/.: .., .. . , .:2000.
.. ,
15.
.. - . 2, 4-10
I, III, IX.
.. - , -
11-14, V, VII, VIII.
.. - , -
1, 3, IV.
.. - , -
VI.
.. , II.
,
..
,
..
, 2000
² ². , : , , -, . .
[1]. , , , , +1, . , , , 1, 2 1 [1].
[7], - [16]. [11], . [6], [8] [25].
[20], [23]. [2].
1.
, , .
, , .
- , , .
2.
1. , 1, ,
c1
c2
c3
c4
11
12
13
14
b1
a21
a22
a23
a24
b2
a31
a32
a33
a34
b3
, , , , , ² ² .
Q-1
, .
H = Q-1
B
- , , , .
2. , , , , ( ). , , .
.
" " . , . ² ² , ( , ); , .
1, 2, 3 [10, c. 21].
3. 2, (a1
,..., am
), - (b1
,..., bn
) =(ij
), i =; j =
b1
b2
. . . bn
a1
c11
c12
. . . c1n
a2
c21
c22
. . . c2n
. . . . . . . . . . . . . . . . . . . .
am
cm1
cm2
. . . cmn
, .
4. , 700 . ., , 3 ( 100 .).
5. . , 4.
6. , 5. . , .
7. . 8-9 , .
8. . , .
9. . , .
10. " ", 2. "" , =3 , , .
11. . .
12. ( ). , , .
13. . - , .
14. 6. .
15. , 7.
16. : - m0
, - m1
, m2
c s1
, s2
. 8.
17. , 7. . , :
1. (2,1/2)(0,1/4)(14,1/8))(6,1/8) 2. (2,1/2)(4,1/4)(18,1/8))(8,1/8)
3. (4,1/4)(0,1/4)(6,1/3))(12,1/6) 4. (6,1/4)(2,1/4)(14,1/3))(4,1/6)
:
1. (2,0,14,6) 2.(2,4,18,8) 3. (4,0,6,12) 4.(6,2,14,4)
, . , : 1.(2,1/2)(0,1/4)(14,1/8)(6,1/8), . . (1/2,1/4,1/8,1/8). :
) .
) , , , (l ).
) 4- : , , .
) .
) , , , - (, ).
) - , , , , - .
) ( !) .
18. - T Xt
, Kt
, Lt
(t = 1, , T) ( ), :
) , 1, 2, 3
;
) 1, 2, 3
) .
3.
5-8
, 1, 2, 4, 6 .
, :
, - -
1 2 3 4 5 6 7 8 9 10 11
- , ,, ,,
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
. . .
. - , .
. 9.
. c , .
4.
, . .
, n
, m
. , , . , , , n
k
. , .
:
i
(i
=1,2, , m
);
j
(j
=1,2, , n
);
aij
i
- j
- ;
bi
i
- ;
xi
j
- ;
(x1
, x2
, , xn
) .
(x1
, x2
, , xn
), , . x1
i
- ai1
x1
, x2
ai2
x2
.. x1
, x2
, , xn
i
-
i
- , .. £ bi
. m
, :
(1)
, , , :
x1
³0, x2
,
³0,, xn
³0.
(2)
j
j- . (1
, 2
, , n
) :
z = c1
x1
+ c2
x2
+ + cn
xn
.
(3)
(1
, 2
, , n
) , (3) .
(1), (2) (3) . (1), (2), , (3) . . A , B C :
, , C=(c1
, , cn
)
, , .
, , ,
, ,
:
(4)
:
(5)
(6)
. (5), (6) OPQRS
. Z
- grad Z
=(6,9) ( ). Z
R
. x1
=3, x2
=2,
36.
, , . ,
(7)
,
:
(x1
, x2
, x3
, x4
)
z = 36x1
+ 14x2
+ 25x3
+ 50x4
(8)
(9)
x1
³ 0, x2
³ 0, x3
³ 0, x4
³ 0. (10)
. (9) 5
, 6
, 7
(11)
. (11),
1
³0, 2
³0, , 5
³0, , 7
³0. (12)
, (8) .
, (11) , . 1
, 2
, 3
, 4
,
x1
=0, x2
=0, x3
=0, x4
=0, x5
=208, x6
=107, x7
=181 (13)
x1
=0, x2
=0, x3
=0, x4
=0 (14)
.
(8) , , . , . , . (11)
(15)
1
=2
=3
=0 4
. ,
.. 0 £ 4
£
4
4
=181/5, , (15). (11)
1
=0, 2
=0, 3
=0, 4
=; x5
=27; x6
=; x7
=0 (16)
,
(11), (11) 4
, , ,
34
=5. , (11)
x1
+ 2x2
+ 2x3
+ x5
- x7
= 27
x1
+ x2
- x3
+ x6
- x7
= (17)
x1
+ x2
+ x3
+ x4
+ x7
=
1
, 2
, 3
, 7
, , (16),
1
=0, 2
=0, 3
=0, 4
=. (18)
, , .. . (8) 1
, 2
, 3
, 7
. (17) 4
(8).
(19)
, (18) , , , , , z 1
. 1
(17) ,
(20)
1
(17), . , (11) , (19) , 1
, . , ( 4
(8)).
