ArchimedeanCopula …downloads.hindawi.com/journals/mpe/2020/6284245.pdfby Zadeh and their various...

Research ArticleArchimedean Copula-Based Hesitant Fuzzy InformationAggregation Operators for Multiple Attribute Decision Making

Ju Wu12 Lianming Mou12 Fang Liu 12 Haobin Liu12 and Yi Liu 123

1Data Recovery Key Laboratory of Sichuan Province Neijiang Normal University Neijiang 641000 Sichuan China2School of Mathematics and Information Sciences Neijiang Normal University Neijiang 641000 Sichuan China3Numerical Simulation Key Laboratory of Sichuan Province Neijiang Normal University Neijiang 641000 Sichuan China

Correspondence should be addressed to Yi Liu liuyiyl126com

Received 3 April 2020 Revised 2 June 2020 Accepted 9 June 2020 Published 3 July 2020

Academic Editor Francesc Pozo

Copyright copy 2020 Ju Wu et al is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In view of the good properties of copulas and their effective use in various fuzzy environments the goal of the current study is todevelop a series of aggregation operators for hesitant fuzzy information based on Archimedean copula and cocopula which areapplied to the MADM problems Firstly operational laws of hesitant fuzzy elements on the basis of copulas and cocopulas aredefined which can show the relevance between hesitant fuzzy values Secondly four aggregation operators (AC-HFWA AC-GHFWA AC-HFWG and AC-GHFWG) under hesitant fuzzy environment are developed according to the proposed operationallaws e properties of these operators are also studied in detail including idempotence monotonicity boundedness etcSubsequently five special cases of copula are also given and the special forms of aggregation operator are obtained In the end anexample is used to illustrate the application of the proposed approach in MADM problems e influences of different generatedfunctions and parameters are shown and the feasibility of the proposed method is validated through comparative analyses

1 Introduction

Multiple attribute decision making (MADM) also known aslimited scheme multiobjective decision is to select the op-timal alternatives or ranking decision making problems in thecase of considering multiple attributes It is a vital part ofmodern decision science its theories and methods have beenwidely utilized in engineering technology economy man-agement military and many other fields One of the mostimportant tasks of MADM is to fuse the attribute values givento each alternative by the decisionmaker and then summarizethe decision makerrsquos opinion on each alternative In thisprocess a primary issue is to describe the values of criteriaFor this issue many experts proposed to adopt fuzzy setsMADM problems with different kinds of fuzzy informationare handled by utilizing fuzzy set (FS) [1] which is proposedby Zadeh and their various extensions including the intui-tionistic fuzzy set (IFS) [2] interval-valued intuitionistic fuzzyset (IVIFS) [3] hesitant fuzzy set (HFS) [4 5] Pythagoreanfuzzy set (PFS) [6] neutrosophic set (NS) [7] and so on

In the numerous extensions of the FS IFS as one of themost important was introduced by Atanassov [2] Because itprovides a membership degree (MD) a nonmembershipdegree (NMD) and a hesitancy degree (HD) to each ele-ment IFS is better at handling uncertainty and vaguenessthan FS Since its emergence IFS has attracted more andmore researchersrsquo attention However when giving themembership degree of an element the difficulty of estab-lishing the membership degree is not because we have amargin of error or some possibility distribution on thepossibility values but because we have several possiblevalues For such cases Torra and Narukawa [4] proposedhesitant fuzzy set (HFS) and indicated that the envelope of ahesitant fuzzy element (HFE) is an intuitionistic fuzzy value(IFV) So all the operations on IFS can be suitable for HFSand many research studies of IFS can be extended to HFS

e aggregation operator which fuses multiple infor-mation sources plays a key role in the realization of col-lective opinions in MADM In order to deal withinformation in different fuzzy environments various

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6284245 21 pageshttpsdoiorg10115520206284245

aggregation operators are proposed Weighted average (WA)operator and weighted geometry (WG) operator are the mostcommonly used integration operators in classical decisionscience theory In the process ofMADM they have been deeplystudied by scholars [8ndash12] which have been extended to theintegration of different types of decision information such asordered weighted averaging operator (OWA) and orderedweighted geometry operator (OWG) Based on the definedoperations for IFS Xia and Xu [13] presented eight hesitantfuzzy aggregation operators such as hesitant fuzzy weightedaveraging (HFWA) operator hesitant fuzzy weighted geo-metric (HFWG) operator and so on According to the op-erators mentioned above many scholars investigated manyoperators to solve MCDM problems under hesitant fuzzyenvironment [14ndash21] Qin et al [22] developed some hesitantfuzzy aggregation operators based on Frank operations such asHFFWA operator HFFOWA operator and so on Yu et al[23] studied a set of hesitant fuzzy Einstein aggregation op-erators such as HFECOA operator HFECOG operatorHFEPWA operator and HFEPWG operator Using thetechnique of obtaining values in the interval Du et al [24]proposed the generalized hesitant fuzzy harmonic mean op-erators including GHFWHMoperator GHFOWHMoperatorand GHFHHM operator Li and Chen [25] presented two newaggregation operators belief structure hesitant fuzzy inducedordered weighted averaging operator and belief structurehesitant fuzzy induced ordered weighted geometric operatorAlthough the research and application of the integration op-erator have beenwell developed the decision problem based onthe integration operator has certain complexity so it is nec-essary to conduct in-depth research on it and explore newinformation integration methods

In the aforementioned aggregation operators under hesitantfuzzy environment the operational laws of any two HFEs arebuilt on the t-norms (TCs) and t-conorms (TCs) CommonlyTNs are applied to integrate MD of fuzzy sets while copulas aretools to deal with probability distributions Besides there existalso TNs which are copulas and vice versaus the applicationof copulas in fuzzy sets has important practical significanceCopulas [26] can not only reveal the dependence among at-tributes but also prevent information loss in the midst of ag-gregation ere are two distinguishing features of copula (1)copulas and cocopulas are flexible because decision makers canselect different types of copulas and cocopulas to define theoperations under fuzzy environment and the results obtainedfrom these operations are closed (2) copula functions are flexibleto capture the correlations among attributes in MADMs Basedon the two obvious characteristic copulas have been applied tosome MADMs In the light of Archimedean copula Tao et al[27] studied a new computational model for unbalanced lin-guistic variables Chen et al [28] defined new aggregation op-erators in linguistic neutrosophic set based on copula andapplied them to settle MCDM problems

In this paper based on the current research the copulasare generalized to the HFS and two kinds of hesitating fuzzyinformation integration operators based on copulas areproposed which are applied to the MADM problems Forthe goals the structure of this work is arranged as followsSome notions on hesitant fuzzy set and copulas are reviewed

firstly in Section 2 e hesitant fuzzy weighted averagingoperator-based Archimedean copulas (AC-HFWA) aredefined in Section 3 before AC-HFWA is given the op-erations of hesitant fuzzy elements based on Archimedeancopula are also defined After AC-HFWA is given thegeneralized hesitant fuzzy weighted averaging operator-based Archimedean copulas (AC-GHFWA) are introducedand the properties of AC-HFWA and AC-HFWG are in-vestigated along with the different cases e hesitant fuzzyweighted geometry operator-based Archimedean copulas(AC-HFWG) are defined in Section 4 before AC-HFWG isgiven the operations of hesitant fuzzy elements based onArchimedean copula are also defined After AC-HFWG isgiven the generalized hesitant fuzzy weighted geometryoperator-based Archimedean copulas (AC-GHFWG) areintroduced and the properties of AC-HFWA and AC-HFWG are investigated along with the different cases InSection 5 the algorithm of MADM with hesitant fuzzyinformation based on AC-HFWAAC-HFWG is con-structed firstly next case analysis will be carried out andsome comparisons with existing approaches in the hesitantfuzzy environment and merits of the proposed MADMapproach based on AC-HFWAAC-HFWG operators areanalysed and the conclusion will be obtained in Section 6

2 Preliminaries

In this section we will retrospect the related concepts of HFSand copula and cocopula these notions are the basis of thiswork

21 Hesitant Fuzzy Sets

Definition 1 (see [5]) Let S be a finite reference set Ahesitant fuzzy set G on S in terms of a function when appliedto S returns a subset of [0 1] denoted by

G langs g(h)rang |foralls isin S1113864 1113865 (1)

where g(h) is a collection of numbers hi from [0 1] in-dicating the possible membership degrees of foralls isin S to G Wecall g(h) a hesitant fuzzy element (HFE) and G the set of allHFEs

To compare the HFEs the comparison laws are definedas follows [5]

Definition 2 (see [5]) For a HFE g(h) cupgi1 hi1113864 1113865 μ(g)

(1g)1113936gi1hi is called the score function of g(h) where g is

the number of possible elements in g(h)For two HFEs g1(h) and g2(h)

If μ(g1)gt μ(g2) then g1 ≻g2If μ(g1) μ(g2) then g1 g2

22 Copulas and Cocopulas

Definition 3 (see [26]) A two-dimensional functionΩ [0 1]2⟶ [0 1] is called a copula if the followingconditions are met

2 Mathematical Problems in Engineering

(1) Ω(m 1) Ω(1 m) m Ω(m 0) Ω(0 m) 0(2) Ω(m1 n1) minus Ω(m2 n1) minus Ω(m1 n2) +Ω(m2 n2)ge 0

where m m1 m2 n1 n2 isin [0 1] and m1 lem2 n1 le n2

Definition 4 (see [29]) A copula Ω is named as an Archi-medean copula if there is a strictly decreasing and con-tinuous function ς(δ) [0 1]⟶ [0infin] with ς(1) 0 andσ from [0infin] to [0 1] is defined as follows

σ(δ) ςminus 1(δ) δ isin [0 ς(0)]

0 δ isin [ς(0) +infin)1113896 (2)

For all (δ ε) isin [0 1]2 we have

σ(δ ε) σ(ς(δ) + ς(ε)) (3)

If Ω is strictly increasing on [0 1]2 ς(0) +infin and σcoincides with ςminus 1 on [0 +infin] then Ω is written as [30]

Ω(δ ε) ςminus 1(ς(δ) + ς(ε)) (4)

and the function ς is called a strict generator andΩ is called astrict Archimedean copula

Definition 5 (see [31]) LetΩ be a copula and the cocopula isintroduced as follows

