What is the maximum possible profit?
Question:A company produces two types of products, A and B. Each product requires different amounts of raw material and labor. The company has a limited supply of 500 units of raw material and 600 hours of labor. The production of one unit of product A requires 2 units of raw material and 3 hours of labor, while the production of one unit of product B requires 4 units of raw material and 2 hours of labor. Each unit of product A generates a profit of $50, and each unit of product B generates a profit of $40.
Determine:
How many units of product A and product B should the company produce to maximize profit?
What is the maximum possible profit?
Constraints:
Raw material constraint:
2
x
+
4
y
500
2x+4y500
Labor constraint:
3
x
+
2
y
600
3x+2y600
Non-negativity constraint:
x
0
,
y
0
x0,y0
Where:
x
x is the number of units of product A produced.
y
y is the number of units of product B produced.
Sample Solution
Linear Programming Problem
Objective Function: Maximize Profit: P = 50x + 40y
Constraints:
- Raw material constraint: 2x + 4y ≤ 500
- Labor constraint: 3x + 2y ≤ 600
- Non-negativity constraints: x ≥ 0, y ≥ 0
Graphical Solution
Step 1: Plot the constraints
- Raw material constraint: 2x + 4y = 500 => y = (500 - 2x) / 4
- Labor constraint: 3x + 2y = 600 => y = (600 - 3x) / 2
Step 2: Identify the feasible region
The feasible region is the area where all constraints are satisfied.
Step 3: Find the corner points
- (0, 0)
- (0, 125)
- (100, 75)
- (200, 0)
Step 4: Evaluate the objective function at each corner point
- P(0, 0) = 50(0) + 40(0) = 0
- P(0, 125) = 50(0) + 40(125) = 5000
- P(100, 75) = 50(100) + 40(75) = 8500
- P(200, 0) = 50(200) + 40(0) = 10000
Solution
The company should produce 200 units of product A and 0 units of product B to maximize profit. The maximum possible profit is $10,000.