: 5

: 5

" "

2001

" "

.

Ƞ ...

ɠ

Ƞ

,

.

,

(1)

(x1, x2, x3, x4),

(2)

:

(3)

蠠 (4)

. (3)

5,6, 7

頠 (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.

,

蠠 .. 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

,

, ,

, 21=4.

,

1.

~

(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 + 2x7 = 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

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

( , xj&#179;0),

,

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 &#179;36.

, :

31 + 22 + 83&#179;32

41 + 73&#179;10

1 + 22 &#179;13

, 1031 + 1482 +

1583.

1, 2, 3, . ,

, - ,

, ,

.

,

: (1, y2,

y3) f = 1031 +

1482 + 1583 (1)

, ,

, ,

21 + 42 + 23 &#179; 36

31 + 22 + 83&#179;32 (2)

41 + 73&#179;10

1 + 22 &#179;13

y10, y20, y30. (3)

,  頠 頠 (1, 2,

3, 4) (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 .

" "

, .. &#178; &#178;.

. 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&#163;148/3, t3&#163;158/3 (6)

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

: . . 2. &#178;&#178;

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 .

= . ,

.

.

&#229;i =40+60+70=170 ,

&#229;bi = 36+32 +40 +53 =161, ..

.

170-161 = 9 ,

, , ,

,

.

&#178;-

&#178;.

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 &#163; 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) = zmax

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&#215;0 + 2 +4&#215;0 = 2*

y2 = 1, x1= 1, W1 (1;1) = 12 + 2&#215;2 + 2 +4&#215;1 = 11

y2 = 2, x1= 2, W1 (2;2) = 22 + 2&#215;2 + 2 +4&#215;2 = 18

y2 = 3, x1= 3, W1 (3;3) = 32 + 2&#215;3 + 2 +4&#215;3 = 29

y2 = 4, x1= 4, W1 (4;4) = 42 + 2&#215;4 + 2 +4&#215;4 = 42

y2 = 5, x1= 5, W1 (5;5) = 52 + 2&#215;5 + 2 +4&#215;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 &#163; x2 &#163; d2 + y3 蠠 0 &#163; x2 &#163; 2 + y3

(1)

x = 3,

0 &#163; y3 &#163; d3 , .. 0 &#163; y3 &#163; 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 &#163; x2 &#163; 2, .. 2

: 0, 1, 2, 2

2, (2): 2 = 2 - 2

:

蠠 x2 = 0, 2 = 2 , W2 (0,2) = 02 + 2&#215;0 + 2+

F1(2) = 2 + 18 = 20,

x2 = 1, y2 = 2 - 1 = 1, W2 (1,2) = 12 + 5&#215;1 + 2 + F1(1) = 8 +

11 = 19,

x2 = 2, y2 = 2 - 2 =0, W2(2,2) = 22 + 5&#215;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 &#163; x2 &#163; 3, .. 2

: 0, 1, 2, 3, 2

2, (2): 2 = 3 - 2

:

蠠 x2 = 0,  y2 = 3-0 = 3, W2 (0,1) = 02 + 2&#215;0 + 2 + 3&#215;1 + F1(3) = 5+

29 = 34,

x2 = 1, y2 = 3-1 = 2, W2 (1,2) = 12 + 2&#215;1 + 2 + 3&#215;1 +F1(2) = 8 + 18 = 26,

x2 = 2, y2 = 3-2 = 1, W2(2,1) = 22 + 2&#215;2 + 2 + 3&#215;1 + F1(1) = 13 +11 =

24,

x2 = 3, y2 = 3-3 = 0, W2 (3,1) = 32 + 2&#215;3 + 2 + 3&#215;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 &#163; x2 &#163; 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&#215;0 + 2 + 3&#215;2 + F1(4) = 8+

42 = 50,

x2 = 1, y2 = 4-1 = 3, W2 (1,2) = 12 + 2&#215;1 + 2 + 3&#215;2 +F1(3) = 11 + 29 =

40,

x2 = 2, y2 = 4-2 =2, W2(2,2) = 22 + 2&#215;2 + 2 + 3&#215;2 + F1(2) = 16 + 18 =

34,

x2 = 3, y2 = 4-3 = 1, W2 (3,2) = 32 + 2&#215;3 + 2 + 3&#215;2 +F1(1) = 23 + 11 =

34*,

x2 = 4, y2 = 4-4 = 0, W2(4,2) = 42 + 2&#215;4 + 2 + 3&#215;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 &#163; x2 &#163; 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&#215;0 + 2 + 3&#215;3 + F1(5) = 11+

57 = 68,

x2 = 1, y2 = 5-1 = 4, W2 (1,3) = 12 + 2&#215;1 + 2 + 3&#215;3 +F1(4) = 14 + 42 =

56,

x2 = 2, y2 = 5-2 = 3, W2(2,3) = 22 + 2&#215;2 + 2 + 3&#215;3 + F1(3) = 19 + 29 =

48,

x2 = 3, y2 = 5-3 = 2, W2 (3,3) = 32 + 2&#215;3 + 2 + 3&#215;3 +F1(2) = 26 + 18 =

44*,

x2 = 4, y2 = 5-4 = 1, W2(4,3) = 42 + 2&#215;4 + 2 + 3&#215;3 + F1(1) = 35 + 11 =

46.

x2 = 5, y2 = 5-4 = 0, W2(5,3) = 52 + 2&#215;5 + 2 + 3&#215;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&#163;y4&#163;0; x=y4; 0 &#163; x3 &#163; d3 + y4 &#8594; 0 &#163; x3 &#163; 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&#215;0 +2 +2&#215;0 +F2(3)=2

+44=46

x3=1 y3=2 W3(1;0)=12 + 2&#215;1 +2+2&#215;0 + F2(2)=5

+34=39

x3=2 y3=1 W3(2;0)=22 + 2&#215;2 +2+2&#215;0 +

F2(1)=10+22=32*

x3=3 y3=0 W3(3;0)=32 + 2&#215;3 +2+2&#215;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)

, , 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 &#179; d1 2 + 2 &#179;d2

3 + 3 &#179; d3

3 + 0 &#179; 3 0 + 3

&#179; 2 1 + 2 &#179; 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 =&#229; 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&#162;, r&#162;) (`Q, r) `Q&#162; &#179;`Q r&#162; &#163; r.

, ,

.

, ,

, .

,

(`Q, r) ,

. , j (Q)= 2&#215;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 .

1. . . .

.. -.: "", 1999.

2. .., ..

. -.: -, 1999.

3. .., .., ..

. , .,1987.

4. .. . -.:

, 1998.

5. ..

. -.: , 1998.

6. .., .., .., ..

. -: , 1979.

7. .., .., ..

. , ., 1981.

8. .., .., .. . , .,

1986.

9. .., .., .. . , .,

1990.

10. .., .. . -.:

,1998.

11. .. ., ., 1980.

12. ..

. -.: , 1976.

13. .. ,

. -.: , 1968.

14. ..

. , ., 1973.

15. .., ..

. , ., 1986.

16. .. .

. , .,1993.

17. .. . -.:-, 1998.

18. .. . -: -, 1999.

19. .. . -: ,

1998.

20. .. . -: , 1999.

21. .., .. . , .,

1989.

22. ., . .

-.: , 1970.

23. .., .. :

. -.: -., 1994.

24. .. . -: , 1985.

25. .., .., ..

. .: , 1998.

26. . . .: , 1985.


 
2012 , , .