Linear Programming

6 Marks Questions

1. A housewife wishes to mix together two kinds of food, X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of 1 kg of food is given below:

Vitamin A Vitamin B Vitamin C

Food X Food Y 2 2

1 kg of food X costs ₹ 6 and 1 kg of food Y costs ₹ 10. Formulate the above problem as a linear programming problem and find the least cost of the mixture which will produce the diet graphically. What value will you like to attach with this problem? Value Based Question; Delhi 2014C

Let the quantity of food X be x kg and the quantity of food Y be y kg.

... The objective function is to minimise,

<math>Z = 6x + 10y</math>.

Subject to the constraints are

<math>x + 2y \ge 10</math>

<math>2x + 2y \ge 12</math>

...(i)

...(ii)

...(iii)

...(iv)

<math>3x + y \ge 8</math>

<math>x, y \ge 0</math>

On considering the constraints as equations, we get

<math display="block">x + 2y = 10</math>

<math>x + y = 6</math>

...(v)

...(vi)

...(vii)

...(viii)

<math>3x + y = 8</math> and <math>x = 0, y = 0</math>

Firstly, draw the graph of line <math>x + 2y = 10</math> (1)

х 0 10

y 5 0 (1) On putting <math>(0, 0)</math> in the inequality <math>x + 2y \ge 10</math>, we get

<math>0 + 2(0) \ge 10 \Rightarrow 0 \ge 10</math>

[which is false]

So, the half plane is away from the origin.

Secondly, draw the graph of line <math>2x + 2y = 12</math>

X 0 6 0