linear programming
posted by alia on .
Exeter Mines produces iron ore at four different mines; however, the ores extracted at each mine are different in their iron content. Mine 1 produces magnetite ore, which has a 70% iron content; mine 2 produces limonite ore, which has a 60% iron content; mine 3 produces pyrite ore, which has a 50% iron content; and mine 4 produces taconite ore, which has only a 30% iron content. Exeter has three customers that produce steel Armco, Best, and Corcom. Armco needs 400 tons of pure (100%) iron, Best requires 250 tons of pure iron, and Corcom requires 290 tons. It costs $37 to extract and process 1 ton of magnetite ore at mine 1, $46 to produce 1 ton of limonite ore at mine 2, $50 per ton of pyrite ore at mine 3, and $42 per ton of taconite ore at mine 4. Exeter can extract 350 tons of ore at mine 1; 530 tons at mine 2; 610 tons at mine 3; and 490 tons at mine 4. The company wants to know how much ore to produce at each mine in order to minimize cost and meet its customers' demand for pure (100%) iron.
a. Formulate a linear programming model for this problem.
b. Do any of the mines have slack capacity? If yes, which one(s)?
c. If Exeter Mines could increase production capacity at any one of its mines, which should it be? Why?
d. If Exeter were to decide to increase capacity at the mine identified in (b), how much could it increase capacity before the optimal solution point (i.e., the optimal set of variables) would change?
e. If Exeter were to determine that it could increase production capacity at mine 1 from 350 tons to 500 tons, at an increase in production costs to $43 per ton, should it do so?

Unit Costs:
Magnetite: $37 => 0.7 ton of iron =>$52.86/ton
limonite: $46 => 0.6 ton => $76.67/ton
pyrite: $50 => 0.5 ton => $100/ton
taconite: $42=> 0.3 ton => $140/ton
Production:
magnetite: 350*0.7=245 tons
limonite: 0.6*530=318 tons
pyrite: 0.5*610=305 tons
taconite: 0.3*490=147 tons
Total = 1015 tons
Demand:
400+250+290=940 tons
Therefore there is excess capacity.
The given data does not give transportation costs to supply which customer, so if they are all equal, the least expensive unit price would prevail.
If further data are given, the answer may change.