Additions of interactive fuzzy numbers

This paper provides an account of an approach to modeling unknown data by means of fuzzy-set theory, and addresses the problem of deriving the uncertainty, on a sum of variables whose values lie within fuzzy intervals. The first part is an extensive presentation of the theoretical background of the approach: the extension principle is stated in terms of possibility of an event; the concept of variable interaction is investigated at length. Section II gives new results regarding the effective practical computation of additions of fuzzy numbers. Its originality lies in the introduction of interaction which enables to control the growth of uncertainty, in calculations. Moreover, the problem of computing mathematical expectations with fuzzy probabilities is solved. The results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed. Typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities. More generally, fuzzy arithmetic can be an important tool for sophisticated, computationally tractable sensitivity analysis in systems modeling, computer-aided design and operations research.