. .
, (). . , , , .
.
1) .
max (min) (1.1)
(1.2)
, .
(1.1) (1.2) , .
, .
(1.1) (1.2) :
max(min)
b,
.
m n- (1.2), b m 1- - (1.2), n 1- - .
.
max (1.3)
= b, (1.4)
(1.5)
.
(1.1)-(1.2) (1.3)-(1.5), :
, .. : , (1.3) : .
, (1.5) , .
, (1.5) , .
, b, (1.4) : + = b, .
, b, (1.4) : = b, .
. .
: x y. 25 10 12 b. 37 15 20 b. 30 40 b. .
b |
Z .
Z = 25 + 37 min.
:
,
-
;
- ;
- ;
|
|
- , , .. .
.
.
:
(2.1)
(2.2)
(2.3)
, .
(2.1)-(2.3) 3 (n = 2), , :
) , , ( , );
) grad f(x1, 2,,n) = - f(x1, 2,,n) ( n=2: grad f(x,y) = )
- grad f(x,y) - f(x,y);
) grad f(x,y) .
( (2.1)-(2.3)):
1. , (2.2). .
2. grad f(x,y) (2.1) (- grad f(x,y)) .
3. .
4. (2.1) , , (n = 2).
5. . .
6. .
.
, .
, . , .
, .
. , .
:
(2.4)
(2.5)
(2.6)
. : ( ).
( (2.4)-(2.6)):
1. (2.5)
(2.7)
( ), +1 . .
2. :
, . (2.8)
3. (2.8). :
|
|
(2.9)
4. (2.6), , , (2.10):
(2.10)
(2.11)
5. 1-6 ( (2.1)-(2.3)) (2.9)-(2.11).
6. (2.8). (2.4)-(2.6).
1. ( ) : ( 4 5 . .. ), : - . 12 - 8 . 1 5 . - 2 . , 1 . 2 . - 4 . . , ?
. . x () y (). .. , .
, 5 x . -, 2 y . 12 . .
: ;
..
, .
.
.1.
: 2 1 , .
.
2. , . 3000 . 20 , 60 . , 2 , . 100 . 6 , . .
1 2 , , . W , 1 . , z ,
.
W ,
,
, , .
,
,
.
, ,
; ; .
,
(2.4)
(2.5)
, , . .
20 1+60 2=3000
1-2 2=0
1=100 (2.6)
1=0
2=0
|
|
. 2.
(2.5) OPRS, (. 2.2).
, , , , OPRS. P, ( z) OPRS.
|
|
. , ,
, (60;30). , 60 . 1200 30 . 1800 .
. . , , , .
.
.
F = 1+2 → max
1-2 ≤ -2 1-2+3 = -2
1-22 ≥ -13 1-22- 4 = -13
31-2 ≤ 6 31-2+5 = 6
1 ≥ 0, 2 ≥ 0 1≥ 0, 2 ≥ 0
) ;
) , 1 2 ( ), 3, 4 5 .
:
-1+2-2 = 3
1-22+13 = 4
-31+2+6 = 5
1 ≥ 0, 2 ≥ 0 F = 1+2
) :
-1 | -2 | b | |
3 | -1 | -2 | |
4 | -1 | ||
5 | -1 | ||
F | -1 | -1 |
x1, x2 + - . , . ( ) , , . (: -2) , . (: -1), ( ). . (min = 2). , , . , , (-1);
) .
) (.. 2 3);
-1 | -3 | b | |
2 | -1 | -1 | |
4 | |||
5 | -1 | ||
F | -2 | -1 |
) (-1) 1/ (1:(-1) = -1);
) ( (-1)) ;
) ( (-1)) ;
) :
-
= ----------------------------------------------------------------,
- ( ),
- ( ).
) , . , .. -2. . min , 2 - ;
|
|
) (.. 1 5); ;
-5 | -3 | b | |
2 | 0,5 | -1,5 | |
4 | -0,5 | 2,5 | |
1 | 0,5 | -0,5 | |
F | -2 |
) :
-5 | -4 | b | |
2 | 0,6 | ||
3 | -0,2 | 0,4 | |
1 | 0,2 | ||
F | 0,6 | 0,8 |
:
1 = 5, 2 = 9, 3 = 2 F = 14 0,6 *5 0,8*4,
Fmax=14, =(1, 2, 3, 4, 5) = (5,9,2,0,0).
: Fmax=14, 1 = 5, 2 = 9.
: , . .