Fuzzy modelling: 1. Concept of fuzzy set - motivation, definition, basic notions. Basic and generalized operations on fuzzy sets. 2. Vertical and horizontal representation of fuzzy sets. Extension principle and its application. 3. Fuzzy relation, projection and cylindrical extension, separability. Composition of fuzzy relations. 4. Binary fuzzy relations and their properties. Fuzzy equivalence and fuzzy ordering. 5. Fuzzy numbers - definition, important classes of fuzzy numbers. Calculation on fuzzy numbers - fuzzy extension of real functions, fuzzy arithmetic. 6. Ordering and metrics on fuzzy numbers. Fuzzy scales. 7. Linguistic variable and linguistic scale, linguistic variables derived from linguistic scales. Linguistic approximation. 8. Fuzzy controllers - schema, fuzzy rule base, fuzzy inference system (Mamdani, Novak and Sugeno algorithms), defuzzification. 9. Application of fuzzy sets in multiple criteria decision-making and evaluation - motivation, fuzzy extension of crisp methods, "solver of multiple-criteria evaluation tasks". 10. Application of fuzzy sets in decision making under risk - motivation, discrete fuzzy random variable, fuzzy expected value and fuzzy variation, fuzzy decision matrices, fuzzy decision trees. Decision making theory: 1. Decision making under risk. Risk analysis. Monte Carlo method. 2. Decision matrix. Rules of decision-making under risk. 3. Indeterministic theory of utility, construction of utility function for risk for given decision criterion, rule of expected utility. 4. Game theory: general formulation of the task, game in normal form. Antagonistic conflict of two players, matrix games. 5. Game theory: non-antagonistic conflict of two players, double matrix games. 6. Game theory: conflicts with a larger number of decision-makers, characteristic function.
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