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PureMathematicsnØêÆ,2023,13(4),1135-1141
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.134118
ÄuSugenoÿÝŒ˜—ChoquetÈ©
ooo|||___
1,2∗
§§§MMM€€€€€€
1,2
§§§"""ûûû
1,2
§§§………ÖÖÖ
1,2
1
žh“‰ŒÆêƆÚOÆ§#õžw
2
žh“‰ŒÆA^êÆïĤ§#õžw
ÂvFϵ2023c320F¶¹^Fϵ2023c421F¶uÙFϵ2023c428F
Á‡
ÄuŒ˜—kS\²þ(SUOWA)Žf´\²þ(WA)ŽfÚkS\²þ(OWA)Žfí2,©
3ÄuSugenoÿÝŒ˜—ChoquetÈ©Ä:þ,(ÜÄuŒ˜— kS \²þŽfA:,?ØÄ
uSugenoÿÝŒ˜—ChoquetÈ©ƒ'5Ÿ,léõêâ?1•Ð/8¤"
'…c
SugenoÿݧChoquetÈ©§SUOWAŽf
TheChoquetIntegralofSemi-Uninorm
BasedonSugenoMeasures
QiaoxiaLi
1,2∗
,SusuXu
1,2
,ZhenXin
1,2
,YuheYang
1,2
1
SchoolofMathematicsandStatistics,YiliNormalUniversity,Yining Xinjiang
2
InstituteofAppliedMathematics, YiliNormalUniversity,Yining Xinjiang
Received:Mar.20
th
,2023;accepted:Apr.21
st
,2023;published:Apr.28
th
,2023
Abstract
Thesemi-uninormorderedweightedaveraging(SOWA)operatorsisgeneralizeofweightedmeans
andOWAoperators.Inthispaper,theChoquetIntegralofsemi-uninormbasedonSugeno
∗ÏÕŠö"
©ÙÚ^:o|_,M€€,"û,…Ö.ÄuSugenoÿÝŒ˜—ChoquetÈ©[J].nØêÆ,2023,13(4):
1135-1141.DOI:10.12677/pm.2023.134118
o|_
measuresarediscussedandcombinethecharacteristicsofthethesemi-uninormorderedweighted
averagingoperators.
Keywords
SugenoMasures,ChoquetIntegral,SUOWAOperators
Copyright
c
2023byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.có
