Question

Blacktop Refining extracts minerals from ore mined at two different sites in Montana. Each ton of ore type 1 contains 20% copper, 20% zinc, and 15% magnesium. Each ton of ore type 2 contains 30% copper, 25% zinc, and 10% magnesium. Ore type 1 costs $90 per ton while ore type 2 costs $120 per ton. Blacktop would like to buy enough ore to extract at least 8 tons of copper, 6 tons of zinc, and 5 tons of magnesium in the least costly manner.
a. Formulate an LP model for this problem.
b. Sketch the feasible region for this problem.
c. Find the optimal solution.

Answer 1

Answer:

a) Minimize $Cost=90x_1+120x_2$

subject to

$0.2x_1+0.3x_2\geq8$

$0.2x_1+0.25x_2\geq6$

$0.15x_1+0.1x_2\geq5$

$x_1\geq0\\x_2\geq0$

b) Attached

c) The optimum value that minimizes cost is x1=28 and x2=8.

Step-by-step explanation:

The objective function is the cost of extraction and needs to be minimized.

The cost of extraction is the sum of the cost of extraction of ore type 1 and the cost of extraction of ore type 2:

$Cost=90x_1+120x_2$

Being x1 the tons of ore type 1 extracted and x2 the tons of ore type 2.

The constraints are the amount of minerals that need to be in the final mix

Copper:

$0.2x_1+0.3x_2\geq8$

Zinc

$0.2x_1+0.25x_2\geq6$

Magnesium

$0.15x_1+0.1x_2\geq5$

Of course, x1 and x2 has to be positive numbers.

$x_1\geq0\\x_2\geq0$

The feasible region can be seen in the attached graph.

The orange line is the magnesium constraint. The red line is the copper constraint. The green line is the zinc constraint.

The optimal solution is found in one of the intersection points between two constraints that belong to the limits of the feasible region.

In this case, the cost can be calculated for the 3 points that satisfies the conditions.

The optimum value that minimizes cost is x1=28 and x2=8.