2. OR-1 20151 Chapter 3. Pitfalls

3. OR-1 20152  Selection of leaving variable: a)No restriction in minimum ratio test : can increase the value of the entering variable indefinitely while satisfying the constraints (including nonnegativity), hence problem is unbounded Ex)

4. OR-1 20153 b) In case of ties in the minimum ratio test : Ties (=1/2) Some basic variables have value 0 after pivot.

5. OR-1 20154 In the next iteration,

6. OR-1 20155  Terminology: degenerate solution ( 퇴화해 ) : basic feasible solution with one or more basic variables having 0 values. degenerate iteration : simplex iteration that does not change the current basic solution (only basis changes).  Observations:  If we have a nondegenerate b.f.s., the simplex iteration is nondegenerate. We move to a different point and the objective value strictly increases.  Given a nondegenerate b.f.s., we must have ties in the minimum ratios so that we have a degenerate solution after the pivot.  A degenerate iteration occurs only if we have a degenerate solution, but the converse is not true (i.e. we may have a nondegenerate iteration although we have a degenerate solution).

7. OR-1 20156 Geometric meaning of a degenerate iteration x 1 =0 x 6 =0 x 5 =0 x 2 =0 x 1 =0 x 6 =0 x 5 =0 x 2 =0 A A

8. OR-1 20157

9. 8  Getting out of degenerate iterations: x 1 =0 x 6 =0 x 5 =0 x 2 =0 ( x 3 =0, x 5 =0 used) A x30x30

10. OR-1 20159  Degenerate iteration is the process of identifying the same point (solution) using different defining equations (different nonnegativity constraints). If we are lucky enough to obtain defining equations that correctly guides the moving direction, we move to a different point with a nondegenerate pivot.  If we have a degenerate solution, pivot may continue indefinitely (Example in text p.31, pivoting rule is largest coefficient for entering variable and smallest subscript for leaving variable in case of ties. Then we have the initial dictionary again after 6 pivots.)  Terminology:  Cycling : appearance of the same dictionary (tableau) again in the simplex iterations.

11. OR-1 201510  In practice, cycling hardly occurs. However, during the degenerate iterations, the algorithm stalls and it may hamper the performance of the algorithm. Such phenomenon is of practical concern and affects the performance of the algorithm (especially, for problems with some special structures and large problems).  We also need mechanisms to avoid cycling for any problem instances. Otherwise, the simplex method may not terminate finitely.  Cycling is the only reason that simplex method may fail to terminate (i.e. simplex method terminates in a finite number of iterations as long as cycling is avoided).

12. OR-1 201511

13. OR-1 201512

14. OR-1 201513  The proof of the theorem shows that if we have the same basis, then the dictionaries (tableaus) are the same. Since there are only a finite number of ways to choose the basis, the simplex method terminates finitely if the same basis (the same dictionary) does not appear again, i.e. cycling is avoided.

15. OR-1 201514 Avoid cycling

16. OR-1 201515  Examples of smallest subscript rule: Ties (=1/2) Ties (=0)