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 6 grains of codeine. It is found by users that it requires at least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing immediate effect. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a LPP and use graphical and simplex/Big-M method.
Answers
Answer:
the least number of pills a patient should take to get immediate relief. = 10
Step-by-step explanation:
Pill A - 2 Grain of Aspirin , 5 grain of bicarbonate , 1 grain of Codeine
Pill B - 1 Grain of Aspirin , 8 Grains of bicarbonate , 6 Grain of codeine
User requirement - 12 grain of asprin , 74 grains of bicarbonate & 24 Grains of codeine
A ≥ 0
B ≥ 0
2A + B ≥ 12
5A + 8B ≥ 74
A + 6B ≥ 24
Min ( A + B) =
for A= 0
B ≥ 12 from 2A + B ≥ 12
B ≥ 10 from 5A + 8B ≥ 74
B ≥ 4 from A + 6B ≥ 24
so B ≥ 12
for B = 0
A ≥ 6 from 2A + B ≥ 12
A ≥ 15 from 5A + 8B ≥ 74
A ≥ 24 from A + 6B ≥ 24
so A ≥ 24
Similarly
B = 1 , A ≥ 18
B =2 A ≥ 12
B = 3 A ≥ 10
B = 4 A ≥ 9
B = 5 A ≥ 7
B = 6 A ≥ 6
B = 7 A ≥ 4
B =8 A ≥ 2
B = 9 A ≥ 2
B = 10 A ≥ 1
B = 11 A ≥ 1
B = 12 A ≥ 0
from table above we can see
B = 8 & A = 2 (min) willl give A + B = 10 min possible values
the least number of pills a patient should take to get immediate relief. = 10
8 + 4 = 12 grains of asprin
64 + 10 = 74 grains of bicarbonate
48 + 2 = 50 grains of codeine