ti, Ii,Fi)(i = 1, 2) be two

SVNNs, then generalized intersection and generalized unionoperational are defined as below:

α1 ∩C,C∗ α2 =(C(T1, T2),C∗(I1, I2),C∗(F1,F2)

), (6)

α1 ∪C,C∗ α2 =(C∗(T1, T2),C(I1, I2),C(F1,F2)

), (7)

in which C and C∗ denotes Archimedean copula and co-copula, respectively.

According to above-mentioned analysis, the operationlaws of SVNNs on the basis of ACC can be depicted asfollows:

(1) α1 ⊕C α2 =(C∗(T1, T2),C(I1, I2),C(F1,F2)

)(8)

=

1− ρ−1(ρ(1− T1)+ ρ(1− T2)),ρ−1(ρ(I1)+ ρ(I2)),ρ−1(ρ(F1)+ ρ(F2))

;(2) α1 ⊗C α2 =

(C(T1, T2),C∗(I1, I2),C∗(F1,F2)

)(9)

=

ρ−1(ρ(T1)+ ρ(T2)),1− ρ−1(ρ(1− I1)+ ρ(1− I2)),1− ρ−1(ρ(1− F1)+ ρ(1− F2))

;

(3) κα=(1−ρ−1(κρ(1− T )), ρ−1(κρ(I)), ρ−1(κρ(F))

);

(10)

(4) ακ =(ρ−1(κρ(T )), 1− ρ−1(κρ(1− I)),

1− ρ−1(κρ(1− F))). (11)

Theorem 1: For SVNNs αi = (Ti, Ii,Fi)(i = 1, 2), thenfor κ > 0, α1⊕C α2, α1⊗C α2, κα1 and ακ1 are still SVNNs.

Proof: By utilizing fundament definition of ρ andSVNN, we have 0 ≤ 1−T ≤ 1 and 0 = ρ(1) ≤ ρ(1−T ) ≤ρ(0) = +∞. Because ρ and ρ−1 are strictly decreasing func-tion, then 0 = ρ−1(+∞) ≤ ρ−1(κρ(1− T )) ≤ ρ−1(0) = 1.Hence, 0 ≤ 1 − ρ−1(κρ(1− T )) ≤ 1. Analogously, we canprove 0 ≤ ρ−1(κρ(I)) ≤ 1, 0 ≤ ρ−1(κρ(F)) ≤ 1.Accordingly, 0 ≤ 1 − ρ−1(κρ(1− T )) + ρ−1(κρ(I)) +ρ−1(κρ(F)) ≤ 3, i.e., κα is a SVNN. Analogously, we canprove α1 ⊕C α2, α1 ⊗C α2 and ακ1 are SVNNs.In the next, we will discuss several relations of the pre-

sented operational laws.Theorem 2: Let α = (T , I,F), αi = (Ti, Ii,Fi)

(i = 1, 2) are three SVNNs and κ, κ1, κ2 ≥ 0, then

(1) α1 ⊕C α2 = α2 ⊕C α1;

(2) α1 ⊗C α2 = α2 ⊗C α1;

(3) κ(α1 ⊕C α2) = κα1 ⊕C κα2;

(4) (α1 ⊗C α2)κ = ακ1 ⊗C ακ2 ;

(5) κ1α ⊕C κ2α = (κ1 + κ2)α;

(6) ακ1 ⊗C ακ2 = ακ1+κ2 .

For details of theorem proving, please refer to Appendix A.Theorem 3: Let α = (T , I,F), αi = (Ti, Ii,Fi)

(i = 1, 2) are three SVNNs and κ ≥ 0, then the followingconclusions are valid:

(1) (αc)κ = (κα)c;

(2) κ(αc) = (ακ )c;

(3) αc1 ⊕C αc2 = (α1 ⊗C α2)c;

(4) αc1 ⊗C αc2 = (α1 ⊕C α2)c.

where αc = (F , 1− I, T ).For details of theorem proving, please refer to Appendix B.In what follows, we will explore several particular cases of

operation laws for SVNNs with respect to the different formcopulas.

VOLUME 8, 2020 35499


Definition 7: Let α = (T , I,F), αi = (Ti, Ii,Fi)(i = 1, 2) are three SVNNs and κ ≥ 0, then the operationlaws with respect to different generating function ρ(t) aredefined as follows.For the details expression forms, please refer toAppendix C.

IV. THE SVN AGGREGATION OPERATORS BASEDON THE ACCIn this part, we shall develop several AOs based on ACCoperations of SVNNs in Section 3.

A. SVNPA OPERATORDefinition 8: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs. The SVNPA operator is a mapping:8n→ 8 and

SVNPA(α1, α2, · · · , αn) = (⊕C)ni=1ηiαi, (12)

where ηi = ((1 + Ri)/(∑n

i=1(1 + Ri))), Ri =∑j=1,j 6=i Sup(αi, αj) and Sup(αi, αj) be symbol of the support

for αi from αj.Theorem 4: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs, then the fusion value produced bySVNCPA operator is still a SVNN and

SVNCPA(α1, α2, · · · , αn)

= (⊕C)ni=1ηiαi

=

(1− ρ−1

(n∑i=1

ηiρ(1− Ti)

), ρ−1

(n∑i=1

ηiρ(Ii)

),

ρ−1

(n∑i=1

ηiρ(Fi)

)). (13)

For details of theorem proving, please refer to Appendix D.In what follows, we shall investigate several anticipated

theorems of SVNCPA operator.Theorem 5 (Idempoyency): Let αi = (Ti, Ii,Fi)(i =

1, 2, · · · , n) be a family of SVNNs. If all αi = α =

(T , I,F), then

SVNCPA(α1, α2, · · · , αn) = α.

Proof: Since αi = α = (T , I,F) for all i, then we have

SVNCPA(α1, α2, · · · , αn)

= (⊕C)ni=1ηiαi

=

1− ρ−1(

n∑i=1

ηiρ(1− Ti)

),

ρ−1

(n∑i=1

ηiρ(Ii)

),

ρ−1

(n∑i=1

ηiρ(Fi)

)

=

1− ρ−1(

n∑i=1

ηiρ(1− T )

),

ρ−1

(n∑i=1

ηiρ(I)

),

ρ−1

(n∑i=1

ηiρ(F)

)

=

(1− ρ−1(ρ(1− T )), ρ−1(ρ(I)), ρ−1(ρ(F))

)= (T , I,F) = α.

Theorem 6 (Monotonicity):Let αi = (Ti, Ii,Fi) and βi =(T ′i , I

′

i ,F′

i

)(i = 1, 2, · · · , n) are two collections of SVNNs.

If Ti ≤ T ′i , Ii ≥ I ′i ,Fi ≥ F ′i for all (i = 1, 2, · · · , n), then

SVNCPA(α1, α2, · · · , αn) ≤ SVNCPA(β1, β2, · · · , βn)

Proof: Since

SVNCPA(α1, α2, · · · , αn)

= (⊕C)ni=1ηiαi

=

1− ρ−1(

n∑i=1

ηiρ(1− Ti)

),

ρ−1

(n∑i=1

ηiρ(Ii)

),

ρ−1

(n∑i=1

ηiρ(Fi)

)

and

SVNCPA(β1, β2, · · · , βn)

= (⊕C)ni=1ηiβi

=

1− ρ−1(

n∑i=1

ηiρ(1− T

′

i

)),

ρ−1

(n∑i=1

ηiρ(I′

i

)),

ρ−1

(n∑i=1

ηiρ(F′

i

))

.

In addition, Ti ≤ T ′i , Ii ≥ I ′i ,Fi ≥ F ′i and ρ(t) is strictlydecreasing function, then we have

ρ(1−Ti)≤ρ(1−T

′

i

), ρ(Ii)≤ρ

(I′

i

)and ρ(Fi)≤ρ

(F′

i

).

Because ρ−1(t) is also strictly decreasing function, then wehave

1− ρ−1(ρ(1− Ti)) ≤ 1− ρ−1(ρ(1− T

′

i

)),

ρ−1(ρ(Ii)) ≥ ρ−1(ρ(I′

i

))and ρ−1(ρ(Fi)) ≥ ρ

−1(ρ(F′

i

)).

Then ACC − SVNWA(α1, α2, · · · , αn) ≤ ACC −SVNNWA(β1, β2, · · · , βn).

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Theorem 7 (Boundedness): The SVNCPA operator isbetween the maximum and minimum operators, that is

min(α, α, · · · , α) ≤ SVNNPA(α1, α2, · · · , αn)

≤ min(β, β, · · · , β).

Proof: According to Theorem 7, we can obtain

SVNCPA(α, α, · · · , α) = α,

SVNCPA(β, β, · · · , β) = β.

