Section 3.3 Graphical Solutions of Linear Programming Problems Theorem 1: Solutions of Linear Programming Problems 1. If a linear programming problem has a solution, then it must occur at a corner point of the feasible set, S , associated with the problem. 2. If the objective function, P , is optimized at two adjacent corner points of S , then it is optimized at every point on the line segment joining the two points (infinitely many solutions). Theorem 2: Existence of a Solution Suppose we are given a linear programming problem with a feasible set S and an objective funtion P = ax + by. 1. If S is bounded then P has both a maximum and a minimum value on S . 2. If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided that the constraints defining S include the inequalities x ≥ 0 and y ≥ 0. 3. If S is empty, then the linear programming problem has no solution; that is, P has neither a maximum nor a minimum value. We say that the problem is infeasible. The Method of Corners 1. Graph the feasible set. 2. If the feasible set is nonempty, find the coordinates of all corner points of the feasible set. 3. Evaluate the objective function at each corner point. 4. Find the corner point(s) that renders the objective function a maximum (or minimum).