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 steelArmco, 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.

Question

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 steelArmco, 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.

Formulate a linear programming model for this problem.
51. Solve the linear programming problem formulated in Problem 48 for Exeter Mines by using the computer.

a. Do any of the mines have slack capacity? If yes, which one(s)?
b. If Exeter Mines could increase production capacity at any one of its mines, which should it be? Why?

Answer 1

To formulate a linear programming model for this problem, we need to define some variables and constraints:

Let x1 be the amount of magnetite ore produced at mine 1 (in tons)
Let x2 be the amount of limonite ore produced at mine 2 (in tons)
Let x3 be the amount of pyrite ore produced at mine 3 (in tons)
Let x4 be the amount of taconite ore produced at mine 4 (in tons)

The objective is to minimize the cost, which can be expressed as:
Cost = 37*x1 + 46*x2 + 50*x3 + 42*x4

Now let's define the constraints:
1. The total iron content should meet the requirements of each customer:
0.7*x1 + 0.6*x2 + 0.5*x3 + 0.3*x4 >= 400 (for Armco)
0.7*x1 + 0.6*x2 + 0.5*x3 + 0.3*x4 >= 250 (for Best)
0.7*x1 + 0.6*x2 + 0.5*x3 + 0.3*x4 >= 290 (for Corcom)

2. The total amount of ore produced should not exceed the capacity of each mine:
x1 <= 350 (mine 1 capacity)
x2 <= 530 (mine 2 capacity)
x3 <= 610 (mine 3 capacity)
x4 <= 490 (mine 4 capacity)

Now that we have formulated the linear programming model, we can use computer software or tools such as Excel Solver or programming languages like Python to solve it.

To answer the specific questions:
a. To determine if any of the mines have slack capacity, we need to check if any of the constraints are non-binding (i.e., not tight). If a constraint is non-binding, it means the mine has some capacity that is not fully utilized. By looking at the constraints, we can see that none of the mines have non-binding constraints. Therefore, none of the mines have slack capacity.

b. To identify which mine to increase production capacity, we need to analyze the shadow prices or dual values associated with the iron content requirements of each customer. The shadow price indicates the rate of change in the objective function value with respect to a unit increase in the right-hand side of the constraint. If the shadow price is high, increasing the resource available will have a significant impact on the objective. We can calculate the shadow prices for each customer requirement and determine which mine's capacity increase will have the highest impact on reducing the cost.