# Solving problems with the Simplex method

• Nov 13th 2011, 01:19 PM
m58
Solving problems with the Simplex method
Hello, I've been having trouble with some problems that involve the Simplex method; one is a word problem.

Problem 1:

An airplane manufacturer is looking to buy fleets of planes from other companies in packages of different sized planes. Firm 1 is offering fleets with of 5 small planes, 5 medium planes, and 10 large planes at a total cost of \$500,000. Firm 2 is offering fleets that consist of 5 small, 10 medium, and 5 large planes at a total cost of \$400,000. Firm 3 is offering fleets of 10 small, 5 medium, and 5 large planes at a total cost of \$300,000. There are also talks of putting together a fleet of at least 550 small planes, 500 medium planes, and 550 large planes. How many packages should the firm buy to minimize costs?

Here's what I have...

x = number of small planes the firm should buy
y = number of medium planes...
z = number of large planes...

My set-up for the simplex method:
Maximize -10x -10y -10z subject to:
x + y + 2z ≤ 5
x + 2y + z ≤ 4
2x + y + z ≤ 3
x ≥ 0, y ≥ 0, z ≥ 0
I got these numbers from reducing the ones in the problem (i.e. -10x came from 550, which has to be negative when maximized, since Simplex doesn't work for minimizing, and then I divided 550 by 55 to get 10, etc).
I'm not sure if I should have reduced the numbers, though... how does it look so far?

Problem 2:

Minimize -x + y subject to:
x + y ≥ 5
x + 2y ≤ 8
7x - 5y ≥ 0
x ≥ 0, y ≥ 0, z ≥ 0

My set-up:
Maximize x - y subject to:
- x - y ≤ -5
x + 2y ≤ 8
- 7x + 5y ≤ 0
x ≥ 0, y ≥ 0, z ≥ 0

If I knew how to post matrices on here, I would show you my work, but typing in the matrix normally makes it look incredibly confusing (nothing lines up, especially once I start to type in negatives). I did get x = 8, y = 56 as my answer, though. However, the answer key says x = 8, y = 0 minimizes the LLP and x = 40/19 and y = 56/19 maximizes the LPP. Would anyone be able to help?