Sum of maximum and minimum value of the objective function $z = 10 x + 7 y$ , subjected to the constraints $0 \leq x \leq 60, 0 \leq y \leq 45, 5 x + 6 y \leq 420$ is

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Solution

Given, $z = 10 x + 7 y$ and constraints are $0 \leq x \leq 60, 0 \leq y \leq 45, 5 x + 6 y \leq 420$
$\Rightarrow 0 \leq x \leq 60, 0 \leq y \leq 45$ and $\frac{x}{84} + \frac{y}{70} \leq 1$
Now, we get common area as marked in figure
Finally,

Corner points $z = 10 x + 7 y$

$(0, 45)$ $315$

$(60, 20)$ $740$ (Maximum)

$(60, 0)$ $600$

$(30, 45)$ $615$

$(0, 0)$ $0$ (Minimum)

Thus, $z_{max} + z_{min} = 740 + 0 = 740$

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