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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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