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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

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sections 4 1 and 4 2

Sections 4.1 and 4.2

The Simplex Method: Solving Maximization and Minimization Problems

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

Maximize

Maximize

s.t.

s.t.

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

Constraints

Objective

Function

choosing a simplex pivot
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
Make a Unit Column
  • Using the row operations (just like Gauss-Jordan), make a unit column.
when are we done
When are we done?
  • Repeat pivots until all entries in the last row are non-negative
interpreting the results
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
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
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
Homework
  • Section 4-1, page 238
    • 11, 13, 15, 21
word problem examples
Word Problem Examples
  • Problem 29
  • Problem 32
homework15
Homework
  • Section 4-1, Page 238
    • 31, 33, 35, 39
standard minimization problem
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
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
Examples
  • Page 257
    • Problem 1
    • Problem 22
homework19
Homework
  • Section 4.2 – Page 257
    • 1- 5 odd
    • 21, 23, 25
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