àÜ$ŽÎ´àÜõ‡‰Æ+•ŠrŒóä.3yk ŒþŽf¥,ChoquetÈ©ÏÙÏ^5ˆüX-‡
Ú[1].Š˜J´ChoquetÈ©í2ü‡Í¶Žf,\þŠŽfÚkS\²þ(OWA)Žf[2]§§
‚3©z¥ª„¦^.¯¢þ, 3&E8¤ž,\²þ(WA) ŽfÚkS\²þ(OWA)Žf´ØÓ.\²
þŽfŠâz‡&E-‡5(á5-)5\3XÚ¥-‡5,kS\²þ(OWA)ŽfKŠâ§‚ƒ
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fÚOWAŽf?˜ÚÿÐ!?ØÚïÄ[6].2015c,LlamazaresJÑÄuŒ˜—kS\²þ(SUOWA)
Žf.XÚïÄQ•Äá5-q•Ä ˜-8¤•{,¿A^uØ(½õá5ûü¥,ÙA:´3&E8¤
Žf¥•)ü|-•þ[7].1974c,FÆöSugenoí2²;VÇÿÝŒŒ\5,±å^‡f
üN5“²;V Ç¥Œ\5^‡,JÑÿÝVg,ù•·‚Jø˜‡Uk£ãy¢¯K¥•
3ƒp•6,ƒp'éy–óä[8–11].8¤ŽfnØ,˜‡-‡SN´8 (&E-(½¯K&E,-
Nyûüö‰Ñûü&E-‡5§Ý,ÙŒ†'X•ªûüJÚ8((J.XÛòü|ÿÝ(Üå
5,éõêâ?1•Ð/8¤(XQ•Äá5-q•Ä ˜-8¤•{),Š•ní2˜—½Œ˜
—´²~•Ä.2019c,ý!o/ÏuSugenoÿÝÚní2˜—ÚŒ˜—½Â[12],JÑÄ
uSugenoÿÝŒ˜—ChoquetÈ©±9†ƒƒ'éornessÿÝ½Â9Ù5Ÿ[13],¿éÙ?1Þ~`
².©1˜Ü©Â8ÿÝ!Œ˜—½Â;1Ü©ŠâÄuSugenoÿÝŒ˜—ChoquetÈ©
½Â9Ù¦5Ÿ,éÄuSugenoÿÝŒ˜—ChoquetÈ©Ù¦5Ÿ?˜Ú!ØÚïÄ.
2.½Â9`²
X•š˜8Ü,A´dXf8 ¤σ−“ê,¡8¼êµ:A→[0,∞)•5ÿÝ(Sugenoÿ
Ý),´•[8–11]:
(1)µ(∅) = 0;
(2)µ(X) = 1;
(3)éuA,B∈A,eA⊆B,Kµ(A) ≤µ(B).
DOI:10.12677/pm.2023.1341181136nØêÆ
o|_
½½½ÂÂÂ1.1X•k•8,µ´AþSugenoÿÝ.½Âx=(x
1
,x
2
,···,x
n
)3XþÄuÿݵ
Choquet驥
(C)
Z
A
xdµ=
n
X
i=1
x
i
(µ(A
i
)−µ(A
i+1
)).
ØJOŽ
(C)
Z
A
xdµ=
n
X
i=1
(x
i
−x
i+1
)µ(A
i
).
Ù¥,X= (1,2,···,n),A⊆X,A
i
= {i,i+1,···,n},i= 1,2,···,n,A
n+1
= ∅,x
1
≥x
2
≥···≥x
n
.
½½½ÂÂÂ1.2[12] ¼êU:[0,1]
2
→[0,1].¡U´˜‡Œ˜—,´•Uéz‡CþäküN5…•3ü
e∈[0,1],=é?¿x∈[0,1],•3e∈[0,1],kU(e,x) = U(x,e) = x.
±e•ü‡ëYŒ˜—µ
~1.1.
U
¯
P
(x,y) =



