-11-

3.

3.1

. , , .

,

.

:

max=xv+...+xv+xv=V nn2211 (3.1)

bx+...+x+x

.......................................

bx+...+x+x

bx+...+x+x

mnmn22m11m

2nn2222121

1nn1212111

(3.2)

)n,...,2,1=i(0 x i (3.3)

. (3.1) , (3.2)

, (3.3)

.

,

.

(3.2)

. (3.3)

x1, x2,..., xn

. ,

,

.

3.2,

b1, b2, ..., bn

.

-12-

3.2

.

, (

)

.

.

0,4 /m3 0,2 /m3 .

0,6 /m3 0,4 /m3

.

4 .

.

15

.m3.

,

.

.

,

. . .

:

max=x4+x6=V 21 (3.4)

40x2+x4 21 (3.5)

15x+x 21 (3.6)

0 x

0 x

2

1 (3.7)

x1 x2

.

-13-

.

x1 x2, (. 3.1),

(3.5), (3.6) (3.7).

,

(3.4).

.

, ,

V

. V

. V,

.

0 5

5

10

10

15

15

20

A

B

x1

x2

3.1

x1+x2=15

4x1 +

2x2 =

40

6x1+4x2=V

.

, . ,

(4),

( )

V.

x1=5, x2=10

-14-

. 5 .

m3 10 . m3 ,

, ( 7 . )

40 . .,

15 . m3.

3.3 1

1 , , .

,

, 1

, .

1

( . 3.2),

.

, , 4

8)

0 5

5

10

10

15

15

20

A

B

x1

x2

x1 +x

2 =15

4x1 +

2x2 =

40

1x

1+4x2=V (

1

-15-

1 > 8, ( 3.2)

(x1 = 10, x2 = 0), .

), 1 < 4,

( 3.2) (x1 = 0, x2 =

15, . ).

, 1, .

,

( 1 > 8,

4 < 1 < 8 1 < 4).

.

1 = 4 1 = 8

. (..

1 = 4) ,

.

.

.

. 1 .

, .

.

3.4 2

,

(. 3.4, 3.5, 3.6 3.7) :

-16-

. .

max=x4+x6=V 21 (3.8)

40=x+x2+x4 321 (3.9)

15=x+x+x 421 (3.10)

0 x

0 x

0 x

0 x

4

3

2

1

(3.11)

x3 x4,

(3.5) (3.6) (3.9) (3.10).

x3 x4

, .

,

, ,

, .

.

2 .

(. 3.8, 3.9, 3.10, 3.11) 4

x1, x2, x3 x4, 2 x1, x2

.

. 4

-17-

. ,

, n

( ) m

, n-m

.

3.5

.

H

,

. ,

,

.

.

.

.

.

.

2 , .

. ,

,

, .

,

,

.

-18-

.

.

-18-

.

.

3.6

. .

+xv+...+xv+xv=V kk2211

max=x.0+...+x.0+x.0+ m+k2+k1+k (3.12)

b=x+...+0+0+x+...+x+x

..............................................................

b=0+...+x+0+x+...+x+x

b=0+...+x+x+...+x+x

mm+kkmk22m11m

22+kkk2222121

11+kkk1212111

(3.13)

(3.13) xk+1, xk+2, ..., xk+m,

.

,

n-m=k+m-m=k . , ,

:

mm+k22+k11+kk21 b=x,...,b=x,b=x,0=x=...=x=x

. .

(3.13)

:

-19-

m

2

1

mk

2k

1k

k

2

1

mk2m1m

k22221

k11211

b

.

.

.

b

b

x

.

x

x

x

.

x

x

1...00...

............

....10...

0...01...

(3.14)

(3.14). m

, m

,

.

. (3.14)

m21mk2k1kk21 b,...,b,b,0,...,0,0x,...,x,x,x,...,x,x

.

,

,

. ..

x1 xk+1,

(3.14) 11,

i1/11. ( i1 21, 31,...,

m1). (3.14) :

11

1m1m

11

2112

11

1

mk

2k

1k

k

2

1

11

1m

11

1mk1mk

11

1m122m

11

21

11

21k1k2

11

211222

1111

k1

11

12

bb

.

.

.

bb

b

x

.

x

x

x

.

x

x

1...00...0

............

