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.