Graphical LP Examples

Optimization · Axiom Academy

Solving a linear program graphically to find the optimal solution A furniture company makes chairs and tables. Each chair requires 2 hours of carpentry and 1 hour of finishing, yielding 40 profit. Each table requires 3 hours of carpentry and 2 hours of finishing, yielding 70 profit. Available resources: 120 hours of carpentry and 80 hours of finishing. Goal: Maximize profit by determining how many chairs and tables to produce. Excellent work! You've completed this graphical LP example. Here's what we learned: Formulation: Converting a real-world problem into mathematical form requires identifying decision variables, an objective function, and constraints. Graphical Method: For 2-variable LPs, we can visualize the feasible region and identify corner points (vertices) where optimal solutions occur. Corner Point Theorem: The optimal solution to a linear program always occurs at a vertex of the feasible region, so we only need to evaluate these points. Constraint Activity: At the optimal solution (24, 28), both the carpentry and finishing constraints are active (tight), meaning all available resources are used. This graphical approach provides intuition for the simplex method, which solves larger LPs by moving from vertex to vertex algebraically!

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