Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Optimal Profit A company makes two models of doghouses. The times (in hours) required for assembling, painting, and packaging are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2.5 & 3 \\ \hline \text { Painting } & 2 & 1 \\ \hline \text { Packaging } & 0.75 & 1.25 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 2500 hours, and 1500 hours, respectively. The profits per unit are $$\$ 60$$ for model \(\mathrm{A}\) and $$\$ 75$$ for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?

Step-by-Step Solution

Answer

To obtain the optimal production level for each doghouse model and the optimal profit, solve the formulated linear programming problem. The solution will provide the number of units for Models A and B to produce, and the maximum possible profit.

Step 1: Formulate the Constraints

From the given data, the time taken by Model A and Model B to assemble, paint, and package can be turned into three inequalities that will serve as constraints for our linear programming problem. \n\n1. Assembling time: \(2.5X + 3Y \leq 4000\) \n2. Painting time: \(2X + Y \leq 2500\) \n3. Packaging time: \(0.75X + 1.25Y \leq 1500\) \n\nAlso add the non-negativity constraints: \(X, Y \geq 0\)

Step 2: Create the Objective Function

The objective is to maximize the company's profit. The company makes \$60 profit for each unit of Model A and \$75 for each unit of Model B. So, the objective function \(\(Z = 60X + 75Y\)\) should be maximized.

Step 3: Solve the Linear Programming Problem

Solving the above constraints and objective function using a Linear Programming method (like the graphical method or the simplex method), will give the values of \(X\) and \(Y\) that will maximize the profit \(Z\).

Step 4: Calculate the Optimal Profit

Substitute the obtained values of \(X\) and \(Y\) into the profit function to calculate the maximum profit.

Key Concepts

Constraints

In linear programming, constraints are the conditions or limits that the solution must satisfy. They are expressed as inequalities that define the feasible region in which the solution lies. For the doghouse production problem, constraints are based on the available time for assembling, painting, and packaging:

• Assembling time: The inequality \(2.5X + 3Y \leq 4000\) represents the total hours for assembling both models. The sum of the time taken by Model A and B must not exceed 4000 hours.
• Painting time: The inequality \(2X + Y \leq 2500\) limits the total time spent on painting to 2500 hours or less.
• Packaging time: The inequality \(0.75X + 1.25Y \leq 1500\) ensures that the packaging process does not exceed 1500 hours.

These constraints are crucial as they determine the boundaries within which the company must operate to achieve optimal production levels. Additionally, non-negativity constraints \(X, Y \geq 0\) indicate that the number of units produced cannot be negative.

Objective Function

The objective function in a linear programming problem is what you aim to optimize, usually either maximizing or minimizing a particular quantity. In our doghouse example, the objective is to maximize profit. The profit function is defined by the profits per unit of Model A and B:

• The profit from each Model A sold is \\(60, while each Model B sold contributes \\)75.

Thus, the objective function is: \[ Z = 60X + 75Y \] where:

• \(X\) is the number of Model A units produced.
• \(Y\) is the number of Model B units produced.

The goal is to find the values of \(X\) and \(Y\) that maximize \(Z\), staying within the constraints set.

Optimal Profit

Once the values of \(X\) and \(Y\) that maximize the objective function have been determined, they can be substituted back into the profit function to find the optimal profit. This optimal point represents the highest level of profit the company can achieve given the constraints. For example:

• If the solution to the problem gave \(X = 500\) and \(Y = 300\), these values would be substituted into the profit equation: \(Z = 60(500) + 75(300)\).
• Calculating this would yield the total profit: \(Z = 30000 + 22500 = 52500\).

This optimal profit value indicates the maximum number of profits the company can obtain while adhering to all the constraint limits.

Simplex Method

The Simplex Method is a popular algorithm used in linear programming to find the optimal solution to a problem. It is particularly useful for problems involving more than two decision variables. The method works by moving along the edges of the feasible region defined by the constraints, seeking the vertex that optimizes the objective function.

• It starts from an initial feasible solution and iteratively moves to a better solution.
• The process continues until no further improvements are possible.
• Although it can be complex de, it guarantees finding either the optimal solution or determining that no feasible solution exists.

For the doghouse example, the Simplex Method would systematically explore the feasible region defined by the constraints, checking vertices until it finds the combination of \(X\) and \(Y\) that maximizes profits.