I'm struggling on determining when to assign an artifical variable a positive or a negative value. The example I have at hand is:
Max: $x_1+x_2$
St.
$$\begin{align}3x_1+2x_2\le5\\ x_1-x_2\le1\\ -2x_1-x_2\le7\\ x_1,x_2\ge0.\end{align}$$
I'm trying to determine how much to increase the right hand side of the third constraint so that my program becomes feasible.
So I would do
Max $x_1+x_2$
St.
$$\begin{align}3x_1+2x_2+x_3&\le5\\ x_1-x_2+x_4&\le1\\ -2x_1-x_2+x_5-a&\le-7\\ x_1\ge 0, x_2&\ge0\\ x_3,x_4,x_5&\le0.\end{align}$$
Should my a be negative because i want to increase the negative $7$? or should it be positive? How do i know when to use a positive artificial variable or negative?
There is no need to change the constraints.
Answer: $x_1 = 0$ and $x_2 = 5/2$, giving the maximum sum $5/2$.
Here's a figure showing the constraints: