Linearno programiranje

1.

1.

, .

,

. ,

,

, , ,

( ),

(, ,

, , .)

. [2]

:

-

.

, ,

.

aj aaa oa aa eo ooo

e a aeaa aea

eja. Aaa eeo a oa ooo o

aa oaaj eje.

ae eae e a oe ao a e ajoe aoa

a ea e e a aeaa ao o

ao, eeja, oe eje. eae eeo a oe

"eja " oj aa eee eaaa.

1 Management Science-

* : *

2

eee a aa a a oajaa o ee.

a aa ooa aea a eo ooa,

eee eeo a oe oje ae eao, oo

ooa oj ooaa a o a oo

aoa a.

Oo e aja e e eao oaae ()

eo o, a aoe j ee aaja

. ae aaae o aea aee ee aa:

o e e aaa aa oeee

e. e eo a ee,

eo aa oaaj eje, eo

aa eee eaaa a aeao

aj ooa aoa. ee

aaa je aaja aja ee e. [3]

aeao , o ae e oj jo a.

ja a e jo aa ej aaa

ejo oao. Oa eaa j e

oe a oj je oeo oe ee eo, j.

oe e a.

.

.

,

.

.

.

.

* *

3

1.1.

oe aj e ajee ooe:

oe ee aa a ee oe (oo oa ea). Oa ooa e oje

jo a oe . ajaj o

eea a aa oj o o ooo

eeo eo. Ao e a o aoaajo

jo e, oe aja

ooa aoa.

o oaea, eae oaaaj aoa, oeae aa o oj oeo eaoa

ee. a e, oae oo ooa ao

ooa ooo oa ea oe je

oaeo aoo ea eea.

oaj ooja oa eea oja oeo aa. a e, ao eee oo aa ooa,

eae oe e a oo ao a a

o oo aoa oje oaee ooe

ee (e, aooe aaee aa, .).

ja a oaea oea oa ae oo ea

jeaa ejeaa.

1.1.

,

, .

:

1 2 3

(/) 4 3 5 300

(/) 6 4 2 420

(/) 2 6 3

* : *

4

300 .

420 .

.

.

1, 2 3.

:

x1 = 1

x2 = 2

x3 = 3

.

.

1 ,

2 , , x1. 2

3 . , F(X)

F(X) = 2 x1 +6 x2 +3 x3

2 x1 = 1

6 x2 = 2

3 x3 = 3

.

,

. 1,

1 4

. 4x1 . ,

2 3x2, 3 5x3 .

300. ,

:

* *

5

4 x1 + 3 x2 + 5 x3 300

.

1 (x1) 6 -

, 2 (x2) 4 , 3 (x3)

2 . 420

,

6 x1 + 4 x2 + 2 x3 420

, ,

. (

)

:

x1 0, x2 0, x3 0

. ,

, ,

.

,

.

.

=

, -

. [2]

,

:

F(X) = 2 x1 +6 x2 +3 x3

.. ( ):

4 x1 + 3 x2 + 5 x3 300

6 x1 + 4 x2 + 2 x3 420

x1 0, x2 0, x3 0

x1 , x2 x3

F(X).

* : *

6

1.2.

.

.

,

. ,

. -

:

/

/

/

()

3 6 2 15

4 3 2 14

6 , 4 ,

3 .

,

.

:

x1 =

x2 =

x3 =

.

, .

:

in F(X) = 6 x1 + 4 x2 + 3 x3

: 6 x1 = ( ) ,

4 x2 = ( ) ,

3 x3 = ( ) .

* *

7

,

.

. :

3 x1 + 6 x2 + 2 x3 15

:

3 x1 = ( ) ,

6 x2 = ( ) ,

2 x3 = ( ) .

1.1. ,

.

, 15 .

,

,

. ,

,

.

:

4 x1 + 3 x2 + 2 x3 14

.

in F(X) = 6 x1 + 4 x2 + 3 x3

..:

3 x1 + 6 x2 + 2 x3 15

4 x1 + 3 x2 + 2 x3 14

x1 0, x2 0, x3 0

x1, x2 x3,

(

F(X)),

.

* : *

8

1.2.

.

: ,

,

.

,

.

,

x1, x2, ... xj,..., x.

.

.

.

, .

. :

=

=+++++=n

j

jjnnjj xcxcxcxcxcXFMinMax1

2211 ......)()(

:

- F(X) -

- cj - ; ( ) j=1,2,...,n

-

. m

bi ( i=1,2,...,m).

aij i j

(j=1,2,...,n). , :

* *

9

a11 x1 + a12 x2 + ... + a1j xj + ... + a1n xn b1

a21 x1 + a22 x2 + ... + a2j xj + ... + a2n xn b2 .......................................................

ai1 x1 + ai2 x2 + ... + aij xj + ... + ain xn bi .......................................................

am1 x1+ am2 x2+ ... + amj xj+ ... + amn xn bmx1, x2,..., xj,..., xn 0

.

, :

ai1 x1+ ai2 x2+ ...+ aij xj+ ...+ ain xn bi

ai1 x1+ ai2 x2+ ...+ aij xj+ ...+ ain xn= bi

:

Max(Min) =

=+++++=n

j

jjnnjj xcxcxcxcxcXF1

2211 ......)(

..:

a11 x1 + a12 x2 + ... + a1j xj + ... + a1n xn (,=, ) b1a21 x1 + a22 x2 + ... + a2j xj + ... + a2n xn (,=, ) b2 .......................................................

ai1 x1 + ai2 x2 + ... + aij xj + ... + ain xn (,=, ) bi .......................................................

am1 x1+ am2 x2 + ... + amj xj+ ... + amn xn (,=, ) bm x1, x2,..., xj,..., xn 0

1.1.

