«  »

/ , /


: 5


" "

2001

" "

.

Ƞ ...

ɠ

Ƞ

, . ,


(1)

(x1, x2, x3, x4),


(2)

:
(3)


(4)

. (3) 5, 6, 7


(5)

. (5), 1³0, 2³0, ,5³0,, 7³0. (6)

, (2) .

, (5) , . 1, 2, 3, 4,

x1=0, x2=0, x3=0, x4=0, x5=103, x6=148, x7=158 (7)

x1=0, x2=0, x3=0, x4=0(8)

. (2) , , . , . , . (5)


(9)

2=3=4=0 1. ,



.. 0 £ 1 £ 37

1 1 =37, , (9). (5) 堠 1=37, 2=0, 3=0, 4=0; x5=29; x6=0; x7=84 (10)

, (5), (5) 1 , , , ꠠ


, 21=4.

, 1.

~
C

36 32 10 13 0 0 0
x1 x2 x3 x4 x5 x6 x7
0 5 103 2 3 4 1 1 0 0 z0 =
H
0 6 148 4 2 0 2 0 1 0
0 7 158 2 8 7 0 0 0 1
z0 -z 0 - z -36 -32 -10 -13 0 0 0
0 5 29 0 2 4 0 1 -1/2 0
0 1 37 1 1/2 0 1/2 0 0
36 7 84 0 7 7 -1 0 -1/2 1 min (29/2; 64;12)=12
z0 -z 1332-z 0 -14 -10 5 0 9 0 min (-14;-10) = -14
36 5 5 0 0 2 2/7 1 -5/14 -2/7
0 1 31 1 0 -1/2 4/7 0 2/7 -1/14
14 2 12 0 1 1 -1/7 0 -1/14 1/7
z0 -z 1500-z 0 0 4 3 0 8 2 Dj ³0

a`ij=aij (ais/ars)*arj

a`iq=aiq (ais/ars)*arq

b`i=bi - (ais/ars)*br

b`r=br/ars


s=1, r=2


a`12=3-2/4 *2= 2

a`13=4

a`14=1-2/4 *2=0

a`15=1

a`16=0-2/4*1= -2/4

a`17=0

a`32=8-2/4* 2= 7

a`33=7

a`34=0-2/4* 2= -1

a`35= 0

a`36=0-2/4 *1= -2/4

a`37=1

a`21=a21/a21=1

a`22=a22/a21=1/2

a`23=0

a`24=1/2

a`25=0

a`26=1/4

a`27=0

a`41= 0

a`42= -14

a`43= -10

a`44=5

a`45=0

a`46=9

a`47=0

a`11=a`31=0

b`1=103-148/4*2=29

b`2=148/4=37

b`3=158-148/4*2=84


(5)


2x2 + 4x3 + x5 - 1/2x6 = 29

x1 + 1/2x2 + 1/2x4 + 1/4x6 = 37 (11)

7x2 + 7x3 - x4 -1/2x6 + x7 = 84

2, 3, 4, 6, , (10), 󠠠 1=37, 2=0, 3=0, 4=0. (12)

(2) -361 - 142 - 103 - 134 = 0 z (13)

(5).



(14)

, (5) 1. (14) D1= -36. 21=4 1 (5), . 1 (2). , (14). , (14) 9 ; 젠

-142 - 103 + 54 - 96 = 1332 z (15)

, (14)


(16)

(11) (5) (10) (12), (16) . , (5) , z=1332+14x2+10x3-5x4-9x6 , 2, .

, Dj - xj (16), . (16) 򠠠 min(Dj<0) = min(-14,-10) = -14 = D2. 2 (11) , 
(17)

2 (11), . 32=7.