....10...0

0...01

...1

(3.15)

(3.15)

-20-

mk1kk21 x,...,x,x,...,x,x

11

1m1m

11

2112

11

1

bb,....,

bb,0,0,...,0,

b

. x1 xk+1.

.

,

.

3.7

.

V

,

,

V .

.

,

.

(3.13)

,

, V

. xi

(3.13) yi. (3.13) :

-21-

mmkmkkkmk222m111m

22k2kkkk222221121

11k1kkkk122121111

byx...00yx...yxyx

.....................................................................................................

b0...yx0yxa...yx)yx(

b0...0yxyx...yxyx

(3.13)

:

0y...00y...yy

............................................................

00...y0y...yy

00...0yy...yy

mkkmk22m11m

2kkk2222121

1kkk1212111

(3.16)

. (3.16) yi

(3.13).

yi (3.16)

xi+yi (3.13).

xi+yi V1

kkk2221111 yxv...yxvyxvV

mkmk1k1k yx0...yx.0

. xi+yi

mk1kkk2211 y.0...y.0yv...yvyvV (3.17)

xi (i = 1

k) , . xi

yi V .

x1 y1 (.

y1 = 1), y2 = y3 = ... = yk = 0, (3.16)

-22-

1mmk

212k

111k

y

................

y

y

(3.18)

. (3.17)

:

1m11k211 .0....00.v...0.vvV (3.19)

,

xj yj (. yj = 1),

, . (3.16)

mjmk

j22k

j11k

y

...............

y

y

(3.20)

. (3.17)

s

ssjjj vvV (3.21)

s

, . .

vj,

, .

xj vj

,

, ( vj)

.

-23-

.

, , ,

.

xj. ..

. (3.14)

x1, v1 > v2, v3,..., vk.

xk+1,xk+2,...,xk+m. xk+1,

x1 xk+1, . (3.14)

. (3.15). . (3.15) x1=(b1/11).

xk+2,

. (3.14) . (3.15),

x1=(b2/21). xk+1

x1=(bi/1).

[x1 = (b1/i1)]

.

, . (bi/i1),

. ,

(bi/i1) .

x1,

.

.

,

. .

. (3.14)

, ,

.

,

-24-

. ,

,

.

-24-

. ,

,

.

3.8 Simplex

Simplex,

:

1

( )

.

,

. .

xk+1, xk+2,..., xk+m ,

.

.

.

2 . (3.21) =1, Vj

. (3.21)

. .

vj .

, s

. ,

vk+1=vk+2=...=vk+m=0, V1=v1, V2=v2, ....., Vk=vk.

v1, v2,..., vk,

-25-

vk+1,...., vk+m .

3 Vj.

,

Vj

. v1>v2,v3,....,vk

x1 .

x1 x2 ...... xk xk+1 xk+2 ...... xk+m

1 xk+1 1

1

12 ...... 1k 1 0 ...... 0 b1

11

1b

2 xk+2 2

1

22 ...... 2k 0 1 ...... 0 b2

21

2b

3 xk+3 3

1

32 ...... 3k 0 0 ...... 0 b3

31

3b

... ... ... ... ...... ... ... ... ...... ... ... ...

m xk+m m

1

m

2

...... m

k

0 0 ...... 1 bm

1m

mb

v1 v2 ...... vk 0 0 ...... 0 V

v1 v2 ...... vk - - ...... - Vj

4 b1, b2,...., bm

. (

, b1, b2,..., bm 11, 21,..., m1).

b1/11, b2/21,..., bm/m1

.

5 ,

-26-

.

b2/21

xk+2.

x1 x2 ...... xk xk+1 xk+2 ...... xk+m

1 xk+1 0 21

11

2212

-

...... 21

11

k2k1

-

1 21

11

-

...... 0 21

11

2-1

bb

21

11

2212

21

11

2-1

-

bb

2 x1 1 21

22

...... 21

k2

0 21

1

...... 0 21

2

b

21

22

21

2

b

3 xk+3 0 21

31

2232

-

...... 21

31

k2k3

-

0 21

31

-

...... 0 21

31

23

b-b

21

31

2232

21

31

23

-

b-b

... ... ... ... ...... ... ... ... ...... ... ... ...

m xk+m 0 211m

222m

-

...... 21

1m

k2mk

-

0 21

1m

-

...... 1 21

1m

2-m

bb

21

1m

222m

21

1m

2m

-

b-b

v1 v2 ...... vk 0 0 ...... 0 v

-

21

22

122

v-v=V

...... 21

k2

1-kk

vv=V

- - ...... - vj

II

6 ( )

.

