Maths: Master Sensitivity Analysis in Linear Programming for IB Math

Linear programming is the art of finding the best outcome—maximum profit, minimum cost—under a set of rigid limits. In this case, a furniture company juggles assembly and finishing hours to decide how many Standard and Deluxe chairs to make, with constraints like 2x + 3y ≤ 120 and x + 2y ≤ 70. The feasible region, a polygon formed by these lines and the axes, holds every possible production plan, but the optimal one always sits at a vertex—where two constraints meet. What makes this topic powerful is sensitivity analysis: how does the solution shift when a profit coefficient changes? Here, the objective function P = 50x + ky lets us test whether the current best point (the intersection of both constraints) remains optimal as k varies. By evaluating P at each vertex—origin, intercepts, and the intersection—you see that the winning corner depends on the slope of the profit line relative to the constraint slopes. If k tilts too far, another vertex takes over. This connection between geometry and algebra reveals not just *where* the optimum is, but *why* it stays or moves—turning a static answer into a dynamic understanding of trade-offs.