Siltronic produces silica from sand and gravel at four different mines; however, the sand and gravel extracted at each mine are different in their silica content. Mine 1 produces golden sand and gravel , which has a 70% silica content; mine 2 produces white ore, which has a 60% silica content; mine 3 produces gray ore, which has a 50% silica content; and mine 4 produces brown ore, which has only a 30% silica content. Siltronic has three customers that produce computer chipsIntel, AMD, and Apple. Intel needs 400 tons of pure (100%) silica, AMD requires 250 tons of pure silica, and Apple requires 290 tons. It costs $37 to extract and process 1 ton of golden sand and gravel at mine 1, $46 to produce 1 ton of white sand and gravel at mine 2, $50 per ton of gray sand and gravel at mine 3, and $42 per ton of brown sand and gravel at mine 4. Siltronic can extract 350 tons of sand and gravel 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 sand and gravel to produce at each mine in order to minimize cost and meet its customers demand for pure (100%) silica.

Formulate a linear programming model for this problem. (Hint: The decision variables will have two indices, such as x_11,x_12, etc. and there will be some equality constraints.)

1. Solve the linear programming model by using the computer.
2. Do any of the mines have slack capacity? If yes, which one(s)?
3. If Siltronic could increase production capacity at any one of its mines, which should it be? Why?
4. If Siltronic decided 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?

If Siltronic 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?

Please answer in an excel format with formulas to enter to get correct answers!