What is the maximum possible profit?

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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.

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