Then

α ≤ SVNCPA(α1, α2, · · · , αn) ≤ β.

Accordingly,

min(α, α,· · ·, α)≤SVNNPA(α1, α2,· · ·, αn)≤min(β,β,· · ·,β).

Theorem 8 (Commutativity): Let αi = (Ti, Ii,Fi) andβi =

(T ′i , I

′

i ,F′

i

)(i = 1, 2, · · · , n) are two collections of

SVNNs. Then

SVNCPA(α1, α2, · · · , αn) = SVNCPA(β1, β2, · · · , βn).

Proof: Since αi = (Ti, Ii,Fi) is any permutation of βi =(T ′i , I

′

i ,F′

i

)(i = 1, 2, · · · , n), then we have

SVNCPA(α1, α2, · · · , αn) = SVNCPA(β1, β2, · · · , βn).

Theorem 9: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be afamily of SVNNs and ξ ≥ 0, then

SVNCPA(ξα1, ξα2, · · · , ξαn)

= ξSVNCPA(α1, α2, · · · , αn).

Proof: Becauseξαi =

(1− ρ−1(ξρ(1− Ti)), ρ−1(ξρ(Ii)), ρ−1(ξρ(Fi))

),

then


=

1− ρ−1(

n∑i=1

ηiρ(ρ−1(ξρ(1− Ti))

)),

ρ−1

(n∑i=1

ηiρ(ρ−1(ξρ(Ii))

)),

ρ−1

(n∑i=1

ηiρ(ρ−1(ξρ(Fi))

))

=

1− ρ−1(

n∑i=1

ηiξρ(1− Ti)),

ρ−1

(n∑i=1

ηiξρ(Ii)),

ρ−1

(n∑i=1

ηiξρ(Fi)

)

=

1− ρ−1(ξ

n∑i=1

ηiρ(1− Ti)),

ρ−1

(ξ

n∑i=1

ηiρ(Ii)),

ρ−1

(ξ

n∑i=1

ηiρ(Fi)

)

.

Besides,

ξSVNCPA(α1, α2, · · · , αn)

=

1− ρ−1(ξρ

(ρ−1

(n∑i=1

ηiρ(1− Ti)

))),

ρ−1

(ξρ

(ρ−1

(n∑i=1

ηiρ(Ii)

))),

ρ−1

(ξρ

(ρ−1

(n∑i=1

ηiρ(Fi)

)))

=

1− ρ−1(ξ

n∑i=1

ηiρ(1− Ti)),

ρ−1

(ξ

n∑i=1

ηiρ(Ii)),

ρ−1

(ξ

n∑i=1

ηiρ(Fi)

)

.

Accordingly, SVNCPA (ξα1, ξα2, · · · , ξαn) = ξ

SVNCNWA (α1, α2, · · · , αn).Theorem 10: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be a

family of SVNNs and w = (w1,w2, · · · ,wn)T be the weightvector of (α1, α2, · · · , αn) with wi ≥ 0,

∑ni=1 = 1. If γ =

(T , I,F) be a SVNN, then

SVNCPA(α1 ⊕C γ, α2 ⊕C γ, · · · , αn ⊕C γ )

= SVNNPA(α1, α2, · · · , αn)⊕C γ.

Proof: Since

αi ⊕C γ =

1− ρ−1(ρ(1− Ti)+ ρ(1− T )),ρ−1(ρ(Ii)+ ρ(I)),ρ−1(ρ(Fi)+ ρ(F))

.Then

SVNCPA(α1 ⊕C γ, α2 ⊕C γ, · · · , αn ⊕C γ )

=

1− ρ−1(

n∑i=1

ηiρ(ρ−1(ρ(1− T1)+ ρ(1− T ))

)),

ρ−1

(n∑i=1

ηiρ(ρ−1(ρ(I1)+ ρ(I))

)),

ρ−1

(n∑i=1

ηiρ(ρ−1(ρ(F1)+ ρ(F))

))

VOLUME 8, 2020 35501


=

1− ρ−1(

n∑i=1

ηi(ρ(1− T1)+ ρ(1− T ))

),

ρ−1

(n∑i=1

ηi(ρ(I1)+ ρ(I))

),

ρ−1

(n∑i=1

ηi(ρ(F1)+ ρ(F))

)

.

In addition,

SVNCPA(α1, α2, · · · , αn)⊕C γ

=

1− ρ−1(

n∑i=1

ηiρ(1−Ti)

),

ρ−1

(n∑i=1

ηiρ(Ii)

),

ρ−1

(n∑i=1

ηiρ(Fi)

)

⊕c(T , I,F)

=

1−ρ−1(ρ

(ρ−1

(n∑i=1

ηiρ(1−Ti)

))+ρ(1−T )

),

ρ−1

(ρ

(ρ−1

(n∑i=1

ηiρ(Ii)

))+ ρ(I)

),

ρ−1

(ρ

(ρ−1

(n∑i=1

ηiρ(Fi)

))+ ρ(F)

)

=

1− ρ−1(

n∑i=1

ηi(ρ(1− T1)+ ρ(1− T ))

),

ρ−1

(n∑i=1

ηi(ρ(I1)+ ρ(I))

),

ρ−1

(n∑i=1

ηi(ρ(F1)+ ρ(F))

)

.

Accordingly, SVNCPA(α1 ⊕C γ, α2 ⊕C γ, · · · , αn ⊕Cγ ) =

SVNCWA(α1, α2, · · · , αn)⊕C γ .Theorem 11: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be a

family of SVNNs and w = (w1,w2, · · · ,wn)T be the weightvector of (α1, α2, · · · , αn) with wi ≥ 0,

∑ni=1 = 1. If ξ ≥ 0

and γ = (T , I,F) be a SVNN. Then,

SVNCPA(ξα1 ⊕C γ, ξα2 ⊕C γ, · · · , ξαn ⊕C γ )

= ξSVNCPA(α1, α2, · · · , αn)⊕c γ.

Proof: From the Theorem 12, we have


= SVNCPA(ξα1, ξα2, · · · , ξαn)⊕C γ

Furthermore, according to Theorem 11, we have


= ξSVNCPA(α1, α2, · · · , αn).

Accordingly,


= ξSVNCPA(α1, α2, · · · , αn)⊕C γ.

which finishes the proof of Theorem 13.Theorem 12: Let A = {αi = (Ti, Ii,Fi)|i = 1, 2, · · · , n}

and B ={βi =

(T ′i , I

′

i ,F′

i

)|i = 1, 2, · · · , n

}are two col-

lections of SVNNs, and w = (w1,w2, · · · ,wn)T be theweight vector of (α1, α2, · · · , αn) with wi ≥ 0,

∑ni=1 = 1.

Then

SVNCPA(α1 ⊕C β1, α1 ⊕C β1, · · · , αn ⊕C βn)

= SVNCPA(α1, α2,· · ·, αn)⊕CSVNCWA(β1, β2,· · ·, β).

Proof: Since

αi ⊕C βi =

1− ρ−1

(ρ(1− Ti)+ ρ

(1− T

′

i

)),

ρ−1(ρ(Ii)+ ρ

(I′

i

)),

ρ−1(ρ(Fi)+ ρ

(F′

i

)).

then we have

SVNCPA(α1 ⊕C β1, α1 ⊕C β1, · · · , αn ⊕C βn)

=

1−ρ−1(

n∑i=1

ηiρ(ρ−1

(ρ(1−Ti)+ρ

(1− T

′

i

)))),

ρ−1

(n∑i=1

ηiρ(ρ−1

(ρ(Ii)+ ρ

(I′

i

)))),

ρ−1

(n∑i=1

ηiρ(ρ−1

(ρ(Fi)+ ρ

(F′

i

))))

=

1− ρ−1(

n∑i=1

ηi

(ρ(1− Ti)+ ρ

(1− T

′

i

))),

ρ−1

(n∑i=1

ηi

(ρ(Ii)+ ρ

(I′

i

))),

ρ−1

(n∑i=1

wi(ρ(Fi)+ ρ

(F′

i

)))

.

Besides, SVNCPA(α1, α2, · · · , αn

)⊕C SVNCPA(

β1, β2, · · · , β), as shown at the bottom of the next page.

