Івченко І.Ю. Математичне програмування
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71
10. -
#, $. !< $
’ = $
’ , " .
@ =< < $! :
1) (0;–1); (4;0)
2) (0,7); (–7;0)
3) (0;1); (2;0)
4) (0;3); (2;0).
:
G
= (3; 3) (. . 3).
,
2
,
1
(2)
4
3
1
1 2 3 4
(2) (3)
(1)
%
7
&. 3. /
*') — ’.
I ! 7.
8
= (0; 3)
F = 9.
1. O ! , ’ ?
2. 6, * ’ -
& .
3. $ ' &?
72
1.
6’ &:
F = x
1
+ 2x
2
o
max
–3x
1
+ 2 x
2
+ x
3
= 9
3x
1
+ 4x
2
- x
4
= 27
2x
1
+ x
2
+ x
5
= 14
x
1
, …, x
5
> = 0
2. F = x
1
+ x
2
o
min
3x
1
+ x
2
– x
3
= 8
x
1
+ 2x
2
– x
4
= 6
x
1
– x
2
+ x
5
= 3
x
1
, …, x
5
> = 0
3. F = 5x
1
– 3x
2
o
min
3x
1
+ 2x
2
– x
3
= 6
2x
1
– 3x
2
– x
4
= –6
x
1
– x
2
+ x
5
= 4
4x
1
+ 7x
2
+ x
6
= 8
x
1
, …, x
5
> = 0
4.
! !, ' ’-
:
max
21
xjx
,
,1
132
21
21
t
d
jxjx
xxj
j — & (, & +1 '
).
73
9
4(3 (3C (
1.
8 (543
4(3 (3C (
’ :; .
— , — $=.
) :
1. " (-).
2. .
3. =.
O> < <.
> =
’ :;. * $
- ’
= $ .
"
. A ! "
". $ @ 1947 .
O :
1. ?! $ $ .
2. '>! , # !
, ! -
.
3. ?! $
" , $= .
* ! ’ :;, $
< ! ,
$> $=
< <.
? ;.;.;
. .
C ( !
> ).
max N L (X) =
¦
n
j 1
c
j
x
j
(9.1)
:
¦
n
j 1
A
j
x
j
= B (j = m,1), (9.2)
74
¸
¸
¸
¹
·
¨
¨
¨
©
§
m
b
b
BA
...
1
0
— $=
¸
¸
¸
¹
·
¨
¨
¨
©
§
mj
j
j
a
a
A ...
1
— , # i = m,1
x
j
j 0 (j = n,1). (9.3)
;, # m < n, # (9.2) :C, $-
<< m.
C! $.
; X = (x
1
, …, x
n
) :; (9.1)—(9.3) (<
) ! , # A
j
, -
(x
j
> 0), :C.
% , # -
(m).
( m :C -
, >! A
j
, # >
.
, # > $ -
, $, " $ $ .
J# $ $" ,
! =, " — =.
2.
3 (339
C! =
!< :; ".
:; [5] >! $. ;$-
($ ) < < n
),...,,(
21 n
x
x
x
:
¦
n
j
jjn
xcxxxLxL
1
21
),...,,()( : (9.4)
$=,
n
x
x
x
,...,,
21
:
,,1,
1
1
mibxa
n
j
ijij
d
¦
(9.5)
75
¦
n
j
ijij
mmibxa
1
1
,,1, (9.6)
1
,1,0 njx
j
t , ),(
1
nn d (9.7)
: > (9.4) — $ —
> > , $
),...,,(
21 n
x
x
x
& , # > (9.5)—(9.7) — -
> .
>
)(
ij
a
A
, ! , >
> .
'-
¸
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
¨
©
§
mj
j
j
j
a
a
a
A
"
2
1
,
! a
ij
, # x
j
(9.5)—
(9.6), (9.4)—(9.7).
'
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
©
§
m
b
b
b
BA
"
2
1
0
$= < .
'-
),...,,(
21 n
cccC
(9.4)—(9.7).
'-
),...,,(
21 n
x
x
x
# , # ! (9.5)—
(9.7), ! .
? (9.4)—(9.7) =
.
),()(max xcxL o , (9.8)
BA
n
j
jj
x
d
¦
1
, (9.9)
0tX , (9.10)
76
d
>, # «=» " m
1
, «k» — " m – m
1
, # !
(9.9), $ :
),()(max xcx Do , (9.11)
BAx
d
, (9.12)
0tx . (9.13)
;
),...,,(
**
2
*
1
*
n
xxxX ! -
= < (9.4)—(9.7), -
! $ ’ (# -
$ (9.4)). @ X
*
$- X "
)()(
*
xLx tD (9.14)
?
