1
$\begingroup$

I was given a problem where a home computer table sells for $36$ dollars and uses $6$ board ft of lumber, $2$ finishing hrs, and $2$ carpentry hrs. Should the company manufacture any home computer tables?

Given LP:

$\max$ $z$ = $60x_1$ + $30x_2$ + $20$$x_3$

$st.$

$8x_1$ + $6x_2$ + $1$$x_3$ $\leq 48$ (Lumber)

$4x_1$ + $2x_2$ + $1.5$$x_3$ $\leq 20$ (Finishing)

$2x_1$ + $1.5x_2$ + $0.5$$x_3$ $\leq 8$ (Carpentry)

$x_1$ , $x_2$, $x_3$ $ \geq 0$


The only solution the textbook gave was that it is not possible to build home computer table. However, I didn't understand why it is. I did the tableau to show some of my findings of the final result


Problem given from textbook


\begin{array}{rrrrrrrr|r|r} row & z & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & BV \\ \hline 0 & 1 & -60 & -30 & -20 & 0 & 0 & 0 & z= 0 \\ \hline 1 & 0 & 8 & 6 & 1 & 1 & 0 & 0 & s_1 = 48 \\ 2 & 0 & 4 & 2 & 1.5 & 0 & 1 & 0 & s_2 = 20 \\ 3 & 0 & 2 & 1.5 & 0.5 & 0 & 0 & 1 & s_3 = 8 \end{array}

\begin{array}{rrrrrrrr|r|r} row & z & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & BV \\ \hline 0 & 1 & 0 & 15 & -5 & 0 & 0 & 30 & z= 240 \\ \hline 1 & 0 & 0 & 0 & -1 & 1 & 0 & -4 & s_1 = 16 \\ 2 & 0 & 0 & -1 & 0.5 & 0 & 1 & 2 & s_2 = 4 \\ 3 & 0 & 1 &-0.75 & 0.25 & 0 & 0 & 1/2 & s_3 = 4 \end{array}

\begin{array}{rrrrrrrr|r|r} row & z & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & BV \\ \hline 0 & 1 & 0 & 5 & 0 & 0 & 10 & 10 & z= 280 \\ \hline 1 & 0 & 0 & -2 & 0 & 1 & 2 & -8 & s_1 = 16 \\ 2 & 0 & 0 & -2 & 1 & 0 & 2 & -4 & s_2 = 4 \\ 3 & 0 & 1 & 1.25 & 0 & 0 & -0.5 & 1.5 & s_3 = 4 \end{array}

$x_1$ $= 2$ , $x_3$ $= 8$ $and$ $Max$ $ z$ $= 280$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.