max(x,y)if(x,y) ∈[1/n,1]
2
,
nxyotherwise.
U
T
L
(x,y) =



max(x,y)if(x,y) ∈(1/n,1]
2
,
max(x+y−1/n,0)otherwise,
duU
¯
P
ÚU
T
L
Ñ´ëY,Kþ¡ü‡Œ˜—Œ±¤µ
U
T
L
(x,y) =



max(x,y)if(x,y) ∈(1/n,1]
2
,
max(x+y−1/n,0)otherwise,
U
¯p
(x,y) =



max(x,y)if(x,y) ∈(1/n,1]
2
,
nxyotherwise.
½½½ÂÂÂ1.3[13]X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,Œ˜—U∈
˜
U
1
n
.éuA⊆X,½Â
υ
U
µ
(1)
,µ
(2)
: 2
X
→R•
υ
U
µ
(1)
,µ
(2)
(A) = |A|U

µ
(1)
(A)
|A|
,
µ
(2)
(A)
|A|

.
eA=∅,Pυ
U
µ
(1)
,µ
(2)
(∅) =0.|A|L«8ÜAÄê.Ù ¥,
˜
U
1
n
= {U∈U
1
n
|U(
1
k
,
1
k
) ≤
1
k
,∀k∈X}.w,/,
U
1
n
i
⊆
˜
U
1
n
.
N´OŽ:
(1)υ
U
µ
(1)
,µ
(2)
(∅) = 0;
(2)υ
U
µ
(1)
,µ
(2)
(X) = |X|U(
µ
(1)
(X)
|X|
,
µ
(2)
(X)
|X|
) = 1;
(3)υ
U
µ
(1)
,µ
(2)
üNCXˆυ
U
µ
(1)
,µ
(2)
(A) = max
B⊆A
υ
U
µ
(1)
,µ
(2)
(B)÷vüN5.
3.ÄuSugenoÿÝŒ˜—ChoquetÈ©9A5
½½½ÂÂÂ2.1[13]X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,U∈
˜
U
1
n
.x=(x
1
,x
2
,···x
n
)Äu
SugenoÿÝŒ˜—ChoquetÈ©C
U
µ
(1)
,µ
(2)
: R
n
→R½Â•
C
U
µ
(1)
,µ
(2)
(x) =
n
X
i=1
(ˆυ
U
µ
(1)
,µ
(2)
(A
[i]
)−ˆυ
U
µ
(1)
,µ
(2)
(A
[i−1]
))x
[i]
,(1)
DOI:10.12677/pm.2023.1341181137nØêÆ
o|_
Ù¥A
[i]
= {[1],[2],···,[i]},A
[0]
= ∅,x
[i]
(i= 1,2,···,n)L«x¥1i‡Œê,=x
[1]
≥x
[2]
≥···≥x
[n]
.
ØJOŽ
C
U
µ
(1)
,µ
(2)
(x) =
n
X
i=1
ˆυ
U
µ
(1)
,µ
(2)
(A
[i]
)(x
[i]
−x
[i+1]
).
R
n
þ˜‡\•þq´•q= (q
1
,q
2
,···,q
n
) ∈[0,1]
n
…
n
P
i=1
q
i
= 1.WL«R
n
þ¤k\•þ8Ü.
52.1d(1)ªŒ•,p=(p
1
,p
2
,···,p
n
),ω=(ω
1
,ω
2
,···,ω
n
)´ü‡-•þ,µ
(1)
(A)=
P
i∈A
p
i
,µ
(2)
(A)=
|A|
P
i=1
ω
i
ž,C
U
µ
(1)
,µ
(2)
(x)òz•Äu-•þp,ωŒ˜—kS\²þ(SUOWA)Žf
S
U
p,ω
:
S
U
p,ω
(x) =
n
X
i=1
(ˆυ
U
p,ω
(A
[i]
)−ˆυ
U
p,ω
(A
[i−1]
))x
[i]
.
52.2d(1)ªŒ•,p= (p
1
,p
2
,···,p
n
),η= (
1
n
,
1
n
,···,
1
n
)´ü‡-•þ,µ
(1)
(A) =
P
i∈A
p
i
,µ
(2)
(A) =
|A|/nž,C
U
µ
(1)
,µ
(2)
(x)òz•Äu-•þp\²þŽfM
p
:
M
p
(x) =
n
X
i=1
p
i
x
i
.
52.3d(1)ªŒ•,η= (
1
n
,
1
n
,···,
1
n
),ω= (ω
1
,ω
2
,···,ω
n
)´ü‡-•þ,µ
(1)
(A) = |A|/n,µ
(2)
(A) =
|A|
P
i=1
ω
i
ž,C
U
µ
(1)
,µ
(2)
(x)òz•ÄuωkS\²þŽf(OWAŽf)O
ω
:
O
ω
(x) =
n
X
i=1
ω
i
x
[i]
.
½½½ÂÂÂ2.2[13] X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,U∈
˜
U
1
n
.ÄuŒ˜—SugenoÿÝ
ChoquetÈ©ornessÿÝorness(C
U
µ
(1)
,µ
(2)
)½Â•
orness(C
U
µ
(1)
,µ
(2)
) =
1
n−1
n−1
X
t=1
1

n
t

X
T⊆X|T|=t
υ
U
µ
(1)
,µ
(2)
(T).(2)
5552.4d(2)ªŒ•, p= (p
1
,p
2
,···,p
n
),ω= (ω
1
,ω
2
,···,ω
n
)´ü‡-•þ,µ
(1)
(A) =
P
i∈A
p
i
,µ
(2)
(A) =
P
|A|
i=1
ω
i
ž, orness(C
U
µ
(1)
,µ
(2)
) òzÄu-•þp,ωŒ˜—kS\²þŽf(SUOWA Žf)ornessÿ
Ýorness(S
U
p,ω
) :
orness(S
U
p,ω
) =
1
n−1
n−1
X
t=1
1