Ωlowast(δ ε) 1 minus Ω(1 minus δ 1 minus ε) (5)

IfΩ is a strict Archimedean copulaΩlowast is also changed tobeΩlowast(δ ε) 1 minus Ω(1 minus δ 1 minus ε) 1 minus ςminus 1

(ς(1 minus δ) + ς(1 minus ε)) (6)

In order to introduce some new operations based oncopulas and cocopulas mentioned above the followingconclusion is given firstly

Theorem 1 For forallδ ε isin [0 1] then 0leΩ(δ ε)le 1 0leΩlowast(δ ε)le 1

Proof If 0le δ le εle 1 then 0le 1 minus εle 1 minus δ le 1 As ς isstrictly decreasing and ς(1) 0 ς(0) +infin

0le ς(ε)le ς(δ)le +infin

0le ς(1 minus δ)le ς(1 minus ε)le +infin(7)

So

ς(δ)le ς(δ) + ς(ε)le 2ς(δ)le +infin

ς(1 minus ε)le ς(1 minus δ) + ς(1 minus ε)le 2ς(1 minus ε)le +infin(8)

We have

0le ςminus 1(ς(δ) + ς(ε))le δ le εle 1 minus ςminus 1

(ς(1 minus δ) + ς(1 minus ε))le 1

(9)

us eorem 1 holds

Definition 6 Let δ ε isin [0 1] the algebra operations basedon copula and cocopula are defined as follows

(1) δ oplus ε Ωlowast(δ ε) 1 minus ςminus 1(ς(1 minus δ) + ς(1 minus ε))

(2) δ otimes ε Ω(δ ε) ςminus 1(ς(δ) + ς(ε))

(10)

It is easy to verify that oplus and otimes satisfy associative lawthat is for forallδ ε ] isin [0 1]

(δ oplus ε)oplus ] δ oplus (εoplus ])

(δ otimes ε)otimes ] δ otimes (εotimes ])(11)

Theorem 2 For forallδ isin [0 1] ρge 0 we have ρδ 1 minus ςminus 1

(ρς(1 minus δ)) δρ ςminus 1(ρς(δ))

3 Archimedean Copula-Based Hesitant FuzzyWeighted Averaging Operator (AC-HFWA)

In this part we will put forward the Archimedean copula-based HF weighted averaging operator (AC-HFWA) BeforeAC-HFWA is introduced the new operations of HFE basedon copula will be defined and then some properties of AC-HFWA are also investigated

31 NewOperations forHFEs Based onCopulas We will givea new version of operational rules based on copulas andcocopulas

Definition 7 Let g1(h) cupg1m11 h1m1

1113966 1113967 g2(h) cupg2m21 h2m2

1113966 1113967and g(h) cupgi1 hi1113864 1113865 be three HFEs and ρge 0 the noveloperational rules of HFEs are given as follows

g1 oplusg2 cuph1m1ising1

h2m2ising2

1 minus ςminus 1 ς 1 minus h1m11113872 1113873 + ς 1 minus h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

g1 otimesg2 cuph1m1ising1

h2m2ising2

ςminus 1 ς h1m11113872 1113873 + ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

ρg cuphiising

1 minus ςminus 1 ρς 1 minus hi( 1113857( 11138571113868111386811138681113868 i 1 2 g1113966 1113967

gρ

cuphiising

ςminus 1 ρς hi( 1113857( 11138571113868111386811138681113868 i 1 2 g1113966 1113967

(12)

Mathematical Problems in Engineering 3

From the above definition the following conclusions canbe easily drawn

Theorem 3 Let g1 g2 and g3 be three HFEs anda b c isin R+ then we have

(1) g1 oplusg2 g2 oplusg1

(2) g1 oplusg2( 1113857oplusg3 g1 oplus g2 oplusg3( 1113857

(3) ag1 oplus bg1 (a + b)g1

(4) a bg1 oplus cg2( 1113857 abg1 oplus acg2

(5) a bg1( 1113857 abg1

(6) g1 otimesg2 g2 otimesg1

(7) g1 otimesg2( 1113857otimesg3 g1 otimes g2 otimesg3( 1113857

(13)

e algorithms can be used to fuse the HF informationand investigate their ideal properties which is the focus of thefollowing sections

32 AC-HFWA In this section the AC-HFWA will be in-troduced and the proposed operations of HFEs based oncopula as well as the properties of AC-HFWA are investigated

Definition 8 Let G g1 g2 gn1113864 1113865 be a set of n HFEs andΦ be a function on G Φ [0 1]n⟶ [0 1] thenΦ(G) cup Φ(g1 g2 gn)1113864 1113865

Definition 9 Let gi(h) cupgi

mi1 himi| i 1 2 n1113966 1113967 ωi be

the weight vector of gi with ωi isin [0 1] and 1113936ni1 ωi 1

Archimedean copula-based hesitant fuzzy weighted aver-aging operator (AC-HFWA) is defined as follows

AC minus HFWA g1 g2 gn( 1113857 ω1g1 oplusω2g2 oplus middot middot middot oplusωngn

(14)

Theorem 4 Let gi(h) cupgi

mi1 himi| i 1 2 n1113966 1113967 ωi be

the weight vector of gi with ωi isin [0 1] and 1113936ni1 ωi 1 then

AC minus HFWA g1 g2 gn( 1113857 oplusni1

ωigi cuphimiisingi

1 minus τminus 11113944n

i1ωiτ 1 minus himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (15)

Proof For n 2 we have

AC minus HFWA g1 g2( 1113857 ω1g1 oplusω2g2

cuph1m1ising1

h2m2ising2

1 minus ςminus 1 ω1ς 1 minus h1m11113872 1113873 + ω2ς 1 minus h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883 (16)

Suppose that equation (15) holds for n k that isAC minus HFWA g1 g2 gk( 1113857 ω1g1 oplusω2g2 oplus middot middot middot oplusωkgk

cuphimiisingi

1 minus τminus 11113944

k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(17)

en

AC minus HFWA g1 g2 gk gk+1( 1113857 opluski1

ωigi oplusωk+1gk+1

cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

oplus cuphk+1mk+1isingk+1

1 minus τminus 1 ωk+1τ 1 minus hk+1mk+11113872 11138731113872 111387311138681113868111386811138681113868 mk+1 1 2 gk+11113882 1113883

cuphimiisingi


k


1113872 1113873 + ωk+1τ 1 minus hk+1mk+11113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

cuphimiisingi


k+1


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(18)


Equation (15) holds for n k + 1 us equation (15)holds for all n


mi1 himi| i 1 2 n1113966 1113967 ωi be


(1) (Idempotency) If g1 g2 middot middot middot gn h AC minus HFWA(g1 g2 gn) h

(2) (Monotonicity) Let glowasti (h) cupglowasti

mi1 hlowastimi| i 1 2 1113966

n if himile hlowastimi

AC minus HFWA g1 g2 gn( 1113857leAC minus HFWA g

lowast1 glowast2 g

lowastn( 1113857

(19)

(3) (Boundedness) If hminus mini12n himi1113966 1113967 and h+

maxi12n himi1113966 1113967

hminus leAC minus HFWA g1 g2 gn( 1113857le h

+ (20)

Proof (1) AC minus HFWA(g1 g2 gn)

cup 1 minus ςminus 1(1113936ni1 ωiς1113864 (1 minus h)) | h isin gi i 1 2 n h

(2) If himile hlowastimi

ς(1 minus himi)le ς(1 minus hlowastimi

) and 1113936ni1

ωiς(1 minus himi)le 1113936

ni1 ωiς(1 minus hlowastimi

)en ςminus 1(1113936

ni1 ωiς(1 minus himi

))ge ςminus 1(1113936ni1 ωiς(1minus

hlowastimi)) and 1 minus ςminus 1(1113936


))le 1 minus ςminus 1

(1113936ni1 ωiς(1 minus hlowastimi

))So AC minus HFWA(g1 g2 gn)leAC minus HFWA(glowast1 glowast2 glowastn )

(3) Suppose hminus mini12n himi1113966 1113967 and h+

maxi12n himi1113966 1113967

erefore 1 minus h+ le 1 minus himi 1 minus hminus ge 1 minus himi

for all i

and mi

Since ς is strictly decreasing ςminus 1 is also strictlydecreasing

en ς(1 minus hminus )le ς(1 minus himi)le ς(1 minus h+) foralli 1 2

n and so

1113944

n

i1ωiς 1 minus h

minus( )le 1113944

n

i1ωiς 1 minus himi

1113872 1113873le 1113944n

i1ωiς 1 minus h

+( 1113857 (21)

at is ς(1 minus hminus )le 1113936ni1 ωiς(1 minus himi

)le ς(1 minus h+)erefore ςminus 1(ς(1 minus h+))le ςminus 1(1113936


))leςminus 1(ς(1 minus hminus )) h0 le 1 minus ςminus 1(1113936


))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be

the weight vector of gi with ωi isin [0 1] and 1113936ni1 ωi 1 e

Archimedean copula-based generalized hesitant fuzzy av-eraging operator (AC-GHFWA) is given by

AC minus GHFWAθ g1 g2 gn( 1113857

ω1gθ1 oplusω2g

θ2 oplus middot middot middot oplusωng

θn1113872 1113873

1θ oplusn

i1ωig

θi1113888 1113889

1θ

(22)

Especially when θ 1 the AC-GHFWA operator be-comes the AC-HFWA operator

e following theorems are easily obtained from e-orem 4 and the operations of HFEs


mi1 himi| i 1 2 n1113966 1113967 ωi be


AC minus GHFWAθ g1 g2 gn( 1113857 cuphimiisingi

minus ςminus 1 1θ

ς 1 minus ςminus 11113944

n

i1ωi ς 1 minus ςminus 1 θς himi

1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(23)

Similar to Deorem 5 the properties of AC-GHFWA canbe obtained easily


mi1 himi| i 1 2 n1113966 1113967 ωi be


(1) (Idempotency) If g1 g2 middot middot middot gn h ACminus

GHFWAθ(g1 g2 gn) h (2) (Monotonicity) Let glowasti (h) cupg

lowasti

mi1 hlowastimi| i 1113966

1 2 n if himile hlowastimi


leAC minus GHFWAθ glowast1 glowast2 g

lowastn( 1113857

(24)

