Ашихмин И.В. МЕТОД КОМБИНИРОВАНИЯ ПАРНЫХ

advertisement
05.13.01
“
,
”
-
- 2006
!
"
!
#
$
%
,
c
"
&
-
$
'
$
"
!
( '
26
"
"
, 117312,
"
)
!
. 60-
2006 ".
11
&.002.086.02
:
%
, 9.
*
!
.
2006 ".
+
"
&.002.086.02
. .#
*
)'
,
"
–
,
*
.
. # *
" ,
.&
'
,#!
,
"
"
.
.
"
"
)
,
)
:
.
–
"
)
%
,
*
!
"
,
"
*
,
,
%
)'
*
.
"
"
-
-
,
.
"
.
*
*
" ,
,
)
)
,
)
*
'
,.(
&*.
0.
"
, !.,..
, #.1
, %. .,
, .&. "
, ..2
"
,
, $./.
. .#
" .
)
*
( ##!)
'
. .
,
, %.
"
,
" , /.!
, $.! ,
.
, . .
,#!
)
,
. %
"
,#!.
,
" )'
. 0
*
1
,
"
"
,#!
)' "
4
, -
"
3
,
,#!.
)
"
"
"
"
"
,#!,
- "
##!.
&
)'
•
*
:
'
)'
.
"
•
#
"
,#!.
"
,#!.
•
!
•
!
,#!
*
!
•
,
"
,
"
.
"
"
"
*
*
,#!.
•
!
,#!
3
.
•
!
##!
.
"
,
"
,
,
"
"
"
*
,
"
"
"
,
2
,
,
.
.% #
1.
#
(.%
),
*
"
2.
3.
4.
"
,#!.
5. #
" /
6. #
*
,#!.
.
"
,#!.
-
"
,#!
.
3
,#!
,
.
*
7.
.% #
)'
UniComBOS,
.
1. !
"
"
"
,#!.
2.
,#!
. &
* '
,
,#!.
3. !
* )' "
-
,
4. !
'
5. #
)
,#!
"
*
,#!
)'
,
"
,#!,
)
"
.
'
.
"
"
*
.
3
.% #
2001-2003 "
##! UniComBOS
«#
+
". 5
UniComBOS
0
) ,,
,
.
.
"
/
,
"
!
2.21
"
«
(2001-2005
»
, 1
,
"
!
»
"
),
% 0
1.2
!
"
00290 !
"
#
!
61964.2003.1.
"
«1
»,
1
01-01-00514, 04-01, "
*
'
!
*
*
"
'
« ##!
#
, 2003; 58«#
'
"
2003; 17*
)
*
7
(MCDM
2004),
( %
2004)»,
» (AIS’05), &
*
«
#
*
7-
*
"
*
» (ISDSS’03), +
,
!
/
»,
,!
,
"
, .
, 2004;
«
, +
, 2004;
«
0-2005), #
» (
4
,!
, 2005;
-(
,
!
!
, 2005;
;
+
1
". 5
,
.
.
#
11
!
'
,
.
.
,
,
' "
2.3
3
3.1
.
!
&
,
(65
*
,
*
.
)
17
5
" ,
.!
)
*
"
136
*
)'
)
,
.%
. %
1
,
* )
*
,
,
.
)
#
*
*
"
*
"
'
. !
"
.
)'
,
.
0
"
,
"
)
"
,
"
*
"
.%
)
5
'
,
,#!.
)
3
.
,
"
#
)
.
" ,
“
*
-
)
,
3
,#!,
”
,#!,
.
)
-
,#!
,#!,
)
“
" )
)
3
)'
,
.
”, “ * ”
,#!. %
(“
”). #
“'
,
, . .
”
)
"
)'
),
)
)
, -
. .
.
3
,#!,
"
)'
,
.
"
)' "
)' "
,
"
.
"
.% #
#
"
(.%
"
),
"
"
(
"
"
*
,
3
,
,
,
).
.
"
,
(
. 8
*
)
(
"
"
"
,
A.
.
,
6
)
,#!
)'
. %
"
C = {C 1 ,...,C k }
, C (a ) –
j
K = {1,..,k }
{
) Cj,
a∈A
*
–
*
6
.
}
S j = s1j ,s2j ,...,smj j , j ∈ K ,
*
)
,
(
*
)
,
C j (a ) . #
Sj =
):
-
a∈A
"
.