n
t

X
T⊆X|T|=t
υ
U
p,ω
(T).
5552.5d(2)ªŒ•, η= (
1
n
,
1
n
,···,
1
n
),ω= (ω
1
,ω
2
,···,ω
n
)´ü‡-•þ, µ
(1)
(A) =
|A|
n
,µ
(2)
(A) =
P
|A|
i=1
ω
i
ž,orness(C
U
µ
(1)
,µ
(2)
)òz•ÄuωkS\²þŽf(OWAŽf)ornessÿÝorness(O
ω
) :
orness(O
ω
) =
1
n−1
n−1
X
i=1
(n−i)ω
i
.
½½½nnn2.1X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,éu¤kj∈X,e
P
j
i=1
µ
(2)
≤j/n…
min
i∈X
µ
(1)
+min
i∈X
µ
(2)
≥1/n,Kéu?¿T⊆X,¦|T|= t≥1,k
DOI:10.12677/pm.2023.1341181138nØêÆ
o|_
v
U
T
L
µ
(1)
,µ
(2)
(T) =
X
i∈T
µ
(1)
(i)+
t
X
i=1
µ
(2)
(i)−
t
n
½½½nnn2.2X•k•8, µ
(1)
´Xþ˜‡SugenoÿÝ,et≥1,K
X
T⊆X
|T|=t
X
i∈T
µ
(1)
(i) =
n−1
t−1
!
n
X
i=1
µ
(1)
(i) =
n−1
t−1
!
=
n
t
!
t
n
?˜Ú,éuj∈X,
X
T⊆X\{j}
|T|=t
X
i∈T
µ
(1)
(i) =
n−2
t−1
!
n
X
i=1
i6=j
µ
(1)
(i) =
n−2
t−1
!

1−µ
(1)
(j)

=
n−1
t
!
t

1−µ
(1)
(j)

n−1
.
½½½nnn2.3X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,éu¤kj∈X,e
P
j
i=1
µ
(2)
≤j/n…
min
i∈X
µ
(1)
+min
i∈X
µ
(2)
≥1/n,Kéut≥1,k
X
T⊆X
|T|=t
v
U
T
L
µ
(1)
,µ
(2)
(T) =
n
t
!
t
X
i=1
µ
(2)
(i)
?˜Ú,éuj∈X,k
X
T⊆X\{j}
|T|=t
v
U
T
L
µ
(1)
,µ
(2)
(T) =
n−1
t
!

1−µ
(1)
(j)
n−1
−
1
n

t+
t
X
i=1
µ
(2)
(i)
!
.
y²Šâ½n2.1Ú½n2.2§Œ
X
T⊆X
|T|=t
v
U
T
L
µ
(1)
,µ
(2)
(T) =
X
T⊆X
|T|=t
X
i∈T
µ
(1)
(i)+
X
T⊆X
|T|=t
t
X
i=1
µ
(2)
(i)−
X
T⊆X
|T||=t
t
n
=
n
t
!
t
n
+
n
t
!
t
X
i=1
µ
(2)
(i)−
n
t
!
t
n
=
n
t
!
t
X
i=1
µ
(2)
(i),
?˜Ú,Œ
X
T⊆X\{j}
|T|=t
v
U
T
L
µ
(1)
,µ
(2)
(T) =
X
T⊆X\{j}
T|=t
X
i∈T
µ
(1)
(i)+
X
T⊆X\{j}
|T|=t
t
X
i=1
µ
(2)
(i)−
X
T⊆X\{j}
T|=t
t
n
=
n−1
t
!

1−µ
(1)
(j)
n−1
−
1
n

t+
t
X
i=1
µ
(2)
(i)
!
.
DOI:10.12677/pm.2023.1341181139nØêÆ
o|_
½½½nnn2.4X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,éu¤kj∈X,e
P
j
i=1
µ
(2)
≤j/n…
min
i∈X
µ
(1)
+min
i∈X
µ
(2)
≥1/n,Kéut≥1,k
orness(S
U
T
L
µ
(1)
,µ
(2)
) = orness(O
w
) ≤0.5
y²Šâ½Â2.2†52.5,Œ
orness

S
U
T
L
µ
(1)
,µ
(2)

=
1
n−1
n−1
X
t=1
1
n
t
!
X
T∈N
|T|=t
v
U
I
1
µ
(1)
,µ
(2)
(T)
=
1
n−1
n−1
X
t=1
t
X
i=1
w
i
= orness(O
w
) ≤0.5,
½½½nnn2.5X •k•8,µ
(1)
,µ
(2)
´X þü‡SugenoÿÝ,éu¤kj ∈X,k
P
j
i=1
µ
(2)
≤
j/n…
P
j
i=1
µ
(2)
/j≤j/n,eT⊆X…|T|= t≥1,K
v
U
¯p
µ
(1)
,µ
(2)
(T) = tU
¯
P
P
i∈T
µ
(1)
(i)
t
,
P
t
i=1
µ
(2)
(i)
t
!
=
n
t
X
i∈T
µ
(1)
(i)
!
t
X
i=1
µ
(2)
(i)
!
.
½½½nnn2.6X•k•8,µ
(1)
´Xþ˜‡SugenoÿÝ,éu¤kj∈X,
P
j
i=1
µ
(2)
≤j/n.et≥1,K
X
T⊆X
|T|=t
v
U
˜
P
µ
(1)
,µ
(2)
(T) =
n
t
!
t
X
i=1
µ
(2)
(i),
?˜Ú§éuj∈X,k
X
T⊆X\{j}
|T|=t
v
U
˜
P
µ
(1)
,µ
(2)
(T) =
n
n−1

