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The problem in the image asks to formulate a linear programming model that maximizes the utility of a workshop, given specific capacity constraints for casting ("fundición") and finishing ("acabado"). The available capacities are:

- Casting: 700 hours per month
- Finishing: 1000 hours per month

To solve this linear programming problem, we need additional information such as:
- The number of products being produced
- The time required in casting and finishing for each product
- The profit or utility generated by each product.

In the absence of this information, I will outline the general steps to set up a linear programming model:

### Steps to Formulate the Model:
1. **Decision Variables**: Define the decision variables that represent the quantities of each product to be produced.
   - Let \( x_1 \) represent the number of units of product 1 to be produced.
   - Let \( x_2 \) represent the number of units of product 2 to be produced.
   
2. **Objective Function**: The objective is to maximize the total utility (profit).
   - Suppose the profit per unit for product 1 is \( p_1 \) and for product 2 is \( p_2 \), the objective function would be:
   \[
   \text{Maximize } Z = p_1 \cdot x_1 + p_2 \cdot x_2
   \]
   
3. **Constraints**:
   - Casting constraint: If product 1 requires \( a_1 \) hours of casting and product 2 requires \( a_2 \) hours of casting, the casting capacity constraint would be:
   \[
   a_1 \cdot x_1 + a_2 \cdot x_2 \leq 700
   \]
   - Finishing constraint: If product 1 requires \( b_1 \) hours of finishing and product 2 requires \( b_2 \) hours of finishing, the finishing capacity constraint would be:
   \[
   b_1 \cdot x_1 + b_2 \cdot x_2 \leq 1000
   \]
   
4. **Non-Negativity**: The number of units produced cannot be negative, so:
   \[
   x_1 \geq 0, \quad x_2 \geq 0
   \]

Once the parameters (time requirements and profits) are known, this model can be solved using methods like the **Simplex method** or graphical methods (if two variables are involved).

Do you have more details about the products or their utility, so we can make the model more specific?

Here are some relative questions:
1. What are the profits associated with each product being produced in the workshop?
2. How many types of products are involved in the manufacturing process?
3. How much time does each product spend in casting and finishing operations?
4. Are there any additional constraints, such as material availability or budget limits?
5. Would you prefer solving this problem using software tools like Excel or by hand?

**Tip**: Always gather all available resources (time, cost, material limits) when setting up linear programming models to ensure accurate results.

Si la capacidad disponible tanto de fundición como de acabado son respectivamente 700 y 1,000 horas de fuerza de trabajo por mes, plantear el modelo de programación lineal que permita maximizar las utilidades obtenidas en el taller.

Learn how to create a linear programming model to maximize workshop profits, given monthly constraints of 700 hours for casting and 1,000 hours for finishing. The solution includes key concepts like objective function maximization, casting and finishing constraints, and non-negativity conditions.

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