Type-1 OWA operators in aggregating multiple sources of uncertain information: properties and real world applications

The type-1 ordered weighted averaging (T1OWA) operator has demonstrated the capacity for directly aggregating multiple sources of linguistic information modelled by fuzzy sets rather than crisp values. Yager's OWA operators possess the properties of idempotence, monotonicity, compensativeness, and commutativity. This paper aims to address whether or not T1OWA operators possess these properties when the inputs and associated weights are fuzzy sets instead of crisp numbers. To this end, a partially ordered relation of fuzzy sets is defined based on the fuzzy maximum (join) and fuzzy minimum (meet) operators of fuzzy sets, and an alpha-equivalently-ordered relation of groups of fuzzy sets is proposed. Moreover, as the extension of orness and andness of an Yager's OWA operator, joinness and meetness of a T1OWA operator are formalised, respectively. Then, based on these concepts and the Representation Theorem of T1OWA operators, we prove that T1OWA operators hold the same properties as Yager's OWA operators possess, i.e.: idempotence, monotonicity, compensativeness, and commutativity. Various numerical examples and a case study of diabetes diagnosis are provided to validate the theoretical analyses of these properties in aggregating multiple sources of uncertain information and improving integrated diagnosis, respectively.