max z = 3x1 + 5x 5x2
≤
4
(1)
2x2 ≤ 12
(2)
3x1 + 2x2 ≤ 18
(3)
≥
0
(4)
x2 ≥
0
(5)
x1
x1
...
x1 x2
max z = 3x1 + 5x2 .
x1 , x2 ≥ 0.
x1 ≤ 4.
2x2 ≤ 12 3x1 + 2x2 ≤ 18.
max z = 3x1 + 5x2
x1
≤
4
2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1
≥
0
x2
≥
0
(x1 , x2 )
3x1 + 2x2 ≤ 18 3x1 + 2x2 = 18
x1
≤
4
(1)
2x2 ≤ 12 (2) 3x1 + 2x2 ≤ 18 (3) x1
≥
0
(4)
x2
≥
0
(5)
n
R
n n
R
ak x1 + ak x2 + . . . + ak xn ≤ bk 1
n
2
n
R
ak1 x1 + ak2 x2 + . . . + akn xn = bk . n
R
x ∈
ak1 x1 + ak2 x2 + . . . + akn xn ≤ bk , k = 1, . . . , p . 3
R
x1 + x2 + x3 ≤ 1 x1 , x2 , x3 ≥ 0
n
R
OABC
O
3
R
z = 3x1 + 5x2 .
z = k.
3x1 + 5x2 = k.
z = 10, 20
x∗ = (2, 6).
36.
z = 3x1 + 2x2
(2, 6)
(4, 3)
x∗ x∗ x∗
z y∗
x∗ = y ∗
n
R
ak1 x1 + ak2 x2 + . . . + akn xn = bk , k ∈ I ⊂ {1, . . . , p}.
n
R
n−1
0. OABC
ABC x1 + x2 + x3 = 1
AB x1 + x2 + x3 = 1
x3 = 0 n=3
C
0
n n=2
m
m=3
c j j
bi
i
aij
i
j
...
n
a11
a12
...
a1n
b1
a21
a22
...
a2n
b2
am1
am2
amn
bm
c1
c2
...
cn
max z = 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 xn ≤ bm ,
n
m+n
x1
≥ 0,
x2
≥ 0,
xn
≥ 0. n
max z = cT x,
(m × n), b
A c
(n × 1),
Ax ≤ b, x
≥ 0.
(m × 1)
.
(n × 1)
x x
S S = {x ∈ Rn | Ax ≤ b, x ≥ 0}.
x∗ ∈ S
∀x ∈ S, cT x∗ ≥ cT x,
max z = 4x1 + 3x2
3x1 + 4x2 ≤ 12 7x1 + 2x2 ≤ 14 x1 , x2 ≥
0
max z = 5x1 + 4x2
x1 + x2 ≤ 20 2x1 + x2 ≤ 35
−3x1 + x2 ≤ 12 x1 , x2 ≥
0
max z = 3x1 + 5x2
≤
x1
4
2x2 ≤ 12 3x1 + 2x2 ≤ 18 x1 x2
≥
0
≥
0
(0, 0)
(0, 6) (0, 6) (2, 6) (0, 6) (4, 3)
max z = cT x,
(m × n), b
A
(n × 1),
c
Ax ≤ b, x
≥0
(m × 1) (n × 1). m+n
x n m
n
3x1 + 2x2 ≤ 18.
3x1 + 2x2 + x3 = 18, x3 x3 ≥ 0.
