Davide Cherubini - PhD Thesis - UniCA Eprints

6.2 Graphs and Network FlowsWhere BS and F S represent the Backward Star and the Forward Star respectively.The matrix formulation of this equation can be directly derived usingthe definition of Incidence Matrix:Ex = b (6.5)A flow x ij is feasible if it verifies the capacity constraints:l ij ≤ x ij ≤ u ij (6.6)where l ij and u ij are the lower and upper values of capacity for arc (i, j). Thelower bound is often set to zero.The objective function can be written as:cx = ∑c ij · x ij (6.7)(i,j)∈AEquations 6.7, 6.5 and 6.6 define the Minimum Cost Flow Problem (MCF)as:min cx (6.8)Ex = b (6.9)0 ≤ x ≤ u (6.10)or, in its extended formulation:∑min ∑c ij · x ij (6.11)x ji −(i,j)∈A∑(j,i)∈BS(i) (i,j)∈F S(i)x ij = b i i ∈ N (6.12)0 ≤ x ij ≤ u ij (i, j) ∈ A (6.13)The MCF problem can be easily solved using the classic Linear Programmingtechniques. However, as shown in the next sections, the MCF can represent eithera Maximum Flow problem or a Shortest Path problem allowing the use of thespecific algorithms developed for those classes of problems.46