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PureMathematicsnØêÆ,2023,13(4),1040-1048
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.134109
ÄuSugenoÿÝŒ˜—ChoquetÈ©
A5
ooo|||___
1,2∗
,………ÖÖÖ
1,2
,"""ûûû
1,2
1
žh“‰ŒÆêƆÚOÆ,#õžw
2
žh“‰ŒÆA^êÆïĤ,#õžw
ÂvFϵ2023c320F¶¹^Fϵ2023c421F¶uÙFϵ2023c428F
Á‡
©3ÄuSugenoÿÝŒ˜—ChoquetÈ©Ä:þ,(ÜÄ uŒ˜—kS\²þŽ fA:,?Ø
ÄuSugenoÿÝŒ˜—ChoquetÈ©âÊŠ!ÄûÚUO•ê.
'…c
SugenoÿݧChoquetÈ©§âÊŠ§ÄûÚUO•ê
TheCharacteroftheChoquetIntegralof
Semi-UninormBasedonSugenoMeasures
QiaoxiaLi
1,2∗
,YuheYang
1,2
,ZhenXin
1,2
1
SchoolofMathematicsandStatistics,YiliNormalUniversity,Yining Xinjiang
2
InstituteofAppliedMathematics, YiliNormalUniversity,YiningXinjiang
Received:Mar.20
th
,2023;accepted:Apr.21
st
,2023;published:Apr.28
th
,2023
Abstract
Inthispap er,theShapley,thevetoandfavorindicesoftheChoquetIntegralofsemi-uninorm
basedonSugenomeasuresarediscussedandcombinethecharacteristicsofthethesemi-uninorm
orderedweightedaveragingoperators.
∗ÏÕŠö"
©ÙÚ^:o|_,…Ö,"û.ÄuSugenoÿÝŒ˜—ChoquetÈ©A5[J].nØêÆ,2023,13(4):
1040-1048.DOI:10.12677/pm.2023.134109
o|_
Keywords
UgenoMeasures,ChoquetIntegral,TheShapley,TheVetoandFavorIndices
Copyright
c
2023byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.có
àÜŽf´ˆ‡ ‰Æ+•rŒóä,ChoquetÈ©ÏÙÏ^5ˆüX-‡Ú.Š˜J´ChoquetÈ
©[1]í2ü‡ ͶŽfx,\²þ£WA)ŽfÚkS\²þ(OWA)Žf[2].¯¢þ,3&E8¤ž,\
²þ(WA)ŽfÚkS\²þ(OWA)Žf´ØÓ.\²þŽfŠâz‡&E-‡5(á5-)5\3
XÚ¥-‡5,kS\²þ(OWA)ŽfKŠâ§‚ƒé ˜-‡5( ˜-)5\Ù3XÚ¥-‡
5.üö3Åì<!Ü6››ì!å!÷¿Ý¯K!õOKà8¯K•¡[3–5]?1!ØÚïÄ.,
éõÆö3ØÓµeòOWAŽf3ØÓDŠ8¤XÚþ?˜ÚÿÐ!?ØÚïÄ[6,7].2015c,Llamazares
JÑÄuŒ˜—kS\²þŽf(SUOWAŽf)[8],ÏLŒ˜—òü|-•þéX å5,XÚïÄ
Q•Äá5-q•Ä ˜-8¤•{,¿A^uØ(½õá5ûü¥,ÙA:´3&E8¤Žf¥•)ü|
-•þ[9].1974c,FÆöSugenoí2²;VÇÿÝŒŒ\ 5,±å^‡füN5“²
;VÇ¥Œ\5^‡,JÑÿÝVg,ù•·‚Jø˜‡Uk£ãy¢¯K¥•3ƒp•6,ƒ
p'éy–óä[10,11].2019c,ý!o/ÏuSugenoÿÝÚní2˜—ÚŒ˜—½Â,JÑ
ÄuSugenoÿÝŒ˜—ChoquetÈ©±9†ƒƒ'éornessÿÝ½Â9Ù5Ÿ[12],¿éÙ?1Þ~`
².©éÄuSugenoÿÝŒ˜—ChoquetÈ©âÊŠ!ÄûÚUO•ê?1!؆ïÄ.
2.½Â9`²
X•š˜8Ü,A´dXf8¤σ−“ê,¡8¼êµ:A→[0,∞)•5ÿÝ(Sugenoÿ
Ý),´•[10,11]:
(1)µ(∅) = 0;
(2)µ(X) = 1;
(3)éuA,B∈A,eA⊆B,Kµ(A) ≤µ(B).
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
).
DOI:10.12677/pm.2023.1341091041nØêÆ
o|_
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[?]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[?]X•k•8,µ
(1)
,µ
(2)
´Xþü‡Sugeno ÿÝ,U∈
˜
U
1
n
.x= (x
1
,x
2
,···x
n
) ÄuSugeno
ÿÝŒ˜—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)
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]
.
DOI:10.12677/pm.2023.1341091042nØêÆ
o|_
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[?] 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
.
DOI:10.12677/pm.2023.1341091043nØêÆ
o|_
½½½ÂÂÂ2.3[9] X•k•8, µ´XþIONþ, j∈X.j'uµâÊŠ½Â•µ
φ(µ,j) =
X
T⊆X\j}
(n−t−1)!t!
n!
(µ(T∪{j})−µ(T)).
Ù,˜‡Lˆª•µ
φ(µ,j) =
1
n
n−1
X
t=0
1
n−1
t
!
X
T⊆X\{j}
|T|=t
(µ(T∪{j})−µ(T)).
½½½ÂÂÂ2.4[9] X•k•8, µ´XþIONþ,Kj∈X.j'uµÄûÚUOŠ½Â•µ
veto(C
µ
,j) = 1−
1
n−1
X
T⊆X\{j}
1
n−1
t
!
µ(T)
favor(C
µ
,j) =
1
n−1
X
T⊆X\{j}
1
n−1
t
!
µ(T∪{j})−
1
n−1
ØJOŽ,,˜«Lˆª•µ
veto(C
µ
,j) = 1−
1
n−1
n−1
X
t=0
1
n−1
t
!
µ(T)
favor(C
µ
,j) =
1
n−1
n−1
X
t=0
1
n−1
t
!
X
T⊆X\{j}
|T|=t
µ(T∪{j})−
1
n−1
.
ÚÚÚnnn2.1[9] X•k•8, µ´XþIONþ,Kk
veto(C
µ
,j)+favor(C
µ
,j) = 1+
nφ(µ,j)−1
n−1
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?¿j∈X,k
φ

