Modeling and Optimization of Scalar Flows on Networks

Detailed models based on partial differential equations characterizing the dynamics on single arcs of a network (roads, production lines, etc.) are considered. These models are able to describe the dynamical behavior in a network accurately. On the other hand, for large scale networks often strongly simplified dynamics or even static descriptions of the flow have been widely used for traffic flow or supply chain management due to computational reasons. In this paper, a unified presentation highlighting connections between the above approaches are given and furthermore, a hierarchy of dynamical models is developed including models based on partial differential equations and nonlinear algebraic equations or even combinatorial models based on linear equations. Special focus is on optimal control problems and optimization techniques where combinatorial and continuous optimization approaches are discussed and compared.