Network optimization

Academic Year 2019/2020 - 1° Year
Teaching Staff: Gabriella COLAJANNI
Credit Value: 6
Scientific field: MAT/09 - Operational research
Taught classes: 35 hours
Exercise: 12 hours
Term / Semester:

Learning Objectives

The objectives of the course Networks and Supernetworks are as follows:

  • to determine paths of minimum and maximum length starting from a root node;
  • to formulate equilibrium traffic problems in the dynamic case using networks, including also capacity constraints, additional restrictions and delay terms;
  • to evaluate the importance of the single components of a network;
  • to build a multi-tiered network for production and distribution problems, for electrical models and in the case of merger of companies;
  • to apply theoretical models to business cases.

Course Structure

The course will be taught through lectures and exercises in the classroom and at the computer labs.

Detailed Course Content

Graph theory:

Graphs and digraphs: Definitions and preliminary notions, associated matrices. Kruskal's algorithm and its variant. Dijkstra's algorithm and its variant. Ford algorithm. Bellman-Kalaba’s algorithm. The traveling salesman problem.


• Traffic networks in the static case: the model; Wardrope’s principle; model with capacity constraints. Traffic networks in the dynamic case: the model; equilibrium conditions; variational formulations; existence theorems; model with additional constraints. Traffic networks with delay terms. Directional derivative: definition and properties. Subdifferential of a convex function: definition and properties. Subgradient method, discretization method. Braess’ paradox in the static case and in the dynamic case. The efficiency of a network: Latora-Marchiori measure and Nagurney-Qiang measure. Importance of the components in a network. Applications to Braess network. Identification of critical elements in networks.

• Spatially distributed networks of economic markets in the static case in the presence of production and demand excesses. Variational formulation and Lagrangian theory.

• Horizontal mergers: the models before and after the merger; associated optimization problems; synergy. Models with environmental interests.

• Variational inequalities for auction problems: the model, equilibrium conditions and equivalent variational formulations.


• Supply chain networks with three levels of decision-makers: economic model in the presence of manufacturers, retailers and consumers with e-commerce; optimality conditions and equivalent variational inequality for the representatives of all levels and for the entire chain. Dynamic case: model with production and demand excesses.

• Networks with critical needs with external sources: optimization problem and variational formulation.

• Supply chains for food: optimality conditions and variational formulation.

• Electricity supply chain networks: the model with electric power producers, energy providers, transmission service providers and demand markets; optimality conditions and equivalent variational formulation for the representatives of all levels and for the entire network. Presentation of the model with non-renewable fuel suppliers and optimality conditions.

• closed loop supply chains: direct chain and reverse chain. Behavior of raw material suppliers, producers, retailers, demand markets, the recovery centers. Variational formulation.

• The model of cybercrime in financial services.

Textbook Information

  1. L. Daboni, P. Malesani, P. Manca, G. Ottaviani, F. Ricci, G. Sommi, “Ricerca Operativa”, Zanichelli, Bologna, 1975.
  2. P. Daniele, “Dynamic Networks and Evolutionary Variational Inequalities", Edward Elgar Publishing, 2006.
  3. A. Nagurney, J. Dong, "Supernetworks", Edward Elgar Publishing, 2002.
  4. Papers on STUDIUM