Corner Point Method: Concepts, Formulas and Examples

The corner point method is the simplest reliable way to solve a linear programming problem once the feasible region is known. You list every corner point (vertex) of the feasible region, substitute each one into the objective function, and pick the best value.

It matters because it is backed by the fundamental theorem of linear programming: if an optimum exists, it occurs at a corner point of the feasible region. That single fact converts optimisation into a short mechanical checklist that never misses the answer.

Key Concepts

• Corner point (vertex): a point where two boundary lines of the feasible region meet.
• Convex polygon: the feasible region of a two-variable LPP, possibly unbounded.
• Fundamental theorem: a linear objective over a convex polygon attains its maximum and minimum at corner points.
• Multiple optimal solutions: when the objective line is parallel to an edge, every point on that edge is optimal.

Formulas

• Objective function: Z = ax + by, evaluated at each corner.
• Corner from two lines: solve a₁x + b₁y = c₁ and a₂x + b₂y = c₂ simultaneously.
• Standard corners: always include axis intercepts and the origin if they satisfy the constraints.
• Comparison rule: maximum = largest Z among corners; minimum = smallest Z among corners.

Worked Example

Minimise Z = 2x + 3y subject to x + y ≥ 5, 2x + y ≥ 7, x ≥ 0, y ≥ 0. The boundary lines are x + y = 5 and 2x + y = 7. Their intersection: subtracting x + y = 5 from 2x + y = 7 gives x = 2, and then y = 3, so one corner is A(2, 3). The axes give more corners: with x = 0, the constraint 2x + y ≥ 7 gives y = 7, so B(0, 7); with y = 0, the constraint x + y ≥ 5 gives x = 5, so C(5, 0). Check feasibility: A(2, 3) gives 2 + 3 = 5 ≥ 5 and 2(2) + 3 = 7 ≥ 7; B(0, 7) gives 0 + 7 = 7 ≥ 5 and 0 + 7 = 7 ≥ 7; C(5, 0) gives 5 + 0 = 5 ≥ 5 and 2(5) + 0 = 10 ≥ 7 — all feasible. Evaluate Z = 2x + 3y: at A, Z = 2(2) + 3(3) = 4 + 9 = 13; at B, Z = 2(0) + 3(7) = 21; at C, Z = 2(5) + 3(0) = 10. The minimum is Z = 10 at (5, 0). The region extends upward without bound, but the minimum is still attained at a corner, exactly as the theorem promises.

Common Misconceptions

• Only line intersections count as corners. Axis intercepts like (0, 7) and (5, 0) are corners too; skipping them loses the true optimum.
• The best value must be unique. If the objective line coincides with an edge, every point on that edge gives the same optimum — infinitely many optimal solutions.
• Corners from infeasible intersections. Always verify each candidate satisfies every constraint before evaluating Z.

Real-World Applications

• Cost minimisation in manufacturing, such as cheapest raw-material blends meeting quality standards.
• Exam-standard solution technique for every LPP in Class 12 mathematics.
• Foundation of the simplex method used by optimisation software for thousands of variables.

Find the corners, evaluate Z at each, and choose the best — the corner point method is short, exact, and guaranteed by the fundamental theorem of linear programming.