The optimization engine behind supply chains and airline scheduling. Constraints carve out a region, and the best answer always hides at a corner.
Linear ProgrammingLive
optimize over a feasible region
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Linear programming maximizes a linear objective subject to linear constraints. The constraints carve out a convex feasible region (shaded), and the optimum always sits at a corner. Slide the objective coefficients and watch the optimal vertex jump between corners — the geometric idea behind the simplex algorithm that runs global logistics and finance.
Reading this result: Maximizing 3·x + 2·y lands the optimum at the corner (6.0, 4.0) worth 26.0 — a linear objective over a convex region always peaks at a vertex, so sliding c₁/c₂ just makes the solution hop between corners.
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How it works
Linear programming maximizes a linear objective subject to linear inequality constraints. Those constraints define a convex feasible region, and a fundamental theorem guarantees the optimum lies at one of its vertices. Sliding the objective coefficients moves the optimal corner — the geometric intuition behind the simplex method used across operations research.
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The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.