# Adding slack nodes to min cost network flows

I have the following question. I want to clarify couple of points. As you can see, total demand and total supply does not match, we do not have enough demand. What I want to ask is:

1. Do we need to add a slack node to provide that gap? If yes, would adding one slack node be enough? I mean, if we had a gap in supply I believe we had to, but for this case, I could not be sure.

2. If we add a slack node, what cost value we assign for the arc that carries the "needed" supply/demand?

3. Does adding a slack node change anything about the other arcs, do we need to remake our calculations?

I am asking for both insufficient demand and supply cases.

My attempt: I am not very sure about my model as I couldn't figure out how the slack node should be but, I thought $$x_{12}=3$$ , $$x_{24}=4$$ and $$x_{32}=2$$. ($$x_{ij}=$$flow from node i to node j). $$z=15$$.

Linear program I created is as follows:

$$Min$$ $$x_{12}+3x_{13}+4x_{24}-2x_{32}+3x_{34}$$

$$s.t$$ $$x_{12}+x_{34}=4$$

$$-x_{12}+x_{24}=-1$$

$$x_{13}-x_{32}-x_{34}=2$$

$$x_{24}+x_{34}=-4$$