Linear programming is a process that takes linear inequalities relating to a situation and finds the optimal value possible given those constraints. It involves graphing the inequalities to form a feasibility region and then finding the corner points to test in the optimization equation to determine the highest and lowest possible values. For example, maximizing profits given material and labor limitations. The document then works through an example problem, graphing the constraints, finding the corner points, and testing them in the optimization equation to determine the maximum and minimum values of z.

1. Linear Programming: Introduction
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Linear Programming: Introduction (page 1 of 5)
Sections: Optimizing linear systems, Setting up word problems
Linear programming is the process of taking various linear inequalities relating to some situation, and
finding the "best" value obtainable under those conditions. A typical example would be taking the
limitations of materials and labor, and then determining the "best" production levels for maximal profits
under those conditions.
In "real life", linear programming is part of a very important area of mathematics called "optimization
techniques". This field of study (or at least the applied results of it) are used every day in the
organization and allocation of resources. These "real life" systems can have dozens or hundreds of
variables, or more. In algebra, though, you'll only work with the simple (and graphable) two-variable
linear case.
Siemens answers:
The general process for solving linear-programming exercises is to graph the inequalities (called the The digital factory
"constraints") to form a walled-off area on the x,y -plane (called the "feasibility region"). Then you Answers for industry.
www.siemens.com/answers
figure out the coordinates of the corners of this feasibility region (that is, you find the intersection
points of the various pairs of lines), and test these corner points in the formula (called the
"optimization equation") for which you're trying to find the highest or lowest value.
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Find the maximal and minimal value of z = 3x + 4y subject to the following Side Rail Type, Wide
constraints: Rail Type
Standard,Wide
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Solution
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The three inequalities in the curly braces are the constraints. The area of the plane that they Unified Storage Tune
mark off will be the feasibility region. The formula "z = 3x + 4y " is the optimization equation. I SQL Database upto
need to find the (x, y) corner points of the feasibility region that return the largest and smallest 80% Faster.
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values of z .
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My first step is to solve each inequality for the more-easily graphed equivalent forms: Material
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2. Linear Programming: Introduction
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To find the corner points -- which aren't always clear from the graph -- I'll pair the lines (thus
Tutoring ($$)
forming a system of linear equations) and solve:
This lesson may be printed out for
y = –( 1 / 2 )x + 7 y = –( 1 / 2 )x + 7 y = 3x your personal use.
y = 3x y=x–2 y=x–2
–( 1 / 2 )x + 7 = 3x –( 1 / 2 )x + 7 = x – 2
3x = x – 2
–x + 14 = 6x –x + 14 = 2x – 4 2x = –2
14 = 7x 18 = 3x Ads by Google
x = –1
2=x 6=x Math Tutor Algebra
y = 3(–1) = –3 Linear Programming
y = 3(2) = 6 y = (6) – 2 = 4
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corner point at (2, 6) corner point at (6, 4) corner pt. at (–1, –3)
So the corner points are (2, 6) , (6, 4) , and (–1, –3) .
Somebody really smart proved that, for linear systems like this, the maximum and minimum
values of the optimization equation will always be on the corners of the feasibility region. So, to
find the solution to this exercise, I only need to plug these three points into "z = 3x + 4y ".
(2, 6): z = 3(2) + 4(6) = 6 + 24 = 30
(6, 4): z = 3(6) + 4(4) = 18 + 16 = 34
(–1, –3): z = 3(–1) + 4(–3) = –3 – 12 = –15
Then the maximum of z = 34 occurs at (6, 4) ,
and the minimum of z = –15 occurs at (–1, –3) .
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Cite this article as: Stapel, Elizabeth. "Linear Programming: Introduction." Purplemath. Available from
http://www.purplemath.com/modules/linprog.htm. Accessed 14 November 2011
Copyright © 2006-2011 Elizabeth Stapel | About | Terms of Use Feedback | Error?
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