I'm somewhat stuck on an example in Linear Programming. I managed to wrap my head around maximisation for a problem with $\le$ constraints, using both graphical and simplex solutions. However, I have encountered issues with minimisation and $\ge$ constraints.
For minimising, the notes say
"Minimising $2x + 4y$ is the same as maximising $-2x – 4y$. Therefore minimising a function is the same as maximising the negative of the function. So the only change in Simplex is that the coefficients of the objective are entered as positive values, rather than negative values".
To find the pivot column, do I still look for the most negative value? What if the function has no negative values in the objective row?
Secondly, for $\ge$ constraints, I know to add artificial variables, but I'm not 100% sure how to proceed after forming the tableau.
The exact problem I'm having difficulty with is:
MINIMISE $\ 2x + 5y, \text{ s.t.}\\ 3x + 4y \ge 12,\\ 5x + 2y \ge 8, \\ x, y \ge0$
I know I begin by adding (or subtracting in this case) slack and artificial variables, i.e. $$3x + 4y - s_{_1} + a_{_1}= 12 \\ 5x + 2y - s_{_2} + a_{_2} = 8$$
Then forming a tableau
$$\begin{array}{ccccccc} & x & y & s_{_1} & s_{_2} & a_{_1} & a_{_2}\\ 12 & 3 & 4 & -1 & 0 & 1 & 0\\ 8 & 5 & 2 & 0 & -1 & 0 & 1\\ 0 & 2 & 5 & 0 & 0 & 0 & 0 \end{array}$$
From here I'm unsure of how to choose the pivot column/pivot itself. Any help would be greatly appreciated.
EDIT: Solved it, will provide details on request, different method to one provided in answers.
