Exercise 31.2
Solve the following linear programming problems graphically :
Question 1
Maximize Z=4x+y, subject to constraints :
$\begin{aligned}&x+y \leq 50 \\&3 x+y \leq 90 \\&x \geq 0, y \geq 0\end{aligned}$
Sol :
Question 2
Maximize Z=3x+4y, subject to the constraints :
$x+y \leq 4$
$x \geq 0, y \geq 0$
Sol :
Question 3
Minimize Z=200x+500y, subject to the constraints :
$x+2 y \geq 10$
$3 x+4 y \leq 24$
$x \geq 0, y \geq 0$
Sol :
Question 4
Minimize Z=-3x+4y, subject to constraints :
$x+2 y \leq 8$
$3 x+2 y \leq 12$
$x \geq 0, y \geq 0$
Sol :
Question 5
Maximize Z=3x+2y, subject to constraints :
$\begin{aligned} x+2 y & \leq 10 \\ 3 x+y & \leq 15 \\ x, y & \geq 0 \end{aligned}$
Sol :
TYPE-II
Question 6
Maximize Z=-x+2y, subject to constraints :
$\begin{aligned}x+y & \geq 5 \\x+2 y & \geq 6 \\x & \geq 3 \\y & \geq 0\end{aligned}$
Sol :
Question 7
Minimize Z=18x+10y, subject to constraints :
$\begin{aligned}4 x+y & \geq 20 \\2 x+3 y & \geq 30 \\x, y & \geq 0\end{aligned}$
Sol :
TYPE-III
Question 8
Minimize and maximize Z=x+2y, subject to constraints.
$\begin{aligned} x+2 y & \geq 100 \\ 2 x-y & \leq 0 \\ 2 x+y & \leq 200 \\ x, y & \geq 0 \end{aligned}$
Sol :
Question 9
Minimize and maximize Z=5x+10y, subject to constraints :
$\begin{aligned} x+2 y & \leq 120 \\ x+y & \geq 60 \\ x-2 y & \geq 0 \\ x, y & \geq 0 \end{aligned}$
Sol :
Question 10
Minimize Z=x+2y, subject to constraints :
$2 x+y \geq 3$
$x+2 y \geq 6$
$x, y \geq 0$
Show that the minimum of Z occurs at more than two points.
Sol :
Question 11
Maximize Z=x+y, subject to constraints :
$\begin{aligned}x-y & \leq-1 \\-x+y & \leq 0 \\x \geq 0, y & \geq 0\end{aligned}$
Sol :
Question 12
Minimize Z=3x+2y, subject to constraints :
$\begin{aligned}2 x+y & \geq 8 \\x+2 y & \geq 10 \\5 x+7 y & \leq 38 \\x \geq 0, y & \geq 0\end{aligned}$
Sol :
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