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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.

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 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.

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.