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Unformatted text preview: Lecture 9: Special Cases in Linear Programming September 15, 2010 Four “special cases” ● Degeneracy ● Multiple solutions ● Unbounded solution ● Infeasibility Degeneracy • A tie for the minimum ratio test can happen. • Then in the next iteration, at lease one basic variable will be zero. The solution is called a degenerate solution. • Degeneracy reveals that there is at least one redundant con straint in the model, namely, the feasible region does not change if that constraint is dropped. • Degeneracy might cause cycling , namely, the objective value does not improve at several successive iterations. In fact, geometrically, the algorithm stays at a single vertex during these iterations, but, algebraically, the basis changes form iteration to iteration. 2 Degeneracy: an Example Consider the following example: max z = 3 x 1 + 9 x 2 s.t. x 1 + 4 x 2 ≤ 8 x 1 + 2 x 2 ≤ 4 x 1 ,x 2 ≥ The simplex iterations: Iteration Basic x 1 x 2 x 3 x 4 Solution z3 9 x 2 enters x 3 1 4 1 8 x 3 leaves x 4 1 2 1 4 1 z 3 4 9 4 18 x 1 enters x 2 1 4 1 1 4 2 x 4 leaves x 4 1 2 0  1 2 1 2 z 3 2 3 2 18 (optimum) x 2 1 1 2 1 2 2 x 1 1 0 1 2 3 Degeneracy: an Example At iteration 1 and 2, the objective does not change although the basis actually changed. In the graph, we can see that the constraint x 1 + 4 x 4 ≤ 8 is actually redundant....
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 Spring '08
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 Linear Programming, Optimization, multiple solutions

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