Air force is experimenting with three types of bombs P, Q and R in which three
kinds of explosives, viz. A, B and C will be used. Taking the various factors into
account, it has been decided to use at the maximum 600 kg of explosive A, at least 480
kg of explosive B and exactly 540 kg of explosive C. Bomb P requires 3, 2, 2kg, bomb
Q requires 1, 4, 3 kg and bomb R requires 4, 2, 3 kg of explosives A, B and C
respectively. Bomb P is estimated to give the equivalent of a 2 ton explosion, bomb Q,
a 3 ton explosion and bomb R, a 4 ton explosion respectively. Under what production
schedule can the Air Force make the biggest bang
Answers
Given : Air force is experimenting with three types of bombs P, Q and R in which three kinds of explosives, viz. A, B and C will be used.
Bomb P is estimated to give the equivalent of a 2 ton xplosion, bomb Q, a 3 ton explosion and bomb R, a 4 ton explosion respectively.
To Find : Under what production schedule can the Air Force make the biggest bang
Solution:
Explosion = 2P + 3Q + 4R to maximize
3P + Q + 4R ≤ 600 Eq1
2P + 4Q + 2R ≥ 480 Eq2
2P + 3Q + 3R = 540 => 2P = 540 - 3Q - 3R
Substitute P in Eq1 and Eq2
=> 3(540 - 3Q - 3R)/2 + Q + 4R ≤ 600
=> 1620 - 9Q - 9R + 2Q + 8R ≤ 1200
=> 7Q + R ≥ 420
2P + 4Q + 2R ≥ 480
=> 540 - 3Q - 3R + 4Q + 2R ≥ 480
=> R - Q ≥ 60 => R ≥ 60
Explosion = 2P + 3Q + 4R
P = 45 Q = 45 , R = 105 90 + 135 + 420 = 645
P = 180 Q = 0 , R = 60 360 + 0 + 240 = 600
P = 0 Q = 60 , R = 120 0 + 180 + 480 = 660
Maximum is at
P = 0 Q = 60 , R = 120
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