Comparing Polyinterval Alternatives: Collective Risk Estimation

Abstract

Procedures for calculating preference and risk indicators that were previously applied to mono interval objects are proposed within the collective-risk estimating method in the case of pairwise comparison for polyinterval objects, generalized interval, and fuzzy objects. The procedures are based on the defuzzification of interval estimates of preference and risk indicators related to mono intervals at alpha-levels in the case of fuzzy polyinterval objects and on the presentation of generalized interval estimates as a probabilistic mixture of the distributions forming such an estimate. Some differences and relationships in approach of generalized interval estimations and fuzzy approach for comparing alternatives are studied. It is established that generalized uniform distributions of probability in the approach of generalized interval estimates are obtained if we use the defuzzification methods for uniform distributions on alpha levels of fuzzy objects discussed in the paper. The manner is shown in which the defuzzification procedures lead to one-numeric estimates for the interval characteristics of fuzzy objects, similar to the numerical characteristics of distribution functions of probability theory, mathematical expectation, variance, and mean semideviation. Depending on the defuzzification method, different probability distributions in the formalism of generalized interval estimates can be obtained from uniform distributions on alpha-levels of fuzzy objects. However, the entire variety of probability distributions that arise in the last formalism is not exhausted by the distributions obtained in this manner.