A manufacturer has a maximum of 240, 360, and 180 kilograms of wood, plastic and steel available. The company produces two products, A and B. Each unit of A requires 1, 3 and 2 kilograms of wood, plastic and steel respectively; each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively, and each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively. The profit per unit of A and B is $4.00 and $6.00 respectively.
Identify all constraints.
Identify all applicable corner points of the feasibility region.
How many units of A and B should be manufactured in order to maximize profits? What would the maximum profit be?
To solve this problem, we need to set up and solve a linear programming problem. Let's go step by step.
Step 1: Identify all constraints.
The constraints in this problem are the limits on the available resources:
- Wood constraint: The maximum amount of wood available is 240 kilograms.
- Plastic constraint: The maximum amount of plastic available is 360 kilograms.
- Steel constraint: The maximum amount of steel available is 180 kilograms.
Step 2: Define the decision variables.
Let's assume we need to decide the number of units to produce for products A and B. Let's denote the number of units of A as 'x' and the number of units of B as 'y'.
Step 3: Formulate the objective function.
The objective of this problem is to maximize profit. The profit per unit of A is $4.00 and the profit per unit of B is $6.00. Therefore, the objective function can be defined as:
Profit = 4x + 6y
Step 4: Formulate the constraints.
Based on the resource constraints mentioned above, we can formulate the following constraints:
- Wood constraint: 1x + 3y ≤ 240 (wood constraint)
- Plastic constraint: 3x + 4y ≤ 360 (plastic constraint)
- Steel constraint: 2x + y ≤ 180 (steel constraint)
- Non-negativity constraint: x ≥ 0, y ≥ 0 (we cannot produce negative units)
Step 5: Identify all applicable corner points of the feasibility region.
To find the corner points, we need to solve the system of linear constraints.
Solving the equations, we find four corner points of the feasibility region: (0,0), (120,0), (100,60), and (0,60).
Step 6: Determine the optimal solution.
Plug in the corner points into the objective function, Profit = 4x + 6y.
1. For (0,0), Profit = 4(0) + 6(0) = $0
2. For (120,0), Profit = 4(120) + 6(0) = $480
3. For (100,60), Profit = 4(100) + 6(60) = $760
4. For (0,60), Profit = 4(0) + 6(60) = $360
The highest profit is obtained at (100,60), producing 100 units of A and 60 units of B, with a profit of $760.
Therefore, in order to maximize profits, the manufacturer should produce 100 units of A and 60 units of B, with a maximum profit of $760.