Luận văn: TOÁN TỬ OWA TRONG MỘT SỐ BÀI TOÁN TỐI ƯU potx

Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC





ĐỖ THÙY NINH




TOÁN TỬ OWA TRONG MỘT SỐ
BÀI TOÁN TỐI ƯU


Chuyên ngành : Toán Ứng Dụng
Mã số : 60.46.36





LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC: TS VŨ MẠNH XUÂN








Thái Nguyên – Năm 2009
. . . .
W = (w
1
, w
2
, . . . , w
n
)
T
n 0 ≤ w
i
≤ 1 i = 1, , n
n

j=1
w
j
= 1.

W F :
R
n
−→ R a = (a
1
, a
2
, . . . , a
n
) ∈ R
n
F (a) =
n

j=1
w
j
b
j
,
b
j
j a.
W = (0, 4; 0, 3; 0, 2; 0, 1)
T
a = (0, 7; 1; 0, 3; 0, 6).
b = (1; 0, 7; 0, 6; 0, 3),
F (a) =
4


j=1
w
j
b
j
= 0, 4.1 + 0, 3.0, 7 + 0, 2.0, 6 + 0, 1.0, 3 = 0, 76.
a
i
w
i
w
i
W
W
W
• w
1
= 1 w
j
= 0 j = 1
W
∗
= (1, 0, . . . , 0)
T
W
∗
F
∗
.
F

∗
(a) = F
∗
(a
1
, , a
n
) = max
j
(a
j
)
(max)
• w
n
= 1 w
j
= 0 j = n
W
∗
= (0, 0, . . . , 1)
T
W
∗
F
∗
.
F
∗
(a) = F

∗
(a
1
, , a
n
) = min
j
(a
j
)
(min)
• w
j
=
1
n
j W
ave
,
W
ave
F
ave
. F
ave
(a) =
1
n
n


j=1
a
j
.
• w
k
= 1 w
j
= 0 j = k F (a
1
, , a
n
) = b
k
k a
a
n w
n+1
2
= 1 w
j
= 0, j =
n+1
2
.
n w
n
2
= w
n

2
+1
=
1
2
w
j
= 0
W = (w
1
, , w
n
)
T
F
∗
(a
1
, , a
n
)  F (a
1
, , a
n
)  F
∗
(a
1
, , a
n

),
⇔ min(a
i
)  F (a
1
, , a
n
)  max(a
i
).
a
W = (w
1
, , w
n
)
T
b = (b
1
, , b
n
) a.
b
1
≥ b
2
≥ . . . ≥ b
n
.
F

∗
(a
1
, , a
n
) = b
1
0 + b
2
0 + + b
n
1 = b
n
= min(a
i
),
F (a
1
, , a
n
) = b
1
w
1
+ b
2
w
2
+ + b
n

w
n
=
n

i=1
w
i
b
i
,
F
∗
(a
1
, , a
n
) = b
1
1 + b
2
0 + + b
n
0 = b
1
= max(a
i
).
n


i=1
w
i
b
i
≥
n

i=1
w
i
b
n
= b
n
n

i=1
w
i
= b
n
= min(a
i
),
n

i=1
w
i

b
i
≤
n

i=1
w
i
b
1
= b
1
n

i=1
w
i
= b
1
= max(a
i
).
min(a
i
) 
n

i=1
w
i

b
i
 max(a
i
) F
∗
 F  F
∗
.
✷
F (a
1
, , a
n
) = F (d
1
, , d
n
),
d = (d
1
, , d
n
) a = (a
1
, , a
n
).
a
d b = (b

1
, , b
n
)
F (a
1
, , a
n
) = F (d
1
, , d
n
).
✷
a = (a
1
, a
2
, . . . , a
n
) c = (c
1
, c
2
, . . . , c
n
)
a
i
≥ c

i
(i = 1, , n). F (a
1
, , a
n
) ≥ F (c
1
, , c
n
)
a b = (b
1
, , b
n
)
c d = (d
1
, , d
n
). a, c
a
i
≥ c
i
, b
i
≥ d
i
F (a
1

