# OPERATIONS RESEARCH

**Academic Year 2023/2024**- Teacher:

**Patrizia DANIELE**

## Expected 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.

**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 dispensatory 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.

## 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 | Symplex Method | 1 |

2 | Search of the basis in 2 phases | 1 |

3 | Geometry of Linear Programming | 1 |

4 | Duality in LP | 1 |

5 | Optimal solution of the dual problem | 1 |

6 | Interpretation of the dual problems | 1 |

7 | Sensitivity Analysis | 1 |

8 | Branch & Bound Method | 1 |

9 | Knapsack problem | 1 |

10 | Theorem of the projection on a convex closed set | 2 |

11 | Existence and Uniqueness theorems on the solutions of a Variational Inequality | 2 |

12 | Traffic networks | 2 |

13 | Smith Theorem | 2 |

14 | Direct Method for the solution of VI | 2 |

15 | Projection Method | 2 |

16 | Lagrange Theory | 2 |

17 | KKT multipliers | 2 |

18 | Von Neumann Theorem | 1 |

19 | Game theory and dual problems | 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.

Final grades will be assigned taking into account the following criteria:

**Rejected**: Basic knowledges have not been acquired. The student is not able to solve simple exercises.

**18-23**: Basic knowledges have been acquired. The student solves simple exercises and has sufficient communications skills and making judgements.

**24-27**: All the knowledges have been acquired. The student solves all the proposed exercises making few errors and has good communications skills and making judgements.

**28-30 cum laude**: All the knowledges have been completely acquired. The student applies knowledge and has excellent communications skills, learning skills and making judgements.

### 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.