, . (8)
-361
-142
-253
-504
=0 z (21)
(11).
(22)
, (11) 4
. (22) D4
=-50. 34
=5 4
(11), . 4
(8). , (22). , (22) 10 ;
-61
-42
-53
-104
= 1810 z (23)
, (22)
x1
+ 2x2
+ 2x3
+ x5
- x7
= 27
x1
+ x2
- x3
+ x6
- x7
= (24)
x1
+ x2
+ x3
+ x4
+ x7
=
-6x1
- 4x2
- 5x3
+10x7
= 1810 - z
(17) (11) (16) (18), (24) (19) . , Dj
- xj
(24), . (19) , , .. (24) min(Dj
<0) = min(-6, -4, -5) = -6 = D1
1
, , (20) 11
=1.
, (17), (24), .
x1
+ 2x2
+ 2x3
+ x5
- x7
= 27
3x2
- x3
- x5
+ x6
+ x7
= 13 (25)
- x2
- x3
+ x4
- x5
+ x7
= 20
8x2
+ 7x3
+ 6x5
+ 4x7
= 1972 - z
(25) (11)
x1
=27, x2
=0, x3
=0, x4
=20, x5
=0, x6
=13, x7
=0 (26)
..
x1
=27, x2
=0, x3
=0, x4
=20 (27)
:
5
=0
6
=13 (28)
7
=0
(25) . z
z = 1972 - 82
- 73
- 65
-47
(29)
( , xj
³0), ,
x2
=0, x3
=0, x5
=0, x7
=0 (30)
, (27)
zmax
= 1972 (31)
, , , .
, 1.
1 36 14 25 50 0 0 0 |
|
|
|
|
x1
x2
x3
x4
x5
x6
x7
|
0 |
5
|
208 |
4 3 4 5 1 0 0 |
z0
= H
|
0 |
6
|
107 |
2 5 0 2 0 1 0 |
|
0 |
7
|
181 |
3 1 2 5 0 0 1 |
0 |
z0
-z |
0 - z |
-36 -14 -25 -50 0 0 0 |
|
0 |
5
|
27 |
1 2 2 0 1 0 -1 |
|
0 |
6
|
173/5 |
4/5 23/5 -4/5 0 0 1 -2/5 |
50 |
4
|
181/5 |
3/5 1/5 2/5 1 0 0 1/5 |
|
z0
-z |
1810-z |
-6 -4 -5 0 0 0 10 |
|
36 |
1
|
27 |
1 2 2 0 1 0 -1 |
0 |
6
|
13 |
0 3 -12/5 0 -4/5 1 2/5 |
Dj
³0 |
50 |
4
|
20 |
0 -1/5 -4/5 1 -3/5 0 4/5 |
z0
-z |
1972-z |
0 8 7 0 6 0 4 |
(22) (24) (25). .
. , D3
=7 3
, ( ), 7 .
, .
, 2
=0, 3
=0. , . , . :
, . ,
(x1
=0, x4
=0) (x1
=0, x4
=) (x1
=27, x4
=20)
( "" ).
5.
.
, . (), - , , , "" 1
, 2
, 3
. : 1
, 2
, 3
.
1
, 2
, 3
, , . , .
, ,
, , 4 , 2 3 ( ). 1
, 2
, 3
41
+ 22
+ 33
, .. , . 36 . , ,
41
+ 22
+ 33
³ 36.
, . 31
+ 52
+ 3
, 14 .
31
+ 52
+ 3
³ 14
.. .
, 2081
+ 1072
+ 1813
. 1
, 2
, 3
, . , , - , , , .
, :
(1
, y2
, y3
)
f = 208y1
+ 107y2
+181y3
(1)
, , , ,
4y1
+ 2y2
+ 3y3
³ 36 3y1
+ 5y2
+ y3
³ 14
4y1
+ 2y3
³ 25
5y1
+ 2y2
+ 5y3
³ 50
y1
0, y2
0, y3
0. (3)
, (1
, 2
, 3
, 4
) (y1
, y2
, y3
)
x 1
(4y1
+ 2y2
+ 3y3
- 36) = 0 y1
(4x1
+3x2
+ 4x3
+ 5x4
- 208) = 0
x 2
(3y1
+ 5y2
+ y3
- 14) = 0 y2
(2x1
+5x2
+ 2x4
- 107) = 0
x 3
(4y1
+ 2y3
- 25) = 0 y3
(3x1
+ x2
+ 2x3
+ 5x4
- 181) = 0 .
x 4
(5y1
+ 2y2
+ 5y3
- 50) = 0
, 1
>0, x4
>0.
4y1
+ 2y2
+ 3y3
- 36 = 0
5y1
+ 2y2
+ 5y3
- 50 = 0
, , ,
2
=0,
4y1
+ 3y3
- 36 = 0
5y1
+ 5y3
- 50 = 0
1
=6, 3
=4.
,
1
=6; 2
=0; 3
=4, (4)
1972.
, (4) . . , 3
=4 , 4 .
6. " "
, .. ² ². . T(t1
,t2
,t3
)- . ,
H + Q-1
T 0.