2x1 + x2 ≥ 4,
2x1 + x2 − x3 = 4, x3 ≥ 0.
max z = 3x1 + 5x2
+x3
x1 2x2
= +x4
3x1 +2x2 x1 ,
x2 ,
4
= 12 +x5 = 18
x3 ,
x4 ,
x5 ≥
0
(3, 2) (3, 2, 1, 8, 5)
5
R
m=3 n+m =2+3=5
x1
n=2
+x3
x1 2x2
= +x4
3x1 +2x2
4
= 12 +x5 = 18
x2 n+ m
n
R
m x3 , x4
x5
x3 = 4 x4 = 12 x5 = 18. n m
x1 = 0,
x2 = 0
x3 = 4, x4 = 12, x5 = 18
(0, 0), (0, 6), (2, 6), (4, 3), (4, 0) (0, 9), (4, 6)
(6, 0)
(x1 , x2 )
(x3 , x4 , x5 )
x1 , x2
(0, 0)
(4, 12, 18)
x1 , x4
(0, 6)
(4, 0, 6)
x1 , x5
(0, 9)
(4, −6, 0)
x4 , x5
(2, 6)
(2, 0, 0)
x3 , x4
(4, 6)
(0, 0, −6)
x3 , x5
(4, 3)
(0, 6, 0)
x2 , x3
(4, 0)
(0, 6, 6)
x2 , x5
(6, 0)
(−2, 12, 0)
x1 =
0, x1 = 0,
x2 =
0, x2 = 6,
x3 =
4, x3 = 4,
x4 = 12, x4 = 0, x5 = 18, x5 = 6
x1 = 0, x1 = 2, x2 = 0, x2 = 6
bi ≥ 0, i = 1, . . . n ,
(x1 , x2 ) = (0, 0).
+x3
x1 2x2 3x1 +2x2
+x4
=
4
=
12
+x5 = 18,
x3 = 4 x4 = 12 x5 = 18.
(0, 0, 4, 12, 18).
z = 3x1 + 5x2 = 3 × 0 + 5 × 0 = 0,
z = 3x1 + 5x2 . 3
x1 z 5
x2 z
z
(ox2 )
(0, 6)
x4
x2
x1 x2 x3 = 4
≥0
x4 = 12 − 2x2 ≥ 0 x5 = 18 − 2x2 ≥ 0
12 =6 2 18 = 9. x2 ≤ 2 x2 ≤
x4 x2 = 6
z −3x1 −5x2 +x3
x1 2x2 3x1 +2x2 x2
x4
+x4
=
0 (0)
=
4 (1)
=
12 (2)
+x5 = 18. (3) x2
x4
1
x2 1/2 (2 ) = (2) ×
1 2
x2
(3 ) = (3) − (2) (0 ) = (0) + 5 × (2 ).
+x3
x1 x2 3x1
= 30 (0 )
+ 52 x4
z −3x1
=
4
=
6 (2 )
−x4 +x5 =
6 (3 )
+ 12 x4
x1
x4
z = 30 x3 = 4 x2 = 6 x5 = 6.
(x1 , x2 , x3 , x4 , x5 ) = (0, 6, 4, 0, 6)
z = 30.
(1)
x2
(3 ) = −1 × (3) + (2).
(0 ) 5 z = 30 + 3x1 − x4 . 2 x1
x1
z
5 z = 30 + 3x1 − x4 . 2 x1
x1 x3 = 4 − x1 x2 = 6 x5 = 6 − 3x1 . x1 = 2
x5
x1
x5 + 32 x4
z
+x5 = 36
x3 + 13 x4 − 13 x5 = x2 x1
+ 12 x4
2
=
6
− 13 x4 + 13 x5 =
2.
(2, 6, 2, 0, 0).
3 z = 36 − x4 − x5 . 2
x∗1 = 2 x∗2 = 6
z ∗ = 36.