Hence, The Theorem 12 is proved.In the following, we will discuss some particular SVN

operators based on different generator ρ(t).Case 1: If ρ(t) = (− ln t)σ with σ ≥ 1, then we can

obtain the Gumbel SVNPA (G-SVNCPA)operator,

G − SVNCPA(α1, α2, · · · , αn)

=

1− e−

(n∑i=1ηi(− ln(1−Ti)σ )

) 1σ

,

e−

(n∑i=1ηi(− ln(Ii)σ )

) 1σ

,

e−

(n∑i=1ηi(− ln(Fi)

σ )

) 1σ

. (14)

35502 VOLUME 8, 2020


Case 2: If ρ(t) = t−σ −1 with σ ≥ 1, then we can obtainthe Clayton SVNPA (C-SVNPA) operator,

C − SVNCPA(α1, α2, · · · , αn)

=

1−

(n∑i=1

ηi((1− Ti)−σ − 1

)+ 1

)− 1σ

,(n∑i=1

ηi(I−σi − 1

)+ 1

)− 1σ

,(n∑i=1

ηi(F−σi − 1

)+ 1

)− 1σ

. (15)

Case 3: If ρ(t) = ln(e−σ t−1e−σ−1

)with σ 6= 0, then we can

obtain the Frank SVNCPA (F-SVNCPA) operator,

F − SVNCPA(α1, α2, · · · , αn)

=

1+1σln

(1+

(e−σ − 1

) n∏i=1

(e−σ(1−Ti) − 1e−σ − 1

)ηi),

−1σln

(1+

(e−σ − 1

) n∏i=1

(e−σIi − 1e−σ − 1

)ηi),

−1σln

(1+

(e−σ − 1

) n∏i=1

(e−σFi − 1e−σ − 1

)ηi)

.

(16)

Case 4: If ρ(t) = ln(1−σ (1−t)

t

)with σ ∈ [−1, 1],

then we can obtain the Ali-Mikhail-Haq SVNCPA

(AMH-SVNCPA) operator,

AHM− SVNCPA(α1, α2, · · · , αn)

=

n∏i=1(1− σTi)ηi −

n∏i=1(1− Ti)ηi

n∏i=1(1− σTi)ηi − σ

n∏i=1(1− Ti)ηi

,

(1− σ)n∏i=1

Iηiin∏i=1(1− σ(1− Ii))ηi − σ

n∏i=1

Iηi,

(1− σ)n∏i=1

Fηii

n∏i=1(1− σ(1− Fi))

ηi − σn∏i=1

Fηi

. (17)


t

)with σ ∈ [−1, 1], then

we can obtain the Joe SVNPA (J-SVNCPA)operator,

J − SVNCPA(α1, α2, · · · , αn)

=

1−

(1−

n∏i=1

(1− T σ

i)ηi) 1

σ

,(1−

n∏i=1

(1− (1− Ii)σ

)ηi) 1σ

,(1−

n∏i=1

(1− (1− Fi)

σ)ηi) 1

σ

. (18)

SVNCPA(α1, α2, · · · , αn)⊕C SVNCPA(β1, β2, · · · , β)

=

1− ρ−1(

n∑i=1

ηiρ(1− Ti)

),

ρ−1

(n∑i=1

ηiρ(Ii)

),

ρ−1

(n∑i=1

wiρ(Fi)

)

⊕c

1− ρ−1(

n∑i=1

wiρ(1− T

′

i

)),

ρ−1

(n∑i=1

ηiρ(I′

i

)),

ρ−1

(n∑i=1

ηiρ(F′

i

))

=

1− ρ−1(ρ

(ρ−1

(n∑i=1

ηiρ(1− Ti)

))+ ρ

(ρ−1

(n∑i=1

ηiρ(1− T

′

i

)))),

ρ−1

(ρ

(ρ−1

(n∑i=1

wiρ(Ii)

))+ ρ

(ρ−1

(n∑i=1

ηiρ(I′

i

)))),

ρ−1

(ρ

(ρ−1

(n∑i=1

ηiρ(Fi)

))+ ρ

(ρ−1

(n∑i=1

wiρ(F′

i

))))

=

1− ρ−1(

n∑i=1

ηi

(ρ(1− Ti)+ ρ

(1− T

′

i

))),

ρ−1

(n∑i=1

ηi

(ρ(Ii)+ ρ

(I′

i

))),

ρ−1

(n∑i=1

ηi

(ρ(Fi)+ ρ

(F′

i

)))

.

VOLUME 8, 2020 35503


B. WEIGHTED AND OWSVNPA OPERATORDefinition 9: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs. WSVNCPA operator is a mapping:8n→ 8 and

WSVNCPA(α1, α2, · · · , αn) = (⊕C)ni=1ϑiαi, (19)

where ϑi = (εi(1 + Ri)/(n∑i=1εi(1 + Ri))), Ri =∑

j=1,j 6=iSup(αi, αj) and ε be the weight vector of αi with 0 ≤

εi ≤ 1,∑n

i=1 εi = 1.Theorem 13: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced byWSVNPA operator is still a SVNN and

WSVNCPA(α1, α2, · · · , αn)

= (⊕C)ni=1ϑiαi

=

1− ρ−1(

n∑i=1

ϑiρ(1− Ti)

),

ρ−1

(n∑i=1

ϑiρ(Ii)

),

ρ−1

(n∑i=1

ϑiρ(Fi)

)

. (20)

The proof is similar to Theorem 4, so we omit it here.Definition 10: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs. OWSVNCPA operator is a mapping:8n→ 8 and

OWSVNCPA(α1, α2, · · · , αn) = (⊕C)ni=1ϑiας (i), (21)



εi ≤ 1,∑n

i=1 εi = 1, and ς is permutation of (1, 2, · · · , n)such that ας (i−1) ≤ ας (i) for i = 2, 3, · · · , n.Theorem 14: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced byOWSVNCPA operator is still a SVNNs and

OWSVNCPA(α1, α2, · · · , αn)

= (⊕C)ni=1ϑiαi

=

1− ρ−1(

n∑i=1

ϑiρ(1− Tς (i)

)),

ρ−1

(n∑i=1

ϑiρ(Iς (i)

)),

ρ−1

(n∑i=1

ϑiρ(Fς (i)

))

. (22)

The proof is similar to Theorem 4, so we omit it here.

Obviously, the presented WSVNCPA and OWSVNCPAoperators still possess the theorems proposed inTheorem 5-12.

C. SVNPG OPERATORDefinition 11: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be a

family of SVNNs. SVNCPG operator is a mapping:8n→ 8

and

SVNCPG(α1, α2, · · · , αn) =((⊗C)

ni=1

)αηii , (23)

where ηi = ((1+Ri)/(n∑i=1

(1+Ri))),Ri =∑

j=1,j 6=iSup(αi, αj)

and Sup(αi, αj) be symbol of the support for αi from αj.Theorem 15: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced bySVNCPA operator is still a SVNN and

SVNCPG(α1, α2, · · · , αn)

=((⊗C)

ni=1

)αηii

=

ρ−1

(n∑i=1

ηiρ(Ti)

),

1− ρ−1(

n∑i=1

ηiρ(1− Ii)

),

1− ρ−1(

n∑i=1

ηiρ(1− Fi)

)

. (24)

The proof is similar to Theorem 4, we omit it here.On the basis of Theorem 15, it can prove that the following

theorems are contented by the presented SVNPG operator.Theorem 16: Let αi, βi(i = 1, 2, · · · , n) be two families

of SVNNs. ξ ≥ 0 and γ = (T , I,F), then we have(T1) If αi=α=(T , I,F), then SVNCPG(α1, α2, · · · , αn)= α;

(T2) SVNCPG(ξα1, ξα2, · · · , ξαn) = ξSVNCPG(α1, α2,

· · · , αn);

(T3) SVNCPG(α1 ⊗c γ, α2 ⊗c γ, · · · , αn ⊗c γ ) =

SVNCPG(α1, α2, · · · , αn)⊗c γ ;(T4) SVNCPG(ξα1 ⊗c γ, ξα2 ⊗c γ, · · · , ξαn ⊗c γ ) =

ξSVNCPG(α1, α2, · · · , αn)⊗c γ ;(T5) SVNCPG(α1 ⊗c β1, α1 ⊗c β1, · · · , αn ⊗c βn) =

SVNCPG(α1, α2, · · · , αn)⊗c SVNCPG(β1, β2, · · · , β).In the following, we will discuss several particular SVNPG

operators based on different generator ρ(t).Case 6: If ρ(t) = (− ln t)σ with σ ≥ 1, then we can

obtain the Gumbel SVNPA (G-SVNCPG)operator,

G− SVNPG(α1, α2, · · · , αn)

=

e−

(n∑i=1ηi(− ln(Ti)σ )

) 1σ

,

1− e−

(n∑i=1ηi(− ln(1−Ii)σ )

) 1σ

,

1− e−

(n∑i=1ηi(− ln(1−Fi)

σ )

) 1σ

. (25)

35504 VOLUME 8, 2020


Case 7: If ρ(t) = t−σ −1 with σ ≥ 1, then we can obtainthe Clayton SVNCPA (C-SVNCPG)operator,

C − SVNPG(α1, α2, · · · , αn)

=

(n∑i=1

ηi(T −σi − 1

)+ 1

)− 1σ

,1−

(n∑i=1

ηi((1− Ii)−σ − 1

)+ 1

)− 1σ

,1−

(n∑i=1

ηi((1− Fi)

−σ− 1

)+ 1

)− 1σ

.

