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The problem presented is an example of a linear programming problem, commonly referred to as a "diet problem." The goal is to find the optimal mix of feeds that satisfies the nutritional requirements of horses at the lowest cost.

### Problem Summary:
1. **Objective**: Minimize the cost of feeding the horses while meeting all the dietary requirements.
2. **Feeds Available**:
   - **Oat Product**: $0.09 per pound
   - **Enriched Grain**: $0.14 per pound
   - **Mineral Product**: $0.17 per pound
3. **Dietary Requirements**: There are five nutrients (A, B, C, D, E) with specified minimum daily requirements.
   - For example, nutrient A requires at least 6 units daily.
4. **Nutrient Content**: Each type of feed provides a different amount of each nutrient, given in units per pound.

### Mathematical Formulation:

Let:
- $ x_1 $ = pounds of Oat Product
- $ x_2 $ = pounds of Enriched Grain
- $ x_3 $ = pounds of Mineral Product

**Objective Function** (Minimize Cost):
\[ \text{Minimize } Z = 0.09x_1 + 0.14x_2 + 0.17x_3 \]

**Constraints**:
1. **Nutrient A Requirement**:
   \[ 2x_1 + 3x_2 + 1x_3 \geq 6 \]
2. **Nutrient B Requirement**:
   \[ 0.5x_1 + 1x_2 + 0.5x_3 \geq 2 \]
3. **Nutrient C Requirement**:
   \[ 3x_1 + 5x_2 + 6x_3 \geq 25 \]
4. **Nutrient D Requirement**:
   \[ 1x_1 + 1.5x_2 + 2x_3 \geq 8 \]
5. **Nutrient E Requirement**:
   \[ 0.5x_1 + 0.5x_2 + 1.5x_3 \geq 5 \]

**Non-negativity Constraints**:
\[ x_1, x_2, x_3 \geq 0 \]

### Next Steps:
- You can solve this problem using methods like the Simplex algorithm or a linear programming solver (e.g., in Python using `scipy.optimize.linprog`).

Would you like me to solve this problem for the optimal feed mix or provide further details on the formulation? 

Here are some related questions:
1. How would you set up this problem in Python for numerical solution?
2. What are the graphical solution methods for linear programming problems with two variables?
3. How do you interpret shadow prices in linear programming?
4. What is the significance of slack variables in linear programming?
5. How does sensitivity analysis work in the context of linear programming?

**Tip**: Always ensure that your constraints accurately reflect the problem requirements, as incorrect constraints lead to incorrect solutions.

The Battery Park Stable feeds horses and aims to minimize the overall daily cost of feeding while meeting nutritional requirements for horses. There are three available feed mixes: oat product, enriched grain, and mineral product, each providing different amounts of nutrients A, B, C, D, E. The goal is to minimize the cost while satisfying the minimum daily nutritional requirements.

This problem focuses on minimizing the cost of feeding horses while ensuring they meet all dietary requirements using three feed options. Learn how to set up the linear programming model, define the objective function, and solve the constraints using optimization techniques.

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