Fundamentals of Operational Research

Academic year
Subject Area
Biomedical Engineering
Language of Instruction
Other Languages of Instruction
Mode of Delivery
ECTS Credits
2nd Cycle Studies - Mestrado

Recommended Prerequisites

Linear Algebra, Calculus

Teaching Methods

Theoretical and methodological concepts are presented in tutorial lectures, being motivated by real-world problems and illustrated with application examples.

Software (commercial and public domain) packages are used to obtain solutions to the mathematical models, thus freeing the students for the more creative tasks of problem formulation, model building and critical analysis of results.

Learning Outcomes

Providing the students with methodological and application competences in the context of optimization in engineering problems, in order to enable them to identify types of problems, develop adequate mathematical models that include the essential characteristics of those problems, and apply algorithms to generate the optimal solutions for the models. Special attention is paid to the use of software packages to obtain the optimal solutions, as well as sensitivity analysis of optimal solutions in face of changes in the model data and parameters.

Work Placement(s)



0. Origin and nature of Operational Research (OR).

1. Linear Programming (LP). Problem formulation and development of PL mathematical models. The simplex method. Duality theory. Sensitivity analysis. The goal programming model.

2. Special LP problems. The transportation problem. The assignment problem. The transshipment problem.

3. Network optimization problems. The shortest path problem. The Dijkstra algorithm. The Floyd algorithm. Minimum spanning tree. The Prim algorithm. Shortest path with fixed costs in nodes. Maximum flow problem. Max flow-min cut theorem. The Ford-Fulkerson algorithm. The minimum cost flow problem.

4. Non-linear programming (NLP). Types of NLP problems. Complementarity. NLP problems without constraints. NLP problems with constraints. The Karush-Kuhn-Tucker conditions. The modified simplex method for quadratic programming.

Assessment Methods

Resolution Problems: 30.0%
Exam: 70.0%


- Hillier, F. S., G. J. Lieberman. "Introduction to Operations Research", McGraw-Hill, 2010 (9th ed.).

- Tavares, L. V., R. C. Oliveira, I. H. Themido, F. N. Correia. “Investigação Operacional”, McGraw-Hill Portugal, 1996.

- Bronson, R., G. Naadimuthu. "Investigação Operacional", Colecção Schaum (2ª. Ed.), McGraw-Hill Portugal, 2001.

- Clímaco, J., C. H.Antunes, M. J. Alves. "Programação Linear Multiobjectivo", Imprensa da Universidade de Coimbra, 2003.

- Chang, Y.L. "WinQSB, Decision Support Software for M/OM (ver 2.0)", Wiley, 2003.

- Antunes, C. H., L. V. Tavares (Coord.). "Casos de Aplicação da Investigação Operacional", McGraw-Hill, 2000.

- Oliveira, R., J. S. Ferreira (Coord.), “Investigação operacional em ação: casos de aplicação”, Imprensa da Universidade de Coimbra, 2014.