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: