Davide Cherubini - PhD Thesis - UniCA Eprints

6.4 LP modelsgraph G with the minimal cost (objective function to be minimized), respectingthe constraints declared by the next two equations defined per single commodity,which are the flow conservation 6.25 , the capacity constraint 6.26, and the lastequation 6.27 indicating the mutual capacity constraint stating that the total sumof all commodities flowing through arc (i, j) have to be less than its maximumtotal capacity.Relaxing these last constraints, it is possible to formulate the problem ina different way simply noting that the k resulting problems are Shortest Pathproblems.This new formulation is called the arc-path formulation:∑p∈P (h)∑p:(i,j)∈fmin ∑ c p x p (6.28)p∈Px f = d h ∀h ∈ K (6.29)x p ≤ u ij ∀(i, j) ∈ A (6.30)x p ≥ 0 ∀p ∈ P (6.31)where P (h) is the set of possible paths for each commodity h and P = ∪ h P (h).The two formulations are basically equivalent and the choice of one of them isstrictly dependent on the problem at hand.6.4 LP modelsIn the following, three LP models that aim to minimize the maximum traffic loadin a telecommunication network and, in the meantime, to avoid congestion in caseof failure (single failure is considered, as it is the most probable) are presented.These models are structured in two layers: the first one, common to all models,solves the well-known “General Routing Problem” (GRP) [43], which is basicallya Multicommodity Min-Cost Flow problem, using the IS-IS routing protocol anda complementary set of LSP tunnels.51