v
U
T
L
µ
(1)
,µ
(2)
,j

= µ
(1)
(j),
veto

S
U
T
L
µ
(1)
,µ
(2)
,j

=
nµ
(1)
(j−1)
2(n−1)
+1−orness

S
U
T
L
µ
(1)
,µ
(2)

,
favor

S
U
T
L
µ
(1)
,µ
(2)
,j

=
nµ
(1)
(j−1)
2(n−1)
+orness

S
U
T
L
µ
(1)
,µ
(2)

.
DOI:10.12677/pm.2023.1341091044nØêÆ
o|_
y²Šâ½Â2.3,Œ
φ

v
U
T
L
µ
(1)
,µ
(2)
,j

=
1
n
n−1
X
t=0
1
n−1
t
!
X
T⊆X\{j}
|T|=t

µ
(1)
(j)+µ
(2)
(t+1)−
1
n

=
1
n
n−1
X
t=0

µ
(1)
(j)+µ
(2)
(t+1)−
1
n

= µ
(1)
(j).
?˜Ú,k½Â2.4,ŒÄûÚUOŠ•µ
veto

S
U
T
L
µ
(1)
,µ
(2)
,j

= 1−
1
n−1
n−1
X
t=1
1
n−1
t
!
X
T⊆X\{j}
|T|=t
v
U
T
L
µ
(1)
,µ
(2)
(T)
= 1−
1
n−1

1−µ
(1)
(j)
n−1
−
1
n

n−1
X
t=1
t+
n−1
X
t=1
t
X
i=1
µ
(2)
(i)
!
= 1−
1−nµ
(1)
(j)
2(n−1)
−orness

O
(2)
µ

=
nµ
(1)
(j−1)
2(n−1)
+1−orness

S
U
T
L
µ
(1)
,µ
(2)

