Question

For an LPP, minimise $z=2x+y$subject to constraints $5x+10y\le 50,x+y\ge 1,y\le 4$ and $x,y\ge 0$, then $z$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. $12$

Answer 1

The correct option is B

$1$

Step 1. Find the endpoints of the constraints $\mathbf{5}\mathit{x}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{\le }\mathbf{50}$.

$\begin{array}{rcl}5x+10y& \le & 50\\ \frac{5}{50}x+\frac{10}{50}y& \le & 1\\ \frac{x}{10}+\frac{y}{5}& \le & 1\end{array}$

for the value of $y=0$

$x=10$

and for the value of $x=0$

$y=5$

Therefore, the points of line $5x+10y\le 50$ are $\left(0,5\right)$ and $\left(10,0\right)$.

Find the endpoints of the constraints $x+y\ge 1$.

$\begin{array}{rcl}x+y& \ge & 1\end{array}$

Therefore, the points of line $x+y\ge 1$ are $\left(0,1\right)$ and $\left(1,0\right)$.

Find the endpoints of the constraints$\begin{array}{rcl}y& \le & 4\end{array}$.

$\begin{array}{rcl}y& \le & 4\end{array}$

Therefore, the points of line $y\le 4$ is $\left(0,1\right)$.

Find the endpoints of the constraints $\begin{array}{rcl}x+y& \ge & 0\end{array}$.

$\begin{array}{rcl}x+y& \ge & 0\end{array}$

Therefore, the points of line $x+y\ge 0$ are $\left(0,0\right)$ and $\left(0,0\right)$.

Step 2. Draw the graph by using points and locate common points of all constraints.

Step 3. Calculate the values of $z$ at points $\left(1,0\right)\mathbf{,}\mathbf{}\left(0,1\right)\mathbf{,}\mathbf{}\left(0,4\right)\mathbf{,}\mathbf{}\left(2,4\right)$ and $\begin{array}{l}\left(10,0\right)\end{array}$.

At point $\left(1,0\right)$:

$z=2·1+0=2$

At point $\left(0,1\right)$:

$z=2·0+1=1$

At point $\left(0,4\right)$:

$z=2·0+4=4$

At point $\left(2,4\right)$:

$z=2·2+4=8$

At point $\left(10,0\right)$:

$z=2·10+0=20$

Therefore, the minimum value of $\mathit{z}$ is $\mathbf{1}$.