10.8 Linear Programming 2011
February 16, 2011
10.8 Linear Programming
Objectives: • Set up linear programming problems.
• Solve linear programming problems.
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10.8 Linear Programming 2011
February 16, 2011
Linear Programming
Linear Programming was originally used for solving problems involving resource allocation of goods and materials for the U.S. Air Force during World War II.
Now, linear programming is used to solve various problems like optimizing airline scheduling and minimizing costs for businesses.
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10.8 Linear Programming 2011
February 16, 2011
Definitions:
Linear programming is a technique that identifies the minimum or maximum value of some quantity.
This quantity is measured with an objective function.
Limits on variables in the objective function are constraints, written as linear inequalities.
The constraints form a system of inequalities and the
feasible region contains all the points that satisfy those constraints.
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10.8 Linear Programming 2011
February 16, 2011
Example #1: Given the objective function z = 3x + 2y and the following constraints, find the values of x and y that maximize z.
Constraints: x > 0 , y > 0 , x + y < 7 , ­3x + 2y > ­ 6
Step 1: Graph the constraints.
Step 2: Find the vertices.
A (0, 0)
B (2, 0)
C (4, 3)
D (0, 7)
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10.8 Linear Programming 2011
February 16, 2011
Example #1: Given the objective function z = 3x + 2y and the following constraints, find the values of x and y that maximize z.
Step 3: Evaluate z at each vertex.
z = 3x + 2y
A (0, 0)
z = 3(0) + 2(0) = 0
B (2, 0)
z = 3(2) + 2(0) = 6
C (4, 3)
z = 3(4) + 2(3) = 18
D (0, 7)
z = 3(0) + 2(7) = 14
Therefore, (4, 3) maximizes z at 18.
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10.8 Linear Programming 2011
February 16, 2011
Homework: page 820 (1 ­ 21)
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