To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three 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 lb) |
Food II( per lb) |
Minimum daily requirementfor the nutrient |
||||

Calcium | 10 | 5 | 20 | |||

Protein | 5 | 4 | 20 | |||

Calories | 2 | 6 | 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.

#### Solution

Let the person takes *x* *lbs* and *y* *lbs* of food I and II respectively that were taken in the diet.

Since, per lb of food I costs Rs 60 and that of food II costs Rs 100.

Therefore, *x* lbs of food I costs Rs 60*x** *and *y* lbs of food II costs Rs 100*y*.

Total cost per day = Rs (60*x** *+ 100*y*)

Let Z denote the total cost per day

Then, Z = 60*x** *+ 100*y*

Total amount of calcium in the diet is \[10x + 5y\]

Since, each lb of food I contains 10 units of calcium.Therefore, *x* lbs of food I contains 10*x* units of calcium.

Each lb of food II contains 5 units of calciu.So,*y* lbs of food II contains 5*y* units of calcium.

Thus, *x* lbs of food I and *y* lbs of food II contains 10*x* + 5*y* units of calcium.

But, the minimum requirement is 20 lbs of calcium.

*x*lbs of food I contains 5

*x*units of protein.

Each lb of food II contains 4 units of protein.So,

*y*lbs of food II contains 4

*y*units of protein.

Thus,

*x*lbs of food I and

*y*lbs of food II contains 5

*x*+ 4

*y*units of protein.

But, the minimum requirement is 20 lbs of protein.

*x*lbs of food I contains 2

*x*units of calories.

Each lb of food II contains units of calories.So,

*y*lbs of food II contains 6

*y*units of calories.

Thus,

*x*lbs of food I and

*y*lbs of food II contains

*x*+ 6

*y*units of calories.

But, the minimum requirement is 13 lbs of calories.

So,

**Hence, the required LPP is as follows:

Min

*Z*= 60

*x*+ 100

*y*

subject to

\[5x + 4y \geq 20\]

\[2x + 6y \geq 13\]

\[x, y \geq 0\]