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.
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