:

:
:
:

/.: .., .. . , .: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) .

9
(1), (2) (3) . (1), (2), , (3) . .

A , B C :

, , C=(c1 , , cn )

, , .

, , ,

, ,

:

(4)

:

(5)

(6)

10
. (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 ,

11
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)

12
, , .. . (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

13
(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.

14
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. , . , . :

15
, .

,

(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 , . , , - , , , .

16
, :

(1 , y2 , y3 )

f = 208y1 + 107y2 +181y3 (1)

, , , ,

(2)
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.

17
, (4) . . , 3 =4 , 4 .

6. " "

, .. ² ². . T(t1 ,t2 ,t3 )- . ,

H + Q-1 T 0.

,

T (t1 , 0, t3 ),

W = 6t1 + 4t3 (1)

(, , )

(2)

, 1/3

(3)

t1 0, t3 0. (4)

(2) (3) :


(6)
(5)

: (1) (5), (6) (4).

: . . 1. ²²

t1 =, t2 =0, t3 =

18
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)

19
.

(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.

20

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. .

21

- . 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 . .

22
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 .
23

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.

24

. , 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) .

.

25
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)

26
.. 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)

(27)
k=1,
(30)
(29)
(28)

,

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

27
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 )

28
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.

29
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


30
2
K=2
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

31
4

K=3
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

32

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 , , , , .

.

33
, , .. , S,

= ( - )-1 = S

, .

11.

, . , . (..) :

.., .. . .. .. , .. . , .

, . : .

. . , .

?

.

.

, .

, ,

, , - , .. :

34
, , , - , , . , , , . , ,

:

- ( ) , . , .

, .

. , .

. . 1.

. 2

, , . , . . 1. , ; , , . 2.


35
. 2 , . . x 1- . 1- , 2-

. . , . 3. . 3 , . . 1- 2- , 1-

. . . : 3,5), 2- .

12.

, - .

. , .. ( , ).

?

. .

- , Q. `Q - .. Q: , pi qi . () - . s r. ,

36
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 - , (. .):

`Q
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- - .

37
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 .

38
() *. 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 .

39
. : 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)

, .

40
14.

, (, ) . , , - . -e , - , , , . ( ). ? . . , , . ?

, , -e . . , , .. . .. - , , .

, -e , , - . .

1.

. ,

. .

"" . . , , 15 . , , .

. - ?

( ). -e , , .. .

. , ,

41

, , 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- .

. - ,

42

, . , .

. 7/6, 3- .

, (..):

4 .

, , .Q3

. ,

. . Q1

, . Q2

.Q4

.

3- .

, , . , , , . , , .. , 3- .

, , . , . :

. , 3- , 4- .

. .

, . - - .

15. -

- , , , , .

43
, " " [ ].

, (, 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
44
) , .

. ,

,

- ( ) ,

- ( ) 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 ;

45

(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 .., .

, ( ) .

46
1996 1998 ..

(. .)

1996-1998 ..

(. .)

) ( ) .

,

,

a K , a L .

- , ( ) .

47

.

(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 .. .

48
: 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%).