Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Optimal Profit The costs to a store for two models of Global Positioning System (GPS) receivers are $$\$ 80$$ and $$\$ 100$$. The $$\$ 80$$ model yields a profit of $$\$ 25$$ and the $$\$ 100$$ model yields a profit of $$\$ 30 .$$ Market tests and available resources indicate the following constraints. \- The merchant estimates that the total monthly demand will not exceed 200 units. \- The merchant does not want to invest more than $$\$ 18,000$$ in GPS receiver inventory. What is the optimal inventory level for each model? What is the optimal profit?

Step-by-Step Solution

Answer

The optimal numbers of the $80 and $100 models and the corresponding optimal profit would be determined using a method for solving linear programming problems under the given set of constraints.

Step 1: Defining the Variables

Let \(X\) represent the number of units of the $80 model and \(Y\) the number of units of the $100 model. Now the objective is to maximize profit.

Step 2: Constructing the Objective Function

The profit from each $80 model is $25 and from each $100 is $30. Therefore, the total profit \(P\) can be represented as: \(P = 25X + 30Y\) which is what we aim to maximize.

Step 3: Constructing the Constraints

The constraints from the problem are: 1. The total monthly demand will not exceed 200 units: \(X + Y \leq 200\).2. The store does not want to invest more than $18,000: \(80X + 100Y \leq 18000\).3. As the number of units cannot be negative: \(X \geq 0\) and \(Y \geq 0\).

Step 4: Solving the Linear Programming Problem

Using a method suitable for solving linear programming problems such as graphical method or Simplex method, determine the values of \(X\) and \(Y\) that optimize the objective function under the given constraints.

Step 5: Determine the Optimal Profit

Once the values of \(X\) and \(Y\) that optimize the profit are found, these can be substituted back into the profit function to find the optimal profit.

Key Concepts

Objective Function

In linear programming, the objective function is what you are trying to maximize or minimize. It's a mathematical representation of the goal you're aiming to achieve. In this particular exercise, the objective function represents the total profit from selling two types of GPS receivers.

To set it up, you need to consider the profit each unit generates. Here, the \(80 model provides a profit of \)25, and the \(100 model offers \)30. Hence, if you let \(X\) be the number of \(80 models and \(Y\) be the number of \)100 models, the objective function can be written as:

\[P = 25X + 30Y\]

The job is to find the values of \(X\) and \(Y\) that yield the maximum possible profit \(P\). This formula is what guides the entire optimization process, clearly showing what needs to be achieved.

Constraints

When working with linear programming, constraints are the limitations or restrictions that must be considered during the optimization process. They define the feasible region where solutions are possible.

In this exercise, three main constraints play a role.

• The total number of GPS receivers demanded in a month, \(X + Y \), can't be more than 200 units.
• The total investment in GPS inventory, given by \(80X + 100Y\), should not exceed $18,000.
• The number of units of each type can't be negative, so \(X \geq 0\) and \(Y \geq 0\).

These constraints need to shape the strategy for finding the optimal number of each model to stock.

Profit Maximization

Profit maximization is a key goal for businesses looking to improve their bottom line. In this context, it involves adjusting the number of each GPS model to maximize total profit.

Consider how different combinations of the models could affect profit, given the constraints. By analyzing how changes in \(X\) and \(Y\) affect the profit function \(P = 25X + 30Y\), and ensuring these stay within the specified limits, you can determine the optimal mix of models for sale that brings in the highest profit.

This process of finding the best solution is the essence of profit maximization in linear programming.

Optimization Problem

An optimization problem in the realm of linear programming is about finding the most effective way to achieve a specific outcome within set constraints. Essentially, it seeks the best possible solution from a pool of feasible options.

In our GPS receiver example, solving the optimization problem involves analyzing the objective function (profit maximization) subject to given constraints on total demand and inventory investment.

Methods like the graphical approach or Simplex method are applied to pinpoint the values of \(X\) and \(Y\) that achieve this.

Understanding and applying these methods helps one efficiently arrive at an optimal decision, perfectly aligning with the needs and limits of the business situation.