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.

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

Starting Solutions (cont’d) Starting basis:

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

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

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.

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.

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

Optimization of the Simplex Tableau … xn xn+1 xn+m RHS 1 -c1 ... -cn -1 a11 a1n b1 xn+2 a21 a2n b2 am1 amn bm Phase I Objective Phase II Objective

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 .

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.

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.

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

Degenerate Linear Programs (contd.) Starting Solution z x1 x2 s1 s2 RHS 1 -5 -2 6 -1 z x1 x2 s1 s2 RHS 1 -7 5 2 -1 6 Solution after the first iteration. It is a degenerate iteration.

Degenerate Linear Programs (contd.) z x1 x2 s1 s2 RHS 1 -7 5 2 -1 6 Solution after the first iteration z x1 x2 s1 s2 RHS 1 3.5 1.5 21 0.5 -0.5 3 Solution after the second iteration. It is a nondegenerate iteration.

Degenerate Linear Programs (contd.) BFS and Extreme Points BVs BFS Extreme Point x1, x2 x1=x3=3, s1=s2=0 D x1, s1 x1=0, s1=6, x2=s2=0 C x1, s2 x1=6, s2=-6, x2=s1=0 Infeasible x2, s1 x2=0, s1=6, x1=s2=0 x2, s2 x2=6, s2=6, s1=x1=0 B s1, s2 s1=6, s2=0, x1=x2=0 B 6 5 4 D 3 2 1 A C 1 2 3 4 5 6

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.