(3) (Boundedness) If hminus mini12n himi1113966 1113967 and

h+ maxi12n himi1113966 1113967

hminus leAC minus GHFWAθ g1 g2 gn( 1113857le h

+ (25)

33 Different Forms of AC-HFWA We can see from e-orem 4 that some specific AC-HFWAs can be obtained whenς is assigned different generators

Case 1 If ς(t) (minus lnt)κ κge 1 then ςminus 1(t) eminus t1κ Soδ oplus ε 1 minus eminus ((minus ln(1minus δ))κ+((minus ln(1minus ε))κ)1κ δ otimes ε 1 minus eminus ((minus ln(δ))κ+((minus ln(ε))κ)1κ


AC minus HFWA g1 g2 gn( 1113857 cuphimiisingi

1 minus eminus 1113936

n

i1 ωi minus ln 1minus himi1113872 11138731113872 1113873

λ1113874 1113875

1λ 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mi 1 2 gi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (26)

Specifically when κ 1 ς(t) minus ln t thenδ oplus ε 1 minus (1 minus δ)(1 minus ε) δ otimes ε δε and the AC-HFWA becomes the following

HFWA g1 g2 gn( 1113857 cuphimiisingi

1 minus 1113945

n

i11 minus himi

1113872 1113873ωi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

GHFWAθ g1 g2 gn( 1113857 cuphimiisingi

1 minus 1113945n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

1θ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(27)

ey are the HF operators defined by Xia and Xu [13] Case 2 If ς(t) tminus κ minus 1 κgt 0 then ςminus 1(t)

(t + 1)minus (1κ) So δ oplus ε 1 minus ((1 minus δ)minus κ + (1 minus ε)minus κ

minus 1)minus 1κ δ otimes ε (δminus κ + εminus κ minus 1)minus 1κ


1 minus 1113944n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus 1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (28)

Case 3 If ς(t) minus ln((eminus κt minus 1)(eminus κ minus 1)) κne 0 thenςminus 1(t) minus (1κ)ln(eminus t(eminus κ minus 1) + 1) So δ oplus ε 1 +

(1κ)ln(((eminus κ(1minus δ) minus 1)(eminus κ(1minus ε) minus 1)(eminus κ minus 1)) + 1)δ otimes ε minus (1κ)ln(((eminus κδ minus 1)(eminus κε minus 1)(eminus κ minus 1)) + 1)


1 +1κln 1113945

n

i1e

minus κ 1minus himi1113872 1113873

minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (29)

Case 4 If ς(t) ln((1 minus κ(1 minus t))t) minus 1le κlt 1 thenςminus 1(t) ((1 minus κ)(et minus κ)) So δ oplus ε 1 minus ((1 minus δ)

(1 minus ε)(1 minus κδε)) δ otimes ε (δε(1 minus κ(1 minus δ)(1 minus ε)))


1113937ni1 1 minus κhimi

1113872 1113873ωi

minus 1113937ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

minus κ1113937ni1 1 minus himi

1113872 1113873ωi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (30)

Case 5 If ς(t) minus ln(1 minus (1 minus t)κ) κge 1 thenςminus 1(t) 1 minus (1 minus eminus t)1κ So δ oplus ε (δκ + εκ minus δκεκ)1κδ otimes ε 1 minus ((1 minus δ)κ + (1 minus ε)κ minus (1 minus δ)κ(1 minus ε)κ)1κ


1 minus 1113945n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (31)


4 Archimedean Copula-Based Hesitant FuzzyWeighted Geometric Operator (AC-HFWG)

In this section the Archimedean copula-based hesitantfuzzy weighted geometric operator (AC-HFWG) will beintroduced and some special forms of AC-HFWG op-erators will be discussed when the generator ς takesdifferent functions

41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be


Archimedean copula-based hesitant fuzzy weighted geo-metric operator (AC-HFWG) is defined as follows

AC minus HFWG g1 g2 gn( 1113857 gω11 otimesg

ω22 otimes middot middot middot otimesg

ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be


AC minus HFWG g1 g2 gn( 1113857 cuphimiisingi

ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)


AC minus HFWG g1 g2( 1113857 gω11 otimesg

ω22

cuph1m1ising1h2m2ising2

ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)

Suppose that equation (33) holds for n k that is

AC minus HFWG g1 g2 gk( 1113857 cuphimiisingi

ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en

AC minus HFWG g1 g2 gk gk + 1( 1113857 otimesk

i1gωi

i oplusgωk+1k+1 cup

himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be


(1) If g1 g2 middot middot middot gn h AC minus HFWG(g1 g2

gn) h (2) Let glowasti (h) cupg

lowasti

mi1 hlowastimi| i 1 2 n1113966 1113967 if himi

le hlowastimi

AC minus HFWG g1 g2 gn( 1113857leAC minus HFWG glowast1 glowast2 g

lowastn( 1113857 (37)

(3) If hminus mini12n himi1113966 1113967 and h+ maxi12n himi

1113966 1113967

hminus leAC minus HFWG g1 g2 gn( 1113857le h

+ (38)

Proof Suppose hminus mini12n himi1113966 1113967 and h+

maxi12n himi1113966 1113967



en ς(h+)le ς(himi)le ς(hminus ) foralli 1 2 n

1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi

)le 1113936ni1 ωiς(hminus ) ς(h+)le

1113936ni1 ωiς(himi

)le ς(hminus ) so ςminus 1(ς(hminus ))le ςminus 1(1113936ni1

ωiς(himi))le ςminus 1(ς(h+))hminus le ςminus 1(1113936

ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be


the generalized hesitant fuzzy weighted geometric operatorbased on Archimedean copulas (AC-CHFWG) is defined asfollows

AC minus GHFWGθ g1 g2 gn( 1113857 1θ

θg1( 1113857ω1 otimes θg2( 1113857

ω2 otimes middot middot middot otimes θgn( 1113857ωn( 1113857

1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)

Especially when θ 1 the AC-GHFWG operator be-comes the AC-HFWG operator


mi1 himi| i 1 2 n1113966 1113967 ωi be


AC minus GHFWGθ g1 g2 gn( 1113857 cuphimiisingi

1 minus τminus 1 1θ

τ 1 minus τminus 11113944

n

i1ωi τ 1 minus τminus 1 θτ 1 minus himi

1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be


(1) If g1 g2 middot middot middot gn h AC minus GHFWGθ(g1 g2

gn) h

(2) Let glowasti (h) cupglowasti


le hlowastimi

AC minus GHFWGθ g1 g2 gn( 1113857leAC minus GHFWGθ glowast1 glowast2 g

lowastn( 1113857 (41)


1113966 1113967

hminus leAC minus GHFWGθ g1 g2 gn( 1113857le h

+ (42)

42 Different Forms of AC-HFWG Operators We can seefrom eorem 8 that some specific AC-HFWGs can beobtained when ς is assigned different generators

Case 1 If ς(t) (minus lnt)κ κge 1 then ςminus 1(t) eminus t1κ Soδ oplus ε 1 minus eminus ((minus ln(1minus δ))κ+((minus ln(1minus ε))κ)1κ δ otimes ε eminus ((minus ln δ)κ+((minus ln ε)κ)1κ


eminus 1113936

n

i1 ωi minus ln himi1113872 1113873

κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)

Specifically when κ 1 and ς(t) minus lnt then δ oplus ε

1 minus (1 minus δ)(1 minus ε) and δ otimes ε δε and the AC-HFWGoperator reduces to HFWG and GHFWG [13]

HFWG g1 g2 gn( 1113857 cuphimiisingi

1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

GHFWGθ g1 g2 gn( 1113857 cuphimiisingi

1 minus 1 minus 1113945n

i11 minus 1 minus himi

1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)


Case 2 If ς(t) tminus κ minus 1 κgt 0 then ςminus 1(t)


minus 1)minus (1κ) δ otimes ε (δminus κ + εminus κ minus 1)minus 1κ


1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)

Case 3 If ς(t) minus ln((eminus κt minus 1)(eminus κ minus 1)) κne 0 thenςminus 1(t) minus (1κ)ln(eminus t(eminus κ minus 1) + 1) So δ oplus ε 1+

(1κ)ln(((eminus κ(1minus δ) minus 1)(eminus κ(1minus ε) minus 1)(eminus κ minus 1)) + 1)δ otimes ε minus (1k)ln(((eminus κδ minus 1)(eminus κε minus 1)(eminus κ minus 1)) + 1)


minus1κln 1113945

n

i1e

minus κhimi minus 11113872 1113873ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi

1113937ni1 1 minus κ 1 minus himi

1113872 11138731113872 1113873ωi

minus κ1113937ni1 h

ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)



1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)

43 De Properties of AC-HFWA and AC-HFWG It is seenfrom above discussion that AC-HFWA and AC-HFWG arefunctions with respect to the parameter which is from thegenerator ς In this section we will introduce the propertiesof the AC-HFWA and AC-HFWG operator regarding to theparameter κ

Theorem 12 Let ς(t) be the generator function of copulaand it takes five cases proposed in Section 42 then μ(AC minus

HFWA(g1 g2 gn)) is an increasing function of κμ(AC minus HFWG(g1 g2 gn)) is an decreasing function ofκ and μ(AC minus HFWA(g1 g2 gn))ge μ(ACminus

HFWG(g1 g2 gn))

Proof (Case 1) When ς(t) (minus lnt)κ with κge 1 By Def-inition 6 we have

μ(AC minus HFWA(κ)) 1

g1g2 middot middot middot gn

1113944

g1

m111113944

g2

m21middot middot middot 1113944

gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

μ(AC minus HFWG(κ)) 1


1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)


Suppose 1le κ1 lt κ2 according to reference [10](1113936

ni1 ωia

κi )1κ is an increasing function of κ So

1113944

n

i1ωi minus ln 1 minus himi

1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1

le 1 minus eminus 1113936

n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)

erefore μ(AC minus HFWA(κ1))le μ(AC minus HFWA(κ2))μ(AC minus HFWG(κ1))ge μ(AC minus HFWG(κ2)) Because κge 1

eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n

i1 ωi minus ln 1minus himi1113872 11138731113872 11138731113872 1113873

+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)

So 1 minus eminus (1113936n

i1 ωi(minus ln(1minus himi))κ)1κ ge eminus (1113936

n

i1 ωi(minus ln himi)κ)1κ

at is μ(AC minus HFWA(g1 g2 gn))ge μ(ACminus

HFWG(g1 g2 gn))

