In the following figure, the shaded region represents the system of constraints:

$A$ diet of a sick person must contain at least $4000$ units of vitamins,$50$ units of proteins,and $1400$ calories. Two foods $A$ and $B$ are available at a cost of ₹ $4$ and ₹ $3$ per unit respectively. If one unit of $A$ contains $200$ units of vitamins,$1$ unit of protein,and $40$ calories,while one unit of food $B$ contains $100$ units of vitamins,$2$ units of protein,and $40$ calories,formulate the problem so that the diet is the cheapest.

Find the point at which the objective function $Z = x + y$ attains its maximum value subject to the constraints $x + 2y \leq 70$, $2x + y \leq 95$, $x \geq 0$, and $y \geq 0$.

The vertices of the feasible region for the constraints $x+y \leq 4$,$x \leq 2$,$y \leq 1$,$x+y \geq 1$,$x, y \geq 0$ are

$A$ dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain at least $8$ $units$ of vitamin $A$ and $10$ $units$ of vitamin $C$. Food $'I'$ contains $2$ $units/kg$ of vitamin $A$ and $1$ $unit/kg$ of vitamin $C$. Food $'II'$ contains $1$ $unit/kg$ of vitamin $A$ and $2$ $units/kg$ of vitamin $C$. It costs Rs $50$ per $kg$ to purchase Food $'I'$ and Rs $70$ per $kg$ to purchase Food $'II'$. Formulate this problem as a linear programming problem to minimise the cost of such a mixture.

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?