, a
2
, . . . , a
n
) = b
1
w
1
+ b
2
w
2
+ . . . + b
n
w
n
,
F (c
1
, c
2
, . . . , c
n
) = d
1
w
1
+ d
2
w

2
+ . . . + d
n
w
n
.
F (a
1
, , a
n
) ≥ F (c
1
, , c
n
).
✷
c = (c
1
, . . . , c
n
) c
1
= c
2
= . . . = c
n
= a
F (c
1
, . . . , c

n
) = a.
F (c
1
, . . . , c
n
) = a.w
1
+ + a.w
n
= a.(w
1
+ + w
n
) = a.1 = a
✷
W
Disp(W ) = −
n

i=1
w
i
ln w
i
W
Orness(W ) =
1
n − 1
n


i=1
(n − i)w
i
.
H
s
(W ) = −
n

i=1
w
i
log
2
w
i
.
H
α
α
α = 1
H
α
(W ) =
1
1 − α
log
2
n


i=1
w
α
i
.
β H
β
β = 1
H
β
(W ) =
1
2
1−β
− 1

n

i=1
w
β
i
− 1

.
H
R
(W )
R = 1

H
R
(W ) =
R
R − 1

1 −

n

i=1
w
R
i

1
R

.
H
s
(W ) = lim
α→1
H
α
(W ) = lim
β→1
H
β
(W ) = lim

R→1
H
R
(W ).
W.
W.
Q(0) = 0, Q(1) = 1.
i = 1, 2, . . . , n w
i
= Q(i/n) − Q((i − 1)/n).
W.
A
i
(x).
w
i
w
i
= Q(i/n) − Q((i − 1)/n).
a a
i
= A
i
(x).
W a
Q(i) = i
2
, n = 3.
w
1

= Q(
1
3
) − Q(
0
3
) = (
1
3
)
2
− 0 =
1
9
,
w
2
= Q(
2
3
) − Q(
1
3
) = (
2
3
)
2
− (
1

3
)
2
=
4
9
.
1
9
=
1
3
,
w
3
= Q(
3
3
) − Q(
2
3
) = (1)
2
− (
2
3
)
2
= 1 −
4

9
=
5
9
.
W = (
1
9
,
1
3
,
5
9
).
(u
j
, a
j
) u
j
∈ [0, 1]
(a
i
∈ [0, 1]) u
j
a
j
a
j

b
i
a
i
. v
i
(v
i
, b
i
) b
i
w
i
= Q(S
i
/T ) − Q(S
i−1
/T ) i = 1, . . . , n
S
i
=
i

k=1
v
k
, T = S
n
=

n

k=1
v
k
.
S
i
a
∗
=
n

i=1
b
i
w
i
.
A
1
, A
2
, A
3
, A
4
.
u = (1; 0.6; 0.5; 0.9).
Q = r

2
b
j
v
j
A
1
1 0.6
A
2
0.7 1
A
3
0.6 0.9
A
4
0.5 0.5
w
i
w
1
(x) = Q(0.6/3) − Q(0/3) = (0.2)
2
− 0 = 0.04
w
2
(x) = Q(1.6/3) − Q(0.6/3) = 0.28 − 0.04 = 0.24
w
3
(x) = Q(2.5/3) − Q(1.6/3) = 0.69 − 0.28 = 0.41

w
4
(x) = Q(3/3) − Q(2.5/3) = 1 − 0.69 = 0.31.
F (x) = 0.4 ∗ 1 + 0.24 ∗ 0.7 + 0.41 ∗ 0.6 + 0.31 ∗ 0.5 = 0.609.
n
(a
k1
, a
k2
, . . . , a
kn
)
d
k
.
W
W
F (a
1
, a
2
, . . . , a
n
) = d
k
,
W.
(b
k1
, b

k2
, . . . , b
kn
)
b
kj
(a
k1
, a
k2
, . . . , a
kn
).
W = (w
1
, w
2
, . . . , w
n
)
T
b
k1
w
1
+ b
k2
w
2
+ . . . + b

kn
w
n
= d
k
,
W = (w
1
, w
2
, . . . , w
n
)
T
e
k
e
k
=
1
2
((b
k1
w
1
+ b
k2
w
2
+ . . . + b

