Carnell Electronics, Inc. - Production Optimization

Problem Definition

Carnell Electronics produces two calculator models: X15 and Y10. The company wants to maximize profit given production constraints in three departments:

Molding: 50,000 minutes available monthly
X15 requires 1 min, Y10 requires 1 min
Assembly: 200,000 minutes available monthly
X15 requires 5 min, Y10 requires 2.5 min
Soldering: 300,000 minutes available monthly
X15 requires 3 min, Y10 requires 8 min
Profit Contribution:
X15: $10 per unit, Y10: $8 per unit

Linear Programming Formulation

Let:

Objective Function (Maximize): Z = 10x + 8y

Subject to Constraints:

  1. Molding: x + y ≤ 50,000
  2. Assembly: 5x + 2.5y ≤ 200,000
  3. Soldering: 3x + 8y ≤ 300,000
  4. Non-negativity: x ≥ 0, y ≥ 0

Graphical Solution

The feasible region is defined by the intersection of all constraints:

We find the corner points of the feasible region:

Point X (X15 units) Y (Y10 units) How determined
A 0 0 Origin
B 40,000 0 Assembly constraint (y=0)
C 30,000 20,000 Intersection of Molding and Assembly
D 20,000 30,000 Intersection of Molding and Soldering
E 0 37,500 Soldering constraint (x=0)

Objective Function Evaluation

We calculate the profit at each corner point:

Point X15 units Y10 units Profit Calculation Total Profit
A 0 0 10(0) + 8(0) $0
B 40,000 0 10(40,000) + 8(0) $400,000
C 30,000 20,000 10(30,000) + 8(20,000) $460,000
D 20,000 30,000 10(20,000) + 8(30,000) $440,000
E 0 37,500 10(0) + 8(37,500) $300,000

Optimal Solution

The maximum profit of $460,000 is achieved at point C with:

Constraint utilization at optimal solution:

Department Available Minutes Used Minutes Slack Minutes
Molding 50,000 30,000 + 20,000 = 50,000 0
Assembly 200,000 5(30,000) + 2.5(20,000) = 200,000 0
Soldering 300,000 3(30,000) + 8(20,000) = 250,000 50,000