*
)
k-
k
S = ∏S j ,
j =1
a∈ A
.. *
*)
(
C (a )
A
*
A. %
#
*
(C ( a ) ,C ( a ) ,...,C ( a ) ) ,
1
2
k
1
j
'
k
C ,...,C . %
,
A⊆S.
A,
A.
Sj
C,
0
*
,#!.
"
,
.!
*
ωj: Qj = Sj
)
Q = ∏Q j ,
"
Sj,
"
{ω } ,
j
S = ∏S j .
*
j∈K
!
j∈K
)
)
a ∈Q
J ⊆K. %
a
7
J
*
)
,
j-
ω ,
j∈J ,
a,
jj
j∈K\ J .
–
)
,
,
. . "
,
,
,
&
P I,
( a ,b ) ∈ P ,
a
( a ,b ) ∈ I ,
a
,#!
*
b,
Q:
b
,
R=P
*
I. #
* '
( x, y ) ,
*
)P
I,
ωj,
ω . #
)'
P–
-
j"
j-
j
. .
)
"
*
,
P, I
)
R
:
"
(
I – -
(
),
(1)
),
(2)
(3)
(4)
).
,
R–
(
P I =∅,R = P I .
.
)
1. .
,
).
)
C
),
*
C* ⊂ C
-
*
"
,
C* ,
*
)'
C \ C* .
#
P
)
I,
)'
:
( a ,b ) ∈ P ( c ,d ) ∈ P ,
( a ,b ) ∈ I ( c,d ) ∈ I ,
"
∀j ∈ K ,( a j = b j )
(c
j
(5)
(6)
= d j ) , (a j ≠ b j )
a j ,b j ,c j ,d j – j-
cj = a j ∧d j = bj,
a, b, c
8
d.
#
2.
,#!
P,I ,R
,
)
)
(1)-(6).
P⊆P
I⊆I.
,#!
&
.
aJ
3
)
J = { j1 , j2 ,..., js } ⊆ K ,
a j1 ,...,a js
b j1 ,...,b js (0
)'
.%
K\J
" )
. 7
,#! a J
P
bJ ,
* '
1).
bJ ,
( a J ,b J ) ,
aJ
(a
I
aJ ,
( b J ,a J ) .
0
,#!
bJ , "
,
J
,b J ) ,
bJ
P
aJ
1.
bJ
C j1
C j2
…
C js
a j1
a j2
…
a js
b j1
b j2
…
b js
#
,#!
. ,#!
*
"
*
,
.
)
)
,#!.
"
"
-
,
,
.% #
.
(
.
&
,
9
3
,#!,
)'
,
"
"
,
,#!
!
.
'
3
,#!
)
,#!. +
-,
*
,
#
*
' "
,
,#!
"
,
,
)
*
7
.
"
,
.
.
)
,
2
3
'
,#! "
. ,#!
*
-
"
*
"
,#!
.
"
.
)
/
*
,
. 7
*
*
, '
*
.% #
*
*
.
*
,#!
3
*
" ,#!. +
,
,#!,
'
"
"
"
'
,
"
*
. $
,#!,
,
)'
"
,
'
*
,#!. # )
)
,
*
,
,
.
.% #
"
,
,#!
10
"
*
)
-
3
*
"
.
.% #
)'
:
•
*
,
.
•
+
.
•
,#!
)
#
•
,
,#!
,
,#!
,#!. .
#
*
"
.
.
)
)
,
)
•
.
!
3
"
#
.
,#!
*
.
-
•
•
+
#
-
)
"
,#!
"
'
,#!
*
"
'
.
"
,#!.
•
"
,
)
"
,
)'
"
)
,#!.
*
,#!,
. #
*
.
,#!
"
I ( a ,b )
R ( a ,b )
( a ,b ) ∈ P , ( a , b ) ∈ I
3)
"
P ( a ,b ) ,
.
)'
( a ,b ) ∈ R .
,
11
*
+
(1)-(6),
)
*
3)
Ω PI ,
)
R
' )
*
.
#
P ⊆ P,I ⊆ I ,
#
{
*
* )'
-
(
P = ( a1 ,b1 ) ,..., aN P ,bN P
)} ,
-
-
{
(
I = ( c1 ,d1 ) ,..., c N I ,d N I
P,
,#!.
