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Question

Solve the following linear programming problem graphically:
Minimize $Z = 200 x + 500 y$
Subject to the constraints:
$x + 2 y \geq 10$
$3 x + 4 y \leq 24$
$x \geq 0, y \geq 0$

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Solution

Minimise $Z = 200 x + 500 y$
Subject to
$x + 2 y \geq 10$ , $3 x + 4 y \leq 24$ , $x \geq 0, y \geq 0$

$x + 2 y = 10$
$\begin{matrix} x & 0 & 10 y & 5 & 0 \end{matrix}$

$3 x + 4 y = 24$
$\begin{matrix} x & 0 & 8 y & 6 & 0 \end{matrix}$
$\begin{matrix} Corner points & Value of Z = 200 x + 500 y (0, 6) & 3000 (0, 5) & 2500 (4, 3) & 2300 \end{matrix}$

$∴$ Minimum value of $z$ is $2300$ at $(4, 3)$ .

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M a x i m u m Z = 4 x + y S u b j e c t t o x + y \leq 50 3 x + y \leq 90 x \geq 0, y \geq 0

M i n i m i z e Z = 200 x + 500 y S u b j e c t t o x + 2 y \geq 10 3 x + 4 y \leq 24 x \geq 0, y \geq 0

M a x, a n d M i n . Z = 3 x + 9 y S u b j e c t t o x + 3 y \leq 60 x + y \geq 10 x \leq y x \geq 0, y \geq 0

M i n i m i z e Z = 3 x + 2 y S u b j e c t t o x + y \geq 8 3 x + 5 y \leq 15 x \geq 0, y \geq 0

M i n i m i z e Z = x + 2 y S u b j e c t t o 2 x + y \geq 3 x + 2 y \geq 6 x \geq 0, y \geq 0

M i n . a n d M a x . Z = 5 x + 10 y S u b j e c t t o x + 2 y \leq 120 x + 2 y \geq 60 x - 2 y \geq 0 x \geq 0, y \geq 0

Z = - 3 x + 4 y

x + 2 y \leq 8

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