Fuzzy Models of Rough Mathematics

Abstract

The study shows that the introduced known formal description of rough sets can be interpreted in terms of fuzzy sets. This makes it possible to solve many problems of rough mathematics by the developed apparatus of fuzzy mathematics. The authors suggest a way of describing rough numbers with the help of membership functions of fuzzy numbers. The study specifies the chosen type of membership functions and the method of calculating their parameters. The algebra of fuzzy numbers is adapted to perform operations with numbers that are described roughly. The obtained elements are formulae for calculating the expected values and variations of rough numbers. These correlations are simplified for the most realistic special cases. A possibility is considered for solving roughly given optimization problems. A procedure is described for reducing an optimization problem with rough parameters to a usual problem of mathematical programming. An example is given on solving a linear programming problem whose parameters are determined roughly. The rough problem parameters are described with functions of an (L-R) type. It is suggested that the problem should be solved on the basis of the introduced complex criterion. The numerical value of the criterion takes into account the extent of closeness of the obtained result to the modal solution as well as the level of compactness of the membership function of the resulting value of the objective function