:
)} ,
"
NP -
-
NI -
I,
*
{
Ω
(
)
(
= P ( a1 ,b1 ) ,...,P aN P ,bN P ,I ( c1 ,d1 ) ,...,I c N I ,d N I
PI
)} .
%
,
P
I. 0
"
"
.
-
*
,
.
,
,
-
,
.
)
.+
)'
*
:
a ,b ∈ Q
1.
PI
Ω ,Ω
!
P ( a,b ) : Ω ,Ω
( a ,b ) ∈ P ;
I ( a ,b ) : Ω ,Ω |− I ( a ,b ) ,
( a ,b ) ∈ I .
PI
"
PI
|− P ( a ,b ) ,
PI
"
"
PI
-
PI
*
)
*
,#!,
3)
)
,
,#!
"
.
2.
PI
PI
Ω ,Ω ,
"
!
,
PI
Ω ,Ω
PI
!#
|− ,
.
$ %
#
,#!
3
.
-
,
"
"
'
NP-
"
,
. .
,
12
'
. # *
.
-
,#! 3
)
,
"
'
modus ponens,
)
.
% '
,
.# "
,
"
,#!
'
3
,
)'
,#!
,#!,
'
Ω mp .
*
Ω
' )
Γ
mp
Ω |− Γ .
modus ponens
a ∈Q
#
δ (a) ,
-
'
"
,
'
δ (a)
|Q j | ,
M=
j∈K
a:
T
δ ( a ) = (δ11 ( a ) ,..,δ m1 ( a ) ,δ m1 +1 ( a ) ,δ12 ( a ) ,..,δ1k ( a ) ,..,δ mk ( a ) ,δ mk +1 ( a ) ) ,
1
"
δi ( a ) =
j
1
k
1,
i ≤ m j a j = sij
0,
i = mj +1
#
a j ≠ sij
D
E,
P
,#! (
(
(
E = (δ ( c ,d ) ,δ ( c ,d ) ,...,δ ( c
D = δ ( a1 ,b1 ) ,δ ( a2 ,b2 ) ,...,δ aN P ,bN P
1
1
2
2
NI
,d N I
δ ( a ,b ) = δ ( a ) − δ ( b ) .
)'
k
:
13
I ),
)) ,
)) ,
)'
:
a ,b ∈ Q "
1.
&
P ( a,b )
mp
Ω ,Ω PI
!
# !
modus ponens Ω mp ,Ω PI|− mp P ( a ,b ) ,
Dy + Ez = δ ( a ,b ) , y ≥ 0 ,
y
(7)
z
NP
,
NI
y≠0.
mp
PI
Ω ,Ω |−
mp
,
I ( a,b ) :
"
I ( a ,b ) ,
y=0.
(7)
P ( a ,b )
(
"
NP.
. &
- "
*
(7). 2
"
,
*
. 7
(7)
*
'
*
)
,
'
,
I ( a ,b ) .
P ( a ,b ) ,
,) )
I ( a ,b )
,#!,
)'
)
"
"
*
,
.
Ω mp ,Ω PI
*
2.
&
Ω mp ,Ω PI ,
!
,
Dy + Ez = 0 , y ≥ 0 ,
y z
y≠0.
(8)
NP
#
*
)
*
,
NI
3
,
*
,
,#!
3
. &
*
14
3
)
,
)
,
"
)'
. . *
-
,#!,
,
,
)' " "
,
,
'
#
3
3
3
3
-
.
)'
. &
-
)'
.
#
b = ( s21 ,s22 ,s23 ) ,
( s ,s
1
1
( s ,s
1
2
b
2
2
2
1
,s23 ) . &
((ω ,s ,s ) ,(ω ,s ,s ) ) .
2.
3.
4.
5.
6.
3
1
1
2
2
3
" ,
,
(
:
2
3
a = ( s11 ,s12 ,s13 )
,s13 )
,
1
P ( a ,b ) , "
)'
a
1.
3
,#!
