# Applications of Linear Programming

• Aug 11th 2006, 08:50 AM
classicstrings
Applications of Linear Programming
I'm having trouble setting up the variables and constraints in these questions. Wondering if someone can help? Thanks!

http://img216.imageshack.us/img216/6991/2zq1.jpg
• Aug 11th 2006, 08:36 PM
Soroban
Hello, classicstrings!

Here's the second one . . .

Quote:

A furniture maker produces cupboards and bookshelves.
Both products use wooden planks, sawing time, sanding time, and assembly time.
The cupboards use 20 m of wood, 40 minutes of sawing, 60 minutes of sanding,
and 10 minute of assembly time for a profit of $280. The bookshelves use 10 m of wood, 30 minutes of sawing, 90 minutes of sanding, and 30 minutes of assembly time for a profit of$340.

There are 220 m of wood, 480 minutes of sawing time, 1080 minutes of sanding time,
and 330 minutes of assembly time available.

How many of each item should be made for maximum profit?

Let $\displaystyle x$ = number of cupboards, $\displaystyle y$ = number of bookshelves. .$\displaystyle x \geq 0,\;y \geq 0$ [1]
A chart helps to organize the information . . .
Code:

                | wood + saw | sand | assem | - - - - - - - - + - - -+ - - + - - -+ - - - +   cupboards (x) |  20x | 40x |  60x |  10x  | - - - - - - - - + - -  + - - + - - -+ - - - + bookshelves (y) |  10y | 30y |  90y |  30y  | - - - - - - - - + - -  + - - + - - -+ - - - +   available    |  220 | 480 | 1080 |  330  | - - - - - - - - + - - -+ - - + - - -+ - - - +

Wood: .$\displaystyle 20x + 10y \:\leq \:220\quad\Rightarrow\quad 2x + y \:\leq \:22$ [2]

Sawing: .$\displaystyle 40x + 30y \:\leq \:480\quad\Rightarrow\quad 4x + 3y \:\leq \:48$ [3]

Sanding: .$\displaystyle 60x + 90y \:\leq \:1080\quad\Rightarrow\quad 2x + 3y \:\leq \:36$ [4]

Assembly: .$\displaystyle 10x + 30y \:\leq \:330\quad\Rightarrow\quad x + 3y\:\leq\:33$ [5]

[1] places us in Quadrant 1.

[2] Graph the line: $\displaystyle 2x + y \:=\:22$. .It has intercepts: $\displaystyle (11,0),\;(0,22)$. .
. . .Shade the region below the line.

[3] Graph the line: $\displaystyle 4x + 3y \:=\:48$. .It has intercepts: $\displaystyle (12,0),\;(0,16)$
. . .Shade the region below the line.

[4] Graph the line: $\displaystyle 2x + 3y \:=\:36$. .It has intercepts: $\displaystyle (18,0),\;(0,12)$
. . .Shade the region below the line.

[5] Graph the line: $\displaystyle x + 3y \:=\:33$. .It has intercepts: $\displaystyle (33,0),\;(0,11)$
. . .Shade the region below the line.

The final region is a hexagon. .Its vertices are (clockwise from the origin):
. . $\displaystyle (0,0),\;(0,11),\;(3,10),\;)6,8),\;(9,4),\;(0,11)$

Test them in the profit function: $\displaystyle P \:= \:280x + 340y$
. . to see which one produces maximum profit.

• Aug 13th 2006, 01:09 AM
classicstrings
Hey Soroban! You have done the harder one for me, and I have gone through it a couple of times myself after, I have done the first one by looking @ how you did them. Cheers!