:

Max F(X)=c1x1 + c2x2 + c3x3

..:

a11 x1 + a12 x2 + a13 x3 b1

a21 x1 + a22 x2 + a23 x3 b2 a31 x1 + a32 x2 + a33 x3 b3

x1, x2, x3 0

:

c1 = 2, c2 = 6, c3 = 3 a11 = 4, a12 = 3, a13 = 5, b1 = 300

a21 = 6, a22 = 4, a23 = 2, b2 = 420

* : *

10

, :

Max(Min) =

=n

j

jj xcXF1

)(

..: i

n

j

jij bxa ),,(1

==

; ),...,2,1( mii = 0jx

, 1.1. :

Max =

=3

1

)(j

jj xcXF

..: =

3

1j

ijij bxa , 2,1=i ; xij 0, j = 1,2,3.

cj, aij

bi [2].

1.3.

. -

-

.

, , n

( ) n- .

,

.[1]

-

,

.

, -

()

. 1.1.

x1 0

* *

11

1. x1 0 x2 0,

.

D .

,

,

.

- .

1.2. x2 0

1.3.

( min F(X))

1.4.

( max F(X)))

1.3 max F(X)

, A B.

:

(/) 1 2 9

(/) 5 5 25

(/) 4 6

* : *

12

3 A

.

:

x F(X)= 4 x1 + 6 x2 ()

..: x1 + 2 x2 9 ()

5 x1 + 5 x2 25 ( )

x1 3 ( A)

x1, x2 0

1.5.

,

.

.

( 1.5).

-

,

.

. -

x1=0 x2 = 9, x2=0 x1=4,5. 1.5.

.

* *

13

,

ABCDE

1.6.

2.

x1,x2

-

.

1.6.

. 1.6. R (

) S ( )

, P Q

.

ABCDE

(x1, x2) (

).

F(X). , F(X)

,

F(X).[2]

1.7.

F(X)

.

F(X) = 12: 4x1+6x2=12,

F(X)1.

F(X) 20,

F(X)2, F(X)3,

F(X)4. 1.7.

2

x1 x2 - x1, x2 0.

* : *

14

1.7.

. , F(X)

. (j=1,2,...,n),

F(X), .

x2 F(X) x1, :

1123

2

6

)(

6

4

6

)(x

XFx

XFx ==

3

2=jk

kj - F(X).

,

.

F(X)4 > F(X)3 > F(X)2 > F(X)1 F(X)1

, .

F(X)4 (x1,

x2) (

ABCDE).

F(X)3 B ( x1=1 x2 =4)

F(X)

. , F(X) B.

, x1 x2

( ) :

x1 + 2 x2 = 9 ()

5 x1 + 5 x2 = 25 ( )

x1 +2x2 = 9 x1 +2x2 = 9 5- x2+2x2 = 9 x2 = 9-5=4 x2 =4

x1+ x2 = 5 x1 = 5- x2 x1 = 5- x2 x1 = 5- x2 x1 =1

x1 = 2 x2 = 4 F(X) (-

) :

F(X) = 4 x1 + 6 x2 = 4 (1) + 6 (4) =28.

a . [2]

* *

15

.

(

).

,

n

(n- ).

n . 1.8.

F(X)

( ).

1.4. min F(X)

min F(X)= 2 x1+ 3 x2 ()

..: 4x1+ 8x2 40 ( )

6x1+ 8x2 48 ( )

x1, x2 0

1.3.

, ABC. -

,

.

, mx F(X) F(X)

. min F(X)

, F(X)

( 1.9.).

1.10.

F(X). F(X)

.

F(X)

, , F(X)3.

* : *

16

B .

B .

1.9.

4x1+ 8x2 = 10 ( )

6x1+ 8x2 = 4 ( )

: x2 = 3, x1 = 4 F(X)=2(4)+3(3)=17.

1.10.

* *

17

: -

, -

. :

1.5.

F(X)

.

,

F(X)

.

F(X) . [2]

x F(X)=2 x1+ 2 x2

..: x1+ 2 x2 9

5 x1+ 5 x2 25

x1 3

x1, x2 0

,

-

1.11.,

A B

-

-

( F(X)3).

-

A B e-

-

.

1.11.

* : *

18

,

.

1.6.

-

-

-

( 1.12.).

x F(X)= 3 x1 + 2 x2

..: 4 x1+ 3 x2 12

x1 4

x2 6

x1, x2 0 1.12.

:

, F(X)

,

.

1.6.

-

-

( 1.13.).

x F(X)= x1 + x2

..: - 2x1+ x2 2 - x1+ x2 3

x1, x2 0

1.13.

* *

19

1.4.

1.4.1.

e e e e 1947.

. e e e e ee e

e e e.

e e ee

e e e.

( )

, ,

.

,

.

, , F(X)

(

).[2]

D .

, ,

,

.

F(X) ,

,

,

. ,

,

. .

,

. ()

, .

* : *

20

1.7. ( )

,

( 1.3., 1.7.). F(X):

x F(X)= 4 x1 + 6 x2

..: x1 + 2 x2 9

5 x1 + 5 x2 25

x1 3

x1, x2 0

,

, .

je

.

