4.1 Slack Variables and the Simplex Tableau

1. 4.1 Slack Variables and the Simplex Tableau • Standard Form • Slack Variable • Solutions Using Slack Variables • Group I and Group II Variables • Simplex Tableau • Solution Corresponding to Simplex Tableau

2. Standard Form • A linear programming problem is in standard form if: • The objective function is to be maximized; • Each variable is constrained to be greater than or equal to 0; • All other constraints are of the form • [linear polynomial] < [nonnegative constant].

3. Example Standard Form • The following is in standard form:

4. Slack Variable • A nonnegative variable that “takes up the slack” between the left-hand side of an inequality and the right-hand side is called a slack variable. The slack variable changes an inequality into an equation.

5. Example Slack Variable • Use slack variables to change the following into a system of equations.

6. Example Slack Variable - Answer • For u, v, w> 0 and M as large as possible: • 6x + 3y< 96 becomes 6x + 3y + u = 96, • x + y< 18 becomes x + y + v = 18, • 2x + 6y< 72 becomes 2x + 6y + w = 72, • and • Maximize 80x + 70y becomes -80x - 70y + M = 0 for M as large as possible.

7. Infinite Number of Solutions • The system of equations using the slack variables have an infinite number of solutions since there are more variables than equations.

8. Example Infinite Number of Solutions • Find two solutions to the following system of equations with u, v, and w> 0.

9. Example Solutions - Answer (1) • Write the augmented matrix for the system If x = y = 0, then a solution is u = 96, v = 18, w = 72 and M = 0.

10. Example Solutions - Answer (2) • If we pivot about a11, we have If y = u = 0, then another solution is x = 16, v = 2, w = 40 and M = 1280.

11. Group I and Group II Variables • In the preceding solutions, two of the variables were chosen to be 0 and the other 4 were solved for using the equations. The variables chosen to be 0 are Group I variables. The variables solved for once the Group I variables were chosen are called Group II variables.

12. Example Group I & II Variables • In the two solutions from the previous example, identify the Group I and Group II variables.

13. Simplex Tableau • You will notice that both matrices from which a solution was obtained in the previous example had the columns of the identity matrix present in some order. Such a matrix is called a simplex tableau.

14. Example Simplex Tableau • Find the columns of the identity matrix in the following two simplex tableaux.

15. Solution Corresponding to Simplex Tableau • The solution corresponding to a given simplex tableau is obtained by choosing those variables whose columns correspond to the columns of the identity matrix as Group II variables. The other variables are Group I variables.

16. Example Solution from Simplex Tableau • Find the solution corresponding to Suppose the columns correspond to x, y, u, v, w and M. Choose u = w = 0, then x = 8, y = 9, v = 0and M = 7.

17. Summary Section 4.1 - Part 1 • A linear programming problem is in standardform if the linear objective function is to be maximized, every variable is constrained to be nonnegative, and all other constraints are of the form [linear polynomial] < [nonnegative constant].

18. Summary Section 4.1 - Part 2 • To form the initial simplex tableau corresponding to a linear programming problem in standard form: • Step 1: For each constraint of the form [linear polynomial] < [nonnegative constant], introduce a slack variable and write the constraint as an equation.

19. Summary Section 4.1 - Part 3 • Step 2: Introduce a variable M to represent the quantity to be maximized, and form the equation -[objective function] + M = 0. • Form the augmented matrix corresponding to the system of linear equations from steps 1 and 2, with the equation from step 2 at the bottom. This matrix is the initial simplex tableau.