Davide Cherubini - PhD Thesis - UniCA Eprints

6.4 LP modelsmalized as follows:z = min(u max ) (6.38)∑x f ij · isf + ∑ flow f ij ≤ u max · c ij ∀(i, j) ∈ A (6.39)f∈F f∈F⎧⎪⎨∑flow f ji −∑−d f + is f ∀i ∈ N : i = I(f)flow f ij = d⎪ f − is f ∀i ∈ N : i = E(f) (6.40)⎩j:(j,i)∈Aj:(i,j)∈A0 otherwiseflow f ij ≥ 0 ∀(i, j) ∈ A, ∀f ∈ F (6.41)is f ∈ [0, d f ] ∀f ∈ F (6.42)While the objective function 6.38 remains unaltered, the capacity constraints6.39 and the flow conservation equations 6.40 become different. While the formeris really close to 6.33 because it is defined for each arc and only the MPLS“variable” has changed, the latter assumes a brand new form, being defined foreach node.Commodity per source aggregationIn order to reduce the number of variables and to make the model easier to solve,the commodities have been grouped based on their source node as follows:flowij h =∑flow f ij (6.43)f:I(f)=hWhere I(f) identifies the ingress node for commodity f.Let us consider the example in figure 6.1 where the commodities A → B,A → C, and A → D are replaced by a single commodity “A” which is the sumof them.Considering F (h) as the set of commodities having as ingress the generic node h,56