A firm manufactures headache pills in two sizes A and b. Size A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine ; size B contains 1 grain of aspirin, 8 grains of bicarbonate and 66 grains of codeine. It has been found by users that it requires at least 12 grains of aspirin, 7.4 grains of bicarbonate and 24 grains of codeine for providing immediate effects. Determine graphically the least number of pills a patient should have to get immediate relief. Determine also the quantity of codeine consumed by patient.
The above LPP can be presented in a table below,
Hence mathematical formulation of LPP is,
Max Z = x + y
Subject to constraints,
2x + y 
12
5x + 8y 
7.4
x + 66y 
24
x, y 
0 [ Since production can not be less than zero]
The corner points are B(0,12), P(5.86, 0.27), E(24,0)
The values of Z at these corner points are as follows:
The minimum value of Z is 6.13 but the region is unbounded so check whether x + y 
6.13
Clearly, it can be seen that it does not has any common region.
So, x = 5.86, y = 0.27
This is the least quantity of pill A and B.
Codeine quantity = x + 66y = 5.86 + (66
0.27) = 24(approx).