,

. xk+2

x1.

7

,

.

21, 22,...,2k, 0, 1,..., 0, b2 21

x1.

-27-

8

,

. ,

.

9 . .

2 ,

Vj

.

.

vj ,

( ).

, vj , ,

,

. .

.

3.9

. .

:

max=xv+...+xv+xv=V kk2211 (3.22)

1kk1212111 b x+...+x+x (3.23)

2kk2222121 b x+...+x+x (3.24)

b x+...+x+x

.......................................

b x+...+x+x

mkmk22m11m

3kk3232131

(3.25)

-28-

0 x

........

0 x

0 x

k

2

1

-28-

0x

........

0x

0x

k

2

1

(3.26) (2

. (3.1), (3.2) (3.3),

(3.23) (3.24) ,

(3.25) .

.

(3.23), (3.24), (3.25) (3.26)

:

1m+k3+k2+k1+kkk1212111 b=x.0+...+x.0+x.0+xx+...+x+x (3.27)

2m+k3+k2+k1+kkk2222121 b=x.0+...+x.0+xx.0+x+...+x+x (3.28)

b=x+...+x.0+x.0+x.0+x+...+x+x

......................................................................................

b=x.0+...+x+x.0+x.0+x+...+x+x

mm+k3+k2+k1+kkmk22m11m

3m+k3+k2+k1+kkk3232131

(3.29)

0x

.......

0x

0x

.......

0x

0x

m+k

1+k

k

2

1

(3.30)

. (27) (28) . (13),

xk+1 xk+2,

. , (23) (24)

.

x1=0, x2=0,...,

xk=0, xk+1=-b1, xk+2=-b2, xk+3=b3,..., xk+m=bm. b1,

-29-

Simplex. .

b2,..., bm ,

xk+1 xk+2 .

(3.30)

,

, . ,

.

() .

(3.22) (3.27),

(3.28), (3.29) (3.30) :

+...+x.0+x.0+xv+...+xv+xv=V 2+k1+kkk2211

max=x.M-x.M-x.0+ 2+m+km+km+k (3.31)

+...+x.0+x.0+x-x+...+x+x 3+k2+k1+kkk1212111

12+m+k1+m+km+k b=x.0+x+x.0+ (3.32)

+...+x.0+x-x.0+x+...+x+x 3+k2+k1+kkk2222121

22+m+k1+m+km+k b=x+x.0+x.0+ (3.33)

b=x.0+x.0+x+...+x.0+

+x.0+x.0+x+...+x+x

...............................................................

b=x.0+x.0+x.0+...+x+

+x.0+x.0+x+...+x+x

m2+m+k1+m+km+k3+k

2+k1+kkmk22m11m

32+m+k1+m+km+k3+k

2+k1+kkk3232131

(3.34)

0x,x,x,...,x,x,x,...,x,x 2+m+k1+m+km+k2+k1+kk21 (3.35)

-30-

(3.32) (3.33) xk+m+1 xk+m+2

. (3.31)

.

.

. ,

.

,

, -xk+m+1, -

Mxk+m+2 -.

, .

,

.

.

Simplex,

. , .

(3.32), (3.33) (3.34),

x1=0, x2=0, ..., xk=0, xk+1=0, xk+2=0, xk+3=b3, ..., xk+m=bm, xk+m+1=b1,

xk+m+2=b2, (3.35),

.

Simplex.

Simplex

. .

,

, (3.35)

-31-

Simplex.

3.10

.

.

3.106

2,2.106 m

3,

1,5.106 2,8.106 m3 .

,

2:1.

(

) : 0,32 /m3,

0,24 /m3, ,

0,26 /m3, 0,2 /m3,

.

, .

.

x1 x2 ( m3

100.000)

, x3 x4

,

:

min=x2,0+x26,0+x24,0+x32,0=V 4321

min=x5+x5,6+x6+x8=V 4321 (/ 25) (3.36)

-32-

15 x+x 31 ( ) (3.37)

28x+x 42 ( ) (3.38)

30x+x 21 ( )

22x+x 43 ( ) (3.39)

0x2+x- 31 ( )

0 x i (i = 1, 2, 3, 4) (3.40)

. .