, . , -

3 () :

x1 + 2 x2 + x3 = 9

5 x1 + 5 x2 + x4 = 25

x1 + x5 = 3

x1, x2 , x3, x4, x5 0

,

F(X):

x F(X)= 4 x1 + 6 x2 + 0 x3 + 0 x4 + 0 x5

( -

). , :

)

x1 x2 ()

. :

x1- A , . .

x2- B , . .

3

() .

.

* *

21

) () .

: x3, x4 x5

, . .

, ,

,

. ,

,

, . -

,

.

:

x3 - ,

x4 - ,

x5 - A - A.

1.4.2.

:

1. a;

2. , 4.

,

, ;

3.

.

.[1]

.

T0 () ,

.

.

4 . ,

. F(X)

* : *

22

5 ,

(x1 = 0 x2 = 0).

:

T0

C B X0 c1 c2 ... cn cn+1=0 cn+2=0 ... cn+m=0

x1 x2 ... xn xn+1 xn+2 ... xn+m

cn+1=0 Xn+1 b1 a11 a12 ... a1n 1 0 ... 0

cn+2=0 Xn+2 b2 a21 a22 ... a2n 0 1 ... 0

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

cn+m=0 Xn+m bm am1 am2 ... amn 0 0 ... 1

Fj - cj 0 1c 2c ... nc 0 0 ... 0

: (Xn+1 = b1, Xn+2 = b2, ..., Xn+m = bm).

x1 = 0 x2 = 0) : F(X)=0.

:

T0,

,

. . :

1.

2. .

1. : C, B X0 :

C - F(X), . ( max F(X)= 4x1 + 6x2,

: c1 = 4 c 2= 6).

B - . :

, , (+1)

,

,

5 (x1 = 0 x2 = 0).

,

.

* *

23

B, (+1).

, B

, .

: x3, x4 x5.

X0 - , B, ( ,

9, 25 3).[1]

2. : Xj, j=1,2,...,n Xn+i, (i=1,2,...,m),

() .

.

F(X) (

c1= 4, c2 =6, c3,4, 5 = 0). ,

, ()

, Fj-cj.

:

x1 + 2 x2 + x3 = 9.

, (9) X0,

Xj Xn+i (1) x1, 2

x2 (1) x3, (0); x4 x5

).

.

Fj-cj

, .

, Fj-cj F(X)

(

, F(x)1F(x)optimalno).

X0 Fj-cj

, ()

Fj-cj .

, T0,

.

, , Fj-cj

, ,

, :

* : *

24

Fj-cj X0 ( F(X)) C

, , X0

. , , ,

C, , ,

Fj-cj X0,

: F(X)= 0 x3+0 x4 + 0 x5 = 0.

Fj-cj a Xj Xn+i C ,

, Xj Xn+i, .

,

, .

C

, Fj-cj a xj,

, ,

(

x1

: F1-c1= 44105010 =++ -c1).

: ?

,

maxF(X) minF(X).

:

) maxF(X): Fj-cj,

, .

Fj-cj 0 , .

Fj-cj

.

.

) minF(X): Fj-cj, -

, .

Fj-cj 0 , .

Fj-cj

, .

* *

25

,

,

T0:

T0

C B X0 4 6 0 0 0

x1 x2 x3 x4 x5

0 X3 4,5 0,5 1 0,5 0 0

9 1 2= 1 0 0 0 X4 25 5 5 0 1 0

0 X5 3 1 0 0 0 1

Fj-cj 0 -4 -6 0 0 0

, ( -

),

:

x1 = x2 = 0

( ), -

,

x3 = 9, x4 = 25, x5 = 3

.

,

A (x1 = 0) B (x2 = 0).

,

, F(X) = 0.

1.4.3.

- .

:

, , ;

, , ;

, , ;

* : *

26

-, , .

() (..) - , ,

.

() (..) - B, ,

.

- .. .. .

.

,

.

. , ,

Fj-cj 0 , (F1-c1= -4 F2-c2= -6), .

.

) maxF(X): , ,

Fj-cj X0.

,

, .

F(X)

.

) minF(X): , ,

Fj-cj, X0.

F(X)

.

T0,

, : x3,

* *

27

x4 x5. x1 x2. T0

: x3, x4 x5 ,

x1 x2 .

, Fj-cj, -

, , :

a. F1-c1 = - 4, X1, ,

, F(X) .

4 , .

x1,

.

b. F2-c2 = -6, X2, ,

, F(X) .

6 , .

x2,

.

( )

.

, F(X)

,

,

, F(X).

, ,

,

( )

Fj-cj.

x2,

: 4411 ==CF 6622 ==CF .

x2,

x? x2 T0 , x2, -

.

, x2

.

* : *

28

, , .

: x3, x4 x5.

:

:

c. , ,

.

X2 x2. ()

( ,

).

d. X0 -

:

0,minmin 0 >

=

ij

j

i

j

aa

b

X

X

(j=1,2,...,n,n+1,...,n+m)

(i=1,2,...,m)

{ } 5,45;5,4min5

25,

2

9min,minmin

22

2

12

1

2

0 ==

=

=

a

b

a

b

X

X

, ,

. T0,

x3.

.

, x3 ,

, x2.

x3 x2. , X3 X2

, X3,

C.

,

, ( T0).

, (

X0 .. X3), B

4,5 x2

0 4,5.

* *

29

(1), ,

() , ,

T0. B, ,

x3 , x2

. C

( B) .

X0:

, (),

( x2=4,5) -

X2.

,

B.

B, 1 .

4,5 B (x2=4,5),

,

B, 4,5 .

Ka 5,415,4 == .

(Kr) 4,5 ,

(Kr) (Ka) ,

B, -

x3 ,

:

X3: x3 = Kr - Ka

x3 = 9 - 4,5 = 4,5 .

x3,

A,

4,5.

