Knowledge and understanding: capability of understanding (a) the conceptual approach of Operations Research as a tool for formulating, solving and evaluating decision-making problems related to complex systems, (b) the methodologies for the formalization of quantitative models and algorithmic solutions, and (c) the theoretical aspects underlying the solution techniques, their mathematical justifications and their implications and applicative potentialities.

Applying knowledge and understanding: capability of actually applying the solution techniques and algorithms, by executing the necessary procedures to attain the solution of numerical problems and being able to critically analyze the solutions obtained.

Making judgments: capability of applying the acquired knowledge to autonomously formulate quantitative models and solve the associated optimization problems, by also manually executing the appropriate solution algorithms.

Communication skills: capability of introducing decision-making problems and their possible solutions, both in written and oral form, and to critically discuss the validity and the limits of formulations and solutions.

Learning skills: capability of gathering information from textbooks, scientific papers and other material for the formulation and autonomous solution of decision-making problems.

Knowledge of the linear algebra formalism (vectors, matrices, their operations and space representations).

No strict prerequisite.

1. Introduction to Operations Research
Decision-making problems: Analysing and solving complex problems in order to optimise solutions.
Areas of application:Business management, resource planning and management, production, logistics and transport, economics, finance, health services, public administration.

2. Linear Programming (LP)
Properties and characteristics: Optimisation of a linear objective function with linear constraints.
Model formulation: Decision variables, objective, constraints.
Geometric interpretation: Representation in two dimensions.
Possible outcomes: Feasibility, unconstrained, single or multiple optimality.
Algebraic formulation: Standard form with slack and surplus variables.
Simlex method: Basic solution, in-base and out-of-base variables, optimality criterion, improvement.

3. Duality in LP
Dual variables and dual constraints: Correspondence between variables and constraints of primal and dual problems.
Dual objective: Objective function of the dual problem.
Duality theorems: Weak and strong duality, complementary slackness theorem.

4. Postoptimality
Sensitivity analysis: Impact of parameter variations on optimal results, variation of right-hand side terms and coefficients of the objective.

5. Integer programming
Graphical examples: Problems with integer variables.
Branch & bound algorithm: Systematic exploration of solutions.
Gomory cuts: Techniques for improving solutions.

F. S. Hillier and G. J. Liebermann: Ricerca Operativa, 9th Ed. McGraw-Hill

1. Introduction to Operations Research
Decision-making problems: Operations research focuses on analysing and solving complex decision-making problems, using mathematical models and algorithms to optimise solutions.

Areas of application:
- Operations management: Optimisation of operational processes.
- Resource planning and management: Efficient allocation of available resources.
- Production: Improvement of production processes.
- Logistics and transport: Optimisation of transport routes and networks.
- Economics and finance: Investment management and economic analysis.
- Health services: Planning and management of health resources.
- Public administration: Efficiency in public services.

2. Linear Programming (LP)
Properties, characteristics and applicability of LP: LP is used to optimise a linear objective function subject to linear constraints.

Formulation of a LP model:
- Decision variables: They represent the available choices.
- Objective: The function to be optimised (maximise or minimise).
- Constraints: Limitations or requirements that must be satisfied.

Geometric interpretation: In two dimensions, the LP problem can be represented graphically with admissible regions and level lines of the objective function.

Possible outcomes of a LP problem:
- Feasible: Existence of at least one solution that satisfies all constraints.
- Unconstrained: The objective function can be improved indefinitely.
- Single and multiple optimality: Presence of one or more optimal solutions.

Algebraic formulation of LP:
- Standard form: All constraints are equations with non-negative variables.
- Slack and surplus variables: Added to convert inequalities into equations.
- Canonical form: A specific form of LP used for the simplex method.

Simplex method:
- Base solution: A solution that satisfies the equality constraints.
- In-base and out-of-base variables: Active and inactive variables in the basic solution.
- Optimality criterion: Conditions for determining whether a solution is optimal.
- Improvement: Techniques for improving sub-optimal solutions.

3. Duality in LP
Dual variables and dual constraints: Each variable in the primal problem corresponds to a constraint in the dual problem and vice versa.
Dual objective: The objective function of the dual problem.

Duality theorems:
- Weak duality: The solution of the dual problem provides an upper or lower bound to the solution of the primary problem.
- Strong duality: The optimal solutions of the primal and the dual problem are equivalent.
- Complementary Slackness Theorem: Relationship between the optimal primal and dual solutions.

4. Post-optimality
Post-optimality and sensitivity analysis: Study of the impact of parameter variations on optimal results.
Variation in right-hand side terms: Changes in the terms of the constraints.
Variation in the coefficients of the objective function: Changes in the function to be optimised.

5. Integer programming
Examples and graphical illustration: Problems in which the decision variables must take integer values.
Branch & bound algorithm: Method for solving integer programming problems by systematically exploring possible solutions.
Gomory cuts: Techniques for improving integer programming solutions.

Lectures on theory topics at the blackboard and/or with the aid of slides projected in the classroom.

Exercises on theory topics either by means of examples on the blackboard or using the computer (Excel solver).

Students may be requested to bring their own laptop.

Teaching materials are available on http://moodle2.units.it and Teams

The final examination consists of a written test involving the completion of five exercises and, where appropriate, the answering of theory questions. It tests both the students’ knowledge of the topics covered during the course and their ability to understand and judge independently.

The exercises include
- Formulating (and possibly solving) a continuous linear programming model (syllabus item 1)
- Graphical solution of a continuous linear programming problem (syllabus item 2)
- Simplex algorithm (syllabus item 2)
- Duality, Complementarity, Sensitivity (syllabus items 3 and 4)
- Integer programming (syllabus item 5)





The maximum score for each individual exercise varies from 4 to 8 depending on the difficulty. The total score (sum of the individual scores) is always 32.

The examination is considered passed by those who obtain a score greater than or equal to 18. Those who obtain a score above 30 (i.e. 31 or 32) are awarded the grade of '30 cum laude'.

Should the student be required to produce content (of any kind) in order to be admitted or to participate in an examination (projects, reports, exercises, tests), any use of Large Language Model tools (ChatGPT and the like) must be explicitly stated. This requirement must also be complied with in the case of partial use.

Regardless of the assessment method used, the lecturer reserves the right to evaluate the student's actual contribution to each type of content produced through an oral examination.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs).