Question

If $z=5x+2y$ subject to the following constraints : $x-2y\le 2,3x+2y\le 12,-3x+2y\le 3,x\ge 0,y\ge 0,$ then which of the following is/are true?

• A. $z_{max}=19$
• B. $z_{max}=21$
• C. $z_{min}=0$
• D. $z_{min}=10$

Answer 1

Answer: (A), (C)

$z_{min}=0$

The subject of constraints is :
$x-2y\le 2\cdots \left(i\right)$
$3x+2y\le 12\cdots \left(ii\right)$
$-3x+2y\le 3\cdots \left(iii\right)$
$x\ge 0,y\ge 0\cdots \left(iv\right)$

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 $z=5x+2y$

$\begin{array}{cc}\text{Corner Point}& \text{Corresponding value of z}\\ B(2,0)& 10\\ G(\frac{7}{2},\frac{3}{4})& 19\\ H(\frac{3}{2},\frac{15}{4})& 15\\ J(0,\frac{3}{2})& 3\\ O(0,0)& 0\end{array}$
Hence,
$z_{max}=19$ at the point $G(\frac{7}{2},\frac{3}{4})$
and $z_{min}=0$ at the point $O(0,0)$