An 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 maximum of 600kg of explosive A, atleast 480kg of explosive B and exactly 540kg of explosive C. Bomb P requires 3, 2, 2kg, bomb Q requires 1, 4, 3kg and bomb R requires 4, 2, 3kg of explosives A, B and C respectively. Bomb P is expected 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? wich explosive have excess amont ans how much?

Let x, y, z be the number of bombs made of type P, Q & R respectively.

P requires: 3 kg A, 2kg B, 2kg C; Q: 1kg A, 4kg B, 3 kg C; R: 4kg A, 2kg B, 3 kg C

Explosives available:

A: max. 600kg; B: min. 480kg; C: 540 kg

Constraints:

Explosive A: 3x + 1y + 4z ≤ 600

Explosive B: 2x + 4y + 2z ≥ 480

Explosive C: 2x + 3y + 3z = 540

Bomb P gives a 2 ton explosion, Q 3 ton, & R 4 ton. This has to be maximized.

Maximize: Z = 2x +3y + 4z

Introducing slack, surplus and artificial variables into the constraints:

3x + 1y + 4z ≤ 600 → 3x + 1y + 4z + $S_1$ = 600

2x + 4y + 2z ≥ 480→ 2x + 4y + 2z – $S_2 + A_1$ = 480

2x + 3y + 3z = 540→2x + 3y + 3z + $A_2$= 540

Using big M method, the maximization function becomes:

$Z = 2x +3y + 4z + 0S_1– 0S_2 – MA_1– MA_2$

Iteration ends here, since all the values are less than, or equal to zero.

Number of bombs: P (x) = 0, Q (y) = 60, R (z) = 120

Explosion possible = 660 ton