Question

{F} The furniture company inexpensive tables and chairs. The production process of each is similar in the painting department each table take 4 hours of carpentry and 2 hours in the painting department. Each chair requires 3 hours of carpentry and 1 hour painting department. During the current product period 240 hours of carpentry time are available and 100 hrs in the painting is available. Each table sold yields of profit of $7 and chair produced sold for $5 profit. Find the best combination of table and chairs to manufacture in order to reach the maximum number of profit?

Answer 1

Let $x_1$ = number of tables

$x_2$ = number of chairs

$z$ = profit

Maximize $z=7x_1+5x_2$

Subject to constraints:

$4x_1+3x_2\le240$ ( Carpentry constraints)

$2x_1+x_2\le100$ (Painting constraints)

$x_1,x_2$ $\ge0$ ( Non-negativity constraints)

Initial system of equations

$-7x_1-5x_2+z=0\\4x_1+3x_2+s_1=240\\2x_1+x_2+s_2=100$

Where $s_1$ and $s_2$ are slack variables

$\begin{matrix} x_1& x_2&s_1&s_2&z&& \\ 4&3&1&0&0&240\\2&1&0&1&0&100\\-7&-5&0&0&1&0 \end{matrix}$

Pivot column is $1^{st}$

Test ratio:

$\frac{240}{4}=60$ $\frac{100}{2}=50$

Pivot row is $2^{nd}$

Using Gaussian Elimination

$R_2÷2\to R_2\>$ $-4R_2+R_1\to R_1$

$7R_2+R_3\to R_3$

$\begin{matrix} x_1&x_2&s_1&s_2&z&&\\ 0&1&1&-2&0&40\\1&0.5&0&0.5&0&50\\0&-1.5&0&3.5&1&350 \end{matrix}$

Pivot column is $2^ {nd}$

Test ratio:

$\frac{40}{1}=40$ $\frac {50}{0.5}=100$

Pivot row is $1^{st}$

Using Gaussian elimination

$-½R_1+R_2\to R_2\\1.5R_1+R_3\to R_3$

$\begin{matrix} x_1&x_2&s_1&s_2&z&&\\0&1&1&-2&0&40\\1&0&-0.5&0.5&0&30\\0&0&1.5&0.5&1&410 \end{matrix}$

$\therefore x_1=$ Number of tables $=40$

$x_2=$ Number of chairs$=30$

$z=$ Profit$= \$ 410$