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
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