Question

Optimal Profit A company makes two models of a patio furniture set. The times for assembling, finishing, and packaging model \(\mathrm{A}\) are 3 hours, \(2.5\) hours, and \(0.6\) hour, respectively. The times for model \(\mathrm{B}\) are \(2.75\) hours, 1 hour, and \(1.25\) hours. The total times available for assembling, finishing, and packaging are 3000 hours, 2400 hours, and 1200 hours, respectively. The profit per unit for model \(\mathrm{A}\) is $$\$ 100$$ and the profit per unit for model \(\mathrm{B}\) is $$\$ 85 .$$ What is the optimal production level for each model? What is the optimal profit?

Short answer

The optimal production level for both models and the corresponding maximum profit will be given by the solution of the linear programming problem in Step 3.

Step-by-step solution

Set Up The Variables

Let \(X\) represent the units of model \(\mathrm{A}\) and \(Y\) represent the units of model \(\mathrm{B}\). We need to optimize the total profit which is \(100X + 85Y\).

Set Up The Constraints

Given the time availability, we can express three different constraints:1. Assembly: \(3X + 2.75Y \leq 3000\)2. Finishing: \(2.5X + Y \leq 2400\)3. Packaging: \(0.6X + 1.25Y \leq 1200\) All of these need to be satisfied.

Solve The Linear Programming Problem

Use a graphing method or a Linear Programming solver to find the values of \(X\) and \(Y\) that maximize the profit function subject to the constraints. This generally involves plotting the constraint equations on a graph, identifying the feasible region, and then finding the point in this region that gives the highest value of the profit function. Remember that \(X\) and \(Y\) cannot be negative.

Calculate The Optimal Profit

After finding the optimal point, substitute the obtained values of \(X\) and \(Y\) into the profit function to find the maximum profit.

Key concepts

Optimization

Optimization in the context of linear programming involves finding the best solution, often the maximum or minimum value, of a given objective function. In this exercise, the objective is to maximize the profit from two models of patio furniture (Models A and B). The profit function is expressed as \(100X + 85Y\), where \(X\) and \(Y\) are the number of units produced for each model.

The first step in optimization is to define your objective clearly, which here is to maximize profit. Once the objective is defined, the next step involves formulating it as a mathematical expression — in this case, the total profit equation.

Optimization also requires analyzing how this profit function behaves under certain limits and conditions, often known as constraints. This helps us strategize the most effective way to allocate resources and production schedules to either maximize or minimize our desired variable.

Constraints

Constraints are the conditions or limitations placed upon a problem that must be satisfied for a solution to be considered valid. In linear programming, constraints are typically equations or inequalities that represent limitations in resources, time, or other factors.

For our exercise, the constraints are based on available hours for assembling, finishing, and packaging. Each model requires a specific amount of time for these tasks, and there is only a limited number of hours available to use for each task:

• Assembly: \(3X + 2.75Y \leq 3000\)
• Finishing: \(2.5X + Y \leq 2400\)
• Packaging: \(0.6X + 1.25Y \leq 1200\)

These equations show us the maximum hours each activity can take. Thus, the constraints serve to ensure that we do not exceed the available resources, making them crucial in determining feasible production levels.

Feasible Region

The feasible region in linear programming is the set of all possible solutions that satisfy all the constraints of a problem. It is typically represented graphically, showing the area where all inequality constraints overlap.

In the problem given, by plotting each constraint as a line on a graph, we can illustrate the feasible region as a polygon where each side is a part of the constraint equation lines. Only points within this polygon area are potential solutions because they don't violate any constraints.

The corners of the feasible region, known as vertices, are especially important to inspect, as these points often provide the maximum or minimum values for optimization problems. Finding these vertices is crucial, as linear programming dictates that the optimal solution will be located at a vertex of the feasible region.

Profit Maximization

Profit maximization focuses on determining the production pattern that yields the highest possible profit within the given constraints. In the context of linear programming, this involves solving the objective function, which, here, is to maximize \(100X + 85Y\).

Once the feasible region is plotted, the next step is to evaluate the profit function at each vertex of the feasible region. The vertex that provides the highest value of the profit function corresponds to the optimal solution.

In practical terms, this means adjusting the number of units produced of each model under the given resource constraints to achieve the most profitable outcome. By substituting the values of \(X\) and \(Y\) obtained from this optimal point back into the profit function, we can calculate the maximum profit, making sure that it adheres to all specified constraints.