Acme Babyfoods Mixes Two Strengths Of Apple Juice One Quart

Acme Babyfoods Mixes Two Strengths Of Apple Juice One Quart Of Beginn

Acme Babyfoods mixes two strengths of apple juice. One quart of Beginner’s juice is made from 30 fluid ounces of water and 2 fluid ounces of apple juice concentrate. One quart of Advanced juice is made from 20 fluid ounces of water and 12 fluid ounces of concentrate. Every day Acme has available 30,000 fluid ounces of water and 3,600 fluid ounces of concentrate. Acme makes a profit of 20¢ on each quart of Beginner’s juice and 30¢ on each quart of Advanced juice. How many quarts of each should Acme make each day to get the largest profit? How would this change if Acme made a profit of 40¢ on Beginner’s juice and 20¢ on Advanced juice?

Paper For Above instruction

The problem at hand involves optimizing the production of two types of apple juice blends to maximize profit, considering resource constraints. It presents a classic linear programming problem, where decision variables represent the quantities of each juice type produced daily, and constraints are based on the availability of resources such as water and concentrate. This analysis will explore two scenarios: one with initial profit margins and a second with altered profit margins, to determine the optimal production quantities.

Introduction

Linear programming is an essential tool in operations research and management science, facilitating optimal decision-making under resource constraints (Winston, 2004). In manufacturing contexts, especially, it helps identify the most profitable production mix. The case of Acme Babyfoods, which produces two types of apple juice—Beginner’s and Advanced—demonstrates a typical linear programming problem involving multiple constraints and objective functions.

Problem Formulation

Let:

- \( x_1 \): number of quarts of Beginner’s juice produced

- \( x_2 \): number of quarts of Advanced juice produced

The resource requirements per quart are:

- Beginner’s juice: 30 oz water, 2 oz concentrate

- Advanced juice: 20 oz water, 12 oz concentrate

Resource constraints are:

- Total water available: 30,000 oz

- Total concentrate available: 3,600 oz

Thus, the constraints are:

1. Water constraint: \( 30x_1 + 20x_2 \leq 30,000 \)

2. Concentrate constraint: \( 2x_1 + 12x_2 \leq 3,600 \)

3. Non-negativity: \( x_1, x_2 \geq 0 \)

The profit functions for the two scenarios are:

- Scenario 1: \( P = 0.20x_1 + 0.30x_2 \)

- Scenario 2: \( P = 0.40x_1 + 0.20x_2 \)

The goal is to maximize \( P \) subject to the constraints, analyzing each scenario separately.

Solution to Scenario 1: Initial Profits

Maximize \( P = 0.20x_1 + 0.30x_2 \)

Subject to:

- \( 30x_1 + 20x_2 \leq 30,000 \)

- \( 2x_1 + 12x_2 \leq 3,600 \)

- \( x_1, x_2 \geq 0 \)

First, we identify the feasible region bounded by these inequalities. The next step involves solving the system's corner points—vertices of the feasible region—by considering intersections of the constraint lines.

1. Intersection of the resource constraints:

- Water line: \( 30x_1 + 20x_2 = 30,000 \)

- Concentrate line: \( 2x_1 + 12x_2 = 3,600 \)

Solve for \( x_1 \) and \( x_2 \):

From the concentrate constraint:

\[ 2x_1 + 12x_2 = 3,600 \]

Divide by 2:

\[ x_1 + 6x_2 = 1,800 \]

Express \( x_1 \):

\[ x_1 = 1,800 - 6x_2 \]

Substitute into the water constraint:

\[ 30(1,800 - 6x_2) + 20x_2 \leq 30,000 \]

\[ 54,000 - 180x_2 + 20x_2 \leq 30,000 \]

\[ 54,000 - 160x_2 \leq 30,000 \]

Subtract 54,000:

\[ -160x_2 \leq -24,000 \]

Divide by -160:

\[ x_2 \geq 150 \]

Now, replace \( x_2 \) in:

\[ x_1 = 1,800 - 6(150) = 1,800 - 900 = 900 \]

Check if these points satisfy all constraints:

- \( x_1 = 900 \), \( x_2 = 150 \)

Calculate resource usages:

- Water: \( 30(900) + 20(150) = 27,000 + 3,000 = 30,000 \) OK

- Concentrate: \( 2(900) + 12(150) = 1,800 + 1,800 = 3,600 \) OK

Thus, one vertex of the feasible region is at \( (900, 150) \).

2. Check the axes intercepts for possible maximums:

- \( x_2 = 0 \):

From water constraint:

\[ 30x_1 \leq 30,000 \Rightarrow x_1 \leq 1,000 \]

From concentrate constraint:

\[ 2x_1 \leq 3,600 \Rightarrow x_1 \leq 1,800 \]

So, \( x_1 = 1,000 \), \( x_2 = 0 \)

- \( x_1 = 0 \):

From water constraint:

\[ 20x_2 \leq 30,000 \Rightarrow x_2 \leq 1,500 \]

From concentrate constraint:

\[ 12x_2 \leq 3,600 \Rightarrow x_2 \leq 300 \]

So, \( x_2 = 300 \)

Check the corner point \( (0, 300) \):

Calculate profit:

\[ P = 0.20(0) + 0.30(300) = 90 \text{ cents} \]

Calculate at \( (900, 150) \):

\[ P = 0.20(900) + 0.30(150) = 180 + 45 = 225 \text{ cents} \]

Calculate at \( (1,000, 0) \):

\[ P = 0.20(1000) + 0.30(0) = 200 \text{ cents} \]

Compare these: maximum profit at \( (900, 150) \).

Optimal Production (Scenario 1):

- 900 quarts of Beginner’s juice

- 150 quarts of Advanced juice

- Maximum profit: 225 cents or \$2.25 per day

Solution to Scenario 2: Altered Profits

Maximize \( P = 0.40x_1 + 0.20x_2 \)

Using the same feasible vertices:

1. At \( (900, 150) \):

\[ P = 0.40(900) + 0.20(150) = 360 + 30 = 390 \text{ cents} \]

2. At \( (1,000, 0) \):

\[ P = 0.40(1000) + 0.20(0) = 400 \text{ cents} \]

3. At \( (0, 300) \):

\[ P = 0.40(0) + 0.20(300) = 60 \text{ cents} \]

Clearly, the maximum profit occurs at \( (1,000, 0) \), i.e., producing only Beginner’s juice:

- 1,000 quarts of Beginner’s juice

- 0 quarts of Advanced juice

- Maximum profit: 400 cents or \$4.00 per day

Implications:

The shift in profit margins significantly alters the optimal production strategy. In the initial case, a balanced combination yields maximum profit, whereas with altered profits, the most profitable approach is to produce exclusively the more profitable juice type within resource constraints.

Conclusion

This linear programming analysis illustrates how resource limitations and profit margins influence manufacturing decisions. Under initial profit conditions, a mix of 900 quarts of Beginner’s and 150 quarts of Advanced juices maximizes profit. When profit margins change to favor Beginner’s juice more heavily, the optimal strategy shifts to produce only Beginner’s juice, fully utilizing water constraints. Such models are vital in production planning, enabling companies like Acme Babyfoods to strategically allocate resources for maximum profitability.

References

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