Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $$\$ 45$$, and the profit for each chair is $$\$ 20$$. In a certain week, the company has 3200 board feet of wood available, and 520 labor-hours. How many tables and chairs should Winston manufacture in order to maximize its profits?

Step-by-Step Solution

Answer

To maximize the profits, the Winston Furniture Company should produce 8 tables and 116 chairs, generating a profit of \(P(8, 116) = \$5560\).

Step 1: Consider the variables

Let x be the number of tables and y be the number of chairs produced.

Step 2: Formulate the constraints

We have two main constraints: the amount of wood and the labor-hours available. The wood constraint is given by: \(40x + 16y \le 3200\) And the labor-hours constraint is given by: \(3x + 4y \le 520\)

Step 3: Set up the objective function

The objective function represents the profit. The profit for each table is \(45, and the profit for each chair is \)20. So, the objective function is: \(P(x, y) = 45x + 20y\), where P(x, y) is the profit.

Step 4: Graph the constraints and find the feasible region

Plot the constraints on an x-y plane, and figure out the feasible region (where both constraints are satisfied). This region represents all possible combinations of tables and chairs that can be produced. In this case, you'll find that the feasible region is a polygon.

Step 5: Determine the vertices of the feasible region

Find the vertices of this polygon by solving for the intersection points of the boundary lines: 1. \(40x+16y = 3200\) and \(3x+4y = 520\) 2. \(40x+16y = 3200\) and \(x = 0\) 3. \(3x+4y = 520\) and \(x = 0\) 4. \(3x+4y = 520\) and \(y = 0\)

Step 6: Calculate the profit at each vertex

Use the objective function P(x, y) to calculate the profit at each vertex (determined in the previous step). The highest profit value corresponds to the optimal number of tables and chairs to be produced.

Step 7: Determine the optimal number of tables and chairs

Based on the highest profit value, the optimal number of tables and chairs to be produced can be found. Make sure that the constraints are satisfied and that the solution makes sense in the context of the problem.

Key Concepts

Objective Function

In linear programming, the objective function is akin to the destination on a map—the goal we strive to reach through the most efficient route. For the Winston Furniture Company, their objective is to maximize profit, a common goal for many businesses. To represent this mathematically, the profit function is defined as:
\[ P(x, y) = 45x + 20y \],
where \(x\) and \(y\) represent the number of tables and chairs produced, respectively. The coefficients, 45 and 20, signify the profits for each table and chair. In practice, the task is to find the values of \(x\) and \(y\) that maximize this function, within the context of the given constraints.

Constraints

Imagine trying to fill a container with water, but you're limited by the container's size—this is akin to the role of constraints in linear programming. They define the limits within our quest to optimize the objective function.

• The wood constraint for Winston Furniture is expressed as: \(40x + 16y \leq 3200\), indicating the total board feet of wood available.
• The labor-hours constraint is similarly noted: \(3x + 4y \leq 520\), showcasing the total labor-hours available for production.

These constraints confine the number of tables and chairs that can be manufactured, forming boundaries for our solution. Only those combinations of tables and chairs that respect these constraints are deemed viable candidates for our real-world solution.

Feasible Region

Considering every possible option at a buffet without overfilling your plate is a culinary example of finding the feasible region in linear programming. It's the 'safe zone' where all constraints are respected.

• In graphing, this region is often a shape, like a polygon, on a coordinate plane where any point inside (or on the boundaries) satisfies all constraints simultaneously.
• For the Winston Furniture Company, the feasible region is identified once we graph the constraints on an x-y plane. It's a polygonal area that represents all potential combinations of tables and chairs the company can produce without surpassing the available wood and labor-hours.

Identifying this region is vital, as it contains the potential solutions —the exact numbers of tables and chairs that might lead to maximum profit.

Graphical Method

The graphical method is the compass that navigates through the constraints towards the objective function's peak. In the context of the Winston Furniture Company:

1. We start by plotting the wood and labor-hours constraints on a graph to find where they intersect. These points of intersection outline the feasible region.
2. Next, we determine the vertices of this region because, in linear programming, the optimal solution resides at one of these vertices.
3. We then calculate the profit (our objective function) at each of these vertices to pinpoint the one that yields the highest profit.
4. Finally, the coordinates of this optimum vertex correspond to the ideal number of tables and chairs to manufacture, satisfying both constraints and maximizing profit.

The graphical method visually simplifies what can be a complex set of equations and inequalities, providing an intuitive pathway to finding the best solution.