.

Ka 5,2255,4 == .

* : *

30

X4: x4 = Kr - Ka

x4 = 25 - 22,5 = 2,5 .

A,

(Kr)

:

X5: x5 = Kr = 3, (Ka = 0)

,

, ()

, : i- j- ( -

), ,

i- j- , ,

i-

j- ,

, ( ) .

, :

, ,

, .[1]

x3 x2 (T1), .

X1 T1:

(2,1), X4

X1, ,

T0, ( ,

T0) X1 -

X2

X4 : ( ) .5,25,2555,05 ==

X1:

* *

31

X5: ( ) 105,01 = jj cF : ( )[ ] .13465,0411 =+== cF

: X4 X5 X3

, , ,

, ()

.

.

,

. X5,

X2.

X2 T1:

, ,

( x2),

X2 ,

, ,

, .

, X2 ,

, ,

() X2

X4: ( ) 0515 = X5: ( ) 0010 = jj cF : ( )[ ] .061622 == cF

X3 T1:

X4: ( ) 5,255,00 = jj cF : ( )[ ] .365,0033 == cF

X4 T1: [ ] .000044 == cF X5 T1: [ ] .000055 == cF

Fj-cj X0,

C , , X0 , .

* : *

32

=

=n

i

oii XCxF1

)( , 5,4,2=i

X3: 275,4612 ==bc X4: 05,2024 ==bc X5: 03035 ==bc

:

.27)( =XF -

, :

( )[ ] .2765,40)( ==XF , 1T , :

x2 = 4,5, x4 = 2,5, x5 = 3.

:

x1 = x3 = 0.

, 4,5 B

(x2 =4,5), 2,5

(x4 = 2,5), -

A ( x1=0).

(x3=0).

27 :

F(X) = 27.

: , T1, -

?

, Fj-cj,

, . Fj-cj .0

, T1, Fj-cj,

X1:-1

* *

33

x ? x1 , x1 ,

. , ,

X0

X1:

{ } 13,1,9min1

3,

5,2

5,2,

5,0

5,4minmin

1

0 ==

=

X

X

,

, x4,

x1:

x 4 x1

. x1,

, 2.

5,2= , ( 1).

B, , : x2, x1 x5,

C , .

-

5,2= , , , e

( T1).

T1

C B X0 4 6 0 0 0

x1 x2 x3 x4 x5

6 X2 4,5 0,5 1 0,5 0 0

0 X4 1 1 0 -1 0,4 0

2,5 = 2,5 0 -2,5 1 0 0 X5 3 1 0 0 0 1

Fj-cj 27 -1 0 3 0 0

.

, -

a X2 X5 , (

) . T2:

* : *

34

T2

C B X0 4 5 0 0 0

x1 x2 x3 x4 x5

6 X2 4 0 1 1 -0,2 0

4 X1 1 1 0 -1 0,4 0

0 X5 2 0 0 1 -0,4 1

Fj-cj 28 0 0 2 0,4 0

x4 ,

x1, B : x2,

x1 x5 a , T2, :

x1 = 1, x2 = 4, x5 = 2.

x3 = x4 = 0.

T2,

A (x1 = 1) 1 .

:

x3 = 0, (100%), x4 = 0, (100%), x5 =2, 50% A (

x1 3 ).

, T2, :

( ) ( ).2,0,0,4,1,,,, *5*4*3*2*1* == xxxxxX : ( ) .28* =XF 28 ,

A ( 1*1 =x ) 1 ,

B ( 4*2 =x ) 4 .

,

, -

D,

,

. (

)

* *

35

, ,

. 1.1.7.

,

DCBAOD : ,

, ,

.

O=T0. , , O (0,0)

A B,

.

x1 = x2 = 0, F(X) = 0.

A=T1. T1, - A(0,4,5).

x1 = 0, x2 = 4,5, F(X) = 27.

B=T2. T2, - B(1,4), :

( ) .28,4,1 *2*1 === XFxx 1.8. ( Fj-cj)

: :

K ,

Fj-cj (

) ? :

,

.

, . .

, ,

F(X), . :

T01

C B X0 2 2 0 0

x1 x2 x3 x4

0 X3 3 1 3/4 1/4 0

12 = 4 3 1 0 0 X4 10 -2 5 0 1

Fj-cj 0 -2 -2 0 0

* : *

36

Fj-cj

x1, x2.

(x1 x2)

, .

x1

x? x1

: 34

12minmin

1

0 =

=

X

X

x3

x3 x1.

x1, ,

6 ( T01). x1

,

F(X ) = 0 - [3 (-2)] = 6.

T02

C B X0 2 2 0 0

x1 x2 x3 x4

0 X3 12 4 3 1 0

0 X4 2 -2/5 1 0 1/5

10 -2 = 5 0 1 Fj-cj 0 -2 -2 0 0

, x2 (T02) x? x2

,

{ } 22,4min5

10,

3

12minmin

2

0 ==

=

X

X

x4

x4 x2. x2 2,

F(X ) = 0 - [2 (-2)] = 4

x1

.

* *

37

1.4.4. ( ,= )

,

, .

= .

. , ,

,

()

.

.

, ,

.

(+1)

,

. ,

:

, ( ) .

.

,

.

,

.

, . .

:

- () .

* : *

38

- , . ()

.[1]

)

,

. ,

,

[2].

1.9. I ( ),..., . in F(X)= 2x1 + 3x2

..: 4x1 + 8x2 40

6x1 + 8x2 48

x1, x2 0

, -

, x3 x4

.

