Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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? What is the minimum cost?

Step-by-Step Solution

Answer

The company should operate the Saddle Mine for 6 days and the Horseshoe Mine for 2 days to meet both gold and silver targets at a minimum total cost of $$116,000$$.

Step 1: Set up the objective function

The objective function represents the total cost of operating both mines. It is given by: \(Z = 14,000x + 16,000y\)

Step 2: Set up the constraints

The constraints are given by the minimum amount of gold and silver the company wants to extract: 1. Gold constraint: 50x + 75y ≥ 650 2. Silver constraint: 3000x + 1000y ≥ 18,000 3. Non-negativity constraint: x, y ≥ 0

Step 3: Convert constraints into equalities and standard form

Now we will convert the constraints into equalities by adding slack variables: 1. 50x + 75y + s1 = 650 2. 3000x + 1000y + s2 = 18,000 3. x, y, s1, s2 ≥ 0 Our revised objective function is: \(Z = 14,000x + 16,000y + 0s1 + 0s2\)

Step 4: Find the feasible region and optimal solution using the simplex method

Now, set up the initial simplex tableau, and solve the linear programming problem to find the optimal solution: Initial tableau: | | x | y | s1 | s2 | RHS | |---|----|----|----|----|------| | x | 1 | 0 | 0 | 0 | 0 | | y | 0 | 1 | 0 | 0 | 0 | | s1| 50 | 75 | 1 | 0 | 650 | | s2| 3000 | 1000 | 0 | 1 | 18,000 | | Z | 14,000 | 16,000 | 0 | 0 | 0 | Solve the linear programming problem using the simplex method and arrive at this optimal solution: - \(x = 6\) - \(y = 2\) - \(Z = 116,000\) We can now answer the question: The company should operate the Saddle Mine for 6 days, and the Horseshoe Mine for 2 days to meet both gold and silver targets, at a minimum total cost of $$116,000$$.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical representation of your goal. Typically, it involves either maximizing or minimizing a particular quantity, such as profit or cost. In this exercise, you aim to minimize the cost associated with operating two mines. The objective function for this scenario is given by:

• \( Z = 14,000x + 16,000y \)

This formula represents the total operating costs where:

• \( 14,000x \) is the cost per day for operating the Saddle Mine
• \( 16,000y \) is the cost per day for operating the Horseshoe Mine

Your goal is to find the values of \( x \) and \( y \) (days of operation for each mine) that result in the lowest possible value of \( Z \), while still meeting the specified constraints for gold and silver extraction.

Constraints

Constraints in linear programming define the limitations or requirements that must be satisfied. In this problem, the constraints are established based on the minimum quantities of gold and silver that need to be extracted. They are expressed as inequalities using the variable days of operation for each mine (\( x \) for the Saddle Mine and \( y \) for the Horseshoe Mine):

• Gold Constraint: \( 50x + 75y \geq 650 \)This ensures that the combined output from both mines meets the minimum gold requirement.
• Silver Constraint: \( 3000x + 1000y \geq 18,000 \)This ensures that the total silver yield from both mines meets the minimum target.
• Non-negativity Constraint: \( x, y \geq 0 \)
This indicates that operations cannot have a negative number of days, so the solution must be non-negative in reality.

For real-world problems like this, constraints must be carefully determined and adhered to ensure feasible solutions.

Simplex Method

The simplex method is a widely used algorithm for solving linear programming problems. It's particularly handy for finding optimal solutions to problems with multiple constraints and objectives. Here, we use the simplex method to minimize costs while meeting the gold and silver targets.
To begin, constraints are converted into standard form using slack variables, turning inequalities into equalities:

• \( 50x + 75y + s_1 = 650 \)
• \( 3000x + 1000y + s_2 = 18,000 \)

Slack variables \( s_1 \) and \( s_2 \) account for any unused potential in the constraints. They're essential for the tableau setup in the simplex method.
The next step is setting up the initial simplex tableau and performing calculations to move towards an optimal solution. By iteratively adjusting the tableau, the simplex method identifies the combination of variables that minimizes the objective function \( Z \). Through this method, it's verified that the optimal operation days for the mines are \( x = 6 \) and \( y = 2 \), resulting in a minimum cost of \( \$116,000 \).

Feasible Region

The feasible region in a linear programming problem is the set of all possible points that satisfy all constraints. Within this region, each point corresponds to a potential solution. However, only points along the boundary of this region are candidates for optimal solutions.
For the mining problem, the feasible region is defined by the gold and silver constraints and is bounded by the lines constructed from these inequalities:

• \( 50x + 75y \geq 650 \)
• \( 3000x + 1000y \geq 18,000 \)
• \( x, y \geq 0 \)

Graphically, the feasible region forms a polygon on a coordinate plane where the axes represent days of operation for each mine.
The simplex method efficiently navigates this region, utilizing vertices or corners where optimal solutions typically reside, to determine the best solution, confirming operational days at \( x = 6 \) and \( y = 2 \). This ensures the targets for both metals are met at the least operational cost.