## Minimal cost circulation

I have a problem of finding the minimum cost circulation for a given network, so each arc has a minimum and a maximum flow with an associated cost. I want to show that my derived answer is indeed the correct minimum. I am slightly unsure as to how I can do this.

First I am saying that I have a set of node potentials that satisfies the conditions of

$d_{ij} - \lambda _{i} + \lambda _{j} < 0$ if the flow is at the upper limit, and >0 if the flow is at the lower limit.

I can provide such a set. If this is true then I have the minimum cost circulation. Am I correct here?