![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.
Linear Guides
Miniature
Find the maximal and minimal value of z = 3x + 4y subject to the following Side Rail Type, Wide
constraints: Rail Type
Standard,Wide
Standard-Blocks&Rails
in.misumi-ec.com/
Data Management
Solution
Do More With Your
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.
www.EMCIndia.co.in/vnx
values of z .
IIT JEE Maths
My first step is to solve each inequality for the more-easily graphed equivalent forms: Material
Theory, IIT Solutions
in Video form
Prepared by Topper.
Free Demo
ExponentEducation.com/IIT…
Worksheets &
Exercises
It's easy to graph the system: Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved Tests, Lessons,
Animations, Videos.
http://www.purplemath.com/modules/linprog.htm[11/14/2011 1:30:34 PM]](https://image.slidesharecdn.com/linearprogrammingintroduction-120202074836-phpapp02/85/Linear-programming-introduction-1-320.jpg)
![Linear Programming: Introduction
Free NCERT Solutions,
Join Free!
www.MeritNation.com/Work…
Purplemath:
Linking to this site
Printing pages
Donating
School licensing
Reviews of
Internet Sites:
Free Help
Practice
Et Cetera
The "Homework
Guidelines"
Study Skills Survey
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
Math
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?
Ads by Google Solve Math Equation Ti 83 Math Algebra 2 Equations Radical Equations
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- 1. Linear Programming: Introduction The Purplemath Forums Search powered by FreeFind Helping students gain understanding and self-confidence in algebra Return to the Lessons Index | Do the Lessons in Order | Get "Purplemath on CD" for offline use | Print-friendly page 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. Linear Guides Miniature Find the maximal and minimal value of z = 3x + 4y subject to the following Side Rail Type, Wide constraints: Rail Type Standard,Wide Standard-Blocks&Rails in.misumi-ec.com/ Data Management Solution Do More With Your 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. www.EMCIndia.co.in/vnx values of z . IIT JEE Maths My first step is to solve each inequality for the more-easily graphed equivalent forms: Material Theory, IIT Solutions in Video form Prepared by Topper. Free Demo ExponentEducation.com/IIT… Worksheets & Exercises It's easy to graph the system: Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved Tests, Lessons, Animations, Videos. http://www.purplemath.com/modules/linprog.htm[11/14/2011 1:30:34 PM]
- 2. Linear Programming: Introduction Free NCERT Solutions, Join Free! www.MeritNation.com/Work… Purplemath: Linking to this site Printing pages Donating School licensing Reviews of Internet Sites: Free Help Practice Et Cetera The "Homework Guidelines" Study Skills Survey 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 Math 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) . Top | 1 | 2 | 3 | 4 | 5 | Return to Index Next >> 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? Ads by Google Solve Math Equation Ti 83 Math Algebra 2 Equations Radical Equations http://www.purplemath.com/modules/linprog.htm[11/14/2011 1:30:34 PM]