what do I do if all the elements in the pivot column are negative? Details are as follow:
$$ \begin{aligned} \max &-60000y_1 + 4800y_2 + 900y_3\\ \text{s.t.:}&\begin{cases}-50y_1 +6y_2 +y_3 \leq3\\-75y_1+ 6.75y_2 + y_3 \leq 1.8\end{cases} \end{aligned} $$
The most negative elements in the bottom row is $-60000$ which defines pivot column as the first non-basic var vector $(-50, -75)$. What do we do in this situation since pivot element is not allowed to be negative?
Thank you
Edit:
@callculus: This is what I have come up with. Similar problem.
If you put the max objective function into the table the signs are flipping: $-z=60000y_1-4800y_2-900y_3$ The table is
$$\begin{array}{|m{cm}|m{1cm}|} \hline y_1 & \color{blue}{y_2} & y_3 & s_1 & s_2 & \textrm{RHS}\\ \hline \hline 60000& -4800&-900 & 0 &0&0\\ \hline -50& 6&1&1&0&3 \\ \hline -75&\color{green}{6.75} &1&0&1&\color{orange}{1.8} \\ \hline \end{array}$$
Here $s_1$ and $s_2$ denote the slack variables. The most negative coefficient of the objective function is $-4800$. Thus $y_2$ is the pivot column. And $\min \{\frac{3}{6},\frac{1.8}{6.75}\}=\frac{1.8}{6.75}$. Thus the last row is the pivot row. I think you can go on. The optimal solution is $(y_1^*,y_2^*,y_3^*) = (0.048, 0,5.4)$.