Sections 4.1 and 4.2

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# Sections 4.1 and 4.2 - PowerPoint PPT Presentation

Sections 4.1 and 4.2. The Simplex Method: Solving Maximization and Minimization Problems. Simplex Method. The Simplex Method is a procedure for solving LP problems

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### Sections 4.1 and 4.2

The Simplex Method: Solving Maximization and Minimization Problems

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)
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.
Preparing a Standard Maximization Problem
• Convert the inequality constraints into equality constraints using slack variables.

Maximize

Maximize

s.t.

s.t.

Building a Tableau
• Rewrite the objective function
• Write a tableau

Constraints

Objective

Function

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
Make a Unit Column
• Using the row operations (just like Gauss-Jordan), make a unit column.
When are we done?
• Repeat pivots until all entries in the last row are non-negative
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
The Simplex Method for Maximization Problems
• Convert the constraints to equalities by adding slack variables
• Rewrite the objective function
• Construct the tableau
• Check for completion
• If all entries in the last row are non-negative then an optimal solution is found
• Pivot
• Select the column with the largest negative entry.
• Select the row with the smallest ratio of constant to entry
• Make the selected column a unit column using row operations
• Go to step 4
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
Homework
• Section 4-1, page 238
• 11, 13, 15, 21
Word Problem Examples
• Problem 29
• Problem 32
Homework
• Section 4-1, Page 238
• 31, 33, 35, 39
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.
Solving Standard Minimization Problems
• Convert the constraints to equalities by adding slack variables
• Rewrite the objective function
• Construct the tableau
• Check for completion
• If all entries in the last row are negative then an optimal solution is found
• Pivot
• Select the column with the largest positive entry.
• Select the row with the smallest ratio of constant to entry
• Make the selected column a unit column using row operations
• Go to step 4
Examples
• Page 257
• Problem 1
• Problem 22
Homework
• Section 4.2 – Page 257
• 1- 5 odd
• 21, 23, 25