https://file.aiquickdraw.com/mathbot/file/1726905563284wrckih3h.jpg

This problem involves formulating a **linear programming (LP)** model to maximize the profit for PT Bakery, which produces three types of bread: *pia*, *bolukismis*, and *coklatkeju*. Each type of bread has a set profit, requires different amounts of preparation, mixing, and baking time, and has minimum production constraints. We are tasked with finding the optimal solution under the time constraints for each process.

### Variables:
Let:
- \( x_1 \): number of *pia* produced
- \( x_2 \): number of *bolukismis* produced
- \( x_3 \): number of *coklatkeju* produced

### Objective function:
Maximize the total profit. The profit for each type of bread is given as:
- Profit per *pia* = Rp 150
- Profit per *bolukismis* = Rp 400
- Profit per *coklatkeju* = Rp 600

Thus, the objective function is:
\[
\text{Maximize} \ Z = 150x_1 + 400x_2 + 600x_3
\]

### Constraints:
1. **Preparation time constraint** (penyediaan max: 130 hours):
   \[
   4x_1 + 2x_2 + 6x_3 \leq 130
   \]
2. **Mixing time constraint** (peracikan max: 170 hours):
   \[
   3x_1 + 4x_2 + 9x_3 \leq 170
   \]
3. **Baking time constraint** (pengovenan max: 52 hours):
   \[
   1x_1 + 2x_2 + 1x_3 \leq 52
   \]

### Minimum production constraints:
- *Pia*: \( x_1 \geq 25 \)
- *Bolukismis*: \( x_2 \geq 130 \)
- *Coklatkeju*: \( x_3 \geq 55 \)

### Non-negativity constraints:
\[
x_1, x_2, x_3 \geq 0
\]

### Full Linear Programming Model:
\[
\text{Maximize} \ Z = 150x_1 + 400x_2 + 600x_3
\]
Subject to:
\[
\begin{aligned}
    4x_1 + 2x_2 + 6x_3 &\leq 130 \\
    3x_1 + 4x_2 + 9x_3 &\leq 170 \\
    1x_1 + 2x_2 + 1x_3 &\leq 52 \\
    x_1 &\geq 25 \\
    x_2 &\geq 130 \\
    x_3 &\geq 55 \\
    x_1, x_2, x_3 &\geq 0
\end{aligned}
\]

This is the LP formulation. To find the optimal solution, we could use the **Simplex Method** or a software tool like **Excel Solver** or **Python** with an LP solver.

Would you like to proceed with solving this or need further details on any part?

### 5 related questions:
1. How do we determine which constraints are binding in an LP problem?
2. What is the geometric interpretation of this linear programming problem?
3. How can sensitivity analysis be applied once we find the optimal solution?
4. How would the solution change if the profit for *bolukismis* was increased to Rp 500?
5. What software tools can be used to solve linear programming problems efficiently?

### Tip:
When solving LP problems, ensure all units are consistent (e.g., time in hours, profit in the same currency) to avoid calculation errors.

PT BAKERY memproduksi tiga jenis roti kering, yaitu pia, bolukismis, dan coklatkeju dengan keuntungan masing-masing Rp 150, Rp 400, dan Rp 600. Setiap minggu ditetapkan minimum produksi pia 25 unit, bolukismis 130 unit, dan coklatkeju 55 unit. Proses produksi melibatkan penyediaan bahan, peracikan, dan pengovenan dengan batas waktu maksimal yang ditentukan. Formulasi program linear dan hitung solusi optimalnya.

Solve the PT BAKERY optimization problem using linear programming. Maximize profits from three types of bread with constraints on preparation, mixing, and baking times. Includes minimum production requirements and non-negativity constraints. Suitable for students and professionals familiar with optimization techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation