Starting Solutions and Convergence

1. Starting Solutions and Convergence CONTENTS • The Initial Basic Feasible Solutions • The Two-Phase Method • The Big-M Method • Degeneracy, Cycling, and Stalling Reference: Chapter 4 in BJS book.

2. Starting Solutions • The simplex method assumes the existence of a basic feasible solution • When a basic feasible solution is not readily available, then we need to create one such solution by adding slack, surplus, or artificial variable

3. Starting Solutions (cont’d) Starting basis:

4. Starting Solutions (cont’d) In both cases, the constraint matrix does not contain an identity matrix.

5. Artificial Variables • In order to obtain identity in the constraint matrix, sometimes we must add artificial variables. • The use of artificial variables changes the solution space; hence we must guarantee that these variables will eventually drop to zero

6. Artificial Variables (cont’d) Let P1: and P2: Result 1: If P1 has a feasible solution, them P2 has a feasible solution with xa=0. Result 2: If P2 has a feasible solution with xa=0, then P1 has a feasible solution. Theorem: There is a one-to-one correspondence between feasible solution of P1 and feasible solutions of P2 with xa=0.

7. Two Phase Method Phase I: • If at optimality , then stop; the original problem has no feasible solutions. • If at optimality , then the original problem has a feasible solution (xB) and we go to phase 2.

8. Two Phase Method (cont’d) Phase II: Solve the following LP:

9. Optimization of the Simplex Tableau Phase I Objective Phase II Objective

10. Big M Method Solve the following LP: where M is a very large number. The term can be interpreted as a penalty to be paid by any solution with .

11. Nondegenerate Linear Program z-value of the new BFS = z-value of the current BFS – (value of the entering variable)*(zj – cj) for the entering variable We know that the reduced cost coeff. of the entering variable is positive. RESULT 1: If value of the entering variable > 0, then z-value of the new BFS is strictly less than the BFS of the current BFS. RESULT 2: If value of the entering variable = 0, then z-value of the new BFS is the same as that for the BFS of the current BFS.

12. Nondegenerate Linear Programs (contd.) Nondegenerate LP: We call a LP to be nondegenerate if in each BFS of the LP all the basic variables are positive. RESULT 3: For a nondegenerate LP, the value of the entering variable is always positive. RESULT 4: For a nondegenerate LP, the simplex algorithm never repeats a BFS and terminates within nCm iterations.

13. Degenerate Linear Programs Degenerate LP: We call a LP to be degenerate if it has at least one BFS in which a basic variable is equal to zero. The following LP is degenerate: max z = 5x1 + 2x2 max z = 5x1 + 2x2 s. t. x1 + x2 6 s. t. s. t. x1 + x2 + s1= 6 x1 – x2 0 x1 – x2 + s2= 0 x1, x2 0 x1, x2 , s1, s2 0

14. Degenerate Linear Programs (contd.) Starting Solution Solution after the first iteration. It is a degenerate iteration.

15. Degenerate Linear Programs (contd.) Solution after the first iteration Solution after the second iteration. It is a nondegenerate iteration.

16. Degenerate Linear Programs (contd.) BFS and Extreme Points B 6 5 4 D 3 2 1 A 1 2 3 4 C 5 6

17. Degenerate Linear Programs (contd.) • In a degenerate iteration, the basis changes, but the BFS solution remains unchanged. • In the presence of degeneracy, the objective function value may not increase from one iteration to next. • For very large LPs, degeneracy is a real problem and over 90% of the pivot operations are degenerate iterations. • For degenerate LPs, cycling can occur (that is, simplex algorithm can perform an infinite sequence of iterations without any improvement) and the algorithm may not obtain an optimal solution. • Cycling can theoretically occur and but has never occurred in practice.