Fuzzification of the OWA Operators for Aggregating Uncertain Information with Uncertain Weights

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Yager’s ordered weighted averaging (OWA) operator has been widely applied in various domains. Yager’s traditional OWA operator focuses exclusively on the aggregation of crisp numbers with crisp weights. However, uncertainty prevails in almost every process of real world decision making, and so there is a need to find OWA mechanisms to aggregate uncertain information. In this chapter, we generalise Yager’s OWA operator and describe two novel uncertain operators, namely the type-1 OWA operator and type-2 OWA operator. The type-1 OWA operator is able to aggregate type-1 fuzzy sets, whilst the type-2 OWA operator is able to aggregate type-2 fuzzy sets. Therefore, the two new operators are capable of aggregating uncertain opinions or preferences with uncertain weights in soft decision making. This chapter also indicates that not only Yager’s OWA operator but also some existing operators of fuzzy sets, including the join and meet of type-1 fuzzy sets, are special cases of the type-1 OWA operators. We further suggest the concepts of joinness and meetness of type-1 OWA operators, which can be considered as the extensions of the concepts-orness and andness in Yager’s OWA operator, respectively. Given the high computing overhead involved in aggregating general type-2 fuzzy sets using the type-2 OWA operator, an interval type-2 fuzzy sets oriented OWA operator is also defined. Some examples are provided to illustrate the proposed concepts.