4x1 +8x2 - x3 = 40

6x1 +8x2 -x4 = 48

x1, x2 , x3, x4 0

:

in F(X)= 2x1 + 3x2 + 0x3 + 0x4

..

4x1 + 8x2 - x3 = 40 (1.1.)

6x1 + 8x2 - x4 = 48

0,,, 4321 xxxx . , x1=5, x2=0

(1.1.):

4 x1 + 8 x2 - x3 = 40

4(5) + 4(0) - x3 = 40

20 - x3 = 40

- x3 = 20

, . x1, x2=0,

(1.1.)

x3 :

* *

39

4(0)+8(0) - x3 = 40

x3 = -40

x3

(

x3 (-1)).

.

(-1),

. ,

, (+1),

.

,

,

(+1), . .

x5 x6.

4x1 + 8x2 - x3 + x5 = 40 (1.2.)

6x1 + 8x2 - x4 + x6 = 48

x1, x2, x3, x4, x5, x6 0 x1=5, x2=0 x5=0,

(1.2.)

x3 : 4(5) + 8(0) - x3 + 1(0) = 40 x3 = -20.

x1,x2=0, x3=0

(1.2.) x5

: 4(0) + 8(0) - 1(0) + x5 = 40 x5 = 40.

.

,

. , F(X),

:

x1 x2 (c1=2, c2=3).

x3 x4, F(X), .

* : *

40

x5 x6 (

) M.

, ,

:

minF(X), (+M):

( ) =

+++ ++++=n

j

mnnnjj MxMxMxxcXF1

21 ...min .

maxF(X), (-M):

( ) =

+++ =n

j

mnnnjj MxMxMxxcXF1

21 ...max .

, F(X) :

in F(X)= 2 x1 +3 x2 + 0 x3 + 0 x4 + Mx5 + Mx6 .

Fj-cj

.

, :

( I ) Fj-cj M,

( II ) Fj-cj M.

T0 :

T0

C B X0 2 3 0 0 M M

x1 x2 x3 x4 x5 x6

M X5

5 1/2 1 -1/8 0 1/8 0

40 4 = 8 -1 0 1 0 M X6 48 6 8 0 -1 0 1

jj cF I 0 -2 -3 0 0 0 0

II 88 10 16 -1 -1 0 0

x5=40 x6=48.

: F6-c6 = M - M = 0

* *

41

F(X)= 88 M +0 .

, ,

x1= x2 = x3 = x4 = 0.

Fj-cj, X0, -

M:

F1-c1 = 10 M - 2 10

F2-c2 = 16 M - 3 16

F3-c3 = - M - 0 = - M -1

F4-c4 = - M - 0 = - M -1

F5-c5 = M - M = 0 0

F6-c6 = M - M = 0 0

, Fj-cj.

F1-c1 =10 M - 2 -

M, C, X1.

. ,

M,

:

(4 M + 10 M) - 2 = 10 M - 2.

M , ,

Fj-cj, ,

.

minF(X),

Fj-cj, ,

. x2. -

x2 .

x? x2 .

{ } 56,5min8

48,

8

40minmin

1

0 ==

=

X

X

5x

x5 x1 .

* : *

42

,

,

8= . T0 T1:

T1

C B X0 2 3 0 0 M

x1 x2 x3 x4 x5 x6

3 X2 5 1/2 1 -1/8 0 1/8 0

M X6 4 1 0 1/2 -1/2 -1/2 1/2

8 = 2 0 1 -1 -1 1

jj cF I 15 -1/2 0 -3/8 0 3/8 0

II 8 2 0 1 -1 -2 0

x2 = 5, x6 = 8, .

: F(X)= 8M + 15.

, B,

x6, Fj-cj

X1:2>0 X3:1>0 .

x1

x? x1 .

X1, Fj-cj,

.

{ } 44,10min2

8,

21

5minmin

1

0 ==

=

X

X

X6, .

x6

x6 x1 .

2= .

T2.

, , (

, T1,

2= )

* *

43

.

T2:

T2

C B X0 2 3 0 0 M M

x1 x2 x3 x4 x5 x6

3 X2 3 0 1 -3/8 1/4 3/8 -1/4

2 X1 4 1 0 1/2 -1/2 -1/2 1/2

jj cF I 17 0 0 -1/8 -1/4 1/8 1/4

II 0 0 0 0 0 -1 -1

Fj-cj :

X0,

,

(-) , (+) , .

.

Fj-cj ,

, Fj-cj,

.

1.14. I

* : *

44

,

,

Fj-cj. Fj-cj, ( T2), -

() Fj-cj,

. :

( ) ( ).0,0,0,0,3,4,,,,, *6*5*4*3*2*1* == xxxxxxX : ( ) .17* =XF

, 1.14.

A(4,3).

1.10. II ( ),...,== . x1 x2

,

ax F(X)= 2x1 + x2

..: x1 + x2 = 6

-4x1 + 4x2 = 8

x1, x2 0

,

.

.

,

.

, . .

x3 x4. :

x1 + x2 + x3 = 6

-4x1 + 4 x2 + x4 = 8

x1, x2, x3, x40 F(X) -M:

MaxF(X) = 2x1 + x2 - Mx3 - Mx4 .

() ,

,

, .

T0:

* *

45

T0

C B X0 2 1 - M - M

x1 x2 x3 x4

- M 3X 6 1 1 1 0

- M 4X 2 -1 1 0 1/4

8 -4 = 4 0 1

jj cF I 0 -2 -1 0 0

II -14 3 -5 0 0

: x3 = 6, x4 = 8 x1 = x2 = 0, F(X) = -14 M + 0.

