Question

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of elements C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Answer 1

Let x bags of brand P and y bags of brand Q were mixed.
Number of bags cannot be negative.
$\therefore x,y\ge 0$

The given information can be tabulated as follows:

	Element A	Element B	Element C
P	3	2.5	2
Q	1.5	11.25	3
Requirement	18	45	24

The constraints are
$$\begin{gathered} 3x+1.5y\ge 18 \\ 2.5x+11.25y\ge 45 \\ 2x+3y\ge 24 \end{gathered}$$

Brand P costing Rs 250 per bag and brand Q costing Rs 200 per bag. Therefore, x bags of brand P and y bags of brand Q costs Rs 250x and Rs 200y respectively.
Total cost = Z = 250x + 200y

The mathematical formulation of the given LPP is

Minimize Z = 250x + 200y
subject to

$$\begin{gathered} 2x+y\ge 12 \\ 2x+9y\ge 36 \\ 2x+3y\ge 24 \end{gathered}$$

First we will convert inequations into equations as follows:
2x + y = 12, 2x + 9y = 36, 2x + 3y = 24, x = 0 and y = 0

Region represented by 2x + y ≥ 12:
The line 2x + y = 12 meets the coordinate axes at A₁(6, 0) and B₁(0, 12) respectively. By joining these points we obtain the line 2x + y = 12 . Clearly (0,0) does not satisfies the 2x + y = 12 . So, the region which does not contain the origin represents the solution set of the inequation 2x + y ≥ 12.

Region represented by 2x + 9y ≥ 36:
The line 2x + 9y = 36 meets the coordinate axes at C₁(18, 0) and D₁(0, 4) respectively. By joining these points we obtain the line 2x + 9y = 36. Clearly (0,0) does not satisfies the inequation 2x + 9y ≥ 36. So,the region which does not contain the origin represents the solution set of the inequation 2x + 9y ≥ 36.

Region represented by 2x + 3y ≥ 24:
The line 2x + 3y = 24 meets the coordinate axes at E₁(12, 0) and F₁(0, 8) respectively. By joining these points we obtain the line 2x + 3y = 24. Clearly (0,0) does not satisfies the inequation 2x + 3y ≥ 24. So,the region which does not contain the origin represents the solution set of the inequation 2x + 3y ≥ 24.

Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x ≥ 0, and y ≥ 0.

The feasible region determined by the system of constraints 2x + y ≥ 12, 2x + 9y ≥ 36, 2x + 3y ≥ 24, x ≥ 0 and y ≥ 0 are as follows.



The corner points are B₁(0, 12), G₁(3, 6), H₁(9, 2) and C₁(18, 0).

The value of the objective function at the corner points

Corner points	Z = 250x + 200y
B1	2400
G1	1950
H1	2650
C1	4500

The minimum value of Z is 1950 which is attained at G₁(3, 6).
Thus, for minimum cost is Rs 1950, 3 bags of brand P and 6 bags of brand Q should be mixed.