Problem 31 - Finite Mathematics and Applied Calculus

Question

Nutrition Ruff makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat/ounce. and grain has 2 grams of protein and 2 grams of fat/ounce. \(\mathrm{A}\) bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 ec/ounce and grain costs 1 d/ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost? What are the shadow costs of protein and fat?

Step-by-Step Solution

Answer

To minimize the cost of producing a bag of dog food, Ruff should use 0 ounces of chicken and 75 ounces of grain in each bag, with a total cost of 75. The shadow costs of protein and fat are 10 ec/gram and 1 d/gram, respectively.

Step 1: Establish the Objective function

Firstly, we establish our objective function, which will represent the cost of producing one bag of dog food, using the given costs per ounce for each ingredient. The objective function is given by: $$ C = 10x + y $$ where x is the amount of chicken (in ounces), and y is the amount of grain (in ounces).

Step 2: Establish the Constraints

Now, we set up the constraints that ensure enough protein and fat are in each bag of dog food. We have two inequality constraints: 1. Protein constraint: \(10x + 2y \ge 200\) 2. Fat constraint: \(5x + 2y \ge 150\)

Step 3: Graphical Representation of the Constraints and the Feasible Region

Before solving for the minimum cost, it is useful to graphically represent the constraint inequalities and identify the feasible region. Plot the two constraint lines on a graph and shade the feasible region where all constraints are satisfied. In this case, since both constraints have greater than or equal to signs, the feasible region will be where the intersection of the constraint lines is above and to the right.

Step 4: Find the Corner Points of the Feasible Region

In order to find the minimum cost solution, we need to evaluate the objective function at the corner points of the feasible region. There are 3 corner points to consider, which are formed by the intersections of the constraint lines with each other and the axes: 1. Intersection of protein constraint with x-axis: \(x = 20, y = 0\) 2. Intersection of fat constraint with y-axis: \(x = 0, y = 75\) 3. Intersection of protein and fat constraints: \(x = \frac{50}{3}, y = \frac{250}{3}\)

Step 5: Evaluate the Objective Function at the Corner Points

Now, we need to evaluate the objective function at the three corner points to determine the minimum cost solution: 1. Objective function at point (20, 0): \(C = 10(20) + 0 = 200\) 2. Objective function at point (0, 75): \(C = 10(0) + 75 = 75\) 3. Objective function at point (\(\frac{50}{3}\), \(\frac{250}{3}\)): \(C = 10\left(\frac{50}{3}\right) + \frac{250}{3} = \frac{850}{3}\)

Step 6: Identify the Minimum Cost Solution

Based on the evaluations in Step 5, the minimum cost solution occurs at the point (0, 75), with a cost of 75. Therefore, Ruff should use 0 ounces of chicken and 75 ounces of grain in each bag of dog food to minimize the cost.

Step 7: Calculate the Shadow Costs

The shadow costs represent the change in the objective function (cost) when each constraint is relaxed by one unit, i.e., when protein is increased by 1 gram or fat is increased by 1 gram. In this case, both constraints are binding at the minimum cost solution, so we can calculate the shadow costs directly: - Shadow cost of protein: \(\frac{\partial C}{\partial x} = 10\) - Shadow cost of fat: \(\frac{\partial C}{\partial y} = 1\) Thus, the shadow costs of protein and fat are 10 ec/gram and 1 d/gram, respectively.

Key Concepts

Objective Function

In linear programming, the objective function is crucial as it describes what we aim to optimize. It could be either to maximize or minimize a certain value. In this exercise, we are trying to minimize the cost of dog food. The objective function is expressed as a formula that combines all relevant costs or benefits of the decision variables.

For the dog food exercise, the objective function is given by:

• \( C = 10x + y \)

This equation represents the total cost of producing a bag of dog food. Here, \( x \) is the number of ounces of chicken, and \( y \) is the number of ounces of grain.

The coefficients (10 and 1) represent the cost per ounce for each ingredient. Thus, minimizing \( C \) means finding values of \( x \) and \( y \) that keep the cost as low as possible while satisfying all other requirements (constraints).

Constraints

Constraints in linear programming set the limits within which we must operate. They are restrictions or conditions that the solution must adhere to, and they form equations or inequalities based on available resources or requirements.

In this example, we have two key constraints related to the nutritional content of the dog food:

• Protein Constraint: \(10x + 2y \geq 200 \)
• Fat Constraint: \(5x + 2y \geq 150 \)

These inequalities ensure that each bag of dog food contains at least 200 grams of protein and 150 grams of fat. We need to respect these constraints when determining the quantities of chicken and grain to use. Constraints ensure that solutions are practical and meet specific needs or standards, essential in decision-making processes.

Feasible Region

The feasible region in linear programming is a graphical representation of all possible solutions that meet the constraints. It's the area where all the inequality constraints overlap on the graph, and it helps in visually identifying potential solutions.

For this problem, graphing the constraints will show two lines on a plane. The feasible region will be the area where the solutions satisfy both constraints.

• The regions where the constraint inequalities are valid (e.g., solutions are above or to the right of certain lines)
• The intersection points of these domains

Crucially, the feasible region helps identify corner points, which are potential locations of the optimal solution. At least one corner of this region will be the solution that maximizes or minimizes our objective function.

Shadow Cost

Shadow costs, also known as dual values in linear programming, measure how much the objective function will change with a one-unit change in a constraint's right-hand side. In simple terms, they help us understand the value of adding more resources.

For this exercise, we calculated the shadow costs for the existing constraints:

• Protein Shadow Cost: 10 ec/gram
• Fat Shadow Cost: 1 d/gram

These values indicate that if you could somehow add one more gram of protein, the cost of the dog food would increase by 10 ec, while an additional gram of fat would increase the cost by 1 d.

Shadow costs help businesses make more informed decisions, especially when considering changes to the resources available for production or supply constraints.