Sections 4 1 and 4 2 The Simplex

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 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, s 1=0, s 2 = 0, P=25

The Simplex Method for Maximization Problems 1. 2. 3. 4. Convert the constraints to equalities by adding slack variables Rewrite the objective function Construct the tableau Check for completion a. 5. Pivot a. b. c. 6. If all entries in the last row are non-negative then an optimal solution is found 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 1. 2. 3. 4. Convert the constraints to equalities by adding slack variables Rewrite the objective function Construct the tableau Check for completion a. 5. Pivot a. b. c. 6. If all entries in the last row are negative then an optimal solution is found 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