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 of three types known as 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 100% pure iron.

Formulate a linear programming model for this problem.

a) Do any of the mines have slack capacity? If yes, which ones?

b) If Exeter mines could increase production capacity at any one of its mines, which it should be? Why?

c) If Exeter decided to increase capacity at the mine identified in part (d), how much could it increase before the optimal solution would change?

d) If Exeter determined that it could increase production capacity at mine 1 from 350 tons to 500 tons , at an increase in production cost to $43 per ton, should it do so?