https://www.nicovideo.jp/watch/sm37958547 https://www.youtube.com/watch?v=Nei0nsMNz-I&feature=youtu.be

1. 8

2. 1.

3. 1.
(a)
(b)
(c)

5.
max c1x1 + c2x2 + ⋯ + cnxn
s . t .
a11x1 + a12x2 + ⋯ + a1nxn + λ1 = b1
a21x1 + a22x2 + ⋯ + a2nxn + λ2 = b2
⋮
am1x1 + am2x2 + ⋯ + amnxn + λm = bm
x1 ≥ 0, x2 ≥ 0, ⋯, xn ≥ 0, λ1 ≥ 0, λ2 ≥ 0,⋯, λm ≥ 0
m + n
n
n
(x*1
, ⋯, x*n , λ*1
, ⋯, λ*m)
m + n x*1
, ⋯, x*n , λ*1
, ⋯, λ*m n

6.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 ≤ 18 ⋯(1)
2x1 + 8x2 ≤ 52 ⋯(2)
2x1 + 10x2 ≤ 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x1
x2
9
6 26
6.5 (2, 6)
32.5

7.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 ≤ 18 ⋯(1)
2x1 + 8x2 ≤ 52 ⋯(2)
2x1 + 10x2 ≤ 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x1
x2
9
6 26
6.5 (2, 6)
32.5
x1 = 0
x1 = 0

8.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 ≤ 18 ⋯(1)
2x1 + 8x2 ≤ 52 ⋯(2)
2x1 + 10x2 ≤ 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x1
x2
9
6 26
6.5 (2, 6)
32.5
x1 = 0
x1 = 0

9.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x1
x2
9
6 26
6.5 (2, 6)
32.5
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1 = 0
x1 = 0

10.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x1
x2
9
6 26
6.5 (2, 6)
32.5
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1 = 0
x1 = 0

11.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0

12.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2
x3 = 18 − 3x1 − 2x2
x4 = 52 − 2x1 − 8x2
x5 = 65 − 2x1 − 10x2
f = 10x1 + 20x2

13.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2
x3 = 18 − 3x1 − 2x2 = 18
x4 = 52 − 2x1 − 8x2 = 52
x5 = 65 − 2x1 − 10x2 = 65
f = 10x1 + 20x2 = 0
x1, x2 (x1, x2, x3, x4, x5) = (0, 0, 18, 52, 65)
x1 x2 x3 x4 x5

14.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x1 x2 x3 x4 x5
x1 6 x3 = 0
x2 6.5 x4 = 0, x5 = 0
x2 x4
x2 x5
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0

15.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x1 x2 x3 x4 x5
x1 6 x3 = 0
x2 6.5 x4 = 0, x5 = 0
x2 x4
x2 x5
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0

16.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x1 x2 x3 x4 x5
x1 6 x3 = 0
x2 6.5 x4 = 0, x5 = 0
x2 x4
x2 x5
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0

17.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x2, x3, x4
x2 =
13
2
−
1
5
x1 −
1
10
x5
x3 = 18 − 3x1 − 2x2 = 5 −
13
5
x1 +
1
5
x5
x4 = 52 − 2x1 − 8x2 = −
2
5
x1 +
4
5
x5
f = 10x1 + 20x2 = 130 + 6x1 − 2x5

18.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x2, x3, x4
x2 =
13
2
−
1
5
x1 −
1
10
x5
x3 = 18 − 3x1 − 2x2 = 5 −
13
5
x1 +
1
5
x5
x4 = 52 − 2x1 − 8x2 = −
2
5
x1 +
4
5
x5
f = 10x1 + 20x2 = 130 + 6x1 − 2x5
x2, x5
(x1, x2, x3, x4, x5) = (0, 13/2, 5, 0, 0)

19.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x2, x3, x4
x2 =
13
2
−
1
5
x1 −
1
10
x5
x3 = 18 − 3x1 − 2x2 = 5 −
13
5
x1 +
1
5
x5
x4 = 52 − 2x1 − 8x2 = −
2
5
x1 +
4
5
x5
f = 10x1 + 20x2 = 130 + 6x1 − 2x5
x1, x5
(x1, x2, x3, x4, x5) = (0, 13/2, 5, 0, 0)
x1

20.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x2, x3, x4
x2 =
13
2
−
1
5
x1 −
1
10
x5
x3 = 18 − 3x1 − 2x2 = 5 −
13
5
x1 +
1
5
x5
x4 = 52 − 2x1 − 8x2 = −
2
5
x1 +
4
5
x5
f = 10x1 + 20x2 = 130 + 6x1 − 2x5
x2, x3, x4 x1
x4 x1

21.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x2, x3, x4
x2 =
13
2
−
1
5
x1 −
1
10
x5
x3 = 18 − 3x1 − 2x2 = 5 −
13
5
x1 +
1
5
x5
x4 = 52 − 2x1 − 8x2 = −
2
5
x1 +
4
5
x5
f = 10x1 + 20x2 = 130 + 6x1 − 2x5
x2, x3, x4 x1
x4 x1
x1 x4

22.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5

23.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x4, x5
(x1, x2, x3, x4, x5) = (0, 13/2, 5, 0, 0)

24.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x4, x5
(x1, x2, x3, x4, x5) = (0, 13/2, 5, 0, 0)
x1 x4

25.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x4, x5
(x1, x2, x3, x4, x5) = (0, 13/2, 5, 0, 0)
(x2, x3, x4)
(x1, x2, x3)

26.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x1, x2, x3 x5
x3 x5 1

27.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x1, x2, x3 x5
x3 x5 1

28.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x1, x2, x3 x5
x3 x5 1
x5 x3

29.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x5
x1 = 2 −
2
5
x3 +
1
10
x4
x2 = 6 +
1
10
x3 −
3
20
x4
x5 = 1 −
1
5
x3 +
13
10
x4
f = 10x1 + 20x2 = 140 − 2x3 − 2x4
(x1, x2, x3, x4, x5) = (2, 6, 0, 0, 1)

30.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x1 x2 x3 x4 x5
x1 6 x3 = 0
x2 6.5 x4 = 0, x5 = 0
x2 x4
x2 x5
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0

31.
n
max 10x1 + 20x2
s . t .
3x1 + 2x2 + x3 = 18 ⋯(1)
2x1 + 8x2 + x4 = 52 ⋯(2)
2x1 + 10x2 + x5 = 65 ⋯(3)
x1 ≥ 0, x2 ≥ 0
x3 ≥ 0, x4 ≥ 0, x5 ≥ 0
x4 = 0 x1 = 0
x5 = 0 x1 = 0
x4 = 0 x5 = 0
x1, x2, x3
x1 = −
5
2
x4 + 2x5
x2 =
13
2
+
1
2
x4 −
1
2
x5
x3 = 5 +
13
2
x4 − 5x5
f = 10x1 + 20x2 = 130 − 15x4 + 10x5
x4, x5
(x1, x2, x3, x4, x5) = (0, 13/2, 5, 0, 0)

33. See you next time
8