Homework 2 - Class Notes for DMOR: Set 2 Focus on solving...

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Unformatted text preview: 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 Goal: Solve problems by the simplex method First step: transform any problem to have equality constraints and variables that are non- negative. 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- ˆ x : everywhere you see x , replace it with- ˆ x , and include the constraint ˆ 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 2 x 1- x 2 + 3 x 3 subject to x 1 + x 2- 2 x 4 = 2 2 x 1 + x 3 + x 4 = 6 x 1 ,x 2 ≥ , x 3 ≤ , x 4 unrestricted 2 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 2 x 1 + x 2 ≤ 6 x 1 + x 2 ≤ 4 x 1 ,x 2 ≥ 0 (do not write slacks for these!) 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 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? How do you compute the variable values at each of the extreme points? Then how can we find all extreme points by just looking at the equations, and not the graph itself? 4 Two solution methods? We now have two method 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? If not, suppose we have an optimal solution at a non-extreme point. Because we’re not at an extreme point, there’s a direction in which we can move both forwards and backwards. The objective must be “flat” in this direction (if not, we could improve our objective, and we wouldn’t be at an optimal solution!). Move until we hit a constraint. Repeat until we hit an extreme point. • Does this mean that all optimal solutions exist at extreme points ? No! Method Two : Find all extreme points and compute the best objective function found. The best one is an optimal solution (unless the problem is unbounded, but we can fix that, too)....
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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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Homework 2 - Class Notes for DMOR: Set 2 Focus on solving...

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