Resource Allocation Examples

Mathematical Modeling · Axiom Academy

EXAMPLE Resource Allocation Examples Worked examples demonstrating optimization techniques for allocating limited resources Scenario: A furniture company manufactures tables and chairs. Each table requires 4 hours of carpentry and 2 hours of finishing. Each chair requires 2 hours of carpentry and 1 hour of finishing. Resources: The company has 240 hours of carpentry time and 100 hours of finishing time available per week. Profits: Each table yields a profit of 70, and each chair yields a profit of 50. Goal: Determine how many tables and chairs to produce to maximize weekly profit. x = number of tables to produce per week y = number of chairs to produce per week Constraints: x >= 0, y >= 0 (non-negativity) Objective: Maximize profit P = 70x + 50y Carpentry: 4x + 2y = 240, or y = 120 - 2x (intercepts at (60, 0) and (0, 120)) Finishing: 2x + y = 100, or y = 100 - 2x (intercepts at (50, 0) and (0, 100)) The feasible region is bounded by these constraints and the axes. We evaluate the objective function at each corner point. Corner Points and Objective Values Note: The point (0, 120) violates the finishing constraint (0 + 120 > 100), so it's not feasible. The feasible corner at y-axis is (0, 100). Step 1: Find Intersection of Constraints Set 4x + 2y = 240 and 2x + y = 100 From the second equation: y = 100 - 2x Substitute: 4x + 2(100 - 2x) = 240 4x + 200 - 4x = 240, which gives 200 = 240 (contradiction!) The lines are parallel. The finishing constraint is more restrictive.

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.