,
T (t1
, 0, t3
),
W = 6t1
+ 4t3
(1)
(, , )
, 1/3
(3)
t1
0, t3
0. (4)
(2) (3) :
: (1) (5), (6) (4).
: . . 1. ²²
t1
=, t2
=0, t3
=
519.
1
j
|
36 |
14 |
25 |
50 |
b |
x4+i
|
yi
|
ti
|
4 |
3 |
4 |
5 |
208 |
0 |
6 |
46 5/12 |
aij
|
2 |
5 |
0 |
2 |
107 |
13 |
0 |
0 |
3 |
1 |
2 |
5 |
181 |
0 |
4 |
60 1/3 |
xj |
27 |
0 |
0 |
20 |
1972 |
519 2/3 |
Dj
|
0 |
8 |
7 |
0 |
7
.
. , m
() 1
, 2
,..., m
, n
, b1
, b2
,..., bn
. i- j- ij
. , .
ij
, i- j- .
(1)
:
= (ij
), i = 1,m; j =1,n
(2)
,
(3)
(4)
11
> 0 ,. . . ., xmn
> 0. (5)
. (1
, 2
, 3
) = (54; 60; 63); (b1
, b2
, b3
, b4
) = (41; 50; 44; 30); =
åi
= 55+60+63 = 178 , åbi
= 42+50+44+30 = 166, .. . 178-166 = 12 , , , , , .
²- ².
|
b1
=41 |
b2
=50 |
b3
=44 |
b4
=30 |
b5
=12 |
|
1
=54 |
41 |
13 |
p1
=0 |
a2
=60 |
37 |
23 |
p2
= |
a3
=63 |
*
|
21 |
30 |
12 |
p3
= |
q1
= |
q2
= |
q3
= |
q4
= |
q5
= |
, (3), (4), , , . (3), (4) m + n - 1 , m + n - 1 .
m
)
.
Dij
= m
Aij
- ij
i = 1,m; j = 1,n
Dij
= pi
+ qj
- cij
i = 1,m; j = 1,n (6)
, (3), (4) . , 1
= 0. , .
D11
= 0, p1
+ q1
- c11
= 0, 0+q1
-1 = 0, q1
= 1
D12
= 0, p1
+ q2
- c12
= 0, 0+q2
-4 = 0, q2
= 4
D22
= 0, p2
+ q2
- c22
= 0, 2
+4-6 = 0, 2
= 2
.., : q3
=0, p3
=6, q4
= 1, q5
= -6.
(6) :
D21
= p2
+ q5
- c21
= 2+1-3 = 0
D31
= p3
+ q1
- c31
= 6+1-2 = 5
D32
= 5; D13
= -3; D14
= -1; D24
= -2; D15
= -6; D25
= -4.
max () = 5 =
31 - , , , , - . 31-11-12-22-23-33.
41 |
13 |
41-r |
13+r |
20 |
34 |
|
37 |
23 |
37-r |
23+r |
16 |
44 |
21 |
r |
21-r |
21 |
= 21
:
bj
|
b1
=41 |
b2
=50 |
b3
=44 |
b4
=30 |
b5
=12 |
ai
|
1
=54 |
20 |
34 |
*
|
p1
=0 |
a2
=60 |
16 |
44 |
p2
=2 |
a3
=63 |
21 |
30 |
12 |
p3
=1 |
q1
=1 |
q2
= 4 |
q3
= 0 |
q4
= 6 |
q5
= -1 |
, . 14. 14-11-31-34
20 |
20-r |
r |
20 |
21 |
30 |
21+r |
30-r |
42 |
10 |
rmax
= 20
. , ,
Dij
£ 0 i = 1,m; j = 1,n
,
8. .
- . n . .
. , .
, n , , b . fi
(xi
) j- , xi
. (x1
,x2
, ... , xn
) ,
z = f1
(x1
) + f2
(2
) + ... + fn
(xn
)
x1
+ x2
+ ... + xn
= b
, xj
xj
= 0, 1, 2, 3, ...
fj
(xj
) , , - .
.
. x , , Fk
(x) k , x . x 0 b. x k- xk
, , x - xk
(-1)- , Fk-1
(x - xk
). k fk
(xk
) + Fk-1
(x - xk
). xk
0 x, ,
Fk
(x)=max{f
k
(x
k
) + Fk-1
(x-x
k
)}
0 £x
k
£x
k = 2, 3, 4, ... , n . k=1,
F1
(x) = f1
(x)
. (n=4). 700 . (b=700), 100 . . fj
(xj
) 1, , , 88 , 600 . . , 88 . .
I
. 2. f2
(x2
) F1
(x - x2
) = f1
(x- x2
) - , . 3.
, F3
(x), (x) .. . 6 x= 700. :
Zmax
= 155 . .,
*
4
= 4
(700) = 300 . .
400 . . . 5 ,
x*
3
= 3
(700-x*
4
) = 3
(400) = 200 . .
,
x*2
= 2
(700 - x*4
- x*3
) = 2
(200) = 100 . .
x*1
= 700 - x*4
- x*3
- x*2
= 100 . .
, :
x*1
=100; x*2
=100; x*3
= 200; x*4
= 300.