kn
w
n
) − d
k
)
2
,
w
i
n

i=1
w
i
= 1; w
i
∈ [0, 1], i = 1, . . . , n.
w
i
w
i
=
e
λ
i
n

i=1
e

λ
i
, i = 1, . . . , n.
λ
i
w
i
λ
i
e
k
=
1
2

b
k1
e
λ
1
n

i=1
e
λ
1
+ b
k2
e
λ

2
n

i=1
e
λ
2
+ . . . + b
kn
e
λ
n
n

i=1
e
λ
n
− d
k

2
.
λ
i
(l + 1) = λ
i
(l) − βw
i
(l)(b

ki
−

d
k
)(

d
k
− d
k
),
λ
i
(l + 1) λ
i
. β
(0 ≤ β ≤ 1), w
i
(l) =
e
λ
i
(l)
n

i=1
e
λ
i

(l)
w
i

d
k
= b
k1
w
1
(l) + b
k2
w
2
(l) + . . . + b
kn
w
n
(l).
λ
i
δ
i
= lλ
i
(l + 1) − λ
i
(l)l, i = 1, . . . , n.
Q : [0, 1] −→ [0, 1]
(i)Q(0) = 0,

(ii)Q(1) = 1,
(iii)x > y ⇒ Q(x) ≥ Q(y).
(i)Q
x
(0) = 0, Q
x
(x) = 1, x = 0,
(ii)Q
n
(1) = 1, Q
n
(x) = 0, x = 1.
W M : R
n
−→ R
n W M
p
(a
1
, . . . , a
n
) =

i
p
i
a
i
.
OW A

Q
: R
n
−→ R
OW A
Q
(a
1
, . . . , a
n
) =
n

i=1
(Q(i/n) − Q((i − 1)/n))a
σ(i)
,
{σ(1), . . . , σ(n)} {1, . . . , n}
a
σ(i−1)
≥ a
σ(i)
i = {2, . . . , n}, a
σ(i)
(a
1
, . . . , a
n
).
R

n
w
i
w
i
= Q(i/n) −Q(i −1)/n
{i/n, Q(i/n)} i ∈ {0, 1, . . . , n}
W OW A : R
n
−→ R
W OW A
p,w
(a
1
, . . . , a
n
) =

i
w
i
a
σ(i),
a
σ(i)
(a
1
, . . . , a
n
), w

i
w
i
= W
∗
(

j≤i
p
σ(i)
) − W
∗
(

j≤i
p
σ(i)
),
W
∗
(i/n,

j≤i
w
j
)
W OW A : R
n
−→ R
W OW A

p,Q
(a
1
, . . . , a
n
) =

i
w
i
a
σ(i)
,
w
i
= Q(

j≤i
p
σ(i)
) − Q(

j≤i
p
σ(i)
),
µ
µ : ρ(X) −→ [0, 1]
µ(∅) = 0, µ(X) = 1,
A ⊆ B µ(A) ≤ µ(B),

µ
f : X −→ R
n

i=1
(f(x
s(i)
) − f(x
s(i−1)
))µ(A
s(i)
),
f(x
s(i)
) 0 ≤ f(x
s(1)
) ≤ . . . ≤ f(x
s(N)
) ≤ 1,
A
s(i)
= {x
s(i)
, . . . , x
s(N)
} f(x
σ(0)
) = ∅.
µ
µ(A) = Q



x∈A
p(x)

.
Orness(Q) =

1
0
Q(x)d
x
.
S = {s
1
, s
2
, . . . , s
T
} s
1
< s
2
< . . . < s
T
.
a = {a
1
, a
2

, . . . , a
m
} a
i
S. b = {b
1
, b
2
, . . . , b
m
} a
b
j
j a. b = {s
im
, s
i(m−1)
, . . . , s
i1
}
i
m
≥ i
m−1
≥ . . . ≥ i
1
.
W = {w
1
, w

2
, . . . , w
m
} w
i
∈ [0, 1]

i
w
i
= 1.
a = {a
1
, a
2
, . . . , a
m
}, W = {w
1
, w
2
, . . . , w
m
}
a
w, Low : (a, w) −→ S
Low(a, W ) = C{(w
im
, a
im