(26)

Case 8: If ρ(t) = ln(e−σ t−1e−σ−1

)with σ 6= 0, then we can

obtain the Frank SVNCPA (F-SVNCPG)operator,

F − SVNPG(α1, α2, · · · , αn)

=

−1σln

(1+

(e−σ − 1

) n∏i=1

(e−σTi − 1e−σ − 1

)ηi)

1+1σln

(1+

(e−σ − 1

) n∏i=1

(e−σ(1−Ii) − 1e−σ − 1

)ηi),

1+1σln

(1+

(e−σ − 1

) n∏i=1

(e−σ(1−Fi) − 1e−σ − 1

)ηi)

.

(27)


t

)with σ ∈ [−1, 1],

then we can obtain the Ali-Mikhail-Haq SVNCPA (AMH-SVNCPG)operator,

AHM − SVNPG(α1, α2, · · · , αn)

=

(1− σ)n∏i=1

T ηii

n∏i=1(1− σ(1− Ti))ηi − σ

n∏i=1

T ηi

n∏i=1(1− σIi)ηi −

n∏i=1(1− Ii)ηi

n∏i=1(1− σIi)ηi − σ

n∏i=1(1− Ii)ηi

,

n∏i=1(1− σFi)

ηi −

n∏i=1(1− Fi)

ηi

n∏i=1(1− σFi)

ηi − σn∏i=1(1− Fi)

ηi

. (28)

Case 10: If ρ(t) = ln(1−σ (1−t)

t

)with σ ∈ [−1, 1],

then we can obtain the Joe SVNCPA (J-SVNCPG)

operator,

J − SVNPG(α1, α2, · · · , αn)

=

(1−

n∏i=1

(1− (1− Ti)σ

)ηi) 1σ

1−

(1−

n∏i=1

(1− Iσi

)ηi) 1σ

,

1−

(1−

n∏i=1

(1− Fσ

i)ηi) 1

σ

. (29)

D. WEIGHTED AND OWSVNCPG OPERATORDefinition 12: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs. WSVNPG operator is a mapping:8n→ 8 and

WSVNCPG(α1, α2, · · · , αn) = (⊕C)ni=1(αi)

ϑi , (30)



εi ≤ 1,∑n

i=1 εi = 1.Theorem 17: Let A = {αi = (Ti, Ii,Fi)|i = 1, 2, · · · , n}

be a collection of SVNNs, then the fusion value produced byWSVNCPG operator is still a SVNN and

WSVNCPG(α1, α2, · · · , αn)

= (⊕C)ni=1(αi)

ϑi

=

ρ−1

(n∑i=1

ϑiρ(Ti)

),

1− ρ−1(

n∑i=1

ϑiρ(1− Ii)

),

1− ρ−1(

n∑i=1

ϑiρ(1− Fi)

)

. (31)

Definition 13: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) bea family of SVNNs. OWSVNCPA operator is a mapping:8n→ 8 and

OWSVNCPG(α1, α2, · · · , αn) = (⊕C)ni=1ϑiας (i), (32)


j=1,j6=iSup(αi, αj) and ε be the weight vector of αi with 0 ≤

εi ≤ 1,∑n

i=1 εi = 1, and ς is permutation of (1, 2, · · · , n)such that ας (i−1) ≤ ας (i) for i = 2, 3, · · · , n.Theorem 18: Let A = {αi = (Ti, Ii,Fi)|i = 1, 2, · · · , n}

be a collection of SVNNs, then the fusion value produced by

VOLUME 8, 2020 35505


OWSVNCPG operator is still a SVNN and

OWSVNCPG(α1, α2, · · · , αn)

= (⊕C)ni=1ϑiαi

=

ρ−1

(n∑i=1

ϑiρ(Tς (i)

)),

1− ρ−1(

n∑i=1

ϑiρ(1− Iς (i)

)),

1− ρ−1(

n∑i=1

ϑiρ(1− Fς (i)

))

. (33)

Obviously, the presented WSVNCPA and OWSVNCPAoperators still possess the theorems proposed in Theorem 16.

V. GENERALIZED WEIGHTED SVN PA OPERATORSIn this part, several generalized power operators includingGWSVNPA, GOWSVNPA, GWSVNPG and GOWSVNPAoperators are developed for aggregating the set of SVNNs8.Definition 14: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs. GWSVNPA operator is a mapping:8n→ 8 and

GWSVNCPA(α1, α2, · · · , αn) =((⊕C)

ni=1ϑiα

qi

)1/q, (34)



εi ≤ 1,∑n

i=1 εi = 1.Remark 1: Particularly, if we assign χ = 1, then

the GWSVNCPA operator will degenerate into WSVNCPAoperator.Theorem 19: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced by

WSVNCPA operator is still a SVNN and (35), as shown atthe bottom of this page.

Proof: Sinceαqi =

(ρ−1(qρ(T )), 1−ρ−1(qρ(1−I)), 1−ρ−1(qρ(1−F))

),

we can acquire

ϑiαqi =

1− ρ−1

(ϑiρ

(1− ρ−1(qρ(Ti))

)),

ρ−1(ϑiρ

(1− ρ−1(qρ(1− Ii))

)),

ρ−1(ϑiρ

(1− ρ−1(qρ(1− Fi))

)).

and

(⊕C)ni=1ϑiα

qi

=

1− ρ−1(

n∑i=1

(ϑiρ

(1− ρ−1(qρ(Ti))

))),

ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Ii))

))),

ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Fi))

)))

.

Accordingly, we have((⊕C)

ni=1ϑiα

qi

)1/q, as shown at thebottom of this page.Definition 15: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs. GOWSVNCPA operator is a mapping:8n→ 8 and

GOWSVNCPA(α1, α2, · · · , αn) =((⊕C)

ni=1ϑiα

qς (i)

)1/q,

(36)



GWSVNPA(α1, α2, · · · , αn) =

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(Ti))

))))),

1− ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Ii))

))))),

1− ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Fi))

)))))

.

(35)

((⊕C)

ni=1ϑiα

qi

)1/q=

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(Ti))

))))),

1− ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Ii))

))))),

1− ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Fi))

)))))

.

35506 VOLUME 8, 2020


εi ≤ 1,∑n

i=1 εi = 1, and ς is permutation of (1, 2, · · · , n)such that ας (i−1) ≤ ας (i) for i = 2, 3, · · · , n.Remark 2: Particularly, if we assign χ = 1, then the

GOWSVNCPA operator will degenerate into OWSVNCPAoperator.Theorem 20: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced byWSVNCPA operator is still a SVNN and (37), as shown atthe bottom of this page.Definition 16: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs. GWSVNCPG operator is a mapping:8n→ 8 and

GWSVNCPG(α1, α2, · · · , αn) =1q

((⊕C)

ni=1ϑiα

qi

), (38)



εi ≤ 1,∑n

i=1 εi = 1.Remark 3: Particularly, if we assign χ = 1, then the

GWSVNCPG operator will degenerate into WSVNCPGoperator.Theorem 21: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced byWSVNPG operator is still a SVNN and (39), as shown at thebottom of this page.Definition 17: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n) be

a family of SVNNs. GWSVNCPG operator is a mapping:8n→ 8 and

GOWSVNCPG(α1, α2, · · · , αn) =1q

((⊕C)

ni=1ϑiα

qς (i)

),

(40)



εi ≤ 1,∑n

i=1 εi = 1, and ς is permutation of (1, 2, · · · , n)such that ας (i−1) ≤ ας (i) for i = 2, 3, · · · , n.Remark 4: Particularly, if we assign χ = 1, then the

GOWSVNCPG operator will degenerate into OWSVNCPGoperator.Theorem 22: Let αi = (Ti, Ii,Fi)(i = 1, 2, · · · , n)

be a family of SVNNs, then the fusion value produced byGOWSVNCPG operator is still a SVNN and (41), as shownat the bottom of the next page.

Obviously, the presented GWSVNCPA, GOWSVNCPA,GWSVNCPG and GOWSVNCPG operators also have thesame theorems whichWSVNCPA andWSVNCPG operatorshave.

VI. THE APPLICATION TO DEVELOPED MADMAPPROACHUnder this part, an approach is proposed to cope withMADMissues by utilizing the developed aggregation operator undersing-valued neutrosophic set context.

