OPERATIONS RESEARCH
Academic Year 2022/2023 - Teacher: Patrizia DANIELEExpected Learning Outcomes
The student will acquire the ability to formulate, in mathematical terms, problems related to profit maximization and cost minimization, optimization of resources, traffic network equilibria and games between two players.
In particular, the course of Operations Research has the following objectives:
- formulating a management problem in mathematical terms;
- solving linear optimization problems using numerical algorithms;
- formulate linear programming and binary programming problems;
- finding the equilibrium distribution in a traffic network by means of the solution to a variational inequality;
- finding the solution to a game between two players.
Knowledge and understanding: the aim of the course is to be able in transforming real life situations in mathematical problems.
Applying knowledge and understanding: students will be able to suggest optimal solutions to real life problems.
Making judgments: students will be able to analyze the data.
Communication skills: students will be able to communicate their experience and knowledge to other people.
Learning skills: students will have acquired the ability to learn, even autonomously, further knowledge on the problems related to applied mathematics.
Course Structure
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.
Required Prerequisites
Attendance of Lessons
Detailed Course Content
Linear Programming: the simplex method, duality, sensitivity analysis (about 26 hours).
Linear Integer Programming: the Branch & Bound method (about 4 hours).
Linear Integer Programming 0-1: the knapsack problem (about 4 hours).
Variational Inequalities: the projection on a convex closed set, existence and uniqueness results for the solution to a variational inequality (about 8 hours).
Traffic networks: Wardrop's principle, characterization by means of variational ineqaulities, direct method for the computation of the equilibrium solution, projection method (about 14 hours).
Game theory: pure and mixed strategies, Von Neumann Theorem (about 8 hours).
Elements of Nonlinear Optimization: Lagrange theory and KKT multipliers (about 9 hours).
Textbook Information
- F.S. Hillier, G.J. Lieberman, "Introduction to Operations Research", McGraw-Hill, 2001.
- Papers on STUDIUM http://studium.unict.it
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Il metodo del simplesso | 1 |
| 2 | La ricerca della base in due fasi | 1 |
| 3 | La geometria della PL | 1 |
| 4 | Dualità in PL | 1 |
| 5 | Calcolo della soluzione ottima del problema duale | 1 |
| 6 | Interpretazione duale dei problemi di PL | 1 |
| 7 | Analisi di sensitività | 1 |
| 8 | Il metodo del Branch & Bound | 2 |
| 9 | Il problema dello zaino | 2 |
| 10 | Teorema della proiezione su un convesso chiuso | 3 |
| 11 | Teoremi di esistenza e unicità delle soluzioni di una disequazione variazionale | 3 |
| 12 | Reti di traffico | 3 |
| 13 | Teorema di Smith | 3 |
| 14 | Metodo diretto per il calcolo della soluzione di una disequazione variazionale | 3 |
| 15 | Metodo delle proiezioni | 3 |
| 16 | Teoria Lagrangiana | 3 |
| 17 | Moltiplicatori KKT | 3 |
| 18 | Teorema di Von Neumann | 1 |
| 19 | Riduzione di un gioco ad una coppia di problemi duali | 1 |
Learning Assessment
Learning Assessment Procedures
Usually, a self-assessment test is scheduled halfway through the course, which consists in carrying out some exercises related to the formulation and resolution of Linear Programming problems.
The final exam consists of an oral test during which the candidate is also invited to solve a numerical exercise. The final grade is established on the basis of the written and oral answers given by the candidate during the final interview.
Verification of learning can also be carried out remotely, should the conditions require it.
Examples of frequently asked questions and / or exercises
The three cases of the simplex method.
The fundamental theorem of duality.
The characterization of the projection on a convex.
The first theorem of existence of solutions of a variational inequality.
Von Neumann's theorem.
The Branch & Bound method.
Smith's theorem.
The knapsack problem
Frequent exercises:
some exercises assigned in previous years can be found on STUDIUM http://studium.unict.it.