Base Knowledge
Elementary concepts of programming and probability theory.
Teaching Methodologies
The course content is delivered through guided theoretical exposition, with emphasis on the mathematical formalisation of the fundamental concepts (networks, queueing systems and image processing).
The practical component focuses on problem-solving in class, computer-based simulation, and critical discussion of results, promoting active and collaborative learning.
In-class tasks integrate the critical use of Artificial Intelligence tools, encouraging validation of results, identification of limitations, and reflection on ethical implications in engineering contexts.
The course fosters the development of the mathematical competences proposed by Niss: thinking, reasoning, modelling, problem-solving, and communicating mathematically, bridging theoretical rigour with practical application to real engineering problems.
Learning Results
By the end of this course unit, students should be able to:
Think and reason mathematically about network optimisation problems, analysing different algorithms (shortest/longest path, maximum/minimum capacity, maximum capacity among shortest paths) and their conditions of applicability.
Model mathematically real engineering situations in the context of queueing systems (M/M/1, M/M/S, M/M/1/k, M/M/S/k, finite capacity models, combinations of models), translating stochastic processes into formal structures.
Represent and manipulate mathematical entities in binary images through mathematical morphology, using symbols and formal operations (dilation, erosion, opening, closing, gradient, thickening/thinning).
Pose and solve mathematical problems applied to concrete engineering contexts, combining formal rigour with the interpretation of results.
Communicate in, with and about mathematics, clearly presenting reasoning, results and implications in technical reports and collaborative settings.
Make use of digital tools and technologies, including optimisation algorithms, simulation software and image processing, critically and responsibly, recognising their limitations and ethical implications.
Program
I– Network Optimization: Representation of a network in the form of emerging and immersive arcs; Algorithms for the shortest path, longest path, maximum capacity, minimum capacity, shortest maximum capacity, maximum capacity in the set of shortest routes. Application to concrete problems.
II- Queueing theory: Structure and elementary concepts; Queues modeling; Life and death processes; Fundamental relations of waiting lines; Classification of waiting lines; Models with one or more servers of unlimited and limited lengths; Combination of models; Application to concrete problems.
III– Image Processing – Mathematical Morphology: Basic operations on binary images: dilation and erosion; Properties; Conditional Dilatation; Morphological Gradient; Opening and closing an image: Properties; Hit and miss; Fattening and Thinning (thickening/thinning).
IV – Neural Networks: basic concepts
Curricular Unit Teachers
Deolinda Maria Lopes Dias RasteiroInternship(s)
NAO
Bibliography
– Martins, E.Q.V., Pascoal, M.M.B., Rasteiro, D.M.L.D., Santos, J.L.E.. (Junho 1999). The Optimal Path Problem, Investigação Operacional, Vol 19, no 1, pp. 43-60. (available at curricular unit Moodle webpage)
– Dias Rasteiro, D.M.L. , 9 – Shortest path problem and computer algorithms. (2020). Editor(s): Jesús Martín-Vaquero, Michael Carr, Araceli Queiruga-Dios, Daniela Richtáriková, In Mathematics in Science and Engineering, Calculus for Engineering Students, Academic Press. Pages 179-195, ISSN 00765392, ISBN 9780128172100, https://doi.org/10.1016/B978-0-12-817210-0.00016-3. (available at curricular unit Moodle webpage)
– Syllabus elaborated by the lecturer. (available at curricular unit Moodle webpage)
– Hillier, F. , Lieberman, G. . (2004). “Introduction to Operations Research”, McGraw Hill. Location at ISEC’s Library: 3-9-56 (ISEC) – 09160
Additional bibliography:
– Romão, M. C. , Pinto, L. S. , Simões, O. , Valente, J. e Vaz Pato, M. . (2011). Investigação Operacional – Exercícios e Aplicações, Verlag Dashofer.
– Nielsen, M. A. (2018). Neural Networks and Deep Learning [misc]. Determination Press
– Mezzadri, D. The Paradox of Ethical AI-Assisted Research. J Acad Ethics (2025). https://doi.org/10.1007/s10805-025-09671-7 (pdf available online)
– Lucas J. Wiese, Indira Patil, Daniel S. Schiff, Alejandra J. Magana, AI ethics education: A systematic literature review, Computers and Education: Artificial Intelligence, Volume 8, 2025, 100405, ISSN 2666-920X, https://doi.org/10.1016/j.caeai.2025.100405. (pdf available online)