155 . .
f1
(x*1
) + f2
(x*2
) + f3
(x*3
) + f4
(x*4
) = z max
2
x - x2
|
0 100 200 300 400 500 600 700 |
x2 |
F1
(x - x2
)
f2
(x2
)
|
0 20 34 46 53 55 60 60 |
0 |
0 |
0 20* 34 46 53 55 60 60 |
100 |
18 |
18 38* 52* 64 71 73 78 |
200 |
29 |
29 49 63 75 82 84 |
300 |
45 |
45 65* 79 91 98 |
400 |
62 |
62 82* 96 108 |
500 |
78 |
78 98* 112* |
600 |
90 |
90 110 |
700 |
98 |
98 . |
3
x |
0 100 200 300 400 500 600 700 |
F2
(x) |
0 20 38 52 65 82 98 112 |
`(x) |
0 0 100 100 300 400 500 500 |
4
x - x3
|
0 100 200 300 400 500 600 700 |
x3 |
F2
(x - x3
)
f3
(x3
)
|
0 20 38 52 65 82 98 112 |
0 |
0 |
0 20 38 52 65 82 98 112 |
100 |
25 |
25* 45* 63* 77 90 107 123 |
200 |
41 |
41 61 79* 93 106 123 |
300 |
52 |
52 72 94* 112 126 |
400 |
74 |
74 94* 112* 126* |
500 |
82 |
82 102 120 |
600 |
88 |
88 106 |
700 |
90 |
90 . |
5
x |
0 100 200 300 400 500 600 700 |
F3
(x) |
0 25 45 63 79 94 112 126 |
(x) |
0 100 100 100 200 400 400 400 |
6
|
x - x4
|
0 100 200 300 400 500 600 700 |
x4 |
F3
(x - x4
)
f4
(x4
)
|
0 25 45 63 79 94 112 126 |
0 |
0 |
126 |
100 |
30 |
142 |
200 |
52 |
146 |
300 |
76 |
155* |
400 |
90 |
153 |
500 |
104 |
149 |
600 |
116 |
141 |
700 |
125 |
125 . |
9.
. , n . . , , , , . n . :
xj
- , j - ;
yj
- j ( , j - );
dj
- , j - ;
fj
(xj
,yj+1
) - j - .
, y1
yn+1
.
,
(x1
, x2
, ..., xn
) (1)
xj
+ yj
- dj
= yj+1
j = 1,n (2)
(3)
xj
³ 0, yj
³ 0, j = 1,n (4)
, , j yj+1
0 £ yj+1
£ dj+1
+ dj+2
+ ... + dn
(5)
.. xj
j , yj+1
,
yj+1
. , (2) (4) , xj
0 £ xj
£ dj
+ yj+1
(6)
, xj
, yj
, .. .
(1)-(6) .
.
x k - x = yk+1
(7)
Fk
(x) k (5)
(8)
x1
,...,xk
,
xj
+ yj
- dj
= yj+1
j = 1, k-1 (9)
xk
+ yk
- dk
= x (10)
,
(11)
yk
(k-1) , (10),
yk
= x + dk
- xk
(12)
(13)
xk
, , (6)
0 £ xk
£ dk
+ x (14)
, ,
0 £x£ dk+1
+ dk+2
+ ... + dn
(15)
k
k = 2, 3, 4, ... , n (16)
k=1,
F1
(x = y2
) = min f1
(x1
, x) (17)
x1
x1
= x + d1
- y1
(18)
0£x£ d2
+ d3
+ ... + dn
(19)
.. y1
x x1
, . , ( k = n) xn
* ,
(20)
fj
(xj
, yj+1
) .
jj
(xj
) = axj
2
+ bxj
+ c
jj
(xj
) - () xj
j;
hj
- , j j+1.
j
fj
(xj
, yj+1
) = jj
(xj
) + hj
yj+1
= axj
2
+ bxj
+ c + hj
yj+1
. (21)
:
(22)
k = 2, 3, ... , n (23)
0 £ yk+1
£ dk+1
+ dk+1
+ ... + dn
(24)
0 £ xk
£ dk
+ yk+1
(25)
yk
= yk+1 +
dk
- xk
(26)
k=1,
,
Wk
(xk
, yk+1
) = axj
2
+ bxj
+ c + hk
yk+1
+ Fk-1
(yk
) (31)
(22)
Fk
(x=yk+1
) = min Wk
(xk
, yk+1
) (32)
xk
xk
, (25)..
.
() : d1
=3 , d2
=2, - d3
=4 . 2 , .. y1
=2. h1
=1, h2
=3, h3
=2. xj
j-
jj
(xj
) = xj
2
+ 5xj
+ 2 (33)
.. =1; b=5; =2. , , , .
:
d1
d2
d3
a b c h1
h2
h3
y1
1 2 4 1 5 2 1 3 2 2
,
F1
(x = y2
), F2
(x = y3
), ..., Fk
(x = yk+1
), ... 1
(x= y2
), 2
(x = y3
), ..., `k
(x = yk+1
), ...
k = 1. (27)
(34)
, (28) x = 2
0 2
d2
+ d3
0 y2
2 + 4
..
