secn9 - 9 Degeneracy In Section 5 we introduced the idea of...

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Unformatted text preview: 9 Degeneracy In Section 5, we introduced the idea of a “degenerate” tableau, by which we mean at least one of the numbers ¯ b i on the right-hand side of the body of the tableau is zero. Thus a tableau and the corresponding basis are degenerate when one of the basic variables takes the value zero at the corresponding basic feasible solution. In the course of the simplex method, if we have to choose between sev- eral possible leaving indices, then the next tableau will be degenerate. To illustrate, consider the following simple linear programming problem: maximize 2 x 1 + x 2 subject to x 1- x 2 ≤ 1 x 1 ≤ 1 x 2 ≤ 1 x 1 , x 2 ≥ . After introducing slack variables, we get the initial tableau z- 2 x 1- x 2 = x 1- x 2 + x 3 = 1 x 1 x 4 = 1 x 2 + x 5 = 1 . The basic variables are x 3 , x 4 , x 5 , the corresponding basic feasible solution is [0 , , 1 , 1 , 1] T , and the corresponding objective value is 0. The current values of the basic variables are all nonzero, so this tableau is nondegenerate. Starting the simplex method, we could choose x 1 as the entering variable, and compute the ratios in the usual way. basic index 3 4 5 ratio 1 1 1 1 We now have a choice of the leaving variable: either x 3 or x 4 . This poses no difficulty in itself. We could choose x 3 , for example, in which case after the pivot, we arrive at the following tableau: z- 3 x 2 + 2 x 3 = 2 x 1- x 2 + x 3 = 1 x 2- x 3 + x 4 = x 2 + x 5 = 1 ....
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secn9 - 9 Degeneracy In Section 5 we introduced the idea of...

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