Download Applications of Linear Programming

Avon High School
Section: 8.6
ACE COLLEGE ALGEBRA II - NOTES
Linear Programming
Mr. Record: Room ALC-129
Semester 2 - Day 9
Basics of Linear Programming
A region consisting of the overlapping parts of two or more graphs of inequalities in a system is sometimes
called the region of feasible solutions or just the feasible region.
Sometimes a feasible region will be bounded, or
enclosed on all sides by boundary lines. Sometimes a
feasible region will be unbounded.
A corner point is a point in the feasible region where two boundary lines intersect.
Linear Programming Terminology
Objective Function: A function for which we are trying to find a MAXIMUM or MINIMUM value.
Usually the objective function represents something like “profit” or “cost”.
Constraints:
A set of restrictions on the problem, represented by a system of linear inequalities.
Feasible Region:
Graphically, the set of all points that satisfy all of the constraints. (the “overlap”)
Corner Points:
The vertices of the feasible region, where two boundary lines intersect.
Corner Point Theorem:
Example 1
If an optimum value (either a maximum or a minimum) of the objective function
exists, then it will occur at one or more of the corner points of the feasible
region.
Finding max/min Value of the Objective Function
Find the value of the objective function at each corner of the
graphed region to the right. What is the maximum value of
the objective function? What is the minimum value of the
objective function?
Objective Function
.
z  3x  2 y
Example 2
Graphing a Linear Inequality in Two Variables
Find the maximum value of the objective function z  3 x  5 y subject to the constraints
x  0, y  0, x  y  1, x  y  6.
Linear Programming Procedure
1.
2.
3.
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5.
(If necessary) Write the objective function and all necessary constraints.
Graph the feasible region.
Identify all corner points.
Find the value of the objective function at each corner point.
For a bounded region, the solution is given by the corner point producing the optimum value of
the objective function.
6. For an unbounded region, check that a solution actually exists. If it does, it will occur at a corner
point.
Applications of Linear Programming
Example 3
A 4-H member raises goats and pigs. She wants to raise no more than 16 animals, including no more
than 10 goats. She spends $25 to raise a goat and $75 to raise a pig, and she has $900 available for the
project. The 4-H member wishes to maximize her profits. Each goat produces $12 in profit and each
pig $40 in profit. How many of each type of animal should she raise?
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Example 4
A banquet hall offers two types of tables for rent: 6-person rectangular tables at a cost of $27 each
and 10-person round tables at a cost of $46 each. You would like to rent the hall for a wedding
banquet and need tables for 230 people. The room can have a maximum of 39 tables and the hall only
has 15 rectangular tables available. How many of each type of table should be rented to minimize cost?
What is the minimum cost?
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