Fj-cj X2:-5

* : *

46

x? x1 .

: 22

4minmin

1

0 =

=

X

X

x3

x3 x1 .

T2

C B X0 2 1 - M - M

x1 x2 x3 x4

2 X1 2 1 0 1/2 -1/8

1 X2 4 0 1 1/2 1/8

jj cF I 8 0 0 3/2 -1/8

II 0 0 0 1 1

B, . , -

Fj-cj X0, X1 X2

, , , X3 X4,

. Fj-cj

Fj-cj.

Fj-cj

-

, -

.

:

4;2 *2*

1 == xx .

:

( ) .8* =XF -

je

1.15. 1.15.

* *

47

)

1.11. I . ( ),..., . 1x 2x -

ax F(X)= x1 + x2

..: x1 + x2 5 - x1 + 3x2 3

x1, x2 0 x3 x4, ,

,

MaxF(X) = x1 + x2 + 0 x3 + 0x4 .

..:

x1 + x2 + x3 = 5

- x1 + 3x2 - x4 = 3

x1, x2, x3, x40

.

x4 (-1).

.

,

(+1)

( ).

x5, (+1), -

. , ,

(-M).

:

MaxF(X) = x1 + x2 + 0 x3 + 0x4 -M x5

..: x1 + x2 + x3 = 5

- x1 + 3x2 - x4 + x5 = 3

x1, x2, x3, x4, x5 0 :

* : *

48

T0

C B X0 1 1 0 0 - M

x1 x2 x3 x4 x5

0 X3 5 1 1 1 0 0

- M X5 1 -1/3 1 0 -1/3 1/3

3 -1 = 3 0 -1 1

jj cF I 0 -1 -1 0 0 0

II -3 1 -3 0 1 0

:

x3 = 5, x5 = 3, x1 = x2 = x4 = 0, F(X) = - 3M + 0.

,

x2, X2

Fj-cj X2:-3

* *

49

X1, X2, X3 X4 , X5 ( -

) .

Fj-cj,

, Fj-cj.

Fj-cj, :

X1:-4/3

* : *

50

x1 x4 .

T1':

T1'

C B X0 1 1 0 0

x1 x2 x3 x4

1 X1 12 4 0 3 1

3 1 0 3/4 = 1/4 1 X2 2 0 1 1/4 -1/4

jj cF 5 0 0 1 0

T2'

C B X0 1 1 0 0

x1 x2 x3 x4

1 X4 12 4 0 3 1

1 X2 5 1 1 1 0

jj cF 5 0 0 1 0

:

( ) ( ).12,0,5,0,,, **4**3**2**1** == xxxxX : ( ) .5=XF ,

, 1.16.

* *

51

1.16.

( T2) x1

x2 (x3 = x4 = 0). T2' x2

x4, -

.

,

, D. O (0,0)

T0. A (0,1) T1. B (3,2), CB,

T2, C (0,5), T2'.

:

C (0,5) B (3,2) D,

. ,

CB.

1.12. III .: ( ),...,, = . x1, x2 x3

-

,

ax F(X)= 3 x1 + 2 x2 + x3

..: x1 + 2x2 8 2 x1 + x2 + x3 = 20

x2 2 x1, x2, x3 0

* : *

52

x4 x5, -

, x6 ,

ax F(X)= 3 x1 + 2 x2 + x3 +0 x4 +0 x5 -M x6

..:

x1 + 2x2 + x4 = 8

2 x1 + x2 + x3 = 20

x2 - x5 + x6 = 2

x1, x2, x3, x4, x5, x60

, 6 x3

(+1),

( ).

x4, x3 x6 ,

. :

T0

C B X0 3 2 1 0 0 - M

x1 x2 x3 x4 x5 x6

0 X4 8 1 2 0 1 0 0

1 X3 20 2 1 1 0 0 0

- M X6 2 0 1 0 0 -1 1

2 0 1= 0 0 -1 1

jj cF I 20 -1 -1 0 0 0 0

II -2 0 -1 0 0 1 0

:

x4 = 8, x3 = 20, x6 = 2.

Fj-cj, X2:-1

* *

53

, : { } 22,20,4min1

2,

1

20,

2

8minmin

2

0 ==

=

X

X

x6:

26 xx . T1. T1

,

Fj-cj : X0, X1, X2, X3,

X4 X5 , , X6

.

Fj-cj, X6

,

Fj-cj. Fj-cj

X1:-1

* : *

54

T2

C B X0 3 2 1 0 0

x1 x2 x3 x4 x5

3 X1 4 1 0 0 1 2

1 X3 10 0 0 1 -2 -3

2 X2 2 0 1 0 0 -1

jj cF 26 0 0 0 1 1

Fj-cj, ,

. O je:

( ) ( ).0,0,10,2,4,,,, *5*4*3*2*1* == xxxxxX : ( ) .26* =XF 1.4.5.

, =.

, . . ,

(-1),

, , .

. ,

,

Xn+1, i=1,2,...,m..

,

,

, (-1)

, . ,

. , i -

ai1 x1 + ai2 x2 + + ain xn = bi

ai1 x1 + ai2 x2 + + ain xn bi

ai1 x1 + ai2 x2 + + ain xn bi

* *

55

. , (-1)

. , (-1),

ai1 x1 + ai2 x2 + + ain xn bi

-ai1 x1 - ai2 x2 - - ain xn - bi.

