Ateneo - UPO

Introduction to Linear programming techniques and its economic applications. Elements of decision theory under risk: Bernoulli criterium, mean-variance criterion, expected utility, certainty equivalent, risk premium, attitude towards risk.

Lecture notes by the teacher, downloadable from the learning portal (DIR). Recommended readings: Bellenzier L., Grassi R., Stefani S., Torriero A., Metodi quantitativi per il Management, Esculapio, 2012. (Chapter 1, Chapter 2 up to Section 2.4 included)Martello S., Speranza M.G:, Ricerca Operativa per l'Economia e l'Impresa, Esculapio, 2012 (Chapters 1 and 2 just reading, Chapter 3, Chapter 4).Castellani G., De Felice M., Moriconi F., Manuale di Finanza II, Il Mulino, 2005 (Chapter 1, Chapter 2 except Section 2.5, Chapter 3 except Section 3.2).

The course is worth 6 ECTS, corresponding to 45 hours of direct teaching. Overall, approximately 30 hours consist of lecture-based teaching (DE) and 15 hours of interactive teaching (DI), with flexible proportions within each lesson.
This course aims to help students achieve: a good understanding of basic mathematical knowledge related to solving and discussing the results of linear programming problems; the acquisition of knowledge regarding individual choice criteria under uncertainty, and judgment skills in selecting and using suitable analytical tools to solve simple problems; the development of the ability to communicate, with sufficient rigor, the logical-deductive paths followed when tackling problems.

Mathematical Methods 1: differential calculus for real-valued functions of one real variable, with a focus on its application to studying monotonicity, convexity/concavity, and finding extrema;Mathematical Methods 2: vectors, matrices, determinant of a square matrix, matrix rank, linear systems and their solution;Statistics: random variable (discrete), probability distribution, cumulative distribution function, mean and variance;Microeconomics: preferences under certainty, utility function, representation of preferences.
The exam of Mathematical Methods 1 is compulsory. It is recommended to take the exams of Mathematical Methods 2 and Statistics.

Lectures are designed to present the syllabus topics (knowledge) and encourage their critical discussion, adopting an integrated approach that combines direct teaching with interactive learning. The course alternates theoretical delivery—preceded by motivational examples—with practical exercises aimed at applying the concepts introduced. For each topic, exercises and questions are proposed, and their solutions or answers are discussed collectively in class to encourage contributions and foster active engagement from all students. Digital tools (DIR platform, Wooclap) are used to support both lectures and self-study.

Any useful information on the course and supplementary didactic resources can be found on the web page of the course at the URL: https://dir.uniupo.it/. Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/services- students-physical-or-learning-disabilities. Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The examination is oral and consists of questions/exercises designed to assess:knowledge, understanding, and the ability to apply techniques for analyzing and discussing the results of a linear programming problem;knowledge, understanding, and the ability to apply decision criteria to real-world individual choice problems under conditions of risk, along with the ability to present the results achieved using appropriate and sufficiently rigorous technical language.It is mandatory to register for the exam through the student portal: once the maximum number of exam sessions/attempts permitted by the current Teaching Regulations has been reached, the system will not allow further registrations.
In addition to the official exam sessions, a partial exam and a final exam in written form are scheduled.
The first exam, lasting one hour and thirty minutes, includes:an exercise on formulating an LP (Linear Programming) problem (3 points);an exercise on solving an LP problem using the simplex method (7 points);an exercise on sensitivity analysis (4 points);one or more theoretical questions on the linear programming section (4 points).The score is assigned by evaluating both the correctness of the execution and the ability to present the results obtained in a clear and rigorous manner.
To take part in the first exam, it is necessary to register using the form published in due time by the Professor on the course's DIR page.
The second (final) exam, accessible by obtaining a score of ≥8 in the first exam, lasts one hour and thirty minutes and consists of:two exercises related to individual choices under risk (12 points);one or more theoretical questions on the section regarding choices under risk (6 points).The score is assigned by evaluating both the correctness of the execution and the ability to present the results obtained in a clear and rigorous manner.
The overall grade is the sum of the marks obtained in the two partial exams, rounded if necessary based on the quality of the tests and/or attendance, up to a maximum of 30 cum laude (if an overall grade of at least 33 is obtained).
No supplementary oral exam is provided.

Linear ProgrammingReal-world problems modelling, variables, constraints, objective function, the feasible region, solutions. Solution methods: graphical analysis (two variables), algebraic method, iterative methods; implementation with Excel. Sensitivity analysis. Introduction to decision-making problems. Introduction to decision theory under risk.Criterion of expected value maximization (MVA), fair price, applications to the theory of insurance. Mean-variance criterion. Preference and indifference relations and their properties. Expected utility theory: compound lotteries, axioms of rationality, representation theorem of Von Neumann - Morgenstern. Principle of maximization of expected utility. Certainty equivalentand risk premium. Estimation of the utility function.Attitude toward risk and its characterizations. Measure of risk aversion. Quadratic approximation of a utility function. Applications in insurance and finance (brief outline).

KNOWLEDGE AND UNDERSTANDING
minimum passing level: sufficient knowledge of the main mathematical concepts and solving methods of linear programming (LP); sufficient knowledge of some important decision-making criteria under risk.
intermediate/advanced level: good knowledge of the main mathematical concepts and solving methods of linear programming (LP); good knowledge of some important decision-making criteria under risk.

APPLYING KNOWLEDGE AND UNDERSTANDING
minimum passing level: sufficient ability to formalize, solve, and analyze solutions to an LP problem in two/three variables—graphically, using simplex tableaux, and with spreadsheets; sufficient ability to solve decision problems using the expected value maximization criterion, the mean-variance criterion, and the expected utility maximization criterion.
intermediate/advanced level: good ability to formalize, solve, and analyze solutions to an LP problem in two/three variables—graphically, using simplex tableaux, and with spreadsheets; good ability to solve decision problems using the expected value maximization criterion, the mean-variance criterion, and the expected utility maximization criterion.

MAKING JUDGEMENTS
minimum passing level: development of basic judgment skills in choosing and using appropriate computational tools to solve the aforementioned problems.
intermediate/advanced level: development of good judgment skills in choosing and using appropriate tools to solve the aforementioned problems.

COMMUNICATION SKILLS
minimum passing level: basic ability to clearly and rigorously communicate the logical-deductive steps followed when addressing problems.
intermediate/advanced level: good ability to clearly and rigorously communicate the logical-deductive steps followed when addressing problems.

LEARNING SKILLS
minimum passing level: sufficient ability to independently use the recommended textbook, lecture notes, and supporting digital resources to review and understand the resolution of standard problems and exercises.
intermediate/advanced level: good independence in exploring concepts not explicitly covered in class and in applying the acquired logical-deductive method to independently tackle quantitative and analytical courses in subsequent years.