*Federal Rent-a-Car* is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 650 small cars, at least 500 medium cars, and at least 650 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible?


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.

*Federal Rent-a-Car* is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 650 small cars, at least 500 medium cars, and at least 650 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible?

Learn to solve a linear programming problem to optimize the fleet purchase for Federal Rent-a-Car. This involves cost minimization while meeting minimum car requirements across three manufacturers, covering small, medium, and large cars. Ideal for students in Operations Research or Business Mathematics.

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