# The simplex method

• Aug 4th 2009, 08:26 AM
chella182
The simplex method
The question goes...

Use the simplex method to solve the following linear programming problem. Maximise

$f(x_1, x_2, x_3)=3x_1+2x_2+4x_3$

subject to the constraints

$3x_1+x_2+4x_3\leq60$
$x_1+2x_2+3x_3\leq30$
$2x_1+2x_2+3x_3\leq600$
$x_1, x_2, x_3\geq0$
• Aug 4th 2009, 09:24 AM
Opalg
This looks like a totally standard exercise in using the simplex method, so where are you having trouble with it?

Just to get you started, you should introduce "slack variables" u,v,w, to convert the inequalities into equations, namely

\begin{aligned}3x_1+x_2+4x_3+u\qquad\qquad &= 60,\\ x_1+2x_2+3x_3 \qquad+ v\qquad&= 30,\\ 2x_1+2x_2+3x_3\qquad\qquad+w &= 600.\end{aligned}

The objective equation is $M = 3x_1+2x_2+4x_3$, which you write as $-3x_1-2x_2-4x_3+M=0$.

Then you write down the simplex tableau, which is just the matrix of coefficients in these equations, and apply the simplex algorithm.

If it's any help to you, I wrote down a systematic description of the simplex algorithm a few years ago when teaching this stuff. You can find a copy of it here (pdf file).
• Aug 4th 2009, 09:52 AM
chella182
My problem is that I don't ever remember being taught this, and it's not in my notes anywhere, so I don't know what the simplex algorithm is or anything. I'll have a look at that .pdf, see if that helps.
• Aug 4th 2009, 07:21 PM
mr fantastic
Quote:

Originally Posted by chella182
My problem is that I don't ever remember being taught this, and it's not in my notes anywhere, so I don't know what the simplex algorithm is or anything.
[snip]

Then you should ask whoever gave it to you why you were given it.