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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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This note was uploaded on 09/27/2010 for the course GE 330 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
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