An Example That Cycles

Constraints = , Variables = .

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Current Dictionary

 

                                     

       

Pick a judge:

Instructions

The initial dictionary is degenerate.

If you use the largest-coefficient rule to pick the entering variable and the smallest-index rule to pick the leaving variable, then the simplex method will cycle back to this initial dictionary in just \(\sf 6 \) pivots.

In the smallest-index rule, we assume that the indices of the \(\sf x \) variables are smaller than the indices of the \(\sf w \) variables.
In other words, ambiguities in the choice of the leaving variable are resolved according to this ordering:

\(\sf x_1, \; x_2, \; x_3, \; \ldots, \; x_n, \; w_1, \; w_2, \; \ldots, \; w_m \)