#### Question

A diet of a sick person must contain at least 48 units of vitamin A and 64 units of vitamin B. Two foods F_{ 1} and F_{2} are available. Food F_{1} costs Rs. 6 per unit and food F_{2} costs Rs. 10 per unit. One unit of food F_{1} contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F_{2} contains 8 units of vitamin A and 12 units of vitamin B.Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutritional requirements.

#### Solution

Let x and y be two different types of food.

Thus, our objective function is minimise the cost

Z = 6x + 10y, subject to the constraints,

6x + 8y ≥48

7x + 12y ≥ 64

Plotting the above lines in a graph, we have,

Thus, the region above ABC is unbounded.

Let us check the value of the function at the corner points A, B and C

Corner point | Value of Z = 6x + 10y |

(0,6) | Z = 0 + 10 x 6 = 60 |

(4,3) | Z = 6 x 4 + 10 x 3 = 54 |

(64/7,0) | Z = 6 x 64/7+ 10 x 0 = 54.85 |

Minimum of the function is at 4, 3

Minimum cost of the optimum diet is Rs. 54