So eorem 12 holds under Case 1

Case 2 When ς(t) tminus κ minus 1 κgt 0 Firstly



1113944

g1

m111113944

g2


gn

mn11 minus ςminus 1

1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)

Secondly because ς(κ t) tminus κ minus 1 κgt 0 0lt tlt 1(zςzκ) tminus κ ln t tminus κ gt 0 lntlt 0 and then (zςzκ)lt 0ς(κ t) is decreasing with respect to κ

ςminus 1(κ t) (t + 1)minus (1κ) κgt 0 0lt tlt 1 so (zςminus 1zκ)

(t + 1)minus (1κ) ln(t + 1)(1κ2) (t + 1)minus (1κ) gt 0 ln(t + 1)gt 0

(1κ2)gt 0 and then (zςminus 1zκ)gt 0 ςminus 1(κ t) is decreasingwith respect to κ

us 1 minus ςminus 1(1113936ni1 ωiς(1 minus himi

)) is decreasing with re-spect to κ and ςminus 1(1113936

ki1 ωiς(himi

)) is decreasing with respectto κ Suppose 1le κ1 lt κ2 we have

μ(AC minus HFWA(κ1))le μ(AC minus HFWA(κ2)) andμ(AC minus HFWG(κ1))ge μ(AC minus HFWG(κ2))

Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n

i1 ωiln 1minus himi1113872 1113873

+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)

at is eorem 12 holds under Case 2 Case 3 When ς(t) minus ln((eminus κt minus 1)(eminus κ minus 1)) κne 0 Firstly



1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)

Secondlyς(κ t) minus ln((eminus κt minus 1)(eminus κ minus 1)) κne 0 0lt tlt 1

If ς minus lnς1 ς1 ((ςt2 minus 1)(ς2 minus 1)) ς2 eminus κ κne 0

0lt tlt 1Because ς is decreasing with respect to ς1 ς1 is decreasing

with respect to ς2 and ς2 is decreasing with respect to κς(κ t) is decreasing with respect to κ

ςminus 1(κ t) minus

1κln e

minus te

minus κminus 1( 1113857 + 11113872 1113873 κne 0 0lt tlt 1

(56)

Suppose ςminus 1 minus lnψ1 ψ1 ψ1κ2 ψ2 eminus tψ3 + 1

ς3 eminus κ minus 1 κne 0 0lt tlt 1

Because ςminus 1 is decreasing with respect to ψ1 ψ1 is de-creasing with respect to ψ2 and ψ2 is decreasing with respectto ψ3 ψ3 is decreasing with respect to κ en ςminus 1(κ t) isdecreasing with respect to κ



ki1 ωiς(himi

)) is decreasing with respectto κ

Suppose 1le κ1 lt κ2erefore μ(AC minus HFWA(κ1))le μ(AC minus HFWA(κ2))

and μ(AC minus HFWG(κ1))ge μ(AC minus HFWG(κ2))Lastly when minus prop lt κlt + prop

minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

le limκ⟶minus prop

ln 1113937ni1 eminus κhimi minus 11113872 1113873

ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop

1113937ni1 eminus κhimi minus 11113872 1113873

ωi 1113937ni1 eminus κhimi minus 11113872 1113873

ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi

1113872 1113873 eminus κhimi eminus κhimi minus 11113872 11138731113872 11138731113872 1113873

minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi

eminus κhimi minus 1⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)

So minus (1κ)ln(1113937ni1 (eminus κ(1 minus himi

) minus 1)ωi + 1) minus (1k)

ln(1113937ni1 (eminus κhimi minus 1)ωi + 1)le 1113936

ni1 ωi(1 minus himi

) + 1113936ni1 ωihimi

1

at is eorem 12 holds under Case 3

Case 4 When ς(t) ln((1 minus κ(1 minus t))t) minus 1le κlt 1 Firstly




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)

Secondly because ς(κ t) ln((1 minus κ(1 minus t))t) minus 1le κlt 1 0lt tlt 1 so (zςzκ) (t(t minus 1)(1 minus κ(1 minus t)))t(t minus 1)lt 0

Suppose ς1(κ t) 1 minus κ(1 minus t) minus 1le κlt 1 0lt tlt 1(zς1zκ) t minus 1lt 0 ς1(κ t) is decreasing with respect to κ

So ς1(κ t)gt ς1(1 t) tgt 0 then (zςzκ)lt 0 and ς(κ t)

is decreasing with respect to κAs ςminus 1(κ t) (1 minus κ)(et minus κ) minus 1le κlt 1 0lt tlt 1

(zςminus 1zκ) ((et + 1)(et minus κ)2) et + 1gt 0 (et minus κ)2 gt 0

then (zςminus 1zκ)gt 0 ςminus 1(κ t) is decreasing with respect to κus 1 minus ςminus 1(1113936


)) is decreasing with respectto κ and ςminus 1(1113936

ki1 ωiς(himi

)) is decreasing with respect to κSuppose 1le κ1 lt κ2 We have μ(AC minus HFWA(κ1))le μ(AC minus HFWA(κ2)) and μ(AC minus HFWG(κ1))ge μ(ACminus HFWG(κ2))

Lastly because y xωi ωi isin (0 1) is decreasing withrespect to κ (1 minus h2

i )ωi ge 1 (1 minus (1 minus hi)2)ωi ge 1 then

1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)

When minus 1le κlt 1 we have

1113937ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi

+ 1113937ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi

1113937ni1 1 + 1 minus himi

1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)


erefore μ(AC minus HFWA(g1 g2 gn))ge μ(ACminus

HFWG(g1 g2 gn)) at is eorem 12 holds underCase 4

Case 5 When ς(t) minus ln(1 minus (1 minus t)κ) κge 1Firstly



1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)

Secondly because ς(κ t) minus ln(1 minus (1 minus t)κ) κge 1 0lt tlt 1 (zςzκ) ((1 minus t)κ ln(1 minus t)(1 minus (1 minus t)κ))ln(1 minus t)lt 0

Suppose ς1(κ t) (1 minus t)κ κge 1 0lt tlt 1 (zς1zκ)

(1 minus t)κ ln(1 minus t)lt 0 ς1(κ t) is decreasing with respect to κso 0lt ς1(κ t)le ς1(1 t) 1 minus tlt 1 1 minus (1 minus t)κ gt 0

en (zςzκ)lt 0 ς(κ t) is decreasing with respect to κςminus 1(κ t) 1 minus (1 minus eminus t)1κ κge 1 0lt tlt 1 so (zςminus 1zκ)

(1 minus eminus t)1κln(1 minus eminus t)(minus (1κ2)) (1 minus eminus t)1κ gt 0

ln(1 minus eminus t) lt 0 minus (1κ2)lt 0 then (zςminus 1zκ)gt 0 ςminus 1(κ t) isdecreasing with respect to κ



ni1 ωiς(himi


Lastly suppose 1le κ1 lt κ2 erefore μ(AC minus HFWA(κ1))le μ(AC minus HFWA(κ2)) μ(AC minus HFWG(κ1))ge μ(ACminus HFWG(κ2))

Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)


i1 ωi(minus ln(1minus himi))κ)1κ ge eminus ((1113936

n

i1 ωi(minus ln(himi))κ))1κ

μ(AC minus HFWA(g1 g2 gn))ge μ(AC minus HFWG(g1 g2

gn))at is eorem 12 holds under Case 5

5 MADM Approach Based on AC-HFWAand AC-HFWG

From the above analysis a novel decisionmaking way will begiven to address MADM problems under HF environmentbased on the proposed operators

LetY Ψi(i 1 2 m)1113864 1113865 be a collection of alternativesand C Gj(j 1 2 n)1113966 1113967 be the set of attributes whoseweight vector is W (ω1ω2 ωn)T satisfying ωj isin [0 1]

and 1113936nj1 ωj 1 If DMs provide several values for the alter-

native Υi under the attribute Gj(j 1 2 n) with ano-nymity these values can be considered as a HFE hij In thatcase if two DMs give the same value then the value emergesonly once in hij In what follows the specific algorithm forMADM problems under HF environment will be designed

Step 1 e DMs provide their evaluations about thealternative Υi under the attribute Gj denoted by theHFEs hij i 1 2 m j 1 2 nStep 2 Use the proposed AC-HFWA (AC-HFWG) tofuse the HFE yi(i 1 2 m) for Υi(i 1 2 m)Step 3 e score values μ(yi)(i 1 2 m) of yi arecalculated using Definition 6 and comparedStep 4 e order of the attributes Υi(i 1 2 m) isgiven by the size of μ(yi)(i 1 2 m)

51 Illustrative Example An example of stock investmentwill be given to illustrate the application of the proposedmethod Assume that there are four short-term stocksΥ1Υ2Υ3Υ4 e following four attributes (G1 G2 G3 G4)

should be considered G1mdashnet value of each shareG2mdashearnings per share G3mdashcapital reserve per share andG4mdashaccounts receivable Assume the weight vector of eachattributes is W (02 03 015 035)T

Next we will use the developed method to find theranking of the alternatives and the optimal choice


Step 1 e decision matrix is given by DM and isshown in Table 1Step 2 Use the AC-HFWA operator to fuse the HEEyi(i 1 4) for Υi(i 1 4)

Take Υ4 as an example and let ς(t) (minus ln t)12 in Case1 we have

y4 AC minus HFWA g1 g2 g3 g4( 1113857 oplus4

j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)

Step 3 Compute the score values μ(yi)(i 1 4) ofyi(i 1 4) by Definition 6 μ(y4) 06944 Sim-ilarlyμ(y1) 05761 μ(y2) 06322 μ(y3) 05149Step 4 Rank the alternatives Υi(i 1 4) and selectthe desirable one in terms of comparison rules Sinceμ(y4)gt μ(y2)gt μ(y1)gt μ(y3) we can obtain the rankof alternatives as Υ4≻Υ2 ≻Υ1 ≻Υ3 and Υ4 is the bestalternative

52 Sensitivity Analysis

521De Effect of Parameters on the Results We take Case 1as an example to analyse the effect of parameter changes onthe result e results are shown in Table 2 and Figure 1

It is easy to see from Table 2 that the scores of alternativesby the AC-HFWA operator increase as the value of theparameter κ ranges from 1 to 5 is is consistent witheorem 12 But the order has not changed and the bestalternative is consistent

