# Linear programming

1. Mar 21, 2006

### dnt

i believe this type of question falls under linear programming but im not sure.

anyways, can someone help me out with this question:

An office manager needs to buy new filing cabinets. Cabinet A costs $15, takes up 6 ft^2 of space and holds 8 ft^3 of files. Cabinet B costs$10, takes up 9 ft^2 of space and holds 14 ft^3 of files. He has \$60 to spend and the office has room for no more than 63 ft^2 of cabinets. How many of each can he buy to maximize the storage capacity? What is the maximum storage capacity?

can someone help me set up the problem? do i set up variables for price, space, and files?

2. Mar 21, 2006

### Hurkyl

Staff Emeritus
Sure! Variables are good. Do those variables capture every aspect of the problem, though?

I suspect that the number of type A and the number of type B would be good variables too.

To be honest, though, I wouldn't really do this problem algebraically: I would do it by brute force. There are only 5 possibilities to consider: the number of type A that he buys.

3. Mar 21, 2006

### PPonte

This is indeed a linear programming problem.
You can reduce the variables to only two:

x - number of Cabinets A
y - number of Cabinets B

The other "pseudovariables" can be easily deduced by the two above:
The space occupied by x Cabinets A is equal to 6x ft2 and the space occupied by y Cabinets B is equal to 9y ft2.

Hope I could help.
See- http://en.wikipedia.org/wiki/Linear_Programming
http://people.brunel.ac.uk/~mastjjb/jeb/or/lp.html

Last edited by a moderator: Mar 21, 2006
4. Mar 21, 2006

### Hurkyl

Staff Emeritus
But because you can doesn't mean you should! I think a lot of time people are being confused because they're trying to do too much in their head! In those cases, making more variables can make things cleaner. They can always be eliminated later when solving the problem, once set up.

5. Mar 21, 2006

### PPonte

Ok. It's your opinion and I understand it. But I think it's easier to resolve the problem if we reduce it in the begining just to two variables. In linear programming problems you determine the optimal solution usually by visualizing the figure that result from the intersection of various inequations. If you have two variables the representation would be a polygon represented in a two-dimensional coordinate system, if you have three it will need to represented in a three-dimensional coordinate system and so on.

6. Mar 22, 2006

### dnt

thanks for the help guys. im starting to get it. if use the variable A and B to represent how many of each he buys, i have teh following equations:

15A + 10B = 60 (money)
6A + 9B = 63 (space)
8A + 14B = ??? (file storage)

what do i do with these equations? how do i solve them and figure out the max values?