To maintain his health a person must fulfil certain minimum daily requiremens for several kinds of nutrients. Assuming that there are only thee kinds of nutrients-calcium, protein and calories and the person’s diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below: , Food I (per Ib), Food II (per Ib), Minimum daily requirement for the nutrient Calcium Protein Calories, 10 5 2, 5 4 6, 20 20 13 Price (Rs.), 60, 100, What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.

Explanation

Let the quantity of foods chosen be ‘x’ and ‘y’
Cost of food `X = 0.6x `
Cost of food `Y = y`
Cost of diet `= 0.6x + y`
Now,
`=> 10x + 4y ≥ 20`
i.e. the minimum daily requirement of calcium in the diet is `20` units.
`=> 5x + 6y ≥ 20`
i.e. the minimum daily requirement of protein in the diet is `20` units.
`=>2x + 6y ≥ 12`
i.e. the minimum daily requirement of calories in the diet is `12` units.
Hence, mathematical formulation of the LPP is as follows:
Find `‘x’ and y’` such that `text(Minimises) Z = 0.6x + y `
Subject to the following constraints:
`(i) 10x + 4y ≥ 20 `
`(ii) 5x + 6y ≥ 20 `
`(iii) 2x + 6y ≥ 12`
` (iv) x, y ≥ 0 `
`(∵` quantity cant be negative)
The feasible region is unbounded
Z is smallest at `C ( 8/ 3 , 10 /9 )`
so, The minimum value of Z is Rs. `2.712 `