A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain atleast 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.
Answers
Answer:
Suppose the dietician mixes x kg of food I and y kg of food II.
The first condition of the mixture containing 8 units of vitamin A is given by 2x+y≥8
Also, atleast 10 units of vitamin C is given by x+2y≥10
The total cost of the mixture is given by 5x+7y
So, our problem becomes to maximise 5x+7y, subject to the conditions 2x+y≥8 ...(1) and x+2y≥10 ...(2)
Equation (1) multiplied by 2 gives 4x+2y≥16 and when subtracted with equation (2), we get 3x≥6 or x≥2
When substituted in equation (1), we obtain y≥4
The minimum cost of such a mixture thus becomes 10+28=Rs.38.
However, the maximum cost can go upto infinity.
Formation of LPP
To Maximize: 5x+7y
Constraints: 2x+y≥8
x+2y≥10
x≥0,y≥0
Verifying values at corner points:
Corner Points (0,8) (2,4) (10,0)
Value of 5x+7y 56 38 50
Minimum at (2,4), and It can attain infinity as it's maximum value.