522 De Effect of Different Types of Generators on theResults in AC-HFWA e ordering results of alternativesusing other generators proposed in present work are listed inTable 3 Figure 2 shows more intuitively how the scorefunction varies with parameters

e results show that the order of alternatives obtainedby applying different generators and parameters is differentthe ranking order of alternatives is almost the same and thedesirable one is always Υ4 which indicates that AC-HFWAoperator is highly stable

523 De Effect of Different Types of Generators on theResults in AC-HFWG In step 2 if we utilize the AC-HFWGoperator instead of the AC-HFWA operator to calculate thevalues of the alternatives the results are shown in Table 4

Figure 3 shows the variation of the values with parameterκ We can find that the ranking of the alternatives maychange when κ changes in the AC-HFWG operator With

the increase in κ the ranking results change fromΥ4 ≻Υ1 ≻Υ3 ≻Υ2 to Υ4 ≻Υ3 ≻Υ1 ≻Υ2 and finally settle atΥ3 ≻Υ4 ≻Υ1 ≻Υ2 which indicates that AC-HFWG hascertain stability In the decision process DMs can confirmthe value of κ in accordance with their preferences

rough the analysis of Tables 3 and 4 we can find thatthe score values calculated by the AC-HFWG operator de-crease with the increase of parameter κ For the same gen-erator and parameter κ the values obtained by the AC-HFWAoperator are always greater than those obtained by theAC-HFWG operator which is consistent with eorem 12

524 De Effect of Parameter θ on the Results in AC-GHFWAand AC-GHFWG We employ the AC-GHFWA operatorand AC-GHFWG operator to aggregate the values of thealternatives taking Case 1 as an example e results arelisted in Tables 5 and 6 Figures 4 and 5 show the variation ofthe score function with κ and θ

e results indicate that the scores of alternatives by theAC-GHFWA operator increase as the parameter κ goes from0 to 10 and the parameter θ ranges from 0 to 10 and theranking of alternatives has not changed But the score valuesof alternatives reduce and the order of alternatives obtainedby the AC-GHFWG operator changes significantly

53 Comparisons with Existing Approach e above resultsindicate the effectiveness of our method which can solveMADM problems To further prove the validity of ourmethod this section compares the existingmethods with ourproposed method e previous methods include hesitantfuzzy weighted averaging (HFWA) operator (or HFWGoperator) by Xia and Xu [13] hesitant fuzzy Bonferronimeans (HFBM) operator by Zhu and Xu [32] and hesitantfuzzy Frank weighted average (HFFWA) operator (orHFFWG operator) by Qin et al [22] Using the data inreference [22] and different operators Table 7 is obtainede order of these operators is A6 ≻A2 ≻A3 ≻A5 ≻A1 ≻A4except for the HFBM operator which is A6 ≻A2 ≻A3 ≻A1 ≻A5 ≻A4


(1) From the previous argument it is obvious to findthat Xiarsquos operators [13] and Qinrsquos operators [22] arespecial cases of our operators When κ 1 in Case 1of our proposed operators AC-HFWA reduces toHFWA and AC-HFWG reduces to HFWG Whenκ 1 in Case 3 of our proposed operators AC-HFWA becomes HFFWA and AC-HFWG becomesHFFWG e operational rules of Xiarsquos method arealgebraic t-norm and algebraic t-conorm and the

operational rules of Qinrsquos method are Frank t-normand Frank t-conorm which are all special forms ofcopula and cocopula erefore our proposed op-erators are more general Furthermore the proposedapproach will supply more choice for DMs in realMADM problems

(2) e results of Zhursquos operator [32] are basically thesame as ours but Zhursquos calculations were morecomplicated Although we all introduce parameters

Table 1 HF decision matrix [13]

Alternatives G1 G2 G3 G4

Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09

Table 2 e ordering results obtained by AC-HFWA in Case 1

Parameter κ μ (Υ1) μ (Υ2) μ (Υ3) μ (Υ4) Ranking order

1 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ312 05720 06160 05249 06678 Υ4 ≻Υ2 ≻Υ1 ≻Υ32 06084 06621 05498 07156 Υ4 ≻Υ2 ≻Υ1 ≻Υ325 06258 06823 05626 07365 Υ4 ≻Υ2 ≻Υ1 ≻Υ33 06399 06981 05736 07525 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

1 12 2 3 35 50

0102030405060708

Score comparison of alternatives

Data 1Data 2

Data 3Data 4

Figure 1 e score values obtained by AC-HFWA in Case 1

Table 3 e ordering results obtained by AC-HFWA in Cases 2ndash5

Functions Parameter λ μ (Υ1) μ (Υ2) μ (Υ3) μ (Υ4) Ranking order

Case 21 06059 06681 05425 07256 Υ4 ≻Υ2 ≻Υ1 ≻Υ32 06411 07110 05656 07694 Υ4 ≻Υ2 ≻Υ1 ≻Υ33 06650 07351 05851 07932 Υ4 ≻Υ2 ≻Υ1 ≻Υ3





which can resize the aggregate value on account ofactual decisions the regularity of our parameterchanges is stronger than Zhursquos method e AC-HFWA operator has an ideal property of mono-tonically increasing with respect to the parameterand the AC-HFWG operator has an ideal property of

monotonically decreasing with respect to the pa-rameter which provides a basis for DMs to selectappropriate values according to their risk appetite Ifthe DM is risk preference we can choose the largestparameter possible and if the DM is risk aversion wecan choose the smallest parameter possible

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4

ndash1 ndash05 0 05 105

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)

Figure 2 e score functions obtained by AC-HFWA


Table 4 e ordering results obtained by AC-HFWG





Case 4minus 09 04866 04803 04732 05317 Υ4 ≻Υ1 ≻Υ2 ≻Υ3025 04689 04547 04627 05051 Υ4 ≻Υ1 ≻Υ3 ≻Υ2075 04507 04290 04511 04804 Υ4 ≻Υ3 ≻Υ1 ≻Υ2


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)

minus1 minus05 0 05 1025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)

Figure 3 e score functions obtained by AC-HFWG

Table 5 e ordering results obtained by AC-GHFWA in Case 1

Parameter κ Parameter θ μ (Υ1) μ (Υ2) μ (Υ3) μ (Υ4) Ranking order

11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

Table 6 e ordering results obtained by AC-GHFWG in Case 1


11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508

Parameter kappaParameter theta

Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)

Parameter kappaParameter theta 2

46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)

Figure 4 e score functions obtained by AC-GHFWA in Case 1

24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)

Figure 5 e score functions obtained by AC-GHFWG in Case 1

Table 7 e scores obtained by different operators

Operator Parameter μ (A1) μ (A2) μ (A3) μ (A4) μ (A5) μ (A6)

HFWA [13] None 04722 06098 05988 04370 05479 06969HFWG [13] None 04033 05064 04896 03354 04611 06472HFBM [32] p 2 q 1 05372 05758 05576 03973 05275 07072HFFWA [22] λ 2 03691 05322 04782 03759 04708 06031HFFWG [22] λ 2 05419 06335 06232 04712 06029 07475AC-HFWA (Case 2) λ 1 05069 06647 05985 04916 05888 07274AC-HFWG (Case 2) λ 1 03706 04616 04574 02873 04158 06308


6 Conclusions

From the above analysis while copulas and cocopulas aredescribed as different functions with different parametersthere are many different HF information aggregation op-erators which can be considered as a reflection of DMrsquospreferences ese operators include specific cases whichenable us to select the one that best fits with our interests inrespective decision environments is is the main advan-tage of these operators Of course they also have someshortcomings which provide us with ideas for the followingwork

We will apply the operators to deal with hesitant fuzzylinguistic [33] hesitant Pythagorean fuzzy sets [34] andmultiple attribute group decision making [35] in the futureIn order to reduce computation we will also consider tosimplify the operation of HFS and redefine the distancebetween HFEs [36] and score function [37] Besides theapplication of those operators to different decision-makingmethods will be developed such as pattern recognitioninformation retrieval and data mining

e operators studied in this paper are based on theknown weights How to use the information of the data itselfto determine the weight is also our next work Liu and Liu[38] studied the generalized intuitional trapezoid fuzzypower averaging operator which is the basis of our intro-duction of power mean operators into hesitant fuzzy setsWang and Li [39] developed power Bonferroni mean (PBM)operator to integrate Pythagorean fuzzy information whichgives us the inspiration to research new aggregation oper-ators by combining copulas with PBM Wu et al [40] ex-tended the best-worst method (BWM) to interval type-2fuzzy sets which inspires us to also consider using BWM inmore fuzzy environments to determine weights

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

Sichuan Province Youth Science and Technology InnovationTeam (No 2019JDTD0015) e Application Basic ResearchPlan Project of Sichuan Province (No2017JY0199) eScientific Research Project of Department of Education ofSichuan Province (18ZA0273 15TD0027) e ScientificResearch Project of Neijiang Normal University (18TD08)e Scientific Research Project of Neijiang Normal Uni-versity (No 16JC09) Open fund of Data Recovery Key Labof Sichuan Province (No DRN19018)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[4] V Torra and Y Narukawa ldquoOn hesitant fuzzy sets and de-cisionrdquo in Proceedings of the 2009 IEEE International Con-ference on Fuzzy Systems pp 1378ndash1382 IEEE Jeju IslandRepublic of Korea August 2009

[5] V Torra ldquoHesitant fuzzy setsrdquo International Journal of In-telligent Systems vol 25 pp 529ndash539 2010

[6] R R Yager and A M Abbasov ldquoPythagorean membershipgrades complex numbers and decision makingrdquo Interna-tional Journal of Intelligent Systems vol 28 no 5 pp 436ndash452 2013

[7] R Sahin ldquoCross-entropy measure on interval neutrosophicsets and its applications in multicriteria decision makingrdquoNeural Computing and Applications vol 28 no 5pp 1177ndash1187 2015

[8] J Fodor J-L Marichal M Roubens and M RoubensldquoCharacterization of the ordered weighted averaging opera-torsrdquo IEEE Transactions on Fuzzy Systems vol 3 no 2pp 236ndash240 1995

[9] F Chiclana F Herrera and E Herrera-viedma ldquoe orderedweighted geometric operator properties and application inMCDM problemsrdquo Technologies for Constructing IntelligentSystems Springer vol 2 pp 173ndash183 Berlin Germany 2002

