Sections 4.1 and 4.2

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

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

Sections 4.1 and 4.2