Your friend Justin is planning to start a new business producing and distributing nutrition bars in Miami, Florida. After some market research and several tasting runs, he has created four types of nutrition bars, as shown in Table 1. Each bar requires different amounts of protein powder and sugar.

The current cost per gram of protein powder is $0.23 and the cost per gram of sugar is $0.32. The monthly demand for bars in the area is estimated in 1405 units in total (considering any and all types of bars).

Justin asked for your help to determine the quantity of each nutrition bar type to produce to ensure the lowest cost while fulfilling the demand.

Table 1: Protein powder and sugar per type of bar

Type of Bar	Protein Powder (in grams)	Sugar (in grams)
Soy	6	9
Nuts	9	4
Yogurt	6	9
Peas	12	3
What would be the minimum cost (in $) to fulfill the monthly demand?

To determine the minimum cost of fulfilling the monthly demand for the nutrition bars, we can model this as a linear programming (LP) problem. We need to find the quantity of each type of bar to produce while minimizing the total cost of protein powder and sugar.

### Problem Details Recap

We have the following information:

1. **Protein powder cost:** $0.23 per gram
2. **Sugar cost:** $0.32 per gram
3. **Total monthly demand:** 1405 units (combined across all bar types)
4. **Bar details:**

| Type of Bar | Protein Powder (grams) | Sugar (grams) |
|-------------|------------------------|---------------|
| Soy         | 6                      | 9             |
| Nuts        | 9                      | 4             |
| Yogurt      | 6                      | 9             |
| Peas        | 12                     | 3             |

### Decision Variables
Let's define the decision variables:
- $ x_1 $: Quantity of Soy bars to produce
- $ x_2 $: Quantity of Nuts bars to produce
- $ x_3 $: Quantity of Yogurt bars to produce
- $ x_4 $: Quantity of Peas bars to produce

### Objective Function
The goal is to minimize the total cost. The cost is determined by the amount of protein powder and sugar required for each type of bar. The objective function is:

\[
\text{Minimize Cost} = 0.23 \times (\text{total grams of protein powder}) + 0.32 \times (\text{total grams of sugar})
\]

Breaking it down for each bar:

- **Soy bar:** $ 6x_1 $ grams of protein powder and $ 9x_1 $ grams of sugar
- **Nuts bar:** $ 9x_2 $ grams of protein powder and $ 4x_2 $ grams of sugar
- **Yogurt bar:** $ 6x_3 $ grams of protein powder and $ 9x_3 $ grams of sugar
- **Peas bar:** $ 12x_4 $ grams of protein powder and $ 3x_4 $ grams of sugar

Total grams of protein powder: $ 6x_1 + 9x_2 + 6x_3 + 12x_4 $

Total grams of sugar: $ 9x_1 + 4x_2 + 9x_3 + 3x_4 $

Therefore, the objective function is:

\[
\text{Minimize Cost} = 0.23(6x_1 + 9x_2 + 6x_3 + 12x_4) + 0.32(9x_1 + 4x_2 + 9x_3 + 3x_4)
\]

### Constraints
1. **Total demand constraint:** The sum of all bars produced should equal the total demand:
   \[
   x_1 + x_2 + x_3 + x_4 = 1405
   \]

2. **Non-negativity constraint:** The quantities of each bar cannot be negative:
   \[
   x_1, x_2, x_3, x_4 \geq 0
   \]

### Formulating and Solving the Linear Program
We can use linear programming techniques to solve this problem. I'll set up and solve it to find the minimum cost.The minimum cost to fulfill the monthly demand of 1,405 nutrition bars is approximately **$4,706.75**.

Would you like details on how this was calculated or have any further questions?

Here are 8 related questions:
1. What is the optimal quantity of each type of bar to produce?
2. How does changing the cost of protein powder or sugar affect the minimum cost?
3. What would happen if the demand increased or decreased?
4. Can this solution be adjusted if Justin wants to prioritize certain bar types?
5. How do linear programming techniques apply to other business problems?
6. What are alternative methods to minimize production costs?
7. How does the sensitivity analysis of cost parameters work?
8. What are the limitations of using linear programming in this scenario?

**Tip:** Linear programming can handle many constraints and variables, making it useful for optimizing resources in various industries.

Learn how to minimize production costs of nutrition bars using linear programming. Solve a real-world problem involving protein powder and sugar costs, with a monthly demand of 1,405 units. Detailed step-by-step solution provided.

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