2
= 0, 1, 2, 3, 4, 5, 6.
, , x1
, (29)
0 1
3 + 2
, 1
, d1
= 3, 1
= 2. ,
1
+ 1
- d1
= 2
, x= 2
x1
= y2
+ d1
- y1
= y2
+ 3 - 2 = y2
+1 (35)
. , 2
1
F1
(x = y2
) = W1
(x1
, y2
)
2
0 6 (35), y2
= 0, x1
= 0+1 = 1, W1
(1;0) = 12
+ 5×1 + 2 + 1×0 = 8
y2
= 1, x1
= 1+1 = 2, W1
(2;1) = 22
+ 5×2 + 2 + 1×1 = 17
.. F1
(x ) . 1
1
x = y2
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
F1
(x = y2
) |
8 |
17 |
28 |
41 |
56 |
73 |
92 |
x1
(x=y2
) |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
. k = 2
F2
(x = y3
) (32)
(37)
2
, , (25),
0 £ x2
£ d2
+ y3
0 £ x2
£ 2 + y3
(38)
x = 3
, , (15),
0 £ y3
£ d3
, .. 0 £ y3
£ 4 (39)
2
(37) 2
3
x2
+ y2
- d2
= y3
y2
= y3
+ d2
- x2
= y3
+ 2 - x2
(40)
0 4, W2
(x2
, x), F2
(x ) 2
(x ).
, x = 3
= 2. , (38),
0 £ x2
£ 4,
.. 2
: 0, 1, 2, 3, 4, 2
2
, (40):
2
= 4 - 2
:
x2
= 0, y2
= 4-0 = 4, W2
(0,2) = 02
+ 5×0 + 2 + 3×2 + F1
(4) = 8 + 56 = 64,
x2
= 1, y2
= 4-1 = 3, W2
(1,2) = 12
+ 5×1 + 2 + 3×2 + F1
(3) = 14 + 41 = 55,
x2
= 2, y2
= 4-2 =2, W2
(2,2) = 22
+ 5×2 + 2 + 3×2 + F1
(2) = 22 + 28 = 50,
x2
= 3, y2
= 4-3 = 1, W2
(3,2) = 32
+ 5×3 + 2 + 3×2 + F1
(1) = 32 + 17 = 49*,
x2
= 4, y2
= 4-4 = 0, W2
(3,2) = 42
+ 5×4 + 2 + 3×2 + F1
(0) = 44 + 8 = 52.
W2
F2
(2), .. F2
(x = y3
= 2) = min W2
(x2
,2) = min (64, 55, 50, 49, 52) = 49,
x2
2
,
`2
(x = y3
= 2) = 3
x = 3
= 3, ,
F2
(x = y3
= 3) = 63; `2
(x = y3
= 3) = 3.
F2
(x = y3
) . 2, . 3.
3
x= 3
|
0 |
1 |
2 |
3 |
4 |
F2
(x= y3
) |
24 |
36 |
49 |
63 |
78 |
(x= y3
) |
2 |
2 |
3 |
3 |
4 |
. k=3 F3
(x = y4
):
x = 4
= 0, . . 4.
F3
(x = y4
) = min W3
(x3
,0) = min (80, 71, 65, 62, 62) = 62,
x3
3
,
`3
(x = y4
= 0) = 3 `3
(x = y4
= 0) = 4.
, , .
= 3 = 4.
,
= 3.
. , ,
3
+ 3
- d3
= y4
3 + 3
- 4 = 0,
3
= 1.
(3)
, ,
2
+ 2
- d2
= y3
|
|
|
xk
|
yk
= yk+1
+ dk
- xk
|
W
k
(xk
, yk+1
) =
j
k
(xk
) + hk
yk+1
+ Fk-1
(yk
)
|
0 £ y3
£ d3
|
x = y3
|
0 £ x2
£ d2
+ y3
|
x2
|
y2
= y3
+ d2
- x2
|
W2
(x2
, y3
) = a + bx + c + h2
y3
+ F1
(y2
) |
0 £ y3
£ 4 |
x = y3
|
0 £ x2
£ 2 + y3
|
x2
|
y2
= y3
+ 3 - x2
|
|
y3
= 0 |
0 £ x2
£ 2 |
x2
= 0
x2
= 1
x2
= 2
|
y2
= 2-0 = 2
y2
= 2- 1 = 1
y2
= 2-2 = 0
|
W2
(0;0) = 02
+ 5×0 + 2 + 3×0 + F1
(2) =2+28 =30
W2
(1;0) = 12
+ 5×1 + 2 +3×0 + F1
(1)=8+17 =25
W2
(2;0) = 22
+5×2 + 2 + 3×0 +F1
(0) =16+8=24*
|
y3
= 1 |
0 £ x2
£ 3 |
x2
= 0
x2
= 1
x2
= 2
x2
= 3
|
y2
= 3 - 0 = 3
y2
= 3-1 = 2
y2
= 3-2 = 1
y2
