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Question
Vitamins A and B are found in two different foods F1 and F2. One unit of food F1contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F2 contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F1 and F2 cost Rs 50 and 25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F1 and F2 at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.
Solution
Let x and y units of food F1 and food F2 were mixed.
Clearly, x ≥ 0 and y ≥ 0
One unit of food F1 contains 2 units of vitamin A and one unit of of food F2 contains 4 units of vitamin A. Therefore, x and y units of food F1 and food F2 respectively contains 2x and 4y units of vitamin A.
It is given that the minimum daily requirements for a person of vitamin A is 40 units.
Hence, 2x+ 4y ≥ 40
One unit of food F1 contains 3 units of vitamin B and one unit of food F2 contains 2 units of of vitamin B. Therefore, x and y units of F1 and F2 respectively contains 3x and 2y units of vitamin B.
It is given that the minimum daily requirements for a person of vitamin B is 50 units.
Hence, 3x+ 2y ≥ 50
One unit of food F1 and food F2 cost Rs 50 and 25 respectively. Therefore, x and y units of food F1 and food F2 costs Rs 50x and Rs 25y respectively.
Let Z denote the total cost
Then, Z = Rs (50x + 25y)
Hence, the required LPP is
Minimize Z = 50x + 25y
subject to
2x+ 4y ≥ 40
3x+ 2y ≥ 50
x ≥ 0,y ≥ 0
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