Assuming that 2 = {21,22, · · · ,2m} is a set of alter-natives. K = {K1,K2, · · · ,Kn} is a collection of attributeswith the weight vector being τ = (τ1, τ2, · · · , τn)T with τj ∈[0, 1](j = 1, 2, · · · , n) and

∑nj=1 τj = 1. A manager assesses

those alternatives and provides assessment information underthe attributes Kj by the form of SVNNs gij =

(Tij, Iij,Fij

),

where Tij, Iij,Fij ∈ [0, 1] and 0 ≤ Tij + Iij + Fij ≤ 3.The assessment information of all alternatives = are writ-ten as a decision-matrix G, as shown at the bottom of thenext page.

GOWSVNCPA(α1, α2, · · · , αn) =((⊕C)

ni=1ϑiα

qς (i)

)1/q

=

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1

(qρ(Tς (i))

)))))),

1−ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ(1− ρ−1

(qρ(1−Iς (i))

)))))),

1−ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1

(qρ(1−Fς (i))

))))))

.

(37)

GWSVNCPG(α1, α2, · · · , αn) =1q

((⊕C)

ni=1ϑiα

qi

)

=

1− ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(1− Ii))

))))),(

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(Ti))

)))))),(

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1(qρ(Ti))

))))))

. (39)

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Aiming at the above-mentioned MADM issues, we designan approach on the basis of the presented aggregation oper-ators and entropy weight method to cope with it in nextsection

A. THE SOLVING OF ATTRIBUTES WEIGHTSEntropy notion was originally brought forward into infor-mation theory by Shannon, and it has been wide-rangingused in economic development, knowledge management andother domains. In information theory, entropy is a measureof uncertainty. The greater the uncertainty is, the greaterthe entropy is and the more information it possesses; thesmaller the uncertainty is, the smaller the entropy is and theless information it possesses. As a valid tool to determinatethe weights of attributes, entropy measures are employed invaries decision problems under different evaluation contexts.Due to the influence of knowledge level, personal experi-ence and decision setting, the attributes weights provided bydecision makers are frequently subjective, which will makethe finally decision resultant ambiguous. Accordingly, aimingat the decision problems with completely unknown attributeweight, the necessity of weight determination is bound toattract attention for us. In the following, an weight determi-nation algorithm based on entropy measure under the SVNcontext.

Step i: Establish the score matrix S =(S(gij)

)m×n of

Q =(gij)m×n

S =(S(gij)

)m×n =

S(g11) S(g12) · · · S(g1n)S(g21) S(g22) · · · S(g2n)...

......

...

S(gm1) S(gm2) · · · S(gmn)

Step ii: Normalizing the matrix S as:

S =(S(gij)

)m×n=

S(g11) S(g12) · · · S(g1n)S(g21) S(g22) · · · S(g2n)...

......

...

S(gm1) S(gm2) · · · S(gmn)

where S(gij) =

S(gij)∑mi=1 S(gij)

; (i = 1, 2, · · · ,m; j =1, 2, · · · , n).

Step iii:Obtain the attribute weights τj(j = 1, 2, · · · , n) by

τj =1− Ej

n∑j=1

(1− Ej

) = 1− Ej

n−n∑j=1

Ej

, (42)

where Ej = − 1lnm

m∑i=1

S(gij) · ln S(gij).

B. SVN MADM APPROACHStep 1: Normalize the decision matrix G = (gij)m×n to

G = (gij)m×n by the following method.

gij =(Tij, Iij, Fij

)(43)

=

{ (Tij, Iij,Fij

), Kj is benefit type(

Fij, 1− Iij, Tij), Kj is cost type.

Step 2: Determinate the attribute weights τj(j =

1, 2, · · · , n) by the following formula;

τj =1− Ej

n∑j=1

(1− Ej

) = 1− Ej

n−n∑j=1

Ej

, (44)

where Ej = − 1lnm

m∑i=1

S(gij) · ln S(gij).

GOWSVNCPG(α1, α2, · · · , αn) =1q

((⊕C)

ni=1ϑiα

qi

)

=

1− ρ−1(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1

(qρ(1− Tς (i))

)))))),(

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1

(qρ(Iς (i))

))))))),(

ρ−1

(1qρ

(1− ρ−1

(n∑i=1

(ϑiρ

(1− ρ−1

(qρ(Fς (i))

)))))))

. (41)

G =(gij)m×n

=

(T11, I11,F11) (T12, I12,F12) · · · (T1n, I1n,F1n)

(T21, I21,F21) (T22, I22,F22) · · · (T2n, I2n,F2n)...

......

...

(Tm1, Im1,Fm1) (Tm2, Im2,Fm2) · · · (Tmn, Imn,Fmn)

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Step 3: Compute the supports

Sup(gij, gi`

)= 1− d

(gij, gi`

)(45)

i = 1, 2, · · · ,m; j, ` = 1, 2, · · · , n; j 6= `.

where d = 13

(| Tij − Ti` | + | Iij − Ii` | + | Fij − Fi` |

)is

the normalized Hamming distance.Step 4: Compute the total support gij

T(gij)=

n∑i=1,j 6=`

Sup(gij, gi`

). (46)

Step 5: Acquire the weight $ij corresponding assessmentvalue gij

$ij =τj(1+ T

(gij))

n∑j=1

(τj(1+ T

(gij))) . (47)

where$j ∈ [0, 1](j = 1, 2, · · · , n) and∑n

i=j$j = 1.Step 6: Fuse the overall comprehensive assessment

value gi for alternative Hi by ACC-SVNPWA operator andACC-SVNPWG operator

gi = WSVNCPA(gi1, gi2, · · · , gin) (48)

Or

gi = WSVNCPG(gi1, gi2, · · · , gin) (49)

Step 7: Calculate the cosine similarity measure of the com-prehensive assessment value.

Step 8: Acquire the order relation of alternatives and selectoptimal one(s).

VII. EMPIRICAL EXAMPLEA. EXAMPLE APPLICATIONIn order to solve the problem of poor Internet connec-tion in remote areas, the effective implementation of goodsand services tax (GST) operation plan should be effec-tively improved [35]. The government plans to work withstate-owned enterprises to provide private mobile serviceproviders. To this end, the Indian government has publisheda global bidding announcement in newspapers to select sup-pliers for these projects, and has considered five requiredattributes, namely, technical expertise (A1), service quality(A2), bandwidth (A3), Internet speed (A4) and customer ser-vice (A5). These importance attributes are considered com-pletely unknown. The five suppliers (alternatives) preparing

for bidding are2i(i = 1, 2, · · · , 5). The assessment informa-tion provided by managers in the form of SVNN are listed inthe bottom of this page.

Then, the objective of government is to choose the optimalsupplier to provide the best Internet services for its citi-zens. The optimal selection process by utilizing the presentedapproach is summarized below.

Step 1: The normalization is omitted because all attributesare considered as the benefit type.

Step 2: Computing the attributes weight by the Eq (44):τ1 = 0.2292, τ2 = 0.2499, τ3 = 0.1348, τ4 = 0.1958,τ5 = 0.1903.Step 3: Compute the supports Sup

(gij, gi`

)Sup

(g1j, g1`

)

=

0.0000 0.9833 0.8000 0.9000 0.93330.9833 0.0000 0.8000 0.9000 0.86670.8000 0.8000 0.0000 0.9000 0.86670.9000 0.9000 0.9000 0.0000 0.96670.9333 0.8667 0.8667 0.9667 0.0000

,Sup

(g2j, g2`

)

=

0.0000 0.9667 0.8667 0.8333 0.96670.9667 0.0000 0.9000 0.8667 0.93330.8667 0.9000 0.0000 0.9667 0.90000.8333 0.8667 0.9667 0.0000 0.86670.9667 0.9333 0.9000 0.8667 0.0000

,Sup

(g3j, g3`

)

=

0.0000 0.9333 0.8333 0.9000 0.90000.9333 0.0000 0.9000 0.9000 0.90000.8333 0.9000 0.0000 0.9333 0.86670.9000 0.9000 0.9333 0.0000 0.86670.9000 0.9000 0.8667 0.8667 0.0000

,Sup

(g4j, g4`

)

=

0.0000 0.9333 0.8333 0.9000 0.90000.9833 0.0000 0.8000 0.9000 0.86670.9333 0.8000 0.0000 0.9000 0.86670.9000 0.9000 0.9000 0.0000 0.96670.9000 0.8667 0.8667 0.9667 0.0000

,Sup

(g5j, g5`

)

=

0.0000 0.9667 0.8333 0.9000 0.76670.9667 0.0000 0.8000 0.9333 0.73330.8333 0.8000 0.0000 0.8667 0.93330.9000 0.9333 0.8667 0.0000 0.80000.7667 0.7333 0.9333 0.8000 0.0000

.

