https://www.nicovideo.jp/watch/sm38021687 https://youtu.be/Fe1_Teh5XEk

1. 10

2. 1.

3. 1.
(a)

5.
m + n
n
n

6.
m + n
n
n
max 10x1 + 20x2
s . t .
2x1 + 8x2 ≥ 52 ⋯(1)
x1 + x2 ≤ 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0
x1
x2
9
6 26
6.5
5
5

7.
m + n
n
n
max 10x1 + 20x2
s . t .
2x1 + 8x2 = 52 + x3 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0

8.
m + n
n
n
max 10x1 + 20x2 − Mx5
s . t .
2x1 + 8x2 + x3 − x5 = 52 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0,x5 ≥ 0
x3, x4 x1, x2
x3 = 2x1 + 8x2 − 52
x4 = 5 − x1 − x2
f = 10x1 + 20x2
x1, x2
(x1, x2, x3, x4) = (0, 0, − 52, 5)
x3
x3 = 2x1 + 8x2 − 52 + x5, x5 ≥ 0
x5 = 52 − 2x1 − 8x2 + x3

9.
m + n
n
n
max 10x1 + 20x2 − Mx5
s . t .
2x1 + 8x2 + x3 − x5 = 52 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0,x5 ≥ 0
x4, x5 x1, x2, x3
x4 = 5 − x1 − x2
x5 = 52 − 2x1 − 8x2 + x3
f = 10x1 + 20x2 − M(52 − 2x1 − 8x2 + x3)
= (10 + 2M)x1 + (20 + 8M)x2 − 52M − Mx3
x1, x2, x3
(x1, x2, x3, x4, x5) = (0, 0, 0, 5, 52)
x1, x2

10.
m + n
n
n
max 10x1 + 20x2 − Mx5
s . t .
2x1 + 8x2 + x3 − x5 = 52 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0,x5 ≥ 0
x4, x5 x1, x2, x3
x4 = 5 − x1 − x2
x5 = 52 − 2x1 − 8x2 + x3
f = (10 + 2M)x1 + (20 + 8M)x2 − 52M − Mx3
x1, x2, x3
(x1, x2, x3, x4, x5) = (0, 0, 0, 5, 52)
x1, x2
x2 x4

11.
m + n
n
n
max 10x1 + 20x2 − Mx5
s . t .
2x1 + 8x2 + x3 − x5 = 52 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0,x5 ≥ 0
x2, x5 x1, x3, x4
x2 = 5 − x1 − x4
x5 = 12 + 6x1 + 8x4 + x3
f = 100 − (10 + 6M)x1 − (20 + 8M)x4 − Mx3 − 12M
x1, x3, x4
(x1, x2, x3, x4, x5) = (0, 5, 0, 0, 12)
x1, x3, x4

12.
m + n
n
n
max 10x1 + 20x2 − Mx5
s . t .
2x1 + 8x2 + x3 − x5 = 52 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0,x5 ≥ 0
x2, x5 x1, x3, x4
x2 = 5 − x1 − x4
x5 = 12 + 6x1 + 8x4 + x3
f = 100 − (10 + 6M)x1 − (20 + 8M)x4 − Mx3 − 12M
x1, x3, x4
(x1, x2, x3, x4, x5) = (0, 5, 0, 0, 12)
x1, x3, x4
x5 = 12 > 0

13.
m + n
n
n
max 10x1 + 20x2 − Mx5
s . t .
2x1 + 8x2 + x3 − x5 = 52 ⋯(1)
x1 + x2 + x4 = 5 ⋯(2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0,x5 ≥ 0
x2, x5 x1, x3, x4
x2 = 5 − x1 − x4
x5 = 12 + 6x1 + 8x4 + x3
f = 100 − (10 + 6M)x1 − (20 + 8M)x4 − Mx3 − 12M
x1, x3, x4
(x1, x2, x3, x4, x5) = (0, 5, 0, 0, 12)
x1, x3, x4
x5 = 12 > 0

15. See you next time
10