MATH 141 501
Section 3.3 Lecture Notes
So far, we have learned how to graphically solve systems of linear inequalities,
and to formulate linear programming problems. Now we combine these two skills
to solve linear programming problems. Consider the following example.
Maximize P = 3x + 2y
subject to 2x + 3y ≤ 12
2x + y ≤ 8
x ≥ 0, y ≥ 0
The solution of the system of linear inequalities is given below. It is called
the feasible set or feasible region for the problem. Each point in S is called a
feasible solution.
WE WANT THE SOLUTION WHICH MAKES P A MAXIMUM!
This is called the optimal solution. How do we find it?
1
Solving Linear Programming Problems
We use the following facts.
(We just accept these as true, and we won’t explain why.)
Theorem 1 (Solution(s) of Linear Programming Problems)
If a linear programming problem has a solution, then it must occur at a corner
point of the feasible set S for the problem.
If the objective function P is optimized at two adjacent corner points of S, it
is optimized at every point on the line segment joining these corner points. In
this case, there are infinitely many solutions to the problem.
Theorem 2 (Existence of a Solution)
Given a linear programming problem with a feasible set S and a linear objective
function P = ax + by
a. If S is bounded, then P has both a maximum and a minimum value on S.
b. If S is unbounded and both a and b are nonnegative, then P has a minimum value on S, as long as the constraints defining S include the inequalities x ≥ 0 and y ≥ 0.
c. If S is the empty set, then the linear programming problem has no solution.
2
Method of Corners
The procedure we will use to solve linear programming problems is known
as The Method of Corners.
Theorem 2 (Existence of a Solution)
1. Graph the feasible set S.
2. Find the coordinates of all corner points of the feasible set.
3. Evaluate the objective function at each corner point.
4. Find the vertex that gives the optimal value of the objective function.
(a) If there is only one, then this is the unique solution to the problem.
(b) If the objective function is optimized at two adjacent corner points
of S, there are infinitely many solutions. The solutions are all points
on the line segment between these two vertices.
3
Example 1
from ( Finite Mathematics for the Managerial, Life, and Social Sciences,
11th edition, by Tan).
Maximize P = x + 1.2y
subject to 2x + y ≤ 180
x + 3y ≤ 300
x ≥ 0, y ≥ 0
Solution:
Step 1: Find the feasible set (shown below).
Step 2: Identify the corner points: (0,0), (90,0), (48,84), (0,100).
4
Example 1, continued
Step 3: Find the values of P at the vertices (corner points).
Vertex
(0,0)
(90,0)
(48,84)
(0,100)
P = x + 1.2y
Step 4: Find the solution and interpret the results.
5
Example 2
based on example from ( Finite Mathematics for the Managerial, Life,
and Social Sciences, 11th edition, by Tan).
Minimize 5x + 7y
subject to 4x + y ≥ 240
2x + 3y ≥ 425
x + 3y ≥ 300
x ≥ 0, y ≥ 0
Step 1: Find the feasible set (shown below.)
Theorem 2 (Existence of a Solution)
Given a linear programming problem with a feasible set S and a linear objective
function P = ax + by
b.
If S is unbounded and both a and b are nonnegative,
then P has a minimum value on S, as long as the
constraints defining S include the inequalities x ≥ 0
and y ≥ 0.
6
Example 2, continued
Step 2: Find the corner points (vertices) of the feasible set.
Step 3: Evaluate the corner points in the objective function.
Vertex
(0,240)
(30,120)
(120,60)
(300,0)
C = 5x + 7y
Step 4: Solve the problem (and interpret, if necessary.)
7
Infinitely Many Solutions
Example: ( based on Homework problem from Finite Mathematics for
the Managerial, Life, and Social Sciences, 11th edition, by Tan).
Find the maximum and minimum value(s) of the objective function Z on
the feasible set S.
Step 1: Find feasible region and vertices.
(Can find immediately from picture.)
Step 2: Find the values of objective function at corner points.
Corner Point
(0,6)
(4,10)
(12,8)
(15,0)
Value of Z
24
44
44
15
8
Infinitely Many Solutions, continued
Step 3: Find solutions.
Theorem 1 (Solution(s) of Linear Programming Problems)
If a linear programming problem has a solution, then it must occur at a corner
point of the feasible set S for the problem.
If the objective function P is optimized at two
adjacent corner points of S, it is optimized at every point on the line segment joining these corner
points. In this case, there are infinitely many solutions to the problem.
is a maximum value!
is a minimum value.
9
No Solution
There are two ways that a linear programming problem can have no solution.
Unbounded We are asked to maximize an objective function whose feasible
region is unbounded.
Infeasible The feasible region is empty.
In practice, it is
10