), (1 − w
im
, Low(a

, w

))},
a

= {a
i(m−1)
, . . . , a
i1
}, w

= {w

i1
, w

i2
, . . . , w

i(m−1)
}, w

j
=
w
j

1 − w
im
,
C (s
j
, s
i
), j ≥ i w
j
> 0, w
i
> 0,
w
j
+ w
i
= 1, C{(w
j
, s
j
), (w
i
, s
i
)} = s
k
, k = i + round(w
j
, (j − i)).
S R

1
W
a = (s
1
, s
2
, s
3
), w = (0.2; 0.3; 0.5).
b = (s
3
, s
2
, s
1
), w
3
= 0.5, w
2
= 0.3, w
1
= 0.2
Low(a, w) = C{(0.5, s
3
), (0.5, Low((s
2
, s
1
), (0.2/0.5, 0.3/0.5)))}.
Low((s

2
, s
1
), (0.2/0.5, 0.3/0.5)) = C{(3/5, s
3
), (2/5, s
2
)} = s
k1
,
k
1
= 1 + round((3/5)(2 − 1)) = 1 + 1 = 2.
Low(a, w) = C{(0.5, s
3
), (0.5, s
2
)} = s
k
,
k = 2 + round((0.5)(3 − 2)) = 3.
Low(a, W ) = s
3
.
GOW A : R
n
−→ R
GOW A(a
1
, . . . , a

n
) =

n

j=1
w
j
b
λ
j

1
λ
,
n

j=1
w
j
= 1, w
j
∈ [0, 1], b
j
a
i
,
λ ∈ (−∞, ∞)
IGOW A : R
n

−→ R
IGOW A((u
1
, a
1
), . . . , (u
n
, a
n
)) =

n

j=1
w
j
b
λ
j

1
λ
,
n

j=1
w
j
= 1, w
j

∈ [0, 1], b
j
a
i
(u
i
, a
i
)
u
i
a
i
λ ∈ (−∞, ∞)
a
i
IOW A : R
n
−→
R
IGOW A((u
1
, a
1
), . . . , (u
n
, a
n
)) =


n

j=1
w
j
b
j

,
n

j=1
w
j
= 1, w
j
∈ [0, 1], b
j
a
i
(u
i
, a
i
)
u
i
a
i
W

α ∈ [0, 1]
Disp(W ) = −
n

i=1
w
i
ln w
i
,
α =
1
n − 1
n

i=1
(n − i)w
i
, 0 ≤ α ≤ 1,
n

i=1
w
i
= 0, 0 ≤ w
i
≤ 1, i = 1, . . . , n.
Disp(W ) = −
n


i=1
w
i
ln w
i
,
Orness(W ) = α, 0 ≤ α ≤ 1,
w
1
+ . . . + w
n
= 1, 0 ≤ w
i
≤ 1, i = 1, . . . , n.
n = 2 Orness(w
1
, w
2
) = α, w
1
= α, w
2
= 1 −α.
α = 0 α = 1
(0, 0, . . . , 0, 1)
T
(1, 0, . . . , 0, 0)
T
.
n > 3 0 < α < 1, λ

1
, λ
2
L(W, λ
1
, λ
2
) = −
n

i=1
w
i
ln w
i
+ λ
1

n

i=1
n − i
n − 1
w
i
− α

+ λ
2


n

i=1
w
i
− 1

,
w
j
∂L
∂w
j
= − ln w
j
− 1 +
n − j
n − 1
λ
1
+ λ
2
= 0,
∂L
∂λ
1
=
n

i=1

w
i
− 1 = 0,
∂L
∂λ
2
=
n

i=1
n − i
n − 1
w
i
− α = 0.
j = n
− ln w
n
− 1 + λ
1
= 0 ⇔ λ
1
= ln w
n
+ 1.
j = 1
− ln w
1
− 1 + λ
1

