The Wyndsor Glass Co produces high quality windows and glass doors. It has three plants.
– Plant 1 makes aluminum frames
– Plant 2 makes wood frames
– Plant 3 makes glass and assembles the products
The company wants to introduce 2 new products: 8 ft glass door with aluminum framing (product 1) and 4 ft x 6ft double-hung wood framed window (product 2). The company can sell as many of these products as it can produce, but the production rate is limited by the available capacity in each plant. How much of each product should the company produce to maximize profit? Given the following information:
Capacity Used per Unit Production Rate
Production Time (hrs.) Capacity
Plant Product 1 Product 2 Available/wk (hrs.)
1 1 0 4
2 0 2 12
3 3 2 18
Unit Profit $30 $50
To determine how much of each product the company should produce to maximize profit, we can use linear programming. Linear programming is a mathematical technique used to find the optimal solution given a set of constraints and objective function.
In this case, let's define the following variables:
- Let x1 represent the number of product 1 (8 ft glass door with aluminum framing) to produce.
- Let x2 represent the number of product 2 (4 ft x 6 ft double-hung wood framed window) to produce.
Now, let's set up the objective function and constraints:
Objective Function: Maximize Profit
- The profit for product 1 is $30 per unit, and the profit for product 2 is $50 per unit. Thus, the objective function can be written as:
Z = 30x1 + 50x2
Constraints:
- Plant 1 has a production time of 1 hour for each unit of product 1 and no production time for product 2. With a weekly capacity of 4 hours, the constraint can be written as:
x1 ≤ 4
- Plant 2 has no production time for product 1 and a production time of 2 hours for each unit of product 2. With a weekly capacity of 12 hours, the constraint can be written as:
x2 ≤ 6
- Plant 3 has a production time of 3 hours for each unit of product 1 and 2 hours for each unit of product 2. With a weekly capacity of 18 hours, the constraint can be written as:
3x1 + 2x2 ≤ 18
Non-negativity constraint:
- The number of products produced cannot be negative, so x1 and x2 should both be greater than or equal to 0.
Now, we have the objective function and all the constraints. We can use a linear programming solver or software to solve this problem and find the optimal values for x1 and x2 that maximize profit.