.
favor

S
U
T
L
µ
(1)
,µ
(2)
,j

= 1−veto

S
U
T
L
µ
(1)
,µ
(2)
,j

+
nφ

v
U
T
L
µ
(1)
,µ
(2)
,j

−1
n−1
= −
nµ
(1)
(j−1)
2(n−1)
+orness

S
U
T
L
µ
(1)
,µ
(2)

+
nµ
(1)
(j−1)
n−1
=
nµ
(1)
(j−1)
2(n−1)
+orness

S
U
T
L
µ
(1)
,µ
(2)

.
½½½nnn2.2X•k•8,µ
(1)
,µ
(2)
´Xþü‡SugenoÿÝ,éu¤kj∈X,
P
j
i=1
µ
(2)
≤j/n.e
v
U
¯p
µ
(1)
,µ
(2)
´XþIONþ,Kéuj∈X,k
φ

v
U
˜p
µ
(1)
,µ
(2)
,j

=
1
n−1
1−µ
(1)
(j)+

nµ
(1)
(j−1)

n
X
i=1
n
X
t=i
1
t
!
µ
(2)
(i)
!
,
veto

S
U
˜w
µ
(1)
,µ
(2)
,j

= 1−
n
n−1

1−µ
(1)
(j)

orness

S
U
˜
~p
µ
(1)
,µ
(2)

,
favor

S
U
ep
µ
(1)
,µ
(2)
,j

=1−veto

S
U
˜p
p,w
,j

+
n
(n−1)
2
·
1−µ
(1)
(j)+

nµ
(1)
(j)−1

n
X
i=1
n
X
t=i
1
t
!
µ
(2)
(i)
!
−
1
n−1
.
DOI:10.12677/pm.2023.1341091045nØêÆ
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yyy²²²Šâ½Â2.4Ú52.5,éuj∈X,·‚Œ
veto

S
U
ep
µ
(1)
,µ
(2)
,j

= 1−
1
n−1
n−1
X
t=1
1
n−1
t
!
X
T⊆X\|{j}
|T|=t
v
U
ep
µ
(1)
,µ
(2)
(T)
= 1−
1
n−1
n
n−1

1−µ
(1)
(j)

n−1
X
t=1
t
X
i=1
µ
(2)
(i)
= 1−

1−µ
(1)
(j)

n
n−1
orness(O
w
)
= 1−

1−µ
(1)
(j)

n
n−1
orness

S
U
˜
P
µ
(1)
,µ
(2)

.
Ón,éut≥1,Œ
X
T⊆X\{j}
|T|=t
v
U
¯
P
µ
(1)
,µ
(2)
(T∪{j}) =
X
T⊆X\{j}
T|=t
n
t+1
X
i∈T
µ
(1)
(i)+µ
(1)
(j)
!
t+1
X
i=1
µ
(2)
(i)
!
=
n
t+1
t+1
X
i=1
µ
(1)
(i)
!
n−1
t
!


t

1−µ
(1)
(j)

n−1
+µ
(1)
(j)


.
5¿,t= 0ž,±e(Ø•¤á,
X
T⊆X\{j}
|T|=0
v
U
¯
P
µ
(1)
,µ
(2)
(T∪{j}) = v
U
¯p
µ
(1)
,µ
(2)
({j}) = nµ
(1)
(j)µ
(2)
(1).
Ïd,Šâ½Â2.4,Œ
favor

S
U
¯p
µ
(1)
,µ
(2)
,j

=
1
n−1
n−1
X
t=0
n
t+1
t+1
X
i=1
µ
(2)
(i)
!


t

1−µ
(1)
(j)

n−1
+µ
(1)
(j)


−
1
n−1
=
n
(n−1)
2
n−1
X
t=0
(n−1−t)µ
(1)
(j)+t
t+1
t+1
X
i=1
µ
(2)
(i)
!
−
1
n−1
=
n
(n−1)
2
n
X
t=1
(n−t)µ
(1)
(j)+t−1
t
t
X
i=1
µ
(2)
(i)
!
−
1
n−1
=
n
(n−1)
2
n
X
i=1
n
X
t=i
(n−t)µ
(1)
(j)+t−1
t
!
µ
(2)
(i)−
1
n−1
.
?˜Ú
DOI:10.12677/pm.2023.1341091046nØêÆ
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n
X
i=1
n
X
t=i
(n−t)µ
(1)
(j)+t−1
t
!
µ
(2)
(i)
=
n
X
i=1
n
X
t=i


