Question

Minimise and maximise $z=5x+2y$ subject to the following constraints :

$x-2y\le 2$

$3x+2y\le 12$

$-3x+2y\le 3$

$x\ge 0,y\ge 0$

Answer 1

The subject of constraints is :

$x-2y\le 2...\left(i\right)$

$3x+2y\le 12...\left(ii\right)$

$-3x+2y\le 3...\left(iii\right)$

$x\ge 0,y\ge 0...\left(iv\right)$

On solving (i), we have $A(0,-1),B(2,0)$

On solving (ii), we have $C(0,6),D(4,0)$

On solving (iii), we have $E(-1,0),F(1,3)$

It is observed that the feasible region OBGHJ is bounded.

Thus, we use corner point method to determine the maximum and minimum value of z, where $z=5x+2y$

$\begin{array}{cc}\text{Corner Point}& \text{Corresponding value of z}\\ B(2,0)& 10\\ G(\frac{7}{2},\frac{3}{4})& 19\\ H(\frac{3}{2},\frac{15}{4})& 15\\ J(0,\frac{3}{2})& 3\end{array}$

Hence, $z_{max}=19$ at the point $G(\frac{7}{2},\frac{3}{4})$

and $z_{min}=3$ at the point $J(0,\frac{3}{2})$