Q =(gij)5×5 =

(0.5, 0.3, 0.4) (0.5, 0.2, 0.3) (0.2, 0.2, 0.6) (0.3, 0.2, 0.4) (0.3, 0.3, 0.4)(0.7, 0.1, 0.3) (0.7, 0.2, 0.3) (0.6, 0.3, 0.2) (0.6, 0.4, 0.2) (0.7, 0.1, 0.2)(0.5, 0.3, 0.4) (0.6, 0.2, 0.4) (0.6, 0.1, 0.2) (0.5, 0.1, 0.3) (0.6, 0.4, 0.3)(0.7, 0.3, 0.4) (0.7, 0.2, 0.2) (0.4, 0.5, 0.2) (0.5, 0.2, 0.2) (0.4, 0.5, 0.4)(0.4, 0.1, 0.3) (0.5, 0.1, 0.2) (0.4, 0.1, 0.5) (0.4, 0.3, 0.6) (0.3, 0.2, 0.4)

.

VOLUME 8, 2020 35509


TABLE 2. The fusion result by WSVNPA operator.

TABLE 3. The fusion result by WSVNPG operator.

TABLE 4. The cosine similarity measure of alternatives based on diverse copulas.

Step 4: Compute the total support T(gij)

T(gij)=

3.5667 3.5000 3.3667 3.6667 3.63333.6333 3.6667 3.6333 3.5333 3.66673.5667 3.6333 3.5333 3.6000 3.53333.4667 3.4333 3.4333 3.5000 3.23333.6000 3.4000 3.6000 3.3667 3.5000

.Step 5: Deduce the weight $ corresponding assessment

value gij,

$ =($ij)5×5

=

0.2504 0.2690 0.1408 0.2186 0.12120.2498 0.2744 0.1469 0.2088 0.12000.2485 0.2749 0.1451 0.2138 0.11760.2513 0.2720 0.1467 0.2163 0.11360.2559 0.2668 0.1505 0.2075 0.1194

.Step 6: Fuse the overall comprehensive assessment value

by WSVNPA and WSVNPG operator, the fused result areshown in Table 2 and Table 3.

Step 7: Computing the cosine similarity measure of theaggregation values, which are listed in Table 4.

Step 8: The order relation of alternatives can be concludedin the Table 5.

Furthermore, if we utilize other propounded operatorsincluding SVNPA, SVNPG, OWSVNPA and OWSVNPGon the basis of different copulas to cope with the example,the corresponding aggregation results deduced by differentoperators and cosine similarity of every alternatives 2i(i =1, 2, 3, 4, 5) are summarized in Table 6. From it, we canderive that although the ranking of alternatives are a littlediverse, the optimal programme is 22.

B. SENSITIVITY ANALYSISUnder this subsection, we will explore the effect of parame-ters σ and q on decision results.

In order to analyze the influence of parameter q on decisionresults, we select the Gumbel type copula to conduct ananalysis with respect to different values of q. On the basis ofthese values, the propounded operators such as GWSVNPA,GOWSVNPA, GWSVNPG and GOWSVNPG operator areapplied to fuse the each evaluation information of alternativeand their corresponding cosine similarity degree with theorder relation are listed in Table 7. From it, we can derivethe following conclusions: (1) the cosine similarity degreesof alternatives based on diverse values of q are different;(2) although the final decision resultants of alternatives areslightly diverse, the optimal programme are the same; (3) the

35510 VOLUME 8, 2020


TABLE 5. The order relation of alternatives based upon different copulas.

TABLE 6. The decision results based on the propounded operators.

parameter q can be regarded as a risk preference attitude ofmanagers, i.e., the parameter q has a better control capabilityfor the comprehensive evaluation value of alternatives.

To analyze the effect of diverse parameter value of σ ,we select the Gumbel-WSVNPA operator to perform theparameter analysis. The values of parameter σ changefrom 1 to 10 incrementing by 1 and the the correspondingparameter analysis resultants are summarized in Table 8 andFigure 1. From those, although the ranking orders of pro-grammes with respect to different value of σ are slightlydifferent, the best programme are the same. Furthermore,when 1 ≤ θ ≤ 6, the ranking relation of alternatives is22 � 24 � 23 � 25 � 21, when σ > 6, the rankingrelation of alternatives is 22 � 24 � 23 � 21 � 25.

In summary, we can derive that the sorting orders of alterna-tives are relatively stable with respect to different values ofσ . Accordingly, the propounded approach can validly avoidinterferences and acquire the best alternative.

C. COMPARISON ANALYSISIn what follows, we perform a comparison explorationbetween the propounded MADM technique with the exist-ing approaches to testify the effectiveness and to reveal thesignificant merits of the propounded approach. we comparethe propounded approach with diverse approaches developedin [33]–[35], [44], [45]. The derived ranking order results aredepicted in Table 9. From it, we can obtain that the decisionresults acquired from the existing methods are basically the

VOLUME 8, 2020 35511


TABLE 7. The decision results of generalize aggregation operators based on diverse parameter values.

TABLE 8. The decision results of G-WSVNPA operator based on different values of σ .

FIGURE 1. The resultants of diverse σ values.

same as those of the presented approach. Accordingly, theeffectiveness of the propounded method can be elaborated.Besides, we perform several characteristic analysis betweenthe presented approach with the prevailing methods, the anal-ysis are displayed in Table 10.

Next, we further conduct a detailed analysis between thepropounded method with the existing approach proposed by[44] and [33].

X Compared with the method based on the SVNHWAand SVNHWG operator propounded by Liu [33], the finaldecision results are the same as the propoundedmethod. Theircommon advantage is that they all have a universal parameterthat makes the process of information fusion more flexible.However, the method proposed by [33] can not clear up theawkward arguments on the process of sorting alternatives.On the contrary, the propounded approach has the ability toeliminate the influence of singular data. Consequently, thedeveloped method is more universal and rational in view ofthese aggregation operators.

X Compared with the approach on the basis of theSVNWA operator propounded by Ye [44], the SVNWA oper-ator can aggregate vague information and the computationalprocess is not relatively complex. However, it has the fol-lowing defects (i) the SVNWA operator based on AlgebraicT-norm and T-conorm lacks of flexibility and robustness;(ii) the SVNWA operator does not take into considerationthe impact of extreme information provided by managerand the correlation of fusion information. Nevertheless, the

35512 VOLUME 8, 2020


TABLE 9. Comparative with previous operators under SVN environment.

TABLE 10. Characteristic comparison with existing operators under SVN environment.

novel method propounded in this paper can not only takeinto account the interrelationship among the input data, butalso eliminate the effect of extreme information and the pre-sented method; on the other hand, the presented method givea general and flexible aggregation function because of thedifferent copulas and flexible parameters. Particularly, whenthe parameter σ = 1 in the type Gumbel-copula, it reducesto Algebraic T-norm and T-conorm. Accordingly, the pro-pounded method on the basis of those presented operators ismore universal and rational to cope with decision problems.

All in all, based on the comparatives and analysing, thepropounded method in this study has the following supe-riorities (i) it supplies a generalized aggregation functionbased on copula and gives more selections for managers tochose appropriate function to fuse information; (ii) it can takeinto consideration the correlation among multiple aggregatedarguments and has a flexibility for aggregation process bythe general parameter; (iii) it can highlight the importanceof the aggregated information and vanish the influence ofembarrassed data by taking diverse weight obtain by thesupport degree.

VIII. CONCLUSIONIn this paper, we first set up several novel operationallaws of SVN numbers on the basis of copulas, and explorecorresponding properties and several representative casesbased on different copulas. Secondly, we propound a series ofpower aggregation operators namely SVNCPA, WSVNCPA,

OWSVNCPA and their geometric operators. Meanwhile,we extend those operators to their generalized version.We also explore several attractive properties and some par-ticular cases of those operators. Besides, we put forward annovel algorithm to deal with MADM problem with SVNinformation on the basis of the propounded operators. Finally,the efficacy of the propounded algorithm is confirmed by aactual application and the advantages are shown by compar-ing with other previous approaches. The main superiorities ofthe presented approach are (i) it can portray inconsistent andincomplete information in several practical issues throughsingle-valued neutrosophic numbers; (ii) it can remove theinfluence of awkward information according to power aver-aging operator; (iii) it can make the decision process moreflexible and make decision makers have more freedoms tochoice appropriate integration operator. In the future, we willgeneralize the Archimedean copula and co-copula to otherfuzzy environment such as Hesitant fuzzy linguistic termset [71], 2-tuple linguistic neutrosophic [72], [73] and soforth, and research the application of emergency manage-ment strategy selection and venture investment decision insingle-valued neutrosophic environment.