= 3-3 = 0
|
W2
(0;1) = 02
+ 5×0 + 2 + 3×1 + F1
(3) = 5+41=46
W2
(1;1) = 12
+ 5×1 + 2+ 3×1 + F1
(2)=11+28 =39
W2
(2;1) = 22
+ 5×2 + 2 + 3×1 + F1
(1)=19+17 =36*
W2
(3;1) = 32
+ 5×3+ 2 + 3×1 + F1
(0)=29+8 =37
|
y3
= 2 |
....................... |
........ |
............................ |
............................................................. |
y3
= 3 |
0 £ x2
£ 5 |
x2
= 0
x2
= 1
x2
= 2
x2
= 3
x2
= 4
x2
= 5
|
y2
= 5 - 0 = 5
y2
= 5 - 1 = 4
y2
= 5 - 2 = 3
y2
= 5 - 3 = 2
y2
= 5 - 4 = 1
y2
= 5 - 5 = 0
|
W2
(0;3) = 02
+ 5×0 + 2 + 3×3 + F1
(5) = 11+73=84
W2
(1;3) = 12
+ 5×1 + 2+ 3×3 + F1
(4)=17+56 =73
W2
(2;3) = 22
+ 5×2 + 2 + 3×3 + F1
(3)=25+41 =66
W2
(3;3) = 32
+ 5×3+ 2 + 3×3 + F1
(2)=35+28 =63*
W2
(4;3) = 42
+ 5×4 + 2 + 3×3 + F1
(1)=47+17 =64
W2
(5;3) = 52
+ 5×5+ 2 + 3×3 + F1
(0)=61+8 =69
|
y3
= 4 |
0 £ x2
£ 6 |
x2
= 0
x2
= 1
x2
= 2
x2
= 3
x2
= 4
x2
= 5
x2
= 6
|
y2
= 6 - 0 = 6
y2
= 6 - 1 = 5
y2
= 6 - 2 = 4
y2
= 6 - 3 = 3
y2
= 6 - 4 = 2
y2
= 6 - 5 = 1
y2
= 6 - 6 = 0
|
W2
(0;4) = 02
+ 5×0 + 2 + 3×4 + F1
(6) = 14+92=106
W2
(1;4) = 12
+ 5×1 + 2+ 3×4 + F1
(5)=20+73 =93
W2
(2;4) = 22
+ 5×2 + 2 + 3×4 + F1
(4)=28+56 =84
W2
(3;4) = 32
+ 5×3+ 2 + 3×4 + F1
(3)=38+41 =79
W2
(4;4) = 42
+ 5×4 + 2 + 3×4 + F1
(2)=50+28 =78*
W2
(5;4) = 52
+ 5×5+ 2 + 3×4 + F1
(1)=64+17 =81
W2
(6;4) = 62
+ 5×6+ 2 + 3×4 + F1
(0)=80+8 =88
|
|
|
|
xk
|
yk
= yk+1
+ dk
- xk
|
W
k
(xk
, yk+1
) =
j
k
(xk
) + hk
yk+1
+ Fk-1
(yk
)
|
0 £ y4
£ 0 |
x = y4
|
0 £ x3
£ d3
+ y4
|
x3
|
y3
= y4
+ d3
- x3
|
W3
(x3
, y4
) = a + bx3
+ c + h3
y4
+ F2
(y3
) |
y4
= 0 |
x = y4
|
0 £ x3
£ 4 |
x3
|
y3
= y4
+ 4 - x3
|
|
y4
= 0 |
0 £ x3
£ 4 |
x3
= 0
x3
= 1
x3
= 2
x3
= 3
x3
= 4
|
y3
= 4-0 = 4
y3
= 4- 1 = 3
y3
= 4-2 = 2
y3
= 4-3 = 1
y3
= 4-4 = 0
|
W3
(0;0) = 02
+ 5×0 + 2 + 2×0 + F2
(4)=2+78=80
W3
(1;0)= 12
+ 5×1 + 2 + 2×0 + F2
(3)=8+63=71
W3
(2;0)= 22
+ 5×2 + 2 + 2×0 + F2
(2)=16+49=65
W3
(3;0) = 32
+ 5×3 + 2 + 2×0 + F2
(1)=26+36=62*
W3
(4;0)= 42
+ 5×4 + 2 + 2×0 + F2
(0)=38+24=62*
|
5
|
|
|
|
3 |
, . |
1
= 2 |
2
= 1 |
3
= 1 |
1
= 2 |
, . |
1
= 2 |
2
= 2 |
3
= 3 |
1
+ 2
+ 3
= 7 |
, . |
d1
= 3 |
d2
= 2 |
d3
= 4 |
d1
+ d2
+ d3
= 9 |
( ), . |
2
= 1 |
3
= 1 |
4
= 0 |
, . |
j(1
)=16 |
j(2
)=16 |
j(3
)=26 |
j(1
) + j(2
) + j(3
) = 58 |
, . |
h1
2
=1 |
h2
3
=3 |
0 |
h1
2
+ h2
3
= 4 |
2 + 2
- 2 = 1,
2
= 1;
(2) 1
(x)
.
,
1
= 2
2
= 3
3
= 3,
62 .