[10] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[11] R R Yager and Z Xu ldquoe continuous ordered weightedgeometric operator and its application to decision makingrdquoFuzzy Sets and Systems vol 157 no 10 pp 1393ndash1402 2006

[12] J Merigo and A Gillafuente ldquoe induced generalized OWAoperatorrdquo Information Sciences vol 179 no 6 pp 729ndash7412009

[13] M Xia and Z Xu ldquoHesitant fuzzy information aggregation indecision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[14] G Wei ldquoHesitant fuzzy prioritized operators and their ap-plication to multiple attribute decision makingrdquo Knowledge-Based Systems vol 31 no 7 pp 176ndash182 2012

[15] M Xia Z Xu and N Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroup Decision and Negotiation vol 22 no 2 pp 259ndash2792013

[16] G Qian H Wang and X Feng ldquoGeneralized hesitant fuzzysets and their application in decision support systemrdquoKnowledge-Based Systems vol 37 no 4 pp 357ndash365 2013

[17] Z Zhang C Wang D Tian and K Li ldquoInduced generalizedhesitant fuzzy operators and their application to multipleattribute group decision makingrdquo Computers amp IndustrialEngineering vol 67 no 1 pp 116ndash138 2014

[18] F Li X Chen and Q Zhang ldquoInduced generalized hesitantfuzzy Shapley hybrid operators and their application in multi-attribute decision makingrdquo Applied Soft Computing vol 28no 1 pp 599ndash607 2015

[19] D Yu ldquoSome hesitant fuzzy information aggregation oper-ators based on Einstein operational lawsrdquo InternationalJournal of Intelligent Systems vol 29 no 4 pp 320ndash340 2014

[20] R Lin X Zhao H Wang and G Wei ldquoHesitant fuzzyHamacher aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 27 no 1 pp 49ndash63 2014


[21] Y Liu J Liu and Y Qin ldquoPythagorean fuzzy linguisticMuirhead mean operators and their applications to multi-attribute decision makingrdquo International Journal of IntelligentSystems vol 35 no 2 pp 300ndash332 2019

[22] J Qin X Liu and W Pedrycz ldquoFrank aggregation operatorsand their application to hesitant fuzzy multiple attributedecision makingrdquo Applied Soft Computing vol 41 pp 428ndash452 2016

[23] Q Yu F Hou Y Zhai and Y Du ldquoSome hesitant fuzzyEinstein aggregation operators and their application tomultiple attribute group decision makingrdquo InternationalJournal of Intelligent Systems vol 31 no 7 pp 722ndash746 2016

[24] H Du Z Xu and F Cui ldquoGeneralized hesitant fuzzy har-monic mean operators and their applications in group de-cision makingrdquo International Journal of Fuzzy Systemsvol 18 no 4 pp 685ndash696 2016

[25] X Li and X Chen ldquoGeneralized hesitant fuzzy harmonicmean operators and their applications in group decisionmakingrdquo Neural Computing and Applications vol 31 no 12pp 8917ndash8929 2019

[26] M Sklar ldquoFonctions de rpartition n dimensions et leursmargesrdquo Universit Paris vol 8 pp 229ndash231 1959

[27] Z Tao B Han L Zhou and H Chen ldquoe novel compu-tational model of unbalanced linguistic variables based onarchimedean copulardquo International Journal of UncertaintyFuzziness and Knowledge-Based Systems vol 26 no 4pp 601ndash631 2018

[28] T Chen S S He J Q Wang L Li and H Luo ldquoNoveloperations for linguistic neutrosophic sets on the basis ofArchimedean copulas and co-copulas and their application inmulti-criteria decision-making problemsrdquo Journal of Intelli-gent and Fuzzy Systems vol 37 no 3 pp 1ndash26 2019

[29] R B Nelsen ldquoAn introduction to copulasrdquo Technometricsvol 42 no 3 p 317 2000

[30] C Genest and R J Mackay ldquoCopules archimediennes etfamilies de lois bidimensionnelles dont les marges sontdonneesrdquo Canadian Journal of Statistics vol 14 no 2pp 145ndash159 1986

[31] U Cherubini E Luciano and W Vecchiato Copula Methodsin Finance John Wiley amp Sons Hoboken NJ USA 2004

[32] B Zhu and Z S Xu ldquoHesitant fuzzy Bonferroni means formulti-criteria decision makingrdquo Journal of the OperationalResearch Society vol 64 no 12 pp 1831ndash1840 2013

[33] P Xiao QWu H Li L Zhou Z Tao and J Liu ldquoNovel hesitantfuzzy linguistic multi-attribute group decision making methodbased on improved supplementary regulation and operationallawsrdquo IEEE Access vol 7 pp 32922ndash32940 2019

[34] QWuW Lin L Zhou Y Chen andH Y Chen ldquoEnhancingmultiple attribute group decision making flexibility based oninformation fusion technique and hesitant Pythagorean fuzzysetsrdquo Computers amp Industrial Engineering vol 127pp 954ndash970 2019

[35] Q Wu P Wu L Zhou H Chen and X Guan ldquoSome newHamacher aggregation operators under single-valued neu-trosophic 2-tuple linguistic environment and their applica-tions to multi-attribute group decision makingrdquo Computers ampIndustrial Engineering vol 116 pp 144ndash162 2018

[36] Z Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11pp 2128ndash2138 2011

[37] H Xing L Song and Z Yang ldquoAn evidential prospect theoryframework in hesitant fuzzy multiple-criteria decision-mak-ingrdquo Symmetry vol 11 no 12 p 1467 2019

[38] P Liu and Y Liu ldquoAn approach to multiple attribute groupdecision making based on intuitionistic trapezoidal fuzzypower generalized aggregation operatorrdquo InternationalJournal of Computational Intelligence Systems vol 7 no 2pp 291ndash304 2014

[39] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

[40] Q Wu L Zhou Y Chen and H Chen ldquoAn integratedapproach to green supplier selection based on the intervaltype-2 fuzzy best-worst and extended VIKOR methodsrdquoInformation Sciences vol 502 pp 394ndash417 2019


aggregation operators are proposed Weighted average (WA)operator and weighted geometry (WG) operator are the mostcommonly used integration operators in classical decisionscience theory In the process ofMADM they have been deeplystudied by scholars [8ndash12] which have been extended to theintegration of different types of decision information such asordered weighted averaging operator (OWA) and orderedweighted geometry operator (OWG) Based on the definedoperations for IFS Xia and Xu [13] presented eight hesitantfuzzy aggregation operators such as hesitant fuzzy weightedaveraging (HFWA) operator hesitant fuzzy weighted geo-metric (HFWG) operator and so on According to the op-erators mentioned above many scholars investigated manyoperators to solve MCDM problems under hesitant fuzzyenvironment [14ndash21] Qin et al [22] developed some hesitantfuzzy aggregation operators based on Frank operations such asHFFWA operator HFFOWA operator and so on Yu et al[23] studied a set of hesitant fuzzy Einstein aggregation op-erators such as HFECOA operator HFECOG operatorHFEPWA operator and HFEPWG operator Using thetechnique of obtaining values in the interval Du et al [24]proposed the generalized hesitant fuzzy harmonic mean op-erators including GHFWHMoperator GHFOWHMoperatorand GHFHHM operator Li and Chen [25] presented two newaggregation operators belief structure hesitant fuzzy inducedordered weighted averaging operator and belief structurehesitant fuzzy induced ordered weighted geometric operatorAlthough the research and application of the integration op-erator have beenwell developed the decision problem based onthe integration operator has certain complexity so it is nec-essary to conduct in-depth research on it and explore newinformation integration methods

In the aforementioned aggregation operators under hesitantfuzzy environment the operational laws of any two HFEs arebuilt on the t-norms (TCs) and t-conorms (TCs) CommonlyTNs are applied to integrate MD of fuzzy sets while copulas aretools to deal with probability distributions Besides there existalso TNs which are copulas and vice versaus the applicationof copulas in fuzzy sets has important practical significanceCopulas [26] can not only reveal the dependence among at-tributes but also prevent information loss in the midst of ag-gregation ere are two distinguishing features of copula (1)copulas and cocopulas are flexible because decision makers canselect different types of copulas and cocopulas to define theoperations under fuzzy environment and the results obtainedfrom these operations are closed (2) copula functions are flexibleto capture the correlations among attributes in MADMs Basedon the two obvious characteristic copulas have been applied tosome MADMs In the light of Archimedean copula Tao et al[27] studied a new computational model for unbalanced lin-guistic variables Chen et al [28] defined new aggregation op-erators in linguistic neutrosophic set based on copula andapplied them to settle MCDM problems

In this paper based on the current research the copulasare generalized to the HFS and two kinds of hesitating fuzzyinformation integration operators based on copulas areproposed which are applied to the MADM problems Forthe goals the structure of this work is arranged as followsSome notions on hesitant fuzzy set and copulas are reviewed

firstly in Section 2 e hesitant fuzzy weighted averagingoperator-based Archimedean copulas (AC-HFWA) aredefined in Section 3 before AC-HFWA is given the op-erations of hesitant fuzzy elements based on Archimedeancopula are also defined After AC-HFWA is given thegeneralized hesitant fuzzy weighted averaging operator-based Archimedean copulas (AC-GHFWA) are introducedand the properties of AC-HFWA and AC-HFWG are in-vestigated along with the different cases e hesitant fuzzyweighted geometry operator-based Archimedean copulas(AC-HFWG) are defined in Section 4 before AC-HFWG isgiven the operations of hesitant fuzzy elements based onArchimedean copula are also defined After AC-HFWG isgiven the generalized hesitant fuzzy weighted geometryoperator-based Archimedean copulas (AC-GHFWG) areintroduced and the properties of AC-HFWA and AC-HFWG are investigated along with the different cases InSection 5 the algorithm of MADM with hesitant fuzzyinformation based on AC-HFWAAC-HFWG is con-structed firstly next case analysis will be carried out andsome comparisons with existing approaches in the hesitantfuzzy environment and merits of the proposed MADMapproach based on AC-HFWAAC-HFWG operators areanalysed and the conclusion will be obtained in Section 6

2 Preliminaries

In this section we will retrospect the related concepts of HFSand copula and cocopula these notions are the basis of thiswork

