Question 1205784
## Formulating the LP Model

### Decision Variables:
Let's define the decision variables as follows:

* **S**: Tons of Spring produced
* **A**: Tons of Autumn produced
* **W**: Tons of Winter produced
* **C**: Tons of Cotton purchased
* **Wo**: Tons of Wool purchased
* **Si**: Tons of Silk purchased

### Objective Function:
The objective is to maximize the total profit. The profit per ton for each product is the selling price minus the production cost and the material cost.

**Maximize Profit = 55S + 52A + 55W - 30C - 45Wo - 50Si**

### Constraints:
1. **Demand Constraints:**
   * S ≤ 3300
   * A ≤ 3600
   * W ≤ 4000

2. **Material Constraints:**
   * C ≥ 0.55S + 0.45A + 0.3W
   * Wo ≥ 0.3S + 0.4A + 0.5W

3. **Non-negativity Constraints:**
   * S, A, W, C, Wo, Si ≥ 0

This linear programming model can be solved using various optimization techniques, such as the simplex method or interior point methods. By solving this model, we can determine the optimal production quantities for each product and the optimal material purchases to maximize the factory's profit.