3
2
P
*
2
3
1 2
( s ,s1 ,ω ) ,( s2 ,s3 ,ω 3 )
1
1
P ( a ,b )
0 "
)'
P ( s11 ,s12 ,ω 3 ) ,( s12 ,s32 ,ω 3 ) ,
:
(
)
P ( ( s ,s ,ω ) ,( s ,s ,ω ) ) ⊃ P ( ( s ,s ,s ) ,( s ,s ,s ) ) ,
P ( ( s ,s ,s ) ,( s ,s ,s ) ) ,
P ( (ω ,s ,s ) ,(ω ,s ,s ) ) ,
P ( (ω ,s ,s ) ,(ω ,s ,s ) ) ⊃ P ( ( s ,s ,s ) ,( s ,s ,s ) ) ,
P ( ( s ,s ,s ) ,( s ,s ,s ) ) ,
1
1
2
1
1
1
2
1
3
3
1
1
2
1
2
2
3
2
3
3
2
3
3
1
1
2
2
3
2
1
2
3
3
1
1
2
2
3
2
2
3
3
1
1
2
2
2
2
1
3
1
1
2
2
3
3
1
1
2
2
3
3
1
1
2
2
2
3
2
3
1
1
1
2
1
1
3
2
15
)'
)
7.
(
) (
⊃ P ( ( s ,s ,s ) ,( s ,s ,s ) ) ,
P ( ( s ,s ,s ) ,( s ,s ,s ) ) .
1
1
8.
)
P ( s11 ,s12 ,s13 ) ,( s21 ,s32 ,s13 ) ∧ P ( s21 ,s32 ,s13 ) ,( s21 ,s22 ,s23 ) ⊃
1
1
2
1
2
1
3
1
3
1
1
2
1
2
2
2
2
2
(
3
2
1
4
2
P,
),
)
)
5–
2,
)
6
' )
8
3, 6
modus
5,
7
P, ,
,
modus ponens
7.
*
3
«
' )
4
3
1
ponens
3
2
)
3
,"
"
sij
*
:
s11 ,s12
,
C1 ,C 2
)'
C j ,#!
s21 ,s32 »;
) C3
*
«#
*
s13 . #
C 1 ,C 2 ,C 3
s11 ,s12 ,s13
:
s21 ,s32 ,s13 »;
«
,
C 2 ,C 3
s22 ,s23 »;
s32 ,s13
) C1
*
«#
*
s21 . #
s21 ,s32 ,s13
:
C 1 ,C 2 ,C 3
s21 ,s22 ,s23 »;
«(
C 1 ,C 2 ,C 3
1
2
2
3
s21 ,s32 ,s13 ,
)
1
2
3
C ,C ,C
3
1
s ,s ,s
1
2
2
2
3
2
s ,s ,s .
s11 ,s12 ,s13
, "
,
s11 ,s12 ,s13
,
s21 ,s22 ,s23 ».
16
, "
C1 ,C 2 ,C 3
*
"
3
,#!
*
"
-
.
.
"
"
) " ,
. 4
3
,
,
a
"
3
–
,#!,
,
,#!,
,#!
,
"
*
.
"
"
"
.
"
,#!
,
"
3
,#!
.
"
u,
a ∈Q
*
-
.
(
*
"
P,I .
u
) u ( a ) = (δ ( a ) ,v* )
a
δ (a)
v*
"
"
)'
"
:
("
M
, v ) → min
N
N
DT v + # ≥ " P , ET v = 0 , # ≤ p* " P , v ,# ≥ 0 ,
"
(
v = v11 ,v12 ,...,vm1 1 ,vm1 1 +1 ,v12 ,...,vij ,...,vmk −k −11 ,vmk −k −11 +1 ,v1k ,...,vmk k −1 ,vmk k ,vmk k +1
-
"M
M, # –
"
NP
–
M
-
"
p*
"
.
'
N
DT v + # ≥ " P , ET v = 0 , # ≤ p" P , v ,# ≥ 0 .
17
–
NP ,
,0–
p → min ,
N
T
'
,
NP
-
-
)
#
N
" P -DT v
v.
#
"
a
,
a
" ,
b
u (a) − u (b) .
"
%
,
)'
*
' )
( a′,b′ )
)
,#!
1.
'
) *
A
*
(
PI
mp
mp
,
A, . .
)
P ( b ,a ) .
{
B = a ∈ A | ¬∃b ∈ A,Ω PI
)
2.
"
%
}
Ω mp|− mp P ( b ,a )
a∈B
b′
a∈B \{a′}
Ω mp|− mp I ( b′,a′) ,
B := B \ {b′}
5.
.
B \ {a′}
u ( b′) : b′∈ arg max u ( a )
"
Ω PI
a
B
a′
u ( a′) : a′ ∈ arg max u ( a )
)
3.
4. 7
A,
.
