In this paper, we consider the max-min fair resource allocation problem as applied to elastic flows. We are interested in computing the optimal max-min fair rate allocation. The proposed approach is a linear programming based one and allows the computation of optimal routing paths with regard to max-min fairness, in stable and known traffic conditions. We consider non-bounded access rates, but we show how the proposed approach can handle the case of upper-bounded access rates. A proof of optimality and some computational results are also presented