GE330_lect6 - Lecture 6 Special Cases in LP February 9,...

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Lecture 6 Special Cases in LP February 9, 2009
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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
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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 0 The simplex iterations: Iteration Basic x 1 x 2 x 3 x 4 Solution 0 z -3 -9 0 0 0 x 2 enters x 3 1 4 1 0 8 x 3 leaves x 4 1 2 0 1 4 1 z - 3 4 0 9 4 0 18 x 1 enters x 2 1 4 1 1 4 0 2 x 4 leaves x 4 1 2 0 - 1 2 1 0 2 z 0 0 3 2 3 2 18 (optimum) x 2 0 1 1 2 - 1 2 2 x 1 1 0 -1 2 0 3
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Degeneracy: an Example At iteration 1 and 2, the objective does not change although
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GE330_lect6 - Lecture 6 Special Cases in LP February 9,...

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