The minimum value of the objective function $z = 4x + 6y$ subject to the constraints $x + 2y \geq 80$,$3x + y \geq 75$,and $x, y \geq 0$ is:

The objective function $Z = 4 x_1 + 5 x_2$,subject to $2 x_1 + x_2 \geq 7$,$2 x_1 + 3 x_2 \leq 15$,$x_2 \leq 3$,$x_1, x_2 \geq 0$ has minimum value at the point

The minimum value of $z = 3x + 5y$, subject to constraints $x \leq 80$, $y \geq 60$, $x + y \leq 200$ and $x, y \geq 0$ occurs at the point:

Two godowns $A$ and $B$ have grain capacity of $100$ quintals and $50$ quintals respectively. They supply to $3$ ration shops,$D$,$E$ and $F$ whose requirements are $60, 50$ and $40$ quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in $Rs$)
From/To $A$ $B$
$D$ $6$ $4$
$E$ $3$ $2$
$F$ $2.50$ $3$

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

The Linear Programming Problem ($L$.$P$.$P$.) to minimize $z = 30x + 20y$ subject to the constraints $x + y \leqslant 8$,$x + 2y \geqslant 4$,$6x + 4y \geqslant 12$,$x \geqslant 0$,and $y \geqslant 0$ has:

The minimum value for the $LPP$ $Z = 6x + 2y$,subject to $2x + y \geq 16$,$x \geq 6$,$y \geq 1$ is