Fuzzy logic, which is based on fuzzy sets theory introduced by Zadeh in 1965, provides a rich and meaningful addition to standard logic. The applications which may be generated from or adapted to fuzzy logic are wide-ranging and provide the opportunity for modeling under conditions which are imprecisely defined. In this article we develop a fuzzy model for assessing student groups' knowledge and skills. In this model the students' characteristics under assessment (knowledge of the subject matter, problem solving skills and analogical reasoning abilities) are represented as fuzzy subsets of a set of linguistic labels characterizing their performance, and the possibilities of all student profiles are calculated. In this way, a detailed quantitative/qualitative study of the students' group performance is obtained. The centroid method and the group's total possibilistic uncertainty are used as defuzzification methods in converting our fuzzy outputs to a crisp number. According to the centroid method, the coordinates of the center of gravity of the graph of the membership function involved provide a measure of the students' performance. Techniques of assessing the individual students' abilities are also studied and examples are presented to illustrate the use of our results in practice.

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In the present paper we use principles of fuzzy logic to develop a general model representing several processes in a system's operation characterized by a degree of vagueness and/or uncertainty. For this, the main stages of the corresponding process are represented as fuzzy subsets of a set of linguistic labels characterizing the system's performance at each stage. We also introduce three alternative measures of a fuzzy system's effectiveness connected to our general model. These measures include the system's total possibilistic uncertainty, the Shannon's entropy properly modified for use in a fuzzy environment and the "centroid" method in which the coordinates of the center of mass of the graph of the membership function involved provide an alternative measure of the system's performance. The advantages and disadvantages of the above measures are discussed and a combined use of them is suggested for achieving a worthy of credit mathematical analysis of the corresponding situation. An application is also developed for the Mathematical Modelling process illustrating the use of our results in practice.

The assessment of a system’s performance is a very important task, enabling its designer/user to correct its weaknesses and make it more effective. Frequently, in practice, a system’s assessment is performed under fuzzy conditions, e.g., using qualitative instead of numerical grades, incomplete information about its function, etc. The present review summarizes the author’s research on building assessment models for use in a fuzzy environment. Those models include the measurement of a fuzzy system’s uncertainty, the application of the center of gravity defuzzification technique, the use of triangular fuzzy or grey numbers as assessment tools, and the application of the fuzzy relation equations. Examples are provided of assessing human (students and athletes) and machine (case-based reasoning systems in computers) capacities, illustrating our results. The outcomes of those examples are compared to the outcomes of the traditional methods of calculating the mean value of scores assigned to the system’s components (system’s mean performance) and of the grade point average index (quality performance) and useful conclusions are obtained concerning their advantages and disadvantages. The present review forms a new basis for further research on systems’ assessment in a fuzzy environment.

In this paper we develop a fuzzy model for the description of the process of Analogical Reasoning by representing its main steps as fuzzy subsets of a set of linguistic labels characterizing the individuals' performance in each step and we use the Shannon-Wiener diversity index as a measure of the individuals' abilities in analogical problem solving. This model is compared with a stochastic model presented in author's earlier papers by introducing a finite Markov chain on the steps of the process of Analogical Reasoning. A classroom experiment is also presented to illustrate the use of our results in practice.