Dual Simplex Method

Question

Suppose that in a Linear Programming problem in the dual Simplex Method there is a first element (in the first column) negative. If there are in that pivot row some negative numbers we take $\max$ from the first row and that pivot row. But what if all the elements in the pivot row are positive, what pivot do we use, i.e. which column? Consider this table $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline -4& 0 & 1&5&16&0&4&0 \\ \hline -12& 0 & 8&-1&-7&0&-3&1 \\ \hline 1& 1 & 1&1&1&1&1&1 \\ \hline \end{array}$$ We take -3 in the second row. OTHER new initial TABLE: $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline -4& 0 & 1&5&16&0&4&0 \\ \hline -12& 0 & 8&1&7&0&3&1 \\ \hline -1& 1 & 1&1&1&1&1&-1 \\ \hline \end{array}$$ What do we take here? Note that the first and last element in the last row have the opposite signs in compared with the previous table.

Answer 1

In a problem that you use dual simplex to solve it, if you have a negative RHS and all the elements in that row are non-negative, then your original problem is infeasible and your dual problem is unbounded.