,

(3.36).

,

-1.

(3.37), (3.38) (3.39)

5

, (3.37) (3.38)

2 ,

(). x5 x6

x7 x8 (3.37) (3.38).

(3.39) x9, x10 x11.

.

maxx.0x.0x.0Mx-Mx-x.0x.0x5-x5,6-x6-x8-V 1110987654321 (3.41)

0=x+x.0+x.0+x.0+x.0+x.0+x.0+x.0+x.2+x.0+x

22=x.0+x+x.0+x.0+x.0+x.0+x.0+x+x+x.0+x.0

30=x.0+x.0+x+x.0+x.0+x.0+x.0+x.0+x.0+x+x

28=x.0+x.0+x.0+x+x.0+x-x.0+x+x.0+x+x.0

15=x.0+x.0+x.0+x.0+x+x.0+x-x.0+x+x.0+x

1110987654321

1110987654321

1110987654321

1110987654321

1110987654321

(3.42)

-33-

0x i (i = 1, 2, ..., 11) (3.43)

x7 x8 .

( . (41), (42) (43))

Simplex

:

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

1 x7 1 0 1 0 -1 0 1 0 0 0 0 15

2

x8 0 1 0 1 0 -1 0 1 0 0 0 28 28

3 x9 1 1 0 0 0 0 0 0 1 0 0 30

4 x10 0 0 1 1 0 0 0 0 0 1 0 22 22

5 x11 -1 0 2 0 0 0 0 0 0 0 1 0 0/0

-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -43M

-8+M -6+M -6,5+M -5+M -M -M - - - - -

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

1 x7 1 0 1 0 -1 0 1 0 0 0 0 15

2 x8 0 1 -1 0 0 -1 0 1 0 -1 0 6 6

3 x9 1 1 0 0 0 0 0 0 1 0 0 30 30

4 x4 0 0 1 1 0 0 0 0 0 1 0 22

5 x11 -1 0 2 0 0 0 0 0 0 0 1 0 0/0

-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -21M-110

-8+M -6+M -1,5 - -M -M - - - - -

-34-

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

1 x7 1 0 1 0 -1 0 1 0 0 0 0 15 15

2 x2 0 1 -1 0 0 -1 0 1 0 -1 0 6

3 x9 1 0 1 0 0 1 0 -1 1 1 0 24 24

4 x4 0 0 1 1 0 0 0 0 0 1 0 22 22

5 x11 -1 0 2 0 0 0 0 0 0 0 1 0 0

-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -15M-146

-8+M - -7,5+M - -M -6 - -M+6 - -1 -

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

1 x7 1,5 0 0 0 -1 0 1 0 0 0 -0,5 15 10

2 x2 -0,5 1 0 0 0 -1 0 1 0 -1 0,5 6

3 x9 1,5 0 0 0 0 1 0 -1 1 1 -0,5 24 16

4 x4 0,5 0 0 1 0 0 0 0 0 1 -0,5 22 44

5 x3 -0,5 0 1 0 0 0 0 0 0 0 0,5 0

-8 -6 -6,5 -5 0 0 -M -M 0 0 0 -15M-146

1,5M-11,75 - - - -M -6 - -M+6 - -1 -0.5M+3,75

-35-

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

1 x1 1 0 0 0 -0.67 0 0.67 0 0 0 -0.33 10

2 x2 0 1 0 0 -0.33 -1 0.33 1 0 -1 0.33 11

3 x9 0 0 0 0 1 1 -1 -1 1 1 0 9

4 x4 0 0 0 1 0.33 0 -0.33 0 0 1 -0.33 17

5 x3 0 0 1 0 -0.33 0 0.33 0 0 0 0.33 5

8 -6 -6,5 -5 0 0 -M -M 0 0 0

- - - - -7,835 -6 -M+7,835 -M+6 - -1 -0,165

A . (3.41), (3.42)

(3.43) : x1=10, x2=11, x3=5, x4=17, x9=9,

.

1.106 m3 0,5.106

m3 . 1,1.106 m3 1,7.106

m3 . (1,054.106 .)

.