APPENDIXESAPPENDIX ATHE PROOF OF THEOREM 2Proof: (1) and (2) are evident, the proofs are omitted here,we prove the others:

VOLUME 8, 2020 35513


(3) According to Eq (8 and Eq (10),

κ(α1 ⊕C α2)

= κ(1− ρ−1(ρ(1− T1)+ ρ(1− T2)),

ρ−1(ρ(I1)+ ρ(I2)), ρ−1(ρ(F1)+ ρ(F2)))

=

(1−ρ−1

(κρ(1−

(1−ρ−1(ρ(1−T1)+ρ(1−T2))

))),

ρ−1(κρ(ρ−1(ρ(I1)+ρ(I2))

)),

ρ−1(κρ(ρ−1(ρ(F1)+ ρ(F2))

)))=

(1− ρ−1

(κρ(ρ−1(ρ(1− T1)+ ρ(1− T2))

)),

ρ−1(κ(ρ(I1)+ ρ(I2))),ρ−1(κ(ρ(F1)+ ρ(F2)))

)=

(1− ρ−1(κ(ρ(1− T1)+ ρ(1− T2))),

ρ−1(κ(ρ(I1)+ ρ(I2))),ρ−1(κ(ρ(F1)+ ρ(F2)))

).

Moreover,

κα1 ⊕C κα2

=

(1− ρ−1(κρ(1− T1)), ρ−1(κρ(I1)), ρ−1(κρ(F1))

)⊕C

(1−ρ−1(κρ(1−T2)), ρ−1(κρ(I2)), ρ−1(κρ(F2))

)=

(1− ρ−1

(ρ(1−

(1− ρ−1(κρ(1− T1))

))+ ρ

(1−

(1− ρ−1(κρ(1− T2))

))),

ρ−1(ρ(ρ−1(κρ(I1))

)+ ρ

(ρ−1(κρ(I2))

)),

ρ−1(ρ(ρ−1(κρ(F1))

)+ ρ

(ρ−1(κρ(F2))

)))=

(1− ρ−1

(ρ(ρ−1(κρ(1− T1))

)+ ρ

(ρ−1(κρ(1− T2))

)), ρ−1(κ(ρ(I1)

+ ρ(I2))), ρ−1(κ(ρ(F1)+ ρ(F2))))

=

(1− ρ−1(κ(ρ(1− T1)+ ρ(1− T2))),

ρ−1(κ(ρ(I1)+ ρ(I2))),ρ−1(κ(ρ(F1)+ ρ(F2)))

).

Accordingly, κ(α1 ⊕C α2) = κα1 ⊕C κα2 holds.(5)

κ1α ⊕C κ2α

=

(1−ρ−1(κ1ρ(1−T )), ρ−1(κ1ρ(I)), ρ−1(κ1ρ(F))

)⊕C(1−ρ−1(κ2ρ(1−T )), ρ−1(κ2ρ(I)), ρ−1(κ2ρ(F))

)=

(1− ρ−1

(ρ(1−

(1− ρ−1(κ1ρ(1− T ))

))+ ρ

(1−

(1− ρ−1(κ2ρ(1− T ))

))),

ρ−1(ρ(ρ−1(κ1ρ(I))

)+ ρ

(ρ−1(κ2ρ(I))

)),

ρ−1(ρ(ρ−1(κ1ρ(F))

)+ ρ

(ρ−1(κ2ρ(F))

)))=

(1− ρ−1

(ρ(ρ−1(κ1ρ(1− T ))

)+ ρ

(ρ−1(κ2ρ(1− T ))

)), ρ−1(κ1ρ(I)+ κ2ρ(I)),

ρ−1(κ1ρ(F)+ κ2ρ(F)))

=

(1− ρ−1((κ1 + κ2)ρ(1− T )),

ρ−1((κ1 + κ2)ρ(I)), ρ−1((κ1 + κ2)ρ(F))).

Furthermore

(κ1 + κ2)α

=

(1− ρ−1((κ1 + κ2)ρ(1− T )), ρ−1((κ1 + κ2)ρ(I)),

ρ−1((κ1 + κ2)ρ(F))).

Hence, κ1α ⊕C κ2α = (κ1 + κ2)α is kept.Similarly, (4) and (6) can be proved by the same method,

which finishes the proof of Theorem 2.

APPENDIX BTHE PROOF OF THEOREM 3Proof:

(1) (αc)κ

= (F , 1− I, T )κ

=

(ρ−1(κρ(F)), 1−ρ−1(κρ(I)), 1−ρ−1(κρ(1− T ))

)= (κα)c.

(2) κ(αc)

= κ(F , 1− I, T )=

(1− ρ−1(κρ(1− F)), ρ−1(κρ(1− I)), ρ−1(κρ(T ))

)= (ακ )c.

(3) αc1 ⊕C αc2

= (F1, 1− I1, T1)⊕C (F2, 1− I2, T2)=

(1− ρ−1(ρ(1− F1)+ ρ(1− F2)), ρ−1(ρ(1− I1)

+ ρ(1− I2)), ρ−1(ρ(T1)+ ρ(T2)))

=

(ρ−1(ρ(T1)+ ρ(T2)), ρ−1(ρ(1− I1)+ ρ(1− I2)),

1− ρ−1(ρ(1− F1)+ ρ(1− F2)))c

= (α1 ⊗C α2)c.

(4) αc1 ⊗C αc2

= (F1, 1− I1, T1)⊗C (F2, 1− I2, T2)=

(ρ−1(ρ(F1)+ ρ(F2)), ρ−1(ρ(I1)+ ρ(I2)),

1− ρ−1(ρ(1− T1)+ ρ(1− T2)))

=

(1− ρ−1(ρ(1− T1)+ ρ(1− T2)), ρ−1(ρ(I1)

35514 VOLUME 8, 2020


+ ρ(I2)), ρ−1(ρ(F1)+ ρ(F2)))c

= (α1 ⊕C α2)c.

APPENDIX CSeveral operational laws based upon diverse copulas.1. If ρ(t) = (− ln t)σ , σ ≥ 1, then we have

(1) α1 ⊕C α2 =

(1− e−((− ln(1−T1))σ+(− ln(1−T2))σ )

1σ,

e−((− ln(I1))σ+(− ln(I2))σ )1σ,

e−((− ln(F1))σ+(− ln(F2))

σ )1σ

);

(2) α1 ⊗C α2 =

(e−((− ln(T1))σ+(− ln(T2))σ )

1σ,

1− e−((− ln(1−I1))σ+(− ln(1−I2))σ )1σ,

1− e−((− ln(1−F1))σ+(− ln(1−F2))

σ )1σ

);

(3) κα =(1− e−(κ(−ln(1−T ))

σ )1σ, e−(κ(−ln(I))

σ )1σ,

e−(κ(−ln(I))σ )

1σ

);

ακ =

(e−(κ(−ln(T ))σ )

1σ,

1− e−(κ(−ln(1−I))σ )

1σ, 1− e−(κ(−ln(1−F))

σ )1σ

).

2. If ρ(t) = t−σ − 1 with σ ≥ −1, σ 6= 0, then we have

(1) α1 ⊕C α2

=

(1−

((1− T1)−σ + (1− T2)−σ − 1

)− 1σ ,(

(I1)−σ + (I2)−σ − 1)− 1

σ ,((F1)

−σ+ (F2)

−σ− 1

)− 1σ

);

(2) α1 ⊗C α2 =

(((T1)−σ + (T2)−σ − 1

)− 1σ ,

1−((1− I1)−σ + (1− I2)−σ − 1

)− 1σ ,

1−((1− F1)

−σ+ (1− F2)

−σ− 1

)− 1σ

);

(3) κα =(1−

(κ((1− T )−σ − 1

)+ 1

)− 1σ ,(

κ(I−σ − 1

)+ 1

)− 1σ ,(

κ(F−σ − 1

)+ 1

)− 1σ

);

(4) ακ =((κ(T −σ − 1

)+ 1

)− 1σ ,

1−(κ((1− I)−σ − 1

)+ 1

)− 1σ ,

1−(κ((1− F)−σ − 1

)+ 1

)− 1σ

).

