Question

A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F₁ and F₂ are available. Food F₁ costs Rs 4 per unit and F₂ costs Rs 6 per unit. One unit of food F₁ contains 3 units of Vitamin A and 4 units of minerals. One unit of food F₂ contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

Answer 1

Let x units of food F₁ and y units of food F₂ are required to be mixed.

Cost = Z = 4x + 6y ........(i) is to be minimised subject to following constraints.

3x + 6y ≥ 80 .........(ii)

4x + 3y ≥ 100 .......(iii)

x ≥ 0, y ≥ 0 ...(iv)

To solve the LPP graphically, the graph is plotted as shown.

The shaded regions in the graph is the feasible solution of the problem. The corner points are

The value of Z at corner point is given as.

Since, feasible region is unbounded therefore a graph of 4x + 6y < 104 is drawn which is shown in figure by dotted line.

Also, since there is no point common in feasible region and region 4x + 6y < 104 .

Hence, for minimum cost Rs 104, 24 units of food F₁ and 43 units of food F₂ is required.