Questions: What are the following solutions, through charts and graph constraints, through the LINGO Software: A baker wants to determine how many desserts and cakes should be produced daily to maximize profit. The baker has 40 kg of dough mix daily to make both products. Each dessert requires 300 g of dough mix, while each cake needs 1200 g. On average, the bakery has orders for at least 30 desserts and 10 cakes daily from other businesses. Additionally, the bakery sells desserts and cakes to walk-in customers. The baker has a small oven that can operate for 15 hours per day to bake the products. Each cake takes half an hour to bake, while desserts take 12 minutes (ie, 0.2 hours). However, no more than one unit of each product can be baked in the oven at a time. The baker has storage space that can hold a maximum of 30 cakes to keep both products fresh. Please consider that both types of products are stored in the same storage space and that 5 desserts occupy the space of one cake. Finally, desserts profit 4 each, and cakes profit 9 each. a) Please formulate an optimization model to address the problem above. b) Use the graphical method to find the feasible region and optimal production plan. c) How many desserts and cakes should be produced? d) If the baker were to make one more cake (i.e, 11 instead of 10), would this increase profit? Yes or no, and why? e) The baker can buy 10 kg more of dough mix for 10, would you recommend proceeding with the transaction?

What are the following solutions, through charts and graph constraints, through the LINGO Software: A baker wants to determine how many desserts and cakes should be produced daily to maximize profit. The baker has 40 kg of dough mix daily to make both products. Each dessert requires 300 g of dough mix, while each cake needs 1200 g. On average, the bakery has orders for at least 30 desserts and 10 cakes daily from other businesses. Additionally, the bakery sells desserts and cakes to walk-in customers. The baker has a small oven that can operate for 15 hours per day to bake the products. Each cake takes half an hour to bake, while desserts take 12 minutes (ie, 0.2 hours). However, no more than one unit of each product can be baked in the oven at a time. The baker has storage space that can hold a maximum of 30 cakes to keep both products fresh. Please consider that both types of products are stored in the same storage space and that 5 desserts occupy the space of one cake. Finally, desserts profit 4 each, and cakes profit 9 each. a) Please formulate an optimization model to address the problem above. b) Use the graphical method to find the feasible region and optimal production plan. c) How many desserts and cakes should be produced? d) If the baker were to make one more cake (i.e, 11 instead of 10), would this increase profit? Yes or no, and why? e) The baker can buy 10 kg more of dough mix for 10, would you recommend proceeding with the transaction?
Transcript text: What are the following solutions, through charts and graph constraints, through the LINGO Software: $\mid$ A baker wants to determine how many desserts and cakes should be produced daily to maximize profit. The baker has 40 kg of dough mix daily to make both products. Each dessert requires 300 g of dough mix, while each cake needs 1200 g . On average, the bakery has orders for at least 30 desserts and 10 cakes daily from other businesses. Additionally, the bakery sells desserts and cakes to walk-in customers. The baker has a small oven that can operate for 15 hours per day to bake the products. Each cake takes half an hour to bake, while desserts take 12 mimutes (ie, 0.2 hours). However, no more than one unit of each product can be baked in the oven at a time. The baker has storage space that can hold a maximum of 30 cakes to keep both products fresh. Please consider that both types of products are stored in the same storage space and that 5 desserts occupy the space of one cake. Finally, desserts profit $\$ 4$ each, and cakes profit \$9 each. a) Please formulate an optimization model to address the problem above. b) Use the graphical method to find the feasible region and optimal production plan. c) How many desserts and cakes should be produced? d) If the baker were to make one more cake (i.e, 11 instead of 10 ), would this increase profit? Yes or no, and why? e) The baker can buy 10 kg more of dough mix for $\$ 10$, would you recommend proceeding with the transaction?
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Solution

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Solution Steps

Step 1: Define Variables

Let \( x \) be the number of desserts produced daily, and \( y \) be the number of cakes produced daily.

Step 2: Formulate Constraints
  1. Dough mix constraint: \( 0.3x + 1.2y \leq 40 \)
  2. Minimum orders constraint: \( x \geq 30 \) and \( y \geq 10 \)
  3. Oven time constraint: \( 0.2x + 0.5y \leq 15 \)
  4. Storage space constraint: \( \frac{x}{5} + y \leq 30 \)
Step 3: Formulate Objective Function

Maximize profit: \( P = 4x + 9y \)

Final Answer

The optimization model is: \[ \begin{align_} \text{Maximize} \quad & P = 4x + 9y \\ \text{subject to} \quad & 0.3x + 1.2y \leq 40 \\ & x \geq 30 \\ & y \geq 10 \\ & 0.2x + 0.5y \leq 15 \\ & \frac{x}{5} + y \leq 30 \\ & x, y \geq 0 \end{align_} \]

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