(16), .

a`ij=aij (ais/ars)*arj

a`iq=aiq (ais/ars)*arq

b`i=bi - (ais/ars)*br

b`r=br/ars


s=1, r=2


a`11=0

a`13=4-2/7*7=2

a`14=0+2/7 *1=2/7

a`15=1

a`16= -5/14

a`17=0-2/7*1=-2/7

a`21=1

a`23= -1/2

a`24=4/7

a`25=0

a`26=2/7

a`27= -1/14

a`31= a31/a32=0

a`32=1

a`33= a33/a32=1

a`34= -1/7

a`35= 0

a`36=-1/14

a`37=1/7

a`41= 0

a`42= -14+2*7=0

a`43= 4

a`44=3

a`45=0

a`46=8

a`47=2

a`12=a`22=0

b`1=29-84/7*2=5

b`2=37-84/7*1/2=31

b`3=84/7=12


2 x3 + 2/7 x4 + x5 5/14 x6 2/7 x7 = 5

x1 - x3 +
x4 + 2/7 x6 1/14 x7 = 31 (18)

x2 + x3 - 1/7 x4 1/14 x6 + 1/7 x7 = 12

4 x3 + 3 x4 + 8 x6 + 2 x7 = 1500 - z


(18) (5)

x1=37, x2=0, x3=0, x4=0, x5=29, x6=0, x7=84 (19)

.. x1=37, x2=0, x3=0, x4=0 (20)

:

5=5

6=0 (21)

7=0

(18) , 2. (18) . z 堠

z = 1500 - 4 x3 - 3 x4 - 8 x6 - 2x7 (22)

( , xj³0), ,

x3=0, x4=0, x6=0, x7=0 (23)

, (20) zmax = 1500 (24)

, , , .

. , D3=4 3 , ( ), 4 .

, x3=0, x4=0. , . , . :




, (x1=0, x2=0) (x1=37, x2=0) (x1=31, x2=12) .

.

, , - , , , "" 1 , 2 , 3 . : 1, 2, 3 .


1, 2, 3 , , . , .

, ,

, , 2 , 4 2 ( ). 1, 2, 3 21 + 42 + 23, .. , . 36 . , , 21 + 42 + 23 ³ 36.

, :

31 + 22 + 83³32

41 + 73³10

1 + 22 ³13

, 1031 + 1482 + 1583 . 1, 2, 3, . , , - , , , .

, : (1, y2, y3) ⠠ f = 1031 + 1482 + 1583 (1)

, , , ,

21 + 42 + 23 ³ 36

31 + 22 + 83³32 (2)

41 + 73³10

1 + 22 ³13

y1
0, y2
0, y3
0. (3)

,  頠 頠
(1, 2, 3, 4) 蠠
(y1, y2, y3)



x 1 (21 + 42 + 23 - 36) = 0 y1 (2x1 +3x2 + 4x3 + x4 - 103) = 0

x 2 (31 + 22 + 83 - 32) = 0 y2 (4x1 +2x2 + 2x4 - 148) = 0

x 3 (41 + 73- 10) = 0 y3 (2x1 +8x2 + 7x3 - 158) = 0 .

x 4 (1 + 22 - 13) = 0

, 1>0, x2>0.


2y1 + 4y2 + 2y3 - 36 = 0

3y1 + 2y2 + 8y3 - 32 = 0

, , , 1=0,


4y2 + 2y3 - 36 = 0

2y2 + 8y3 - 32 = 0

򠠠 2=8, 3=2.

, ⠠ 1=0; 2=8; 3=2, (4)

1500.

, (4) . . , 3=2 , 2 .

" "

, .. ² ². . T(t1,t2,t3)- . , 堠 H + Q-1T
0.

, 𠠠 T (0, t2, t3), 蠠 W = 8t2 + 2t3 (1) (, , )

(2)

,   1/3 ࠠ


(3)

蠠 t2
0, t3
0. (4)

(2) (3) :


(5)

(3) t2£148/3, t3£158/3 (6)

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

: . . 2. ²²

t1=0, t2=14, t3=0 112.

e 2.

j 36 32 10 13 b x4+i yi ti
2 3 4 1 103 5 0 0
aij 4 2 0 2 148 0 8 14
2 8 7 0 158 0 2 0
xj 31 12 0 0 1500 112
Dj 0 0 4 3

, 3 () 40; 60; 70 , 4 , 36; 32; 40; 53 . =




. , .

.

åi =40+60+70=170 , åbi = 36+32 +40 +53 =161, .. . 170-161 = 9 , , , , , .

²- ².


b1 =36 b2 =32 b3 =40 b4 =53 b5 =9

1 =40 36 4 p1 =0
a2 =60 28 32 p2 =
a3 =70 * 8 53 9 p3 =
q1 = q2 = q3 = q4 = q5 =

:

L= 36* 2 + 4 *3 + 28 *2 + 32 + 8* 7+ 53 =281

, (3), (4) . , 1 = 0. ,
.

D11 = 0, p1 + q1 - c11 = 0, 0+q1 -2 = 0, q1 = 2

D12 = 0, p1 + q2 - c12 = 0, 0+q2 -3 = 0, q2 = 3

D22 = 0, p2 + q2 - c22 = 0, 2 +3-2 = 0, 2 = -1

.., : q3=2, p3=5, q4= -4, q5= -5.

(6) :

D21 = p2 + q5 - c21 = -1+2-4 = -3

D31 = p3 + q1 - c31 = 5+2-2 = 5

D32 = 1; D13 = -2; D14 = -5; D24 = 0; D15 = -5; D25 = -6.

max (
) = 5 =

31 - , , , , - . 31-11-12-22-23-33.

36 4 36-r 4+r 28 12


28 32 28-r 32+r 20 40
8 r 8-r 8


= 8

:


bj
b1 =36 b2 =32 b3 =40 b4 =53 b5=9
ai

1 =40
28 12 * p1 =0
a2 =60 20 40 p2 = -1
a3 =70 8 53 9 p3 =0
q1 =2 q2 = 3 q3 = 2 q4 = 1 q5=0

, .



D13 = -2; D14 = 0; D15 = 0; D21 = -3; D24 = -2; D25 = -1; D32 = -4; D33 = -5,

.. Dij £ 0 i = 1,m; j = 1,n





:

L= 28* 2 + 12 *3 + 20 *2 + 40 + 8* 2+ 53 =241 .

.

(n=4). 700 . (b=700), 100 . . fj(xj) 1, , , 50 , 600 . . , 50 . . I

. 2. f2(x2) F1(x - x2) = f1(x- x2) - , . 3.

, F3(x), (x) .. . 6 x= 700.

2

x - x2 0 100 200 300 400 500 600 700

x2 F1(x - x2)

f2(x2) 0 15 24 30 36 40 43 45

0

0 0 15 24 30 36 40 43 45

100 18 18* 33* 42* 48 54 58 61

200 26 26 41 50* 56 62 66

300 34 34 49 58* 64* 70*

400 39 39 54 63 69

500 42 42 57 66

600 44 44 59

700 46 46

3

x 0 100 200 300 400 500 600 700

F2(x) 0 18 33 42 50 58 64 70

` (x)

0 0 100 100 200 300 300 300

4

x - x3 0 100 200 300 400 500 600 700

x3 F2(x - x3)

f3(x3) 0 18 33 42 50 58 64 70

0

0 0 18* 33 42 50 58 64 70

100 16 16 34* 49* 58 66 74 80

200 27 27 45 60* 69 77 85

300 37 37 55 70* 79* 87*

400 44 44 62 77 86

500 48 48 66 81

600 50 50 68

700 56 56

5

x 0 100 200 300 400 500 600 700

F3(x) 0 18 34 49 60 70 79 87

(x)

0 0 100 100 200 300 300 300

6

x - x4 0 100 200 300 400 500 600 700

x4 F3(x - x4)

f4(x4) 0 18 34 49 60 70 79 87

0 0 87

100 10 89*

200 17 87

300 23 83

400 29 78

500 34 68

600 38 56

700 41 41 .

: Zmax = 89 . .,

 *4 = 4 (700) = 100 . .

600 . . . 5 ,  x*3 = 3 (700-x*4) = 3 (600) = 300 . .

, 젠 x*2 = 2 (700 - x*4 - x*3) = 2 (300) = 100 . .

x*1 = 700 - x*4 - x*3 - x*2 = 200 . .

, :

x*1 =200; x*2 =100; x*3 = 300; x*4 = 100.

89 . .

: f1(x*1) + f2(x*2) + f3(x*3) + f4(x*4) = z max

24+18+37+10=89

Ƞ

.

() : d1=3 , d2=2, - d3=3 . 3 , .. y1=3. h1=4, h2=3, h3=2. xj j- 頠 jj(xj) = xj2 + 2xj + 2

.. =1; b=5; =2. , , , .

:

d1 d2 d3 a b c h1 h2 h3 y1

3 2 3 1 2 2 4 3 2 3

, F1 (x = y2), F2 (x = y3), ..., Fk (x = yk+1), ... 젠 1 (x= y2), 2 (x = y3 ), ..., ` k (x = yk+1), ...

k = 1.

x = 2

0 2 d2 + d3 0 y2 2 + 3 .. 2 = 0, 1, 2, 3, 4, 5.

x1, 0 1 d1 + 2 0 1 3 + 2

1 + 1 - d1 = 2 , x= 2

x1 = y2 + d1 - y1 = y2 + 3 - 3 = y2

. , 2 1 F1(x = y2) = W1 (x1, y2)

2 0 6 ,

y2 = 0, x1 = 0, W1 (0;0) = 02 + 2×0 + 2 + 4×0 = 2*

y2 = 1, x1 = 1, W1 (1;1) = 12 + 2×2 + 2 + 4×1 = 11

y2 = 2, x1 = 2, W1 (2;2) = 22 + 2×2 + 2 + 4×2 = 18

y2 = 3, x1 = 3, W1 (3;3) = 32 + 2×3 + 2 + 4×3 = 29

y2 = 4, x1 = 4, W1 (4;4) = 42 + 2×4 + 2 + 4×4 = 42

y2 = 5, x1 = 5, W1 (5;5) = 52 + 2×5 + 2 + 4×5 = 57

F1(x ) . 1

1

x = y2 0 1 2 3 4 5

F1 (x = y2)

2 11 18 29 42 57

x1(x=y2) 0 1 2 3 4 5

. k = 2 F2(x = y3)

2,

0 £ x2 £ d2 + y3 蠠 0 £ x2 £ 2 + y3 (1)

x = 3,

0 £ y3 £ d3 , .. 0 £ y3 £ 3

2 2 3 젠 x2 + y2 - d2 = y3 򠠠 y2 = y3 + d2 - x2 = =y3 + 2 - x2 (2)

0 3, W2 (x2, x), F2(x ) 2(x ).

x = 3 = 0. , (1), 0 £ x2 £ 2, .. 2 : 0, 1, 2, 2 2, (2): 2 = 2 - 2

:

蠠 x2 = 0, 2 = 2 , W2 (0,2) = 02 + 2×0 + 2 + F1(2) = 2 + 18 = 20,

x2 = 1, y2 = 2 - 1 = 1, W2 (1,2) = 12 + 5×1 + 2 + F1(1) = 8 + 11 = 19,

x2 = 2, y2 = 2 - 2 =0, W2 (2,2) = 22 + 5×2 + 2 + F1(0) = 16 + 2 = 18*,

W2 F2 (0), ..

F2 (x = y3 = 0) = 18,

2, ` 2 (x = y3 = 0) = 2

x = 3 = 1. , (1), 0 £ x2 £ 3, .. 2 : 0, 1, 2, 3, 2 2, (2): 2 = 3 - 2

:

蠠 x2 = 0,  y2 = 3-0 = 3, W2 (0,1) = 02 + 2×0 + 2 + 3×1 + F1(3) = 5 + 29 = 34,

x2 = 1, y2 = 3-1 = 2, W2 (1,2) = 12 + 2×1 + 2 + 3×1 + F1(2) = 8 + 18 = 26,

x2 = 2, y2 = 3-2 = 1, W2 (2,1) = 22 + 2×2 + 2 + 3×1 + F1(1) = 13 +11 = 24,

x2 = 3, y2 = 3-3 = 0, W2 (3,1) = 32 + 2×3 + 2 + 3×1 + F1(0) = 20 + 2 = 22*,

W2 F2 (1), ..

F2 (x = y3 = 1) = min W2 (x2,1) = 22,

2, ` 2 (x = y3 = 1) = 3

x = 3 = 2. , (1), 0 £ x2 £ 4, .. 2 : 0, 1, 2, 3, 4, 2 2, (2): 2 = 4 - 2

蠠 x2 = 0,  y2 = 4-0 = 4, W2 (0,2) = 02 + 2×0 + 2 + 3×2 + F1(4) = 8 + 42 = 50,

x2 = 1, y2 = 4-1 = 3, W2 (1,2) = 12 + 2×1 + 2 + 3×2 + F1(3) = 11 + 29 = 40,

x2 = 2, y2 = 4-2 =2, W2 (2,2) = 22 + 2×2 + 2 + 3×2 + F1(2) = 16 + 18 = 34,

x2 = 3, y2 = 4-3 = 1, W2 (3,2) = 32 + 2×3 + 2 + 3×2 + F1(1) = 23 + 11 = 34*,

x2 = 4, y2 = 4-4 = 0, W2 (4,2) = 42 + 2×4 + 2 + 3×2 + F1(0) = 32 + 2 = 40.

