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).

1. Find the maximum and/or minimum value(s) of the objective function on the feasible set S.

Z = 5x+ 6y

2. Solve the linear programming problem by the method of corners.

Maximize P = 3x+ 5y

subject to 2x+ y 16

2x+ 3y 24

y 7

x � 0

y � 0

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3. Solve the linear programming problem by the method of corners.

Maximize P = x+ 2y

subject to x+ 2y 4

2x+ 3y � 12

x � 0

y � 0

4. Solve the linear programming problem by the method of corners.

Minimize C = 3x+ 6y

subject to x+ 2y � 40

x+ y � 30

x � 0

y � 0

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Solutions