1−µ
(1)
(j)

n−1
t
!
t
X
i=1
µ
(2)
(i).
y²Šâ½n2.2†½n2.5,Œ
X
T⊆X
|T|=t
v
U
˜
P
µ
(1)
,µ
(2)
(T) =
n
t
t
X
i=1
µ
(2)
(i)
!
X
T⊆X
|T|=t
X
i∈T
µ
(1)
(i)
=
n
t
!
t
X
i=1
µ
(2)
(i).
l
X
T⊆X\{j}
|T|=t
v
U
~
P
µ
(1)
,µ
(2)
(T) =
n
t
t
X
i=1
µ
(2)
(i)
!
X
T⊆X\{j}
|T|=t
X
i∈T
µ
(1)
(i)
=
n
n−1

1−µ
(1)
(j)

n−1
t
!
t
X
i=1
µ
(2)
(i).
DOI:10.12677/pm.2023.1341181140nØêÆ
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ë•©z
[1]Grabisch,M.(1995)FuzzyIntegralinMulticriteriaDecisionMaking.FuzzySetsandSystems,69,279-298.
https://doi.org/10.1016/0165-0114(94)00174-6
[2]Yager,R.R.(1988) OnOrdered WeightedAveragingAggregation OperatorsinMuiticriteria DecisionMak-
ing.IEEETransactionsonSystems,Man,andCybernetics,18,183-190.https://doi.org/10.1109/21.87068
[3]Torra,V.(1996)WeightedOWAOperatorsforSynthesisofInformation.ProceedingsoftheFifthIEEE
InternationalFuzzySystems,2,966-971.https://doi.org/10.1109/FUZZY.1996.552309
[4]Yager,R.R.(1993)FamiliesofOWAOperators.FuzzySetsandSystems,59,125-148.
https://doi.org/10.1016/0165-0114(93)90194-M
[5]Llamazares,B.(2015)ConstructingChoquetIntegral-BasedOperatorsThatGeneralizeWeightedMeans
andOWAOperators.InformationFusion,23,131-138.https://doi.org/10.1016/j.inffus.2014.06.003
[6]ýO,o¦x.ÄušŒ\ÿÝOWAŽf488¤9Ù 8¤ìO[J].ñÜ“‰ŒÆÆ,g,‰Æ‡,
2018,46(3):48-54.
[7]Llamazares,B.(2016)SUOWAOperators:ConstructingSemi-UninormsandAnalyzingSpecificCases.
FuzzySetsandSystems,278,119-136.https://doi.org/10.1016/j.fss.2015.02.017
[8]Çl,ê².©ÛÆÄ:[M].®:I“ó’ч,1991.
[9]Grabisch, M., Murofushi,T. and Sugeno, M. (2000)Fuzzy Measuresand Integrals:Theory and Application.
Physica-Verlag,Heidelberg.
[10]Llamazares,B.(2020)OntheRelationshipbetweentheCrescentMethodandSUOWAOperators.IEEE
TransactionsonFuzzySystems,28,2645-2650.https://doi.org/10.1109/TFUZZ.2019.2934937
[11]M²m,Çl.ÿ݆È©nØ[M].®:‰Æч,1998.
[12]Liu,H.W.(2012)Semi-UninormsandImplicationsonaCompleteLattice.FuzzySetsandSystems,191,
72-82.https://doi.org/10.1016/j.fss.2011.08.010
[13]ýO,o|_.ÄuSugenoÿÝŒ˜—kS\²þŽf9Ù488¤ìO[J].XÚ†êÆ,
2019,33(6):11-28.
DOI:10.12677/pm.2023.1341181141nØêÆ

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