. 5 ,
1
+1
³ d1
2
+2
³ d2
3
+3
³ d3
2 + 2 ³ 3 1 + 2 ³ 2 1 + 3 ³ 4
1
+ 1
+ 2
+ 3
= d1
+ d2
+ d3
2 + 2 + 2 + 3 = 3 + 2 + 4
j(1
) + j(2
) + j(3
) + h1
2
+ h2
3
= F3
(y4
=0)
16 + 16 + 26 + 1 + 4 = 62
, 4 , .
10
.
n . . j- xj
, yj
, .
ajk
- j- , k- . aij
, . X(x1
, , xn
), (1
, , n
). ,
( - ) = = ( - )-1
.
( - )-1
, , , , .
.
, , .. , S, = ( - )-1
= S
, .
11.
,
. , . (..) :
.., .. . .. .. , .. . , .
, . : .
. . , .
?
.
.
, .
, ,
, , - , .. :
, , , - , , . , , , . , , :
- ( ) , . , .
, .
. , .
. . 1.
, , . , . . 1. , ; , , . 2.
. 2 , . . x 1- . 1- , 2-
. . , . 3. . 3 , . . 1- 2- , 1-
. . . : 3,5), 2- .
12.
, - .
. , .. ( , ).
?
. .
- , Q. `Q - .. Q: , pi
qi
. () - . s r. ,
D[Q] = M [(Q - `Q)2
] = M [Q2
] - `Q2
. Q1
, Q2
, Q3
, Q,4
. `Qi
ri
.
, :
Q1
|
: |
5 |
2 |
8 |
4 |
`Q1
= 29/6 4.81 |
r1
1.77 |
1/2 |
1/6 |
1/6 |
1/6 |
Q2
|
: |
2 |
3 |
4 |
12 |
`Q2
= 25/6 4.16 |
r2
3.57 |
1/2 |
1/6 |
1/6 |
1/6 |
Q3
|
: |
8 |
5 |
3 |
10 |
`Q3
= 7 |
r3
2.30 |
1/2 |
1/6 |
1/6 |
1/6 |
Q4
|
: |
1 |
4 |
2 |
8 |
`Q4
= 17/6 2.81 |
r4
2.54 |
1/2 |
1/6 |
1/6 |
1/6 |
, `Q r.
`Q1
=å qi
pi
= 5*1/2+2*1/6+8*1/6+4*1/6=29/6
j
r1
= M [Q2
1
] - (Q1
)2
; M [Q2
1
] = 25*1/2+4*1/6+64*1/6+16*1/6=159/6;
Q2
1
= 841/36; D [Q1
] = (159*6-841)/36 = 113/36;
`Q r - , (. .):
4 . (`Q, r), , - . , . (`Q¢, r¢) (`Q, r) `Q¢³`Q r¢£ r. 1- 2-, 3- 2- 3- 4-. 1- 3- - 3- , ., , . , , , .
, (`Q, r) , . , j (Q)= 2×Q - r . :
j (Q1
)= 2*4.81-1.77 = 7.85; j (Q2
)= 4.75; j (Q3
)= 11.70; j (Q4
)= 3.08
, 3- - , 4- - .
13.
.
, , : , , .. . .
: . E . E , m
.
V . , V=D[E]= M[( E- m
)2
] s =.
, , .
xi
- , i- . Ei
- ( , ) i- , . Vij
i- j - ( Kij
). mi
- Ei
si
= , Vii
- Ei
. i- si
.
, , . ( , - ), , , Ep
, mp
=M[Ep
]=. D[Ep
]= . . D[Ep
] Vp
. , .
: , . " ", .
:
xi
,
Vp
= ,
,
mp
, ..
mp
=.
xi
- , :
=1 .
() *. x*
i
>0 , x*
i
i- . x*
i
<0 , "short sale". , xi
³ 0 . "short sale" ? x*
i
< 0 , , , - i- ( , ). . !
( ), .
m0
- , x0
- . mr
- Vr
, sr
- (), , (1-x0
) . mp
=x0
m0
+(1-x0
)mr
, Vp
=(1-x0
)2
Vr
sp
=(1-x0
) sr
(, ). x0
,
mp
= m0
+sp
(m -m0
)/ sr
,
.. .
. 1 n .
x0
m0
+ = mp
x0
+ = 1
.
V - , X=(xi
), M=(mi
) - - xi
, i- , i=1,.., n. I - n- -, 1. xi
.
V-1
- , V . , , (
-), , , , V-1
(M-m0
I) - - n . , mp
. , mp
. , mp
. X*
mp
, , X*
mp
, x0
.
. : 2 4 10 2 4 . ? "short sale" ? . , m0
=2, M=, V=. mp
. V . : V-1
= . :
.
, X*
=((m
-2)/5). , (m
-2)/10 . , x*
0
=1-(m
-2)/5 . , "short sale" , x*
0
< 0, .. m
> 7 .
, ,
(1) :
.
, .. ,
, , ..
, :
, , (2):
(3)
, .
14.
, (, ) . , , - . -e , - , , , . ( ). ? . . , , . ?
, , -e . . , , .. . .. - , , .
, -e , , - . .
1.
. ,
. .
"" . . , , 15 . , , .
. - ?
( ). -e , , .. .
. , ,
, , 2,2,3,1 . 3 . , 3- .
