A diet of sick person contains at least 48 units of vitamin A and 64 uints of vitamin B. Two foods `F_(1) andF_(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 of food `F_(2)` contain 8 units of vitamin A and 12 units of vitamin B. Formulate the LPP, for the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutrition requirements

Explanation

Let x units of food ` F _ 1 ` and y units of food ` F _ 2 ` be included in the diet of the sick person.
Then their total cost is ` z = ₹ ( 6x + 10 y ) `
This is the objective function which is to be minimized.
The constraints are as per the following table :

From this table, the constraints are
`6x + 8y ge 48, 7 x + 12 y ge 64 `
Also, the number of units of foods ` F _ 1 and F _ 2 ` cannot be negative
` therefore x ge 0 , y ge 0 `
` therefore ` the mathematical formulation of given LPP is
Minimize ` z = 6x + 10 y `, subject to
` 6x + 8 y ge 48, 7 x + 12 y ge 64, x ge 0, y ge 0 `
First we draw the lines AB and CD whose equations are ` 6 x + 8 y = 48 and 7x + 12 y = 64 ` respectively.

The feasible region is shaded in the figure.
The vertices of the feasible region are ` C (( 64)/(7), 0)`, P and B `(0, 6) `.
P is the point of intersection of the lines
` 6 x + 8 y = 48 " " `... (1)
and ` 7 x + 12 y = 64 " " `...(2)
Multipyting equation (1) by 3 and equation (2) by 2, we get,
` 18 x + 24 y = 144 `
` 14 x + 24y = 128 `
On subtracting, we get,
` 4x = 16`
` therefore x = 4`
` therefore ` from ` (1), 6(4) + 8 y = 48 `
` therefore 8y = 24 `
` therefore y = 3 `
` therefore P = ( 4, 3 ) `
The values of the objective function ` z = 6x + 10 y ` at these vertices are
` z (C) = 6(( 64)/(7)) + 10 (0) = (384) /(7) = 54.85 `
` z (P) = 6 ( 4) + 10 (3 ) = 24 + 30 = 54 `
` z ( B) = 6(0) + 10 ( 6) = 60 `.
` therefore ` the minimum value of `z ` is 54 at the point (4, 3 ).
Hence, 4 units of food ` F _ 1 ` and 3 units of food ` F _ 2 ` should be included in the diet of the sick person to meet the minimal nutritional requirements, in order to have the minimum cost of ₹ 54.