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secn10 - 10 Cycling In Section 9 we considered the worrying...

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10 Cycling In Section 9, we considered the worrying possibility that the simplex method might fail to terminate because of cycling : a sequence of pivots that lead us back to an earlier tableau. In this case, unless our rules for choosing the entering and leaving variables change, we would then enter an infinite loop. Finding examples where cycling occurs is not very easy, although it can happen. Consider the following linear program (from [2]): maximize 10 x 1 - 57 x 2 - 9 x 3 - 24 x 4 subject to 0 . 5 x 1 - 5 . 5 x 2 - 2 . 5 x 3 + 9 x 4 0 0 . 5 x 1 - 1 . 5 x 2 - 0 . 5 x 3 + x 4 0 x 1 1 x 1 , x 2 , x 3 , x 4 0 . As usual, we introduce slack variables, which will also serve as the initial basic variables in the tableau z - 10 x 1 +57 x 2 +9 x 3 +24 x 4 = 0 0 . 5 x 1 - 5 . 5 x 2 - 2 . 5 x 3 +9 x 4 + x 5 = 0 0 . 5 x 1 - 1 . 5 x 2 - 0 . 5 x 3 + x 4 + x 6 = 0 x 1 + x 7 = 1 . Notice that the tableau is degenerate. The corresponding basic feasible so- lution is [0 , 0 , 0 , 0 , 0 , 0 , 1] T , and the objective value is 0. We will use a simple and natural pivoting rule, as follows. Whenever we have a choice between several possible entering indices, we will choose from among those with the largest reduced cost. Whenever we have a choice among several possible leaving indices, we will choose the smallest. Following our pivoting rule, x 1 enters and x 5 leaves, giving the new tableau z - 53 x 2 - 41 x 3 +204 x 4 +20 x 5 = 0 x 1 - 11 x 2 - 5 x 3 +18 x 4 +2 x 5 = 0 4 x 2 +2 x 3 - 8 x 4 - x 5 + x 6 = 0 11 x 2 +5 x 3 - 18 x 4 - 2 x 5 + x 7 = 1 . This pivot was degenerate: the basis and tableau have changed but the basic feasible solution and objective value have not. 56
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Using this same pivoting rule, we now perform a sequence of degenerate pivots, each time changing the basis and tableau but leaving the basic feasible solution and objective value unchanged. First, x 2 enters and x 6 leaves: z - 14 . 5 x 3 +98 x 4 +6 . 75 x 5 +13 . 25 x 6 = 0 x 1 +0 . 5 x 3 - 4 x 4 - 0 . 75 x 5 +2 . 75
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