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 70% iron content; mine 2 produces limonite ore, which has 60% iron content; mine 3 produces pyrite ore, which has 50% iron content; and mine 4 produces taconite ore, which has only 30% iron content. Exeter has 3 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) Solve the linear programming model formulated for Exeter Mines by using the computer.
(c) Do any of the mines have slack capacity? If yes, which one(s)?
(d) If Exeter Mines could increase production capacity at any one of its mines, which should it be? why?
(e) If Exeter decided to increase capacity at the mine identified in (d), how much could it increase capacity before the optimal solution point (ie., the optimal set of variables) would change?
(f) If Exeter determined 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?