Problem 11 - 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?

Step-by-Step Solution

Answer

In order to meet the management's target with minimum costs, Saddle Mine should operate for 4 days and Horseshoe Mine should operate for 5 days. The total cost in this case will be minimized at \(86,000. \)

Step 1: Formulate the variables

Let x be the number of days Saddle Mine operates and y the number of days Horseshoe Mine operates.

Step 2: Define the objective function

Our objective is to minimize the total cost of operating the two mines. The cost function is given by: \[C(x,y) = 14000x + 16000y\]

Step 3: Write down the constraints

We have two types of constraints: the target for gold production and the target for silver production. These constraints can be written as inequalities: 1. Gold constraint: \(50x + 75y \geq 650\) 2. Silver constraint: \(3000x + 1000y \geq 18,000\) Additionally, the number of days each mine operates must be non-negative: 3. Non-negativity constraint: \(x \geq 0\) 4. Non-negativity constraint: \(y \geq 0\)

Step 4: Solve the linear optimization problem

The given problem is a linear optimization problem consisting of an objective function to minimize and a set of inequality constraints. There are various methods to solve such a problem (e.g. using the simplex method or graphical method). For simplicity, we will use the graphical method and find the feasible region.

Step 5: Plot the inequality constraints

Graphically represent each constraint on a coordinate plane with x and y axes representing the number of days for Saddle Mine and Horseshoe Mine, respectively. Since we are looking for the minimum cost, we should be interested in the intersection points of these inequalities as they form the vertices of the feasible region. In this case, these intersection points represent the optimal combination of days to operate both mines.

Step 6: Test the vertices of the feasible region

Calculate the cost at each intersection point (vertex of the feasible region) and choose the combination of x and y with the minimum cost. After this analysis and calculation, we can determine the number of days that each mine should be operated to reach the management's target at a minimum cost.

Key Concepts

Objective Function

Understanding the objective function is like identifying the goal line in a game of soccer; it's the target we're aiming for. In linear optimization problems, the objective function represents the mathematical expression of the goal we want to achieve. Specifically, the purpose is either to maximize or minimize this function. For the Perth Mining Company, the goal is to minimize the total cost of operating the mines.

The cost function is formulated as a linear equation involving our variables, which in this case are the number of days each mine operates, denoted as x for Saddle Mine and y for Horseshoe Mine. The equation \[C(x,y) = 14000x + 16000y\] defines the total cost and setting our sights on the lowest possible value of this function will dictate our strategy in allocating operational days between the mines.

Constraint Inequalities

Just as a video game restricts you within the boundaries of its virtual world, the constraint inequalities of a linear optimization problem limit the feasible solutions to a specific set of possibilities. These constraints, often practical limitations or requirements, are expressed in the form of inequalities that our variables must satisfy.

In the context of the mines, two resources – gold and silver – come with their own constraints reflecting the minimum targets set by the company: at least 650 oz of gold and 18,000 oz of silver. This translates into two key inequalities:

• Gold constraint: \(50x + 75y \geq 650\)
• Silver constraint: \(3000x + 1000y \geq 18,000\)

Additionally, a realistic scenario prevents us from considering negative operational days, leading to a pair of non-negativity constraints: \(x \geq 0\) and \(y \geq 0\).

These constraints define the 'playable area' in our optimization game and ensure we’re mining not just efficiently, but also effectively.

Feasible Region

Enter the feasible region — the 'safe zone' in a game of tag where you can't be tagged. Within the scope of our linear optimization problem, the feasible region is the graphical representation of all possible solutions that satisfy the given constraints. It's the convergence of areas on a graph that adhere to each of the inequality constraints including the non-negativity requirements.

What we're looking for is a set of values for x and y that will not only keep the mines’ cost to a minimum but also meet or exceed the production targets for gold and silver. Each point in the feasible region represents a scenario where the mines operate for a certain number of days, meeting the established criteria of the problem.

Graphical Method

Moving on to the graphical method, imagine it as drawing up a battle plan on a map. This technique is an intuitive way to solve a linear optimization problem involving two variables, such as our example with the Perth Mining Company. By plotting the constraint inequalities on a graph, we create lines that divide the plane into sections.

We then identify where these lines intersect, which frames our feasible region. The optimal solution lies at one of the vertices (corner points) of this feasible region. By evaluating the objective function at each of these vertices, we can pinpoint the exact combination of x and y that awards us the victory — the minimum operational cost while reaching the gold and silver targets.