1.4.6.

x1 x2

ax F(X)= x1 + x2

..: -2 x1 + x2 2 - x1 + x2 3

x1, x2 0. ,

MaxF(X) = x1 + x2 + 0 x3 + 0x4 .

..

-2x1 + x2 + x3 = 2

- x1 + x2 + x4 = 3

x1, x2, x3, x40 :

T0

C B X0 1 1 0 0

x1 x2 x3 x4

0 X3 2 -2 1 1 0

2 -2 = 1 1 0 0 X4 3 -1 1 0 1

jj cF 0 -1 -1 0 0

x3=2 x4 = 3,

: F(X )= 0.

Fj-cj, X1:-1

* : *

56

, : { } 23,2min1

3,

1

2minmin

2

0 ==

=

X

X

x3: x3 x2 .

T1:

T1

C B X0 1 1 0 0

x1 x2 x3 x4

1 X2 2 -2 1 1 0

0 X4 1 1 0 -1 1

1 = 1 0 -1 1

jj cF 4 -3 0 2 0

:

x2 = 2, x4 =1, x1 = x3 = 0, F(X) = 2.

Fj-cj, X1:-3

* *

57

Fj-cj ( T2) -

X3:-10:

a13 = -1 23 = -1)

.

(x1 x2) ,

x3.

, . ( ).

.

. ,

.

1.1.17.

1.17.

D . F(X),

, , ,

,

.

, . .

, ,

.[1]

* : *

58

1.4.7.

, =.

, .

. , (-1),

,

, . .

, ,

Xn+1, i=1,2,...,m. [1]

,

,

, (-1)

, . ,

. , i -

ai1 x1 + ai2 x2 + + ain xn = bi

ai1 x1 + ai2 x2 + + ain xn bi

ai1 x1 + ai2 x2 + + ain xn bi

. , (-1)

. , (-1),

ai1 x1 + ai2 x2 + + ain xn bi

-ai1 x1 - ai2 x2 - - ain xn - bi.

1.5.

.

.

()

.[4]

* *

59

.

,

, .

,

.

, .

, ,

, ,

.

, ,

.

, ,

. m

n, . n m

mn . .

.

:

() F(X),

() G(Y),

yi, i=1,2,..., m, ,

, xj, j =1, 2 ,..., n,

, , G(Y)

,

cj -, ,

.

,

:

* : *

60

1.

2. .[1]

1.5.1.

, , ,

.

7 :

Max F(X) = c1 x1 + c2 x2 +...+ cn xn

.. a11 x1 + a12 x2 ...+ + a1n xn b1 a21 x1 + a22 x2 ...+ + a2n xn b2 ...............................................

am1 x1 + am2 x2 ...+ + amn xnbm xj 0, j=1,2,..., n (1.1.3) :

=

=n

j

jj xcXF1

)(max

..: =

n

j

ijij bxa1

, mi ,...,2,1=

xj 0, j = 1,2,..., n , (1.1.3.)

(1.1.4.) ,

, G(Y).

Min G(Y) = b1 y1 + b2 y2 + ... + bm ym

..: a1n y1 + a21 y2 + + am1 ym c1

a1n y1 + a22 y2 + + am2 ym c2

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

a1n y1 + a2n y2 + + amn ym cm

yi 0, i = 1,2,, m (1.1.4.)

: =

=m

i

ii ybYG1

)(min

7 mn bmn

.

* *

61

..: =

m

i

jiij cya1

, nj ,...,2,1=

yi 0, i = 1,2,, m

(aij, bi

cj) .[4]

-

x1 x2 ... xn

y1 11 a12 ... a1n b1

()

y2 21 a22 ... a2n b2

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

ym m1 am2 ... amn bm

c1 c2 ... cn

()

:

1.13.

:

Max F(X) = 4 x1 + 6 x2 :

Min G(Y) = 9 y1 + 25 y2 + 3 y3

..: x1 + 2 x2 9

5 x1 + 5 x2 25

x1 3

x1, x2 0.

..: y1 +5 y2 + y3 4

2y1 +5 y2 6

y1, y2, y3 0.

* : *

62

-

x1 x2

y1 1 2 9

() y2 5 5 25

y3 1 0 3

4 6

()

1.5.2.

,

(-

).

, ,

F(X) (),

()

G(Y), .[1]

, :

1: .

2: ( )

,

, () F(X)

, () G(Y)

Fmin (X) = Gmax (Y)

Fmax (X) = Gmin (Y)

X = ( x1, x2, , xn) , Y = ( y1, y2, , ym ).

* *

63

3: () ,

() , .

4: X

Y , ,

G(Y) , (),

( ) F(X) ,

() , .

Gmin (Y) Fmax (X) Fmax (X) Gmin (Y)

Gmax (Y) Fmin (X) Fmin (X) Gmax (Y)

5: xn+1 ,

, yi

, xj , ,

ym+j ,

.

iin yx + jmj yx +

(1.1.5.)

0=+ iin yx 0= + jmj yx

( i = 1,2, , m) (j = 1,2, , n)

,

, .

Gi-bi,

, , , .

Gi-bi0.

, ,

Gi-bi , (

).

,

X0 . ,

B ,

(1.1.5.),

( ).

Y0, ,

(), -

* : *

64

Fj-cj, -

, ,

. [1]

1.14.

:

Min F(X) = 2x1 + 3x2

..: 4 x1 + 8 x 2 40

6 x1 + 8 x 2 48

x1, x2 0.

:

ax G(Y) = 40 y1 + 48 y2

..: 4 y1 + 6 y2 2

8 y1 + 8 y2 3

y1, y2 0.

