Chapter 4-2 - Chapter 4 The Simplex Algorithm (cont'd) 4.11...

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Chapter 4 The Simplex Algorithm (cont’d)
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2 4.11 Degeneracy and the Convergence of the Simplex Algorithm Theoretically, the simplex algorithm can fail to find an optimal solution to an LP. However, LPs arising from actual applications seldom exhibit this unpleasant behavior. The following are facts: 1. If (value of entering variable in new bfs) > 0, then ( z -value for new bfs) > ( z -value for current bfs). 2. If (value of entering variable in new bfs) = 0, then ( z -value for new bfs) = ( z -value for current bfs).
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3 Assume that in each of the LP’s basic feasible solutions all basic variables are positive. An LP with this property is a nondegenerate LP . An LP is degenerate if it has at least one bfs in which a basic variable is equal to zero. Any bfs that has at least one basic variable equal to zero (or equivalently, at least one constraint with a zero rhs) is a degenerate bfs .
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4 Recall: In the application of the feasibility condition , a tie for the minimum ratio (for determining the leaving variable) may be broken arbitrarily for the purpose of determining the leaving variable. When this happens, however, one or more of the basic variables will necessarily equal zero in the next iteration. In this case, we say that the new solution is degenerate . From the practical standpoint, the condition reveals that the model has at least one redundant constraint . Degeneracy has two implications: 1. Cycling or circling 2. Temporarily degenerate
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5 If the bfs is d egenerate, the solution may stay the same z stays the same If the bfs is non-degenerate , the entering variable can strictly increase z will strictly improve
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6 Theorem . If every basic feasible solution is non-degenerate, the simplex method is finite. 1. The number of basic feasible solutions is at most n! / (n-m)! m!, which is the number of ways of selecting m basic variables 2. Each basic feasible solution is different Each has a better cost than the last, assuming non-degeneracy Therefore, the simplex method is finite
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7 Degenerate LP Example max z = 5x1+2x2 s.t. x1+x2 6 x1-x2 0 x1,x2 0 Standard form: z - 5x1-2x2 =0 s.t. x1+x2+s1 =6 x1-x2 +s2=0 x1,x2, s1, s2 0 current Basic z x1 x2 s1 s2 Solution Ratio z 1 -5 -2 0 0 0 s1 0 1 1 1 0 6 6/1=6 s2 0 1 -1 0 1 0* 0/1=0* Initial Tableau NBVs: x1=x2=0; and BVs: s1=6, s2=0; thus, degenerate ; z=0
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8 Apply Simplex Iteration 1 Current Basic z x1 x2 s1 s2 Solution Ratio z 1 0 -7 0 5 0 s1 0 0 2 1 -1 6 6/2=3* x1 0 1 -1 0 1 0 none NBVs: x2=s2=0; and BVs: s1=6, x1=0; z=0 (no change in z) consistent with the Fact 2 . In new bfs, all variables have exactly the same values as they had before the pivot. the new bfs is also degenerate. Current Basic z x1 x2 s1 s2 Solution z 1 0 0 7/2 3/2 21 x2 0 0 1 1/2 -1/2 3 x1 0 1 0 1/2 1/2 3 x1=3, x2=3,s1=s2=0, z=21 Optimal Tableau
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9 So, after all, degeneracy did not prevent the simplex method to find the optimal solution in this example. It just slowed things down a little.
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This note was uploaded on 04/23/2010 for the course QWE 2131 taught by Professor Asda during the Spring '10 term at University of Karachi.

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Chapter 4-2 - Chapter 4 The Simplex Algorithm (cont'd) 4.11...

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