Question

The corner points of the feasible region determined by the system of linear constraints are $(0,10),(5,5),(15,15),(0,20)$. Let $z=px+qy$ where $p,q>0$. Condition on p and q so that the maximum of z occurs at both the points $(15,15)$ and $(0,20)$ is __________.

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

Answer 1

Answer: (D)

$q=3p$

Let $z_0$ be the maximum value of $z$ in the feasible region.

Since maximum occurs at both $(15,15)$ and $(0,20)$$,thevalue$z_{0}$isattainedatboth$(15,15)$and$(0,20)$.

$⟹z_0=p\left(15\right)+q\left(15\right)$ and $z_0=p\left(0\right)+q\left(20\right)$

$⟹p\left(15\right)+q\left(15\right)=p\left(0\right)+q\left(20\right)$

$⟹15p=5q$

$⟹3p=q$

Hence, the answer is option (D).