:

max G(Y) = 40 1 + 48 2 + 0 3 + 0 4

.. 41 +6 2 + 3 = 2

81 + 8 y2 + 4 = 3

1, 2, 3, 4 0

:

d

C YB Y0 40 48 0 0

y1 y2 y3 y4

0 Y3 1/3 2/3 1 1/6 0

2 4 = 6 1 0 0 Y4 3 8 8 0 1

Gi - bi 0 -40 -48 0 0

* *

65

Y1:-40

* : *

66

Gi-bi,

,

. :

0,4/1,8/1 *4*

3

*

2

*

1 ==== yyyy ; 17)(*

max =YG .

,

(1.1.5.),

. T 2d, ,

.

Gi-bi, Y3 Y4,

,.

4*13 = xy

3*24 = xy . y3, ,

() x1,

, y4

x2 ( T2p).

, (1.1.5.), y1

y2 Y0, ,

T2d ()

Fj-cj,

(X3 X4),

, ( T2p).

Gi-bi,

,

, .

X0 .[1]

je 1.1.4.

.

(1.1.6.) :

in F(X)= 2x1 + 3x2 + 0x3 + 0x4+ x5 + x6

..

* *

67

4x1 + 8x2 - x3 + x5 = 40 (1.2.)

6x1 + 8x2 - x4 + x6 = 48

x1, x2, x3, x4, x5, x6 0

, T2p.

T2p

C B X0 2 3 0 0 M M

x1 x2 x3 x4 x5 x6

3 X2 3 0 1 -3/4 -1/4 0 -1

2 X1 4 1 0 1 -1 1 0

jj cF I 17 0 0 -1/8 -1/4 0 0

II 0 0 0 0 0 -1 -1

*

3y *

4y *

1y *

2y T2p :

0,3,4 *4*

3

*

2

*

1 ==== xxxx ; 17)(*

min =XF . T2p ,

, ,

( y3 y3),

(x3 x4),

, (x5

x6). n = 6

m = 2 , m = 4

n = 2 .

, i-

i-

.

, .

Gi-bi

, , , .

Gi-bi 0. ,

,

, Gi-bi,

(), . X0

. ,

* : *

68

B ,

(1.1.5.).

1.18. 1.19.

, Y0 ,

Fj-cj,

, ,

.[1]

1.18.

B

: 4*1 =x 3*

2 =x . : 17)(min =XF .

* *

69

1.19.

B

: 8/1*1 =y 4/1*

2 =y . :

17)( *max =YG .

1.5.3.

(

) 1.2.

:

xj (=1,2,3,,n)

cj

F(X)

bi i (i=1,2,3,,m)

aij

* : *

70

,

y1, y2, , ym.

y0 = b1 y1 + b2 y2 + + bm ym

y0 F(X)

. , y1, y2, , ym Gi-bi

, yi

i (i=1,2,3,,m).

, 1.14.,

, 8/1*1 =y 4/1*

2 =y , :

8/1*1 =y - 1/8,

4/1*2 =y - 1/4.

, :

G(Y) = 40 y1 + 48 y2 = 40 (1/8) + 48 (1/4) = 40/8 + 48/4= 5 +12= 17.

,

yi

(

). ,

.[4]

, yi i,

bi,

.

,

(bi) 1 ( 1.9.).

* *

71

(x1 = 4, x2 = 3) 8

F(X)=17, (i=1,2,3) :

8/1*1 =y 4/1*

2 =y ,

1) x1 = 15/4, x2 = 25/8 F(X)=135/8 (*

18/1)( yXF == )

2) x1 = 7/2, x2 = 13/4 F(X)= 67/4 (*

24/1)( yXF == )

1.20.

bi. A

, (

)

,

. , (

) )

:

4 x1 + 8 x2 = 40 6 x1 + 8 x2 = 48,

b1 = 40

b2 = 48 8.

8

* : *

72

.

,

bi . , bi,

.

, (

9)

. *iy i

()

.

,

.[4]

,

. ,

b1 = 40 b2 = 48

.

:

,

=

=m

i

iijj yaXF1

)(

( =1,2, ... , n).

,

, :

ji

m

i

ij cya =1

9 shadow prices [4, .83]

* *

73

. , :

0iy i (i=1, 2, , m)

. ,

. :

Min i

m

i

i yby =

=1

0

.

() - -

. ( -

) Fj-cj.

, (x1, x2,, xn+m),

ji

m

i

ij cya ==1

, xj > 0 (j=1,2,,n)

(xj>0),

.

yi = 0, xn+i > 0 (i=1,2,,m).

i (yi = 0)

(xn+i > 0).

, " 10

", .

.

,

.[4]

.

10

free good - [4, .84]

* : *

74

.

( )

.

, ( )

, ( ). ,

. , xj ,

,

=

m

i

iij ya1

(i=1,2,,m), (j=1,2,,n)

(

* *

75

.

Fj-cj

( ),

.[4]

1.6.

,

,

..

, ,

, .

(

),

(, , , , .).

, ,

.

.

,

.

,

.

,

: , ,

. :

, , . ,

.

* : *

76

:

[1] .

, ,

, , 1998.

[2] Lee S., Moore L., Taylor B.

Management Science, Allyn and Bacon, , 1990.

[3] Anderson D., Sweeny D., Williams T.

An Introduction to Management Science, Quantitative Approaches

to Decision Making, West Publishing Company, , ,

1988.

[4] Hillier F., Lieberman

Operations Research, Holden - Day, Inc., , ,

1974.