1−µ
(1)
(j)

+
nµ
(1)
(j−1)
t

!
µ
(2)
(i)
=
n
X
i=1

1−µ
(1)
(j)

(n−i+1)+

nµ
(1)
(j−1)

n
X
t=i
1
t
!
µ
(2)
(i)
=

1−µ
(1)
(j)

n
X
i=1
(n−i)µ
(1)
(i)+1
!
+(np
j
−1)
n
X
i=1
n
X
t=i
1
t
!
µ
(2)
(i),
dorness(O
ω
)9orness

S
U
˜p
µ
(1)
,µ
(2)

,Œ
favor

S
U
¯p
µ
(1)
,µ
(2)
,j

=
n
n−1

1−µ
(1)
(j)

orness

S
U
¯p
µ
(1)
,µ
(2)
,j

+
n
(n−1)
2
·
1−µ
(1)
(j)+

nµ
(1)
(j)−1

n
X
i=1
n
X
t=i
1
t
!
µ
(2)
(i)
!
−
1
n−1
=1−veto

S
U
¯
P
µ
(1)
,µ
(2)
,j

+
n
(n−1)
2
·
1−µ
(1)
(j)+

nµ
(1)
(j)−1

n
X
i=1
n
X
t=i
1
t
!
µ
(2)
(i)
!
−
1
n−1
.
ŠâÚn2.1,Œ
φ

v
U
¯p
µ
(1)
,µ
(2)
,j

=
1
n

(n−1)

veto

S
U
µ
(1)
,µ
(2)
,j

+favor

S
U
µ
(1)
,µ
(2)
,j

−1

+1

=
1
n−1
1−µ
(1)
(j)+

nµ
(1)
(j)−1

n
X
i=1
n
X
t=i
1
t
!
µ
(2)
(i)
!
.
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[1]Grabisch,M. (1995)FuzzyIntegral inMulticriteria DecisionMaking. FuzzySetsandSystems, 69,279-298.
https://doi.org/10.1016/0165-0114(94)00174-6
[2]Yager,R.R. (1988)On OrderedWeighted Averaging AggregationOperatorsin MuiticriteriaDecision Mak-
ing.IEEETransactionsonSystems,Man,andCybernetics,18,183-190.https://doi.org/10.1109/21.87068
DOI:10.12677/pm.2023.1341091047nØêÆ
o|_
[3]Yager,R.R.(1993)FamiliesofOWAOperators.FuzzySetsandSystems,59,125-148.
https://doi.org/10.1016/0165-0114(93)90194-M
[4]Dujmovic,J.J.(2008)ContinuousPreferenceLogicforSystemEvaluation.IEEETransactionsonFuzzy
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[5]Jin,L.S.(2015)Some PropertiesandRepresentationMethodsforOrdered Weighted Averaging Operators.
FuzzySetsandSystems,261,60-86.https://doi.org/10.1016/j.fss.2014.04.019
[6]Yager,R.R.(2004)OWAAggregationoveraContinuousIntervalArgumentwithApplicationstoDecision
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[7]Li,D.F.(2011)TheGOWAOperatorBasedApproachto MultiattributeDecisionMakingUsing Intuition-
isticFuzzySets.MathematicalandComputerModelling,53,1182-1196.
https://doi.org/10.1016/j.mcm.2010.11.088
[8]Llamazares,B.(2016)SUOWAOperators:ConstructingSemi-UninormsandAnalyzingSpecificCases.
FuzzySetsandSystems,278,119-136.https://doi.org/10.1016/j.fss.2015.02.017
[9]Llamazares, B.(2018) Closed-FormExpressions forSomeIndicesofSUOWAOperators.InformationFusion,
41,80-90.https://doi.org/10.1016/j.inffus.2017.08.010
[10]Çl,ê².©ÛÆÄ:[M].®:I“ó’ч,1991.
[11]Grabisch, M., Murofushi, T. and Sugeno, M.(2000) Fuzzy Measures andIntegrals:Theory and Application.
Physica-Verlag,Heidelberg.
[12]ýO,o|_.ÄuSugenoÿÝŒ˜—kS\²þŽf9Ù488¤ìO[J].XÚ†êÆ,
2019,33(6):11-28.
DOI:10.12677/pm.2023.1341091048nØêÆ

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