Download presentation

Presentation is loading. Please wait.

Published byTristan Threadgill Modified over 3 years ago

1
Chapter 5 The Simplex Method The most popular method for solving Linear Programming Problems We shall present it as an Algorithm

2
General Structure of Algorithms Initialise Perform a sequence of repetitive steps Check for desired results Stop No Yes Iterate

3
Construct a feasible extreme point Move along an edge to a better extreme point Is this point optimal ? Stop No Yes Iterate

4
Missing Details : Initialisation: – How do we represent a feasible extreme point algebraically? : Optimality Test: – How do we determine whether a given extreme point is optimal? : Iteration: – How do we move a long an edge to a better adjacent extreme point?

5
5.1 initialisation Transform the LP problem given in a form into a form. Transform the LP problem given in a standard form into a canonical form. This involves the introduction of, one for each functional constraint. This involves the introduction of slack variables, one for each functional constraint. Thus if we start with n variables and m functional constraints, we end up with and m functional constraints. Thus if we start with n variables and m functional constraints, we end up with n+m variables and m functional equality constraints.

6
Standard Form opt=max ~ b i ≥ 0, for all i.

7
Canonical Form

8
ObservationObservation The i-th measure the “distance” of the point x=(x 1,...,x n ) from the defining the i-th constraint (This is not a Euclidean distance). The i-th slack variable measure the “distance” of the point x=(x 1,...,x n ) from the hyperplane defining the i-th constraint (This is not a Euclidean distance). Thus, if the i-th slack variable is equal to the point x= (x 1,...,x n ) is. Otherwise it is not. Thus, if the i-th slack variable is equal to zero the point x= (x 1,...,x n ) is on the i-th hyperplane. Otherwise it is not. The “measure” the distance to the hyperplanes defining the respective constraints. The original variables “measure” the distance to the hyperplanes defining the respective non-negativity constraints.

9
ExampleExample x 3,x 4,x 5 are slack variables

10
Why do we do this? If we use the variables as a, we obtain a !!! If we use the slack variables as a basis, we obtain a feasible extreme point !!!

11
5.5.1 Definition A basic feasible solution is a basic solution that satisfies the constraint. A basic feasible solution is a basic solution that satisfies the non-negativity constraint. : Observation: A basic feasible solution is an of the feasible region. A basic feasible solution is an extreme point of the feasible region.Thus: involves constructing a using the. Initialisation involves constructing a basic feasible solution using the slack varaibles.

12
Example basic feasible solution: x =(0,0,40,30,15), namely Initial basic feasible solution: x =(0,0,40,30,15), namely x 1 = 0 x 2 = 0 x 1 = 0 x 2 = 0 x 3 = 40 x 4 = 30 x 5 =15 x 3,x 4,x 5 are slack variables

13
Summary of the Initialisation Step Select the slack variables as basic : Comments: – Simple – Not necessarily good selection: the first basic feasible solution can be (very) far from the optimal solution.

Similar presentations

OK

The Simplex Method. and Maximize Subject to From a geometric viewpoint : CPF solutions (Corner-Point Feasible) : Corner-point infeasible solutions 0.

The Simplex Method. and Maximize Subject to From a geometric viewpoint : CPF solutions (Corner-Point Feasible) : Corner-point infeasible solutions 0.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Appt only Ppt on formal education Ppt on stem cell technology Ppt on digital media broadcasting company Ppt on trial and error supernatural Ppt on art and craft movement pottery Ppt on law against child marriage in saudi Ppt on metro rail in india Ppt on indian textile industries in indonesia Ppt on traction rolling stock manufacturers