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. , (1), 0 £ x2 £ 5, .. 2 : 0, 1, 2, 3, 4, 5, 2 2, (2): 2 = 5 - 2

蠠 x2 = 0,  y2 = 5-0 = 5, W2 (0,3) = 02 + 2×0 + 2 + 3×3 + F1(5) = 11 + 57 = 68,

x2 = 1, y2 = 5-1 = 4, W2 (1,3) = 12 + 2×1 + 2 + 3×3 + F1(4) = 14 + 42 = 56,

x2 = 2, y2 = 5-2 = 3, W2 (2,3) = 22 + 2×2 + 2 + 3×3 + F1(3) = 19 + 29 = 48,

x2 = 3, y2 = 5-3 = 2, W2 (3,3) = 32 + 2×3 + 2 + 3×3 + F1(2) = 26 + 18 = 44*,

x2 = 4, y2 = 5-4 = 1, W2 (4,3) = 42 + 2×4 + 2 + 3×3 + F1(1) = 35 + 11 = 46.

x2 = 5, y2 = 5-4 = 0, W2 (5,3) = 52 + 2×5 + 2 + 3×3 + F1(0) = 46 + 2 = 48.

W2 F2 (3), ..

F2 (x = y3 = 3) = min W2 (x2,3) = 44,

2, ` 2 (x = y3 = 3) = 3

F2 (x = y3) . 2.

2

x= 3 0 1 2 3

F2 (x= y3) 18 22 34 44

(x= y3)

2 3 2 3 3

. k=3 F3 (x = y4):

x = 4 = 0, .

0£y4£0; x=y4; 0 £ x3 £ d3 + y4 → 0 £ x3 £ 3; y3 = y4 + d3-x3= y4+3- x3;

W3(x3, y4) = a + bx3 + c + h3y4 + F2(y3)= +2 x3+2 + 2 y4 + F2(y3)

x3=0 y3=3 W3(0;0)=02 + 2×0 +2 +2×0 +F2(3)=2 +44=46

x3=1 y3=2 W3(1;0)=12 + 2×1 +2+2×0 + F2(2)=5 +34=39

x3=2 y3=1 W3(2;0)=22 + 2×2 +2+2×0 + F2(1)=10+22=32*

x3=3 y3=0 W3(3;0)=32 + 2×3 +2+2×0 +F2(0)=17 +18=35

F3 (x = y4) = min W3 (x3,0) = 32, ` 3 (x = y4 = 0) = 2.

, , . ࠠ = 2.

. , , 3 + 3 - -d3 = y4 2 + 3 - 3 = 0, o 3 = 1. 砠 (2) 頠 젠

, , 2 + 2 - d2 = y3 3 + 2 - 2 = 1, 젠 2 = 0; (1) 1(x) 젠 .

, 1 = 0, 2 = 3, 3 = 2,

32 .

.

5 ,

1 + 1 ³ d1 2 + 2 ³ d2 3 + 3 ³ d3

3 + 0 ³ 3 0 + 3 ³ 2 1 + 2 ³ 3

1 + 1 + 2 + 3 = d1 + d2 + d3 3 + 0 + 3 + 2 = 3 + 2 + 3

j(1) + j(2) + j(3) + h12 + h23 = F3(y4=0)

2 + 17 + 10 + 0 + 3 = 32

۠ 򠠠 3 ࠠ

, . 1 = 3 2 = 0 3 = 1 1 = 3

, . 1 = 0 2 = 3 3 = 2 1+ 2+ 3 = 5

, . d1 = 3 d2 = 2 d3 = 3 d1+ d2+ d3 = 8

( ), . 2 = 0 3 = 1 4 = 0

, . j(1)=2 j(2)=17 j(3)=10 j(1) + j(2) + j(3) = 29

, . h12 = 0 h23 = 3 0 h12 + h23 = 3

ɠ

-

0*80+ 0,1*60 +0,2*70=20

0,4*80 +0*60 +0,1*70=39

0,2*80 +0,3*60 +0,2*70=48

Y - .

