The feasible region for the constraints $x-y \geqslant 0$,$x-5y \leqslant -5$,$x \geqslant 0$,$y \geqslant 0$ is shown by the figure:

The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$, subject to the constraints $x + 3y \leq 60$, $x + y \geq 10$, $x - y \leq 0$, and $x, y \geq 0$ is

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 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:

In $L.P.P.$,the maximum value of the objective function $Z = 6x + 3y$ subject to the constraints $x + y \leq 5$,$x + 2y \geq 4$,$4x + y \leq 12$,$x, y \geq 0$ is:

The maximum value of the objective function $z=2x+3y$ subject to the constraints $x+y \leq 5$,$2x+y \geq 4$,$x \geq 0$,and $y \geq 0$ is