Download presentation

Presentation is loading. Please wait.

Published byBrooklynn Burdsall Modified over 2 years ago

1
Sections 4.1 and 4.2 The Simplex Method: Solving Maximization and Minimization Problems

2
Simplex Method The Simplex Method is a procedure for solving LP problems It moves from vertex to vertex of the solution space (convex hull) until an optimal (best) solution is found (there may be more than one optimal solution)

3
Standard Maximization Problem The objective function is to be maximized. All the variables involved in the problem are nonnegative. Each constraint may be written so that the expression with the variables is less than or equal to a nonnegative constant.

4
Preparing a Standard Maximization Problem Convert the inequality constraints into equality constraints using slack variables. Maximize s.t. Maximize s.t.

5
Building a Tableau Rewrite the objective function Write a tableau Constraints Objective Function

6
Choosing a Simplex Pivot Select a pivot –Select the column with the largest negative entry in the last row (objective function) –Select the row with the smallest ratio of constant to entry

7
Make a Unit Column Using the row operations (just like Gauss- Jordan), make a unit column.

8
When are we done? Repeat pivots until all entries in the last row are non-negative

9
Interpreting the Results Unit Columns (zeros in last row) Non-unit Columns (no zeros in last row) x=1, y=5, s1=0, s2 = 0, P=25

10
The Simplex Method for Maximization Problems 1.Convert the constraints to equalities by adding slack variables 2.Rewrite the objective function 3.Construct the tableau 4.Check for completion a.If all entries in the last row are non-negative then an optimal solution is found 5.Pivot a.Select the column with the largest negative entry. b.Select the row with the smallest ratio of constant to entry c.Make the selected column a unit column using row operations 6.Go to step 4

11
Using the TI-83 Calculator The PIVOT program Enter the tableau into matrix D Run the PIVOT program –Asks to pivot or quit –Select pivot –Asks for row and column –Enter pivot row and column –Continue until an optimal solution is found

12
Calculator Example Problem 12

13
Homework Section 4-1, page 238 –11, 13, 15, 21

14
Word Problem Examples Problem 29 Problem 32

15
Homework Section 4-1, Page 238 –31, 33, 35, 39

16
Standard Minimization Problem The objective function is to be minimized. All the variables involved in the problem are nonnegative. Each constraint may be written so that the expression with the variables is greater than or equal to a nonnegative constant.

17
Solving Standard Minimization Problems 1.Convert the constraints to equalities by adding slack variables 2.Rewrite the objective function 3.Construct the tableau 4.Check for completion a.If all entries in the last row are negative then an optimal solution is found 5.Pivot a.Select the column with the largest positive entry. b.Select the row with the smallest ratio of constant to entry c.Make the selected column a unit column using row operations 6.Go to step 4

18
Examples Page 257 –Problem 1 –Problem 22

19
Homework Section 4.2 – Page 257 –1- 5 odd –21, 23, 25

Similar presentations

OK

MIT and James Orlin © 2003 1 Chapter 3. The simplex algorithm Putting Linear Programs into standard form Introduction to Simplex Algorithm.

MIT and James Orlin © 2003 1 Chapter 3. The simplex algorithm Putting Linear Programs into standard form Introduction to Simplex Algorithm.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on new invention of computer technology Ppt on fire extinguisher types pictures Download ppt on islands of india Ppt on needle stick injury product Ppt on need for conservation of natural resources Ppt on multipurpose river valley projects Ppt on product specification file Ppt on project spark Ppt on working of lcd tv Ppt on centring ig