Question

The corner points of the feasible region determined by the system of linear constraints are $(0,10),(5,5),(25,20)$ and $(0,30)$. Let $Z=px+qy$, where $p,q>0$. Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both the points $(25,20)$ and $(0,30)$ is _______.

• A. $5p=2q$
• B. $2p=5q$
• C. $p=2q$
• D. $q=3p$

Answer 1

Answer: (A)

$5p=2q$

Maximum of $Z$ occurs at $(25,20)$ and at $(0,30)$.

Hence, equating the vales of $Z$ at these points, we get

$25p+20q=30q$

$\therefore 25p=10q$

$\therefore 5p=2q$

This is the required relation.

Also as $p,q>0$, the value of $Z$ is always positive and hence, is greater at $(25,20)$ and at $(0,30)$ than at $(0,10)$ and $(5,5)$.