**Plenary Lecture
A General Fuzzy Framework for Representing a System and Measuring its Performance**

**Professor Michael Gr. Voskoglou**

School of Technological Applications

Graduate Technological Educational Institute (T. E. I.)

263 34 Patras, Greece

E-mail: voskoglou@teipat.gr

**Abstract: **We develop a general model for representing several processes in Mathematics Education, Artificial Intelligence and Management (e.g. learning, mathematical modelling, problem-solving, case-based reasoning, etc) involving fuzziness and uncertainty. To each of the main stages of these processes we correspond a fuzzy subset of the set of the linguistic labels of negligible, low intermediate, high and complete success respectively at this stage and we use the total possibilistic uncertainty, i.e. the sum of strife and non specificity, as a measure of the system’s performance (e.g. students’ capacities, CBR system’s effectiveness, etc). Examples and classroom experiments are also presented illustrating the use of our model in practice. Other fuzzy measures of a system’s performance are also mentioned and used. These measures include the Shannon’s entropy - properly adapted in terms of the Dempster-Shafer mathematical theory of evidence for use in a fuzzy environment - connected to the system’s probabilistic uncertainty and the associated information. They also include the “centroid” method, in which the centre of mass of the graph of the membership function involved provides an alternative measure of the system’s efficiency.

**Brief Biography of the Speaker: **Michael Gr. Voskoglou (B.Sc., M.Sc., M.Phil. , Ph.D. in Mathematics) is currently Professor of Mathematical Sciences at the Graduate Technological Educational Institute of Patras, Greece. He is the author of 8 books (7 in Greek and 1 in English language) and of about 240 papers published in reputed journals and proceedings of conferences of 22 countries in 5 continents, with many references from other researchers. He is also a reviewer of the AMS and member of the Editorial Board or referee in several mathematical journals. His research interests include Algebra, Fuzzy Sets, Markov Chains and Mathematics Education.