( ). . -e ,
. , ,
, , 8,6,5,7 . 5. 3- .
( ). ,
. . 1, , 0, " " ( , ). 2- .
. .
, , . . ? .
. , - ,
, . , , .
, . (1/2, 1/6, 1/6, 1/6).
7, 3- .
. - ,
, . , .
. 7/6, 3- .
, (..):
4 .
, , .Q3
. ,
. .
Q1
, .
Q2
.Q4
.
3- .
, , . , , , . , , .. , 3- .
, , . , . :
. , 3- , 4- .
. .
, . - - .
15. -
- , , , , .
, " " [ ]. , (, Statistica for Windows, Statgraf, SAS), , [ ].
. , : Economic Report of the President, 1995,Wash,1995; Statistical Abstract of the USA, 1995, Wash, 1995, .
, ( 1987 .), ( 1987 .) 1960-1995 ..
.. |
|
(. .)
Xt
|
(. .)
Kt
|
(. .)
Lt
|
1 |
1960 |
1986,9 |
5596,9 |
65,8 |
2 |
1961 |
2035,7 |
5685,6 |
65,7 |
3 |
1962 |
2140,5 |
5849,8 |
66,7 |
4 |
1963 |
2234,2 |
6098,9 |
67,8 |
5 |
1964 |
2357,4 |
6336,1 |
69,3 |
6 |
1965 |
2493,3 |
6621,5 |
71,1 |
7 |
1966 |
2635,7 |
6921,8 |
72,9 |
8 |
1967 |
2705,6 |
7237,0 |
74,4 |
9 |
1968 |
2816,0 |
7434,0 |
75,9 |
10 |
1969 |
2891,0 |
8062,0 |
77,9 |
11 |
1970 |
2889,5 |
8416,8 |
78,7 |
12 |
1971 |
2978,2 |
8596,7 |
79,4 |
13 |
1972 |
3133,2 |
9533,6 |
82,2 |
14 |
1973 |
3298,5 |
9718,1 |
85,1 |
15 |
1974 |
3283,5 |
9455,7 |
86,8 |
16 |
1975 |
3250,2 |
9493,2 |
85,8 |
17 |
1976 |
3414,0 |
9620,9 |
88,8 |
18 |
1977 |
3568,2 |
9755,9 |
92,0 |
19 |
1978 |
3738,8 |
11217,1 |
96,0 |
20 |
1979 |
3848,6 |
12117,0 |
98,8 |
21 |
1980 |
3824,4 |
11691,4 |
99,3 |
22 |
1981 |
3883,1 |
11987,8 |
100,4 |
23 |
1982 |
3794,5 |
10717,1 |
99,5 |
24 |
1983 |
3938,5 |
10849,2 |
100,8 |
25 |
1984 |
4177,5 |
11989,2 |
105,0 |
28 |
1987 |
4544,5 |
13063,7 |
112,4 |
29 |
1988 |
4724,0 |
13382,5 |
115,0 |
30 |
1989 |
4854,2 |
13838,9 |
117,3 |
31 |
1990 |
5002,5 |
15411,8 |
117,9 |
32 |
1991 |
4881,6 |
14295,5 |
116,9 |
33 |
1992 |
4984,1 |
14252,1 |
117,6 |
34 |
1993 |
5139,9 |
14412,5 |
119,3 |
35 |
1994 |
5372,0 |
15319,8 |
123,1 |
36 |
1995 |
5604,1 |
15939,2 |
126,7 |
) , . . ,
,
- ( ) ,
- ( ) t=0 (x
1
- ),
- ,
e
t
.
[ ]
, ( ) ,
= 1854,1 1959 . (. .)
= 96,66 (. .),
t
= 1854,1 + 96,66×t
.
( )
,
(1996) = 1854,1 + 96,66×37 = 5430,6;
(1997) = 5527,3;
(1998) = 5623,9.
= 5071,7 + 290,05t
;
(1996) = 5071,7 + 290,05×37 = 15803,6;
(1997) = 16093,6;
(1998) = 16383,7;
= 60,36 + 1,796t
;
(1996) = 60,36 + 1,796×37 = 126,8;
(1997) = 128,6;
(1998) = 130,4.
. 1960 1995 .. 1996 1998 .., .
, ( ) .
1996 1998 ..(. .)
1996-1998 ..
(. .)
) ( ) .
,
,
a
K
, a
L
.
- , ( ) .
.
(ln X
t
, t
= 1,,T
), (ln K
t
, t
= 1,,T
) (ln L
t
, t
= 1,,T
),
.
" Statistica for Windows"
,
(= 2,248)
.
,
(1996)
(1997) = 5576,7;
(1998) = 5680,1.
, ( ) .
1996-1998 ..
(. .)
) .
1960-1995 .. , 1960-1961 .., 1969-1970 .., 1974-1975 .., 1980-1982 .., 1990-1992 .. .
: 96,7 . ., 290,1 . ., 1,8 . . 1% 0,404%, 1% - 0,803%, .. . , 1998 . 16383,7 . . ( 1995 . 2,8%), 1998 . : 5623,9 . . ( 0,35%), 5680,1 ( 1,4%).
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