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Problem 11

# MINING-PRoDucmoN Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs $$\ 14,000 /$$ day to operate, and it yields 50 oz of gold and 3000 oz of silver each day. The Horseshoe Mine costs $$\ 16,000 /$$ day to operate, and it yields 75 oz of gold and 1000 oz of silver each day. Company management has set a target of at least 650 oz of gold and $$18.000$$ oz of silver. How many days should each mine be operated so that the target can be met at a minimum cost?

Expert verified
To meet the gold and silver extraction targets at a minimum cost, the Saddle Mine should be operated for 4 days and the Horseshoe Mine should be operated for 4 days.
See the step by step solution

## Step 1: Formulate the objective function

We want to minimize the total cost. The cost of operating the Saddle Mine is $$\14,000/day$$ and the cost of operating the Horseshoe Mine is $$\16,000/day$$. The objective function is given by the total cost: Objective function: $$Cost(x_1, x_2) = 14,000x_1 + 16,000x_2$$

## Step 2: Formulate the constraints

We need to meet the targets of at least 650 oz of gold and 18,000 oz of silver. The Saddle Mine yields 50 oz of gold and 3,000 oz of silver each day, and the Horseshoe Mine yields 75 oz of gold and 1,000 oz of silver each day. So the constraints are: 1. Gold constraint: $$50x_1 + 75x_2 \geq 650$$ 2. Silver constraint: $$3000x_1 + 1000x_2 \geq 18,000$$ 3. Non-negativity constraints: $$x_1 \geq 0$$, $$x_2 \geq 0$$

## Step 3: Solve the linear programming problem

Now that we have the objective function and the constraints, we can solve the linear programming problem to minimize the total cost. This can be done using graphical methods, the simplex method, or specialized software like solver in Excel. The optimal solution for this problem is $$x_1 = 4$$ and $$x_2 = 4$$

## Step 4: Interpret the results

To minimize the cost while meeting the gold and silver extraction targets, the Saddle Mine should be operated for 4 days and the Horseshoe Mine should be operated for 4 days.

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