ISTITUZIONI DI RICERCA OPERATIVA

Module MODULO II

The objectives of the course are as follows:

• to formulate traffic equilibrium problems for the deterministic and stochastic cases;
• to model socio-economic problems using network games and variational inequalities;
• to apply optimization methods to the study of network component centrality, with particular reference to the inverse problem;
• 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.

Knowledge and understanding:

At the end of the course, the student, in addition to having acquired the basic knowledge and skills in the field of optimization and mathematical modeling, will demonstrate:

• being able to transform real situations of maximization of profits, minimization of costs and risks, ... into mathematical models;
• possess knowledge and ability to understand texts.

Applying knowledge and understanding:

The theoretical and practical knowledge acquired during the course will allow the student to:

• critically analyze various business situations;
• propose optimal solutions to complex problems;
• identify the essence of a problem and apply general principles to specific cases.

Making judgements:

The student, by virtue of the acquired training, also of an analytical-quantitative type, will be able to critically analyze and interpret the data provided.

Communication skills: 

At the end of the course the student will be able to:

• pass on their experience and knowledge to others;
• confronting others, especially in the development of projects in which you work in a team.

Learning skills:

• The student will acquire the ability to learn, even independently, additional knowledge on applied mathematics problems. These skills will allow him to face and solve concrete optimization problems.

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

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Learning assessment may also be carried out on line, should the conditions require it.

The basic concepts of Linear Algebra (vectors and matrices), Mathematical Analysis I and II (differentiability, convexity of sets and functions, topology, ...), Operations Research (concept of network and variational inequality) and Optimization (minimum problems, subdifferentials,...) are required.

This knowledge is to be understood as important.

• Variational inequalities and traffic networks: Theoretical overview of variational inequalities in finite- and infinite-dimensional settings; numerical methods for variational inequalities: the projection method and the extragradient method. Applications of variational inequalities to traffic network equilibrium problems: deterministic and stochastic cases.

• Network Games: Network game models and their economic interpretation; formulation of network games using variational inequalities and numerical solution.

• Network component importance measures. Inverse centrality problem: The inverse problem for network Eigenvector Centrality. Optimization methods for directed networks. Brief overview of the undirected case.

• Applications in MATLAB: MATLAB syntax basics; implementation of the projection method and the extragradient method.

Networks:

• 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:

• 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.

• 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.

Matlab applications.

1. Joachim Gwinner, Baasansuren Jadamba, Akhtar A. Khan, Fabio Raciti, "Uncertainty Quantification in Variational Inequalities", Chapman and Hall/CRC, 2021
2.  M.O. Jackson, Y. Zenous, "Supernetworks", Chapter 3 in Handbook of Game Theory with Economic Applications, 2015, vol. 4, pp 95-163, Elsevier
3. P. Daniele, “Dynamic Networks and Evolutionary Variational Inequalities", Edward Elgar Publishing, 2006.
4. A. Nagurney, J. Dong, "Supernetworks", Edward Elgar Publishing, 2002.
5. Papers on STUDIUM

For the first module of the course, students may prepare a term paper assigned by the instructor. In this case, the exam will consist of a detailed discussion of the paper, including references to related course concepts (though formal proofs will not be required).

Alternatively, students may take an oral exam covering the topics discussed during the course; the majority of these topics are covered in the lecture notes provided on Studium/Moodle. The notes also contain certain proofs that were not presented in class and will not be included in the exam.

Elements of the MATLAB language will be explained during lectures, accompanied by brief programming examples. Regarding the oral exam, students may be asked how to use the `fmincon` or `quadprog` commands to optimize a multivariable function, but they will not be asked to perform MATLAB programming exercises.

The final exam consists of an oral test during which the candidate demonstrates that she/he has assimilated the topics covered in the course.

Verification of learning can also be carried out electronically, should conditions require it.

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or compensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.

Sufficient conditions for operator monotonicity in finite dimensions

Projection method

Variational formulation of Nash equilibria

Variational formulation of network games

Network node centrality measures

The inverse problem of eigenvector centrality

Present the merger model between two companies with and without environmental interests.

Present the mathematical model of auction sales.

Present layered networks with three levels of decision makers.

Examine the behavior of producers.

Present supply chain networks in case of critical needs with external sources.

Present electricity supply chain networks with and without non-renewable fuel suppliers.

Present closed-loop supply chain networks with material recycling and examine the behavior of recovery centers.