)(
*
xD !
< .
J# ! ) , #
M
x
L )(max , $-
X L(X), , # L(X) $-
= .
J# ! :
¦
o
n
j
jj
xcxL
1
)(max , (9.15)
¦
n
j
ijij
mibxa
1
,1, , (9.16)
njx
j
,1,0 t , (9.17)
! > [1].
(9.15)—(9.17) ! > (9.4)—(9.7),
(9.4)—(9.7) = (9.15)—(9.17) [1],
" (9.6) $>
),1(0
1
mmlx
ln
t
, >-
"
),1(
1
nnjx
j
, (9.4)—(9.6), $ " <
jj
xx
cc
c
, ),1(0,
1
nnjxx
jj
t
ccc
.
J# $ (9.4), =
<, # L(X) L(X). =-
$ 4.
77
y
= –f (x)
y
= f (x)
X
x
y
x
y
x
&. 4. ; < < <
)).(max()(min
xLxL (9.18)
% , -
= $ < . ; #
’ < , $
),...,,,...,,(
**
1
**
2
*
1
*
1
mmnnn
xxxxxX
, ’ <
(9.4)—(9.7) $
),...,,(
**
2
*
1
*
n
xxxX [1]. ! =
! -
.
;
),...,,(
21 n
x
x
x
X !
(9.15)—(9.17), #
)0( !
si
x
-
> -= A
si
.
A
j
! m- ,
! $" m -
.
% $ , # (9.16) -
>! m, $ .
) m -= , # >!
A
si
, 0!
si
x
, $ -
X. x
si
, ’ $, > $-
. J# $
$" 0, ! -
=, = — =.
. J# (9.16) ! $ ,
)(
x
D $= , L(X) . (9.15)—
(9.17) !
[1].
A ! "
, $"
m
n
C .
78
) ( ’ :;)
:;.
3.
8 (543 (
& :;,
¦
o
n
j
jj
xcxL
1
)(max (9.19)
¦
n
j
ijij
mibxa
1
,,1, (9.20)
0t
j
x (9.21)
( , # (9.20) -
= # m > n.
C
),...,,(
21 n
x
x
x
X = -
(9.19)—(9.21) $
mix
si
,1,0 ! . )-
(9.20) "
BxA
n
j
jj
¦
1
(9.22)
%
mi
si
A ,1, $ X.
K I
x
$ $
8, $
},...,,{
21 mx
sssI (9.23)
0
j
x , #
x
Ij .
)" (9.22) = $ :
¦
n
j
si
BA
1
(9.24)
?
000
, BAxx
isi
. (9.25)
79
% (9.24) !
¦
m
i
isi
xAA
1
00
. (9.26)
)" (9.26) = -
a
0
$ A
si
.
},...,,{}{
211 smss
m
isi
AAAA
(9.27)
$ m- , - A
j
=
$ :
njxAA
ij
m
i
sij
,1,
1
¦
, (9.28)
),1( mix
ij
! A
j
$ -
),1( miA
si
.
;!> (9.26) (9.28), = "
njxAA
m
j
ijsij
,0;
1
¦
, (9.29)
> :
¦
m
i
ijksikj
njmkxaa
1
,0,,1, . (9.30)
' :
¦
=
m
i
ijsij
njxc
1
,1, , (9.31)
njcZ
jjj
,1, ' , (9.32)
c
si
— ! < (9.19) $ -
X. ; Z
j
j
' > ,
$ 8 $
m
isi
A
1
}{
.
(%# $'+ $)
8 ! ’ (9.19)—(9.21), #
0t'
j
nj ,1 [5].
!"#$
J# $ < ,
8 $
nj
j
,1,0 d' . J#
80
j
' $>> (9.32), , #
! " .
'
),...,,(
21 m
OOO / :
¦
O
m
k
sikksi
mica
1
,1 . (9.33)
% =
¦¦¦ ¦¦ ¦
O O O =
m
i
m
i
m
k
m
k
m
i
m
k
kkjkijksiijkksiijsij
axaxaxc
111 11 1
,)()( (9.34)
$
¦
O =
m
k
kkjj
nja
1
,1, . (9.35)
J# Z
j
$>! (9.35), , # -
! .
)" (9.34)
. % , #
X, > < = -
.
;
j
' > A
j
. J#
,
= > .
!'( 1
(9.29) !, # !
l
is
x
(
l
Sj )
$
l
s
A
$ $>>
:
¯
®
z
mili
li
x
l
is
,1,#,0
#,1
. (9.36)
!'( 2
(9.31), (9.32) (9.36) !, #
x
Ij 0 '
j
, $
$
0 '
si
.