. , - .
,
Z = 100 1 + 50 x2
,
71+ 22 700,
21 + 42 480,
21 + 22 300,
x1 0, x2 0.
, , . 4.3.
A |
B |
C |
. 4.3.
, ( Z = 11 500), . , , . : 71+ 22 = 700 21 + 22 = 300.
, , , 1 700 840 . 1* 71+ 22 = 840. 1, . : .
. 4.3 , () B, , Z. , , -, ( ), -, (). , 1 , ? , . 1 1 . , , 1050, () C . Z = 15 000. , , .
, ( ), , , . . ( ) ().
, 2 ( ), (). () , , , . 2 . , 2 . . 4.3, 2 (). 2 A, . D 1 2. Z , , 3 ().
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, .
() .
.
.
() . () , , . , , .
4.1.
2 . 2- , 1 . 4.1. 1 3 . ., 2 . . ( ), .
. 4.1.
(x1) | (x2) | ||
1 | |||
2 |
x1 ( ) , x2 ( ) .
Z = 3 x1 + 2 x2 (4.1)
, ,
x1 +3 x2 6 (4.2)
2x1 + x2 8
, :
x1 0, x2 0. (4.3)
, : (4.1), (4.2) (4.3). , :
x1opt = 3.6
x2opt = 0.8
, :
Zmax = 12,4 . .
. , , , . , , , , , .. , . , , , , , . , . () , .
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1, 2. , 1 1 (. . 1 ), 2 2 (. . 1 ).
, , , , .
3 . ., , , 3 . . 1 2 (. 4.1), :
1 + 2 2 3
, :
31 + 2 2
,
y1 0, y2 0.
-, , . 6 8 , 1, 2
Z* = 61 + 8 2.
,
Z* = 6 1 + 8 2 => min (4.4)
1 + 2 2 3. (4.5)
31 + 2 2
y1 0, y2 0. (4.6)
(. 4.7), :
y1opt = 0,2 . . 1
y2opt = 1,4 . . 2
Z*min = 6y1+ 82 = 12,4 . .
Y2
Y1 |
y1opt = 0,2 y2opt = 1,4 |
.4.7.
, () ()
Zmax = 12,4 . . Z*min = 12,4 . . (4.7)
.
ylopt = 0,2 y2opt =1,4 . ( , ).
, (4.7).
, (), . .
4.1 (. . 4.2).
. 4.2.
ij | bi | yj | ||
1 | y1 | |||
2 | y2 | |||
- | x1 | x2 |
1,2 | y1,y2 | |
() Z = 3x1 + 2x2 max | () Z* = 6y1 + 8y2 min | |
x1 +3 x2 6 2x1 + x2 8 x1, x2 0 | 1 + 2 2 3 31 + 2 2 y1 0, y2 0. |
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.
1. , .
2. .
3. .
4. , .
5. .
. 4.3.
. 4.3.
x1,x2,,xn | y1,y2,,ym |
Z = c1x1+c2x2+..+cnxn max | Z = b1y1+b2y2+..+bmym min |
a11x1+a12x2+.+a1nxn b1 .. am1x1+am2x2++amnxn bm x1 0, x2 0, ., xn 0 | a11y1+a21y2+.+am1ym c1 .. a1ny1+a2ny2++amnyn cn y1 0, y2 0, ., ym 0 |
, . , , , . , () .
, , :
Zmax = Z*min (4.8)
4.1. .
Z*min = b1y1 + b2y2 +.+bmym
b1, b2,,bm , , y1, y2, ..,ym , , (4,8) , ( ) .
Zmax = Z*min = b1y1 + b2y2 +.+bmym
Zmax = b1y1 + b2y2 +.+ bmym (4.9)
, (4.9) Zmax, - bi . , (4.9) :
;
.
(4.9) , , 4.3.
, , b1 b1, . ( )
Zmax = b1 y1
, , :
yi = (4.10)
(4.10) .
, ( ) .
:
1. .
2. , ( ) .
(4.9) (4.10). , . . . , 1975 .
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