Question 1205784: A factory makes three products called Spring, Autumn, and Winter, from three materials containing Cotton, Wool and Silk. The following table provides details on the sales price, production cost and purchase cost per ton of products and materials respectively.
Sales price Production cost Purchase price
Spring $60 $5 Cotton $30
Autumn $55 $3 Wool $45
Winter $60 $5 Silk $50
The maximal demand (in tons) for each product, the minimum cotton and wool propor- tion in each product is as follows:
Demand min Cotton proportion min Wool proportion
Spring 3300 55% 30%
Autumn 3600 45% 40%
Winter 4000 30% 50%
a) Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints. There is no penalty for the shortage.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! ## 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.
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