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To formulate this problem as a Linear Programming (LP) model, we need to maximize the daily revenue subject to the constraints imposed by the department capacities and production times for each cabinet style.

### Decision Variables:
Let:
- $ x_1 $ = Number of French Provincial cabinets produced per day
- $ x_2 $ = Number of Danish Modern cabinets produced per day

### Objective Function:
Maximize the total revenue from producing both cabinet types. The revenue per cabinet is provided in the table:
- French Provincial: $28 per cabinet
- Danish Modern: $25 per cabinet

Thus, the objective function to maximize is:
\[
Z = 28x_1 + 25x_2
\]
Where $ Z $ is the total daily revenue.

### Constraints:

1. **Carpentry Capacity**:
   The department has 360 hours available per day for carpentry, and the time required for each type of cabinet is:
   - French Provincial: 3 hours/cabinet
   - Danish Modern: 2 hours/cabinet

   So the carpentry constraint is:
   \[
   3x_1 + 2x_2 \leq 360
   \]

2. **Painting Capacity**:
   The department has 200 hours available per day for painting, and the time required for each type of cabinet is:
   - French Provincial: 1.5 hours/cabinet
   - Danish Modern: 1 hour/cabinet

   So the painting constraint is:
   \[
   1.5x_1 + x_2 \leq 200
   \]

3. **Finishing Capacity**:
   The department has 125 hours available per day for finishing, and the time required for each type of cabinet is:
   - French Provincial: 0.75 hours/cabinet
   - Danish Modern: 0.75 hours/cabinet

   So the finishing constraint is:
   \[
   0.75x_1 + 0.75x_2 \leq 125
   \]

4. **Minimum Production Constraint**:
   The company has a contract to produce a minimum of 300 cabinets per week (or 60 cabinets per day) for each type of cabinet, which gives us:
   \[
   x_1 \geq 60
   \]
   \[
   x_2 \geq 60
   \]

5. **Non-negativity**:
   \[
   x_1 \geq 0, \quad x_2 \geq 0
   \]

### LP Model:

Maximize: 
\[
Z = 28x_1 + 25x_2
\]

Subject to:
\[
3x_1 + 2x_2 \leq 360
\]
\[
1.5x_1 + x_2 \leq 200
\]
\[
0.75x_1 + 0.75x_2 \leq 125
\]
\[
x_1 \geq 60
\]
\[
x_2 \geq 60
\]
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

This completes the formulation of the LP problem. 

Would you like me to solve this LP using software or a spreadsheet, or would you prefer more details on how to approach the solution manually?

### Follow-up Questions:
1. How would we interpret the shadow price in this problem?
2. What happens if the company increases the available carpentry hours?
3. How can sensitivity analysis help understand this LP model?
4. What is the economic interpretation of binding and non-binding constraints in this context?
5. Can we alter the net revenue per cabinet and how would that affect the optimal solution?

### Tip:
Always make sure that your constraints represent the actual limits and operational capacity in a production optimization problem to avoid unrealistic solutions.

Learn how to formulate and solve a linear programming problem to maximize the daily revenue from producing French Provincial and Danish Modern cabinets. The problem includes constraints for carpentry, painting, and finishing departments, as well as minimum production requirements for each cabinet type. Suitable for students in undergraduate business or operations research courses.

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