Let the direction mix x kg of food ‘I’ with y kg of food ‘II’. Then, the mathematical model of the LPP is as follows:
Minimize Z = 5x + 7y
Subject to 2x + y ≥ 8
x + 2y ≥ 10
and, x, y ≥ 0
To solve this LPP graphically, we first convert the inequations into equations to obtain the following lines.
2x + y = 8, x + 2y = 10, x = 0, y = 0
The line 2x + y = 8 meets the coordinates axes at A1 (4, 0) and B1 (0, 8). Join these points to obtain the line represented by 2x + y = 8. The region not containing the origin is represented by 2x + y ≥ 8.
The line x + 2y = 10 meets the coordinates axes at A2 (10, 0) and B2 (0, 5). Join these points to obtain the line represented by x + 2y = 10. Clearly, O(0, 0) does not satisfy the inequation x + 2y ≥ 10. So, the region not containing the origin is represented by this inequation.
Clearly, x ≥ 0, y ≥ 0 represent the first quadrant.
Thus, the shaded region in fig. is the feasible region of the LLP. The coordinates of the corner-points of this region are A2 (10, 0), P (2, 4) andB1 (0, 8).
The point P (2, 4) is obtained by solving 2x + y = 8 and x + 2y = 10 simultaneously. The values of the objective function Z = 5x + 7y at the corner points of the feasible region are given in the following table:
Point (x, y)
|
Value of the objective function
Z = 5x + 7y
|
A2 (10, 0)
P (2, 4)
B1 (0, 8)
|
Z = 5 × 10 + 7 × 0 = 50
Z = 5 × 2 + 7 × 4 = 38
Z = 5 × 0 + 7 × 8 = 56
|
Clearly, Z is minimum at x = 2 and y = 4. The minimum value of Z is 38. We observe that open half plane represented by 5x + 7y < 38 does not have points in common with the feasible region. So, Z has minimum value equal to 38 at x = 2 and y = 4.
Hence, the optimal mixing strategy for the dietician will be mix 2 kg of food ‘I’ and 4 kg of food ‘II’. In this case, his cost will be minimum and the minimum cost will be Rs 38.00.