21 Hesitant Fuzzy Sets

Definition 1 (see [5]) Let S be a finite reference set Ahesitant fuzzy set G on S in terms of a function when appliedto S returns a subset of [0 1] denoted by

G langs g(h)rang |foralls isin S1113864 1113865 (1)

where g(h) is a collection of numbers hi from [0 1] in-dicating the possible membership degrees of foralls isin S to G Wecall g(h) a hesitant fuzzy element (HFE) and G the set of allHFEs

To compare the HFEs the comparison laws are definedas follows [5]

Definition 2 (see [5]) For a HFE g(h) cupgi1 hi1113864 1113865 μ(g)

(1g)1113936gi1hi is called the score function of g(h) where g is

the number of possible elements in g(h)For two HFEs g1(h) and g2(h)

If μ(g1)gt μ(g2) then g1 ≻g2If μ(g1) μ(g2) then g1 g2

22 Copulas and Cocopulas

Definition 3 (see [26]) A two-dimensional functionΩ [0 1]2⟶ [0 1] is called a copula if the followingconditions are met






0 δ isin [ς(0) +infin)1113896 (2)


σ(δ ε) σ(ς(δ) + ς(ε)) (3)













So



We have



(9)

us eorem 1 holds




(10)










1113966 1113967 g2(h) cupg2m21 h2m2



h2m2ising2


1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883


h2m2ising2

ςminus 1 ς h1m11113872 1113873 + ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

ρg cuphiising


gρ

cuphiising

ςminus 1 ρς hi( 1113857( 11138571113868111386811138681113868 i 1 2 g1113966 1113967

(12)








(5) a bg1( 1113857 abg1



(13)





mi1 himi| i 1 2 n1113966 1113967 ωi be




(14)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ωigi cuphimiisingi



1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (15)



cuph1m1ising1

h2m2ising2


1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883 (16)


cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(17)

en



cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭



cuphimiisingi


k


1113872 1113873 + ωk+1τ 1 minus hk+1mk+11113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

cuphimiisingi


k+1


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(18)




mi1 himi| i 1 2 n1113966 1113967 ωi be







lowast1 glowast2 g

lowastn( 1113857

(19)


maxi12n himi1113966 1113967


+ (20)





) and 1113936ni1












maxi12n himi1113966 1113967


for all i

and mi



n and so

1113944

n

i1ωiς 1 minus h

minus( )le 1113944

n


1113872 1113873le 1113944n

i1ωiς 1 minus h

+( 1113857 (21)






))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




ω1gθ1 oplusω2g


θn1113872 1113873

1θ oplusn

i1ωig

θi1113888 1113889

1θ

(22)




mi1 himi| i 1 2 n1113966 1113967 ωi be



minus ςminus 1 1θ


n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(23)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti





lowastn( 1113857

(24)


h+ maxi12n himi1113966 1113967


+ (25)






n


λ1113874 1113875

1λ 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mi 1 2 gi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (26)



1 minus 1113945

n

i11 minus himi

1113872 1113873ωi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭


1 minus 1113945n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

1θ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(27)





1 minus 1113944n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus 1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (28)




1 +1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (29)





1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (30)



1 minus 1113945n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (31)




41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be





ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)



ω22


ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)



ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en


i1gωi


himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti


le hlowastimi


lowastn( 1113857 (37)


1113966 1113967


+ (38)


maxi12n himi1113966 1113967




1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References















































0 δ isin [ς(0) +infin)1113896 (2)


σ(δ ε) σ(ς(δ) + ς(ε)) (3)













So



We have



(9)

us eorem 1 holds




(10)










1113966 1113967 g2(h) cupg2m21 h2m2



h2m2ising2


1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883


h2m2ising2

ςminus 1 ς h1m11113872 1113873 + ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

ρg cuphiising


gρ

cuphiising

ςminus 1 ρς hi( 1113857( 11138571113868111386811138681113868 i 1 2 g1113966 1113967

(12)








(5) a bg1( 1113857 abg1



(13)





mi1 himi| i 1 2 n1113966 1113967 ωi be




(14)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ωigi cuphimiisingi



1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (15)



cuph1m1ising1

h2m2ising2


1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883 (16)


cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(17)

en



cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭



cuphimiisingi


k


1113872 1113873 + ωk+1τ 1 minus hk+1mk+11113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

cuphimiisingi


k+1


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(18)




mi1 himi| i 1 2 n1113966 1113967 ωi be







lowast1 glowast2 g

lowastn( 1113857

(19)


maxi12n himi1113966 1113967


+ (20)





) and 1113936ni1












maxi12n himi1113966 1113967


for all i

and mi



n and so

1113944

n

i1ωiς 1 minus h

minus( )le 1113944

n


1113872 1113873le 1113944n

i1ωiς 1 minus h

+( 1113857 (21)






))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




ω1gθ1 oplusω2g


θn1113872 1113873

1θ oplusn

i1ωig

θi1113888 1113889

1θ

(22)




mi1 himi| i 1 2 n1113966 1113967 ωi be



minus ςminus 1 1θ


n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(23)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti





lowastn( 1113857

(24)


h+ maxi12n himi1113966 1113967


+ (25)






n


λ1113874 1113875

1λ 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mi 1 2 gi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (26)



1 minus 1113945

n

i11 minus himi

1113872 1113873ωi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭


1 minus 1113945n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

1θ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(27)





1 minus 1113944n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus 1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (28)




1 +1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (29)





1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (30)



1 minus 1113945n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (31)




41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be





ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)



ω22


ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)



ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en


i1gωi


himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti


le hlowastimi


lowastn( 1113857 (37)


1113966 1113967


+ (38)


maxi12n himi1113966 1113967




1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References

















































(5) a bg1( 1113857 abg1



(13)





mi1 himi| i 1 2 n1113966 1113967 ωi be




(14)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ωigi cuphimiisingi



1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (15)



cuph1m1ising1

h2m2ising2


1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883 (16)


cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(17)

en



cuphimiisingi


k


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭



cuphimiisingi


k


1113872 1113873 + ωk+1τ 1 minus hk+1mk+11113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

cuphimiisingi


k+1


1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(18)




mi1 himi| i 1 2 n1113966 1113967 ωi be







lowast1 glowast2 g

lowastn( 1113857

(19)


maxi12n himi1113966 1113967


+ (20)





) and 1113936ni1












maxi12n himi1113966 1113967


for all i

and mi



n and so

1113944

n

i1ωiς 1 minus h

minus( )le 1113944

n


1113872 1113873le 1113944n

i1ωiς 1 minus h

+( 1113857 (21)






))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




ω1gθ1 oplusω2g


θn1113872 1113873

1θ oplusn

i1ωig

θi1113888 1113889

1θ

(22)




mi1 himi| i 1 2 n1113966 1113967 ωi be



minus ςminus 1 1θ


n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(23)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti





lowastn( 1113857

(24)


h+ maxi12n himi1113966 1113967


+ (25)






n


λ1113874 1113875

1λ 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mi 1 2 gi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (26)



1 minus 1113945

n

i11 minus himi

1113872 1113873ωi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭


1 minus 1113945n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

1θ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(27)





1 minus 1113944n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus 1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (28)




1 +1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (29)





1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (30)



1 minus 1113945n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (31)




41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be





ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)



ω22


ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)



ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en


i1gωi


himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti


le hlowastimi


lowastn( 1113857 (37)


1113966 1113967


+ (38)


maxi12n himi1113966 1113967




1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References













































mi1 himi| i 1 2 n1113966 1113967 ωi be







lowast1 glowast2 g

lowastn( 1113857

(19)


maxi12n himi1113966 1113967


+ (20)





) and 1113936ni1












maxi12n himi1113966 1113967


for all i

and mi



n and so

1113944

n

i1ωiς 1 minus h

minus( )le 1113944

n


1113872 1113873le 1113944n

i1ωiς 1 minus h

+( 1113857 (21)






))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




ω1gθ1 oplusω2g


θn1113872 1113873

1θ oplusn

i1ωig

θi1113888 1113889

1θ

(22)




mi1 himi| i 1 2 n1113966 1113967 ωi be



minus ςminus 1 1θ


n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(23)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti





lowastn( 1113857

(24)


h+ maxi12n himi1113966 1113967


+ (25)






n


λ1113874 1113875

1λ 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mi 1 2 gi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (26)



1 minus 1113945

n

i11 minus himi

1113872 1113873ωi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭


1 minus 1113945n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

1θ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(27)





1 minus 1113944n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus 1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (28)




1 +1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (29)





1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (30)



1 minus 1113945n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (31)




41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be





ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)



ω22


ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)



ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en


i1gωi


himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti


le hlowastimi


lowastn( 1113857 (37)


1113966 1113967


+ (38)


maxi12n himi1113966 1113967




1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References













































n


λ1113874 1113875

1λ 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mi 1 2 gi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (26)



1 minus 1113945

n

i11 minus himi

1113872 1113873ωi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭


1 minus 1113945n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

1θ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

(27)





1 minus 1113944n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus 1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (28)




1 +1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (29)





1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (30)



1 minus 1113945n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ 11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (31)




41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be





ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)



ω22


ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)



ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en


i1gωi


himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti


le hlowastimi


lowastn( 1113857 (37)


1113966 1113967


+ (38)


maxi12n himi1113966 1113967




1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References













































41 AC-HFWG


mi1 himi| i 1 2 n1113966 1113967 ωi be





ωn

n

(32)


mi1 himi| i 1 2 n1113966 1113967 ωi be



ςminus 11113944

n

i1ωiς himi

1113872 11138731113889 | mi 1 2 gi1113897⎛⎝⎧⎨

⎩ (33)



ω22


ςminus 1 ω1ς h1m11113872 1113873 + ω2ς h2m2

1113872 11138731113872 111387311138681113868111386811138681113868 m1 1 2 g1 m2 1 2 g21113882 1113883

(34)



ςminus 11113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (35)

en


i1gωi


himiisingi

ςminus 11113944

k+1

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (36)



mi1 himi| i 1 2 n1113966 1113967 ωi be




lowasti


le hlowastimi


lowastn( 1113857 (37)


