Class_Notes_Part_2

Class_Notes_Part_2 - Class Notes for DMOR: Set 2 Focus on...

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Class Notes for DMOR: Set 2 Focus on solving linear programs by the simplex method, and what kind of information we can obtain from them. 1
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Goal: Solve problems by the simplex method First step: transform any problem to have equality constraints and variables that are nonnegative. This is called “standard form.” We have already learned how to make equality constraints by adding slack variables. What about nonpositive variables? What about “free” (or “unconstrained”) variables, which can take on either sign? • If you have a variable x that is restricted as x ≤ 0, substitute x with a variable − x: everywhere you see x, replace it with − x, and include the constraint x ≥ 0. • If you have a variable x that is unrestricted, create two variables, x + and x , both of which are ≥ 0. Everywhere you see x, replace it with x + − x . Variable x + represents the positive component of x, while x represents the negative component of x. Example: Minimize 2x 1 − x 2 + 3x 3 subject to x 1 + x 2 − 2x 4 = 2 2x 1 + x 3 + x 4 = 6 x 1 , x 2 ≥ 0, x 3 ≤ 0, x 4 unrestricted Standard form: min 2 x 1 x 2 3 x 3 s.t. x 1 + x 2 2 x 4 + + 2 x 4 = 2 2 x 1 x 3 + x 4 + x 4 = 6 all vars nonnegative 2
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Transformation to “standard form” How to transform problems to have equality constraints? Add a (nonnegative) slack variable for ≤ constraints and subtract a (nonnegative) slack variable for ≥ constraints. x 1 + x 2 ≥ 2 2x 1 + x 2 ≤ 6 x 1 + x 2 ≤ 4 x 1 , x 2 ≥ 0 (do not write slacks for these!) x 1 + x 2 s 1 = 2 2 x 1 + x 2 + s 2 = 6 x 1 + x 2 + s 3 = 4 All x, s nonnegative What does this look like graphically? We cannot draw in five dimensions, but we can show x 1 , x 2 , s 1 , s 2 , and s 3 on a two-dimensional drawing. 3
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Extreme Points (aka: corner points, basic feasible solutions) Look at the corner points in this graph. What do you notice about the number of variables that equal to zero? x 1 + x 2 s 1 = 2 2 x 1 + x 2 + s 2 = 6 x 1 + x 2 + s 3 = 4 All x, s nonnegative How do you compute the variable values at each of the extreme points? Set x 1 = x 2 = 0, and solve Set x 1 = s 1 = 0, and solve Set s 2 = s 3 = 0, and solve 10 of these combinations 10 of these combinations 10 of these combinations 10 of these combinations Then how can we find all extreme points by just looking at the equations, and not the graph itself? x 1 + x 2 = 2 2 x 1 + x 2 + s 2 = 6 x 1 + x 2 = 4 x 1 + x 2 = 2 x 2 + s 2 = 2 0 x 1 + 0 x 2 + 0 s 2 = -2 4
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Two solution methods? We now have two methods for solving LP s. Method One: graph it. But what if the problem is not two-dimensional? (Or three-dimensional, if you re an accurate drawer?) Important Theorem : An optimal solution to a linear program, if it exists, will exist at an extreme point. • Why?
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This note was uploaded on 02/25/2010 for the course ESI 6314 taught by Professor Vladimirlboginski during the Fall '09 term at University of Florida.

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Class_Notes_Part_2 - Class Notes for DMOR: Set 2 Focus on...

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