(0, 0)
(0, 6) (2, 6)
• •
• z
•
z,
•
e
•
bs bi = min . ase i|a aie ie>0
s
s
• • •
xe
ase . s.
max z = 4x1 + 3x2
3x1 + 4x2 ≤ 12 7x1 + 2x2 ≤ 14 0
x1 , x2 ≥
max z = 5x1 + 4x2
x1 + x2 ≤ 20 2x1 + x2 ≤ 35
−3x1 + x2 ≤ 12 x1 , x2 ≥
0
max z = 20x1 + 16x2 + 12x3
x1 2x1 +
≤
400
x2 + x3 ≤ 1000
2x1 + 2x2 + x3 ≤ 1600 x1 , x2 , x3 ≥
0
max z = 3x1 + 5x2
≤
x1
2x2 ≤ 12 3x1 +2x2 ≤ 18
≥
0
x2 ≥
0.
x1
z −3x1 −5x2 +x3
x1 2x2 3x1 +2x2
4
+x4
=
0
=
4
= 12 +x5 = 18
z
x1
x2 x3 x4 x5
1 −3 −5
0
0
0
0
0
1
0
1
0
0
4
0
0
2
0
1
0
12
0
3
2
0
0
1
18
•
• •
•
(x1 , x2 , x3 , x4 , x5 ) = (0, 0, 4, 12, 18).
x2
z
x1
x2 x3 x4 x5
1 −3 −5
0
0
0
0
0
1
0
1
0
0
4
0
0
2
0
1
0
12
0
3
2
0
0
1
18
= 6.
min
12 18 , 2 2
−1
x2
0
x1 = 4 + x2 . x2 > 0
x1
1
x4 x4
z
x1
x2 x3 x4 x5
1 −3 −5
0
0
0
0
0
1
0
1
0
0
4
0
0
2
0
1
0
12
0
3
2
0
0
1
18
x2
x4 x4
z
x1
x2
x2 x3
x4 x5
1 −3 −5
0
0
0
0
0
1
0
1
0
0
4
0
0
1
0 1/2
0
6
0
3
2
0
1
18
z
x1 x2 x3
0
x2
x4 x5
1 −3
0
0 5/2
0
30
0
1
0
1
0
0
4
0
0
1
0 1/2
0
6
0
3
0
0
1
6
−1
(x1 , x2 , x3 , x4 , x5 ) = (0, 6, 4, 0, 6) z = 30.
x1
z
x1 x2 x3
x4 x5
1 −3
0
0 5/2
0
30
0
1
0
1
0
0
4
0
0
1
0 1/2
0
6
0
3
0
0
1
6
min
−1
4 6 , 1 3
= 2. x5
z
x1 x2 x3
x4 x5
1 −3
0
0 5/2
0
30
0
1
0
1
0
0
4
0
0
1
0 1/2
0
6
0
3
0
0
1
6
−1
z
x1 x2 x3
x4
x5
1 −3
0
0
5/2
0
30
0
1
0
1
0
0
4
0
0
1
0
1/2
0
6
0
1
0
0 −1/3 1/3
2
x1
z x1 x2 x3
x4
x5
1
0
0
0
3/2
1
36
0
0
0
1
1/3 −1/3
2
0
0
1
0
1/2
0
6
0
1
0
0 −1/3
1/3
2
(x1 , x2 , x3 , x4 , x5 ) = (2, 6, 2, 0, 0) z = 36.
x∗1 = 2 x∗2 = 6 z ∗ = 36.
P 0 = (0, 0)
P 1 = (0, 6)
x2
P 2 = (2, 6) P 2
x j = 0, ∀ j = 1, . . . , n .
z
x1
x2
...
x1
xn+1 xn+2 . . . xn+m
1 −c1 −c2 · · · −cn
0
0
0
0
0 a11
a12
· · · a1n
1
0
···
0
b1
0 a21
a22
· · · a2n
0
1
···
0
b2
0 an1
an2 · · · ann
0
0
···
1
bn
− ce ≤ −c j , ∀ j | − c j < 0.
xe
z
xn
x2
···
xe
···
xn
xn+1 xn+2 . . . xn+m
1 −c1 −c2 · · ·
ce
· · · −c n
0
0
0
0
0 a11
a12
· · · a1e · · · a1n
1
0
···
0
b1
0 a21
a22
· · · a2e · · · a2n
0
1
···
0
b2
0 an1
an2 · · · ane · · · ann
0
0
···
1
bn
