# Homework13 - DOC

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```					Homework 13

1 (4 points) A coal fired power plant carefully mixes coal from a variety of suppliers before burning the
coal to create power. The plant draws coal from three different mines: the Hondu mine, the Mussentuchit
mine, and the Swazey mine. Burning coal from different mines produces different amounts of heat, ash,
and sulfur. The objective is to mix the coal to maximize heat while keeping sulfur emissions below
regulatory limits and minimizing ash (ash cakes the steam lines, causing them to bend, then to break,
eventually causing the plant to stand down while the pipes are repaired). Hondu coal generates 500
BTUs, 5 ounces of sulfur, and 3 pounds of ash per thoushand pounds burned. Mussentuchit coal
generates 450 BTUs, 9 ounces of sulfur, and 6 pounds of ash per thousand pounds burned. Swazey coal
generates 700 BTUs, 10 ounces of sulfur, and 10 pounds of ash per thousand pounds burned.

How much of each kind of coal should b eburned in a given day to maximize the BTUs generated while
keeping total ash emissions below 25 pounds, sulfur emissions below 64 ounces, while keeping in mind
that 20000 pounds of Hondu coal, and unlimited supply of Mussentuchit coal, and 10000 pounds of
Swazey coal are available per day?

maximization function: 500h + 450m + 700s
subject to
3h + 6m + 10s < 25
5h + 9m + 10s < 64
h<=20
s<=10

2. (12 points) Convert the following linear programming problem into slack form:
Maximize 5x_1 + 2x_2 - 9x_3
subject to
4x_1 - 1x_2 -x_3 <= 3
-x_1 + 3x_2 + x_3 <= -12
2x_1 - 3x_2 + x_3 <= 5
5x_1 - 2x_2 + 2x_3 <= 10
x_1, x_2, x_3 >= 0

This is already in standard form, so we only need to convert it to slack form:

z = 5x_1 + 2x_2 - 9x_3
x_4 = 3 - 4x_1 + x_2 + x_3
x_5 = -12 + x_1 - 3x_2 - x_3
x_6 = 5 - 2x_1 + 3x_2 - x_3
x_7 = 10 - 5x_1 + 2x_2 - 2x_3
x_1, x_2, x_3, x_4, x_5, x_6, x_7, z >= 0

3. (12 points) Concisely express the linear programming problem of question 2 using B, N, c, b, A, and v
as described above:

A = 4 -1 -1
-1 3 1
2 -3 1
5 -2 2

b=     3
-12
5
10

c = transpose of 5 2 -9
v=0

B = {4, 5, 6, 7}

N = {1, 2, 3}

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