Iron Mountain Mines produces iron ore at four different mining locations in the Iron Mountain Range area of Colorado; 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. The company supplies three customers that produce steel; Armoco Metals, Bethlehem Steel, and Corocom Industrial Metal Fabricators. In the current period, Armoco requires 400 tons of pure (100%) iron, Bethlehem requires 250 tons of pure iron, and Corocom requires 290 tons of pure iron. It costs $67 to extract and process one ton of magnetite ore, $83 to process a ton of limonite ore, $90 to process a ton of pyrite ore, and $76 to process a ton of taconite ore. In the current period, Iron Mountain Mines can extract and process 350 tons of ore from mine #1, 530 tons from mine #2, 610 tons from mine #3, and 490 tons from mine #4. The company needs to develop an optimal production schedule for ore extraction and processing at minimum cost for the current period.
a. Formulate a linear programming model that can be used to determine the optimal production schedule for the current period to meet customer requirements.
b. Determine the optimal production schedule using the Management Scientist software, including the number of tons of each ore type to be extracted and processed and the total contribution to cost. Determine if slack capacity exists in any of the mines in the optimal schedule.
c. If Iron Mountain Mines could increase production capacity at any one of its mines, determine the mine that should be selected and the level of capacity increase at that mine that could be provided before the optimal solution would change.
d. If Iron Mountain Mines could increase production capacity at mine #1 from 350 tons to 500 tons at an increase in production cost to $77 per ton, determine if the company should do so and provide a justification.
NOTE: Instead of using Management Scientist Software, I would rather you gave me formulations using Excel including complete formulas for the constraints and the max. Please show all your work so that it is clear how you arrived at the answer. Also, in responding to the various sub-parts, I need you to write in clear and understandable English. The more detail and explanation at how you arrived at your answer, the better. This problem is for an MBA level course. Thanks.