To solve this problem, we can formulate a linear programming (LP) model that minimizes costs under given constraints.

Variables

Let:

• $x$: Number of packages bought from Fred Motors
• $y$: Number of packages bought from Admiral Motors
• $z$: Number of packages bought from Chrysalis

Objective Function

The objective is to minimize the total cost: $\text{Minimize } 500,000x + 400,000y + 300,000z$

Constraints

Federal Rent-a-Car needs at least 650 small cars, 500 medium cars, and 650 large cars. Therefore, the constraints are:

1. Small Cars: $5x + 5y + 10z \geq 650$
2. Medium Cars: $5x + 10y + 5z \geq 500$
3. Large Cars: $10x + 5y + 5z \geq 650$

Non-negativity Constraints

$x \geq 0, \, y \geq 0, \, z \geq 0$

Solution Approach

This problem can be solved using linear programming methods such as the Simplex method or by using computational tools.

Would you like a step-by-step breakdown of how to solve this LP problem or need assistance with using a specific tool for solving it?

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Follow-up Questions

1. What is the importance of setting up a linear programming model for cost minimization?
2. How would the solution change if Federal Rent-a-Car needed different numbers of cars?
3. Can this model accommodate additional car categories, such as electric cars?
4. What assumptions are we making in the formulation of this LP model?
5. How would the model adjust if one of the manufacturers increased its package price?

Tip

When setting up LP models, clearly define variables and constraints to avoid errors in larger models.