Feasible Region — Bounded and Unbounded

Once an LPP is formulated, its constraints are graphed exactly as in Std XI: each inequation is a half-plane, and the region satisfying all of them at once is the set of allowable plans.

Feasible and infeasible solutions

Feasible solution and feasible region

A feasible solution is any point $(x,y)$ that satisfies all the constraints and the non-negativity restrictions simultaneously. The collection of every feasible solution is the feasible region — the common (overlapping) region of all the half-planes.

A point that violates even one constraint is an infeasible solution and lies outside the feasible region. Because $x\ge 0,\ y\ge 0$ appear in almost every commercial LPP, the feasible region normally lies in the first quadrant.

Corner (vertex) points

Corner point

A corner point (vertex) of the feasible region is a point where two of its boundary lines intersect. Corner points are found by solving the relevant boundary lines two at a time as simultaneous equations.

Corner points are the heart of the graphical method: the optimum of a linear objective always occurs at one of them (Section 3).

Bounded vs. unbounded regions