1113966 1113967


+ (38)


maxi12n himi1113966 1113967




1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References












































1113936ni1 ωiς(h+)le 1113936

ni1 ωiς(himi





ni1 ωiς(himi

))le h+


mi1 himi| i 1 2 n1113966 1113967 ωi be




θg1( 1113857ω1 otimes θg2( 1113857


1θotimesni1

θgi( 1113857ωi1113888 1113889 (39)



mi1 himi| i 1 2 n1113966 1113967 ωi be





n


1113872 11138731113872 11138731113872 11138731113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭

(40)


mi1 himi| i 1 2 n1113966 1113967 ωi be



gn) h



le hlowastimi


lowastn( 1113857 (41)


1113966 1113967


+ (42)




eminus 1113936

n


κ1113872 1113873

1κ 111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi1113896 1113897 (43)




1113945

n

i1hωi

imi

111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭




1113872 1113873θ

1113874 1113875ωi

⎞⎠

1θ

⎛⎜⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩

(44)






1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References















































1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎨

⎩

⎫⎬

⎭ (46)




(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

1113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (47)




n


1113872 1113873κ

1113872 1113873ωi ⎞⎠

1κ

⎛⎝

11138681113868111386811138681113868111386811138681113868111386811138681113868mi 1 2 gi

⎫⎬

⎭⎧⎪⎨

⎪⎩(48)




HFWG(g1 g2 gn))




1113944

g1

m111113944

g2


gn

mn11 minus e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889



1113944

g1

m111113944

g2


gn

mn1e

minus 1113936n


κ1113872 1113873

1κ

1113888 1113889

(49)



ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References












































ni1 ωia


1113944

n


1113872 11138731113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n


1113872 11138731113872 1113873κ2⎛⎝ ⎞⎠

1κ2

1113944

n

i1ωi minus ln himi

1113872 1113873κ1⎛⎝ ⎞⎠

1κ1

le 1113944n

i1ωi minus ln himi

1113872 1113873κ2⎞⎠

1κ2

⎛⎝

(50)

Furthermore


n


κ11113872 1113873

1κ1


n


κ21113872 1113873

1κ2

eminus 1113936

n


κ11113872 1113873

1κ1

ge eminus 1113936

n


κ21113872 1113873

1κ2

(51)


eminus 1113936

n


κ1113872 1113873

1κ

+ eminus 1113936

n


κ1113872 1113873

1κ

le eminus 1113936

n


+ eminus 1113936

n

i1 ωi minus ln himi1113872 11138731113872 1113873

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(52)



n



HFWG(g1 g2 gn))





1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎞⎠⎞⎠⎛⎝⎛⎝

(53)







ki1 ωiς(himi



Lastly


1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References











































1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ 1113944n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

lt limκ⟶0

1113944

n

i1ωi 1 minus himi

1113872 1113873minus κ⎛⎝ ⎞⎠

minus (1κ)

+ limκ⟶0

1113944

n

i1ωih

minus κimi

⎛⎝ ⎞⎠

minus (1κ)

e1113936n


+ e1113936n

i1 ωiln himi1113872 1113873

1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imi

le 1113944n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(54)




1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(55)






1κln e

minus te


(56)






ki1 ωiς(himi




minus1κln 1113945

n

i1e


+ 1⎛⎝ ⎞⎠



ωi+ 11113872 1113873

minus κ lim

κ⟶minus prop



ωi+ 11113872 11138731113872 1113873 1113936

ni1 ωi minus himi


minus 1

limκ⟶minus prop


ωi


ωi+ 1

⎛⎝ ⎞⎠ 1113944

n

i1ωihimi

eminus κhimi


n

i1ωihimi

minus1κln 1113945

n

i1e


minus 11113888 1113889

ωi

+ 1⎛⎝ ⎞⎠le 1113944

n

i1ωi 1 minus himi

1113872 1113873

(57)





) + 1113936ni1 ωihimi

1






1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References













































1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873))⎛⎝⎛⎝

(58)









ki1 ωiς(himi




1113945

n

i11 + himi

1113872 1113873ωi

+ 1113945

n

i11 minus himi

1113872 1113873ωi ge 2 1113945

n

i11 + himi

1113872 1113873ωi

1113945

n

i11 minus himi

1113872 1113873ωi⎛⎝ ⎞⎠

12

2 1113945

n

i11 minus h

2imi

1113872 1113873ωi⎛⎝ ⎞⎠

12

ge 2

1113945

n

i12 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imige 2 1113945

n

i12 minus himi

1113872 1113873ωi

1113945

n

i1hωi

imi

⎛⎝ ⎞⎠

12

2 1113945n


1113872 11138732

1113874 1113875ωi

⎛⎝ ⎞⎠

12

ge 2

(59)



1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +(1 minus κ) 1113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

⎛⎝ ⎞⎠

le1113937

ni1 1 minus himi

1113872 1113873ωi


1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 11138731113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

21113937

ni1 1 minus himi

1113872 1113873ωi

1113937ni1 1 + himi

1113872 1113873ωi


1113872 1113873ωi

⎛⎝ ⎞⎠ +21113937

ni1 h

ωi

imi


1113872 1113873ωi

+ 1113937ni1 h

ωi

imi

⎛⎝ ⎞⎠

le1113945n

i11 minus himi

1113872 1113873ωi

+ 1113945n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(60)

So


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωi


1113872 1113873ωige

(1 minus κ) 1113937ni1 h

ωi

imi


1113872 11138731113872 1113873ωi


ωi

imi

(61)







1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References
















































1113944

g1

m111113944

g2


gn


1113944

n


1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠



1113944

g1

m111113944

g2


gn

mn1ςminus 1

1113944

k

i1ωiς himi

1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(62)









ni1 ωiς(himi



Because κge 1

1113945

n

i11 minus h

κimi

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945

n


1113872 1113873κ

1113872 1113873ωi⎛⎝ ⎞⎠

1κ

⎛⎝ ⎞⎠

le 1113945n

i11 minus h

1imi

1113872 1113873ωi⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠ + 1 minus 1 minus 1113945n


1113872 11138731

1113874 1113875ωi

⎛⎝ ⎞⎠

1

⎛⎝ ⎞⎠

1113945

n

i11 minus himi

1113872 1113873ωi

+ 1113945

n

i1hωi

imile 1113944

n

i1ωi 1 minus himi

1113872 1113873 + 1113944ωihimi 1113944

n

i1ωi 1

(63)



n

















j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References














































j1ωjgj cup

hjmjisingj


4j1 ωi 1minus hjmj

1113872 111387312

1113874 111387556 111386811138681113868111386811138681113868111386811138681113868111386811138681113868

mj 1 2 gj

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

05870 06139 06021 06278 06229 06470 07190 07361 07286 06117 06367 06257 06496 06450 06675

07347 07508 07437 06303 06538 06435 06660 06617 06829 07466 07618 07551 07451 07419 07574

07591 07561 07707 07697 07669 07808

(64)




















Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References
















































Υ1 02 04 07 02 06 08 02 03 06 07 09 03 04 05 07 08

Υ2 02 04 07 09 01 02 04 05 03 04 06 09 05 06 08 09

Υ3 03 05 06 07 02 04 05 06 03 05 07 08 02 05 06 07

Υ4 03 05 06 02 04 05 06 07 08 09




1 12 2 3 35 50

0102030405060708


Data 1Data 2

Data 3Data 4











1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References













































1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 2

(b)

Y1Y2

Y3Y4

0 2 4 6 8 1005

055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 3

(c)

Y1Y2

Y3Y4


055

06

065

07

075

08

085

09

Parameter kappa

Scor

e val

ues o

f Cas

e 4

(d)

Y1Y2

Y3Y4

1 2 3 4 5 6 7 8 9 1005

055

06

065

07

075

08

085

09

Parameters kappa

Scor

e val

ues o

f Cas

e 5

(e)










1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References


















































1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 1

Y1Y2

Y3Y4

(a)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 2

Y1Y2

Y3Y4

(b)

1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 3

Y1Y2

Y3Y4

(c)


03

035

04

045

05

055

Parameter kappa

Scor

e val

ues o

f Cas

e 4

Y1Y2

Y3Y4

(d)

Figure 3 Continued


1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References











































1 2 3 4 5 6 7 8 9025

03

035

04

045

05

055

Parameter kappaSc

ore v

alue

s of C

ase 5

Y1Y2

Y3Y4

(e)




11 05612 06009 05178 06524 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06317 06806 05722 07313 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06701 07236 06079 07745 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

51 06757 07356 06045 07893 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 06852 07442 06142 07972 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 06892 07479 06186 08005 Υ4 ≻Υ2 ≻Υ1 ≻Υ3

101 07081 07688 06386 08199 Υ4 ≻Υ2 ≻Υ1 ≻Υ35 07112 07712 06420 08219 Υ4 ≻Υ2 ≻Υ1 ≻Υ310 07126 07723 06435 08227 Υ4 ≻Υ2 ≻Υ1 ≻Υ3



11 04744 04625 04661 05131 Υ4 ≻Υ1 ≻Υ3 ≻Υ25 03962 03706 04171 04083 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03562 03262 03808 03607 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

51 03523 03222 03836 03636 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03417 03124 03746 03527 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03372 03083 03706 03481 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

101 03200 02870 03517 03266 Υ3 ≻Υ4 ≻Υ1 ≻Υ25 03165 02841 03484 03234 Υ3 ≻Υ4 ≻Υ1 ≻Υ210 03150 02829 03470 03220 Υ3 ≻Υ4 ≻Υ1 ≻Υ2

24

68 10

24

68

1005

05506

06507

07508


Scor

e val

ues o

f Y1

058

06

062

064

066

068

07

(a)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y2

062

064

066

068

07

072

074

076

(b)

Figure 4 Continued



46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References












































46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y3

052

054

056

058

06

062

064

(c)


46

810

24

68

1005

05506

06507

07508

Scor

e val

ues o

f Y4

066

068

07

072

074

076

078

08

082

(d)


24

68 10

24

68

10025

03035

04045

05055


Scor

e val

ues o

f Y1

032

034

036

038

04

042

044

046

(a)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y2

03

032

034

036

038

04

042

044

046

(b)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y3

035

04

045

(c)


46

8 10

24

68

10025

03035

04045

05055

Scor

e val

ues o

f Y4

034

036

038

04

042

044

046

048

05

(d)






6 Conclusions




Data Availability




Acknowledgments


References











































6 Conclusions




Data Availability




Acknowledgments


References
































































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