Mathodological aspects
1. The representation of fuzzy numbers; the uncertain data are described mathematically by using the flexible parametric forms of the membership function based on Average Cumulative Function. Elicitation procedures are considered for real-valued fuzzy numbers and, via multidimensional copulas, for vector-fuzzy-valued quantities, including the so called interactive fuzzy numbers. Fuzzy calculations are then obtained by relatively simple algorithms with good (and controllable) approximation properties.
2. The Fuzzy Transform allows the construction of fuzzy models for general functions of single and multiple variables; in particular the F-transform setting (extended to general Lp-norm approximation) is adopted to obtain local trends of multiple time series.
Application themes
1. The fuzzy modeling of some classes of problems in investment appraisal and decision making can be performed by the use of the proposed mathematical tools for coping with uncertainty: (i) data-based representation of fuzziness, and (ii) interval and fuzzy calculus. In particular, a “fuzzy” rate of return and a “fuzzy” net value of an investment can be introduced within a related sensitivity analysis.
2. The F-transform setting can be applied to financial time series analysis to represent a time series in terms of an interval-valued or a fuzzy-valued "smooth" function. A number of financial time series, with different time scales and horizons, are processed through a specific software produced in MATLAB and, eventually, in the open-source R language. A deep analysis of the time series opens interesting investigations to the analysis and the forecasting of economic and financial data and a possible setting for fuzzy-valued time series, allowing, among others, a rigorous risk analysis.
3. New methods to compare solutions of fuzzy optimization problems, expressed by interval and fuzzy objective functions and constraints can be built on the basis of the introduced comparison index, based on the generalized Hukuhara difference, together with new developments in fuzzy ranking and decision making for complex real problems is several fields.
