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  4. 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

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Q. 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

Linear Programming

Solution:

Value of $Z = px + qy$ at $(15,15)$ is $15p + 15q$ and
that at $(0, 20)$ is $20 q$.
According to given condition, we must have
$15p + 15 q = 20 q$
$\Rightarrow 15p = 5q$
$\Rightarrow q = 3p$