Which of the following is the correct linear programming model for minimizing production cost in the 3-product problem?

• A. Min Z = 5 X1 + 4 X2 + 3 X3 Subject to X1 + X2 + X3 ≤ 10; 45 X1 + 50 X2 + 55 X3 ≤ 480; X1, X2, X3 ≥ 0
• B. Max Z = 5 X1 + 4 X2 + 3 X3 Subject to X1 + X2 + X3 = 10; 45 X1 + 50 X2 + 55 X3 ≤ 480; X1, X2, X3 ≥ 0
• C. Min Z = 5 X1 + 4 X2 + 3 X3 Subject to X1 + X2 + X3 = 10; 45 X1 + 50 X2 + 55 X3 ≤ 480; X1, X2, X3 ≥ 0
• D. Min Z = 5 X1 + 4 X2 + 3 X3; X1 + X2 + X3 = 10; 45 X1 + 50 X2 + 55 X3 ≤ 480

Answer: (C)

When you minimize cost with several products, you choose how many units of each product to produce (X1, X2, X3) to meet a total demand while staying within available capacity. The total cost is Z = 5X1 + 4X2 + 3X3, so you want to minimize that. The demand constraint requires producing exactly 10 units in total, so X1 + X2 + X3 = 10. The production time constraint uses 45, 50, and 55 minutes per unit for the three products, with 480 minutes available: 45X1 + 50X2 + 55X3 ≤ 480. You also need nonnegativity: X1, X2, X3 ≥ 0. This setup is correct because it enforces meeting the 10-unit demand, respects the time limit, and minimizes cost. Options that use ≤ for the total or that maximize the objective would not meet the problem’s requirements or goal.