Davide Cherubini - PhD Thesis - UniCA Eprints

6.2 Graphs and Network Flowswith a particular “centering” direction that, in the Karmarkar version of thealgorithm, is based on the concept of analytic center.The related problem of Integer Linear Programming (ILP) requires some orall the variables to be integer values. A subset of the ILP is defined when thevariables x assume boolean values [0, 1]. These programs have often the advantageof matching better the “reality” than LPs, but they are much harder to solve. Themost widely used techniques for solving integer programming problems use thesolutions of a series of Linear Programs searching for integer optimal solutions.Linear and integer programming proved to be valuable for modelling manyand diverse types of problems in planning, routing, scheduling, assignment, anddesign. Furthermore, LP and its extensions have been successfully applied totransportation, energy, telecommunications, and manufacturing.6.2 Graphs and Network FlowsMost of the optimization problems can be modelled by means of a graph [25]. Thisstructure often directly derives from the intrinsic nature of the problem (e.g., theurban transport network, power system, hydric pipelines), while in other cases itarises from the model. Generally, in Operations Research, the term “network”denotes a weighted graph G = (N, A) where the “weights” are numeric valuesassociated to nodes and/or arcs of the graph.More precisely:• At each node a real value is associated, which may beo a positive value, representing the amount of “good” (or commodity)exiting from the node (surplus);o a negative value, which represents the amount of commodity enteringin the node (deficit);o a zero value; in such case the node is called transit node.• At each arc the following features are associated44