On a class of defuzzification functionals

Classical convex fuzzy numbers have many disadvantages. The main one is that every operation on this type of fuzzy numbers induces the growing fuzziness level. Another drawback is that the arithmetic operations defined for them are not complementary, for instance: addition and subtraction. Therefore the first author (W. K.) with his coworkers has proposed the extended model called ordered fuzzy numbers (OFN). The new model overcomes the above mentioned drawbacks and at the same time has the algebra of crisp (non-fuzzy) numbers inside. Ordered fuzzy numbers make possible to utilize the fuzzy arithmetic and to construct the Abelian group of fuzzy numbers and then an algebra. Moreover, in turn out, that four main operations introduced are very suitable for their algorithmisation. The new attitudes demand new defuzzification operators. In the linear case they are described by the well-know functional representation theorem valid in function Banach spaces. The case of nonlinear functionals is more complex, however, it is possible to prove general, uniform approximation formula for nonlinear and continuous functionals in the Banach space of OFN. Counterparts of defuzzification functionals known in the Mamdani approach are also presented, some numerical experimental results are given and conclusions for further research are drawn.