)
∃b ∈ A, Ω ,Ω |−
*
b
,#!
B
*
b′ :
)
" 3.
-
( a′,b′ ) ,
)'
P ( a′,b′ ) ,
Ω mp ,Ω PI
#
,
I ( a′,b′ ) ,
*
. .
P ( b′ ,a ′ ) .
*
*
,
.
&
,
"
"
18
,#!,
"
r1+ ,r1− ,...,rt + ,rt −
M,
( v ,r
*
+
1
−
1
−r
).
ri + ,ri − ,i = 1,...,t ,
( gi ,hi ) ,
)
gi ,hi ∈ Q, i = 1,...,t
δ ( gi ,hi ) = ri − ri .
+
"
−
)
*
( v ,r
*
+
i
a′
− r1− )
1
g
( i ,hi ) ,i = 1,...,t ,
* " i = 1,...,t .
P
«1»
+
1
− ri − ) ,i = 2 ,...,t ,
b′
)
#
( gi ,hi )
)'
( v ,r
:
,
«-1»
ri + ,ri −
3
)
,#!
u ( gi ) − u ( hi )
"
( gi ,hi ) .
#
*
,
*
"
)'
,
.
0
*
*
,
7
,
3
&
)
.
+
−
ri = ri = 0
,
.
-
*
3
-
,
*
' )
(
.% #
,#!
"
.
19
.
1
)'
,
,
.
100.00%
80.00%
60.00%
40.00%
%
20.00%
!
51
49
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
9
11
7
5
3
1
0.00%
1&
"
' ,#!
##! UniComBOS,
"
)'
.% #
. %
*
,#!,
"
3
,
"
,
,#!. %
2.
##! UniComBOS
9
)
"
/
##!
UniComBOS
,#!
,
.
"
* )'
,
)
,
.
,
# -
.
"
",
-
*
,#!
. . *
,
*
.
*
,#!,
"
*
20
2
)
)
,
.
$&
%
#
#
#
3
,#!
!
2. .
'
##! UniComBOS
,
!
.
*
,
*
3)
"
,
1,
.
21
1. #
'
"
)'
,
"
)'
,
)
.
2. !
"
.% #
)
,
,#!,
,
)
"
*
.
3. #
,#!
"
"
.
4.
,#!.
5. !
"
*
"
6. #
*
,#!.
*
,#!,
.
7. !
,#!
*
8. #
'
9. #
)
,
"
,
"
-
*
.
"
,#!
. #
.
*
3
,
*
.
10. !
##!
.% #
)'
,
. ., !
1.
».
.: 7
. ., !
2.
3
,
.
.
/. .
"
3
*
. // «
+! , 2001 "., . 51-71.
/. ., 1
7. . +
*
22
. //
0
*
"
"
"
XXI
". .: 1
, 2001, 0.1, . 463-470.
3. Ashikhmin I.V., Furems E.M., Larichev O.I., Roizenson G.V. Decision
Support System UniComBOS to Discreet Multi-Criteria Choice Problems //
DSS in the Uncertainty of the Internet Age. Poland, The Karol Adamiecki
University of Economics in Katowice, 2003, pp. 111-121.
4. ,
%. .,
. ., !
/. ., 1
7. .
#
*
"
3
. //
"
. 2003, :4, . 12-19.
5. Ashihmin I.V., Furems E.M. Decision Support System for the Best
Object Selection with Inconsistency Control. // Abstracts of 58th Meeting
of the European Working Group Multiple Criteria Decision Aiding.
Moscow, URSS, 2003, pp. 5-6.
6.
. .
"
3
3
. //
. 2004, :2, . 11-16.
7.
. ., 1
7. . ##! UniComBOS
"
3
"
. //
2004, :2, . 243-247.
8.
. .
,#!
"
3
3
. //
*
. 0.12, .: 7
+! , 2005, . 7-15.
9.
. ., 1
7. . UniComBOS *
"
3
. //
*
. 0.12, .: 7
+! , 2005, . 16-25.
10.
. ., 1
7. . ##! UniComBOS
"
3
*
)
3
. // 0
*
«
» (IEEE AIS’05)
«
#!» (CAD-2005). 0.1, .: 1
, 2005, . 376-381.
11.
. ., 1
7. .
*
"
"
UniComBOS. // #
*
«
" » (
0-2005): 0
. 0.1, .: . . " ,
2005, . 236-239.
23
Download