3. If ρ(t) = ln(e−σ t−1e−σ−1

)with σ 6= 0, then we have

(1) α1 ⊕C α2

=

(1+

1σln

((e−σ (1−T1) − 1

)(e−σ (1−T2) − 1

)e−σ − 1

+ 1

),

−1σln

((e−σI1 − 1

)(e−σI2 − 1

)e−σ − 1

+ 1

),

−1σln

((e−σF1 − 1

)(e−σF2 − 1

)e−σ − 1

+ 1

));

(2) α1 ⊗C α2

=

(−1σln

((e−σT1 − 1

)(e−σT2 − 1

)e−σ − 1

+ 1

),

1+1σln

((e−σ (1−I1) − 1

)(e−σ (1−I2) − 1

)e−σ − 1

+ 1

),

1+1σln

((e−σ (1−F1) − 1

)(e−σ (1−F2) − 1

)e−σ − 1

+ 1

));

(3) κα

=

(1−

1σln(1+

(e−σ − 1

)1−κ(e−σ(1−T ) − 1)κ)

,

1σln(1+

(e−σ − 1

)1−κ(e−σI − 1)κ)

,

1σln(1+

(e−σ − 1

)1−κ(e−σF − 1)κ))

;

(4) ακ

=

(1σln(1+

(e−σ − 1

)1−κ(e−σT − 1)κ)

,

1−1σln(1+

(e−σ − 1

)1−κ(e−σ(1−I) − 1)κ)

,

1−1σln(1+

(e−σ − 1

)1−κ(e−σ(1−F) − 1)κ))

.

4. If ρ(t) = ln(1−σ (1−t)

t

)with σ ∈ [−1, 1], then we have

(1) α1 ⊕C α2

=

(1−

(1− T1)(1− T2)1− σT1T2

,I1I2

1− σ(1− I1)(1− I2),

F1F2

1− σ(1− FI1)(1− F2)

);

(2) α1 ⊗C α2

=

(I1I2

1− σ(1− I1)(1− I2),

1−(1− I1)(1− I2)

1− σI1I2,

1−(1− F1)(1− F2)

1− σF1F2

);

(3) κα =((1− σT )κ − (1− T )κ

(1− σT )κ − σ(1− T )κ,

(1− σ)Iκ

(1− σ(1− I))κ − σIκ,

VOLUME 8, 2020 35515


(1− σ)Fκ

(1− σ(1− F))κ − σFκ

);

(4) ακ =(

(1− σ)T κ

(1− σ(1− T ))κ − σT κ,

(1− σI)κ − (1− I)κ

(1− σI)κ − σ(1− I)κ,

(1− σF)κ − (1− F)κ

(1− σF)κ − σ(1− F)κ

).

5. If ρ(t) = − ln(1− (1− t)σ ), σ ≥ −1, then we have

(1) α1 ⊕C α2

=

((T σ1 + T σ

2 − T σ1 T σ

2) 1σ ,

SVNCWA(α1, α2) = η1α1(⊕C)η2α2

=

1− ρ−1(η1ρ(1− T1)),ρ−1(η1ρ(I1)),ρ−1(η1ρ(F1))

⊕c 1− ρ−1(η2ρ(1− T2)),

ρ−1(η2ρ(I2)),ρ−1(η2ρ(F2))

=

1− ρ−1

(ρ(ρ−1(η1ρ(1− T1))

)+ ρ

(ρ−1(η2ρ(1− T2))

)),

ρ−1(ρ(ρ−1(η1ρ(I1))

)+ ρ

(ρ−1(η2ρ(I2))

)),

ρ−1(ρ(ρ−1(η1ρ(F1))

)+ ρ

(ρ−1(η2ρ(F2))

))

=

1− ρ−1(η1ρ(1− T1)+ η2ρ(1− T2)),ρ−1(η1ρ(I1)+ η2ρ(I2)),ρ−1(w1ρ(F1)+ η2ρ(F2))

=

(1− ρ−1

(2∑i=1

ηiρ(1− Ti)

), ρ−1

(2∑i=1

ηiρ(Ii)

), ρ−1

(2∑i=1

ηiρ(Fi)

)).

SVNCWA(α1, α2, · · · , αk+1)

= SVNCWA(α1, α2, · · · , αk)⊕C (ηk+1αk+1)

=

1− ρ−1(

k∑i=1

ηkρ(1− Tk)

),

ρ−1

(k∑i=1

ηkρ(Ik)

),

ρ−1

(k∑i=1

ηkρ(Fk)

)

⊕C

1− ρ−1(ηk+1ρ(1− Tk+1)),ρ−1(ηk+1ρ(Ik+1)),ρ−1(ηk+1ρ(Fk+1))

=

1− ρ−1(ρ

(ρ−1

(k∑i=1

ηkρ(1− Tk)

))+ ρ

(ρ−1(ηk+1ρ(1− Tk+1))

)),

ρ−1

(ρ

(ρ−1

(k∑i=1

ηkρ(Ik)

))+ ρ

(ρ−1(ηk+1ρ(Ik+1))

)),

ρ−1

(ρ

(ρ−1

(k∑i=1

ηkρ(Fk)

))+ ρ

(ρ−1(ηk+1ρ(Fk+1))

))

=

1− ρ−1(

k∑i=1

ηkρ(1− Tk)+ ηk+1ρ(1− Tk+1)

),

ρ−1

(k∑i=1

ηkρ(Ik)+ ηk+1ρ(Ik+1)

),

ρ−1

(k∑i=1

ηkρ(Fk)+ ηk+1ρ(Fk+1)

)

=

1− ρ−1(k+1∑i=1

ηk+1ρ(1− Tk+1)

),

ρ−1

(k+1∑i=1

ηk+1ρ(Ik+1)

),

ρ−1

(k+1∑i=1

ηk+1ρ(Fk+1)

)

.

35516 VOLUME 8, 2020


1−((1− I1)σ + (1− I2)σ − (1− I1)σ (1− I2)σ

) 1σ ,

1−((1− F1)

σ+(1−F2)

σ−(1−F1)

σ (1−F2)σ) 1σ

);

(2) α1 ⊗C α2

=

(1−

((1−T1)σ + (1− T2)σ−(1−T1)σ (1− T2)σ

) 1σ ,

(Iσ1 + Iσ2 − Iσ1 I

σ2) 1σ ,(Fσ1 + Fσ

2 − Fσ1 F

σ2) 1σ ,

);

(3) κα=((

1−(1− T σ

)κ) 1σ , 1−

(1−

(1−(1− I)σ

)κ) 1σ ,

1−(1−

(1− (1− F)σ

)κ) 1σ

);

(4) ακ=(1−

(1−

(1− (1− T )σ

)κ) 1σ ,(1−

(1−Iσ

)κ) 1σ ,

(1−

(1− Fσ

)κ) 1σ

).

APPENDIX DTHE PROOF OF THEOREM 4Proof: For Theorem 4, the first conclusion is obvious, and

then we prove Eq (13) by mathematical induction on n.For n = 2, we have SVNCWA(α1, α2), as shown at the

bottom of the previous page.Suppose that Eq (13) is kept for n = k , then we have

SVNCWA(α1, α2, · · · , αk)

=

(1− ρ−1

(k∑i=1

ηkρ(1− Tk)

), ρ−1

(k∑i=1

ηkρ(Ik)

),

ρ−1

(k∑i=1

ηkρ(Fk)

)).

Then, when n = k+1, we have SVNCWA(α1, α2, · · · , αk+1),as shown at the bottom of the previous page, i.e., Eq (13) isvalid for n = k + 1. Accordingly, Eq (13) is kept for all n.

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YUAN RONG received the B.S. degree in mathe-matics and applied mathematics from the NeijiangNormal University, Neijiang, China, in 2018. He iscurrently pursuing the master’s degree in appliedmathematics.

He has authored or coauthored six publications.His research interests include fuzzy set theory,aggregation operators, fuzzy decision making, andtheir applications.

35518 VOLUME 8, 2020


YI LIU received the M.S. degree in pure mathe-matics from Sichuan Normal University, Chengdu,China, in 2007, and the Ph.D. degree in computersciences and technology from Southwest JiaotongUniversity, Chengdu, in 2014.

He is currently a Professor with the Data Recov-ery Key Laboratory of Sichuan Province andSchool of Mathematics and Information Sciences,Neijiang Normal University, Sichuan, China.He has authored or coauthored more than 60 publi-

cations. His research interests include aggregation operators, fuzzy decisionmaking, automated reasoning, and their applications.

ZHENG PEI received the M.S. and Ph.D. degreesfrom Southwest Jiaotong University, Chengdu,China, in 1999 and 2002, respectively.

He is currently a Professor with the School ofComputer and Software Engineering, Xihua Uni-versity, Chengdu. He has nearly 70 research arti-cles published in academic journals or conference.His research interests include rough set theory,fuzzy set theory, logical reasoning, and linguisticinformation processing.

VOLUME 8, 2020 35519

Documents

fs.unm.edufs.unm.edu/neut/GeneralizedSingleValuedNeutrosophicPower.pdf · Received January 27, 2020, accepted February 13, 2020, date of publication February 18, 2020, date of current