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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.

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

Calculator Example

- Problem 12

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

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