Chapter 12 — Linear Programming

Chapter 12 of the Class 12 Maths NCERT textbook, "Linear Programming", teaches how to find the optimal (maximum or minimum) value of a linear objective function subject to linear constraints, solved using the graphical Corner Point Method.

• Turning real problems into optimisation — The chapter shows how everyday goals like maximising profit or minimising cost become a linear objective function bounded by linear constraints, teaching students to translate a word problem into a precise mathematical programme with non-negative restrictions.
• The feasible region — All the constraints together carve out a feasible region — a convex area whose every point is a valid solution. Understanding its shape, and whether it is bounded or unbounded, is what decides whether an optimum exists at all.
• The Corner Point Method — Key theorems guarantee the best value sits at a vertex of the feasible region, so the objective function need only be tested at the corner points. The chapter also flags unbounded regions, where a maximum or minimum may simply not exist.

1. 01A Linear Programming Problem seeks the optimal value (maximum or minimum) of a linear objective function Z = ax + by subject to linear constraints and non-negative restrictions.
2. 02The feasible region is the common region satisfying all constraints; it is always a convex region and every point in it is a feasible solution.
3. 03By Theorem 1, the optimal value of the objective function must occur at a corner point (vertex) of the feasible region.
4. 04By Theorem 2, if the feasible region is bounded, the objective function has both a maximum and a minimum, each occurring at a corner point.
5. 05The Corner Point Method involves finding all vertices of the feasible region, evaluating Z at each, and selecting the largest or smallest value; if two corner points yield the same optimal value, every point on the segment joining them is also optimal.
6. 06If the feasible region is unbounded, a maximum or minimum may not exist; this must be verified by checking whether the open half-plane beyond the candidate value shares points with the feasible region.