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The problem you provided is a Linear Programming (L.P.) model designed to minimize costs for producing a dog biscuit mix composed of chicken and liver biscuits, while satisfying nutritional requirements and constraints on production quantities.

Here's how we can formulate the Linear Programming model based on the description:

### Variables:
- \( X \) = number of liver-flavored biscuits in a package
- \( Y \) = number of chicken-flavored biscuits in a package

### Objective Function:
The cost to produce the biscuits is the sum of the costs for liver and chicken biscuits. The cost of making one liver biscuit is 1 cent, and for one chicken biscuit, it is 2 cents. Thus, we want to minimize the total cost:
\[ \text{Minimize: } Z = 1X + 2Y \]
where \( Z \) is the total cost in cents.

### Constraints:
1. **Nutritional Requirements:**
   - The liver biscuit contains 1 unit of nutrient A and 2 units of nutrient B.
   - The chicken biscuit contains 1 unit of nutrient A and 4 units of nutrient B.
   
   The mix must meet or exceed 40 units of nutrient A and 60 units of nutrient B:
   - For nutrient A: 
   \[
   X + Y \geq 40
   \]
   - For nutrient B:
   \[
   2X + 4Y \geq 60
   \]

2. **Maximum Liver Biscuits:**
   The company decided that there should be no more than 18 liver biscuits in a package:
   \[
   X \leq 18
   \]

3. **Non-Negativity Constraint:**
   Both \( X \) and \( Y \) must be non-negative:
   \[
   X \geq 0, \quad Y \geq 0
   \]

### Summary of the L.P. Model:
- **Minimize:**
  \[
  Z = 1X + 2Y
  \]
- **Subject to:**
  \[
  X + Y \geq 40
  \]
  \[
  2X + 4Y \geq 60
  \]
  \[
  X \leq 18
  \]
  \[
  X \geq 0, \quad Y \geq 0
  \]

Would you like help solving this system to find the optimal solution for \( X \) and \( Y \), or would you like to clarify any of the constraints?

### 5 Related Questions:
1. How do we solve a linear programming problem graphically?
2. What happens if there are multiple optimal solutions?
3. How would the constraints change if the cost of chicken biscuits was reduced?
4. What is the significance of non-negativity in a linear programming problem?
5. How can sensitivity analysis be applied to this L.P. model?

### Tip:
When solving linear programming problems graphically, it's useful to first plot the constraints as lines on a graph and identify the feasible region, which will help find the optimal point.

Kaleena Turner Food Processors wishes to introduce a new brand of dog biscuits composed of chicken and liver flavored biscuits that meet certain nutritional requirements. The liver flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B; the chicken flavored biscuits contain 1 unit of nutrient A and 4 units of nutrient B. According to federal requirements, there must be at least 40 units of nutrient A and 60 units of nutrient B in a package of the new mix. The company has also decided that there can be no more than 18 liver flavored biscuits in a package. It costs 1¢ to make 1 liver flavored biscuit and 2¢ to make 1 chicken flavored biscuit. Kaleena wants to determine the optimal product mix for a package of the biscuits to minimize the firm's cost.

This problem involves determining the optimal product mix of liver and chicken flavored dog biscuits to minimize production cost while satisfying federal nutritional requirements. Learn how to set up and solve this linear programming problem with constraints on nutrients and product quantity.

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