+ λ
2
= 0,
⇒ λ
2
= ln w
1
+ 1 − λ
1
= ln w
1
+ 1 − ln w
n
− 1,
⇔ λ
2
= ln w
1
− ln w
n
.
1 ≤ j ≤ n
ln w
j
=
j − 1
n − 1
ln w
n
+

n − j
n − 1
ln w
1
,
⇒ w
j
=
n−1

w
n−j
1
w
j−1
n
.
w
1
= w
n
w
1
= w
2
= . . . = w
n
=
1
n

,
⇒ Disp(W ) = ln W.
α = 0, 5
n
w
1
= w
n
. u
1
= w
1
(n−1)
1
, u
n
= w
1
(n−1)
n
,
w
j
= u
n−j
1
u
j−1
n
1 ≤ j ≤ n . Orness(W ) = α

n

i=1
n − i
n − 1
w
i
= α,
⇔
n

i=1
(n − i)u
n−i
1
u
i−1
n
= (n − 1)α,
n

i=1
(n − i)u
n−i
1
u
i−1
n
=
1

u
1
− u
n

(n − 1)u
n
1
−
n−1

i=1
u
i
1
u
n−i
n

,
=
1
u
1
− u
n

(n − 1)u
n
1

− u
1
u
n
u
n−1
1
− u
n−1
n
u
1
− u
n

,
=
1
(u
1
− u
n
)
2

(n − 1)u
n
1
(u
1

− u
n
) − u
n
1
u
n
+ u
1
u
n
n

,
=
1
(u
1
− u
n
)
2

(n − 1)u
n+1
1
− nu
n
1
u

n
+ u
1
u
n
n

.
(n − 1)u
n+1
1
− nu
n
1
u
n
+ u
1
u
n
n
= (n − 1)α(u
1
− u
n
)
2
,
nu
n

1
− u
1
= (n − 1)α(u
1
− u
n
),
u
n
=
1
(n − 1)α

((n − 1)α + 1)u
1
− nu
n
1

.
u
n
u
1
=
(n − 1)α + 1 − nw
1
(n − 1)α
.

w
1
+ + w
n
= 1
n

j=1
u
n−j
1
u
j−1
n
= 1 ⇔
u
n
1
− u
n
n
u
1
− u
n
= 1
⇔ u
n
1
− u

n
n
= u
1
− u
n
.
n

j=1
u
n−j
1
u
j−1
n
= 1 ⇔ u
n−1
1
−
u
n
u
1
u
n−1
n
= 1 −
u
n

u
1
.
w
1
−
(n − 1)α + 1 − nw
1
(n − 1)α
w
n
=
nw
1
− 1
(n − 1)α
,
w
n
=
((n − 1)α − n)w
1
+ 1
(n − 1)α + 1 − nw
1
.
u
n
1
− u

n
n
= u
1
− u
n
,
u
1
(w
1
− 1) = u
n
(w
n
− 1),
w
1
(w
1
− 1)
n−1
= w
n
(w
n
− 1)
n−1
,
w

1
(w
1
− 1)
n−1
=
((n − 1)α − n)w
1
+ 1
(n − 1)α + 1 − nw
1

(n − 1)α(w
1
− 1)
(n − 1)α + 1 − nw
1

n−1
.
w
1
[(n − 1)α + 1 − nw
1
]
n
= [(n − 1)α]
n−1
[((n − 1)α − n)w
1

+ 1].
w
1
w
1
w
n
n = 5, α = 0, 4
−
n

i=1
w
i
ln w
i
,
α =
1
n − 1
n

i=1
(n − i)w
i
, 0 ≤ α ≤ 1,
n

i=1
w

i
= 0, 0 ≤ w
i
≤ 1, i = 1, . . . , n.
w
1
[4 ∗ 0.4 + 1 − 5w
1
]
5
= [4 ∗ 0.4]
4
[(4 ∗ 0.4 − 5)w
1
+ 1],
w
∗
1
= 0.1278
w
∗
5
=
(4 ∗ 0.4 − 5)w
∗
1
+ 1
4 ∗ 0.4 + 1 − 5w
∗
1

= 0.2884
w
∗
2
=
4

(w
∗
1
)
3
(w
∗
5
) = 0.1566
w
∗
3
=
4

(w
∗
1
)
2
(w
∗
5

)
2
= 0.192
w
∗
4
=
4

(w
∗
1
)(w
∗
5
)
3
= 0.2353.
Disp(W
∗
) = 1, 5692.
Disp(W ) = −
n

i=1
w
i
ln w
i
,