.

h11=4*0 +7*0,1+ 2*0,2=1,1

h21=2*0 +4*0,1+ 1*0,2=0,6

h31=20*0 +13*0,1+ 16*0,2=4,5

h41=0,2*0+0,3*0,1+ 0,2*0,2=0,07

h12=4*0,4 +7*0+ 2*0,1=1,8

h22=2*0,4+4*0+ 1*0,1=0,9

h32=20*0,4+13*0+ 16*0,1=9,6

h42=0,2*0,4 +0,3*0+ 0,2*0,1=0,1

h13=4*0,2+7*0,3+ 2*0,2=3,3

h23=2*0,2+4*0,3+ 1*0,2=1,8

h33=20*0,2+13*0,3+ 16*0,2=11,1

h43=0,2*0,2+0,3*0,3+0,2*0,2=0,17

1,1*80 +1,8*60 +3,3*70=427

0,6*80 +0,9*60 +1,8*70=228

4,5*80 +9,6*60 +11,1*70=1713

0,07*80 +0,1*60 +0,17*70=23,5

S .

Ƞ

. (, 1-). -, .

nj(x) , (, 1-), j- . n1(x)= + 2(1-); n2(x)=2 +3(1-); n3(x)=4 2(1-); n4(x)=5 5(1-). , , nj(x). n1(x), n2(x), n3(x), n4(x)- , 1, 2, 3, 4 .

. - . . n1(x)=n4(x), *=7/11, n=n1(x)=n4(x)=15/11.

, *=(7/11, 4/11), n=15/11.

, . y, (1- y). , , 1*=*>0 , *>0, (1; y*)=n, .. y* +2(1-y*)=15/11, y*=7/11.

: - *=(7/11, 4/11), Q=(7/11;0;0;4/11), n=15/11.

, - . . , .. ( , ). . .

Q1, Q2, Q3, Q4. ri . ( , ri) , , . . :

j(Q) = 2 - r.

Q1 : 2 4 6 18

1/2 1/4 1/8 1/8

Q2 : 0 4 6 12

1/4 1/4 1/3 1/6

Q3 : 2 5 8 14

1/3 1/6

Q4

: 0 1 2 8

1/3 1/3 1/6 1/6

Q1 =å qipi = 2*1/2+4*1/4+6*1/8+18*1/8=5

Q21 = 25

M [Q21] = 4*1/2+16*1/4+36*1/8+324*1/8=51;

Q2 = 1+2+2=5

Q22 = 25

M [Q22] = 16*1/4+36*1/3+144*1/6=40;

Q

Q3 = 2+5=7

Q23 = 49

M [Q23] = 4*1/4+36*1/4+64*1/3+196*1/6=64;

Q4 = 2

Q24 = 4

M [Q24] = 1*1/3+4*1/6+64*1/6=70/6;

`Q r - , (. .):

4 . (`Q, r), , - . , . (`Q¢, r¢) (`Q, r) `Q¢ ³`Q r¢ £ r.

, , . , , , .

, (`Q, r) , . , j (Q)= 2×Q - r . :

j (Q1)= 2*5-5,1 = 4,9; j (Q2)= 2*5-3,9=6,1; j (Q3)= 2*7-3,9=10,1; j (Q4)= 2*2-2,8=1,2

, 3- - , 4- - .

V - ), M=(mi) - - xi , i- , i=1,.., n. I - n- -, 1. xi

.

V-1 - , V . , , ( -), , , , V-1(M-m0I) - - n . , mp. , mp. , mp. X* mp, , X* mp, x0 .

: 3 5 9 3 6 . ? "short sale" ? . , m0 =3, M= , V= . mp.

V . : V-1 = . :

.

, X* =((m-3)9/13)

x*0 =1- 4/26(m-3) 3/26(m-3)=42-7m/26.

, "short sale" , x*0 < 0, .. m > 6 .

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